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THESIS

2cm

 

LIBRARY

Michigan State
University

 

 

 

This is to certify that the

dissertation entitled

Intermediate—energy Coulomb excitation of the neutron-
rich radioactive isotopes 26,28Ne, 28-31Na, 30-34Mg,
34,35A1, 3381 and 34?

presented by
Boris V. Pritychenko

has been accepted towards fulfillment
of the requirements for

Ph.D. Physics
degree in

 

 

'1' “LL

Majdaflifessor

04 17 2000
Date / /

 

MS U i: an Affirmatiw Action/Equal Opportunity Institution 0-12771

 

 

_ f. _‘ ..._.--——

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11/00 cJCIRCJDaIeDmpGS-p.“

INTERMEDIATE-ENERGY COULOMB EXCITAT19138OF
THE NEUTRON-RICH RADIOACTIVE ISOTOPES , NE

28—31NA, 30’34MG, 34’35AL, 33SI AND 3413
By

Boris V. Pritychenko

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

2000

1m.
RAD

ll)?

IlP‘lZlUl‘

Intel‘s-yr
». .

Si‘peru

L,

ABSTRACT

INTERMEDIATE-ENERGY COULOMB EXCITATION OF THE NEUTRON-RICH
RADIOACTIVE Isoropss 26vngs, 28‘31NA, 30‘34MG, 34'35AL, 3381 AND 34F

By

Boris V. Pritychenko

The subject of this thesis is the study of collectivity and deformation in light
neutron-rich radioactive nuclei in the Z ~ 12 and N ~ 20 region. Recent experimental
and theoretical results indicate the existence of strongly deformed nuclei near the
N = 20 shell closure. In a Coulomb excitation experiment conducted at RIKEN, large
values for the reduced transition probability (B(E2T)) and deformation parameter
,32 in 32Mg were reported. These results are in good agreement with shell-model
predictions, which take into account an inversion of the normal V( f7/2)1/(d3/2) shell
ordering.

To achieve a more complete understanding of the nuclear properties of light nuclei,
intermediate-energy Coulomb excitation experiments were conducted at the National
Superconducting Cyclotron Laboratory at Michigan State University. Radioactive nu-
clear beams of 26’28Ne, 28‘31Na, 30’34Mg, 34*35Al, 338i and 34? produced by projectile
fragmentation with energies of E z 50 MeV/ nucleon were directed onto a secondary
197Au target, where Coulomb excitation of the projectile and target took place. The
de—excitation photons were detected in an array of position-sensitive NaI(Tl) detec-
tors, which were selected for in-beam y-ray spectroscopy.

The energies and B(E2;0:8_ —> 21*) values for the lowest J"r = 2+ states in the
neutron-rich radioactive nuclei 26”Ne and 30*32Mg were measured. In addition, a
1.436 MeV state was observed in 32Mg. An upper limit on B(E2T) was established

in 34Mg. The energies of the first excited states, and excitation cross sections were

also mmili
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(lei. onlv {I

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HEEHIS.

also measured for: 28'30'31Na, 31'33Mg, 34'35Al, 33Si and 34?. Data on 31Na and 33Mg
indicate that excited states are highly collective as predicted by the island of inversion
hypothesis. These results imply large collectivities in the Z 2 11 and N Z 20 region.

Due to the mixing ratio-, spin- and parity assignment uncertainties in the odd nu-
clei, only the maximum possible values for B(E1T), B(E2T), B(M IT) and B(M2T)
were extracted. Comparison of these values with recommended upper limits for 7-
strengths in light nuclei allowed us to exclude M2 and often M 1 excitations as possible
explanations. The extracted upper limits on transition intrinsic electric quadrupole
moments in 28'29’30'31Na were compared to reported ground state moment measure-

ments.

To those

who believe

iv

The W“
Cyrlotmn l
the NSCL- l
and l apul“:

Imust I
thank him 1
deal dflft’li’f

I also it
for i9d( hing
ashram a
an enfighter

I am ind
committee.
for my llifff‘
mitt than}:

lWEs w:

‘i‘ ..
ml- Profes:

ACKNOWLEDGMENTS

The research described herein was performed at the National Superconducting
Cyclotron Laboratory (NSCL) and it involved the collaborative efforts of many at
the NSCL. It is a difficult task to express my gratitude to all those esteemed people,
and I apologize to those I may neglected to mention.

I must first thank my advisor, Professor Thomas Glasmacher. I would like to
thank him for his guidance in the many stages my research, for teaching me a good
deal accelerator nuclear physics; for his enthusiasm and most of all, for his patience.

I also would like to express my gratitude to Heiko Scheit and Richard Ibbotson
for teaching me about nuclear data analysis. I would also like to thank them for their
assistance and encouragement during my research. I was very fortunate to find such
an enlightened comrade as Heiko. I believe he made me a better person.

I am indebted to Prof. Vladimir G. Zelevinsky for actively serving on my guidance
committee. He and Alexander Sakharuk provided me with theoretical calculations
for my thesis project and a good deal valuable information about nuclear theory. I
must thank him for his constant help and support during my stay at the N SCL.

I was very fortunate to have an excellent collaboration with Florida State Univer-
sity; Professors Kirby Kemper and Paul Cottle taught me a great deal about physics.

I will remember John Yurkon for assisting me with many difficult aspects of ex-
perimental nuclear physics, and also helping me with my work at the N SCL.

In addition, I will also remember Takashi Nakamura for his valuable insights into
Coulomb excitation and for exposing me to Japanese culture.

I am also thankful to the cyclotron operations group for running the accelerator
during my thesis experiments, and to the computer department for their help and

patience during my data analysis.

Speriul t
thesis. lure
provided ht“

I would

throughout

8- D Mdhf
on my gut

anaesthetist

Special thanks go to Barry Davids and Joann Prisciandaro for proof reading my
thesis. I was also lucky to have Luke Chen and Richard Shomin as office mates. They
provided help and a good environment for productive research.

I would also like to acknowledge the generous moral support of Lulu Rinkleib
throughout this project.

Finally, I would like to express my gratitude to Professors James Linnemann,
S. D. Mahanti and Michael Thoennessen for their help, expertise and for serving
on my guidance committee. The financial support of the NSF is also gratefully

acknowledged.

vi

 

 

I was. but
State [him
of Physics at.
in 1933 for t
reserve of th
worked at th
of Sciences c

Looking
Alexander A
that Puritan:
(’ont'inted tl
hm on At-
Son Of 3 km
Caucasus n1

‘0 ’md auot

Extended Curriculum Vitae

I was born and raised in the Former Soviet Union. In 1979 I attended Tomsk
State University and after one year in Western Siberia transfered to the Department
of Physics and Technology at the Kharkov State University. I was granted a Diploma
in 1985 for experimental nuclear physics and received the rank of lieutenant in the
reserve of the Red Armed Forces. From the summer of 1984 until December 1991 I
worked at the Baksan Neutrino Observatory, Institute for Nuclear Research, Academy
of Sciences of the USSR (Northern Caucasus, Russia).

Looking back, I must admit that I was very lucky to be there and work for Prof.
Alexander A. Pomansky [1], despite the low pay and poor living conditions. To say
that Pomansky became a professor under the leadership of Bruno M. Pontecorvo and
convinced the leadership of the Tyrnyauz district not to support communist hard-
liners on August 20, 1991, this would not be sufficient to describe him. He was the
son of a known Russian painter and was well versed in art and history. Life in the
Caucasus mountains was rough, we basically had two choices: to succeed in physics or
to find another place of employment. The first option was the more interesting, and
from 1988 until I moved to Berkeley in 1991, I worked on the dark matter problem.
Somehow Pomansky was able to predict that I would not return to Russia but never
tried to stop me. I learned a great deal from him, unfortunately, he died in April,
1993 from the heart failure. My first Ph.D. project was never completed.

A few weeks later, I spoke with V.M. Novikov and VP. Spiridonov. They sug-
gested that I become a graduate student in the States and forget about my ”ego”. I
would like to thank “BOBA” and Slava for their help during this difficult moment in
my life. I still remember that day in June of 1993. I was standing on the corner of

Telegraph Avenue and Derby in Berkeley, when I made my final decision to stay in

vii

this count r}:

physics and :

 

problems with
f. T. Avignon
for their help .
look at myself

in my life and

 

lrt Septerulr
serious lnterrtio:
came to realize
which resulted i
requirements. I
work for Prof. l
lreally appreric
it was during tl‘.
insueh friends a

lwould also 1
me with all my :
and can freely at

”Eur
‘ " remark'dlllf

l‘p
n

1449.

this country. I would like to express my gratitude to the Center for Particle Astro-
physics and to Prof. Bernard Sadoulet, for allowing me to stay despite the obvious
problems with my former employers. I also would like to express my gratitude to Prof.
F. T. Avignone (University of South Carolina) and R. E. Lanou (Brown University)
for their help and support. During my stay in Berkeley I was able to get a different
look at myself and learned a lot. Without doubt, it was the most remarkable period
in my life and I have never been the same.

In September of 1994 I became a graduate student at Michigan State with the
serious intention of obtaining a Ph.D., in spite of unfavorable circumstances. I soon
came to realize the difference between my expectations and reality in East Lansing,
which resulted in two unhappy years satisfying the Michigan State University degree
requirements. In the Fall of 1996 I passed my comprehensive exams and started to
work for Prof. T. Glasmacher at the National Superconducting Cyclotron Laboratory.
I really appreciate the fact that Thomas invited me to work with the Gamma group,
it was during this time I was able to recover from some of my problems and confide
in such friends as Heiko Scheit.

I would also like to express my gratitude to this wonderful country, which accepted
me with all my strengths and weaknesses. I have finally come to terms with myself,
and can freely admit that I do not feel the need to prove myself anymore. I’m tired of

many remarkable things which happened to me and would like to have just a normal

life.

viii

CON ’.

 

LIST OF TA
LIST OF F1(

1 lntroduct
1.1 Plush
1.? Single

1.3 C Oller-

N

Intermedi
2-1 Glfllf‘r
2-3 Excite
3.3 Augul
3-1 Exper
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w

Experirner
3-1 EXOllt‘;
3.1.1
3.1.2
3.2 Expert
3.2.1
3.2.2
3.2.3
3.2.4
33 Callbrg
3.3.1
3.3.2
3.3.3
3.3.4
3'4 Effitle
3.4.1
3.4.2
3.4.3
Data‘s

tr

CONTENTS

LIST OF TABLES

LIST OF FIGURES

1

Introduction

1.1 Physics of Atomic Nuclei .........................
1.2 Single—Particle Approach .........................
1.3 Collective Approach ...........................

Intermediate-Energy Coulomb Excitation

2.1 General Description ............................
2.2 Excitation Cross Section .........................
2.3 Angular Distribution ...........................
2.4 Experimental Cross Sections .......................
2.5 Doppler Shift ...............................

Experimental Setup and Main Principles of Data Analysis

3.1 Exotic Nuclear Beams ..........................
3.1.1 Nuclear Fragmentation ......................
3.1.2 Fragment Separation .......................

3.2 Experimental Setup ............................
3.2.1 The NSCL NaI(T1) Array ....................
3.2.2 Zero-Degree Detector .......................
3.2.3 Fragment Identification with Silicon Detector .........
3.2.4 Electronics .............................

3.3 Calibration and Gain Matching of the NaI(T1) Detectors .......
3.3.1 Gain Matching ..........................
3.3.2 Position Calibration .......................
3.3.3 Energy Calibration ........................
3.3.4 Stability of Calibrations .....................

3.4 Efficiency Estimations ..........................
3.4.1 Efficiency for an IsotrOpic Source ................
3.4.2 Efficiency for De—excitation Photons ...............
3.4.3 Photons Absorption in the Target ................

3.5 Data Analysis ...............................

ix

331
332
333
334
335

 

1 Experime
«1.1 Prime
1? ”Ar
13 Ermr

131
132
11 Cool.
111
112
113
111
113
111
113
15 Cort:
131
132

5 Summar
A CIOSS Sr
B Cakulat

C Detecto
Cl 130s.
C2 Elle

3.5.1 Experimental Gates and Particle Groups ............ 49

3.5.2 Calculation of Incoming Flux .................. 50

3.5.3 Time Cut ............................. 51

3.5.4 Photon Multiplicity ........................ 53

3.5.5 Experimental Errors ....................... 53

4 Experimental Results 55
4.1 Primary and Secondary Beams ..................... 55
4.2 36Ar Test Beam .............................. 56
4.3 Even-Even Isotopes of 26"ZE‘Ne and 30,323“ Mg .............. 58
4.3.1 Experimental Results ....................... 58

4.3.2 Quadrupole Moments Calculation ................ 64

4.4 Coulomb Excitation of Sodium Isotopes ................. 68
4.4.1 Experimental Observations for 28'29’30N a ............ 68

4.4.2 Experimental Observations for 31Na ............... 70

4.4.3 Cross Section Corrections for 31N a ............... 72

4.4.4 Shell-Model Calculations for 31Na ................ 75

4.4.5 ECIS Calculations of 31N a .................... 78

4.4.6 Data Interpretation ........................ 81

4.4.7 Intrinsic Quadrupole Moments in 28’29'30'31Na .......... 84

4.5 Coulomb Excitation of 31”Mg, 3435A], 33Si and 34P .......... 86
4.5.1 Odd Isotopes of 31*33Mg and 3435A] ............... 88

4.5.2 N = 19 Isotopes of Silicon and Phosphorus ........... 92

5 Summary 97
A Cross Section Calculations 100
B Calculation of Angular Distributions 110
C Detector Calibrations 117
CI Position Calibrations ........................... 117
C2 Energy Calibrations ............................ 118
C3 Efficiency Calibrations for IsotrOpic Source ............... 119

D ECIS Calculations 126
LIST OF REFERENCES 130

 

LIST

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LIST OF TABLES

3.1
3.2
3.3

4.1
4.2
4.3
4.4
4.5
4.6

4.7

4.8

4.9

4.10

CI

Ratio of energy losses in the silicon detector for different nuclei.

Efficiency for an isotropic source fit to equation 3.12. .........
Fit parameters for equation 3.21 for the absorption cross section of
photons in gold ...............................

Beam parameters for the isotopes with observed 7-transitions.

Experimental parameters and results for even-even nuclei. ......
Possible excitations of the 2321 keV state in 32Mg ............
Experimental parameters and results for sodium nuclei. ........
Charge and matter density distributions in 31Na .............
ECIS calculation. Excitation cross sections (integrated over 9m33.25°)
for states in 31Na from coupled channels calculations with an optical
model parameter set determined for the 17O + 208Pb reaction at 84
MeV/A [83] .................................
Experimental upper limits on reduced transition probabilities for as-
sumed E 1, E 2, M 1 and M 2-transitions in 28'29'30’31Na deduced from the
measured excitation cross sections in Table 4.4. ............
Recommended upper limits for reduced transition probabilities in light
isotopes (21_<_AS44). Recommended 'y-strengths are taken from [70,
71] and Weisskopf (single-particle) estimates of the reduced transition
strengths are extracted from Refs. [31, 84] ................
Coulomb excitation of 31"’3Mg, 34'35Al, 333i and 34F. Spin and parity
assignments for electromagnetic transitions from the shell-model cal—
culations [5] (denoted by *) and [73] (denoted by *). Spin and parity
assignments extracted from systematics [62] (denoted by j). .....
Experimental upper limits for reduced transition probabilities in odd
nuclei. n/a - denotes multipolarities excluded by selection rules in the
cases of known spins and parities. ....................

Description of the calibration y-sources (* - denotes uncertainties at
the 99% confidence level, i - denotes uncertainties with an unknown
confidence level and I - denotes a corrected source strength). .....

xi

48

57
62
63
7O
77

79

81

82

86

87

120

LIST

1.1 Neutr
the hi
tuled.

 

lovers
92‘. .
1.2 hdlsso
hfllou"
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target

31 SCher;
32 Seheur
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LIST OF FIGURES

1.1

1.2

1.3

2.1

3.1
3.2
3.3

3.4

3.5
3.6
3.7

3.8
3.9

Neutron dripline and the island of inversion. Isotopes produced at
the NSCL for this thesis with an experimental yield more than 1 par-
ticle/sec are shown in gray, the slashed boxes represent the island of
inversion and the dashed line represents the calculated neutron dripline
[2]. .....................................
Nilsson diagrams for light nuclei. Each nuclear state is described by the
following parameters: total number of quanta, number of quanta along
the z-axis, projection of orbital momentum on the z-axis and projec-
tion of the total momentum on the z-axis. This particular diagram is
explained in section 4.3.2. ........................
Energy spectra for the rotational nucleus 17"’Hf and the vibrational
nucleus 62Ni. Data is taken from [33] ...................

Classical picture of the Coulomb—projectile trajectory. The projectile
nucleus is deflected by means of electromagnetic interaction with the
target nucleus, which is located at the center 0. ............

Schematic layout of the NSCL facility [46] ................
Schematic illustration of nuclear fragmentation. ............
The A1200 fragment separator of the NSCL. A 4’3Ca primary beam
strikes a 9Be production target and produces a variety of light nuclei.
The radioactive beam of interest is selected by using two sets of dipole
magnets [46]. ...............................
Typical isotope identification pattern. Energy losses in a thin fast
plastic scintillator are plotted vs. time of flight. Light isotopes were
produced by fragmentation of a 48Ca primary beam on a 9Be target at
80 MeV/nucleon (Bp = 3.15 T.m). ...................
Schematic view of the experimental apparatus around the secondary
target. ...................................
One—dimensional position sensing by light division in the position-
sensitive NaI(Tl) crystal ..........................
Schematic time characteristics of the phoswich zero-degree detector. .
Electronics Diagram. ...........................
Position spectrum of a NaI(Tl) detector. ................

3.10 The y-spectrum of 88Y in a NaI(Tl) detector. .............

xii

14

23
25

26

27
28
29

31
34

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3.11

3.12

3.13
3.14

3.15
3.16

Position response of a NaI(Tl) detector. The reconstructed position
(see equation 3.7) is plotted versus geometrical position with respect
to the beam pipe flange for any y-ray from a 60C0 source ........
Position-dependent energy calibration. The left panel shows the mea—
sured energy versus position for an 88Y source before application of
a position-dependent energy calibration. The right panel depicts the
same spectrum after the position-dependent energy calibration.

The 7-spectrum of 8”Y ...........................
The N SCL NaI(Tl) array efficiency. The efficiency for isotropic angular
distribution was measured with 228T h, 22N a and 8”Y 7—sources located
at the target position. ..........................
Absorption cross sections in gold versus photon energy. ........
Bits Spectrum. The particle-singles and particle-7 events are defined
as single-channel histograms ........................

3.17 The time spectrum contains two peaks, the peak on the left is created

3.18

4.1

4.2

4.3

4.4

4.5

by photons from the target or projectile de-excitations in the target
and the peak on the right is due to de—excitations in the phoswich
detector ...................................
”Mg data as a function of multiplicity. The left panels contain time
spectra and the right panels show the corresponding 7-spectra for dif-
ferent multiplicities gated on the ”Mg particle group. The nuclear
fragmentation events (large multiplicity) produce many photons in the
target area and increase the 7-background in the ”Mg data. .....

Experimental results for even-even isotopes. The upper panels show
photon spectra in the laboratory frame. The 547 keV (7/2+ —+ g.s.)
transition in the gold target is visible as a peak, while the (2+ —>
9.3.) transitions in each projectile are very broad. The lower panels
show Doppler-shifted 'y-ray spectra. The 2+ —> 9.3. transitions in each
projectile sharpens. ............................
3Z'Mg gates. The upper gate contains ”Mg beam and the lower gate
has a possible admixture of neutron-stripping events (”Mg + 31Mg +
”Mg data with total energy gates. Doppler-shifted 7-ray spectra for
”Mg and ”Mg + 31Mg + ..........................
Systematic behavior of transition energies and collectivities for the
known even-even isotopes of Ne, Mg and Si. Data is taken from [17,
33, 76] ....................................
Calculations of electric quadrupole moments as a function of the de-
formation parameter 6 for 26”We and 30’32'3‘1Mg. The “experimental”
electric quadrupole moments are shown as bands bounded by dashed
lines corresponding to experimental uncertainties. The bands are lo-
cated at both positive and negative values since the experimental data
cannot distinguish between prolate and oblate deformations. .....

xiii

39

40
41

45

49

51

52

54

59

60

61

65

 

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4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

Experimental results of 28'29’30Na. The upper panel show background
photon spectra in the laboratory frame and the 547 keV (7/2+ —-> 9.3.)
transition in the gold target is visible as a peak. The lower panels show
Doppler-corrected 7-ray spectra ......................
Energy spectrum of ”Na. The upper panel shows photon spectra in
the laboratory frame and the 547 keV (7/2+ —> 9.3.) transition in the
gold target is visible as a peak. The lower panel shows Doppler-shifted
'y-ray spectra and a peak at 350(20) keV becomes visible ........
Equivalent photon numbers versus the incident beam energy for a ”Na
nucleus incident on a 197Au target. The impact parameter distribu-
tion is integrated from bmm = 16.46 fm to infinity corresponding to a
Coulomb excitation reaction. The transition energy in ”Na is assumed
to be E7 = 350 keV. ...........................
ECIS calculation. Shown is the angular distribution of the reaction
197Au(”Na,”Na")197Au exciting the 5/2+ state of ”Na. The dotted
and dashed curves represent the cross sections for the nuclear and
Coulomb excitations, respectively. The solid curve corresponds to the
coherent sum of two excitations ......................
Recommended and experimental upper limits on reduced transition
probabilities for E 1, E2, M 1 and M 2-transitions in 285‘29’30'31Na. . . . .
Intrinsic electric quadrupole moments for the ground and transition
states in sodium isotopes (ignoring possible feeding from the higher-
lying states. Data for the intrinsic electric ground state quadrupole
moments were taken from [85] .......................
Recommended upper limits and experimental results in ”'33Mg, 34’35Al,
33’Si and 34 P. Recommended upper limits are extracted from [70, 71] and
experimental upper limits are deduced using the formalism described
in [35]. ...................................
Energy spectrum of y-rays emitted from the 33Mg+197Au reaction at
54.3 MeV/ nucleon. Upper panel contains photon spectra in the labo—
ratory frame and lower panel contains Doppler-shifted 7-ray spectra.

One-neutron removal in 33Mg. Doppler-shifted 7-spectra for the ”Mg
and 33Mg +”Mg particle gates. .....................
Experimental 7-ray spectra of 33Si and 34P. Upper panels contain pho-
ton spectra in the laboratory frame and lower panels contain Doppler-
shifted y-ray spectra. The 547 keV (7/2+ —+ 9.3.) transition in the gold
target is visible as a peak, while 1010 keV, 1941.5 keV and 429 keV,
625 keV transitions are present in the Doppler-shifted 7-ray spectra of
33Si and 34P, respectively ..........................
Level schemes of 33Si and 3“P. ......................

xiv

69

71

73

80

83

85

89

90

91

93
95

--1r 1

 

 

 

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The stud

dull oeforrua

Chapter 1

Introduction

1.1 Physics of Atomic Nuclei

The experimental study of atomic nuclei is currently motivated by:

a The unique nature of nuclei as the main constituents of matter.

0 The necessity of establishing the limits of nuclear stability, radioactivity and
searching for the neutron and proton driplines [2].

o The search for nuclei with unique physical properties (large spatial neutron or
proton halos [3, 4] and large deformations in neutron-rich isotopes [5]).

o The extension of experimental knowledge of fundamental nuclear processes [6].

o Astrophysical applications (solar neutrino problem, origin of the elements [6, 7]).

0 Industrial and medical applications of nuclear physics [8].

The interplay of nuclear stability and radioactivity shaped the world as we know
it. The study of these phenomena allows us to gain a better understanding of our uni-
verse useful in many different applications: astrophysics, radioactive dating, geology,
industry, power production and nuclear medicine.

The study of the fundamental properties of nuclei such as neutron or proton halos

and deformations in neutron-rich isotopes is part of the search for an understanding

of the lllllllri

 

nur'lear strut
most of the i
called the "r
low energies.
”halo" strett

1“Be. and ”L?

First 9119
shell closure
of sodium 1!“
expected fror
region of high
was explaiuet
for Z < 11 It
worl: of Hart;
bash for undt
[13. 11 13] a
excitation. .\1
first excited 2
leg defortrrati
the National
110115 by Caur
SEiii-E‘s 01 ”Ne
51

M

an]! N
.4] D .. N '
”1% 1r.-

Qieff-IOH am e

of the fundamental properties of nuclear matter. Currently, the center of interest in
nuclear structure has shifted to nuclei far from the line of stability. In halo nuclei,
most of the nucleons occupy standard single-particle states, forming what is usually
called the “core” of the nucleus. This core is relatively inert in nuclear reactions at
low energies. Only a few neutrons (one or two) occupy outer orbitals and create a
“halo” stretched in space far away from the core. The most studied halo nuclei are
11Be and 11Li [4]. The halo phenomenon has also been identified in ”Be, 178 and 19C
[9].

First evidence for the existence of an “island of deformed nuclei” near the N = 20
shell closure was obtained in 1975 by Thibault et al. [10] from mass measurements
of sodium isotopes. They found that ”Na and ”Na are more tightly bound than
expected from the spherically symmetrical 7r(sd)-shell model. (Figure 1.1 shows the
region of light neutron-rich nuclei with N ~ 20.) In the same year this phenomenon
was explained by Campi et al. [11] via the introduction of neutron f7/2 intruder orbits
for Z < 14 nuclei (i.e., an “inversion” of the standard shell ordering) in the frame-
work of Hartree-Fock calculations. Theoretical research in this field [5, 12] created a
basis for understanding the unusual properties of ”Mg, which was studied at CERN
[13, 14, 15] and RIKEN [16]. Using the technique of intermediate-energy Coulomb
excitation, Motobayashi et al. [16] reported the reduced transition probability of the
first excited 2+ state in ”Mg, B(E 2; 0:03. —> 2:”) = 454(78) e2fm4, with a correspond-
ing deformation parameter of 62 z 0.52. A similar result was recently obtained at
the National Superconducting Cyclotron Laboratory (NSCL) [17]. Recent calcula-
tions by Caurier et al. [18] predict that intruder configurations dominate the ground
states of 30Ne, ”Na, and ”Mg, and that they are nearly degenerate with the closed
shell states in ”Ne, ”Na and 33Mg. However, the exact boundaries of the island of

inversion are still not known experimentally [5, 19]. The technique of intermediate-

 

 

 

 

Figure 1.12 X
NSCL for this
gray. the slt'L‘li
the calculated

energy Coulo!
the deduction
citation cross:
intermediate-e

The unust;
Study of these
which the isot
intermediatee
done on the h

15 31110115 1 [7

 

 

 

' I
12 _ ‘28Mg 29Mg 30Mg 31Mg 32Mg 33Mg 34Mg 35Mg 36Mg [ 37Mg 3

 

271% 28Na 29Na 30Na 31Na’ 32Na 33Na 34Na 35Na

 

 

 

10 L '26Ne 27Ne 28Nc 29Ne 30Ne 31Ne 32Ne

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3251:. 26F 27F l 1 29F I 1311:
I I I l
I ___ I l____l l____
s- a a -
I I I
23N
16 18 20 22 24
N

Figure 1.1: Neutron dripline and the island of inversion. Isotopes produced at the
NSCL for this thesis with an experimental yield more than 1 particle/ sec are shown in
gray, the slashed boxes represent the island of inversion and the dashed line represents
the calculated neutron dripline [2].

energy Coulomb excitation is well-suited for this kind of research because it allows
the deduction of reduced transition probabilities (collectivity) from experimental ex-
citation cross sections in a model-independent way [20, 21]. A detailed discussion of
intermediate-energy Coulomb excitation is presented in chapter 2.

The unusual prOperties observed in the island of inversion region motivate the
study of these nuclei. For this thesis, experiments were conducted at the NSCL, in
which the isotopes 26”Ne, 28‘”Na, 3O‘34Mg, ”'35Al, 33Si and 34P were studied via
intermediate-energy Coulomb excitation. Interpretation of the experimental data was

done on the basis of theoretical nuclear models, which are described in general terms

in sections 1.2 and 1.3.

‘Vq

 

1.2 Sir

 

lo the single-l
or Hartree—Ft
model repres-
independent-
is replaced it

singlepartit 1

Where the st
particle stat
the occupant
Pét’tlcle spec
of single-par
”91.2”." 01 tie
them usual '1;
protons and
the glouud :
1361510111511
$3th 01 011)
and 0111511115
ddormation.
a large mitttl

The Hat”.

1” arid

SIT. (‘1
“4A

.
r"):

e L
111: Fr

lllfilhod

1.2 Single-Particle Approach

In the single-particle approach we often employ the widely used shell model [5, 18, 22]
or Hartree—Fock calculations [23, 24, 25, 26, 27]. In its current form [5] the shell
model represents a very powerful tool for predicting the structure of nuclei. In the
independent-particle approximation model the effect of all nucleons in the nucleus
is replaced by an average or mean field. The nuclear Hamiltonian is then a sum of

single-particle terms
n

H0 2 26mm, (1.1)

i=1

where the summation is over all single-particle states. The energy of each single—
particle state is represented by 6(1) and n,- is the number operator which measures
the occupancy (0 or 1) of the single-particle state i. In this model, the nuclear single-
particle spectrum is not smooth; it has relatively large energy gaps between groups
of single-particle states. When each group of states is completely filled, the Fermi
energy of the nucleus is just below one of these large energy gaps. Thus more energy
than usual is required to excite the nucleus. Nuclei fulfilling this condition for either
protons and neutrons are called closed shell (magic) nuclei. With all the orbits filled,
the ground state of the nucleus is tightly bound and spherical in shape. By adding
the residual interaction Vto the independent-particle model (1.1) and truncating the
space of orbitals i, one performs a diagonalization of the total Hamiltonian H 0 + V
and obtains many-body states and their energies. This allows one to describe nuclear
deformation. In the framework of the shell model large deformation typically requires
a large number of basis states.

The Hartree-Fock method is used for determining the average one-body poten-
tial and single-particle energies 6(a) from the nucleon-nucleon interaction [26, 27].

This method allows a self-consistent study of nuclear structure starting from a given

Skyrme type nucleon-nucleon force Vij and determines the average nucleonic proper-

 

ties (hindit-
in each nutl
nucleon-nut?»
to be deternh

For many
to use defort:

10 the X1155“:

singlepartich

 

 

 

where H0 L: th
llllPP-(lllllengg
nucleon spin. r

held.

Here ”on plOVl

coordinate of r'
for huge (1631011
for light defor:
figure 1?. It
penetrate into
number. This
Expert ten:
01cm" ener
' large (leg

on.

1, .
ldtl\.‘Jl 1
-l

ties (binding energies, nuclear radii, density distributions, ...) and the excited states
in each nucleus. Because of the need to describe average nuclear properties, the
nucleon-nucleon force is parameterized with a relatively small number of parameters,
to be determined throughout the nuclear mass region.

For many nuclei, a deformed intrinsic shape is more stable. Then it is possible
to use deformed average potentials. The desire to obtain a simple picture leads us
to the Nilsson model [28, 29, 30, 31]. In the case of axial symmetry the Nilsson

single-particle Hamiltonian is given by
H=H0+H6+al-s+b12, (1.2)

where H0 is the spherical part, generally taken to be the Hamiltonian of an isotropic
three—dimensional harmonic oscillator. land 8 are the orbital angular moment and the

nucleon spin, respectively. The deformation is produced by H; due to a quadrupole

field,

1 167r
H6(I‘i) '2 —6osc§/.ngT1-2 ?Y20(6i) . (1.3)

Here 60“ provides a measure of the departure from a spherical shape and ri is the
coordinate of nucleon z'. a and b are phenomenological parameters for each shell, and
for large deformations, effects due to l - s and l2 are less important. N ilsson orbitals
for light deformed nuclei recently calculated by A. Sakharuk [17] are presented in
Figure 1.2. It is easy to see that for N = 20 and 6 2 0.4, the f7); nuclear orbitals
penetrate into the sd—shell (intruder states). Therefore N = 20 ceases to be a magic
number. This phenomenon explains the existence of the island of inversion.

Experimental observables for the disappearance of the shell gap include:

0 Low energy of the first excited state.

0 Large degree of collectivity (deformation).

0 Relatively large neutron- or proton capture cross sections.

 

3 [301 1/2]
[303 5/2]
[301 312]

i [312 312]
[310 112]
. [303 7/2]

, [202 3/21
[321 1/2]

1358 3%

i [202 5121
[3213/2]
[2111/2]
‘ [3301/2]
. [2113/2]

 

 

 

 

 

 

' [101 1/2]
‘ [220 1/2]
[101 3/2]

 

 

 

 

 

. . r . 1 . r . 1 a [110 1/21
-O.6 -0.4 —0.2 0.0 0.2 0.4 0.6

 

Figure 1.2: Nilsson diagrams for light nuclei. Each nuclear state is described by the
following parameters: total number of quanta, number of quanta along the z-axis,
projection of orbital momentum on the z-axis and projection of the total momentum
on the z-axis. This particular diagram is explained in section 4.3.2.

for an 1
interaction 1
Hamiltonian

three group

 

single-part it lt
nucleons \citl
rium shape c
wide variety
moments in 3
imentally mt

these nuclei

1-3 Cc

‘3 ”111' Obset‘t
motion of m,

nth a Hm;

13018110113]

the filled she

the“
elongate

g

fldllf‘fied at

:1 .-
$1351th

(it'd c
‘ 4910 .— -
mldLll

”ligr‘w
1 When}
A A

For an independent-particle approximation in the deformed basis, the residual
interaction is ignored and the nuclear Hamiltonian is a sum of the single-particle
Hamiltonians over all active nucleons. In this model, all states are separated into
three groups: the core states, the valence states and the empty states. Since the
single-particle Hamiltonian H (r;) is, in part, a result of the interaction of the valence
nucleons with the core, the actual value of the deformation depends on the equilib-
rium shape of the core. The Nilsson model provides a reasonable description of a
wide variety of nuclei. It was successfully applied to calculating intrinsic quadrupole
moments in 26”Ne and 30””Mg [17]. The comparison of the calculated and exper-
imentally measured moments (see equations 4.1 and 4.2) allows us to deduce that

these nuclei likely have a prolate shape.

1.3 Collective Approach

Many observed properties of nuclei can be described in a picture that includes the
motion of many nucleons “collectively”. It is convenient to describe nuclear properties

with a Hamiltonian expressed in terms of macroscopic coordinates of the system.

Rotational Model. From the previous discussion it follows that nuclei away from
the filled shells tend to be deformed. In general, the nuclear shape tends to be prolate,
i.e., elongated along the z-axis, at the beginning of a major shell, and oblate, i.e.,
flattened at the poles, toward the end of a major shell. The deformation has the same
Sign as the quadrupole moment. The appearance of rotational spectra indicates onset
of deformation. In quantum mechanics, rotation can be observed only for asymmetric

(non-spherical) objects. For an axially symmetric object with angular momentum

J = Iw, (1.4)

 

figure 1.31 El
62X] Data is

the rotational

where I is the
smmetry an
can rotate wi

total ar

. gular
hand. The gr

rotational bar

I

where -
. E5 19;).

the ., ‘
. rotational

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

% 4+
2 16+ 2+
3 1- 2+
>2. O+
es
2 14+
+
Lu 3+
2 .. 12+ 0‘“
10+
2+
+
1 «- 8
6+
4+
2+
0 1- 0+ 0+
170Hf 62Ni

Figure 1.3: Energy spectra for the rotational nucleus 170Hf and the vibrational nucleus
”Ni. Data is taken from [33].

the rotational Hamiltonian is
h2

Hro :—
‘ 21

J2, (1.5)

where I is the moment of inertia for rotation around the axis perpendicular to the
symmetry axis and w is the angular velocity. For a given intrinsic state, the nucleus
can rotate with different angular velocities. A group of states, each with different
total angular momentum J but sharing the same intrinsic state, forms a rotational
band. The ground state band for 1"’OHf is presented in Figure 1.3. The energy of a

rotational band is
hz

where EK represents contributions from the intrinsic part of the wave function. In

the rotational model we also can calculate static moments and transition rates. For

 

example' th

 

where h, 15 a
quadrttlh’lf 1

probability it

where tht‘ C
the quantum

intrinsic 511111

Vibrational
the hquid-dn
and volume e
set them into
its size witho
and is called
mode genera:
presence of a
shape of the r
also explained

described in te

it‘l' ,
“”9 (l 9. or i

110.] and RI 1
J

example, the observed quadrupole moment is

3K2 — J(J +1)

(J+1)(2J+3)Q°’ (1'7)

Que =

 

where K is an eigenvalue of J3 (J = K for the ground state) and Q0 is the intrinsic
quadrupole moment, which is a function of deformation. The reduced transition

probability in the limit of the rigid rotator is taken from [30] as
5
B(E2; J,- —> 1,) = me2Q3(J,-K'20|JfK)2, (1.8)

where the Clebsch—Gordan coefficient (J,K20|J,K) represents the conservation of
the quantum number K in the transition between the rotational states of the same

intrinsic structure.

Vibrational Model. A large number of nuclear properties can be explained from
the liquid-drop model as the interplay between surface tension, Coulomb repulsion
and volume energy of the drop. Coulomb or nuclear excitation can excite nuclei and
set them into vibration around the equilibrium. For example, the nucleus can change
its size without changing its shape. Such motion involves the oscillation of density
and is called a breathing mode [31]. To preserve the nuclear shape, the breathing
mode generates 0+ states. Low-lying 0+ states in 16O, 40Ca and 90Zr indicate the
presence of a breathing mode. A second type of vibration is an oscillation in the
shape of the nucleus without changing the density; such collective behavior can be
also explained in the liquid-drop model [30, 31]. The shape of the liquid drop can be

described in terms of a set of shape parameters my:

R(9. ab) = Ro{1+ ZaottWW, cm}. (19)
Au

where R(6, (1)) is the distance from the center of the nucleus to the surface at angles

(9,¢>) and R0 is the radius of the equivalent density sphere. In even-even nuclei,

.1

 

 

spin and p
and 2‘. f
energt' rou
vibrations.

shape can

where the
fluid and .
kinetic ene
113111}: and
form 1. 10

small oscil

Where 5*
01%er /\

radius R (

Fix. 1
31' '4

m: rt.

11%

spin and parity assignments for the ground and first excited states are usually 0+
and 2‘“. Figure 1.3 shows a triplet of levels with J7r equal to 0+, 2+ and 4+ with
energy roughly twice the excitation energy which is a good indication of quadrupole
vibrations. The Hamiltonian for vibrational excitation around spherical equilibrium

shape can be written as [30, 31]

d0)“

1 2 1
H=—C§ —DE
A 2AplaApl+2 ”pldt

 

2
| (1.10)
where the quantity CA is related to the surface and Coulomb energies of the nuclear
fluid and D1 is a quantity having the equivalent role as the mass in nonrelativistic
kinetic energy in mechanics. However, the vibrations can be also described microscop-
ically, and the resulting Hamiltonian still has, in the harmonic approximation, the
form (1.10). If different modes of excitation are decoupled from each other, then for

small oscillations the amplitude a1], undergoes harmonic oscillation with frequency

C i-
w): (17:) , (1.11)

where hwy is a quantum of vibrational energy for multipole A. A shape vibration
of order A, with density vibrations, is characterized by large values of the multipole

moment (related to the total particle density p(r) )

MOM) = [power/tare. «ptdr. (1.12)

The electric multipole moment for a spherical charge distribution is exactly the same
as for a surface deformation in a system with constant density and a sharply defined

radius R (liquid-drop model)
3 A
M(E)\,[J) = 528R CIA” . (113)

Finally, the transition probability for exciting a vibrational quantum is very similar to

the rotational model (see equation 1.8) except for the geometrical (Clebsch-Gordan)

10

 

coefficient

for event

parameter

The (‘0
ons and t1
the expert
first excite

defor run it

coefficient [32]
3

It?

2
B(EA;n,\ = o—mA = 1) = ( ZeR") 1322. (1.14)

For even-even nuclei both models produce an identical result for the deformation
parameter
47r B(E2)

1.321: ?W (1.15)

The collective models were introduced to describe coherent motion of many nucle-
ons and these models successfully predict properties of heavy nuclei. Unfortunately,
the experimental measurement of the transition probability between the ground and
first excited states by itself does not discriminate static and dynamic (vibrational)

deformations.

11

 

 

Intt

Exc

This that
energy C

and limit

2.1

Coulomb
1' o
13‘3- 3-‘1, :

919113711112

”UP-01110
\ ”I“ y .
E's-Titania;

15 339.10“. t

~ t
\V. . ..
‘f’llf‘ftb Fl:

Chapter 2

Intermediate-Energy Coulomb

Excitation

This chapter contains a general description of the semi-classical theory of intermediate-
energy Coulomb excitation. Possible physical implications of the theory are discussed

and limitations are given.

2.1 General Description

Coulomb excitation is a very powerful tool for the study of low-lying nuclear states
[34, 35, 36]. The excitation of target or projectile nuclei occurs by means of the
electromagnetic interaction with another nucleus. Since the interaction strength is
proportional to the charge Z of the projectile nucleus, Coulomb excitation is especially
useful in the collision of heavy ions, with cross sections proportional to Z 2. Coulomb
excitation dominates nuclear excitation in scattering when the bombarding energy
is below the Coulomb barrier or when the distance of the closest approach between
nuclei is larger than the sum of the projectile and target nuclear radii (“touching

Sphere” distance) [37]. This can be experimentally achieved by selecting only events

12

 

with sir
indeper:
and Ode

interest

2.2

The twt
intotion
on its (11
one plat

en
*1

todows

with small scattering angles. The Coulomb excitation cross sections provide a model-
independent way to measure collectivity and deformation parameters for even-even
and odd nuclei [35, 38] with in-beam 7-spectroscopy [21, 39], when the isotope of

interest is short-lived and a stable target cannot be produced.

2.2 Excitation Cross Section

The two-body scattering problem can be reduced [40] to the central-body problem
(motion of a single body in an external field when its potential energy depends only
on its distance r from some fixed point). Consequently the path of a particle lies in
one plane and the classical Lagrangian can be written in polar coordinates r, q) as

follows

L = %m(f2 + Ra?) — up), (2.1)

where 1" and o are the radial and angular velocity of the particle, respectively. In a
central field the path of a particle is symmetrical about a line from the center to the
nearest point in the orbit (0A in Figure 2.1). Here the deflection angle x is equal to

[7r — 2%] and (1)0 is given by

E(M/7(": >317“

 

 

(2.2)

Where is M = mr2qh. For infinite motion it is common to use the initial relative velocity
v00 of the particle and the impact parameter b instead of the constants energy (E) and
angular momentum (M). The impact parameter is the length of the perpendicular
from the center 0 to the direction of 220°, i.e. the distance at which the particle would

pass the center if there were no field or force, and E- — 52mm” and M— - mbvoo. After

13

 

Figure 2.1: Classical picture of the Coulomb-projectile trajectory. The projectile
nucleus is deflected by means of electromagnetic interaction with the target nucleus,
which is located at the center 0.

substitution the equation for $0 becomes

00 (b/T2)d7'
on : , (2.3)
L... \/[1_ (be/r2) — (2U /mv§o)l

 

 

The last integral describes the scattering angle x as a function of b. Assuming that
the scattering angle is a monotonically decreasing function of the impact parameter
we can deduce that only those particles whose impact parameters lie between b(x) and
b(x)+db(x) are scattered at angles between x and x+ dx. The effective cross section
is do = 27rbdb. The dependence of do on the angle of scattering can be introduced as
follows

do = 27rb(x)

db(x)
dx

——l dx. (2.4)

 

The above described formalism can be applied to a Coulomb field. In this case U

= a/r and after integration

1 a/mvofb

(A1 + (a/mvoo2b)2] ’

 

(so = cos—

 

l4

where b2 = (a/m'voo2)2tan2co0; after substituting (150 = %(7r — x)

a 2 1
b2 : (mt) 2) CO” [231 O (2.6)

Differentiation of the last expression with respect to x and substitution into 2.4

 

produces the Rutherford equation

 

0 2cos%
do=7r -.
31

2

v 2 '
ml'oo srn

 

1 27
]. (d

which is often written in the c.m. system as follows

2 .
do = ( a 2) d“) ., (2.8)
2771000 sin4[%x]

 

where dw = 27rsinxdx.

In the semi-classical theory of Coulomb excitation the nuclei are assumed to follow
classical trajectories and the excitation probabilities are calculated in time-dependent
perturbation theory. At low energies one assumes Rutherford trajectories for the
relative motion, while at relativistic energies one assumes straight-line motion. In
intermediate-energy collisions, where one wants to account for recoil and retardation
simultaneously, one should solve the general classical problem of the motion of two
relativistic charged particles. The semi-classical solution can be deduced by using the
relativistic Lagrangian

l

— 2

2_4ae
T

L = —m.0c2{1 — 6:202 + r2d2)} (2.9)

But, even if radiation is neglected, this problem can only be solved if one particle has
infinite mass [41]. This approximation is valid if we take the collision ”Mg + 197Au
as our system. An improved solution may be obtained by use of the reduced mass
mo = mpmt/(mp + mt), where mp and mt are masses of projectile and target nucleus,

respectively [36, 41].

15

For pure Coulomb excitation, where the charge distribution of the two nuclei do
not overlap at any time during the collision, the excitation cross section can be ex-
pressed in terms of the same electromagnetic multipole matrix elements characterizing
the electromagnetic decay of the nuclear states [34]. According to Winther and Alder
[35], in most cases it is sufficient to assume that the relative motion takes place on a
classical Rutherford trajectory, and the cross section for exciting a definite state | f )

from the state [2) is given by

do) (do)
— = _— P,_,,, (2.10)
(d0 CE d0 Ruth

where Pi.” is the probability of excitation from the initial state Ii) to the final
state If). Assuming that the electromagnetic interaction potential V(r(t)) is a time-

dependent perturbation, PH, can be deduced as
Ref 2 Ia,_,f|2 With (2.11)

0..., = 3,; / e‘“fi‘<flV(r(t))li>dt, (2.12)

where wfi- = (E f - E,)/ h. The amplitudes a,.,, can be expressed as a product of two

factors
. A
0H; = 2: Xi—foAM), (2-13)
A
where the excitation strength

(..) ~ Zpe<fIM({ mun/27>

 

' — , 2014
XI—)f h {1): }baA ( )
is a measure of the strength of the interaction,
- 1 and [3 - Up (2 15)
7 — 1 ’3‘ _ C , .

'Up the projectile velocity in the laboratory system, ()0 is the distance of the closest

approach in the collision and the function 3(5) (fA(£) = 1 for E = 0 and fA(€) ~

16

 

adidlmt

to the t

Therein

energy .

to the 51
can be 3

Where I]

Which [5
[[011 [llffl

The c
citation
can be d
OI the 9);

[10m by”

at.
ere 6m

e45 for 6 >> 1) measures the degree of adiabaticity of the process in terms of the

adiabaticity parameter E, which is defined as the ratio of the collision time

b
Tcou = —— (2.16)

Al)

P

to the time of internal motion in the nucleus

_ h
Tm = wfi‘ = -A—E. (2.17)

Therefore Coulomb excitations are possible when 6 < 1. This limits intermediate-

energy Coulomb excitation when v ~ 0.3 c, b ~ 15 fm and

AB
~ ___ , 2.18
E 5 MeV ( )
to the study of low-lying collective states with energies of several MeV. Such limitation
can be avoided by using heavy projectiles (Z, > 1) for excitation of collective states,

where the strength parameter
X“) it: VA(ba)Tcoll/ h a (2.19)

which is measuring the action of the field, is larger than unity. In this case perturba-
tion theory breaks down and multiple excitations may occur [35, 36].

The Coulomb excitation cross sections are usually calculated by integrating the ex-
citation probability from a minimum impact parameter bmm to infinity. Final results
can be deduced by introducing an adiabatic cutoff and integrating the absolute square
of the excitation strength Ix]2 from bmm to bmax, instead of integrating Pi! = [X f (£ )|2

from bmin to infinity:

00 bmax
o=27r / P,,bdb:::27r / |X|2bdb, (2.20)
bmin bmin

where bmax is a function of transition energy AE

7v '7 hv
— = . 2.21
IUf,‘ AE ( )

 

bmax :

17

 

nmpmmi

ofthe red)“
Ugh 2:

thweris

menmhhu.

Whm{J(,
The exa

muhhxlarh
2“,:

Where

GE... (.

[Of “park E

CM¢(

I Htaglifftir

h

a)
"M“dfilbi

1
TL

in? 11 is

This produces an approximate expression for the excitation cross section as a function

of the reduced transition probability B(7rA, O —+ A)

(2.22)

 

 

min 3

(2,62)2 B,(«.\,0 —> A) 2,14, (A -1)‘1 for A 2 2
e 21n (9533:) for A =1

N hc

where 7r is the parity and A is multipolarity. In general, B(7rA, 0 ——> A) depends on

the multipole operator for electromagnetic transitions (.M(7rAp.)) as follows [31]:

MW,

._ 1 _ 2
— 2,1, 1|<Jxl|M(7r/\)IIJ.>| ,

 

where (J,]|.M(7rA)]|J,-) is the reduced matrix element.
The exact expression for the excitation cross section, summed over parities and

multipolarities, was derived by Winther and Alder [35]

Z e2 2 2 M B((vrAJ, ——> 1,) 6
OF” =< fie) ZN ) 62 GM“ (5)

2

g#(€(bmin)) a (223)

 

 

 

«Au

where

Gm" (g) = iw,\(2,\1+6-7i)!! (8.73:): ((3) —1)_5
((A+1)(A+H)Pu (c) __ A(A —mu+1)P" (CD

 

 

2A +1 A“

for electric excitations (7r 2 E) and

Gm" (g) = wamlfyiw ([21:30 (G) -1)_%pp(* (S) (2'24)

for magnetic excitations (7r 2 M), and Pf (1:) are associated Legendre polynomials

 

evaluated for :1: > 1. For )2 < O the following relations are applicable
03A-” = (—1)#GE,\# and 01(1)-” = —(-1)”GM,\p , (2.25)

where p is

)u = M, — M,. (2.26)

18

In the pn
E, = M
prnhahilit
section (TE

Finall

directly p

COPSF‘QUP
model-int

relation l

excitatim‘
Sane fun

he t
(lUITtlI‘L‘c‘tle
selenium

l ~ ,
mu‘ilpole

In the present chapter we denoted k = Eff—

, Z, is the proton number of the projectile,
C

E, = AE is the excitation energy, and B((7rA,I,- ———+ I f) is the reduced transition
probability of the target nucleus. The MATHEMATICA code boris-wi79 .m for cross
section calculations is based on equation 2.23 and shown in Appendix A.

Finally, it is important to notice that the Coulomb excitation cross section is

directly proportional to the reduced transition probability
01;)! OC B(Tf/\,I§ —) If) . (2.27)

Consequently, for given 7rA and pure transitions the B(7rA) value can be extracted in a
model-independent way from a cross section measurement. In the present review the
relation between the cross section and the reduced transition probability for target
excitation was obtained. The cross section for projectile excitation is given by the
same formulas with B, substituted by 8,, and Z, by Z, [35].

The transition between an initial nuclear state J,” and a final state J; is usually
dominated by the lowest multipolarity allowed by angular momentum and parity
selection rules [30, 31]. The angular momentum (triangle) selection rule for the A-th

multipole electromagnetic transition is
Ih—Msxsh+t 9%)
and the parity selection rules are
7mg 2 (—1)A (2.29)

for electric and

7?in = (—1)AJr1 (2.30)

for magnetic transitions.

19

2.3

The 31‘.
cieney.

39] he

5.15:1

depend

In t

é‘flid

Use

57"" 7
‘ld[9(t
. .4‘

2.3 Angular Distribution

The angular distribution of 'y-rays is important for the calculation of detection effi-
ciency. A complete analysis of Coulomb excitation angular distributions is given in

[39], here I will present only final results. The angular distribution W(0) is given by

”3(6) : Z [GAp(S)]2g#(€)(—)p (Ii-Aug)

I: evenm
L.L’

I I k ,———
X { )f {1.}Fk(L,LI,Ifl,If) 215+ 1 Pk(COS(9)) {51,61} , (2.31)

where Pk(cos(6)) are Legendre polynomials, and the Winther and Alder functions

gnaw» 2vr(—7-) [pdeKttetpmz

= 27r/[K,,(I)|2xd:r
E

= «:2[1K...1<ot2—IA’.<€>I2—%Ki+l<e)K.<e> ,

K p(€) is a modified Bessel function and the 7 — 7 correlation function Fk(L, L’, [1, I2)

depends on Clebsch-Gordan and Racah coefficients as follows [39, 42]

 

Fk(L, L’, 11, 12) = (—)’1+’2‘1(/(2k + 1)(2I2 +1)(2L + 1)(2L’ + 1)
L L’ k L L’ k
x(1—10){I21211}' (2'32)
In the Coulomb excitation data analysis we express the angular distributions as

W(6) = Z akPk(cos(6)), (2.33)

I: even

and use MATHEMATICA [43] for the calculation of the coefficients ak as demon-

strated in Appendix B.

20

 

2.4 E

The cross 5

in nuclear p

 

as the ratio
product of c

of target mi

The nut

hunger (.\'

the atomic

lnciden
t0 detern'iiz
gFOHIEIQ' (
matter, A

113.4.

2.4 Experimental Cross Sections

The cross section is an experimentally observable parameter which is often measured
in nuclear physics. The Coulomb excitation cross section a to a bound state is defined
as the ratio of the number of detected de-excitation photons N, from this state to the
product of efficiency cm, the number of incoming beam particles Nb and the number

of target nuclei per unit area N),

N 7 1
a = — . 2.34
6tot Arijt ( )

 

The number of target nuclei per area N, is calculated as a product of Avogadro’s
number (N A = 6.021023 1/mole) and target areal density p (in g/cm2) divided by

the atomic mass A (in g/ mole) of the target material,

N4 'P
N = ‘ .
‘ .4

 

(2.35)

Incident flux calculations are presented in 3.5.2. The efficiency is more difficult
to determine because photons can escape without detection mostly due to imperfect
geometry of the detector and the statistical nature of the interaction of 7-rays with
matter. A detailed discussion on the interaction of 7—rays with matter is presented

in 3.4.

2.5 Doppler Shift

At intermediate energies (21,, ~ 0.3c) relativistic effects become noticeable (7 =
1.0483), photons emitted from the “moving” or projectile frame have different en-
ergies in the “stationary” or laboratory frame. The relation between the 'y-ray energy
in the projectile (Ep) and laboratory (Elab) frames is affected by Doppler shift as
follows

E, = 7E)0b(1 — Bcos(6)ab)), (2.36)

21

where 610,, is the angle between the direction of motion of the particle and the direction
of the 7 ray as measured in the laboratory, [3 is the beam velocity in units of the speed
of light, and 7 is the Lorentz factor. For the photons emitted from target excitation
Doppler correction is not necessary.

Experimental parameters such as spread in beam velocities AB [21, 39] and energy
loss in the target contribute to the uncertainty in the measured y-ray energy (energy
resolution) as follows

2
AEfgf’) _ 008(61616) I, 2 2 x 2 BSID(610b) 2 2
( E122: ) — (1 “’ 6005(61ab) — 5’7 ) A3 + (1 __ BCOS(61ab)) A6 . (2.37)

 

 

 

dopp

22

Ch

Ex
Pr

This c

Ndl . T

(ha-pee
under:
the NE
K1300

arHon

Chapter 3

Experimental Setup and Main

Principles of Data Analysis

This chapter contains a general description of the N SCL facility [44] and the NSCL
NaI(Tl) array [45]. The schematic layout of the NSCL facility is presented in Figure
3.1. While a complete discussion on the experimental beams will be presented in
chapter 4, I will present a short introduction here, which is necessary for a better
understanding of the experimental technique [21]. Primary beams were produced with
the NSCL superconducting electron cyclotron resonance ion source (SECR) and the
K1200 cyclotron. Secondary beams were produced via fragmentation of calcium and

argon primary beams in a 9Be primary target, located at the mid-acceptance target

 

 

 

 

 

 

 

 

 

SECR
l ‘2 ' . 1‘ I . ’J . J17
‘ :1 _. 1. , Tm , ,
11212002! . , \ :   8800
/ *5 . . ‘ . _ _.
/
a '1 7”“, , ["2/ ‘/ 1., ‘ ' , \_/ .‘ r
r/ . , _ ., .7

 

 

 

 

 

 

 

 

 

’ ‘ l " I l "
A 1200 F-line (Transfer Hall)“

Figure 3.1: Schematic layout of the NSCL facility [46].

23

 

position or
target. loce
bachgronnc
this chapte

NallTl) an

3.1 E

There are '
radioactive
of stable isc
and becaus
made of a 1
problem an
the life tiin
[43:1 isotop
Such as the
ertrarmn (
[Primary] I

E

Chimlf'dl 85‘
I. .
16m: {01‘ Illjr

position of the A1200 fragment separator [47] and delivered onto a 197An secondary
target, located in the center of the NSCL NaI(Tl) array. The experimental setup, the
background reduction procedures, and principles of data analysis will be presented in
this chapter. A review of position, energy and efficiency calibrations for the NSCL

NaI(Tl) array will be given.

3.1 Exotic Nuclear Beams

There are ~300 stable and up to 8000 radioactive isotopes in nature. The study of
radioactive isotopes is one of the frontiers in modern physics. The nuclear properties
of stable isotopes are mostly known. Radioactive isotopes often have short half-lives
and because of that it is very difficult or often impossible to manufacture a target
made of a radioactive species for studies. Radioactive nuclear beams overcome this
problem and the only limiting factor is the time of flight, which should be less than
the life time. There are two main methods of radioactive nuclear beam production
[48]: isotope separation on-line (ISOL) and nuclear fragmentation. ISOL facilities
such as the ISOLDE mass separator at CERN use sophisticated chemistry for the
extraction of radioactive nuclei with half-lives up to few msec from the production

(primary) target [49] 1 .

3.1.1 Nuclear Fragmentation

Chemical selectivity and microsecond life times of radioactive isotopes are not prob-
lems for nuclear fragmentation facilities such as MSU, RIKEN, GANIL, and GSI;
because of the quick physical separation of wanted and unwanted isotopes (~10—6

sec). Re-acceleration of the radioactive ions is not necessary since the fragmentation
lThe shortest-lived isotope studied up to date at ISOLDE (”Be) has a half-life time of only 4.3
msec [50].

 

24

 

y—ray Final Observed

   

Projectile Prefragment /0 Fragment
.1 o //o 03‘»
. . . ./ Prefragment
Target De- excitation
0

Figure 3.2: Schematic illustration of nuclear fragmentation.

products have essentially the primary beam velocity. Nuclear fragmentation was first
described by Serber [51] as a peripheral, highly energetic, two-step heavy-ion reaction
in which each step occurs in clearly separated time intervals graphically illustrated
in Figure 3.2. The first step consists of the initial collision between the constituents
of the target and projectile nuclei and occurs within ~10'23 sec. This can create
highly excited objects (prefragments) which lose their excitation energy through the
emission of nucleons (neutrons, protons, small clusters) and '7-rays. The second step
(de-excitation) proceeds slowly relative to the first step (~10'17 sec) and depends on
the excitation energy of the prefragrament. A disadvantage of this beam production

technique is poor quality of beams due to large emittance.

3.1.2 Fragment Separation

The A1200 fragment separator was constructed at the NSCL [47] to separate out the
radioactive beams of interest. Figure 3.3 presents the fragment separator which is
achromatic with two intermediate images between two sets of dipoles that bend in
Opposite directions. Two images allow for the use of degrading wedges and provide
space for momentum measurement. Since nuclear fragmentation beams, such as those
produced in the A1200, may contain more than one isotope it is necessary to identify

them on an event-by-event basis. A typical pattern of isotopes produced by the A1200

25

 

“(a Pin
K.12

figure 3.
8 9Be pit
of intere:
fragment
for tl
time of f.
howled;
the bean;
[Tesla-rite
The TOT
tning the
St‘llztlllau
before tl.
Wllltgh l5
Uniquely
based Oi.
Of The PM
21‘ l5 (‘0.

4 E a Hum

9Be Production Target Achromatic Degrader
(376 mg/cmZ) (Wedge)
. Focal Plane: Slits for
‘4] .. /Q¢Bff\\k Secondary Beam Selection

/
43Ca Primary Beam from
K 1200 cyclotron Momentum Selection
Slits

     

22.50 Magnetic Dipoles

Figure 3.3: The A1200 fragment separator of the NSCL. A 48Ca primary beam strikes
a 9Be production target and produces a variety of light nuclei. The radioactive beam
of interest is selected by using two sets of dipole magnets [46].

fragment separator is presented in Figure 3.4.

For the identification of the isotope of interest several techniques can be used. A
time of flight (TOF) method in combination with an energy loss measurement and
knowledge of the magnetic rigidity (Bp) of the A1200 allows the identification of
the beam particle before the secondary target. The magnetic rigidity is measured in
[Tesla-meter] and is equal to the momentum of the particle divided by its charge2 .
The TOP method is based on the measurement of velocity of the projectile nucleus
using the time signals from two separate detectors. For this purpose a thin plastic
scintillator located after the A1200 focal plane and a PIN silicon detector placed
before the NaI(Tl)-array (secondary target area) can be used. The PIN detector,
which is described in 3.2.3, is also good for energy loss (6E) measurements, which
uniquely define the atomic number of the nuclear fragment. This Z-measurement is
based on the fact that energy losses in the thin absorbers are small and the velocity

of the projectile (B) is roughly constant. The mean energy loss, of the fully stripped

 

0.5
2It is convenient to express magnetic rigidity as Bp = 3.10715 fi} [(%)Z + 93%] [52], where

A is a number of nucleons, q is the charge of the isotope and E is an energy in MeV/nucleon.

26

>

AB (Fast Plastic)

 

 

>
Time of Flight

 

 

Figure 3.4: Typical isotope identification pattern. Energy losses in a thin fast plastic
scintillator are plotted vs. time of flight. Light isotopes were produced by frag-
mentation of a 48Ca primary beam on a 9Be target at 80 MeV/ nucleon (Bp = 3.15

T-m).

27

 

 

figure 3.3:
target.

nut-lens. car

Where are
absorbing)
t0 know 1]
nuclear re;
the ldentif
measure rf

excitation

Thin Fast Plastic Fast-lSlow-
(TOF) Phoswich-Detector
PPACs (TOF, AE, E)

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.5: Schematic view of the experimental apparatus around the secondary
target.

nucleus, can be approximated by the Bethe-Bloch formula [53]:

2
(SE = 27rNar€2mec2p§ (g) x, (3.1)

where are 21rNar82mec2 = 0.1535 MeV chg and p, z and a are properties of the
absorbing material and :1: is the absorber thickness. At the same time, it is important
to know the isotopic composition of the beam after the secondary target because
nuclear reactions can take place in the target. Such information is necessary for
the identification of gamma rays from the isotope of interest. For this purpose, we
measure the energy loss and total energy of each nuclear fragment in the Coulomb
excitation experiments with the plastic phoswich detector (discussed in subsection

3.2.2), located at zero degrees with respect to the beam.

3.2 Experimental Setup

The experimental apparatus (NSCL N aI(Tl) array, silicon PIN, T OF and AE/E and
tracking detectors) was located on the F-line in the transfer hall of the NSCL. A
schematic view of the experimental apparatus is presented in Figure 3.5. The position

and direction of each fragment incident on the 702 and 518 mg/cm2 secondary gold

28

 

Figure 3'6;
Na] .Tl l (‘T
targets W"
ldellllfi‘id

scdttt‘lt‘d 1

3.2.1 '

The NSC]
rings 0[ I
the three)
are Q'lllld'
0.4-5 mint
Optical gl
each wind
phoswich
into a 16:

hr inte
beans. 00
the photo:
one dirnen.
end as. iliu.
each end 0
ti

“3‘ K'llrilli.

 

 

 

PMTl PM 2

 

 

 

 

 

Figure 3.6: One-dimensional position sensing by light division in the position-sensitive
NaI(Tl) crystal.

targets were measured with two parallel-plate avalanche counters (PPAC) [54] and
identified in the phoswich detector. Photons were measured in coincidence with the

scattered beam fragments by the NSCL NaI(Tl) array [45].

3.2.1 The NSCL NaI(Tl) Array

The N SCL N aI(Tl) array consists of 38 detectors, which are arranged in 3 concentric
rings, of 11 (inner), 17 (middle) and 10 (outer) detectors, respectively. The radii of
the three detector rings are 10.8 cm, 16.9 cm and 21.8 cm, respectively. The crystals
are cylindrical, approximately 18.0 cm long and 5.75 cm in diameter and placed into a
0.45 mm thick aluminum shield. A 1 cm thick quartz window is attached to each end.
Optical glue was used to connect 5 cm in diameter photomultiplier tubes (PMTs) to
each window. To shield the NaI(Tl) detector array from photons originating at the
phoswich detector, the PPACs, and natural radioactivity, the entire array was placed
into a 16 cm thick lead shield.

In intermediate-energy Coulomb excitation experiments with radioactive nuclear
beams, position sensitivity of the y-array is necessary for Doppler shift correction of
the photons’ energies and has been presented in section 2.5. For sensing position in
one dimension, long cylindrical crystals can be used with PM tubes positioned at each
end as illustrated in Figure 3.6. For this geometry, the intensity of light measured at

each end of the crystal drops off exponentially with the distance from the origin of

the scintillation light [55]. Thus, the signal amplitude E1(E2) for PMT1(PMT2) is

29

 

given by

 

where E. i
produced .

energy deg»

 

coefficient r

scintillation

lt is (Olth’l
which will

fllld lllf‘ (it:

3.22 1

After pass
Q'lindrica
ZDD] Wh:
OHdaQ- be
“Warm
“39 D051:
follo’d’ing }
‘ Pym.

° SQIET.

. COIN)

given by

 

 

E1 = Egpexp[-a(L/2 + 113)] (3.2)
E2 = Egpexp[—a(L/2 — 113)], (3.3)
0

where E7 is the energy deposited by the 7—ray, P is probability that a light quantum
produced at one end will generate a photoelectron in an adjacent tube, E0 is the
energy deposited per photon created in the scintillator and a is the light attenuation
coefficient and z is the distance between the end of the crystal and the origin of the
scintillation light. The position of the interaction can be found by

E E2

1 2
—— — I . O
:r —— log E1 ~ 10g E1 (3 4)

It is convenient to keep 2 always positive by introducing an additional offset of 2000,
which will be discussed in subsection 3.3.2. By multiplying equations 3.2 and 3.3, we

find the deposited energy E, is independent of the position within the crystal

E
E,=(/E1E2-Foexp[aL/2] ~ (/E,E2. (3.5)

3.2.2 Zero-Degree Detector

After passing through the secondary target, the secondary beams are stopped in a
cylindrical fast-slow plastic phoswich detector (called the zero—degree detector, or
ZDD) which allowed charge identification of the secondary beam particles. The sec-
ondary beams are often run in “cocktails” which contain several species in a single
separator setting. The ZDD and time of flight measurements in the beam line pro-
vide positive isotope identification. In addition, the zero-degree detector serves the
following purposes:

0 Provides a trigger and time signal for particle-7 coincidences.

a Selection of beam particles scattered into laboratory angles 3 6,01,.

0 Counts the number of nuclear fragments during the experiment.

30

 

 

 

Figure 3.

The pht
slow plastit
scintillator
a light guitl
attached is
stalled to (
Vacuum. l
on the fro
were used

Figure

Time, nsec
0 200 400

V

 

\
r

I
N

 

 

Signal, mV
L

I

1

Figure 3.7: Schematic time characteristics of the phoswich zero-degree detector.

The phoswich detector has a diameter of 101.6 mm and is made of 100 mm thick
slow plastic scintillator (Bicron 444) and a thin layer (0.6 mm thick) of fast plastic
scintillator (Bicron 400) is glued to it. The zero-degree detector is viewed through
a lightguide by 2 PMTs (T HORN EMI ElectronTubes 9807B02). The PMTs were
attached by using Tracon F113 epoxy. In addition, a water cooling system was in-
stalled to cool down the PMT voltage dividers [56] because the detector operates in a
vacuum. To improve the light collection, 1.5 pm aluminized polyester foil was placed
on the front part of the detector, and Teflon tape and reflecting paint (Bicron 620)
were used on its side.

Figure 3.7 illustrates that ZDD signals consist of two different time components.
The thin fast plastic is responsible for a sharp (~10 nsec) pulse at the beginning of
the time scale and the slow plastic produces rather broad (~400 nsec) pulse. The
different time responses in combination with pulse shape discrimination allow the use
of one PMT for detection of both signals.

The thickness of the fast plastic was chosen so that fragments would lose up to 20
‘70 of their energy and their velocity would be roughly constant. For example, after
passing through the thin fast plastic scintillator, 32Mg ions at 49 MeV/A will lose
5.8% of their kinetic energy and 30Mg ions with 26.1 MeV/ A will lose 20.7% of theirs.

Equation 3.1 indicates that energy losses in the fast plastic provide an opportunity to

31

new
on the

plane
of the

zered

partit
frag};

with 1

lS-OlOl

3.2.1

The l

fl ATE“;
lite

ti...

measure Z for each fragment. Time of flight (fragment mass) measurements are based
on the start signal from the thin fast plastic scintillator located after the A1200 focal
plane and stop signal from the phoswich as shown in Figure 3.5. The rate capability
of the phoswich is as high as 50,000 - 70,000 ions/ sec. It is convenient to present the
zero-degree detector data as follows:

0 Energy loss in thin fast plastic scintillator versus time of flight (for nuclear
fragment identification and corresponding gates for particle-'7 coincidences).

0 Energy loss in thin fast plastic versus slow plastic scintillator (to test that
particular nuclear fragments are passing through the secondary target, if some of the
fragments are missing the target then the AE / E spectrum has two particle groups
with the same AE but different E values).

0 Energy loss in slow plastic scintillator versus time of flight (for separation of the

isotopically pure secondary beams).

3.2.3 Fragment Identification with Silicon Detector

The basic ideas behind fragment identification in a thin (~300pm) silicon detector
are the same as in the case of the thin fast plastic scintillator. Moreover, silicon
detectors have better energy resolution and the detector response function is more
uniform over the active area of the detector.

The silicon detector data also provides an additional possibility for fragment iden-
tification. From the Bethe-Bloch formula (see equation 3.1), the ratio of energy loss

for two different fragments in the cocktail beam is

6E. 2.5.)?
_ z , 3.6
6E. (2.5, l l

where B1, [32 are fragment velocities and Z1, Z2 are their respective electric charges.

 

The same ratio can be experimentally measured as a ratio of vertical positions of

the particle group centroids, when energy losses are plotted versus time of flight

32

Tc

lxe Fl

idenhh

3.2.4

The be
in239.
in the
In
ondary
CODrert
(“(0.71
S“ 94111;
Si mam

Witlflh i

I

J

f“

‘-
Gimp-

'3 lisec

Allfg ,

~-- I

Dified a

Table 3.1: Ratio of energy losses in the silicon detector for different nuclei.

 

Nuclei Calculated Ratio Measured Ratio

of Energy Losses of Energy Losses

 

 

 

 

 

”Al/”Mg 1.195 1.187(32)
32Mg/29Na 1.216 1.221(34)
29Na/26Ne 1.242 1.250(34)

 

(see Figure 3.4). Table 3.1 demonstrates a successful test of this method of particle

identification with secondary beams from 48Ca.

3.2.4 Electronics

The basic electronics setup for Coulomb excitation experiments was already described
in [39, 45]. In Figure 3.8 I will present the set of nuclear electronics which was used
in the present experiments.

In general, nuclear fragments produce photons while passing through the sec-
ondary target and some of those photons interact with NaI(T 1) crystals. PMTs
convert scintillations (optical signals) into electrical signals, which are directed into
custom built NSCL/MSU fast amplifiers and splitters. The splitter produces two data
streams which are used for recording of timing and energy information. The first data
stream is directed into a constant fraction discriminator (LeCroy 3420 or MSU 1806),
which is used for an event trigger, timing and scaler signals. The second data stream
is directed into a l6-channel CAMAC shaping amplifier with a shaping time of about
5 psec (Pico systems) and signals are digitized in 16-channel CAMAC peak-sensing
ADCs (Phillips Scientific 7164H). The signals from the zero-degree detector are am-

plified and split into three different paths. The first group of signals is directed into

33

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5355:85 owum minus u QB

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

imfiz

 

 

 

Figure 3.8: Electronics Diagram.

34

 
   
    
   

Leading
and part;
mwflwh'
zetedegro
The e\
the Kalil)
mmflwC
Quad hJUT-
cohuideutt
pmdmesa
the zerodo
CAMA'
Gueemue
wmmm
. MEN
‘ Slaw
0 Bran
Way Cables
Thedd
Onto DLT .
the fact ll.-
anSQH
affl'llsfiltlon
of 99.3ng

l

by a multj.C

Leading Edge Discriminator (LED) and used for determination of the beam intensity
and particle-7 coincidences. The second and third groups are delayed and directed
into the first and second QDCs (F ERAS), which are gated on the zero-degree fast and
zero-degree slow gates, respectively.

The event trigger requires two simultaneous signals within 3200 nsec from any of
the N aI(Tl)-crystals and the zero-degree detector. The coincidences between 'y-signals
from the CFDs and particle signals from the phoswich detector were realized in the
Quad four-fold coincidence unit (Phillips Scientific 755), which contains particle-7
coincidence, master-gate and master-gate-live signals. The master-gate—live signal
produces an ADC gate, a trigger—bit gate, a bit gate, a TDC-start signal and gate for
the zero-degree detector QDCs (LeCroy 4300B).

CAMAC crates are read out via crate controllers (Bi-Ra systems). The VME
crate contains the Master processor, the Slave processor and a Branch driver, which
perform following functions:

a Master broadcasts data over ethernet.

o Slave sends commands to CAMAC modules through the branch driver.

0 Branch Driver connects the slave to the CAMAC modules via the branch high-
way cables and crate controllers.

The data is collected by DEC 300 alpha workstations via ethernet and recorded
onto DLT and 8 mm tapes. The dead time of the data acquisition system arises from
the fact that a typical event read-out time is ~300 psec. The front end produces
a veto signal for the master-gate-live unit when it processes an event. Typical data
acquisition rates for the 48Ca secondary beams were ~500 events/ sec with a life time
of 99.8%. The high voltage (~+1400 V) for the NSCL NaI(Tl)-array was provided

by a multi-channel power supply (LeCroy Systems 1440).

35

3.3

Position. e
ysis. The .

descriptitn.

3.3.1 G

 

 

The gain In.
tions. was p
The (‘0llllll'd
CH1 lOllg am
A 0.5 hqu
nith glued
Procedure i
fOllOWSj

° The \
response 51

' The 5'
0f ”'2 l;e\'
equation 3

4‘ ll’plc

3.3 Calibration and Gain Matching of the NaI(Tl)

Detectors

Position, energy and efficiency calibrations are an important part of the data anal-
ysis. The ideas behind the calibrations are the topic of this section and a detailed

description of the corresponding procedure is presented in Appendix C.

3.3.1 Gain Matching

The gain matching of PMTs in the N aI(T1) array, which usually precedes the calibra-
tions, was performed with a collimated 'y-ray source placed in the center of the array.
The collimator consists of two “HeviMet” (tungsten / copper alloy) cylinders each 7.62
cm long and 14 cm in diameter arranged co-axially with a 4.6 mm gap in between.
A 0.5 MBq 60Co source is centered between the two cylinders. An aluminum stick
with glued measuring tape is attached to one of the cylinders. The gain matching
procedure with the collimated source in the center of the array was conducted as
follows:

0 The voltage for each phototube was adjusted to produce a negative ~70 mV
response signal into 50 Ohm impedance.

0 The shaper gains were adjusted to produce simultaneously an energy calibration
of ~2 keV/channel (see equation 3.5) and a position peak in channel ~2000 (see
equation 3.4).

A typical position spectrum is shown in Figure 3.9 and typical gain-matched 7-

spectrum of 88Y is shown in Figure 3.10.

36

 

H
01
O

f 1' T rfi
l

100

ITTfir' I I

l

—

Counts (a.u.)

50

0 w
1000 2000 3000 4000
Channels

 

 

 

 

Figure 3.9: Position spectrum of a N aI(T1) detector.

 

300

I I I I I I l I I T T r T T 1 I'

 

 

 

 

 

 

A .4
>
cu
x 200 —
N
v
E .
da - -l
:3 100' -
o
O
O 1.11.... --~4..LII-_-1
500 1000 1500 2000
Channels

Figure 3.10: The 7-spectrum of 88Y in a NaI(Tl) detector.

37

The p
photo!
calihn

Th
excite
Nal . T
insertt
A refe
positit
detect
data 2

Tl

recon:

Where
Spfftu
Comte
fitted

3.3.2 Position Calibration

The position calibration of the array is required for the DOppler correction of the
photon energy and efficiency calculations. A detailed description of the position
calibration procedure is presented in Appendix CI.

The position calibrations are usually performed before and after the Coulomb
excitation experiments by moving a collimated 7-source through the center of the
NaI(Tl) array and measuring the detector response. The whole collimator can be
inserted into the beam pipe for simultaneous position calibration of the entire array.
A reference location for the NSCL NaI(Tl) array is the beam pipe flange. A typical
position calibration is done in steps of 1.27 cm and signals from both PMTs for each
detector and the geometrical position of the collimated source are recorded by the
data acquisition system.

The SMAUG histogrammer [57] is used to produce 1-dimensional spectra of the

reconstructed position. The reconstructed position was pameterized as
X, = 2000 + 1000-log(Y1/Y2) , (3.7)

where (Y1 / Y2) is the ratio of PMT signals for each detector. A typical position
spectrum is shown in Figure 3.9. The offset of 2000 was added into equation 3.4 for
convenience (to avoid negative numbers for X,). Peaks in the position spectra were
fitted with the GF2 (GeLi Fit) peak fitting program [58]. The detector’s response
was fitted in PHYSICA [59] with a third order polynomial

X, = a + b-Lr + 0:52 + d-ar3, (3.8)

where a is the offset and b,c,d are fit parameters. The parameterization of the detec-
tor position response is appropriate for the center of the detector. This polynomial
also includes the turnover of the curve close to the edges of the detector. However,

edge-related effects (due to non-uniform conditions for light collection) are definitely

38

 

 

 

figure 3.11
equation 3
flange for 2
Present an
from the d

Flgure

POSition,

H J' On p05

Strated in f

 

3500 i 1 L 1

3000 4 —

2500 4 _

2000 *

lSOO a _

2ooo+iooo*mg(Yi/Y2)

lOOO -* _

 

 

 

500 t r l r
38 43 48 53 58 63
Geometrical Position (cm)

Figure 3.11: Position response of a NaI(Tl) detector. The reconstructed position (see
equation 3.7) is plotted versus geometrical position with respect to the beam pipe
flange for any 7-ray from a 60C0 source.
present and the curve flattens there. These regions close to the edges are excluded
from the data analysis.

Figure 3.11 shows the position response for detector #3 plotted versus geometrical
position.

This procedure produces an average position resolution of about 2 cm and an
angular resolution of emitted photons for the inner-ring detectors better than 10°.

This angular uncertainty yields an uncertainty in the reconstructed energy of about

5% (see equation 2.37 ),which is small compare to the intrinsic detector resolution of

8%.

3.3.3 Energy Calibration

Differences between the N aI(Tl)-detectors and the dependence of the uncalibrated en-
ergy on position creates a need for a position-dependent energy calibration as demon-

strated in Figure 3.12 for detector #3.

39

figure 3
sured en
dependet
position-

 

 

Energy (MeV)

   

  

 

 

 

 

 

..~, l - . ..l .‘. l . 1* ',-., ._ . _.' .1 _" .. "I
1400 1800 2200 2600 1400 1800 2200 2600
Reconstructed Position

 

Figure 3.12: Position-dependent energy calibration. The left panel shows the mea-
sured energy versus position for an 88Y source before application of a position-
dependent energy calibration. The right panel depicts the same spectrum after the
position—dependent energy calibration.

Suitable photon sources for the energy calibrations of NaI(Tl) crystals are 22Na
and 88Y, because they have two well-separated and easily-identifiable 7—lines. Among
other 7—sources, 228Th is valuable because of its high-energy (2.614 MeV) and low-
energy (0.24 MeV) 7—lines. In the current thesis, energy calibrations were done by
using 22Na (0.511 and 1.275 MeV) and 88Y (0.898 and 1.836 MeV) 7-sources. Each
detector was split into 20 virtual slices of reconstructed position (see equation 3.7)
with a photon multiplicity of one (meaning that only one detector in the array trig-
gered). A typical 'y—spectrum of 88Y collected for one slice is shown in Figure 3.13.
Experimental details are described in Appendix C.2.

Photoabsorption peaks [53, 55] from the 'y-transitions in the calibration sources
were fit with the GF2 routine. Their centroid positions are plotted for each slice as
measured energy versus known energy and fit in PHYSICA with a linear fit. The linear
fit reflects the fact that the response function of the NaI(Tl) detector is Close to linear

over most of the energy range significant in this work [55]. The immediate impact

40

 

3.3..

The 5
fill me

all he

 

150 f I I I I I I I I I I—r I I l I I

l
I

100

l
l

50

Counts / (2 keV)

 

 

 

 

 

500 1000 1500 2000

Channels

Figure 3.13: The y-spectrum of 88Y collected for one slice.

of the position-dependent energy calibration is demonstrated in the right panel of
Figure 3.12. The energy resolution of a typical detector is usually ~8% at 662 keV.
The position-dependent energy calibrations were tested in-beam by comparing the
experimentally measured (calibrated) energies for transitions in the projectile and
laboratory frames in the 197Au target (E, 2 547.3(5) keV) and 36Ar projectile (E, =
1971.2(2.0) keV) with the known energies: 547.5 keV and 1970.4 keV (see discussion

on the test beam experiment in section 4.2).

3.3.4 Stability of Calibrations

The stability of the array calibrations during the experiment is important for success-
ful measurements. In our case, the NaI(Tl) detectors’ instability is mainly defined by

an instability of the photomultiplier tubes, which have gains on the order of 106. An

41

 

overall gain l

0m0ht
hmmmu
and Ff is tl
shnehmrt
thMr
amenm
Three n
GHQ
Ohm;
'EMH
hme
OWWhm
lmflscond
Open and
hO‘JTS. Th
tahhratior
Coulomb f

[[0111 the n

3.4 E

The followi;
Afiwhm

0

mean

overall gain for a PMT is defined by [55]
Gain 2 00‘”, (3.9)

where a is the fraction of all photoelectrons collected by the multiplier structure, 6
is a number of electrons produced by the first dynode for each incident photoelectron
and N is the number of stages. Equation 3.9 implies that the tube’s gain is a sen-
sitive function of the applied voltage V. For conventional dynodes, 6 varies as some
fractional power of the interdynode voltage so that for a 10-stage tube the overall
gain is typically proportional to V6 — V9.

Three major factors can contribute to PMT gain instability during the experiment:

0 High voltage power supply instability.

0 Temperature and humidity variations.

0 External magnetic fields.

In the present thesis experiments an air conditioner in the transfer hall was not
Operational and the array was in a hot and humid environment. During the calibra-
tions conducted after completion of the experiments a sliding door to the vault was
Open and the temperature change led to 3—5 % PMT gain increase over the next 24
hours. This phenomenon was recognized during the data analysis and only position
calibrations and 22Na—energy calibration data (collected in the first 4 hours after the
Coulomb excitation experiments) were used. The remaining calibration data came

from the measurements performed before and during the experiments.

3.4 Efficiency Estimations

The following is a description of the efficiency calibrations for the N SCL N aI(Tl) array.
A good knowledge of the array’s efficiency cm is important for a precise determination

of excitation cross sections (see equation 2.34).

42

 

341 El

hgnenltl
animah]

u0.Asnm

hramnan
6:6]5.

lnthec
Appendix 1
pmhhn-a
numerha
sunmanx
duuonsg
Compton
hennes.
hEsathr
Kauenng
beaniexf
manna
ahnan
SCaIterin;
underth.

Thee

new“,

I

1 . _ '

3.4.1 Efficiency for an Isotropic Source

In general, the detection efficiency is a function of photon energy and detection geom-
etry (overlap between the angular distribution of the 'y-rays and the detector geome-
try). Assuming that photons are emitted from the center of the target, the efficiency
for a certain fixed angle (6,0) and energy E, is given by

e : e(E,, l9, 0) = # of detected 'y—rays of energy E, with direction (0,05)

 

. .1
# of emitted 7-rays of energy E, into direction (9,¢) (3 0)

In the current thesis, calibrated radioactive sources of 22Na, 88Y and 228Th (see
Appendix C.3), placed in the secondary target position, were used to measure the
position— and energy-dependent detector efficiency. The interaction of 7-rays with
matter has been described by many authors [53, 55] and I will present only a short
summary. At low energies, the photoelectric process (photon absorption by atomic
electrons and further emission of electrons by atoms) dominates the total cross section.
Compton scattering, which reduces the photon energy by producing electron recoils,
becomes dominant in the approximately 0.2-5 MeV energy range. Pair production
has a threshold of 1.022 MeV and is important for high energy photons. Coherent
scattering, which preserves the energy of photons, is important for so-called narrow
beam experiments, where both the source and the detector are well collimated, and
this is not the case for secondary beams. In this work we used the efficiency of total
absorption, i.e. a photon is absorbed in a single crystal due to photoeffect or multiple
scattering effects followed by the photoeffect. Consequently, the number of counts
under the photoabsorption peak indicates the total number of absorbed photons.

The efficiency for an isotropic angular distribution was deduced from the experi-

mentally measured photoabsorption peak efficiencies for different energies as follows

. 1
6(E,)<'S°> = Z7;([.s(15,,a,a) .10. (3.11)

This is integrated over the solid-angle range (Q) efficiency with the weight (= fi)

43

T0 (‘alt'l
tirtnal slid
the indit'id
the data at
talihration
the (if? n
the length
the intern
multiplied
experirner
total arra
The [11985

fithilOli

where A
lor low-
error [ .
fit para

IOOI m

the cor-

Error

{I}
p-

l.

To calculate the efliciency for an isotropic source each detector was split into 10
virtual slices over calibrated position (approximately 18 mm each). Depending on
the individual detector and the experiment, the first and last slice were discarded in
the data analysis and efliciency calibrations due to poor energy resolution. Efficiency
calibration data were analyzed for each slice and photoabsorption peaks were fit using
the GF 2 routine [58]. The total number of counts under each peak, combined with
the length of the measurements and the data acquisition live time was divided by
the intensity for the particular 7-line. This intensity is equal to the source activity
multiplied by the branching ratio for the particular 7-line. This procedure yields an
experimental efficiency for each slice (differential efficiency e(E,, 0, (p) ). Finally, the
total array efficiency (608°)(E,)) was obtained as a sum of the differential efficiencies.
The measured total efficiency of the array was fitted in Physica [59] with an efficiency

function of the form
€(E7)(i80) = WIN-(A 'l' B'logiE'r/EO)))°e$p(C/(109(E7/EO)l) a (3-12)

where A, B and C are fit parameters and E0 = 50 keV. The second factor accounts
for low-energy ( _<_400 keV ) threshold effects. During this fit an additional systematic
error (~5%) associated with the judgment of the fit boundaries, was assumed. The
fit parameters with xz/degree of freedom 2 1.151 are presented in table 3.2. The
root mean square statistical errors are the square roots of the diagonal elements of

the covariance matrix. The covariance matrix for this fit is given in (3.13)

 

 

 

 

03, 0,403 0.00 0.29286 -0.05545 0.34847
036,, a}; 0300 = —0.05545 0.01074 —0.06460 (3-13)
0'ch 6003 02. 0.34847 —0.06460 0.42612

The results of the fit are graphically presented in Figure 3.14. The relatively large

errors for the 88Y measured efficiencies are explained by the large uncertainty (+30%,

44

Table 3.2: Efficiency for an isotropic source fit to equation 3.12.

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Value Root Mean Square Root Mean Square
Statistical Error Total Error of Estimate
of Estimate (Standard Error)
.4 -3.4055 0.54117 0.58071
B 1.3712 0.10365 0.11122
C -4.0446 0.65278 0.70047
0
’[O _, 1 1 1 1 1 1 1 l 1 1 _
d efficienCy(350.0 keV) = 0.2659 _
: efficiency/(885.5 keV) = 0.1442 :
a 228 efficienCy(l436.i keV) = 0.0907 _
‘ Th 22Na 7
>\ 2 _
0
C 1
Q) — h—-
:9 l0 1 ~
L1] j _
-2
TO 2 l l T T l T T I 3 T T
10 10

Energy (keV)

Figure 3.14: The NSCL NaI(Tl) array efliciency. The efficiency for isotropic angular
distribution was measured with 228Th, 22Na and 88Y 7-sources located at the target
position.

45

-0%) of the radioactive source intensity.

The efficiency error is defined by the error propagation equation [60]

Be 2 (96 2 36 06
2 2 2
orE ~_ 0A(—0A) +UB(_8B) +...+20A03(6A)(8B) +..., (3.14)

where 0%, 023, 01403 are diagonal and off-diagonal covariance matrix elements and

66 6:

a, 6—5 are partial derivatives of the efficiency function 3.12.

3.4.2 Efficiency for De-excitation Photons

It was found in section 2.3 that the angular distribution of 7-rays in Coulomb ex-
citation (W(6)) is nonisotropic. Therefore, such angular distributions can lead to
over/underestimations of the detection efficiency. A detailed discussion of this prob-
lem is presented in [39], here I will present a short summary.

The total efficiency can be obtained by folding the detector efficiency e(E,, 0, (p)
with the calculated angular distribution W(0) as follows
1 We) - 415,, 6, <7) cm
Q 4i W(6)dQ

 

etot(E,) = (3.15)

The angular distribution of the 7 rays integrated over all space is normalized to
unity

/W(Q) do = 1. (3.16)

Thus, if one assumes the efficiency to be constant over the range of Q,-

.,,,(E,) = /W<e>e<E.,6.¢>d0
Q
= Z/W(6)6(E,,0,¢)dfl
z 9’,

: 25(E7a9i1055z') /W(6)dQ.
i Q,

46

The efficiency e(E,,6,-,q‘),-) is related to the efficiency for an isotropic source in

equation 3.11 through
6(0) = —e,', . (3.17)

Therefore, the total photopeak efficiency can be expressed in terms of the mea-
sured isotropic efficiency and the calculated angular distribution as follows:
5,0, = Z gamma) / W(6) an, (3.18)
i I 9.-
where eliso)(E,, 6,, 69,-) is obtained from the efficiency calibration and W(6) is defined
by equation 2.31.
Besides an efficiency correction for a particular experimental angular distribution,
the efficiency calculation code presented in Appendix C.3 also calculates an efficiency

correction for photon absorption in the target, which is discussed in section 3.4.3.

3.4.3 Photons Absorption in the Target

Absorption of photons in the secondary target modifies the efficiency in the Coulomb
excitation experiments from the efficiency for an isotropic source. This modification is
especially crucial for low-energy photons and thick targets, such as the E, = 350(20)
keV photons emitted by 31Na in the 702 mg/cm2 197Au target. The present method
closely follows the description found in [61, 62] and assumes that the life time for an
excited state is less than 10’12 sec, i.e. the photon is emitted in the target.
The decrease in intensity of a parallel beam of photons passing through an absorber
of thickness d is given by [61]
I = 102973, (3.19)

where Io and I are the beam intensities before and after passing through the target.
d1/2 is an absorber half-thickness. For a monoisotopic absorbing material (such as

the gold target) the quantity d1 /2 can be expressed in terms of the photon energy E,

47

Table 3.3: Fit parameters for equation 3.21 for the absorption cross section of photons

 

 

 

 

 

 

 

 

in gold.
Parameter Value Root Mean Square Root Mean Square
Statistical Error Total Error of Estimate
of Estimate (Standard Error)
A1 7.6767 0.29609 0.08605
A2 76.891 23.916 6.9503
A3 344.10 49.868 14.492
A4 1763.8 533.73 155.11
A5 79.207 10.083 2.9301
and properties of the material
A-log2
d = , 3.20
”2 NA'P'0(E7l ( )

where N A is Avogadro’s number, p is the density of the absorbing material and 0(E,)
is an atomic cross section. The atomic cross section includes contributions from the
coherent (Rayleigh) and incoherent (Compton) scattering, photoelectric effect and
positron-electron pair production [53, 55, 63].

To estimate the efficiency corrections for a gold target the total absorption cross

sections [63] were fitted with a second order exponential
0(E,) = A1+ A2~exp(-E,/A3) + A4-exp(—E,/A5), (3.21)

where E, is the energy of the emitted photon. A1, A2, A3, A4 and A5 are fit
parameters, which are presented in table 3.3 and 0(E,) is expressed in mb. Figure
3.15 graphically represents this fit (with a total x'z/degree of freedom = 0.0845 and

a confidence level of 99.47 %). The fit results were folded with the photons angular

48

 

 

 

 

E

O 3

6 [O A 1 1 1 1 1 g 1 l 1 1 1 1 1 1 1 _

E 3 sigmo(350.0 keV) = 56.73 b E

v .1 sigmo(885.5 keV) = 13.57 b -

3 ‘ sigmo(1436.‘l keV) = 8.86 b ”

<( d _

C 2

'- TO 1 :-
C j :

g .1 ..

+2 '4 ..

U -l ..

CD

(I) a _

U)

(D “i _—
O 2 _

L —: r-

0 a _

C - _

.9 _

a ‘

L_ O

0 1O 2 I I I I I r I [3 T I I I I I I 4
(D

D 10 10 10

< Energy (keV)

Figure 3.15: Absorption cross sections in gold versus photon energy.

distribution of photons and used for efficiency calculations.
The reliability of this procedure was tested with a 3“’Ar test beam and a 197Au
target, which will be discussed in section 4.2. During this test the known Coulomb

excitation cross sections were reproduced with good accuracy.

3.5 Data Analysis

This section describes the principles of data analysis for the NSCL NaI(Tl) array.

The accuracy of the present analysis procedure is discussed and limitations are given.

3.5.1 Experimental Gates and Particle Groups

The sensitivity of the experiment depends on the separation of Coulomb excitation
events from the radioactive background. To select events originating from the projec-

tile and target de—excitations, only NaI(Tl)-detector signals which coincided within

49

~200 nsec with signals from the phoswich detector were recorded. This group of
events is called particle-7 coincidences. Figure 3.4 demonstrates the projectile iden-
tification in this experiment. It is easy to see that good particle identification is

necessary for experiments with cocktail beams.

3.5.2 Calculation of Incoming Flux

The total number of incoming beam particles (particle-singles) needs to be known for
a cross section calculation. This number of particle-singles is usually recorded with a
down-scale (d/s) factor (a d/s factor of 100 indicates that only one out of 100 events
in the phoswich detector was recorded), which allows an increased live time of the
data acquisition system 3 .

Figure 3.8 demonstrates that phoswich-detector signals are processed by Quad
four-fold-unit which generates trigger bits. Bits spectrum (particle-7, particle-singles)
signals with a gate on the isotope of interest is plotted in Figure 3.16.

The total number of incoming nuclei is equal to the number of down-scaled
particle-singles events multiplied by the down-scale factor, which usually varies from
30 to 500. The total number of counts under the particle-7 peak on the plot provides
the total number of particle-7 coincidences. Particle-7 coincidences can be created
by photons from de—excitations in the target and the phoswich detector. Further
background rejection is achieved by applying a time out (which will be discussed in
subsection 3.5.3), effectively reducing the coincidence window from 200 nsec to a few
nsec.

The down-scaled incoming beam flux can be calculated by applying particle-singles

gate to a particle ID spectrum (see for example Figure 3.4). The number of counts

 

3The recorded number of incoming particles is affected by the live time of the data acquisition
system. However, for cross section calculation (see equation 2.34) this correction is not necessary

because the number of detected de-excitation photons N, is identically affected.

50

 

particle-7

particle—singles d/s H

Counts (a.u.)

 

 

 

 

 

 

L J 1 L l A A L A l L L J L 1 L L A A 1 I L

0 20 40 60 80 100

 

Channels

Figure 3.16: Bits Spectrum. The particle-singles and particle-7 events are defined as
single-channel histograms.
in this gate multiplied by the down-scale factor is also equal to the total number of

incoming nuclei of this particular isotope.

3.5.3 Time Cut

In the current experiments the average target / first-ring detector and target / phoswich
distances (see Figure 3.5) were ~10 cm and ~1 m, respectively. For a typical projectile
velocity of (22,, ~ 0.30) these distances define times of flight for photons emitted
from the target area (~0.3 nsec) and the zero—degree detector (~13 nsec). This
separates target / projectile de-excitations from de-excitations in the ZDD by ~10-15
nsec. To prevent accidental coincidences, the time difference between the detection
of the photon in the N aI(T1) detectors and the detection of the scattered fragment in
the ZDD was recorded for each event. During the data analysis theses two classes of

data can be separated as shown in Figure 3.17.

51

36Ar Data

 

 

 

 

 

 

 

 

Target Peak Gate Phoswich Peak Gate
:> 100 > 100
E E
o 80 o 80 E
N cu
\ 60 \ 60 .
U) (I) ..
s a =
:3 40 :3 4O :-
O O :
U 20 U 20 .-
0:11111111111
1 2 3 4 1 2 3 4
Energy (MeV) Energy (MeV)
100 _v . . .

 

 

 

 

 

50 100 150

Time (nsec)

Figure 3.17: The time spectrum contains two peaks, the peak on the left is created
by photons from the target or projectile de-excitations in the target and the peak on
the right is due to de—excitations in the phoswich detector.

52

3.5.4 Photon Multiplicity

The photon multiplicity is the number of photons emitted in each event. In intermediate-
energy scattering, many different reactions such as Coulomb and nuclear excitations,
neutron removal and nuclear fragmentation take place. Coulomb excitation creates
a few photons due to de-excitation while other nuclear reactions such as projectile
fragmentation produce many 7-rays. Coulomb de-excitation events can be enhanced
by selecting events with low multiplicities.

The impact of different multiplicity gates on the 32Mg 7—data is shown in F ig-
ure 3.18. The figure indicates that, for large multiplicity events the ’y-background

increases and the 885 keV transition in 32Mg becomes less pronounced.

3.5.5 Experimental Errors

The three major sources of errors in the experimental data are

0 A statistical error associated with the number of detected photons.

o The efficiency error, which is 3.5-4.5 % in the 0.3-2.0 MeV energy range.

0 A phoswich detector position uncertainty, which creates an uncertainty in the
scattering angle of 63:31: z O.1°.

Contributions from other errors such as uncertainties in beam velocity, transition
energies, and cross sections for absorption in the target are much smaller. The errors,
considered here, are independent and the total error is equal to the square root of the

sum of quadratures [60].

53

 

mult= 1 32Mg

 

 

 

Counts (a.u.)

 

  

mult>3

 

 

 

 

A ‘41 k;

50 100 150

 

 

Time (nsec) Energy (MeV)

Figure 3.18: 32Mg data as a function of multiplicity. The left panels contain time
spectra and the right panels show the corresponding 'y-spectra for different multi-
plicities gated on the 32Mg particle group. The nuclear fragmentation events (large

multiplicity) produce many photons in the target area and increase the 'y-background
in the 32Mg data.

54

Chapter 4

Experimental Results

The present chapter is dedicated to the description of experimental results. The
experimental data for even-even and odd nuclei were analyzed using the method
discussed in section 3.5 and interpreted on the basis of nuclear models which were
described in chapter 1. Possible physical explanations of the experimental results are

discussed and conclusions are given.

4.1 Primary and Secondary Beams

Primary beams of 48Ca‘“ with an energy of 80 MeV/nucleon and intensity as high
as 8 particle-nA1 and 40Ar12+ having energy of 90 MeV/nucleon and intensity of
80 particle-nA were produced with the NSCL superconducting electron cyclotron
resonance ion source and the K1200 superconducting cyclotron.

Table 4.1 presents the secondary beams of 26”We, 28‘31Na, 30‘34Mg and 3435Al,
33Si and ”P which were made via fragmentation of the calcium or argon primary
beams in a 9Be primary target located at the mid-acceptance target position of the

A1200 fragment separator [47] and delivered onto a 197Au secondary target located

 

ll particle-nA = 10—9 _ 1.6-110fi9 partsicles R: 6.2 . 109 partsicles

 

55

in the middle of the NSCL NaI(Tl) array. Typical beam rates for the particles of
interest ranged between one and a few hundred particles per second. Substantial
amounts of 21,220, 22‘25F, 24*25'27Ne, 26,27Na, ”Mg, 30-33,36,37Al, 32,36,373i and 40,411)
were also produced in the present experiments, however electromagnetic transitions
between band states in these isotopes were not observed in the range of 200 keV
< E, < 4 MeV.

All nuclei described here were subjected to the same data analysis procedure,
nevertheless the data interpretation was different. From the nuclear properties of
even-even nuclei [33] one can predict 0+ and 2+ spin and parity assignment for the
ground and first excited state, respectively. Such spin and parity assignments indicate
E2 electromagnetic transitions (see equations 2.28, 2.29). Odd isotopes provide a
challenge for data interpretation, because spin and parity assignments for the ground
and first excited states are often unknown and electromagnetic transitions are not
uniquely defined (different transitions such as E 1, E2, M 1 and M2 are allowed).
Therefore additional theoretical and experimental considerations provide an avenue

for the odd-nuclei data interpretation.

4.2 36Ar Test Beam

The data analysis procedure was checked for two known nuclei: a 36Ar beam and the
197Au target.

Figure 4.1 contains results for the 36Ar test beam. 36Ar was selected for calibra-
tions because of the well-established adopted value of reduced transition probability
for the first excited state of B(E2;0;3. —-> 2+) = 298(30) e2fm4 [64]. This value is
in excellent agreement with the present experimental result of B (E2; 0:3. ——> 2+) =
286(23) e2fm4. Besides, results of the current measurement are also in agreement

with the results from inelastic electron scattering: B(E2; 0:3. —> 2+) = 280(16) e2fm4

56

Table 4.1: Beam parameters for the isotopes with observed y-transitions.

 

 

 

 

 

 

 

 

 

 

Primary 9Be target Secondary Total beam 197Au target 91,232
beam (mg/cm?) beam particles/106 (mg/cm2) (deg)
“Ca 376 28N6 1.46 702 2.8

29Na 12.96

3“Na 3.30

31Na 1.28

31Mg 8.0

32Mg 13.0

33Mg 1.70

34Mg 0.22

34Al 5.52

35Al 5.35
4°Ar 564 26Ne 39.83 518 3.96

28Na 82.44

3°Mg 98.35

3331 1020.4

3413 106.34

 

57

 

[65].

Target excitations in 197Au with the 36Ar beam were used to find the excitation
cross section in 197'Au. Data analysis of the target excitations was based on the
fact that scattering angle in the c.m. system between projectile and target nuclei
is exactly the same as between target and projectile nuclei and the beam velocity
is known. For this case, Z and A for target and projectile nuclei were interchanged
in the MATHEMATICA reduced transition probability calculation routine, given in
Appendix A. The present experimental value for 197Au of B(E2; 3/2:,. —> 7/2+) =
4899(351) ezfm4 agrees with the adopted value of B(E2; 3/2;: —> 7/2+) = 4988(170)
e2fm4 [66].

4.3 Even-Even Isotopes of 26’28’Ne and 30323‘ng

The following is a presentation of the Coulomb excitation results for 26'28Ne and
30'32’34Mg. The data are discussed using the single-particle approach (presented in

section 1.2). These results have been published in Physics Letters B [17].

4.3.1 Experimental Results

The 'y-ray spectra, both without the Doppler correction (laboratory frame) and with
the Doppler correction (projectile frame) for 26”We and 30'32'34Mg are shown in Figure
4.1. The laboratory frame and projectile frame spectra from the test measurement of
the stable nucleus 36Ar are also included. Photons de-exciting the previously observed
2fL states in 30Mg and 32Mg (at 1482 and 885 keV, respectively) are apparent in the
projectile-frame spectra for those two nuclei. A strong peak occurs in the projectile-
frame spectrum for 26Ne at 1990(12) keV, while a somewhat weaker (though still
clear) peak appears in the corresponding 28Ne spectrum at 1320(20) keV. On the

basis of these observations, we propose that the 2? states occur at these energies

58

 

 
  
 
   

 

 

 

 

 

 

 

 

 

 

300 30Mg 32mg 3‘Mg “Ar {
A E v=0 v=0 v=0 v=0 v=0 3
> 200 ~ -

(D * x20 x0.5 l x20 x005 :
:4 t .
O 100 j 1
EL :

\ 0 b I“! l - ..L l 1

4(5) b

:1 32Mg 34Mg I 3"Ar ~

:03 v=0.34c v=0.32c v=o.31c§
0 x20 x01?

fiLflLLflflll l . . l . . . 1

 

 

12312312312312l3
Energy(MeV)

Figure 4.1: Experimental results for even-even isotopes. The upper panels show
photon spectra in the laboratory frame. The 547 keV (7/2+ —> 9.3.) transition in the
gold target is visible as a peak, while the (2+ —> g.s.) transitions in each projectile
are very broad. The lower panels show Doppler-shifted '7-ray spectra. The 2+ —+ 9.3.
transitions in each projectile sharpens.

59

D

 

Total Energy

 

 

 

 

 

 

 

4»

Time of Flight

Figure 4.2: ”Mg gates. The upper gate contains ”Mg beam and the lower gate has
a possible admixture of neutron-stripping events (”Mg + 31Mg + ).

in 26”We. The results on the 2? state energies in the neon isotopes are consistent
with those reported at a recent conference [67, 68]. Only a few counts appear in the
34Mg spectrum above a Doppler-shifted energy of 800 keV. Below this energy, there
is background due to the Coulomb excitation of the gold target.

It is useful to analyze the Doppler-shifted spectrum of ”Mg in more detail. Along
with the peak from the 2* -—> 0:3. transition another broad peak around 1.45 MeV
is visible. From previous experience, we assume that radioactive background in this
region is due to stripping of neutrons and subsequent de-excitations of 31Mg or 30Mg.
If this assumption is true, then such stripping events can be distinguished because the
total energy of 31 Mg or 30Mg ions is smaller than total energy of ”Mg. To test this,
all 3"’Mg events were separated into two groups: the higher (”Mg) and the lower than
average total energy (”Mg + 31Mg + ) events (see Figure 4.2). The corresponding

7-ray spectra are shown in Figure 4.3.

60

 

60

  

40

1438(12) keV

 

Counts/(40 keV)

 

 

 

Energy (MeV)

Figure 4.3: ”Mg data with total energy gates. Doppler-shifted 7-ray spectra for ”Mg
and ”Mg + 31Mg + .

The important feature in the ”Mg beam 'y-spectrum (top part of Figure 4.3) is a
small peak at 1438(12) keV, an energy which agrees with a 7—ray observed at 1436(1)
keV in the B-decay of 32N a [13, 14, 15]. Klotz et al. [15] determined that the 1436 keV
7-ray is in coincidence with the 885 keV 23‘ —> 0;, y-ray and, therefore, it de-excites
a state at 2321 keV.

For 26”Ne and 30Mg, where the 21* states do not appear to be fed by higher-lying
states, the cross sections for populating the 21‘ states can be determined by using
equation 2.34. Results are listed in Table 4.2. The p0pulation cross sections for the

21" states of these nuclei can then be used to obtain B(E2; 0;, —> 2;“) values using the

formalism of Winther and Alder [35] (see equation 2.23), and these reduced matrix

61

Table 4.2: Experimental parameters and results for even-even nuclei.

 

 

Nucleus Eggfia'gd E(2f) 0 93;,” B(E2;0;,,—>2+)
(MeV/A) (keV) (mb) (deg) (ezfm‘l)
26Ne 41.7 1990(12) 74(13) 4.48 228(41)
28Ne 53.0 1320(20) 68(34) 3.20 269(136)
30Mg 36.5 1481(3) 78(7) 4.56 295(26)
”Mg 57.8 885(9) 80(17) 3.25 333(70)
34Mg 50.6 g 164 3.28 _<_ 670

 

 

 

 

elements are also listed in Table 4.2.

In the case of 3"’Mg, the 21* state is not only populated directly in the intermediate
energy Coulomb excitation reaction but is also fed via the 1436 keV 'y-ray decay from
the 2321 keV state. Therefore, the population cross section for the 2? state is the
difference between the production cross sections for the 885 and 1436 keV 7—rays.
Since the 885 keV transition has E2 (2? —> 0;) character [14], the production cross
section for this state can be unambiguously determined from the experimental yield
to be 107(13) mb. However, there is some uncertainty about the cross section for
production of the 1436 keV 7-ray because the efficiency for detection of a 7-ray
depends on its angular distribution, which in turn depends on the multipolarity of
the transition and the spins of the initial and final states. We do not know the spin
and parity of the 2321 keV state. However, a coupled-channels Coulomb excitation
calculations RELEX [69] allows us to exclude the possibility of a two-step excitation
of the 2321 keV state via 885 keV and 1436 keV transitions and only direct excitation
of the 2321 keV state can occur. The possible J 7’ values for this state are limited to 1‘

and 2+ by requiring that the reduced matrix elements B(A; 035 —-) A") corresponding

62

Table 4.3: Possible excitations of the 2321 keV state in 32Mg.

 

 

Transition Energy 0(1436 keV) B (E A)
(EA) (keV) (mb) (e2fm2")
E1;0+ —> 1‘ 2321 28(11) 0.040(16)
E2; 0+ —> 2+ 2321 26(10) 105(42)

 

 

 

 

 

 

to the observed experimental yield for the 2321 keV state are less than or equal to
the recommended upper limits listed by Endt [70, 71]. Table 4.3 contains possible
reduced transition probabilities of the 2321 keV state assuming one-step excitation
(2321 keV) and two-step de-excitation (1436 and 885 keV) processes.

The results in Table 4.3 indicate that the value of the de-excitation cross section
for production of the 1436 keV 7—ray (feeding cross section) is only slightly affected
by the multipolarity of the transition. Consequently, the 2+, 1“ —> 2+ cross section is
consistent with 26(10) mb. When the feeding cross section is subtracted from the ”y-
ray production cross section for the 885 keV 'y-ray, we obtain a cross section of 80(13)
mb for direct population of the 21+ state, which then yields B (E 2; 0;, —> 2]”) = 333(70)
e2fm4 for ”Mg. This value is 27% lower than the value reported by Motobayashi et
a1. [16].

Motobayashi et 01. did not report the observation of the 1436 keV '7-ray. However,
it is worth noting that the difference between the B(E2; 0;, —> 2?) values obtained
in Ref. [16] and the present work can be accounted for by the feeding correction
applied here. Without the feeding correction, we would have obtained B(E2; 0;, ->
2?) = 440(55) ezfm“, which would be consistent with the result from Ref. [16] of
B(E2; 03; —) 21+) 2 454(78) e2fm4.

The secondary 3‘lMg beam was particularly weak, and the integrated number of

63

beam particles was small. However, we can still draw some conclusions from the '7-ray
spectrum. The Doppler shifted spectrum for 34Mg contains a significant background
below 800 keV which results in part from 7-rays from the gold target. There are also
several counts in the spectrum above 800 keV. The data are not sufficient to identify
the energy of the 2:“ state. However, if we assume the 2? state is located between 0.9
and 1.4 MeV, we can place an upper limit of 670 ezfm4 on B(E2; 09+, —+ 2;“).

The systematic behavior of transition energies and reduced transition probabilities
of even-even isotopes in the island of inversion region is presented in Figure 4.4. The
small transition energy and large B(E2) value in 3"’Mg clearly indicate that this
nucleus is dominated by intruder configurations. sd-shell model calculations [73]
predict a large transition energy E, = 1.677 MeV and a smaller degree of collectivity

of B(E2 T) = 172 e2fm4.

4.3.2 Quadrupole Moments Calculation

While the spherical shell model has been used extensively to study nuclei in the
vicinity of the island of inversion, the deformed shell model, or Nilsson model, provides
another framework for gaining insights about isotopes in this region. If we assume
that the nuclei studied here have static quadrupole deformations with axial symmetry,
we can use the Nilsson model [29] to calculate intrinsic quadrupole moments for oblate
and prolate shapes to see whether the data provide a preference for one shape over
the other. The Nilsson diagram used for the present calculations (generated with
v), = -0.16 and v" = 0, v), = —0.127 and on = —0.0382 parameters used in the
single-particle potential [30] for the sd and pf shells, respectively) is shown in Figure
1.2. The deformation parameter used in the diagram is 6, which is related to the usual
spherical harmonic coefficient 520 (or just 32) by equation 62 z 6/0.95 [30]. With

this Nilsson model, the intrinsic electric quadrupole moments Q0 have been recently

64

 

 

 

 

 

 

 

 

 

Figure 4.4: Systematic behavior of transition energies and reduced transition prob-
abilities for the known even-even isotopes of Ne, Mg and Si. Data is taken from
[17, 33, 76].

65

calculated in Ref. [17] for the nuclei studied here over a range of deformations by

summing over the contributions of the individual protons

Q0 =(167f/5)1/2 :(AIT‘2YQOIA) , (4.1)
A

where /\ are the occupied proton orbitals. The intrinsic quadrupole moments are
graphed as a function of 6 for 26”Ne and 30”*34Mg in Figure 4.5. The figures do
not include quadrupole moment results for the range of small 6 values (—0.1 < 6 <
0.1) where the residual interaction outside of the standard Nilsson model becomes
important. This figure also illustrates the “experimental” intrinsic electric quadrupole

moments extracted from the measured B(E2; 0;s —-> 21+) values via the equation [30]

Q§=(16?W)B(E2-O+ —> 2:). (4.2)

, g,

The bands shown in Figure 4.5 as dashed lines correspond to the ranges of ex—
perimental uncertainty in the present work. Both positive and negative experimental
values are shown in the graphs because our experiment cannot discriminate between
prolate and oblate shapes. For all the nuclei in Figure 4.5, it is clear that the “experi-
mental” quadrupole moments can be reproduced if the nuclei have substantial prolate
deformations (6 2 0.3). However, the quadrupole moments for all oblate deforma-
tion parameters shown have magnitudes which are much larger than the experimental
values. We conclude that if these nuclei have static axially symmetric deformations,
they must be prolate. The present conclusion that ”Mg is prolate is consistent with
the results of Refs. [5, 18] in which the spherical shell model is used.

These results provide a more complete picture of the extent of the island of in-
version and the role of the intruder states outside the boundaries of the island of
inversion. It is clear that the energies and B(E2; 09*, -> 2+) values for the lowest 2+
states in the N = 16 isotope 26Ne and the N = 18 isotope ”Mg can be explained

using the normal Ohw configurations2 , while the energy of the 2? state in 28Ne

 

2Normal (0 hw) configuration denotes the sci-shell while the intruder (2 F162) configuration has

66

 

 

—80'TT‘ —“__—_T__‘T—":

 

L— A—A.+.

.
‘ ‘ ' 28N6 ‘
——1 T " d
- .
g -_ A_.L_-_._‘——A——.—rb——L ‘— A _A 4 ...4‘ -.fi—‘y ._.__‘ A _ A _ J
.

  

Er... . ,,_ .-

     

 

 

 

 

 

 

 

 

 

Figure 4.5: Calculations of electric quadrupole moments as a function of the deforma-
tion parameter 6 for 26”Ne and 30”'34Mg. The “experimental” electric quadrupole
moments are shown as bands bounded by dashed lines corresponding to experimental
uncertainties. The bands are located at both positive and negative values since the
experimental data cannot distinguish between prolate and oblate deformations.

67

suggests strong mixing between the intruder and normal configurations in this nu-
cleus. We also determine the B(E2; 03', —> 2+) value for the lowest 2+ state in ”Mg
to be 27% lower than the value reported by Motobayashi et al. [16]. The deformed
shell model calculations demonstrate that if these nuclei have static axially symmetric

deformations, they must be prolate.

4.4 Coulomb Excitation of Sodium Isotopes

Historically, the first evidence for the existence of the so-called island of deformed
nuclei near the N = 20 shell closure was obtained in 1975 by Thibault et al. [10] from
mass measurements of sodium isotopes. In particular, the authors of Ref. [10] wrote:
“Then the behavior of the experimental data for the sodium isotopes at N = 20
is inconsistent with the classic shell closure effect, and more reminiscent of the be-
havior one observes when entering a region of sudden deformation.” Experimental
observation of the excited states in 28"’9'”’”Na is a main topic of this section. Nu-
clear properties of these isotopes will be presented and possible explanations will be
discussed.

4.4.1 Experimental Observations for 28:291”Na

The cocktail beams used in this thesis work contained 38 different nuclei. This allowed
us to study 28:29:30'31Na fragments simultaneously. Figure 4.6 shows the Coulomb
excitation data for 28'”'”Na [72].

The 1240(11) keV transition is present in the 28N a data. Two peaks are observed
in the Doppler corrected spectrum of ”Na. Additional tests, which were previously
described in section 4.3, indicate that only the 433(16) keV transition belongs to

”Na and the 700(20) keV line originates from the one-neutron-stripping reaction

 

two nucleons from the p f-shell [17].

68

 

400 i
300 I

E
200 ,

100:

 

 

400 *

300 :

Counts/(20 keV)

200 I

100:

 

 

 

 

 

 

Energy (MeV)

Figure 4.6: Experimental results for 2828*”Na. The upper panel show background
photon spectra in the laboratory frame and the 547 keV (7/2+ —> 9.3.) transition in
the gold target is visible as a peak. The lower panels show Doppler-corrected 'y-ray
spectra.

with a = 39.3(18.2) mb and further de-excitation of ”Na. In fact a weak peak with
such an energy can be seen in the 29Na data. Observation of the one—neutron—stripping
reaction in the ”Na-data could be explained by the fact that ”Na has the smallest
neutron separation energy among the sodium isotopes 28129’”’31N a, which are 3520(80)
keV, 4420(120) keV, 2100(130) keV and 4000(190) keV, respectively.

From the magnetic resonance measurements of 28””Na [74] we know that the
spin and parity for the ground state are 1+, 3/2 and 2*, respectively. Spin and
parity assignments for the first excited states in 28’29:”Na can be deduced assuming
the rotational nature of collectivity as 2“, 5/2+ and 3*, respectively. These spin

and parity assignments were used for the angular distribution in the calculation of

excitation cross sections, which are presented in Table 4.4.

69

 

Table 4.4: Experimental parameters and results for sodium nuclei.

 

midtar get max
Nucleus Ebeam E7 a 6m

(MeV / A) (keV) (mb) (deg)

 

”Na 43.11 1240(11) 26(6) 4.52
29Na 59.97 700(20) 26(21) 3.21
”Na 55.56 433(16) 42(14) 3.23
”Na 51.54 350(20) 115(32) 3.24

 

 

 

 

4.4.2 Experimental Observations for ”Na

Very little was known about the nuclear structure of ”Na until the transition energy
and excitation cross section for the first excited state were measured at the NSCL
[72, 75]. Figure 4.7 demonstrates an observation of an excited state in ”Na.

The low excitation energy E, = 350(20) keV and the corresponding large cross
section of a = 115(32) mb provide a clear indication that nuclear properties of ”Na
are very similar to those of ”Mg. This is in agreement with a shell-model calculation
of Caurier et al. [18] and BA. Brown [73], who predicted that ground state of ”Na is
dominated by intruder configurations and the ground, first and second excited states
should have spin and parity assignments 3/2+, 5/2+ and 7/2“, respectively. The
theoretical prediction for the ground state spin in ”Na agrees with an experimental
value which can be deduced from optical measurements of G. Huber et al. [74], who
wrote: “The spin assignment for ”Na is based on a value of the hyperfine structure
and the isotope shift which indicate that I = g is the most probable value”. In
addition the shell-model calculation of Caurier et al. [18] predicts that the transition
energy for the first excited state is ~ 200 keV. The difference between theoretical

and experimental transition energies can be explained by the incomplete knowledge

70

 

- 31 _
30_ Na 4

20 - 547.5 keV -

 

105
L

 

 

 

Counts/(20 keV)
O

 

 

30 — 350(20) keV BlNa _

E v V=O.320‘
20 L. ..
10 - 4

 

 

1
. Mlflilfliimnllfiiunmnfln.

0.5 1.0 1.5 2.0
Energy (MeV)

Figure 4.7: Energy spectrum of ”Na. The upper panel shows photon spectra in
the laboratory frame and the 547 keV (7/2+ —-) 9.3.) transition in the gold target is
visible as a peak. The lower panel shows D0ppler-shifted 7-ray spectra and a peak at
350(20) keV becomes visible.

U I
:-

 

71

of nuclear interaction in this region 3 .
The relatively large error-bars for the Coulomb excitation cross section in Table
4.1 are predominantly statistical in origin due to the low intensity of the ”Na beam.

At the same time the observation of the 350 keV transition in ”Na illustrates that

in-beam 'y-ray spectroscopy is possible with beam rates as low as 3 particles/sec.

4.4.3 Cross Section Corrections for ”Na

The lack of the experimental decay scheme in ”Na introduces additional uncertain-
ties for the excitation cross section. The uncertainties associated with incomplete
knowledge of the decay scheme, absorption of 'y-rays in the secondary gold-target and
conversion electrons are the main topics of the present subsection.

The experimental spin and parity assignment for the first excited state in ”Na
is unknown. Therefore the 5/2+ assignment is based on shell-model calculations for
”Na by BA. Brown (presented in subsection 4.4.4). Equations 2.28, 2.29 and 2.30
indicate that E2 and M1 transitions are possible and the experimental mixing ratio

between them is unknown. The mixing ratio is defined as follows [31]

W(E(/\+1);J.- —> J,)

62 =
W(M/\;Ji—)Jf) ,

 

(4.3)

where W(7r)\) are transition rates, which are directly proportional to reduced transi-
tion probabilities. Lets consider this problem in more detail:

0 The M 1 transition cannot be solely responsible for the excitation of ”Na.
This follows from the experiment because the corresponding B(M 1T) value would
be roughly three times higher than the recommended upper limit [70, 71]. Such a

conclusion is in agreement with the shell-model prediction for B(M 1, 3/ 2+ —> 5/2+)

 

3In the most recent version of shell-model calculations of ”Na, performed by the authors of Ref.

[18], the first excited 5/2+ state is located at 284 keV.

72

 

Figure 4.5
nucleus if
from bmm
transitior
8 0.09 #2
tion stre:
Couloml
and the
PhOIOIl 1
tons for
that the
of “1813,71.
the 5/72-
”36m
' The
1101) for fa
Stare is 19.5

0. 4.3 454 fl

 

 

I... 1010 , .., . . ..--.., . - -....
.8
31 197
8 Na + Au
:5 105 ,
z ” E7=350 keV
c l .
.8 6
o 10 E2
.121 , <
(1..
...) 104 L q
C
0)
To [
.2 102 ~
.2
:1
0"
E15] 100 . n-111u1 . . .....l . . W.

 

 

 

101 102 103 A A “”164
Incident Energy (MeV/A)

Figure 4.8: Equivalent photon numbers versus the incident beam energy for a ”Na
nucleus incident on a 197Au target. The impact parameter distribution is integrated
from bmin = 16.46 fm to infinity corresponding to a Coulomb excitation reaction. The
transition energy in ”Na is assumed to be E7 = 350 keV.
z 0.09 [1%, and corresponding excitation cross section of 0.236 mb . The small transi-
tion strength for the M 1 excitation can be understood from the following argument.
Coulomb excitation is often described as a photo-absorption of virtual photons [42, 77]
and the total excitation cross section is proportional to a product of the equivalent
photon number and photo-absorption cross sections. The numbers of equivalent pho-
tons for the E2 and M1 transitions are 6698780 and 59.73, respectively. Assuming
that the photo-absorption cross sections for M 1 and E2 photons are of similar order
of magnitude, one can conclude that E2 transitions are responsible for excitation of
the 5/2+ state. The spectrum of equivalent photons for M1 and E2 multipolarities
as a function of incident beam energy is presented in Figure 4.8.

o The M1 multipolarity dominates the de—excitation because the Doppler correc-
tion for fast beams (21,, = 0.32c) is applicable only when the life-time of the excited

state is less than 10’10 sec and the decay occurs close to the target. A lifetime of

0.43 nsec for an E2 de—excitation would not allow a successful Doppler-shift proce-

73

dure. The
the de—exr
prediction
ratio wouli
cross sectit
multipolari

Absorpi

 

 

 

of detectior
decay wit hi
state in ”N

0 Assun
for 3:30 ke\'
citation cro
1311198 from
would corre
08:: within .
2.36 is 1.3.9
angle Cortes
% Thus tli
0f 11 7t fOr
mentally 0h
SllOWn in Fl

° If the
target “’lll l.
flight dlStam

that (143,3

dure. The observed peak at 350(20) keV indicates that the DOppler shift works and
the de-excitations have an M1 nature. This is in agreement with the shell model
prediction that 62 = W(E2) / W(M 1) z 0. In this particular case, a different mixing
ratio would not change the angular distribution and thus the value of the excitation
cross section, because the difference between detection efficiencies for the M 1 and E2
multipolarities (0.1%) is much smaller than the efficiency error (4.1%).

Absorption of '7-rays in the Au-target is important for a correct determination
of detection efficiency, especially for low-energy photons. Excited states in ”Na can
decay within the target or after passing it. Consequently the lifetime of the 350 keV
state in ”Na affects the detection efficiency as follows:

0 Assuming that all gamma—emissions occur in the target, the detection efficiency
for 350 keV photons has to be reduced from 18.5% to 13.4%. This increases the ex-
citation cross section from 93(27) mb to 128(37) mb, and the corresponding B (E2T)
values from 394(115) e2fm4 to 543(159) e2fm4, respectively. For E2-transitions this
would correspond to half-live time of 0.3 nsec. In this case 63.2% of the ”Na nuclei de-
cay within 42 mm from the target. The corresponding uncertainty for 010,, in equation
2.36 is 15.9 %. Equation 2.37 indicates that this uncertainty in the photon scattering
angle corresponds to a broadening in the energy resolution of (AE8)] /E,‘g,,’)dopp = 8.5
%. Thus the energy resolution (FWHM) would decrease from the expected resolution
of 11 % for a 350 keV photon [53, 55] to 14 %. This is inconsistent with the experi-
mentally observed energy resolution (FWHM) of 9.3(2.7) % for the 350 keV photon
shown in Figure 4.7. Therefore the decay has predominantly M1 multipolarity.

a If the decay is M 1 in character, then the fraction of nuclei decaying in the
target will be 6.2% (the half-life time for B(M 1T) 2 0.09 pi, is 15.3 psec and the
flight distance in one half-life is 1.47 mm) and we correct for the absorption assuming

that decays occur in the middle of the target. 99% of these nuclei decay within 7

74

half—life times, which correspond to a distance of 10.3 mm from the target. If the
angular distribution was isotropic, we would have to correct for the absorption of half
of these gammas (emitted backwards), using a correction about two times as much
as compared to gammas emitted from the middle of the target. Since the fraction of
nuclei decaying outside the target is 93.8%, and 12% of those go forward more than
backwards, we should not apply the absorption correction to 012-0938 = 0.1126 of the
photons. The absorption correction is ~11.26% relative to the isotropic distribution,
so we need to decrease the efficiency correction, which was 18.5% —>13.43% by 11.26%,
so the average efficiency will change 13.43%-1.1126 = 14.94%. As the result of this,
the cross section will be reduced from 128(37) mb to 115(32) mb or by 10%.

This cross section of 115(32) mb contains contributions from the electron con-
version, nuclear excitations and a possible feeding contribution from the 7/2+ state
which will be described in subsection 4.4.5. Conversion electrons are not a contribut-
ing factor because the beam is fully stripped. The beam can pick up electrons in the
target producing a second-order effect (with an electron conversion coefficient ~4%

[78]) and this correction is not included in the final cross section.

4.4.4 Shell-Model Calculations for ”Na

In the first shell-model calculation of ”Na, which was performed by Caurier et al. [18],
spin and parity assignments and the transition energy of the first excited state were
calculated. These parameters are not sufficient for interpretation of the ”Na Coulomb
excitation experiment [75]. Therefore a complete OXBASH shell-model calculation 4
of ”Na was performed by BA. Brown by using the same sd — pf Hamiltonian and

model space that was used in [80, 81, 76] for the neutron-rich Si, S and Ar isotopes.

4The OXBASH (Oxford-Buenos Aires-MSU) shell model code was developed by BA. Brown, A.

 

Etchegoyen and W.D.M. Rae in 1988 [79].

In this calculation, 2p — 2h (2 ha.) neutron excitations - which are responsible for the
strong deformation in ”Mg - were included in addition to the usual 0 hw configura-
tions. The allowed 2 hw neutron configurations were (d5/2)6(d3/2,31/2)4(f7/2,p3/2)2,
and the allowed proton configurations were dgfl, and dg/2(31/2, d3/2). This calculation
gives two concentrations of E2 strength. One is the first excited J1r = 5/2+ state, as is
observed in the experiment (although the energy is calculated to be 197 keV instead of
the observed value of 350 keV). The second concentration of E2 strength is predicted
to reside in a J1r = "/2+ state at 1.525 MeV. The B(E2 T) value for the J1r = 7/2+
state is predicted to be 45% of the value for the JTr = 5/2+ state. The absolute
value of excitation cross section for 5/2+ state can also be affected by feeding from
de-excitations of the 7/2+ state. The shell model predicts that B(E2, 3/2+ —+ 7/2+)
= 87 e2fm4, which will correspond to 21.7 mb. This will result in 4.1 events at 1.1725
MeV, which is consistent with the experiment 3 :l: 2 counts.

The shell model calculation of ”Na also predicts a static axial quadrupole defor-
mation with two deformation parameters, which reflect charge and matter density
distributions. The rms radii 5 for charge and matter density distributions in ”Na
are presented in Table 4.5.

The first, the “Coulomb deformation” BC, reflects the deformation of the pro-
ton fluid in the nucleus and corresponds to the electromagnetic matrix element
B(E2; I9, —-> 1,). In the rotational model [32], B(E2; I9, —-> If) and BC are related

via the equations

B(E2; I.- —> 1,) = %Q3 < I,K20|I,K >2 (4.5)

 

5The rms radii are related to the “rigid sphere” radii by equation:

R2 = §R2 (4.4)

rms 5 rigid '

Table 4.5: Charge and matter density distributions in ”Na.

 

 

 

 

 

 

 

 

 

Particles Shell model rms radii Rigid sphere radii

(fm) (fm)
Protons 2.9550 3.815
Neutrons 3.3232 4.290
Protons+Neutrons 3.1974 4.128

and
1677 1/2 3 2
Q0 — (’57) 47261? (3, (4-6)

where Q0 is the intrinsic quadrupole moment. The rigid sphere radius R can be
approximated by R = roA1/3, where we take r0 = 1.20 fm. For the 5/2+ state, the
shell model result (B(E2; 3/2 —+ 5/2) = 196 e2fm4) gives a prediction of BC = 0.51.
In the case of the 7/2+ state, the shell model calculation gives B(E2;3/2 —-> 7/2)
= 87.5 e2 f m4, so that BC 2 0.46. Deformation parameters can be expressed as a

function of B(E2 T) values and rigid sphere radii as the following:

W ,/B E2
lfizl = 4 ( ) 1 (4.7)

3 ZeR2 (J,K20|J,K) ’

 

where the corresponding Clebsch-Gordan coefficients for 3/2+ —> 5/2+ and 3/2+ —>
7/2+ transitions are (/18/ 35 and 2/ 7, respectively.

The second deformation parameter in the calculation is the “nuclear deformation
parameter” 6N. While the Coulomb deformation parameter is used to calculate the
electromagnetic interaction between target and projectile, the nuclear deformation
parameter is used to determine the interaction via the nuclear force. In the standard
collective model, the neutron and proton fluids are assumed to have the same defor-
mation. In such a case, the nuclear deformation parameter could be set in a trivial

way: 6N = BC == 6 [16]. However, the present shell model calculations yield results for

77

 

neutron and proton transition multiple matrix elements which are not consistent with
the standard collective model picture. The neutron transition matrix element for the
5/2+ state in ”Na is B(E2; 3/2 —> 5/2) = 252 ez’fm4 and BC = 0.46. In the case of
the 7/2+ state, the shell model calculation gives B(E2; 3/2 —+ 7/2) = 171 ezfm“, so
that 130 = 0.51. Transition matrix elements for the nuclear fluid (protons+neutrons)

can be estimated as 0.5-(proton+neutron) values.

4.4.5 ECIS Calculations of 31Na

The present shell model calculation predicts electromagnetic matrix elements connect-
ing members of the ground state rotational band, but a reaction model is necessary to
translate the shell model predictions into experimental cross sections for the 5/ 2 and
7/ 2 states in the present scattering experiment. These calculations were performed
by EA. Riley, who used the coupled-channels code ECIS88 [82] with an Optical model
parameter set determined for the 17O-l-208Pb reaction at 84 MeV/ nucleon [83] to cal-
culate the angular distributions, and then integrated the angular distribution out to
the maximum scattering angle measured in the experiment (6:32,, = 28°) with an
average beam energy of 51.5 MeV/nucleon. The phenomenological potentials and
input data file for the ECIS calculation are presented in Appendix D.

The coupled-channels calculations using the predicted shell-model calculation de-
formation parameters yield 54 mb for the 5/2 state and 27 mb for the 7/2 state,
see Table 4.6. If 95% of the decays of the 7 / 2 state go to the 5/2 state, the cross
section for producing the 350 keV *y-ray would be 81 mb. Since the experimental
result is 115 :l: 32 mb, we conclude that the shell model calculation does reproduce
the measurement.

One issue relevant to this study is understanding the role of the nuclear interaction

in the scattering reaction measured here. We performed a calculation in which the

 

Table 4.6: ECIS calculation. Excitation cross sections (integrated over 66mg3.25°) for
states in ”Na from coupled channels calculations with an optical model parameter

set determined for the 17O + 208Pb reaction at 84 MeV/ A [83].

 

Transition 0001.: (mud 0m fie BA

 

 

Shell model
3/2+—>5/2+ 49.1 7.8 54.2 0.510 0.470
3/2+—>7/2+ 23.7 5.0 27.3 0.460 0.470

 

Fit to data
3/2"—>5/2+ 64.9 11.7 73.1 0.587 0.587
3/2+—>7/2+ 39.4 7.5 44.6 0.587 0.587

 

 

 

 

nuclear interaction was set to zero (Coulomb only), another in which the Coulomb
interaction was set to zero (nuclear only) and a third in which both interactions were
used. The angular distributions for these calculations for inelastic scattering to the
5 / 2 state are shown in Figure 4.9. We conclude that the nuclear interaction accounts
for z 15% of the cross section for the angular range detected in this experiment.
Coulomb excitation plays the dominant role in this experiment, but scattering via
the nuclear force cannot be neglected.

We can also use the standard rotational model (where the proton and neutron
deformations are equal) to directly extract a quadrupole deformation parameter 32
from the data. For this fit, we assume that 50 = 6N, that the deformation parameters
for the 5/ 2 and 7/ 2 states are equal, and that 95% of the de—excitations of the 7 / 2 state
go to the 5/ 2 state. The result, 62 2 059(8), is close to the deformation parameters

obtained for ”Mg [16, 17]. This result reflects only the experimental uncertainty,

an

 

31Na: 8/2+ —> 5/2+

 

 

 

 

 

 

105 a) Shell Model (season)
104 . ,\
i I 1 \
i V .... \
10:3 it” “xv
A 1‘
:4 \
3 102 t
\
..Q \
8 ‘~
v 1 \
10 ifiiliiiiliiiiiiiifl””liii‘h” .
G l I I
2 105 b) Rotational Model Fit (6,343") _
b
“o
104 'll \ x ‘\ 1"
I i '- J \ 3
l . ...... _\ j
8 ll' v... _
10 5‘1 \\"..
. \ ~.
1! . \ .
o 0 \ 1
103 ‘4 .‘
k c
‘\
101 7 111111111 1111L1L4J_LIJA_LAALIIAII‘JII
O 2 4 6

®cm (deg)

Figure 4.9: ECIS calculation. Shown is the angular distribution of the reaction
197Au(”Na,”Na‘)197Au exciting the 5/2+ state of ”Na. The dotted and dashed
curves represent the cross sections for the nuclear and Coulomb excitations, respec-
tively. The solid curve corresponds to the coherent sum of two excitations.

nn

Table 4.7: Experimental upper limits on reduced transition probabilities for assumed
E 1, E2, M l and M 2-transitions in 28'29'30'31Na deduced from the measured excitation

cross sections in Table 4.4.

 

Nucleus B(El T) B(E2 T) B(Ml T) B(M2 T)

(ezfmz) (e2fm4) (ezfmz) (e2fm4)

 

 

28Na 3 0.0182(40) g 86.7(19.3) g 0.2912(634) g 1234(276)
29Na g 0.0062(45) g 47.7(35.1) g 0.066(50) s 436(321)
”Na 3 0.0236(80) g 186.2(62.8) _<_ 0.205(69) g 1800(608)

 

 

 

31Na 3 0.0452(128) g 543.1(158.6) g 0.540(153) g 5489(1604)

 

an additional theoretical uncertainty which reflects the model dependence can be
introduced. Finally, the deformation parameter is 62 = 0.59 :l: (0.08) (experimental)

:l: (0.06) (theoretical).

4.4.6 Data Interpretation

All experimental excitation cross sections in Table 4.4 were corrected for possible
theoretical and experimental uncertainties due to incomplete knowledge of spin and
parity assignments and mixing ratios in 28'29’30’31Na. These cross sections were used
to deduce the upper limits on reduced transition probabilities in sodium isotopes by
using the usual Winther and Alder formalism [35]. Table 4.7 presents the reduced
transition probabilities for the sodium isotopes.

Table 4.8 presents the recommended upper limits for 7-ray strength for light iso-
t0pes (21$AS44) [70, 64, 71] and the Weisskopf (single-particle) estimates of the
reduced transition strengths [31, 84]. The products of the recommended upper limits

for 7—ray strength and the Weisskopf (single-particle) estimates are equal to the upper

 

Table 4.8: Recommended upper limits for reduced transition probabilities in light iso-
topes (21$AS44). Recommended 'y-strengths are taken from [70, 71] and WeisskOpf
(single-particle) estimates of the reduced transition strengths are extracted from Refs.

[31, 84].

 

 

 

 

Transition Recommended 7-strength Weisskopf (single-particle) estimates
EA or BA I‘.,/I‘w 92mm” or ugvfmm-m

E1 0.1 56.413-10‘2-A2/3

E2 100 $5.883-10‘2-A4/3

E3 100 _<_5.899-10'2-A2

E4 100 $6.238-10‘2-A8/3

M1 10 31.777

M2 3 51.645.A2/3

M3 10 _<_1.64-A4/3

M4 $1.736-A2

 

 

 

 

limits on collectivities. In fact, recommended 7—strengths and corresponding upper
limits on reduced transition probabilities are for transitions from the upper to the

lower levels. The ratio between reduced transition probabilities can be derived as

2J+1

Bm=2JO+1 (l).

 

(4.8)

where Jo and J are spins of the lower and upper levels, respectively. It is often more

convenient to present results for magnetic excitations in units of my instead of e-fm.

 

This can be accomplished with a ratio 1 M; = 2126 = 0.105 e-fm.

Figure 4.10 presents recommended and experimental upper limits on reduced tran-

sition probabilities for E1, E2, M1 and M 2—transitions in 28'29’30'31Na. Comparison

 

Recommended Upper Limits: D

Experimental Results: +

[:1 g g

 

103

B(E2), ezfm4 B(El), ezfmz

 

C]
Hi]
1 l l|||| llllfllfl llllllfll lJJJllfll—LILHILLUJlWfl—LUJW

2

. MN
01
O
CH

B(Ml)

H H H
O O O .... ._.
I I I o o
(.0 N H O H
1 11mm 111mm I I IMWWWWWWW
r +¢+ [3
~ l—H

l 1 1111111

 

 

 

2
Ht

102 E] E]
g a

l l l L

28 29 30 31
Na Na Na Na

Figure 4.10: Recommended and experimental upper limits on reduced transition
probabilities for E 1, E2, M1 and M 2-transitions in 28'29'30'31N a.

B(Mz), Mszm

between recommended and experimental upper limits on reduced transition proba-
bilities indicate that the experimental cross sections are consistent with E 1, E2 and
M 1 (for 28’29Na) nature of excitation and M 2 excitations are completely excluded as

a possible explanation.

4.4.7 Intrinsic Quadrupole Moments in 28293031Na

In the previous section the 31Na data was explained on the basis of shell model
calculations. The same data could have another interpretation if the possible feeding
corrections are ignored. To understand the nature of deformation in sodium we will
compare our data with the measured values of the intrinsic electric ground state
quadrupole moments. The intrinsic electric ground state quadrupole moments can be
extracted from [85] using the formalism described in [30] and our knowledge of the
ground state spin assignments. The same values for the transition moments can be
obtained from our results on B(E 2) (assuming pure E2 excitations) and the formalism
described in [32].

In general, intrinsic electric quadrupole moments values for static deformations
(rigid rotator) are not affected by excitations. On the contrary, for dynamic de-
formations transition intrinsic electric quadrupole moments are always larger than
ground state quadrupole moments. The results (presented in Figure 4.11) suggest
that 28'29’30Na are statically deformed (rotational nuclei). At the same time the ob—
served difference between intrinsic electric quadrupole moments for the ground and
transition states in 31Na would indicate the presence of dynamic deformation (a vibra-
tional nucleus) [72]. The experimental data do not allow us to discriminate between
the two possible interpretations (rotational or vibrational) of collectivity in 31Na. A

measurement of the energy level structure of 31Na would elucidate this question.

. J—

 

T j I I I I f T I I I I I I I I I I I I I I I

A : j
.2 _ m — Ground state value .
v 1250 f :
El . + — TranSItlon state value .
“é - (present work) ‘
O 1000 — —
2 . .
cu - .
F8 .. ..
o. 750 - _
:3 - .
La - .
“U - .
g p -4
0' 500 " 3 g ‘
.6 .- .+
.E 1. -
sq 250 '- —
...) l- ..
C‘. . .
b—d

O I. 1 1 1 I I 1 1 1 1 I 1 L 1 1 I m 1 1 1 I 1 1 1 1 I

 

 

 

Mass Number

Figure 4.11: Intrinsic electric quadrupole moments for the ground and transition
states in sodium isotopes (ignoring possible feeding from the higher-lying states).
Data for the intrinsic electric ground state quadrupole moments were taken from

[85].

85

 

Table 4.9: Coulomb excitation of 31”Mg, 3435A], 338i and 34F. Spin and parity assign-
ments for electromagnetic transitions from the shell-model calculations [5] (denoted
by at), [73] (denoted by *) and spin and parity assignments extracted from the nuclear

systematics [62] (denoted by T).

 

 

 

 

 

 

 

Nucleus Exam-get E, J " a 63?;
(MeV/ A) (keV) (assumed) (mb) (deg)

31Mg 62.08 905(13) 3/23. —> 5/2if 30.7(11.5) 3.24
33Mg 53.82 478(5) 7/2; —+ 3/2it 81.3(25.0) 3.27
34A1 5970 657(9) 4;; —) 3=tt 24.2(9.5) 3.28
35A] 55.83 1023(8) 5/2;;_ —+ 3/2if 30.3(13.5) 3.30
33Si 40.84 1010(7) 3/2;;_ —+ 1/2“ 4.1(080) 3.96
4200(100) mg; —+ 5/2+* 11.6(2.2) 4.62

34F 4438 422(7) 1;, —> 2+* 5.2(2.4) 4.64
627(9) 1;, —> 2+* 6.8(3.0) 3.96

 

4.5 Coulomb Excitation of 3133Mg, 3435A], 338i and
34p

Odd isotopes such as 31’33Mg, 34'35A1 and 33Si, 34F were also produced as 48Ca and
4"Ar fragments and studied. The first four nuclei are located in the island of inversion
region and they are important for understanding the collectivity among odd isotopes
in the region. 33Si and 34F provide a test case for comparison of the N~20 isotones,
because these two nuclei are not dominated by intruder configurations. Results of
the experimental observations are presented in Table 4.9.

In the present data analysis the ground state spin and parity assignments are

86

Table 4.10: Experimental upper limits for reduced transition probabilities in odd

nuclei. n / a - denotes multipolarities excluded by selection rules in the cases of known

spins and parities.

 

 

 

 

 

Nucleus E7 B(El T) B(E2 T) B(Ml T)
(keV) (e2fm2) (e2fm4) (e2fm2)
31Mg 905(13) 3 0.019(7) _<_ 124.6(46.5) g 0.200(74)
33Mg 478(5) 3 0.035(10) _<_ 343.5(1055) g 0.401(120)
34A] 657(9) 3 0.013(5) g 101.1(39.7) g 0.135(52)
35A] 1023(9) 3 0.020(9) g 124.6(55.7) g 0.240(106)
3331 1010(7) n/a 3 16.5(3.2) 3 0.044(8)
34p 422(7) n/a 5 20.2(96) g 0.025(11)
627(9) n/a g 26.5(11.7) s 016(7)

 

 

taken from the shell-model calculations by Warburton et al. [5] and excited states
assignments are deduced from shell-model calculations by Brown [73] and the sys-
tematic behavior of light nuclei [62]. Different spin and parity assignments for the
excited states can change excitation cross sections by 540%.

The observed excitation cross section of 33Mg is much larger than cross sections in
31Mg and 3435A] and comparable to that of 32Mg. This indicates that 33Mg belongs
to the island of inversion, while 3“Mg, and 3435A] lie outside the island. These cross
sections contain possible contributions from nuclear excitation and feeding from the
higher-lying levels. These results are in very good agreement with the theoretical
predictions for light nuclei [5, 12, 18].

Table 4.10 presents the upper limits on reduced transition probabilities for differ-

ent transitions in odd isotopes.

Q7

Excitation cross sections and reduced transition probabilities in 33Si and 34F are
small and consistent with WeisskOpf (single-particle) estimates of the reduced transi-
tion strengths, which are presented in Table 4.8.

Figure 4.12 shows results of the data analysis and the recommended upper limits
[70, 71] for odd isotopes. Comparison between them indicates that this experiment
is consistent with E 1, E2, and M 1 excitations while M2 excitations are completely

excluded as a possible explanation.

4.5.1 Odd Isotopes of 3133Mg and 3435A]

Beams of the neutron-rich radioactive isotopes 31”Mg and 34’35Al have been produced
by fragmentation of a 48Ca primary beam at 80 MeV/nucleon. The energies and
excitation cross sections to the lowest excited states were measured via intermediate-
energy Coulomb excitation on a 197Au target. Study of these nuclei is important
because shell-model calculations of Caurier et al. [18] predict that intruder states can
be also observed in N = 21 nuclei such as 31Ne, 32N a and 33Mg. Figure 4.13 shows the
Coulomb excitation spectrum of 33Mg. This spectrum contains a 484.9(10) keV line,
which was previously observed but not identified as belonging to 33Mg in the fl-decay
of 33Na [14]. Observation of the 484.9(10) keV transition with a large excitation
cross section of 81(25) mb is consistent with the presence of intruder configurations
in 33Mg.

Neutron-stripping reactions in 33Mg were observed as two small peaks with en-
ergies ~900 keV and ~1450 keV, which are equivalent to the transition energies in
32Mg. To prove this hypothesis two different particle gates (see discussion in sub-
section 4.3.1 were created for ions with higher than average (pure 33Mg beam) and
lower than average total energy (”Mg + 32Mg beam). The 7-spectra collected in

coincidence with these gates are presented in Figure 4.14. This data cut shows that

88

 

 

p—s
O
H

Recommended Upper Limits: D

Experimental Results: +

Cl C]
E f E]

10"3 f f

 

B(E2), ezfm4 B(El) ezfmz
E]
D

 

 

 

[3 E]
102 E f f
101 i I
100
l l l l l l l l l l 1
NZ 100 a
3. 50 i
f; % f g D :
V—I
2 10: [:1 T D
v 5E E
m — f i 1
N8 51E i f
(‘19-! 10 f f a;
Z 104
3.
- 103
A
N z
2 10 Cl C]
v E] C] C] [:1
m 101 1 1 1 1 1 1 1 1 1 1 1

 

BlMg 33Mg 34A1 35A] 33SI 3413

Figure 4.12: Recommended upper limits and experimental results in 31’33Mg, 3435.151],
33Si and 34?. Recommended upper limits are extracted from [70, 71] and experimental
upper limits are deduced using the formalism described in [35].

89

 

 

I 33 1
40 : Mg 7
I V=O .

30 7 ‘
547.5 keV ‘

20 f ‘

 

   

10}

 

 

We]

Counts/(20 keV)
O

 

 

 

 

 

 

I 4851 keV 33 I
40— () Mg -
I v=O.330:
30: i
20 _- j
10 j j
L .1

O P I 1 114 Ilnllng'flkfl LII—

l 0.5 1.0 1.5 2.0
Energy (MeV)

Figure 4.13: Energy spectrum of y-rays emitted from the 33Mg+197Au reaction at
54.3 MeV/ nucleon. Upper panel contains photon spectra in the laboratory frame and
lower panel contains Doppler-shifted 7—ray spectra.

90

 

Y I Y T Y Y I 1' V V V I T V V V

1

33Mg 1
V=0.33€

I

40_

fl

 

V fi‘ V V V I

20

 

10E-

 

 

.
1n1crl1.n.mn ‘

 

TI T T T T Y Y I V I V fj ‘
J
i

i 33 32
40 - Mg+ Mg:

Counts/(40 keV)
O

30

' I

 

:I l 885 keV
20}

 

J; 1436 keV

  

.4

l .
. 1an m
1 5 2.0

 

 

 

Obll.11..11.1
0.5 1.0

Energy (MeV)

Figure 4.14: One-neutron removal in 33Mg. Doppler-shifted 7-spectra for the 33Mg
and 33Mg +32Mg particle gates.

91

the 484.9(1.0) keV transition belongs to the 33Mg level scheme and the ~900 keV
and ~1450 keV transitions originate from de—excitations in 32Mg. The partial cross
sections for neutron stripping in 33Mg which lead to the excitation of 885.5 keV and
1436.1 keV transitions in 32Mg are 0' = 98(30) mb and a = 45(25) mb, respectively.

In contrast, 31Mg has a smaller excitation cross section than 32Mg and its value
is comparable with that of another N = 19 nucleus, 30Na. This result provides
experimental indication that the properties of 31Mg can be explained in the context
of 0 Flu) configurations.

Transition energies and excitation cross sections for 3435Al are presented in Table
4.9. The present result for 35Al is in agreement with the previous work of R.W.
Ibbotson et al. [86], while the excitation cross section for 34A] is smaller than that
of another N = 21 isotOpe 33Mg. Comparison of the nuclear properties of 34“351511 and
32'33Mg indicates that the pr0perties of aluminum isotopes can be understood within

the normal 0 hw configurations.

4.5.2 N = 19 Isotopes of Silicon and Phosphorus

Beams of 33Si and 34F were produced via fragmentation of a primary 40Ar beam at
90 MeV/ nucleon. The 'y-ray spectra of 33Si and 34F are presented in Figure 4.15.
The observed excitation cross section of the first excited state in 33Si is substan-
tially smaller than those in 32’3“Si [76] and comparable to its single-particle value
estimate. Shell-model calculations of HA. Brown [73] predict that in 33Si the ground
state is a d3/2 neutron hole and the 1010(7) keV excited state is the 31/2 neutron hole
state. The calculated B(E2 T) = 19 e2fm4 is in very good agreement with the mea-
sured value of B(E2 T) = 17(3) e2fm4. The relatively strong E2 transition strength
arises from the change in the orbital angular momentum of 2 it between the d3/2 and

81/2 nuclear orbitals.

92

6000
5000
4000
,.\3000
c” 2000

CD 1000

800

Counts/(4

600

400

200

 

III IIIII'

IIIII

rrrrrr

 

 

 

11'111

LLJilLA

 

I I I I I I I I ‘

' I

 

llllL

llllllLlJJll—ljl

33..
81.

v=0.29o5

 

 

1

2345

500 ..

400

300

N
O
O

P-I
O
O

Counts/(20 keV)

300

200

100

 

TIWI

I frI

 

 

 

 

 

 

LleLngLllllLLlj

 

0.5 1.0 1.5 2.0

Energy (MeV)

Figure 4.15: Experimental 7—ray spectra of 33Si and 34F. Upper panels contain pho-
ton spectra in the laboratory frame and lower panels contain Doppler-shifted y-ray
Spectra. The 547 keV (7 /2+ —+ 9.3.) transition in the gold target is visible as a
peak, while 1010 keV, 1941.5 keV and 429 keV, 625 keV transitions are present in
the Doppler-shifted 7-ray spectra of 3:‘Si and 34P, respectively.

93

 

The shell model predicts the next concentration of E2 strength to occur at 4
MeV, and this might correspond to the 2+ state built on the ground state in 34Si.
The high-energy part of the 33Si spectrum behaves like a step-function at 4.2-4.3
MeV, which indicates a possible excitation of the 4330(30) keV state in 33Si [62] or a
neutron-stripping reaction and further de-excitation of the second 2+ state in 32Si at
4230.8 keV. The extracted cross section of 11.6(2.2) mb and corresponding B(E2 T)
2 69(13) e2fm4 are in a very good agreement with the theoretical prediction for the
E2 strength at 4 MeV.

A neutron-stripping reaction in 3L’Si with further 1941.5 keV transitions in 32Si
and a corresponding cross section a = 11.7(1.4) mb was observed. The difference in
cross section values for neutron-stripping reactions in magnesium and silicon can be
explained by the neutron separation energies, which are 2070(170) keV and 4483(16)
keV in 33Mg and 33Si, respectively.

According to the shell-model calculations of BA. Brown [73] in 34F both the
ground state and the 429 keV state are members of the multiplet formed by the
coupling of a 31/2 proton to the d3/2 neutron hole. Multiplet members are usually
connected by M 1 transitions, not E2 transitions. The difference between the shell-
model predicted B (E21) = 0.18 ezfm“ and the experimentally measured upper limit
on B(EZT) = 12(6) e2fm“ supports the M1 nature of decay. The M 1 nature of de-
excitation is also in agreement with the observation of the 429 keV transition in the
Doppler-reconstructed energy spectrum of 34F because at intermediate energies the
”P ions will travel up to 28.7 cm before decaying via E2 transition.

The next multiplet up in energy is one formed by the coupling of the d3/2 neutron
hole to a d3); proton. This gives possible 1*, 2+ and 3+ states, and these states
should be connected to the ground state via fairly strong E2 transitions because of

the orbital angular momentum change of 2 h between the d3/2 and 31/2 orbits. In

.04

 

5/2+

 

 

 

 

 

 

 

4200(100) keV

2+

1 625 keV

1+

l/2+ 1607 keV 1178 keV
1010 keV 2
y I 429 keV

3/2+ 1+

 

338i 34p
Figure 4.16: Level schemes of 33Si and 34F.

fact, calculations predict a 1+ state at 1.4 MeV that connects to the ground state via
a 3.3 ezfm4 E2, a 2+ state at 2.2 MeV with B(E2) = 5.8 e2fm4, and a 3+ state at
2.7 MeV with B(E2) = 6.9 e2fm4. Figure 4.16 demonstrates that the 1.4 MeV state
probably corresponds to a previously known 1+ state at 1607.6(2) keV. The relatively
large B (E2) values for the second multiplet are in agreement with an observed cross
section for the 627(9) transition in 34P, which is probably due to de—excitation of the
second excited 2+ state at 2232 keV to the first excited 1+ state at 1607 keV.
Unfortunately our experimental result is in partial disagreement with the shell-
model calculations [73], which predict that the branching ratio from the 2232 keV
state to the 1607 keV state is only 1.9% and in 67.4% of cases this state de-excites via
the 429 keV state. A similar problem exists between the calculated and experimentally
known branching ratios for the decay of the 1607 keV state into the 429 keV state [62],
which are 97% and 64%, respectively. An alternative explanation of the disagreement
between the shell-model model calculations and the experimental observations is a

negative parity state at 625 keV.

95

 

A more complete understanding of the experimental level scheme of 34F is neces-

sary to clarify this problem.

 

96

Chapter 5

Summary

Two intermediate-energy Coulomb excitation experiments with nuclear radioactive
beams produced via nuclear fragmentation of 40Ar and 48Ca primary beams have
been performed at the National Superconducting Cyclotron Laboratory. In the first
experiment the neutron-rich radioactive nuclei of 26Ne, 3°Mg, 28Na, 33Si and 34F were
produced and studied. In the second experiment, neutron-rich radioactive nuclei of
28Ne, 29130131Na, 3’1’32’33'34Mg and 3435A] were investigated.

A primary motivation for these experiments was the study of the nuclear structure
of the N ~ 20 neutron-rich nuclei. Only recently, modern accelerator technologies
provided an opportunity to study this region. The 32Mg measurements, which were
conducted at CERN and RIKEN [13, 14, 15, 16], indicate that this isotope has large
deformation, high collectivity and a low energy of the first excited state. Such a
combination of nuclear properties is explained by the inversion of the normal shell
ordering (the u(f7/2)-neutron orbitals lowering in energy into the V(sd)-shell). Nuclear
theory [5, 18] indicates that intruder configurations are also present in 33Mg, 3132Na
and 3031Ne, which form the so—called island of inversion or island of deformed nuclei.

The scientific objective of this thesis was to create a more complete and com—

prehensive experimental picture of deformed light nuclei in this region. Properties

97

of the first excited states in 15 different nuclei were studied, including 32’33Mg and
31Na. Results for 32Mg are in good agreement with the previous measurement. The
observation of the 1436(1) keV transition indicates feeding, which can reduce the
B(E2) value for 32Mg by 27%. Due to the uncertainty in spin and parity assignments
in odd isotopes, the results on 33Mg are not conclusive. However, the large observed
excitation cross section and transition energy in 33Mg are comparable to that of 32Mg.

The current work is the first study of the excited states in 28’29’w’31Na. Intrin-
sic ground state and transition electric quadrupole moments in sodium isotopes, in
which possible feeding from higher-lying excited states was ignored, were compared.
Agreement between absolute values of the ground and transition state moments in
2839*”Na indicates a rotational nature of deformation while a relatively large tran-
sition quadrupole moment in 31Na could indicate the vibrational nature of the de-
formation. However, this experimental study of the nuclear properties of the first
excited state in 31N a is not sufficient for a definite conclusion.

The study of transition energies and excitation cross sections for the first excited
states in 34'35Al, 31Mg and 30N a experimentally proves that the island of inversion is
centered at N = 20 and Z S 12. Other isotopes which were studied in this thesis,
such as 30’34Mg, 26'23Ne, 338i and 34F provide information about nuclear properties in
the vicinity of the island, such as the energy of the 2? state and collectivity in 28Ne,
which suggest strong mixing between the intruder and normal configurations in this
nucleus.

An ongoing upgrade of the NSCL accelerator facility combined with the devel-
opment of the NSCL segmented Ge—detectors and a new NaI(Tl) array will provide
additional capabilities to complete Coulomb excitation studies of the island of inver-
sion region. At the same time strong progress in this field is expected at RIKEN,

REX-ISOLDE and GANIL, which will result in a better understanding of nuclear

98

 

structure and collectivity in this interesting region.

99

Appendix A

Cross Section Calculations

Cross sections for electric and magnetic transitions were calculated by using the
boris_wi79 .m code, which is based on equation 2.23 and located in the /usr/TruClus
ter/users/prityche/math subdirectory. This code uses definitions and constants
which are located in lusr/TruC1uster/users/prityche directory. Consequently,

constants and definitions are loaded first by using an incl .m file which is located in

/usr/TruC1uster/users/prityche directory.

<</usr/local/math/Packages/Graphics/Graphics.m
<</usr/TruC1uster/users/prityche/constants.m
<</usr/TruCluster/users/prityche/gen_defs.m
<</usr/TruC1uster/users/prityche/em_trans.m
<</usr/TruCluster/users/prityche/lboost.m

LoadInclude=True

Finally, the actual cross section calculation code (boris_wi79 .m code) is presented

below.

(* make sure the constants and definitions are loaded *)

<</usr/TruC1uster/users/prityche/ incl .m

100

ShowParm[t_]:=Do[

Print[t[[11][[2]]," = ",tffiII[[11]].{i,1,Length[t]} I

Xibeta[]:=Block[{temp,tempA,Ecm,mu,gamma,a0} ,
gamma=Eva1uateESqrt[1-BetaBeam‘2]‘-1];

mu = mNucl * Atarget*Aprojectile/(Atarget+Aprojectile);

 

(* reduced mass in MeV *)
tempA = Ztarget*Zprojectile * FineAlpha * HBarC ;
(* define this since we have to use it later on *)
Ecm = gamma mu BetaBeam“2 /2;
a0 = tempA / (2 Ecm) ;
(* half distance of closest approach in head on
collisions (fm) *)
BMin=NEaO CotEThetaMax/2IJ;
(* minimum impact parameter (fm)*)
RMin=N[tempA/(2 Ecm) + Sqrt[(tempA/(2 Ecm))“2 + BMin“2]];
(* distance of closest approach *)
Xi=DeltaEnergy/(HBarC*gamma*BetaBeam) * BMin;
(* adiabaticity parameter xi w/o correction term *)
XiHigherOrder=De1taEnergy/(HBarC*gamma*BetaBeam)*
(BMin + Pi/2 Ztarget Zprojectile FineAlpha *
(* w/ correction term *)
HBarC / (mu gamma BetaBeam“2) );
temp=EvaluateEaO/gamma * DeltaEnergy/

(HBarC*gamma*BetaBeam)];

101

ClearEXiFuncTh];

XiHigherOrder

I
GEElam_,mu_,invbeta_]:=
Block[{GEposmu,posmu},

posmu=AbsImu];

GBposmu=Evaluate[SimplinyI‘(lam+posmu) *

 

Sqrt[16 Pi]/(lam*(2 1am +1)!!) *
Sqrt[(lam-posmu)!/(lam+posmu)!] *
Sqrtfinvbeta‘2 - 1]“-1 *
((1am+1)(lam+posmu)/(2 lam + 1 ) *
(Evaluate[LegendreP[lam-1,posmu,3,xxx
]]/.xxx->invbeta) -
1am * (lam-posmu+1)/(2 lam + 1 ) *
(Evaluate[LegendrePElam+1,posmu,3,xxx
]])/.xxx->invbeta)]];
If[mu>=O,GEposmu,(-1)‘mu * GEposmu,
Print["siw in GE[]"] ]
I
GM [lam_ ,mu_ , invbeta-) :=
Block [{GMposmu , posmu} ,
posmu=AbsImu];
GMposmu=Eva1uate[SimplinyI‘(lam+posmu+1) *
Sqrt[16 Pi]/(lam*(2 lam +1)!!) *
Sqrt[(1am-posmu)l/(lam+posmu)l] *

Sqrt[invbeta“2 - 1]“-1 *

102

( posmu * (Evaluate[LegendreP[1am,posmu,3,xxx

II/.xxx->invbeta))]];

If[mu>=O,GMposmu,(-1) * ((-1)‘mu) * GMposmu,
Print["siw in GM[]"] ]

I
gAlEmu_,xi_]:=Block[{mupos},

mupos=Abs[mu];

Pi xi‘2 ( Abs[Besse1K[mupos+1,xi]]‘2 -
Abs[BesselK[mupos,xi]]“2 -

2 mupos/xi BesselKEmupos+1,xi] BesselKEmupos,xi]) ]

SigmaCEIxi_]:=N[(Ztarget*FineAlpha)‘2 * (DeltaEnergy/HBarC)‘
(2 (LambdaExcitation-1)) *

IfEEorM==1,BeELam,BeMLam] *

Sum[If[Eor ==1,Abs[GEELambdaExcitation,mu,1/BetaBeamJJ‘2,
Abs[GMELambdaExcitation,mu,1/BetaBeamIJ“2] *

gAlEmu,xi],{mu,-LambdaExcitation,LambdaExcitation}],7]

C1ear[ThetaMax,DeltaEnergy,BetaBeam,Atarget,Ztarget,

Aprojectile,Zprojectile,LambdaExcitation,BeELam,BeMLam,EorM]

InputSigmaII:=(

ThetaMax =

Input["Max. sc. angle in CM in Degrees : "J Degree;
DeltaEnergy =

Input["Excitation energy in MeV : "J;

103

BetaBeam =

 

Input["Beam velocity in c : "I;
Atarget =

Input["A of target 2 "I;
Ztarget =

Input["Z of target : "I;
Aprojectile =

Input["A of projectile : "I;
Zprojectile =

Input["Z of projectile : "I;
LambdaExcitation =

Input["Multipolarity of transition : "I;
EorH =

Inputf" Is it electric - 1 or
magnetic - 0 transition? : "I;
If[EorM==1,Print["B(E",LambdaExcitation,") in e‘2 fm‘",
2 LambdaExcitation],
Print["B(M",LambdaExcitation,") in 972 fm7",
2*LambdaExcitationII;
If[EorM==1,BeELam = Inputf" : "I,
BeMLam = Input[" : "II;
If[EorM==1,BeMLam=O,BeELam=OI;
(* Print["B(E",LambdaExcitation,") in e“2 fm“",
2 LambdaExcitation];
BeELam = Inputt" : "l;

Print["B(M",LambdaExcitation,") in muN“2 fm“",

104

2*LambdaExcitation-2];

BeMLam = Input["
)
SigmaList:={

{ThetaMax,
{DeltaEnergy,
{BetaBeam,
{Atarget,
{Ztarget,
{Aprojectile,
{Zprojectile,
{LambdaExcitation,
{BeELam,
{BeHLam,
{EorM,
}

CheckDef[t-]:= Block[{notdef},
notdef=False;

DoI

"ThetaMax"},
"DeltaEnergy"},
"BetaBeam"},
"Atarget"},
"Ztarget"},
"Aprojectile"},
"Zprojectile"},
"LambdaExcitation"},
"BeELam"},
"BeMLam"},

11 EOI‘M" }

If[N[t[[i]]][[1]]==0,Nu11,Null,Print[tlfiJI[[1]]."

is not defined;"];notdef=True]

{i , 1 , Length [1:]

9

105

ll;

notdef]

SigmaE]:=Block[{sigma},
If[CheckDef[SigmaList],InputSigmaIII;
(* start calculation *)
Xibetaf];
sigma = SigmaCEEXiHigherDrder];

sigmaNoCor = SigmaCEin];

PrintE" "];

PrintE" "J;

PrintE" ---------------------------------- n];
Print["Tar89t I A = ",Atarget," Z = ",Ztarget];

Print["Projectile : A = ",Aprojectile," z = ",Zprojectile];
Print [" ---------------------------------- n] ;

Print ["‘I‘hetaMax = " ,ThetaMax] ;

Print["Beam velocity ",BetaBeam," c "I;

",NEBMinJ," fm"];

Print["BMin (impact parm.)
Print["RMin (closest appr.) = ",N[RMin]," fm"];
PrintE" ---------------------------------- "I;
Print["Excitation Energy = ",
DeltaEnergy," MeV "I;
Print["Mu1ipolarity of Transition: ",
LambdaExcitation];
Print["Xi =",N[Xi]," xi w/ higher order =",

N[XiHigher0rder]];

106

If[EorH==1,

Print["B(E",LambdaExcitation,") .
BeELam," 9‘2 fm“",
2 LambdaExcitation],

Print["B(M",LambdaExcitation,") = ".
BeMLam," e‘2 fm‘",
2*LambdaExcitationII;

Print[" --------------------------------- "l;

Print["Projectile excitation cross section:"];

Print["Values in parenthesis are w/o the correction
term for Xi"];

Print["sigma = ",sigma,"(",sigmaNoCor,")"," fm“2 = ",
10 sigma," mb "I;

Print[" ---------------------------------- "J;

sigma

]

Print["Use Sigma[] to calculate Coulomb Excitation
cross sections”]

Print["Use AngDisE] to calculate Angular Distributions"]
Print["Use An[] to calculate Angular Distributions;

only \n changing the variables in AngDisList"]
Print["Use ShowParmESigmaList] and ShowParm[AngDisList]

to see \nwhich variables to change "I

This code allows the calculation of experimental cross sections when Zp, 2;, AP, At,

93,37, vp, energy of the first excited state and reduced transition probability (B(7r, A))

107

are known (see equation 2.23). In practice, the analysis of experimental data produces
excitation cross section which allows the present code to calculate B(7r, A).

The Winther and Alder functions (0“,, (g), gp(§(bm,-n))) can be calculated as
fbuons:
troy.nscl.msu.edu> math
Mathematica 4.0 for Digital Unix

Copyright 1988-1999 Wolfram Research, Inc.

-- Motif graphics initialized --

In[1]:= <<boris_wi79.m
-- File graphics initialized --
loaded constants.m
Use SigmaE] to calculate Coulomb Excitation cross sections
Use AngDisE] to calculate Angular Distributions
Use An[] to calculate Angular Distributions; only
changing the variables in AngDisList
Use ShowParmESigmaList] and ShowParmEAngDisList] to see

which variables to change
In[2] = NEGEE1,1,x]]

1. + x
1.67109 (-1. + x) x Sqrtf ------- ]
-1. + x
0ut[2]= ---------------------------------

2

108

SqrtE-l. + x ]

In[3] := NEGM[2,1,x]]

1. + x
0.289441 (-1. + x) x Sqrtf ------- J
-1. + x
0ut[3]= ----------------------------------
2
Sqrt[-1. + x I
In[4]:= NEgA1[1,x]]
2 2 2

0ut[4]= 3.14159 x (-1. Abs[Besse1K[1., x3] + Abs[BesselK[2., x]] -
2. BesselK[1., x] BesselK[2., x]

> ................................ )

X

In[5]:= Quit

These calculated values reproduce the values tabulated in [35].

109

 

I’Jl

Appendix B

 

Calculation of Angular

Distributions

The angular distribution of Coulomb de—excitation y-rays was calculated by using a
boris_ang_dis .m code which is based on equation 2.31 and located in lusr/TruCluster/
users/prityche/math subdirectory. The boris_ang_dis .m code was written in MATH-

EMATICA and presented below.

<</usr/TruCluster/users/prityche/math/boris_wi79.m

Unprotecth]
F[k_,Jf_,L1-,L2_,Ji_]:=(-1)”(Jf-Ji-1) * Sqrt[(2*L1 + 1) (2*L2 + 1) *
(2*Ji+1)] * ClebschGordan[{L1,1},{L2,-1},{k,0}] *

Racahw [{Ji , .11 ,L1 ,L2} , {11, Jf}]
Clear [Ii , If , Jff , DeltaMixing, ParityChange]

I nputAngDis [] :=(

IfECheckDefISigmaList],InputSigmafII;

110

Ji = Input[" Initial angular momentum : "J;
Jf = Input[" Angular momentum of excited state : "I;
Jff = Input[" Final angular momentum : "J;
DeltaMixing = Input[" mix. ratio for mixed multip. de-exc. : "J;

ParityChange= Input[" Paritychange between If and Jff?
1/0 (= y/n) : "I;
(*EorM = Input[" Is it Electr. or Mag. transition? 1/0 (= y/n) : "];*)
)

AngDisList:={

{Ji, "JiHI,
{Jf, llJfll},
{Jff, "Jff"},

{DeltaMixing,"DeltaMixing"},
{ParityChange,"ParityChange"}
}

CheckInputAngDisE]:=Block[{temp},
If[Abs[Ii-If]>LambdaExcitation,Print["AbsIJi-JfI >
LambdaExcitation"];temp=True];
If[Ji+Jf<LambdaExcitation, Print["Ji+Jf < LambdaExcitation"];
‘temp=TrueI;

temp]
AngDisE] := Block [{kMax,LamMin,LamMax},

IfECheckDefEAngDisListI,InputAngDistI;

IfECheckDefESigmaListI,InputAngDistI;

111

If[NEXiHigherOrder]==O,Null,XibetaII,XibetaIII;

If[CheckInputAngDis[I,Print["Calculation aborted"I;Return[O]];
(* Ji gets excited to If which then decays to Jff;

in most cases we have Ji=Jff *)

(* find out what multipolarities for de-excitation are possible *)
LamMin=AbS [If-JffI ;
If[(LamMin==O)&&(Jf+Jff>0),LamMin=1,Null,Print["siw"I I;

(* there is no no or E0 transition *)
If[ ((Mod[LamMin,2I==0)&&(ParityChange==0)) ||

((ModELamMin,2]==1)&&(ParityChange==1)).

LamMax=LamMin, (* Pure ELamMin transition *)
LamMax=Min[LamMin+1,Jf+JffI,

(* for Mlam transition consider next higher E transition if at all

possilble *)

Print["siw AngularDistrib"II;

(* Print ["LamMax " ,LamMaxI ;

Print ["LamMin ",LamMin]; *)

(* now we can determine the coeff. in front of the P_1m *)

(* first we determine kMax *)
kMax=MaxE2Jf,2LambdaExcitation];

Do[A[kI=Sum[
(* Print["ll " ,11I;
Print["12 ",12I;
Print["mu ",muI; *)
If[11==LamMin,1,DeltaMixing,Print["siw AngularDistrib 2"]]

12f[l2==LamMin,1,DeltaMixing,Print["siw AngularDistrib 3"II *

112

IfEEor ==1,

Abs[GEELambdaExcitation,mu,1/BetaBeamII‘2,

Abs[GMELambdaExcitation,mu,1/BetaBeamII‘2] *

gA1[mu,XiHigher0rder] (-1)‘mu *

ThreeJSymbol[{LambdaExcitation,mu},{LambdaExcitation,-mu},
{k,0}] *SixJSymbol[{Jf,Jf,k},{LambdaExcitation,LambdaExcitation,Ji}]
ka,Jff,l1,12,Jf] Sqrt[2k+1I,

{11 , LamMin , LamMax} , {l2 , LamMin , Laml‘iax} ,

 

(* sum over the possible de-excitation multipolarities *)
{mu,-LambdaExcitation,LambdaExcitation}

(* sum over magnetic QN of excitation *)
I (* close Sum *)

,{k,0,kMax,2}I; (* close do-loop *)

Do[ Print[”a",k," ", NEAIkI/AEOIII ,{k,0,kMax,2}I;

Print["w = 1/(4 Pi) (a0 PO + a2 P2 + ...)\n\n"];
Clear[W,WboostedI;
WEtheta-]:=Evaluate[Sum[ N[A[kI /A[0I /4 /Pi,15I

LegendrePEk,Cos[thetaII,{k,0,kMax,2}II;

(* angular distrib normal. to one *)
Hcontracted[thetacm_]:=W[thetach*DomCmDomLab[thetacm,BetaBeamI;
WboostedEthetalab_]:=Wcontracted[ThetaCm[thetalab,BetaBeam]I;
Print["HEthetaI defined as Angular distribution in

projectile frame "I ;

Print["WboostedEtheta] defined as Angular distribution

in labframe "J;

I

113

Anf]:=(InputAngDis[];AngDis[])

The boris-ang_dis.m code requires information on target and projectile parame-
ters, which are set by the boris-wi79.m code. The boris.wi79.m code is located
in the same directory and discussed in Appendix A. An example of use of the
boris-ang_dis.m for Coulomb cross section calculations for E2T transitions is pre-

sented below.

troy.nsc1.msu.edu> math
Mathematica 4.0 for Digital Unix
Copyright 1988-1999 Wolfram Research, Inc.

-- Motif graphics initialized --

In[1]:= <<boris-ang_dis.m
-- File graphics initialized --
loaded constants.m
Use SigmaE] to calculate Coulomb Excitation cross sections
Use AngDisI] to calculate Angular Distributions
Use An[] to calculate Angular Distributions; only
changing the variables in AngDisList
Use ShowParm[SigmaList] and ShowParmEAngDisListI to see

'which variables to change

InI2]:= SigmaII

ThetaMax is not defined;
[JeltaEnergy is not defined;
BetaBeam is not defined;

Atarget is not defined;

114

Ztarget is not defined;
Aprojectile is not defined;
Zprojectile is not defined;
LambdaExcitation is not defined;
BeELam is not defined;

BeMLam is not defined;

EorM is not defined;

 

Max. sc. angle in CM in Degrees : 3.24

Excitation energy in MeV : 0.35

Beam velocity in c : 0.3195
A of target : 197

Z of target : 79

A of projectile : 31

Z of projectile : 11

Multipolarity of transition : 2

Is it electric - 1 or magnetic - 0 transition? : 1

B(E2) in e‘2 fm‘4

: 100
'Target : A = 197 Z = 79
IProjectile : A = 31 Z = 11
'IhetaMax = 0.0565487
Beam velocity = 0.3195 c

115

BMin (impact parm.) = 16.4614 fm
RMin (closest appr.) = 16.9336 fm
Excitation Energy = 0.35 MeV

Mulipolarity of Transition: 2
Xi =0.0865958 xi w/ higher order =0.0904428

B(E2) = 100 e‘2 fm‘4

 

Projectile excitation cross section:
Values in parenthesis are w/o the correction term for Xi

sigma = 2.35695(2.56844) fm“2 = 23.5695 mb

OutI2I= 2.35695

In[3I:= Quit

Finally, it is worth to mention that this code is good for both EAT and MAT transi-

tions.

116

Appendix C

Detector Calibrations

The present appendix contains a description of the standard calibration procedure of
the NSCL NaI(Tl)-array, which includes position, energy and efficiency calibrations
and the generation of calibrated position, energy and efficiency data files for the

nuclear data analysis.

C.1 Position Calibrations

Usually position calibrations of the NSCL NaI(Tl) array are conducted before and
after each experimental run by using the special collimator described in subsection
3.3.1. A 60Co radioactive source inserted between two discs made of HeviMet can be
installed and moved inside the beam pipe with a calibrated stick (measuring tape is
attached to it). This allows to calibrate all detectors at the same time by moving the
source in 1.27 (0.635) cm-long steps. The duration of a typical calibration run is 5
minutes.

The physical principles of position calibrations are described in subsection 3.3.2.
Here I will present a practical recipe for development of a calibrated position data

file:

117

0 Convert position on the stick into metric system (cm) and add the distance
from the end of stick to the actual position of 60Co radioactive source in the HeviMet
assembly (7.62 cm + 0.23 cm = 7.85 cm).

0 Create Smaug spectra by using the position.def file from nsc1_nuclear:
[96037.boris .position] subdirectory. For each run (certain position on the stick)
one has to dump raw position files.

0 Convert Smaug files into GeLi fit standard and fit it using modified command file
gfl . cmd. Rename gf2 . out files into pos_* . out. Pos_* . out files contain information
about position fit for each detector.

0 Create detNN-fit.dat files by cutting and pasting from pos-* . out files. These
files should have two columns. The first is position in cm the second one is channel
number (see for example det01_fit-example .dat).

0 Start Physica by typing setup physica and after that physica. Start the
fitting routine by typing intall. It will produce files called detNN.out and will
print position curves.

0 Update a maketab1e2.for code (east and west edges of the array) and run it.

It will produce file poscal_cubic2.dat which can be used in the broutines.

C.2 Energy Calibrations

Subsection 3.3.3 contains the description of energy calibrations which include 2-hours
long data collection runs with a radioactive sources (152Eu, 228Th, 88Y and 22Na)
placed in the center of the array and development a calibrated-energy data file.

The calibrated-energy data file is generated as follows:

0 Create Smaug files with cuts over non-calibrated position for each detector using
gates .for file from the nsclmuclear: [96037.boris .energy] subdirectory.

0 Convert Smaug files into GeLi fit standard and fit them, using command file

118

 

gf2.cmd (gf2.for). Rename gf2.out files into 22na.out, 228th.out, 152eu.out
and 88y.out.

0 Run enca1.for, which will create gelieca1_XXX.list, where XXX stands for
the slice number.

0 Run Physica by typingzorichfit geliecaLXXX ‘Energy’. File richf it .pcm
can be edited for linear or quadratic fit.

0 Create 198XXX.dat file and update phyres11.for file. Run physresl1.for,
which will take the fit results from geliecallxx . out and place them into 198XXX . dat.

0 Basically 198XXX.dat can be used for the energy calibration in the broutines.
At the same time one can create en_array.tab files.

0 Create en_array.tab by running trans.for, which will produce en_array.
tab-new. Rename en-array.tab-new into en-array.tab and use it for energy cali-

bration in the broutines. Currently, trans .for is set for 10 slices.

C.3 Efficiency Calibrations for Isotropic Source

Efficiency calibrations (measurements) for isotropic sources are described in subsec-
tion 3.4.1. The efficiency calibration procedure is conducted with the calibrated 7
radioactive sources (”Na, 88Y, 152Eu and 228Th) located in the exact target position.
Nuclear data is collected for each 2-8 hours long calibration runs and analyzed. The
extracted efficiency of the array is presented in Figure 3.14. The experimentally mea-
sured detection efficiency is more preferable than calculated because of the complex
design of the experimental apparatus. The calibration ’7-sources are presented in Ta-
ble C.1. To achieve consistency between different radioactive sources activity of 152Eu
was reduced by 21%.

The development of the calibrated efficiency data file consists of the following

steps:

119

Table C.1: Description of the calibration y-sources (* - denotes uncertainties at the
99% confidence level, T - denotes uncertainties with an unknown confidence level and

1 - denotes a corrected source strength).

 

 

 

 

 

Parent nucleus NSCL source number Source strength“ / Date
22Na _ NSCL-132 10.46 i 0.37/1Ci (ll/1/1988)
88Y NSCL-507 11.5 d: 1.5)uCil (12/15/1996)
1521311 NSCL-139 14.64 :t 0.542501I (11/1/1988)
228Th F9074 11.47 i 0.3711Ci (10/1/1994)
228Th NSCL-503 1.286 :1: 0.03211Ci (6/ 1 / 1995)

 

 

 

 

o The preliminary requirements include good energy and position calibrations,
and good knowledge of the radioactive sources activities, branching ratios, live time
and total time of the measurements.

0 Cut each detector into 10 slices over calibrated position, and plot calibrated
energy with no multiplicity for each slices (152Eu,228Th,88Y and 22Na). It is good to
use 10 keV/ch binnings. This can be done in Smaug using the modified gatec.for
file from the nsclmuclear: [96037.boris.efficiency] subdirectory.

0 Convert Smaug output files into GeLi fit standard.

0 Fit photopeaks for all sources by using the gf2.cmd command file. After that
rename gf2.out files into eff22na.out, eff88y.out ,

0 Run effca1152eu1.for, which create eff152eu1.list. Run effca1th1.for
0 Run eff97013-152eu1.for, eff97013-152eu2.for, which will create

files *.tab1e. The typical order is 152Eu, 228Th, 88Y and 22Na.

0 Look into *.tab1es files and identify the files which are empty or different from

120

 

the rest.
oldeateoverallq,10pos-det.pcm, overallqupos-det.pcm and effgfit4thr.
pcm for the detectors which are different from the rest.

a Start Physica and type syst=0.05 (5% systematic error originates from an
imperfect personal judgment of the fit boundaries). In physica type Qoverall and
after that det01 , . . . this will create file effdetOl .out,

0 Using file effdet01. . . one has to edit 9XXXX-eff .m and genera19xxxx_eff.m.

An example of an efficiency calculations for E2 transitions (“N a) and 11 detectors

(inner ring) is presented below.

troy.nscl.msu.edu> math
Mathematica 4.0 for Digital Unix
Copyright 1988-1999 Wolfram Research, Inc.

-- Motif graphics initialized --

In[1]:= <<genera19701311-eff.m
-- File graphics initialized --
loaded constants.m
Use SigmaE] to calculate Coulomb Excitation cross sections
Use AngDisII to calculate Angular Distributions
Use An[] to calculate Angular Distributions; only
changing the variables in AngDisList
Use ShowParmESigmaList] and ShowParmEAngDisList] to see
which variables to change
Use Effic[] to calculate efficiencies

Function Effwfbeta,energy,i1,i2,HalfThick] defined

121

 

 

In[2]:= detlist={1,2,3,4,5,6,7,8,9,10,11}

0ut[2]= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

In[3]:= EfficEI

File to load efficiency array from? 9701311_n101-eff.m
Please enter Half-Thickness for Au at this energy (mg/cm‘2) : 2552
Please enter actual total thickness of Au target (mg/cm“2) : 702
Ji is not defined;

If is not defined;

Jff is not defined;

DeltaMixing is not defined;

ParityChange is not defined;

ThetaMax is not defined;

DeltaEnergy is not defined;

BetaBeam is not defined;

Atarget is not defined;

Ztarget is not defined;

Aprojectile is not defined;

Zprojectile is not defined;

LambdaExcitation is not defined;

BeELam is not defined;

BeMLam is not defined;

EorM is not defined;

Max. sc. angle in CM in Degrees : 3.24

Excitation energy in MeV : 0.35

122

Beam velocity in c : 0.3195

A of target : 197
Z of target : 79
A of projectile : 31
Z of projectile : 11
Multipolarity of transition : 2
Is it electric - 1 or magnetic - 0 transition? : 1

B(E2) in e“2 fm“4

: 310
Initial angular momentum : 1.5
Angular momentum of excited state : 2.5
Final angular momentum : 1.5

mixing ratio for mixed multip. deexcitation : 0

Paritychange between If and Jff? 1/0 (= y/n) : 0

ClebschGordanzztri: ThreeJSymbol[{1., 1}, {2., -1}, {0, 0}]

is not triangular.

ClebschGordanzztri:

SixJSymbol[{2.5, 2.5, 0}, {2., 1., 1.5}] is not triangular.

ClebschGordanzztri: ThreeJSymbol[{1., 1}, {2., -1}, {0, 0}]

is not triangular.

Generalzzstop: Further output of ClebschGordan: tri

will be suppressed during this calculation.

123

-32

a0 = 1. - 4.93038 10 I

a2 0.134231 + O. I

a4

0. + 0. I

W = 1/(4 Pi) (a0 PO + a2 P2 + ...)

W[theta] defined as Angular distribution in projectile frame
Wboosted[theta] defined as Angular distribution in labframe
The total raw efficiency of the array is 20.0934 %

For projectile peak, efficiency is 17.5029% +/- 0.08441312
The total raw efficiency of the array is 20.0934 2

Including absorption: 10.8893% +/- 0.0552119%

a0 = 1.
a2 = -0.385396
a4 = -0.0240751
a6 = 0.

W = 1/(4 Pi) (a0 P0 + a2 P2 + ...)

W[theta] defined as Angular distribution in projectile frame
Wboosted[theta] defined as Angular distribution in labframe
The total raw efficiency of the array is 15.0157 2

For target (gold) peak, efficiency is 14.0039% +/- 0.0670267%
The total raw efficiency of the array is 15.0157 Z

Without absorption, 0 547.5 keV: 16.8188%

124

 

In[4I:= Quit

Such calculations can be performed for all EA and BA transitions.

125

 

Appendix D

ECIS Calculations

A phenomenological Saxon-Woods nuclear potential (optical potential) is usually de-
fined as follows [87]
V7.0) = (V + z'W)f(7‘), (Doll

where V(r) and W(r) are real and imaginary parts, f(7) = (1 + e$p((r - Ro)/a)‘1

with R0 = 70(A],/3 + A2”), To and a are the potential reduced radius and diffuseness,

respectively. The Coulomb potential Vc(r) (potential between a point charge and
1/3

a uniform charge distribution with a radius RC = rc(A,1,/3 + A, ), where re is the

reduced charge distribution radius) is defined as follows [87]

ZpZteZ/r for 1‘ 2 Re
Vc(7‘) = - (D2)

ZpZtez/2RC [3 — (r/Rc)2] for r < Re
In the current calculation the following optical potential parameters were used [83]:

V=50 MeV, W=57.9 MeV, rv=rw=1.067 fm, av=aw=0.800 fm and r6212 frn.

The input data file for the coupled-channels code ECIS88 [82] with an Optical
model parameters set determined for the 17O+2°8Pb reaction at 84 MeV/ nucleon [83]
is presented below. This particular file was created by L.A. Riley and used for the

calculation of nuclear contribution of the 31Na+197Au reaction at 51.5 MeV/ nucleon

[75].

126

197Au on 31Na Elab = 51.5 MeV/A 5/2 State Coulomb + Nuclear Rot.
tFFFTFFtFFTTFFFTFFFFTFFFFFFFTFFFFFFFFFFFFFFTFFFFFF
FFFFFFFFFFFFFFFFTTTFfFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

25000 500

1.5 00 10145.5000 0.00000 197.0000 31.0000 869.
2.5 00 0.350

2 2 1.5

0.5900

0.5900 0.0 0.0 0.0 0.0 0.5900 0.0
50.0 1.067 0.800

57.9 1.067 0.800

1.200

0.01000 0.01000 3.2400
FIN

//

********************************************************************

197Au on 31Na Elab = 51.5 MeV/A 5/2 State Coulomb (only) Rot.

127

tFFFTFFtFFTTFFFTFFFFTFFFFFFFTFFFFFFFFFFFFFFTFFFFFF
FFFFFFFFFFFFFFFFTTTFfFFFFFFFFFFFFFFFFFFFFFFFFFFFFP

25000 500

1.5 00 10145.5000 0.00000 197.0000 31.0000 869.
2.5 00 0.350

2 2 1.5

0.000

0.000 0.0 0.0 0.0 0.0 0.5900 0.0
50.0 1.067 0.800

57.9 1.067 0.800

1.200

0.01000 0.01000 3.2400
FIN

//

*******************************************************************

197Au on 31Na Elab = 51.5 MeV/A 5/2 State Nuclear (only) Rot.

tFFFTFFtFFTTFFFTFFFFTFFFFFFFTFFFFFFFFFFFFFFfFFFFFF

128

 

FFFFFFFFFFFFFFFFTTTFfFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

25000 500

1.5 00 10145.5000 0.00000 197.0000 31.0000 869.
2.5 00 0.350

2 2 1.5

0.5900

0.5900 0.0 0.0 0.0 0.0 0.0000
50.0 1.067 0.800

57.9 1.067 0.800

1.200

0.01000 0.01000 3.2400
FIN

//

129

0.0

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