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LIBRARY
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This is to certify that the

dissertation entitled

LOW-LYING COLLECTIVE EXCITATIONS IN NEUTRON-RICH
EVEN-EVEN SULFUR AND ARGON ISO’I‘OPES STUDIED VIA
INTERMEDIATE—ENERGY COULOMB EXCITATION AND PROTON
SCATTERING

presented by

Heiko Scheit

 

has been accepted towards fulfillment
of the requirements for

PhD. degree in Physics

lo. 4M

Major profeur

 

 

 

Date December 17, 1998

 

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ma mu

LOW-LYING COLLECTIVE EXCITATIONS IN
NEUTRON-RICH EVEN-EVEN SULFUR AND ARGON
ISOTOPES STUDIED VIA INTERMEDIATE-ENERGY
COULOMB EXCITATION AND PROTON SCATTERING

By

Heiko Scheit

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

1998

ABSTRACT

LOW-LYING COLLECTIVE EXCITATIONS IN NEUTRON-RICH EVEN-EVEN SULFUR
AND ARGON ISOTOPES STUDIED VIA INTERMEDIATE-ENERGY COULOMB
EXCITATION AND PROTON SCATTERING

By
Heiko Scheit

The energies and B(E2; 0;... —> 2f") values for the lowest J’r 2 2+ states in the
neutron-rich radioactive nuclei 38’40'423 and 44’46A1‘ were measured via intermediate-
energy Coulomb excitation. Beams of these nuclei were produced by projectile frag-
mentation with energies per nucleon of E /A z 40 MeV and directed onto a secondary
Au target, where Coulomb excitation of the projectile took place. The subsequently
emitted de-excitation photons were detected in an array of NaI(Tl) detectors, which
allowed the identification of the first excited 2+ states in these nuclei. The ener-
gies of the first excited J’r = 2+ states in 40’428 and 44Ar were established, while a
previously observed state in 46Ar was assigned a definite spin. For all isotopes the
B(E2; 0:3. —> 2?") was established. The results for 40'428 provide the first evidence
of moderate collectivity (52 z 0.29) near N = 28, while the effects of the N = 28
shell closure persist in the Z = 18 nucleus 46Ar (,82 = O.176(17)). The deformation
parameter of 44Ar was measured to be ,82 = 0.241(14).

A proton scattering experiment was also performed on the unstable nuclei 42’44AI‘.
The measured cross sections have been compared to optical model calculations as-
suming rotational and vibrational excitation. The resulting deformation parameters

indicate an isoscalar excitation.

Meiner Familie

gewidmet

iii

ACKNOWLEDGMENTS

First and foremost I would like to thank Thomas Glasmacher. Thanks Thomas!
You were a great boss. You gave me plenty of freedom (especially in my working
hours) and you always knew when I needed a “kick in the butt” to get me going
again. I want to especially thank you for sending me to the summer school in Erice,
Sicily. (I won’t elaborate...)

I would like to thank Walt Benenson, Wolfgang Bauer, Wayne Repko, and Bernard
Pope for serving on my Ph.D. committee.

Special thanks go to Maggie HellstrOm, since she got me started on the Coulomb
excitation experiments. It all sounded so easy in the beginning...

Rich and Sharon, thanks for the nice times at your parties. I really enjoyed a lot;
throwing darts, bottling beer, playing foosball, doing the BBQ, watching Wallace and
Gromit . . . Your house was often the meeting point for the “international crew”, which
I had the pleasure to know: Greg, Heather, Marielle, Roy, Razvan, Kurt, Justine,
Peter, Marcus, Navin, Anu, Andreas, Thomas A., Yorick, Tiina, Theo, Bertram,
Takashi, Valentina, Marilena, ...(I am sure I forgot many more). I wonder who will
keep things together after you leave.

I am thankful to the people that helped me during my graduate studies: Mike
Thoennessen, Paul Cottle, Kirby Kemper, Rick Harkewicz, Rich Ibbotson, Dave Mor-
rissey, Mathias Steiner, Peter Thirolf, Yorick Blumenfeld, Tiina Suomijarvi, Alex
Brown, Gregers Hansen, Jim Brown, Keith Jewell.

Of course there are also my fellow students: Don — thanks for the good deal
on my car, it wasn’t so bad after all, Jon, Mike R, Pat, Declan, Chris — sorry for
going overboard in taking messages (from Sara or Jen? I forgot. . . ), Joelle, Erik,

Joann, Jing, Barry, Raman, Luke, Sally, Gerd ...Maybe we will see each other at

iv

some “boring” physics conference again. (J ac, was it 305 Schuss Village?) Thank you,
Barry, for the times we spent (jointly) together and the many discussions. Daniel and
Jac, thanks for introducing me to IGOR. Rich, Boris, Marcus, and Barry: thanks for
proof reading my thesis. Unfortunately they didn’t check this section.

I thank the cyclotron staff members for their support. I really enjoyed doing the
experiments here at the NSCL. I am also grateful for the generous financial support
during my graduate studies.

The Sandhill Soaring Club also helped me to survive during the last 4 years and
thanks, Jeff, for bringing me in contact with them. I would like to thank everybody
that went to fly with me. It was a pleasure.

“Moi Droog” Boris, I know “In russia, we ...”

I hope I wasn’t too rude to you at times, but I am really glad to have you as a friend.

Well, now I need to say something about Marcus. Without you it would have been
a lot more boring in East Lansing. I will never forget the many evenings and nights
at the RIV and many other establishments in the area. Also, our discussions about
physics and in general (“Did you have meat in the East Germany?”) have been very
enlightening and useful. The analysis of my data without the “Driftellipsenmethode”
would have been much more difficult. Your soccer expertise is very much appreciated.

I would like to thank my family, especially my mother. Their support made this
work possible.

Finally, thanks to Simona, her cell phone, e-mail and the TeleGroup calling card.

 

CONTENTS

LIST OF TABLES
LIST OF FIGURES

1 Introduction
1.1 The Atomic Nucleus ...........................
1.2 Nuclear Models ..............................
1.2.1 Microscopic Models ........................
1.2.2 Collective Models .........................
1.3 Experimental Probes of Nuclear Structure ...............
1.3.1 Coulomb Excitation as an Electromagnetic Probe .......
1.3.2 Proton Scattering as an Hadronic Probe ............
1.4 Heavy sd-Shell Nuclei ...........................

2 Intermediate-Energy Coulomb Excitation
2.1 General Description ............................
2.2 Excitation Cross Section .........................
2.2.1 Approximations ..........................
2.2.2 The Excitation Cross Section ..................
2.2.3 Electric Versus Magnetic Excitations ..............
2.2.4 Equivalent Photon Method ....................
2.3 Basic Parameters .............................
2.3.1 Impact Parameter and Distance of Closest Approach .....
2.3.2 Sommerfeld Parameter ......................
2.3.3 Adiabaticity Parameter ......................
2.3.4 Excitation Strength ........................
2.4 Experimental Considerations .......................
2.4.1 Projectile Excitation .......................
2.4.2 Radioactive Nuclear Beam Production .............
2.4.3 Detection of De-Excitation Photons ...............

3 Coulomb Excitation of 3840423 and 44"‘6Ar
3.1 Introduction ................................
3.2 Experimental Procedure .........................
3.2.1 Secondary Beams .........................

vi

>4

X
nu

I—|
OOCOCDanmI—lI-i

...—l

13
13
15
16
17
18
18
19
19
20
2O
24
25
25
26
28

34
34
35
35

3.2.2 Experimental Setup ........................ 36

3.2.3 Particle Detection ......................... 37

3.3 The NSCL NaI(Tl) Array ........................ 40
3.3.1 Mechanical Setup and Principles of Operation ......... 40

3.4 Electronics ................................. 42
3.4.1 Electronics for the NaI(Tl) Array ................ 44
3.4.2 Calibration and Gain Matching of the Detectors ........ 45

3.5 Analysis .................................. 51
3.5.1 Stability of Calibration ...................... 51
3.5.2 Angular Distributions ...................... 51
3.5.3 Photon Yields ........................... 57
3.5.4 Error Analysis ........................... 57

3.6 Results and Discussion .......................... 60
3.6.1 Observations ........................... 60
3.6.2 Comparison to Theory ...................... 63

4 Direct Reactions 68
4.1 Elastic Scattering ............................. 70
4.1.1 Optical Model ........................... 70
4.1.2 Effective Optical Potentials ................... 71
4.1.3 Microscopic Optical Potentials .................. 72

4.2 Inelastic Scattering ............................ 75
4.2.1 Coupled Channels ......................... 75
4.2.2 Nuclear Deformation ....................... 76
4.2.3 Mn/Mp .............................. 77

5 Proton Scattering of 36“42"‘4Ar 79
5.1 Experimental Setup ............................ 80
5.1.1 Proton Detectors ......................... 81
5.1.2 Beam Particle Tracking ...................... 91
5.1.3 Electronics ............................. 94
5.1.4 Kinematic Reconstruction .................... 96

5.2 Simulation ................................. 100
5.2.1 Examples ............................. 100
5.2.2 Efficiency ............................. 101

5.3 Analysis .................................. 101
5.3.1 Excitation Energy ......................... 101
5.3.2 Problem with the Cross Section ................. 108
5.3.3 Deformation Parameters and M,,/Mp .............. 115

6 Summary 121

vii

A Coulomb Excitation 123

A.1 Semi-Classical Approach and Perturbation Theory ........... 123
A.1.1 Cross Sections ........................... 126
A.1.2 Relation Between Impact Parameter and Deflection Angle . . 127

A.2 Some Formulas .............................. 128
A21 Relations between 62, Q0, B(E2) .............. 128
A22 Constants ............................. 130

B ’y-Ray Angular Distribution Following Relativistic Coulomb Excita-

tion 131

B.1 General Structure ............................. 132

B2 Detailed Derivation ............................ 133
B.2.1 What is (IffofkalH,|Ifo)? .................. 133
B.2.2 The Angular Distribution .................... 136

B3 Angular Distribution Using the Winther and Alder Excitation Ampli-
tudes .................................... 139

BA Summary of Formulas .......................... 143

B.4.1 Angular Momentum Representation of the Rotation Matrix . 143
B.4.2 Some Properties of the Clebsch - Gordan - Coefficients and 31'-

Symbols .............................. 144

B.4.3 Some Properties of the 6 j-symbols ............... 144

B.4.4 In", ................................. 145

B.4.5 7 — '7 Correlation Function .................... 145

C Relativistic Kinematics 146
C.1 Notation and Preliminaries ........................ 146
C2 Collision of Two Particles ........................ 148
C.2.1 Relation Between 06m and 9,0,, .................. 151

C22 Solid Angle Relation ....................... 152

C3 Decay of A into B,{,-=1,2MN} ....................... 154
CA Summary of Formulas .......................... 155
C.4.1 General .............................. 155

C.4.2 Total Energy in the Center of Mass System Em, ........ 155

C.4.3 Individual Energies in the Center of Mass System ....... 155

C.4.4 Velocity of the Center of Mass in the Laboratory 6cm ..... 156

C45 Relation Between 06", and 0M, .................. 156

C.4.6 Solid Angle Relation ....................... 156

C47 Invariant Mass .......................... 157

D Simulation of the Proton Scattering Experiment 158
D.1 Structure Of Computer Code ....................... 158
D2 Description of Input File ......................... 160

E Differential Cross Section 162

viii

F A New Method for Particle Tracking 165

El Traditional Tracking ........................... 165
R2 New Method ................................ 166
F.2.1 Some Beam Physics ........................ 166

F.2.2 Determination of Position and Slope .............. 169

E23 Practical Method ......................... 171

G Fitting of Spectra 173
LIST OF REFERENCES 175

ix

LIST OF TABLES

3.1 Parameters a0, a2, a4 of the angular distribution. ........... 52
3.2 Summary of uncertainties. ........................ 59
3.3 Experimental parameters and results ................... 67
4.1 Becchetti-Greenlees optical potential parameters. ........... 72
5.1 Detector thicknesses in am. ....................... 85
5.2 Experimental Parameters. ........................ 108
5.3 Deformation parameters 62 Obtained with the various ECIS fits. . . . 118
5.4 Summary of properties of argon isotopes ................. 120
0.1 Momenta of particles before and after scattering. ........... 149
D.1 Explanation of input file for simulation .................. 161

LIST OF FIGURES

1.1
1.2
1.3

2.1
2.2
2.3

2.4
2.5
2.6

3.1
3.2
3.3
3.4
3.5
3.6

3.7
3.8

3.9

3.10
3.11
3.12
3.13
3.14
3.15

3.16
3.17
3.18
3.19
3.20

Low-lying energy levels of 2°9Bi. ..................... 4
Typical vibrational (114Cd) and rotational (238U) level schemes. . . . 7
Energies Of the first excited 2+ state E (2?) for even-even isotopes. . . 11
Classical picture of the projectile trajectory. .............. 15
Adiabatic cutoff. ............................. 21
Cross sections for Coulomb excitation of 40S as a function of beam

energy for 40S impinging on 197Am ................... 23
Secondary Beam Production. ...................... 27
A typical experimental setup for intermediate energy Coulomb excitation. 29
Contributions to the photon energy resolution .............. 31
Setup around the secondary target. ................... 36
Arrangement of the position sensitive N aI(Tl) detectors. ....... 37
Illustration of pulse shape discrimination ................. 39
Energy-loss vs. total energy after the secondary Au target. ...... 39
Arrangement of the position sensitive NaI(T1) detectors. ....... 41
Position calculated from the two PMT signals over the actual position

Of the collimated source. ......................... 42
Electronics Diagram. ........................... 43
Lead housing and photon collimator for the position calibration of the

NaI(Tl) detectors .............................. 45
Sample energy calibration curve ...................... 47
Illustration of the position dependence. ................. 47
Typical spectrum illustrating the correction for the Doppler shift. . . 49
Efficiency calibration ............................ 50
Photon angular distributions for pure E2 transitions. ......... 53
Angular Distribution of de-excitation photons .............. 54
The angular distribution in the laboratory system and in the center of

mass ..................................... 55
The observed photon spectrum for 40S. ................. 57
Observed energies of 'y-rays as a function of position without correction. 61
Background subtracted photon spectra .................. 62
Shell model space and interactions. ................... 65
Results. .................................. 66

xi

4.1

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8

5.9

5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22

5.23
5.24
5.25
5.26
5.27
5.28

6.1

C.1
C.2

F.1
F.2
E3
E4

Fit by Becchetti and Greenlees for the proton scattering cross sections. 73

Particle identification. .......................... 80
Setup around the secondary target. ................... 81
Schematic 3D view of the setup around the secondary target ...... 83
Position sensitive silicon detector telescopes. .............. 84
228Th a-source spectrum .......................... 86
Energy calibration of strip detector backside ............... 87
Position calibration of the strip detector. ................ 88
Signals that are registered by the individual detectors in the telescope

as a function of proton energy ....................... 89
Particle ID spectra in the telescopes. .................. 92
Calibrated PPAC position spectrum. .................. 93
Effect of target orientation and beam spot size on the angular resolution. 95
Electronics Diagram. ........................... 97
Kinematics for the reaction 36Ar(p, p’) at E /A = 33.6 MeV ....... 99
Simulated spectra under ideal conditions ................. 102
Simulation including target effects. ................... 103
Simulation including target and detector effects. ............ 104
Actual data ................................. 105
Efficiency. ................................. 106
Excitation energy of 36Ar. ........................ 107
Excitation energy spectra for 42Ar and 4“Ar. .............. 109
Comparison of the measured cross section to previously published data. 111
Relation between the proton laboratory energy and the center of mass

scattering angle ............................... 112
Ratio of the present data to the ones by Kozub. ............ 114
Different fits are obtained depending on the chosen shape of the nucleus.115
Measured cross sections and ECIS fits. ................. 116
Deformation parameters obtained with the various fits. ........ 117
A comparison of the extracted Mn/Mp values. ............. 119
A comparison of the extracted Mn/Mp values. ............. 119
Summary of results ............................. 122
Schematic drawing of a collision of two particles ............. 149
Relation between z,y, and 0 ........................ 153
Setup for particle tracking ......................... 166
Phase-space—ellipse ............................. 167
Comparison of old and new method. .................. 170
Measured excitation energy. ....................... 172

xii

Chapter 1

Introduction

1.1 The Atomic Nucleus

The atomic nucleus is a many-body system and consists of neutrons and protons
which interact mainly via the strong interaction. Protons and neutrons are spin-%
particles and therefore obey Fermi-Dirac statistics. Nuclear sizes are on the order of
10 fm (= 10"14 m).

The structure of the nucleus is extremely complicated, mainly because of the small
size of the nucleus and the strong (but short range) interaction between nucleons,
which renders perturbation theory not applicable in most cases. Typical velocities
of the nucleons in a nucleus are on the order of 0.2 -— 0.3 c, and the corresponding

de-Broglie wave length is approximately

/\ 2 2—7E z 4.5 fm. (1.1)
mv

This wavelength is on the order of the size of the nucleus and hence a quantum

mechanical description of the nucleus is essential.

1.2 Nuclear Models

The previous section hints at the complications one encounters while attempting to

describe the nucleus theoretically. In general one has to solve the SchrOdinger equation
77011 = E\II . (1.2)

The subject of nuclear structure theory is the determination of ”H is, and the resulting
wave-functions look. Even with an exact knowledge of 71, it is impossible to solve
equation 1.2 exactly, except for the lightest nuclei. Therefore one introduces a model
that focuses on certain aspects of the structure and neglects others. A broad range
of nuclear models exists; depending on the physical problem some models are more
convenient than others and offer more insight into the structure of a particular nucleus.
Nuclear models can be divided into two major groups according to the choice of
coordinates used to describe the nucleus.

Seemingly the most natural coordinates are those of the individual nucleons (i.e.

the position r, the spin 5, and isospin 1'), hence the nuclear states are described by

\Il=\Il(r1,S1,7’1,r2,...) a (13)
and the Hamiltonian ’H is
A p2 1 1
7-1:Z—'+—Zv(z',j)+—:v(i,j,k)+..., (1.4)
i=1 2m 2 i,j 61“,“:

where the sums go over all nucleons. These models are called microscopic models.
The other major group Of nuclear models uses shape coordinates. These models
are based on the collective degrees of freedom such as the center of mass R and the

quadrupole moment Q20 of the nucleus

1 A A
R = Z 2 Pi Q20 = 2722360620 (15)
i=1 i=1

These are collective models.

1.2.1 Microscopic Models

The best known microscopic nuclear model is the shell model. In the shell model the
nucleons in the nucleus are considered to move independently of each other in the
central potential produced by all other nucleons. Thus the multitude of interactions
between the nucleons is replaced by an external mean field. Originally this model
was motivated by the observation of magic numbers of protons and neutrons. The
numbers are 2, 8, 20, 28, 50, 82, and 126. It was observed that nuclei in which the
neutron or proton number (or both) corresponds to a magic number were particularly
stable. Discontinuities (peaks or dips) in other quantities e.g., energies of first excited
states, neutron absorption cross sections, and binding energies were also observed.
The properties of many nuclei in the vicinity of closed shells could easily be explained
by the shell model. Figure 1.1 (taken from [1]) shows the excited states of 209Bi. In
the simplest shell model this nucleus consists of an inert 208Pb core (a doubly magic
nucleus), which creates a mean field, plus one proton in the h % shell. Hence the spin
and parity of the ground state is J1r = g-. A number Of excited levels can simply
be explained by putting the extra proton into higher and higher orbits. The spins,
parities, and even the excitation energies of these states can easily be predicted.
However, one can also see in figure 1.1 that there is a group of levels around
2.6 MeV that can not be explained in this simple way. These states correspond to
an excitation in the 208Pb core coupled to the extra 5; proton. The first excited
state in 208Pb is a J 7' = 3" state with an excitation energy of 2.6 MeV. Hence these

states (in 2°9Bi) have an energy of about 2.6 MeV, positive parity, and spins ranging

from g- to 12,—. In order to explain these states within the shell model, the whole
lead nucleus would have to be included into the shell model calculation, and the
resulting wavefunction would correspond to a superposition of many single particle

states. However, this coherent motion of many nucleons can be described more easily

zoo
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Figure 1.1: Low-lying energy levels of 209Bi. The group of states near 2.6 MeV are
excitations of the 208Pb core.

 

 

 

 

 

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in collective models, in which this 3‘ state is considered to be an octupole vibration
of the nuclear surface.

The mean field is only the average smooth part of the interaction between the
nucleons and does not include all interactions. The remaining part is called the
residual interaction. This interaction gives rise to collective behavior within the shell

model.

1.2.2 Collective Models

It was pointed out in the previous section that many features of nuclei in the vicinity
of magic numbers can most easily be described by the shell model. Properties of
nuclei in the mid-shell regions, however, can not be explained in this simple way. The
behavior of these nuclei is best described in a collective model. The best examples of
collective motion are the giant resonances, which occur in all nuclei. These excitations
can be considered to be an oscillation of all neutrons and all protons. The emphasis
is on all, since this is the most extreme contrast to the shell model, where, as in the
209Bi example, only one nucleon is responsible for the excitation.

Besides the giant resonances, nuclei in the mid-shell regions show low-lying states
(much lower in energy than corresponding single particle states), with large matrix
elements that couple these states. A nucleus that exhibits these features is said to
be collective since the large matrix element can only arise if more than one nucleon
participates in a coherent collective motion. The energy spacing between levels is
by itself only an indication of collective motion: One can for instance guess that
neighboring even-even nuclei with the same energy for the first excited state are
similarly collective, but this might be misleading. The best measure of the collectivity
in a transitions between two states is the reduced transition probability B(7r/\), where

7t and /\ denote the parity and multipolarity of the transition operator. The B(7rA)

is proportional to the square of the absolute value of the transition matrix element

between states Ii) and | f)
Bax) o< l<flM(7r/\)li>|2, <16)

and can e.g. be deduced from the lifetimes of nuclear states or from measured exci-
tation cross sections. Equation 1.6 shows why the reduced transition probability is
such a good indicator of collective motion; it depends directly on the wave functions
of the involved states.

Two major mechanisms are responsible for the low-lying collective states: One is
the rotational motion of a statically deformed nucleus and the other is the vibration
of the nuclear surface (without change in matter density within the nucleus). Two
examples of collective nuclei are shown in figure 1.2, where the excited states of the
even-even nuclei 238U and 114Cd are depicted. The level energies for 238U follow a
very distinctive J (J +1) pattern (spacing increases linearly with J) and the spins are
spaced with AJ 2 2. This spectrum is typical of a rigidly-deformed rotating body.
The level scheme of 114Cd shows a completely different structure. One observes a
2+ state at 0.56 MeV and at about twice this energy a set of states with quantum
numbers 0+, 2+, and 4+ are seen. This is typical of a nuclear surface vibration.
The first level corresponds to a one phonon state carrying angular momentum 2 and
positive parity. Two phonons (corresponding to the first harmonic vibration) can
then couple to 0+, 2+, and 4‘“; other values of J are not allowed because the phonons
Obey Bose-Einstein statistics.

In the rotational case mainly electric quadrupole transitions (E2) occur between
the states because of angular momentum and parity selection rules. One can show
that the E2 transition Operator is identical to the quadrupole moment Operator and

therefore the B(E2) value is related to the intrinsic quadrupole moment Q0 of the

 

 

 

 

 

 

spacings that

 

0.777

 

0.5"

 

0.045
O

 

23.”
ROTATIONAL

2* W F.”

 

 

1.2.
2’--—--I.zl

 

“‘ca
VIBRATIONAL

Figure 1.2: Typical vibrational (”4Cd) and rotational (238U) level schemes. Level
spacings that increase linearly with angular momentum indicate a statically deformed
shape, whereas equal spacings indicate a vibrational nature of the excitation.

 

 

nucleus

The factor
and also or
shape will

shape

where .33 i
quadrupol

deformati

Where R9

The d
COllGCllYii
the Perio
ground SI
SH 18 (
Collective
defgrmat:
deformat:
the dOUb]

Time

IOI Collem

nucleus

B(E2) O< Q3 . (1.7)

The factor of proportionality depends on the angular momentum J of the initial state
and also on the quantum number K that characterizes the orientation of the deformed
shape with respect to the quantization axis [2] Assuming a symmetrically deformed
shape

12(6) = Ro(1+ mow, «b» , (1.8)

where ,82 is called the quadrupole deformation parameter. Using the definition of the
quadrupole moment yields the following relation between the B(E2) value and the

deformation parameter (to first order):

 

 

4
52 = §\/3(E2;0+ —> 2+) (1,9)

1
ZeRg ’
where R0 is given by 1.2 fmA1/3.

The deformation parameter given in equation 1.9 is a convenient measure of the
collectivity in a nucleus, which allows the comparison of different nuclei throughout
the periodic table, even if the nucleus is not statically deformed. For instance the
ground state and the first excited state in the proton magic (non-collective) nucleus
122Sn is connected by a B(E2;0‘L —> 2+) Of 1930 ezfm", which is much larger than
collective values for nuclei in the A z 40 region, which are typically 400 ezfm“. The
deformation parameter, on the other hand, for 122Sn is 0.1, which is comparable to
deformation parameters of non-collective nuclei throughout the periodic table (e.g.
the doubly magic 40Ca has a 62 of 0.122).

Typical deformation parameters range from 0.1 for non-collective nuclei to 0.5—0.6

for collective nuclei.

1.3 Experimental Probes of Nuclear Structure

The nucleus can be studied passively, e.g. by Observing the radiation from radioactive
decay, or by studying the nucleus’s electronic environment through the observation
of the hyperfine structure in atomic spectra.

The most fruitful approach, however, is to apply an external field and to study
the reaction of the nucleus to this field. This is done by directing a beam of particles
onto a target and observing the reaction products (other particles, 7 rays, electrons,
etc.).

Depending on the target-projectile combination and the relative velocity, different
fields can be applied and the nucleus can be probed in various ways. The best probes
are the ones that interact the least with the nucleus to be studied, so that it remains
almost undisturbed. In addition, if the probe is weak, perturbative methods can
be applied to extract the structure information. A possible drawback of a weakly

interacting probe is a low reaction cross section.

1.3.1 Coulomb Excitation as an Electromagnetic Probe

The best understood probe available is the electromagnetic interaction. For instance,
electron scattering is used to map the charge distribution deep inside the nucleus,
where no hadronic probe can give reliable results. Another method, which has long
been applied to stable nuclei, is Coulomb excitation, where a nucleus is excited in the
Coulomb field of another nucleus. The energies of the subsequently emitted photons
(conversion electrons or other particles) reveal the spacings between the levels in the
nucleus and the cross sections can be related to nuclear matrix elements connecting

these levels.

1.3.2 Proton Scattering as an Hadronic Probe

Coulomb excitation only probes the charge distribution of the nucleus since only the
protons in the nucleus interact electromagnetically (neglecting the magnetic moments
Of the neutrons). Because of the Pauli principle, the like-nucleon interaction is about
3 times weaker than the unlike-nucleon interaction in a nucleus. Therefore, proton
scattering (p, p’ ) in the energy range of E /A z 10 — 50 MeV, which corresponds to
typical kinetic energies Of nucleons in a nucleus, is mostly sensitive to the neutrons
[3]. Information on both the neutron and the proton motion can be obtained by
a comparison of the hadronic and the electromagnetic probe. The drawback of the
hadronic probe is that the interaction is strong, and the structure information ob-
tained is somewhat model dependent in contrast to the Coulomb excitation results,

which are largely model independent.

1.4 Heavy sd-Shell Nuclei

One of the frontiers in nuclear structure research is the investigation of shell evolution
far away from stability. The nuclei in the region of N 220—28 and Z =14—20 (heavy
sd-shell nuclei) have attracted particular interest for several reasons. These nuclei
are located in between two major neutron shells (N =20 and N =28), which are very
pronounced for the stable isotopes. This is shown in figure 1.3, which includes the
results of the present work and shows the energies of the first excited 2+ states of
the even—even nuclei for N up to 38 and for Z up to 28 are shown. The possibility
of obtaining high-intensity secondary beams of these nuclei makes them particularly
attractive experimentally. In addition, the nuclei in this region are large enough to
show collective properties but are also sufficiently light that the neutron drip line can

be approached. From the theoretical perspective full (no truncation) or almost full

10

 

 

Flgilr

that 1
ll] thf;

 

Figure 1.3: Energies of the first excited 2+ state E (2?) for even-even isotopes. Results
of this thesis work are included. The magic numbers are indicated and one observes
that the energies of magic nuclei are much larger than neighboring nuclei. Changes
in the nuclear structure are expected, however, as the driplines are approached.

11

 

 

 

  
  
  
  
  
  
 

shell model .

The \ms
An introdui
‘2. and the 1
gives an int:
proton scan.

compares the

shell model calculations can be performed.

The unstable nuclei 38’40’428 and 44"“SAr have been studied via Coulomb excitation.
An introduction to Coulomb excitation at intermediate energies is given in chapter
2, and the experimental details and results are presented in chapter 3. Chapter 4
gives an introduction to proton scattering, while the results of elastic and inelastic
proton scattering on 36’42'44Ar are presented in chapter 5. Chapter 6 summarizes and

compares the results obtained.

12

...

 

 

Cha

Inte

Exc:

Following
Valldlly C

Pertainin

2.1

COUIOmb
0f the Dr
OUlSide i
in terms
elicited I;
SGCtiOn It
The Con
are largf']

1
In the
b} eXch 8.119

Chapter 2

Intermediate-Energy Coulomb

Excitation

Following is a general description of intermediate-energy Coulomb excitation. The
validity of the assumptions made is discussed, and an overview over the parameters

pertaining to Coulomb excitation at intermediate energies is given.

2.1 General Description

Coulomb excitation is the excitation of the target nucleus in the electromagnetic field
of the projectile, or vice versa.1 For pure Coulomb excitation, where the nuclei stay
outside the range of the strong force, the excitation cross section can be expressed
in terms of the same multipole matrix elements that characterize the 'y decay Of
excited nuclear states. Therefore, a determination of the Coulomb excitation cross
section leads directly to the determination of basic nuclear structure information.
The Coulomb excitation process, as outlined below, is well understood, and results

are largely model independent.

 

1In the following target excitations will be considered. Projectile excitations can be considered
by exchanging target and projectile quantities (i.e. Z, H Zp,. . . ).

13

 

Keepl”
dear exill
MeV/“ml
E/A S 4
such bfall
fragmelliii
far from 5
MeV/hurl
at the $81
the use of
beaniinte
In the
way to (‘II
don.ist0
onh'eveni
imumdis1
reaethons
thatthe i
Their
matrix eh
(hseussed.

El

lousin

Keeping the bombarding energy below the Coulomb barrier ensures that no nu-
clear excitation can take place. This beam energy is typically of the order of a few
MeV/nucleon. For example for 40S impinging on a 197Au target, a beam energy Of
E/A g 4 MeV is sufficiently small to exclude nuclear excitations. Unfortunately,
such beam energies are impracticable for radioactive beams produced by projectile
fragmentation, the mechanism employed at the NSCL to produce and study nuclei
far from stability. The beams have energies Of several tens or even hundreds of
MeV/ nucleon, and if the beams were slowed down, they would lose quality and flux
at the same time. Furthermore, the long range of the ions at high energies allows
the use of thick secondary targets, which helps to compensate for the potentially low
beam intensity.

In the intermediate energy range, defined here as E /A = 10—100MeV, the easiest
way to ensure the dominance of Coulomb excitation, as compared to nuclear excita-
tion, is to limit the scattering angle of the projectile to small angles. This means that
only events are considered in which the impact parameter is larger than a certain min-
imum distance defined by the maximum scattering angle. Also, events where violent
reactions occur (corresponding to more central collisions) are rejected by requiring
that the incoming beam particle be observed after the target.

The next sections outline how the measured cross sections are related to nuclear
matrix elements. Basic parameters that are involved in Coulomb excitation are also
discussed. The main references are [4, 5]. A description of the experimental setup

follows in chapter 3.

14

 

 

Figure 2.1: Classical picture of the projectile trajectory. The (in this picture stati-
cally deformed) target nucleus can be excited due to the tidal forces exerted by the
electromagnetic field of the projectile.

2.2 Excitation Cross Section

Assuming that the projectile follows a Rutherford trajectory, the Coulomb excitation

(CE) cross section is given by (more details are given in appendix A)

do) (do)
(d9 CE d9 Ruth

where PH; is the probability of excitation from the initial state Ii) to the final state
If). Treating the electromagnetic interaction potential V(r(t)) as a time-dependent

perturbation, PH; is obtained as

00

meta-4|? with ai—Jzleg / e‘“’*‘<f|V(r(t))li>dt- (22)

-00

The amplitudes a,_,, can be expressed as a product of two factors

Cit—+1 = iZXiflffalgl I (2-3)
A

where the excitation strength X is a measure of the strength of the interaction and
the function f (.5) measures the degree of adiabaticity of the process in terms of the
adiabaticity parameter 5. Section 2.3 gives a detailed discussion Of these and other

parameters involved.

15

2.2. 1 Approximations

Winther and Alder [5] obtained a closed formula for the Coulomb excitation cross
section by assuming straight line trajectories and a static target.
As shown in appendix A.1.2, the momentum transfer to the target perpendicular

to the direction of motion is given by

2Z Z e2
77mm 2 Apit) = ——l:—J£—, (2.4)
p

where Z120) is the proton number Of the projectile(target), b is the impact parameter,
’0’, is the (incident) projectile velocity, and at is the target recoil velocity after the
collision. 7 is the relativistic factor and mt is the mass of the target nucleus. For
collisions considered in this work (Zt z 80, Z10 z 20, mt as 200 u up as 0.3 c, and
b = 15 fm) recoil velocities of ’Ut < 0.2%c are obtained. This by itself might seem
quite high, but typical collision times are on the order of 50fm/c z 1.5 - 10‘228 and
the corresponding flight path of the recoiling target nucleus of 0.1 fm is much less
than the nuclear radius of about 7 fm.2 Therefore, the assumption that the target
nucleus remains at rest during the collision process is justified, which allows the use of
a coordinate system with the target nucleus located (fixed) at the origin. A correction
for the small target recoil can be introduced [5].

The deflection angle of the projectile in the laboratory is given by
2
glab = —- b— . (2.5)

Inserting the values given above, deflection angles of only a few degrees are obtained.

Hence the assumption of a straight-line trajectory is justified.

 

2Nuclear radii can be estimated as R = 1.25 fm Ai with A being the mass number of the nucleus.

16

2.2.2 The Excitation Cross Section

The excitation cross section can be obtained by integrating the excitation probability
from a minimum impact parameter bmm, determined by the experimental conditions
(e.g. a maximum scattering angle), to infinity. An approximate result is obtained by
introducing the adiabatic cutoff and integrating the absolute square of the excitation
strength (equation 2.3) [le from bmin to bmax, instead of integrating H, = [X f (§)|2

from bmin to infinity:3

00 bmax
0 = 27r / P,fbdbz 27r / |x|2bdb. (2.6)
bmin bmin

bum can be estimated as

712 _ yl'w ~ 7197

bmar : '23; — AB ~ AE MeV fm , (2.7)

where AE is the energy of the transition and to], = 935-. This leads to an approximate

expression for the excitation cross section of parity it and multipolarity /\

 

 

22 /\—1‘1 for A>2
a“ z (Zte ) B(m,0 —> MMQUA,’ ( ) _ (2.8)

2 min
hc e 2ln(%:l::) for A=1,
where bmaz >> bmm was assumed. B(7r/\, 0 —> A) is the reduced transition probability,

defined as

BOWL—>0) = ZIUIMIIMWAMIJz-MeHz
I‘M]
I

_ , 2
— 2Ji+1|<JfllM(7r/\)IIJ.>| ,

 

where M(7T/\/.l.) is the multipole operator for electromagnetic transitions. The exact
expression for the excitation cross section, summed over parities and multipolarities,

derived in [5], is

Z e2 2 B m, 1,- —> I
OH, = ( £6 ) Zk2(’\'1) t( 62 f)

2

9u(€(bmin)) - (2-9)

 

C
Ge“ (3)

 

 

«Au

 

3i.e. the function f is approximated by a step function.

17

 

 

 

 

 

t
Here k = 7

 

etgt. and B.
The \Tinth-
ll l0iil.i\\' '

transition ["1

 

Thus. the B

2.2.3 E]

Electric and

Therefore f
0370-3) In

(“hem-[59‘

Here k = 5%}, Z, is the proton number of the projectile, E7 2 AE is the excitation en-
ergy, and Bt(7r)\, I,- —-) I f) is the reduced transition probability of the target nucleus.
The Winther and Alder functions G and g are explained in appendix A.

It follows that the excitation cross section is directly proportional to the reduced

transition probability

0'1'_,f OC Bt(7l')\,Ii ——-) If) . (2.10)

Thus, the B(rrA) value can be extracted from a cross section measurement.

2.2.3 Electric Versus Magnetic Excitations

Electric and magnetic fields of a moving charge are related through
22
IE! = -C-|E| (211)

Therefore for intermediate- or high-energy Coulomb excitation, of interest here, (i z
0.3—0.5) magnetic excitations are possible and must be considered if not forbidden
otherwise, e.g. by selection rules. In contrast, in low-energy Coulomb excitation

(E z 0.1) only electric excitations are of importance.

2.2.4 Equivalent Photon Method

In principle, Coulomb excitation can be viewed as the absorption of virtual photons
by the target nucleus. These virtual photons are produced by the moving projectile
and the equivalent photon number (the number of real photons that would have an
equivalent net effect for one particular transition) is related to the Fourier transform
of the time-dependent electromagnetic field produced by the projectile.

One can express the Coulomb excitation cross section as

dd)
0,-_,f : Z/Nfl(w)0§“)(w)—, (2.12)
«A

w

18

 

 

 

 

where t'
photon

The phi:

where p

nuclear

lnsertin

number

Since th
SPCtion
IS USC’d q

COUIOTTT

2.3

2.3.1

AS Shim

rleatpS t
tern. Z

the pffjj

that Str

E

where the spectrum of photons Of multipolarity A is determined by the equivalent
photon number N..,\(w) and the photoabsorption cross section is given by UgfiANw).

The photoabsorption cross section for real photons is given by [6]

0.(7rA)(w) _ (271F)3(A + 1)
‘ A((2A —1)u)2

7

p(e)k2’\+lB(7r/\) , (2.13)

 

where p(e) is the density of final states and is usually given as a 6-function for discrete
nuclear states

p(€) = 6(E, + c — E,) = on — 13,). (2.14)

Inserting equation 2.13 into 2.12 and comparing the result to equation 2.9 gives the

number of equivalent photons of multipolarity 7r). [6]

a... (5)

Since the number of equivalent photons can be determined the photoabsorption cross

2

_ 2il((21+1)!!)2 9.46). (2.15)

NMW) _ ”he (27r)3(/\ +1) ,,

 

 

 

section can be related to the Coulomb excitation cross section and vice versa. This
is used e.g. to derive astrophysically important photodissociation cross sections from

Coulomb excitation results [7].

2.3 Basic Parameters

2.3.1 Impact Parameter and Distance of Closest Approach

As shown in appendix A the equation

2Zth C2 b—1

2
7m,,,vp

Grab = (2-16)

relates the impact parameter b to the deflection angle 0,0,, in the laboratory sys-
tem. Z”. are the proton numbers of the target and projectile, mp is the mass of
the projectile and up is its incident velocity. It was shown in the previous section

that straight-line trajectories are a good approximation, and therefore the distance

19

 

 

 

0i closest n

 

larger lilal;

of Coulont? i

The nut‘lear

of the nude

 

 

 

the sentient».

2.3.2 5.

The South

mp [5 ll)(
Tallies a
Small Cr.
along tt

2.3.3

Iflhe<
Dmitri

“Ti l (h

of closest approach is nearly equal to the impact parameter. The latter has to be
larger than the sum of the two nuclear radii plus 2—4 fm [8] to ensure the dominance

Of Coulomb excitation:
DzbgRt+Rp+A3 with A,~2—4fm. (2.17)

The nuclear radii can be estimated as R = 1.25 fmAi, where A is the mass number
of the nucleus [9]. A minimum distance can be ensured experimentally by limiting

the scattering angle of the projectile to be below a certain maximum scattering angle

0 3 6m, => b 2 bmmwmx). (2.18)

2.3.2 Sommerfeld Parameter

The Sommerfeld parameter 1) compares the physical dimensions of the classical orbit,
in this case the impact parameter b, with the de Broglie wavelength X of the relative

motion of the two particles

b b
n=_:____’ympvp with 7:

_ 3’2
X h and ,6— c . (2.19)

1
W
mp is the projectile mass, 2),, the projectile velocity in the laboratory system. Typical
values are r) z 1000, meaning that a wave packet containing several waves is still
small compared to the dimensions of the trajectory. Such a wave packet will move

along the classical trajectory, justifying the use of the semi-classical approach in the

calculation of the Coulomb excitation cross section.

2.3.3 Adiabaticity Parameter

If the time-dependent perturbation potential changes slowly the nucleus follows the
perturbation adiabatically and no excitation is possible. This is the adiabatic cutoff,

which is schematically illustrated in figure 2.2. This cutoff is parameterized in terms

20

 

 

 

 

Figure :
adiabarj
aHows a
“UC‘leus

the nud‘
“0 9Xcit.
Cases.

 

—>
Vb vb Vb

Figure 2.2: Adiabatic limit: In panel a) the collision time is short enough for the
adiabaticity parameter 5 to be small and excitations are possible. Indicated by the
arrows are the force vectors acting on the nucleus. Because of the orientation of the
nucleus a torque is generated which gives rise to excitations. However, in panel b)
the nucleus is able to follow the motion of the projectile, no torque is generated and
no excitations can occur, even though the field strengths (~ x) are similar in both
cases.

21

of the adiabaticity parameter 5, which is defined as the ratio of the collision time rm“

to the time of internal motion in the nucleus Tnucl

T coll

 

5: (mm

7-nucl

If f is large, because the projectile velocity is low or the impact parameter is large,
then no excitation is possible.

In appendix A it is shown that the electric field component in the x-direction E,c
(perpendicular to the direction of motion) produced by the projectile at the target

position is given by

 

 

 

E b Z
a: 7° 3mmr=——mdm=iL (no
(1+(t/T)2)§ 7v. b2
Therefore 7' defines the collision time
b
Tcoll : _ (2.22)
7UP
and the time scale for the nuclear motion is given as Tnucl = wgl = 3%. Hence
5 AE b
= — == . 2.23
C can 7’01: 717% ( )
For I) ~ 0.3 c and b ~ 15 fm it follows that
AE
N , 2.24
6 5 MeV ( )

6 should be smaller than unity for excitations to occur, and thus low-lying collective
states with energies of several MeV can be excited. Going to even higher beam
energies, e.g. 1) ~ 0.5 c, states in the giant resonance region with excitation energies
of 10 — 20 MeV can readily be excited. Figure 2.3 shows the Coulomb excitation
cross section as a function of beam energy for low-lying collective states and giant
resonance states for dipole and quadrupole excitations for 40S impinging on 197Au.

In both cases the giant resonance states are assumed to exhaust the electromagnetic

22

 

4
10 I IIIIIIII I IjITIIIl I IIFrIIrr r I IIII-

 

 

 

+ +
(lflzl = 0.234)

3 102 - GQR
E f d? ,

I 1 mi

;' 1 hm: 15 fm
10-1 1.....‘llllll L llllllll L l lllllll l lllLlll
10‘ 102 103 10“ 105
Beam Energy E/A (MeV)

Figure 2.3: Cross sections for Coulomb excitation of 4”S as a function of beam en-
ergy for 40S impinging on 197Au, with constant minimum impact parameter cutoff
bmm = 15 fm. The energy region accessible with radioactive beams at the NSCL is
indicated. The rising giant resonance cross section can be traced to the reduction
of the adiabaticity parameter. The drop of the cross section for the low-lying collec-
tive state is caused by a diminishing excitation strength parameter, due to a shorter
collision time.

sum rules [5]. It follows that in the region between 30 and 50 MeV, which is the
energy region where the highest secondary beam yield can currently be achieved at
MSU, only low-lying collective states can be excited. At higher energies, however,
giant resonance states will also be populated [10].

For Coulomb excitation below the Coulomb barrier, where E/A ~ few MeV,
the adiabatic cutoff essentially limits the possible excitation energies to be below 1—
2 MeV. The excitation to higher lying collective states is nevertheless possible, since
the strength parameter (see next section) can become large. In this case first-order

perturbation theory is no longer valid and multiple excitations can occur.

23

2.3.4 Excitation Strength

As mentioned in section 2.2 the excitation amplitude can be expressed as a product

of two factors:
ai—1f = 2: XiiszAlgl- (2-25)
,\

f (g) is a measure of the adiabaticity as a function of the adiabaticity parameter, as

stated in the previous section, and x measures the strength of the excitation from

(A)

state Ii) to state If) Classically, Xi—d is a measure of the strength of the interaction of
the monopole field of the projectile with the A-pole component of the field created by
the target nucleus. The monopole-monopole interaction gives rise to the Rutherford
trajectory (and is treated classically). The next higher order is monopole-dipole
interaction, which is not important for low energy excitations, because nuclei can
not have a static electric dipole moment.4 Dynamic dipole-moments are nevertheless
possible and they give rise to the giant dipole resonance, which can be found in all
nuclei. Most nuclei, though, have a static (intrinsic) quadrupole moment, making
the monOpole-quadrupole interaction the most important for low-lying (collective)
states. Classically the field of the projectile exerts a tidal force (for A = 2) on the two
lobes of the target nucleus and the resulting torque can excite the nucleus (see figure
2.2). x depends on the internal structure of the target nucleus, e.g. its quadrupole

moment for A = 2, the strength of the monopole field of the projectile nucleus,5 and

the duration of the interaction, the collision time Tm“. Hence x can be estimated as

(1.1) ~ VA(b)Tcoll N Zt€(f|M(7rAp)|z')
x (b) N —— ~ A ’
h hyvb

 

(2.26)

where Vy(b) is the monopole—A-pole interaction potential, which has a b‘lHl) depen-

dence.

 

4Parity and time-reversal invariance forbid the existence of static electric dipole moments.

5The quadrupole moment of the target nucleus interacts with the derivative of the external electric
field. The interaction energy is given by -%Qij g—fj [11].

24

The matrix element (f |M(7rAp)|i) can be estimated as the square root of the
B(E2T). A collective transition in the A240 region, which is of interest in this work,
might have a B(E2) = 300 e2fm4 [12]; assuming furthermore v = 0.3 c and b = 15 fm
it follows that, even for collective states and heavy projectiles, values of only about
X z 0.15 are obtained and perturbation theory is justified. X can also be thought Of as
the number of (7rA) quanta that are exchanged in one collision. Using equation A.10
one can also calculate the excitation probability at the minimum impact parameter,
which has the highest probability. Typical values are on the order of 1%.

Hence, Coulomb excitation at intermediate or relativistic energies is mostly a one

step process.

2.4 Experimental Considerations

In the following sections some experimental matters that need to be considered for
a successful experiment will be discussed. Attention will be focused on projectile
excitation of radioactive nuclear beams and the detection of '7 rays as a method of

measuring Coulomb excitation cross sections.

2.4.1 Projectile Excitation

In traditional Coulomb excitation the target nucleus is studied by bombarding it with
heavy ion beams with energies below the Coulomb barrier. Since targets can only
be produced from stable isotopes —— with few exceptions where long-lived isotopes
have been formed into targets — the powerful method of Coulomb excitation has
been limited to stable isotopes and the vast majority of nuclei (only about 10% of all
known nuclei are stable) have been inaccessible by it.

This changed with the construction of radioactive nuclear ion beam facilities,

as described in the next section. Using projectile excitation (i.e. the interest lies

25

in exciting the projectile nucleus instead of the target nucleus; see figure 2.5) one
can study all nuclei that can be made into a beam with sufficient intensity. This
has opened up the possibility to study many previously inaccessible nuclei far from

stability. It is also possible to use a cocktail beam consisting of many different isotopes

if an event-by—event particle identification is possible.

2.4.2 Radioactive Nuclear Beam Production

There are two main methods for the production of radioactive nuclear beams. One
method is known as Isotope Separation On-Line (ISOL). Facilities that are working
or are coming on line in the near future are HRIBF at Oak Ridge, SPIRAL in France,
ISOLDE at CERN, and the REX-ISOLDE experiment, which will accelerate radioac-
tive nuclei produced by ISOLDE. In the ISOL approach a high-energy, light-ion beam
is stopped in a production target, where fragmentation of the target occurs. The re-

sulting fragments diffuse out of the target, are transported to an ion source, and are

then accelerated to the desired energy.
An alternative method employed at MSU, RIKEN in Japan, GANIL in France,

and GSI in Germany is projectile fragmentation. More details on these and other

methods of radioactive beam production are given in [13].

Projectile Fragmentation

The principle of projectile fragmentation is illustrated in figure 2.4. The primary beam
impinges on the target, and even though most of the beam particles pass through the
target Without reacting, a considerable number Of primary beam particles collide with
the target nuclei and fragmentation occurs. A variety of fragments are produced and
a system Of magnetic dipoles and quadrupoles together with an energy degrader (a

piece of material in which the projectile loses energy) is used to reduce the number

26

 

 

   
  
   
 
 
 
 

l—> O O
=’ 0 => 0
.N
O
A’ .\
Momentum Slits
Slits for Beam
Wedge Shaped Selection

  

Production 1‘
,1 Degrader

Target . "' '"
/ Secondary
Prunary Beam B cam

Fragmentation Products

&-' QT- s~%
. p0“ [ w. w. _2_25° Dimes
$115-

 
    

 

 

First Intermediate

Image: Momentum
Focal Plane:

 

Production
Target: usually selection
98c Slits to select
Second Intermediate Image: beam of interest
Achromatic degrader (wedge)

for beam punfication

Figure 2.4: Secondary Beam Production: Illustrated here is prOJectIle fragmentation

on the example of the A1200 fragment separator at the NSCL. The primary beam

strikes the target and several nucleons are removed from the projectile nucleus. Due

to the different energy losses of the fragments and the primary beam in the target,
the resulting fragments have a large (longitudinal) momentum spread. Slits are used
to reduce this spread. The various ions have different energy losses in the degrader
and therefore have different magnetic rigidities after the degrader. The second set of

dipoles spatially separates different ion spec1es

27

of unwanted fragments and to produce a secondary beam. The desired ions are
transported to the experimental area where a secondary target is located, which serves
as the Coulomb excitation target. The production of secondary beams at the N SCL
using the A1200 fragment separator is now very common and the initial identification
and setup takes, depending on the nuclei wanted, from a few hours to at most one
day. For more details on the A1200 fragment separator see [14].

Notable properties of beams produced by projectile fragmentation are: a) high
beam velocities, typically between 30-50% the speed of light, b) potentially low sec-
ondary beam intensities, especially for the most exotic6 nuclei, which are often the
most interesting ones, and c) a poor beam quality, due to the production mechanism;
the large transverse and longitudinal momentum spreads (up to ApH/p” = i1.5%
at MSU) result in a large beam spot size (around one cm) and a considerable beam
emittance.

The time from the production of the secondary isotopes to the arrival at the

experimental station is on the order of a few hundred us to one as. Therefore it is

possible to study extremely short-lived isotopes.

2.4.3 Detection of De-Excitation Photons

After Coulomb excitation to a bound state the excited nucleus decays back to the
ground state by emitting a 7 ray. The measurement of the y-ray energy readily reveals
the spacing between the involved energy levels and the number of detected 'y rays can

be used to determine the Coulomb excitation cross section which is directly related

to transition matrix elements of the nucleus (see section 2.2).

 

6i.e. very neutron or proton rich compared to stable nuclei with the same mass number.

28

 

a“

 

 

Photon Detector

2

 

 

 

 

H

‘29; 5v
Coulomb _

$3 Excitation Y may

5 o

C

O

 

 

 

 

 

 

 

Figure 2.5: A typical experimental setup for intermediate energy Coulomb excitation:
The beam particle is detected in a detector covering only forward angles. When
particle—7 coincidences are measured, this limited extend of the detector ensures the
dominance of Coulomb excitation by limiting the impact parameter to be above a
certain minimum bmin. For each Coulomb excitation at least one 7 ray is emitted,

which can be detected in a suitable detector.

Energy Resolution and Dappler Broadening

The beam velocities of particles produced by projectile fragmentation are very high,
about 30—50% of the speed of light. Thus the 7-ray energy is Doppler shifted consid-

erably, as can be seen from the relation between the photon energy in the projectile

and the laboratory frame, EC", and Elab respectively7

Ecm = VEZabu — ,BCOS(010b)) . (2.27)

6,0,, is the angle between the direction of motion of the particle and the direction of
the 7 ray, ,6 is the beam velocity in units of the speed of light, and 7 is the usual
relativistic factor. Since, in an experiment, a single photon detector covers a range
in solid angle, the width of the full-energy peak8 is increased. In addition the spread

in beam velocities Afl, due to energy loss in the target and the intrinsic beam energy

 

7Since only one photon-emitting particle, the projectile, is considered here the subscript cm refers

to the projectile rest frame and not to the center of mass of projectile and target.
8Events where the total photon energy was measured form the full-energy peak in the energy

spectrum.

 

29

spread, will also contribute to the energy resolution. Using

2 2
AEfab = (on...) A32 + (81%) A62 (2.28)

 

83 80

it follows that the contribution to the energy resolution, due to the Doppler effect is

ABS: 2 COS(0[ab) 2 2 2 ,5 sin (01“) 2 2
lab d

 

 

 

(7) 1 — flcoswzab) 1 - 3 008(91ab)

Figure 2.6 shows that these contributions are quite significant and that an accurate
determination of the photon direction is very important. To the resolutions shown
one should add the intrinsic energy resolution of the detector, which is for high-purity
germanium detectors on the order or 0.1% and for N aI(Tl) detectors on the order of
7%. Figure 2.6 illustrates that the use of a NaI(Tl) detector is almost sufficient,
since the resolution is dominated by the Doppler effect for an angular resolution of
A0 = 10°. In practice, the assumption of Afl = 0.05 is quite high. Even though this
value corresponds to a typical energy loss in the secondary target (see table 3.3), most
of the particles considered in the present work decay outside the target and therefore
have the same speed when emitting the photon, as discussed in the next paragraph.
In order to correct for the Doppler Shift one needs to know not only where the
photon interacts in the detector but also the position where the excited nucleus 7-

decays. The transition rate for E2 transitions is given by (see [9])9

W(E2)=1.23-109-( E7 )5. (B(E2)) .3”, (2.30)

 

MeV e2 f'm4

and the mean lifetime can be obtained as the inverse: T = W‘l. For 40Ar a lifetime
of the first excited state of 1.2 ps is obtained, and for 408, which was measured as
part of this thesis work, the lifetime is 22 ps. The path traveled by the particle in
the laboratory is given by s = 7227. Hence, assuming 22 = 0.30, distances on the

order of 0.1 mm to 1 mm are traversed between excitation and 7 decay. Therefore,

 

9E2 transitions are the only mode of 7 decay relevant to this work.

30

 

 

 

I l I | I
|dE/d[3|*AB —

— AB = 0.05 —

S
I

N-POOO

  
  

 

l I

 

  

 

 

 

 

A r I
be 10 — B=05 |dE/d9|*A9 a
z 8 —- A9 = 100 —4
.2 6 I— _
a 4— —
8 2 -
5‘2 0 I I I I I
10 — [3:05 total

 

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

 

 

 

ON-h-OOO
I
l

 

 

3O 60 90 120 150 180

Olab (deg)

Figure 2.6: Contributions to the photon energy resolution resulting from the energy
spread of the beam (top) and from the detector’s angular resolution (middle). The
quadratic sum of the two contributions is shown in the bottom. Also indicated is
the intrinsic energy resolution of a NaI(Tl) detector. In practice, the assumption of
Afl = 0.05 is quite high. Even though this value corresponds to a typical energy loss
in the secondary target, most of the particles, considered in the present work, decay
outside the target and therefore have the same speed when emitting the photon.

Q

31

as a typical target thickness is on the order of 0.1 mm (20052-3 197Au target), most
particles decay in the target or shortly after the target and the target position can
be taken as the origin of the detected photons. But, since as outlined above, many
nuclei decay outside the target (even if only shortly after it), as in the 408 case here of
interest, one has to be careful about what to assume for the speed of the projectile,
since the speed at the center of the target might not be the optimum value.

As shown above, the accurate determination of the photon direction is very impor-

tant in order to obtain good energy resolution. A discussion on the choice of detector

is given in section 3.3, where the NSCL NaI(Tl) array is discussed.

Cross Section

In order to obtain the cross section one needs to know the number of beam particles
that have been excited. This can be related to the number of photons emitted,
since the nuclei de—excite, after Coulomb excitation, by emitting at least one photon.

Other decay mechanisms, such as internal conversion or particle emission [15], are

very unlikely for the nuclei studied here.

Detection Efficiency. Unfortunately not all 7 rays that are emitted can be de-
tected. A large number will escape, because they either do not hit the detector or
they do not interact while passing through the detector. Therefore one has to know
the 7-ray detection efficiency cm of the detector setup. In general 6 will be a function
of the 7 ray energy and the photon direction. The source of the photons is assumed

to be the center of the target. The efficiency for a certain direction and energy is

defined as

# of photons detected with direction 9
= = 2.
6 EU?” 9) # of photons emitted into direction {2 ’ ( 31)

32

 

 

 

where 0 stands for the direction defined by 9 and o in spherical coordinates. The

detection efficiency, if the photons are emitted isotropically, is given by

étot = /€dfl . (2.32)

This is not the case for photons emitted after Coulomb excitation. However, the angu-
lar distribution of the photons can be calculated, as shown in appendix B. The total
efliciency is obtained by folding the detector efficiency with the calculated angular

distribution W(Q)
f we) - eIEIIm, 9) do
0

{{WM) d0 (2.33)

6tot(E7) :

 

As indicated 6 depends in two ways on the photon direction S2; directly and from the
dependence of the photon energy on Q due to the Doppler effect. The resulting total
efficiency depends, of course, only on the photon energy.

The total number of Coulomb excitations NCE is found by dividing the number

of detected photons N, by the total detection efficiency etc,
NCE = __ (2.34)

The Coulomb excitation cross section a is defined as the ratio of the number of
excitations NCE to the product of the number of incoming beam particles Nb and the

number of target nuclei per unit area Nt

1 N, 1

= — . 2.35
NbNt 6tot NbNt ( )

 

 

O’ZNCE

Therefore the cross section can be determined by measuring the number of emitted

photons.

33

 

 

Chapter 3
Coulomb Excitation of 38’40’423 and

44,46 Ar

3.1 Introduction

T wo of the basic properties of nuclei are their level structure and the transition matrix

elements by which these levels are coupled. This information can be related to the

shape of the nucleus. Nuclei that are spherical or stiff against deformation show large
excitation energies and small coupling strengths. Nuclei that are deformed or soft
against deformation have small excitation energies and larger coupling strengths. As

pointed out in chapter 1, a large coupling strength indicates collective motion, since

a single particle can not give rise to large matrix elements.
A region in the chart of nuclei where little is known about the collectivity of

nuclei is the region of the neutron-rich sulfur and argon isotopes. The structure of
these nuclei is of particular interest as they are located between two major neutron
shell gaps (N =20 and N =28), which are well known for the stable isotones. However,
Sorlin et al. predicted in [16], based on fi-lifetime measurements, a rapid weakening

of the N =28 shell gap below 48Ca. Shortly thereafter Werner et al. [17] published

34

 

 

similar conclusions based on relativistic mean field and Hartree—Fock calculations. In
addition, the nuclei in this region play an important role in the nucleosynthesis of the
heavy Ca, Ti, and Cr isotopes [16].

As discussed in chapters 1 and 2 a direct measurement of the excitation energy
of the first excited J’r = 2+ state together with a transition strength measurement
can give conclusive experimental evidence about the collectivity of these nuclei. The
availability of a very high intensity 48Ca beam at the NSCL at MSU [18] makes the
production of secondary beams of these nuclei possible through projectile fragmen-
tation. The beam intensities (10—2000 particles/s) are sufficiently high to perform
intermediate energy Coulomb excitation experiments. In a first experiment five iso-

topes (38’40'428, 44’46Ar) were studied; the excitation energies of the first excited states

and the corresponding B(EZT) values were determined.

3.2 Experimental Procedure

3.2.1 Secondary Beams

Primary beams of 48Ca13+ and 40Ar12+ with energies up to E/A = 80 MeV and
intensities as high as 5 pnA1 were produced with the NSCL room temperature electron
cyclotron resonance (RTECR) ion source and the K1200 cyclotron. The 48Ca beam
was produced using a new technique discussed in [18]. The secondary sulfur and argon
beams were obtained via the fragmentation of the primary beams in a 379 mg/cm2 9Be
primary target located at the mid-acceptance target position of the A1200 fragment

separator [14]. The rates and purities of the secondary beams are listed in Table 3.3.

35

 

 

[ PMT ] NaI(T1) ] PMT ]

TOF 5 30m ( Position.TOF Km
{Beam PI

 

 

 

Thin Plastic I PPAC PPAC 197Au Target PPAC Fast-ISlow-
2 Phoswich-
(96 or 184 mg/cm ) Detector

Scintillator

Figure 3.1: Setup around the secondary target.

3.2.2 Experimental Setup
Figure 3.1 shows the setup around the secondary target. The time of flight (TOF)
between a thin plastic scintillator located after the A1200 focal plane and a parallel
plate avalanche counter (PPAC) [19] located in front of the secondary target provided
identification of the fragment before interacting in the secondary target as shown in
figure 5.1. A fast—slow phoswich detector, which is described in section 3.2.3, was
located at zero degrees with respect to the beam axis and allowed the identification
of the beam particle after interacting in the target. The target thicknesses for the
various beams are listed in table 3.3. Tracking detectors (PPACS) allowed particle
tracking before and after the target (mainly used during beam tuning). Photons were
detected in coincidence with beam particles by the NSCL NaI(Tl) array, which is

described in section 3.3. Figure 3.2 shows the arrangement of the NaI(Tl) detectors

in the experiment.
11 pnA (particle nA): 10-9 - —w1.6_,10_ P———a"j°‘“ z 6.2 ~ 1099—“;icle

36

NaI(Tl)
Detectors

 

Figure 3.2: Arrangement of the position sensitive NaI(Tl) detectors.

3.2.3 Particle Detection

The forward particle detector serves several purposes: (a) It provides particle identifi-
cation after the target on an event-by-event basis, (b) it ensures that only photons are
analyzed that are emitted from beam particles scattered into laboratory angles of less
than a certain maximum scattering angle. This angle is defined by the radius of the
detector and the distance to the target (in this case 0:33“ 2 4.1°), (c) it provides the
trigger (start detector), ((1) it counts the number of incoming beam particles (needed
for the normalization of the cross section), and (e) it is used to measure photons in
timed coincidence with beam particles, which strongly reduces the background, since
it is possible to distinguish photons emitted from the target from photons emitted

from the zero degree detector.
The beam—particle detector used in this experiment consisted of a 0.6 mm thick

37

fast plastic scintillator (Bicron, BC400) followed by 10 cm of slow plastic scintillator
material (Bicron, BC444). The detector is cylindrical and has a diameter of 101.6
mm (4”). A light guide followed the slow plastic and light was collected by two 5 cm
diameter photomultiplier tubes (PMT). The thickness of the fast plastic was chosen
so that the particles in this experiment lose about 20—30% of their energy in the fast
plastic. The remaining energy is deposited in the slow plastic. The PMT signal, which
was pr0portional to the produced scintillation light, was split into two streams and
in one the full charge is collected (giving the total energy). In the other stream only
the first pulse of charge (due to the fast plastic scintillator) was integrated (giving
the energy loss in the fast plastic). This is illustrated in figure 3.3. The total signal
is a sum of two components. By integrating the total charge (slow gate) the total
energy is determined. An integration of the first part of the signal (fast gate) yields
a signal roughly proportional to the energy loss in the fast plastic scintillator. This
pulse-shape discrimination allowed the identification of the beam particle after the
target. The detector had excellent Z resolution, but the neutron number was not
completely resolved. Nevertheless, events from the breakup of the projectile could be
rejected. Figure 3.4 shows a typical AE—E spectrum.
The detector could tolerate rates as high as 5 - 104 particles/s. The detector was
arranged so that fragments scattered into laboratory angles less than 0:23" = 4.1°
were detected. For all beams, in this work, this opening angle corresponded to an

impact parameter which is about 3—4 fm larger than the sum of the two nuclear radii,

thereby ensuring the dominance of Coulomb excitation.

38

   
 
  

due to fast plastic
I due to slow plastic

Si gnalhei ght

[_] 50ns fast gate
] 250ns J slow gate

Figure 3.3: Illustration of pulse shape discrimination. The total signal is a superpo—
sition of two components, a slow (light grey) and a fast (dark grey) contribution.

 

Energy-loss (fast gate)

 

 

. l 'l ... -..—... I

 

Total Energy (slow gate)

Figure 3.4: Energy-loss vs. total energy after the secondary Au target. The main
contribution is the unreacted 44Ar beam (Z=18). If the beam particle collides with a
gold nucleus fragmentation can occur and the resulting fragments are observed. The

different Z—bands are indicated and clearly resolved.

39

3.3 The NSCL NaI(Tl) Array

3.3.1 Mechanical Setup and Principles of Operation

Each N aI(Tl) crystal is cylindrical, 17.1 cm long and 5.0 cm in diameter and encapsu-
lated in a 2 mm thick aluminum shield. Quartz windows, 0.5 cm thick, are attached
at both ends. A 5 cm diameter photomultiplier tube (PMT) is optically coupled to
each window. For mechanical stability the aluminum shield of the NaI(Tl) crystal is
rigidly connected (epoxied) to a second aluminum pipe which holds the PMT and the
connectors for high voltage (HV) input and signal output. In addition, each PMT is
surrounded by a 0.5 mm thick magnetic shield. A total of 38 detectors are arranged
in an aluminum frame in 3 concentric rings, of 11 (inner), 17 (middle) and 10 (outer)
detectors, oriented co-axially to the particle beam axis around a 150 mm diameter
beam pipe (figure 3.5). The radii of the three detector rings are 10.8 cm, 16.9 cm
and 21.8 cm. As assembled the full array is 60 cm long, 51 cm wide and 55 cm
high. When operated at its standard dedicated setup the array is surrounded by a
16 cm thick layer of lead to shield the scintillators from ambient background and 7
rays originating from the beam particles stopping in the fast-slow phoswich detector
located at zero degrees with respect to the beam axis. The room background 7 rate
is reduced, depending on the threshold settings, by about a factor of 100.
When a “y ray interacts with the crystal a certain number of scintillation photons,
proportional to the deposited energy, are produced, and about half propagate to each

end of the crystal. Assuming an exponential attenuation, the light output on each

side is described by the following formulas:
E1 o< Ee_“(%+x) and E2 oc Ee-”(%_I) , (3.1)

where a: is the distance of the interaction point from the center of the crystal, L

is its total length, it describes the attenuation of the light. E is the total energy

40

   
  

 
 

 
  
  

\%§

a.

    
  

\V

     

 

   

      
  

:\/\%.

 

'/ /%§\\\\\\\\l
\’/////////

  

 

 

 

 

 

 

 

 

 

 

Figure 3.5: Arrangement of the position sensitive NaI(Tl) detectors. The array is
surrounded by a 16 cm thick layer of lead to shield from background radiation.

deposited, and E1 and E2 are the measured signals. The omitted factor of propor-
tionality includes the gain of the PMTs and amplifiers. Exponential attenuation of the
scintillation light along each NaI(Tl) crystal can be achieved in various ways, e.g. by
uniformly coating the crystal surface with a light absorbing substance or by diffusing
the surface of the crystal [20, 21]. Similar detectors have successfully been used in the
APEX experiment at Argonne National Laboratory [21]. It follows from equations

3.1 that one can recover the total energy and the interaction position through:
E oc E1E2 and a: o< log(E1/E2) . (3.2)
A more detailed derivation including the errors on these quantities is given in [20, 21].
There are several effects that change this idealized picture: The assumption of
exponential attenuation of the scintillation light is only valid when the interaction
point is close to the center of the detector, while at the edges solid angle effects (which
lead to a reduction of the light absorption) are more important. This effect can be

41

 

 

 

 

 

 

I I I l I l I I I I 230 _ I I I I I _ 6 7T%
0'6 N...“ 60cc 1 W IDVI - .
... ...... 2m a
0.4 - f ------ 0... ‘3 15° ‘ - 800
""" t. 8 100 .-
0 2 - -- P U 50 _.
. . o O
T“ b. O T T I 600 A
E 0'0 " § ”a. 800 1000 1200 1400 5
LL}— Energy (keV) m.
:3 .,,__ as
.0. -0.2 - a“... _ 400 E
FWHM = 2.3 cm "'1: ..... g
.... o
.04 — . U
‘ ..... {M _ 200
-0.6 - at“.
”0.8M l 1 1 -~_ -_-
0 5 10

 

Position on detector (cm)

Figure 3.6: Position calculated from the two PMT signals over the actual position of
the collimated source. In the central region the relation is linear, while close to the
edges of the detector the curve flattens because the assumption of exponential light
attenuation is no longer valid. The open circles show a sample position spectrum
with the source located at 8 cm. The dotted lines are cubic and Gaussian fits to the
calibration curve and the position spectrum, respectively. The inset shows an energy
spectrum with the source at the same position. The line is a double Gaussian fit with

a quadratic background.

corrected through a cubic position calibration which nicely describes the turnover of

the position calibration curve close to the edges of the detector (figure 3.6).

3.4 Electronics

Figure 3.7 shows the electronics diagram for the Coulomb excitation experiment. The
Master Trigger was defined as a coincidence between a beam particle and a single
N aI (Tl) PMT. In addition to particle-7 coincidences, particles without a coincidence

7 ray (particle singles) were also measured at a rate of every 500th incoming beam

42

 

 

:48 ES:
>w3m

    

 

 

u>5 OE

 

 

 

   

LJ.
030 :52
8.32

 

.33:E:un_a 55.3.“. E3359 u at
5.2—.8530 umam 35164 n Dun.

355.3 amigo. 5:02:40 “are 3 M33}. u (mun.
5:355 3&5 3 DEC. u DOE

352.} £595 5:02.20 12.?0 a. as; H UQ<
5.5250 030 n 00

 

 

Figure 3.7: Electronics Diagram.

43

particle (down-scaled particles), which allowed the determination of the total number

of incident beam particles.

3.4.1 Electronics for the NaI(Tl) Array

A multi-channel high voltage (HV) power supply (LeCroy System 1440) provided typ-
ically +1400 V to each PMT. Signals from the PMTs were fed into shaping amplifiers
(custom built at NSCL/MSU) and the resulting signals were used for discrimination
and energy measurement. The fast output (30 ns shaping time) of each shaper was fed
into a constant fraction discriminator (LeCroy MSU 1806 CFD, LeCroy 3420) whose
output signal was used for the generation of an event trigger, a scaler signal, and a
timing signal. The event trigger required, in addition to a 7 signal, a beam particle
signal in timed coincidence. After a trigger signal had been generated, all PMTS that
belonged to a NaI(Tl) in which at least one PMT fired (according to the bit register,
LeCroy 4448) were read out. The energy was obtained from the slow output of the
shaping amplifier (5ps shaping time) by digitizing the resulting signals by 8-channel
Silena peak sensing ADCs (Silena 4418/V). The trigger, whose signal is correlated
in time with the beam particle signal, initiated a start signal for all time to digital
converters (TDCs) which were stopped by individual PMT signals and therefore mea-
sured the time between detection of the photon and the beam particle. (The TDCs
consisted of a combination of time-to-FERA converters (LeCroy TFC 4303) followed
by charge integrating ADCs (LeCroy FERA 43003). Between the START and STOP
signals the former applied a constant voltage (-50 mV) on the input terminal of the
latter, which resulted in a signal proportional to the time between the START and
the STOP signal.) The TDC range was set to 200 us, which corresponded to the
length of the coincidence window between photons and beam particles. The time

spectra helped to distinguish between photons emitted from the target and photons

44

   

     
  
   
 

 
   
  

     
 
   

 
  

         
    
 

     
     

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Figure 3.8: Lead housing and photon collimator for the position calibration of the
NaI(Tl) detectors. A 0.5 MBq 60Co source was positioned at the top end of the
opening.

emitted by particles stopping in the zero degree detector. The event read-out time

was about 300 ps.

3.4.2 Calibration and Gain Matching of the Detectors

Before and after the experiment a position, energy, and efficiency calibration of the
NSCL NaI(Tl) array was performed.

For the position calibration a collimated 7—ray source was constructed, which
consisted of a 8 mm wide and 15 cm long cylindrical opening in two lead bricks, and
a 0.5 M Bq 60Co source, which was positioned at one end of the opening. Each NaI(Tl)
detector was inserted into a lead housing (shown in figure 3.8) to shield it from ambient

background and the collimated source was initially positioned so that the photons are

directed at the center of the crystal (in longitudinal and transverse direction). Then a

45

first coarse gain adjustment was performed by tuning the PMT HV through a visual

inspection of the PMT signal amplitudes. Afterwards the shaper gains were adjusted

to the desired dynamic range, depending on the expected maximum 7-ray energy

(4 MeV in this case), while, at the same time, keeping the total gain of two signal

processing chains, belonging to the same detector, matched by centering the peak of
the reconstructed position (as defined by equation 3.2) around zero, from which it
follows that E1 = E2. The dynamic range was chosen large enough to account for
the asymmetry in signal height when the 7 ray interacted close to the edges of the
detector, i.e. since we wanted to detect photons in a range of up to 4 MeV we had to
make sure that no component of the signal processing chain saturated if this photon
deposited all its energy close to one PMT. Therefore the range of the shaper was set
about a factor of 1.5 higher, which looks like a 6 MeV range for photons interacting
in the center of the crystal.

After each detector was matched it was moved in steps of 1—2 cm, according to a
measuring tape attached to each detector, to different positions; the data was digitized
and recorded on magnetic tape for later analysis. Figure 3.6 shows the reconstructed
position, as defined in equation 3.2, versus the true position. One can see that in
the central region of the detector the correlation is linear, while at the edges the
assumption of equations 3.1 are no longer valid and the curve flattens. The data

points were fitted with a third order polynomial, which nicely describes the turnover
of the curve close to the edges of the detector.

Photon sources of 88Y, 152Eu, and 228Th were used for an energy calibration. They
provided calibration points from a few 100 keV to 2.6 MeV. Since the raw energy was
not completely independent of position, a position dependent energy calibration was
performed by applying position cuts to the energy spectra (10 cuts per detector). In

between these slices we interpolated to obtain a smoothly varying calibration as a

46

 

    
  

 

I I I I I I
2500— a = —24.58 38y I J
b = 2.03 228
2000 _. c = 4.57e—05 152Eu I Th_l
%
5 1500 — _
>~
P."
d.)
l: 1000 —- _
m ISZEu
Detector #03
500 _ (Central Slice)
0 l I I I I I

 

 

0 200 400 600 800 1000 1200 1400
Channel

Figure 3.9: Sample energy calibration curve. A quadratic fit was used. The photon
sources used for each point are indicated.

 

Energy (MeV)
S '3 g

9
Ln

 

 

 

 

 

40 50 60 70 80 40 50 60 70 80
Position (arb. units)

Figure 3.10: Illustration of the position dependence: The left panel shows the uncal-
ibrated energy over the position for an 88Y source. The right panel shows the same
spectrum after a position dependent energy calibration.

47

function of position. Figure 3.9 shows the energy calibration curve for the central
position slice of detector number 3. The effect of the position dependent energy
calibration is illustrated in figure 3.10. Shown is the measured (reconstructed) energy
over position for an 88Y source spectrum. One can see that after the calibration
the measured energy is independent of position whereas some distortion is present in
the uncalibrated spectrum. The sources of these distortions might have been non-
uniformities in the NaI(Tl) crystal, which changed the light attenuation along the
NaI(Tl) crystal from the ideal exponential behavior (see section 3.3).

The detectors had an average position resolution of about 2 cm and an energy
resolution of about 8% at 662 keV. The 2 cm position resolution translated into a
contribution to the energy resolution, due to the Doppler effect, of 5% for the inner
11 detectors, whereas without the position information the energy resolution would
have been much worse. This is illustrated in figure 3.11. The top panel shows an

energy spectrum without Doppler corrections for the case of 40S on 197Au. A peak at

547.5 keV is visible, which corresponds to a transition in 197Au, which served as the
(stationary) target nucleus. Also visible is a “bump” around 900 keV. The bottom
panel shows the same spectrum, but each event has been Doppler corrected using
the known photon interaction position in the NaI(Tl) detectors. Now one can clearly
see a peak centered at 891 keV which corresponds to the 7 decay of the first excited
state in the radioactive isotope 40S. The measured resolution is 8.7% and only slightly
larger than the resolution obtained from a stationary source.

From the given detector length and the radius of the inner ring of detectors it
follows that for a source located at the center of the array the geometrical efficiency
is close to 27r (assuming an azimuthal coverage of 80%). The photopeak efficiency
for gamma radiation can be determined in two different ways. One method is to

use photon sources that emit two or more 7 quanta in coincidence. Gating on one

48

 

I f I I I I
FWMH = 9.0%

140
120
100

80

I3 = 0.0
547.5 keV

 

 

 

0 keV _

40
20

 

é

 

80—

Counts / (10 keV)
O

 

 

 

 

 

 

400 600 800 1000 1200 1400 1600
Energy (keV)

Figure 3.11: Typical spectrum illustrating the correction for the Doppler shift. The
top panel shows the raw 7-ray spectrum, while the bottom panel shows the same
spectrum corrected for the Doppler shift using the position information of the NaI(Tl)

detectors.

49

 

 

 

 

2'0 I I I I F I I
85<01ab<95
15 (Sdetectors) _
S
3‘ 1.0 -- —1
fl
.2
U
E
m 0.5 — —
(,0 I l l l I I l
600 800 1000 1200 1400 1600 1800 2000

Energy (keV)

Figure 3.12: Efficiency calibration. The used points are from gamma transitions
resulting from the decay of 22Na, 88Y, and 152Eu. The coincidence method applied to

the first two sources gave an absolute calibration, while 152Eu was used for a relative

calibration.

photopeak energy of the cascade in a specific detector and integrating the number
of counts registered by all other detectors in the photopeak of the corresponding
second 7 transition (and taking into account the coincidence ratio between the two
transitions) leads to a value for the absolute photopeak efficiency of the detector
array. The advantage of this method is that it is not necessary to have an absolute
calibration of the photon source. The other method consists of using a calibrated
photon source and applying corrections for the dead time of the data acquisition
system. Both methods were compared and gave consistent results.

Figure 3.12 shows efficiency calibration points from several sources. 22Na, 88Y
provided absolute points through the coincidence method, whereas the points from
152Eu were used as a relative calibration. The functional form of the fit was 5 = A - E}? ,
where A and B are fit parameters with B yielding values around -1 (i.e. e or i). The

photopeak efficiency for the inner 11 detectors, was about 10% at 890 keV and scaled

50

roughly with inverse energy in the region between 0.7 and 2 MeV. The exact values

for the nuclei studied here are listed in table 3.3.

3.5 Analysis

3.5.1 Stability of Calibration

During the course of the experiment unexpected energy shifts over time were ob-
served; only two detectors had a stable energy and position calibration, determined
by comparing source calibrations at the end and the beginning of the experiment.
The most probable causes were gain shifts in the photomultiplier tubes, which have
been used extensively. For subsequent experiments several PMTs were reglued and
loose high voltage and signal leads were resoldered, which improved the gain stability.
Nevertheless, the statistics in the two stable detectors were sufficient to estab-
lish the energy of the first excited states in these nuclei. Since 7-ray peaks due to
transitions in the projectile nucleus were observed in most detectors, an additional
energy calibration was performed in order to move the peaks (corresponding to the
first excited states) to the correct energies, as determined by the two stable detectors.

In all, 8 detectors (all, except the most unstable) were used for the extraction of the

Coulomb excitation cross section.

3.5.2 Angular Distributions

In appendix B it is shown that the angular distribution of the de—excitation photons

is given by
(3.3)

W(0) = z akacosw»,

I: even

51

 

 

 

 

 

 

 

I m a0 02 a4 bm,,,(fm) E/ A (MeV)
1 0 4—11; -1 0 E1

1 1 31; % 0 pure

2 0 L Q _ 12

2 1 :1. 1% 37 pure E2

2 2 El? _ g _ Z transrtlons

2 g -0535 0152 15 40

2 31; -0.663 -0.239 15 200

2 2% —0.405 -0.082 25 40

 

Table 3.1: Parameters a0, 02, 04 of the angular distribution for pure transitions and
distributions in intermediate energy Coulomb excitation. The value for do ensures
the proper normalization.

with

a.=:

ulLlL’

I I 11:
° {if1.1.}Fk(LlL’vaf,If)\/2k+l6L6L,_

 

0mg)

 

2 ,, /\ A k .
g..(o(—) (Mm)

Using equation B.14 one can also obtain the 0,, for pure EAm transitions. Listed in
table 3.1 are the ak’s for pure E1 and E2 transitions together with the calculated
coefficients for actual distributions. Figure 3.13 shows the angular distributions for
pure E2 transitions and the calculated transitions from table 3.1. The similarity of
the actual distributions with the 1:2, m=2 case shows that the main contribution
to the excitation results from maximum angular momentum transfer along the beam
direction, as shown analytically in [5].

The importance of considering the angular distribution is illustrated in figure 3.14.
Shown are the ((1) integrated) angular distributions for isotropic 7 decay and 7 decay
following Coulomb excitation of 40S on a 197Au target at E/A = 40 MeV. Here, the

detector efficiency was approximated as a step function. In this case the measured

cross section would have been overpredicted by 14% if the angular distribution were

52

Pure E2m Transitions

   

l=2,m=:l'1

 

‘97Au ( 408,40? ) '97Au

   

,/
I.
\
E/A = 40 MeV E/A = 200 MeV E/A = 40 MeV
bmin=15 fm bmm=15 fm bmin=25 fm

Figure 3.13: The top row shows the angular distributions for pure E2 transitions.
The quantization axis, which is identical to the beam direction, is going from the
left to the right. The bottom row shows angular distributions for different beam
energies and impact parameters. One can see that the main contribution results from
maximum angular momentum transfer along the beam direction (similarity to the

pure 1:2, m=2 case).

53

197Au ( 403’4OS* ) 197Au
012 I I I I

 

------ isbtropic
0.10 — — actual _

0.08
0.06

W(0) sin(0)

0.04

0.02

 

 

 

0.00 i
0 30 60 90 120 150 180

90m (0)

 

Figure 3.14: Angular Distribution of de-excitation photons. The area under the
curve corresponds to the geometric efficiency. Neglecting the angular distribution of
photons would result in a 14% larger cross section measurement.

not considered. Figure 3.15 shows the angular distribution in the laboratory system.

Folding with the Efficiency
With the angular distribution normalized to unity
/ W(n) do = 1 (3.4)
41r
the total efficiency is obtained as
5,0, = / W(0) {(E,(n), 0) do
n

= Z/W(o)e(E,(n),n)dn

i9;

54

— Center of Mass
......... Laboratory ([3:0284)

 

Figure 3.15: The angular distribution in the laboratory system and in the center of
mass.

55

Ze(E.(0.-).0.) / W10) do.

91
In the last step it was assumed that the efficiency is constant in the range 52,-.2
During the source calibration, using either the coincidence method or a calibrated

photon source, the following quantity is measured:

° 1
6880) = E/dflflifl. (3.5)
n

This is the efficiency integrated over Q (as indicated by the subscript g), with the

weight (= %)corresponding to an isotropic distribution. (If N photons are emitted

(iso)

isotropically then en - N photons are detected within the solid-angle range {2.)

Assuming that 6(0) is constant within 52 it follows that

47]- iso
6(0) = 5.], ). (3.6)
Hence the total photo peak efficiency is given by
2:4 ISO)
e,o,— _ 7r 5],], E)/( W(n (3.7)

where 689?) (E7) is obtained from the efficiency calibration and W(Q) is obtained from
formula 8.17.

In this experiment the whole array was segmented (in software) into 7 regions
(Shrug) defined by cuts in 0 (45—55°,. . . ,115—125°). For each of these regions the
efficiency was experimentally determined and the photon angular distribution was

calculated and integrated. Subsequently formula 3.7 was used to obtain the total

efficiencies, which are listed in table 3.3 for all five isotopes.

 

2In the previous and following expressions Q is used to label both a range in solid angle, as the Q
under the integral sign, and a direction in spherical coordinates Q = (I9, (15), as the Q in the argument
of e() or W(). It should be clear from the context what is meant.

56

 

 

 

 

160 1 I I I I I I w 1
140 197Au (408,408“ ) 197Au _
E 120 — -—
o 100 — I -—
\ 80 — II —
8
G 60 — _
5 4o — ‘ I a
20 II '11 .Ln
' ‘1
0 T 1 1 1 L I
400 600 800 1000 1200 1400 1600 1800 2000
Energy (keV)

Figure 3.16: The observed photon spectrum for 408. A Gaussian fit with quadratic
background was used to determine the peak area.

3.5.3 Photon Yields

The number of emitted photons was obtained by integrating the observed 7-ray spec-
tra. A sample spectrum is shown in figure 3.16. No cuts have been applied, besides
the requirement of the correct isotopes before (AE—TOF) and after (AE—E) the tar-
get. In contrast, the spectra shown in figure 3.18 have been obtained by requiring
a photon multiplicity (fold) of one and a cut was made in the time spectra corre—
sponding to photons emitted from the target. This condition could not be applied
in the cross section determination, since the peaks due to the target and zero degree
detector were not completely resolved. The spectrum in figure 3.16 is fitted well by

a Gaussian peak with quadratic background.

3.5.4 Error Analysis

The uncertainties on the extracted excitation energies and the measured B(E2) values
are calculated by using standard error propagation [22]. The sources for systematic

and statistical errors will be discussed for the example of 403.

57

Excitation Energy

Statistical Errors. The largest contribution to the statistical error was the intrinsic
resolution of the NaI(Tl) detectors, as discussed in sections 2.4.3 and 3.3. Other
contributions were the uncertainty in the angle of the photon and the uncertainty
in the velocity of the 7-emitting particle, due to energy loss in the target and the
initial beam energy spread. Both of these contribute to the energy resolution via
the Doppler shift. All of these uncertainties manifest themselves in the width of the

measured peak in the energy spectrum.

Systematic Errors. The only systematic error, besides the error in the energy
calibration parameters, was the uncertainty in the target position with respect to the

photon detectors, which was estimated to be 5 mm. This includes:
. the position of target with respect to beam-line
0 the position of NaI(Tl) array with respect to beam-line
0 the position of NaI(Tl) detectors in the array (there is a 1—2 mm play)

0 the position of measuring tape on the detectors

The resulting systematic error for the measured energy was 1.25%. In subsequent
experiments this contribution was reduced through the use of a position calibration
device that calibrates the detector position relative to a certain point on the beam
line [23]. Therefore the last three points of the previous list can be replaced by the

uncertainty in positioning of the calibration device, which is very small.

Cross Section

Statistical Errors. The only statistical error was due to the counting of photons in
the photopeak. The numerical value was obtained by fitting the peak in the photon
spectra (figure 3.16) and therefore includes the uncertainty in the background as well

as the uncertainty in the peak.

58

Result

 

 

 

 

Quantlty Contribution Magnitude on Quantity
Energy statistical 0.4%
calibration parameters 5 keV
beam velocity 0.1% <0.01%
target position 5 mm 1.25%
total 1.4%
Efficiency calibration parameters 8 %
0m... 0.1° 0.4 %
beam velocity 0.1% <0.1%
total 8 %
Cross section statistical 5 %
efficiency 8 %
total 10 %
B(E2T) am 01° 5 %
excitation energy 1.4% 2.8 %
beam velocity 0.1% 0.3%
cross section 10 % 10 %
total 12 %

 

 

 

Table 3.2: Summary of uncertainties in the excitation energy, the detection efficiency,
the excitation cross section, and the B(E2T) value for the 40S measurement.

59

Systematic Errors. The largest contribution to the uncertainty of the cross section
came from uncertainties in the determination of the photon detection efficiency. The
uncertainty in the efficiency includes contributions from the fit of the calibration
curve, by far the largest contribution, and contributions due to the calculated angular
distribution, which depends on dmax (see figure 3.13). In subsequent experiments the
error due to the efficiency calibration was reduced by the use of calibrated photon
sources. Here, one does not have to apply gates, which raises the number of counts

by an order of magnitude, and thus reduces the statistical uncertainty.

B(E2T) value

Statistical Errors. The statistical error is the same as that of the cross section

because of the linear relationship between excitation cross section and B(E2) value

(equation 2.9).

Systematic Errors. Besides the systematic errors from the cross section there were
also uncertainties in the calculation of the factor of pr0portionality, the Winther and
Alder functions GM” and g“. Contrary to the cross section the B(E2) value depends
strongly on the maximum scattering angle and this uncertainty has to be considered.

Table 3.2 lists the error sources and the contributions for the 40S case as an
example. For the other cases the systematic errors are very similar and are not

explicitly listed. The final results with total errors are listed in table 3.3.

3.6 Results and Discussion

3.6. 1 Observations

Photons emitted from the fast moving fragments (11 a: 030) could be clearly dis-

tinguished from photons emitted from the stationary target by their Doppler shifts.

60

 

   
 

  

Energy (keV)

‘2‘“: 1”: ,1...‘ J";

 

‘;;‘;:. fit-n" 1.; 1“,." #1:)!

._ .. , _. . ~ hash-J"
40 80 120 160 40“ 80 120 160
Position (mm) Position (mm)

 

Figure 3.17: Observed energies of 7-rays as a function of position without correction
for Doppler shifting (left panel) and with the Doppler correction (right panel) for the
40S+”’7Au reaction. The target was located at 90 mm. The 7-rays near 547 keV are
from the gold target, while those near 890 keV are from the (2fL —) 0;,_) transition
in the projectile.

Figure 3.17 shows the 7-ray energy spectrum as a function of position in the NaI(Tl)
detectors for the 40S+197Au reaction. The left panel shows the 7-ray energies in the
laboratory rest frame; i.e. before any Doppler shift adjustment. In this panel, the
energy of the 547 keV (y —> g.s.) transition from the 197Au target is independent
of position, while the energy observed for the 2f —) 03.3. 7-ray from the projectile
4oS depends on the position and, therefore, on the angle at which it was emitted.
For forward angles a higher energy is observed, whereas for backward angles a lower
energy is measured. The right panel shows the same energy spectrum, but each event
has been corrected for the Doppler shift. Therefore this spectrum corresponds to the
projectile rest frame (0 = 0.27c). In this panel, the energy of the 7-ray from the
projectile is constant, while the energy of the target 7—ray now varies as a function of
position.

The Doppler-corrected, background-subtracted 7-ray spectra for all five nuclei

61

 

zoobfi'V'Il'YI'l'VYv'
D

150 =0

p...

O

O
l

 

 

 

0'
O
I Y t

 

 

 

O

 

 

ass
, v=0.27c v=0.27c v=0.280 v=0.250 v=0.26c
I x5

Counts / ( 1 0 keV)
a:
O

a
O
I

0F
0
IVY—II"

 

20

.
'-
D
0 ‘1“ "
F
allllAlll‘A‘AA“ l L‘IIL‘LALALIA‘IAA ‘L‘L‘Al‘lll‘l ‘Ll‘L l... l I. “ I‘LL

500 1500 500 1500 500 1500 500 1500 500 1500
Energy (keV)

 

  

 

 

 

 

Figure 3.18: Upper panels contain background subtracted photon spectra in the lab-
oratory frame. The 547 keV (7 / 2+ —> 9.3.) transition in the gold target is visible as
a peak, while the (2+ —+ gs.) transitions in each projectile are very broad. Lower
panels contain Doppler-corrected, background-subtracted 7-ray spectra.
studied here are shown in figure 3.18. All five spectra clearly show one photo-
peak associated with each projectile. The measured energies of the 21* states and
B (E2; 03.... —+ 21*) values are listed in Table 3.3. It should be noted that the B(E2 T)
result obtained here for 388 is consistent with the lower limit set on the lifetime of
the 2? state by Olness et al. [24]. In addition, the well-known energy of the 21* state
of 388 [25, 26] was used to check the energy calibration procedure.

N o excited states have been observed previously in 40"""S, but excited states have
been reported for 44"“SAr. Crawley et al. [27] observed states in 44Ar using the

48Ca(3He,7Be) reaction and judged the 2;" state of 44Ar to lie at 1.61 MeV. The

spectra in the study of Crawley et al. are quite difficult to interpret because the

62

background peaks are much larger than those from 44Ar. If the 1.144 MeV state,
proposed here as the 2? state, was populated in the experiment of Crawley et 01., it
would have been obscured by a peak corresponding to an excited state of 7Be. Mayer
et al. [28] reported an energy of 1.55 MeV for the corresponding state in 46Ar from

their work with the 48Ca(14C,1“O) reaction in agreement with the present work.

3.6.2 Comparison to Theory

Self-consistent mean field techniques [17] predict permanent quadrupole deformations
in 401428 of 62 ~ 0.25, only slightly smaller than those measured here (see Table 3.3).
For 44’46Ar, Werner et al. [17] did not provide definitive predictions but instead
showed that their two calculation techniques (Hartree-Fock+Skyrme and relativistic
mean field) give very different answers for these two nuclei. The Hartree-Fock calcu-
lations yield a significant prolate deformation (,32 = +0.17) for 44Ar and an oblate
deformation (62 = —0.13) for 46Ar. On the other hand, the relativistic mean field
calculations yield 52 = —0.13 for 44Ar and ,62 = 0.00 for 46Ar. The experimental
B (E2; 0;“, -+ 2?) results agree better with the Hartree-Fock results, since a non-zero
deformation is measured for 46Ar (52 = 018(2)) and a relatively large deformation is
measured for 44Ar (32 = 024(2)). The effects of the N = 28 major shell gap persist
in 46Ar because it is less deformed than 44Ar and its deformation and the energy of
its first excited state are similar to 50Ti (E(2f) = 1554 keV, ,32 = 0.17, [29]). It would
be of considerable interest to measure the 2;" state of 44S to see whether the N = 28
shell gap is still present even further from the line of stability.

While the shapes of 40’428 can be understood with the mean field calculations of
Werner et al. [17] which attempt to account for changes in single particle binding
energies and residual interactions away from the line of stability, the data for all

nuclei measured here except 46Ar can also be explained with shell model calculations

63

which use empirical interactions obtained from nuclei close to the stability line. These
calculations were carried out in a model space in which the protons occupy the 0d5/2,
0d3/2 and 131/2 (3d) orbitals and the neutrons occupy the 0f7/2, 1123/2, 0f5/2 and 1121/2
(pf) orbitals. For many of the nuclei under consideration the dimension of the full
7r(sd)-1/(p f) model space is too large, and the calculations reported here have been
truncated by leaving out the 0f5/2 and 1121/2 neutron orbitals. With this truncation
the dimension for the 2+ state in 42S is 4335. For some nuclei such as 48Ca and
46Ar, this truncation can be compared to those performed in a model space which
includes the 0f5/2 and 1121/2 orbitals, and the results for the orbital occupations and
excitation energies of the 2+ states are found to be very similar. The Wildenthal sd—
shell interaction [30], the recent F PD6 p f -shell interaction [31] and the WBMB sd — p f
cross-shell interaction [32] was used. This latter cross-shell interaction successfully
accounts for the properties of the N = 20 — 22 nuclei including the intruder state
deformation in 32Mg [32]. The model space used and interactions are illustrated in
figure 3.19. The B(E2) values were calculated using proton and neutron effective
charges of 6,, = 1.356 and en :2: 0.658, respectively, which were chosen to reproduce
the E2 transition strengths of the proton sd—shell transitions in 36S and 38Ar [25] and
neutron p f—shell transitions in 48Ca [33].

In the top two panels of Fig. 3.20, the measured 62 values are compared to the
results of the mean field calculations of Werner et al. and the present shell model
calculations. The mean field calculations slightly underpredict the measured values
for 40,428 and the shell model calculations slightly overpredict ,62 for these nuclei.
However, the shell model calculation predicts that the 62 value of 46Ar is larger than
that of 4‘l‘Ar, contrary to the downward trend in the data, which can be explained
by the persistence of the N = 28 shell closure. The increase in B(E2) for the shell-

model calculation is related to the crossing of the 0d3/2 and 131/2 proton orbitals

64

42

 

 

 

 

16 26
1cm >-<
p—f shell P1/2 *
Fm —0000—
fm [ 00000000- :>
sd-pf

—0000— 20000: (13,2
s- -d shell 6+

Protons Neutrons

Figure 3.19: Shell model space and interactions. The protons occupy the 7r(sd) shell
and the neutrons the u(pf) shell. The model space was truncated in the calculations
and the u0f 5 and lel orbitals have been left out. The arrows indicate the interac-
tions used. The s- and2 p-shells below the sd- shell are not shown and are filled with 8
particles.

65

 

 

 

 

 

 

 

 

0'3? i f i r ‘:

I E a a 1: iii 2

_ 0.2— — — —
6‘; t 8 1: a E :
_ i f: 0 1:1 i
0-1: t r i
0.0- 4 — <> 1

9 r - - :
m 1.5— -— — ' -—
2 _ _ - 2
v _ . A _ 1
>5 - - a
an E 5 5 3:2. 4
2 1.0L— ” — — —
m y— 2:; 7R. .1 l- -

388 408 428 “Ar “Ar

Figure 3.20: The top two panels compare the experimental quadrupole deformation
parameters [62] (solid points) to shell model calculations (stars) described in the text,
relativistic mean field calculations (open diamonds) and Hartree-Fock (open squares)
calculations. The bottom two panels compare the experimental excitation energies
E (2+) (solid points) to the shell model calculations (stars).

observed between 35K (which has a 3/2+ ground state [25]) and 37K (which has a
1/2+ ground state [25]). The bottom two panels of figure 3.20 show that the shell
model calculations successfully reproduce the energies E (21*) in the nuclei reported
here.

The results presented demonstrate that a direct measurement of B(E 2; 0;, —> 21")
is necessary to determine the nuclear collectivity, which is interpreted in terms of the
deformation, and that the energies of the 2? states (without the B(E2) values) are
not sufficient to deduce the deformation on the basis of systematics. For example,

the global systematics of Raman et al. [34] give ,82 = 0.4 from the energies of the 2?

states in 401428. The experimental ,62 deformations are significantly smaller.

66

 

 

 

 

 

 

 

[secondary beam 38 S 4”S 428 ‘4 Ar 46Ar
Energy (MeV/nucleon) 39.2 39.5 40.6 33.5 35.2
Beam purity 0.99 0.65 0.55 0.99 0.99
Typical intensity on target (3") 50,000 17,000 1,800 50,000 27,000
Target thickness/(mg/cmz) 184.1 184.1 184.1 93.5 93.5
Energy loss in target (MeV/ nucleon) 9.1 8.4 7.9 5.1 4.9
Beam velocity (C) 0.269 0.271 0.276 0.254 0.262
023* 4.916 4.958 5.003 5.041 5.085
Efficiency (%) 4.66 7.24 7.28 5.36 3.75
Energy of first excited state (keV) 1286(19) 891(13) 890(15) 1144(17) 1554(26)
0(E2;03'.,_ -> 27,0131, 3 4.1°)(mb) 59(7) 94(9) 128(19) 81(9) 53(10)
B(E2;03’.,. -> 21+) (e2fm4) 235(30) 334(36) 397(63) 345(41) 196(39)
Iflzl 0.246(16) 0.284(16) 0.300(24) 0.241(14) 0.176(17)

 

 

Table 3.3: Experimental parameters and results. The purity of the secondary beam
is for reference only; the secondary fragments were positively identified on an event
by event basis and only desired fragments were analyzed. The energy spread of the

secondary beam was i3%.

In summary, the energies and B(E2; 0:3. —> 2?) values of the 2? states of 381401428

and 44""‘Ar have been measured using intermediate-energy Coulomb excitation. The

isotopes 40’428 are deformed, indicating the presence of a new region of deformed

nuclei near N = 28. The data on the 21* state in 46Ar demonstrate that the N = 28

major shell gap persists at Z = 18. Both the mean field calculations and shell model

calculations using empirical interactions can approximately reproduce the behavior

of the 2? states of 401428.

67

 

Chapter 4

Direct Reactions

In this chapter a short introduction to direct nuclear reactions is given. The main
references are [35, 1, 36].

When a nucleon collides with a nucleus it can penetrate into the nucleus and
form a compound system. These are compound nucleus reactions. On the other
hand, nuclei have a relatively sharp surface, and it is likely that the incoming nucleon
collides with a single nucleon, or a normal mode of nuclear motion, and the residual
particle (which could be the same as the incoming particle as in elastic scattering
or some other reaction product) escapes immediately. This kind of reaction is called
a direct reaction because usually only a single (direct) interaction is involved in the
scattering process.

Since in a direct reaction no intermediate system is formed, the wavefunctions of
the initial and final states overlap, and useful information on the initial configuration
can be obtained from the final state.

In order to extract nuclear structure information from direct reactions, the reaction
dynamics has to be known. Cross sections depend on a nuclear matrix element, which
contains both the wavefunctions of the nuclear states and the effective two—body

interaction Veg mediating the transition between these states. Consequently, a priori

68

knowledge of the effective interaction is necessary in order to use direct reactions as
a spectroscopic tool [37, 38]. Two approaches have been devised to obtain Veg. One
approach is empirical and discussed briefly in section 4.1.2. The other is entirely
theoretical and is outlined in section 4.1.3.

Direct reactions can further be classified as inelastic scattering (e.g. (p, p’)) , as
stripping (e.g. (d, p)) or pick-up (the inverse of stripping, e.g. (p, d)) reactions, and as
knock-out (e.g. (p, p’ p”)) reactions. Each of these reactions has particular advantages
for the study of certain aspects of nuclear structure. Stripping (or pick-up) as well
as knock-out reactions are useful in studying the single particle nature of nuclei; in
particular spectrosc0pic factors can be obtained.

Inelastic scattering, on the other hand, is particularly effective in exciting collective
states. One can imagine that the incoming projectile touches the nuclear surface and
brings the drop (using the liquid drop model) into a state of oscillation, or if the
nucleus is deformed, makes it rotate. The excitation cross section is then dependent
on the degree of collectivity of the nucleus. A deformed nucleus (or a nucleus soft
against vibration) is easy to excite, in contrast to a spherical (closed shell) nucleus.
The results are somewhat model dependent, mainly because the projectile and the
target interact strongly and perturbative methods can hardly be applied. In contrast,
Coulomb excitation (see chapter 2) is well understood and largely model independent,
giving very reliable results.

Coulomb excitation is only sensitive to the protons in the nucleus, and a different
experimental probe is needed to get information on the neutrons. Because of the
Pauli principle, theplike-nucleon interaction is about 3 times weaker than the unlike-
nucleon interaction in a nucleus. Therefore proton scattering (p, p’) in the energy
range of E / Am 10— 50 MeV, which corresponds to typical kinetic energies of nucleons

in a nucleus, is mostly sensitive to the neutrons [3]. Information on both the neutron

69

and the proton motion can be obtained by comparing the two experimental probes.
Section 4.2.3 shows how the ratio of the neutron and proton matrix elements Mn/Mp

can be extracted.

4.1 Elastic Scattering

The simplest interaction between an incident particle and a target nucleus is elastic
scattering. The particle’s direction of motion and / or state of polarization is changed,

without loss of kinetic energy.

4.1.1 Optical Model

Direct or shape elastic scattering occurs when the incident particle interacts with
the nucleus as a whole. This interaction can be described fairly well by an average

nucleon-nucleus interaction as single absorbing potential well: the Optical potential
V(r) = (U + iW)f(r) . (4.1)

U and W are the real and imaginary parts of the potential and f (r) is a Woods-Saxon
form factor. This form is usually not sufficient to describe the elastic scattering data

and surface W0 and spin-orbit V30 terms are introduced.

V(’I‘) = V0(T) — V ' f(.’130) — Z {W ° f($w) - 4WDfid—:;f($0)} (4.2)

 

h 2 1 d
+ (mwc) Vso(L ' 0');$f($30) I
where
—LZ‘€ 62 for r _>_ RC
V00”) = z z e, 2 (4-3)
fil’C—(3—fig) for r3120.

Vc (r) is the Coulomb electrostatic potential of a uniformly charged sphere of radius

RC = r0145. The other terms in equation 4.2 are the real, imaginary, and spin-orbit

70

parts of the optical potential. The form factors f (23,) are given in Woods-Saxon form

1 — ,A1
= _ with :13,- = r——:—: , (4.4)
1 + ext 61,-

 

“1135)

even though other functions can in principle be used. a,- is called the diffuseness
parameter. Approximate values for the parameters are a. z 0.6 fm, r z 1.2 fm,
V 2 50 MeV, W a: 10 MeV, and V30 z 8 MeV (see [36]).

The real part of the optical potential describes the refraction of the incoming
wave and the imaginary part takes into account all non-elastic processes through
absorption.

This potential is inserted into the Schrfidinger equation
V29: + —(E — V)\p = 0, (4.5)

whose solution is expressed in the form of an incoming plane wave plus an outgoing
spherical wave

\II o< e1“ + f(0) , (4.6)

 

and hence the (elastic) differential cross section is given by

do 2
E = If(9)I - (4-7)

Practically one obtains the scattering amplitude f from the phase-shifts 6L

“9) = i]; ZL:(2L + 1)(c2*'6L — 1)PL(cos(6)) . (4.8)

Here, PL(a:) are the Legendre Polynomials. Calculations are done numerically. A

very powerful computer code, which was used in this analysis, is ECIS [39].

4.1.2 Effective Optical Potentials

The usual procedure is to adjust the parameters of the optical potential (potential

depths, radii, and diffusenesses), as given in equation 4.2, until the calculation agrees

71

 

strength V, W (MeV) radius r (fm) diffuseness 0 (fm)
V 54.0 — 0.3218 + 0.4Z/A1/3 + 24(N — Z) /A 1.17 0.75
W 0.22E — 2.7 1.17 0.75
WD 11.8 — 0.25E +12.0(N — Z)/A 1.32 0.51 + 0.7(N — Z)/A
V30 6.2 1.01 0.75

 

 

 

Table 4.1: Becchetti-Greenlees optical potential parameters [40] for elastic proton

scattering.

with the measurement. Often several sets of parameters reproduce the measured
elastic cross section equally well, and the nuclear structure information extracted
is model dependent. However, if data are available for neighboring nuclei, one can
demand that the optical parameters vary slowly as one changes neutron number,
proton number, and beam energy. A comprehensive analysis of a wide range of proton-
scattering data for energies Ep below 50 MeV and A 240 was performed by Becchetti
and Greenlees [40]. The optical parameters in their work were parameterized in terms
of the neutron number N, proton number Z, and the incident lab energy of the proton
E. The Optimum parameters found in [40] are given in table 4.1.

The optical parameters obtained in this way are empirical and based on system-
atics for stable isotopes. Thus it is not clear if these parameters are applicable when
investigating nuclei with large neutron excess. Also, the lowest mass number A that
was included in Becchetti and Greenlees’ fit around a proton laboratory energy of

E, z 30 MeV was 56 which deviates considerably from the nuclei studied here.

4.1.3 Microscopic Optical Potentials

A more fundamental approach is given by Jeukenne, Lejeune, and Mahaux (JLM)
[41]. The authors derived a complex microscopic optical potential based solely on
nuclear matter calculations. The derived potential can reproduce proton elastic scat-

tering angular distributions, provided that the imaginary potential is adjusted by a

72

 

p.
..
..-
I—
.—
p-
n-
I

“‘6

  

    
 

P (9)

Ill"! rIIlIIlI *1

val/gm .An

 

 

 

 

 

 

Figure 4.1: This figure shows the fit by Becchetti and Greenlees for the proton scatter-
ing cross sections on different targets around Ep 2 30 MeV. The lowest mass number
included in this energy range was 56.

73

normalization Aw of about 0.8 [42].
The main idea in the J LM approach is to fold a more or less well known nucleon

nucleon interaction 11 with calculated nuclear densities p resulting in a potential
v0) = / p(r1)u(|r — r1|)dr'[3. (4.9)

The subtleties lie in the determination of the different parts of the optical potential,
such as the real volume and surface terms, and imaginary terms. (see [41]).

With the cost of computing power becoming rapidly decreasing, realistic density
calculations have become possible. Recent theoretical advances give promising results.
For instance Kelley et al. [43] measured the elastic and inelastic proton scattering
cross section on the radioactive isotope 38S. The cross sections were interpreted by
fitting the data using the Becchetti-Greenlees optical parameters and applying the
prescription by Bernstein [3] to obtain the ratio of neutron to proton matrix elements
(see section 4.2.3). The resulting Mn/Mp is 2.06, which is unreasonably large (see
section 4.2.3). A different analysis was performed on the same data by Alamanos
et al. [44]. They used the JLM method with nuclear matter and transition densities
obtained from shell model calculations. The resulting Mn/Mp is 1.58, which is much
closer to the expected value of N /Z = 1.37. A possible source for the discrepancy
of the two methods is that especially for the neutron rich isotopes the neutron and
proton densities, and hence the corresponding potentials, have diflerent root mean
square radii. This difference is not included in standard Woods-Saxon form factors,
which are the same for both neutrons and protons.

One might conclude that the use of an average phenomenological potential of
Woods-Saxon form is inadequate for studies of neutron rich nuclei, since the micro-

scopic approach seems to give more reliable results.

74

4.2 Inelastic Scattering

Inelastic scattering occurs when the projectile interacts with the target leaving either
the target or the projectile in an excited state. Energy is removed from the relative
motion of the two particles and transformed into energy of the intrinsic motion of
either particle.

There are two simple pictures of this process. In terms of the single particle shell
model one can think of a single nucleon lifted into a higher shell, whereas in the
collective picture the scattered particle induces as surface vibration or a rotation if
the nucleus is statically deformed.

In this work we are interested in the second picture and the theoretical interpre-

tation is done using the coupled channel formalism.

4.2.1 Coupled Channels

The Schrédinger equation for the whole system of two colliding particles, represented

by the wavefunction \P(r , If), is [36]
(T- V036) +H(€))‘I’(r,€) = E‘1'(T,€)- (4-10)
The (intrinsic) nuclear states x are defined by

H(€)xa(§) = 6..)(..(€) , (4.11)

where a labels the intrinsic states. The total wavefunction (including the relative

motion) can then be written as a superposition of the intrinsic states

$0.16) 2 Z ¢a(T)Xa(€) a (4.12)

where the sum 62 goes over all states of the nucleus (discrete and continuum). The co-

efficients of the expansion depend on the relative position of the two nuclei. Inserting

75

4.11 and 4.12 into equation 4.10 yields

(T'— E + 6or )(War :0: Vaa’ wa’( T) a (413)

with

— —/5 as W o x. (ode (4.14)
Equation 4.13 constitutes as set of coupled equations for the wavefunction of the
elastic and all inelastic channels.

In practice one can only include a few of the inelastic channels in the calculation
and one compensates for the neglected channels by letting the interaction potential
be complex.

In our case, only two states have been included: the ground state and the first

excited 2+ state.

4.2.2 Nuclear Deformation

The interaction potential V(r, 6) depends on the character of the excited state. For
example, a statically deformed nucleus can be described by a potential that depends

on the orientation of the nucleus
V = W? - R(9, 4)) and RU), a5) = 1200 + 5130099)) , (4-15)
where 6 is the deformation parameter. Hence,
0 dV
V = W? - R0) - flRoYz (0, (Md—7, - (4-16)

The first term is the usual spherical optical potential, and the second term describes

coupling between the incident and outgoing (inelastic) channels.

76

4.2.3 Mn/Mp

In order to compare the neutron to the proton motion, one calculates Mn/Mp, which
is the ratio of the neutron to proton matrix elements, which defined as

Mn(p) = (JillzrfYAmilllle

n(10)

= / pygp)(7‘)7"\+2 dr,
0

where pig”) are the neutron (proton) transition densities. The B(EA) value is related

to the proton matrix element through

lMpl2

. 4.
2J5 + 1 ( 17)

B(EA, J,- —+ J,) =

Mn/Mp can be obtained by a comparison of a hadronic probe (in this case proton
scattering) with an electromagnetic probe (Coulomb excitation) via a description by

Bernstein et al. [3]

 

 

 

Mn bp 6(7) 10’) bnN
— = —— 1 —— — 1
M, 6,, < 6..., + 6,, 2
bp 6(1019’) 50> p’) N
= _ ’ _ 1 1
b. ( 6.... + 6.... z
_ N 500,1”) Z bp 600,100
— Z ( 5cm + an 5cm 1 . (4.18)

Here, bum are the relative sensitivities of the (hadronic) probe to the neutrons (pro-
tons) in the nucleus and 68m and 60ml) are the deformation lengths (6 = 6R) obtained
via the electromagnetic probe and via (p, p’), respectively. The last equation can be
interpreted in the following sense: The ratio of the matrix elements is equal to N /Z
times a factor that is the ratio of the deformations measured via (p, p’) and Coulomb
excitation. Because of the different sensitivities of the probes a correction term has
to be added, which is given as the relative difference of the deformations measured

with the two probes. In other words, if (p, p’) was a probe that is only sensitive to

77

the neutrons then bp = 0 and
Mg = {Y 6(pm’)
M,D Z 66m '

(4.19)

From the collective model one expects a ratio Mn/Mp = N /Z , since the neutron
and proton liquids move in the same way. In contrast, one also expects deviations from
this simple picture, especially in the vicinity of closed shells. A nucleus with a single
closed shell should not yield a value of Mn/Mp = N /Z , since e.g. the neutrons form a
spherical closed shell and hence Mn = 0 [3]. However, core polarization restores the
isoscalar character of the excitation to a large extent and brings Mn/Mp ~ N /Z , even

for single closed shell nuclei [45]. Nevertheless a ratio Mn/Mp that deviates from N /Z

is an indication of the importance of shell effects in the structure of these nuclei.

78

Chapter 5

Proton Scattering of 36’42’44Ar

The availability of high-intensity radioactive beams creates new possibilities in nuclear
structure studies through the use of direct reactions in inverse kinematics. Specifically
the neutron-rich sulfur and argon isotopes have become accessible through projectile
fragmentation of 48Ca, as described in section 3.1.

Proton scattering in inverse kinematics was pioneered in an experiment by Kraus et
al. [33], which measured the proton elastic and inelastic scattering on the fl-unstable
isotope 56Ni. The low energy of the recoiling protons limits the target thickness
to about 2—4 mg/cm2 and high beam intensities, of about 104 s”, are needed to
perform the experiment in a reasonable amount of time (one or two days). Even
with thin targets the energy resolution is only about 800 keV (FWHM). Since the
nuclear states have to be resolved, an additional limit is put on the nuclei that can
be studied. Usually in even-even nuclei the first excited 2+ state is well separated
from the ground state and other excited states. Here, we focused on the even-even
isotopes 42"MAL As a check the N = Z nucleus 36Ar was also investigated [46, 47].
The 36Ar nucleus has equal numbers of protons and neutrons (N = Z) and should
yield a ratio Mn/Mp of unity by isospin symmetry [6].

The secondary beams of 42"MAr were produced as described in section 3.2.1. The

79

 

_
—
—

 

£2 “=-
3 T‘" - :3" ‘fl-f'm '_~_" ‘ ~ .7: ; .—
>~ j a :

an - — _ __
8 _Contaminant 4
[Ll (less than 2% of total) j, . - ..

—

l .;l f a.» 1
Time of Flight

 

 

 

Figure 5.1: The incoming particles were identified according to their time of flight
and energy loss in the fast scintillator of the zero degree detector. The desired beam,
in this case 42Ar, and the contaminants are completely separated.

36Ar beam was produced directly in the K1200 cyclotron, where the beam was in-
jected from the Superconducting Electron Cyclotron Resonance ion source. The beam
energy was adjusted with the help of a variable degrader and the A1200. The beam
energies for all three isotopes were about E /A = 33 MeV. The exact values together

with the target thicknesses and other experimental parameters are listed in table 5.2

on page 108.

5. 1 Experimental Setup

The experimental setup in the 82 vault is schematically shown in figure 5.2. The
incoming beam particles were tracked with two PPACs located 0.8 m and 1.8 m
in front of the target. As in the Coulomb excitation experiment (chapter 3) the
projectiles were identified according to their time of flight (TOF) as shown in figure

5.1.

80

 

— .
— 5‘ ' T“WWW:

(Energy. Position)
TOP 5 ( Position,TOF AB E
I i. :3. 7 W". 7 - I H ’ 7

 

 

 

     

 

30 m .
Thin Plastics I PPAC PPAC Fast-ISlow-
Scintillator Phoswich-
Detector
” _
— SI - Telescope
— (Energy, Position)

 

 

 

Figure 5.2: Setup around the secondary target for the (p, p’) experiment.

The target, a 2.7 mg/cm2 polypropylene (CH2),, foil, was located close to the
center of the NSCL 32” scattering chamber. It was oriented at an angle of 60° with
respect to the beam axis, thus providing an effective target thickness of 3.12 mg/cmz.
The tilt of the target allows the protons, which are scattered toward angles of approxi-
mately 80° in the laboratory system, to easily escape from the target. The kinematics
of the reaction is discussed in section 5.1.4.

The scattered beam particles were detected in a fast-slow phoswich detector and
were identified as described in section 3.2.3, the only difference being that the de-
tector had only one photo multiplier tube and had a smaller diameter of 3”, which
corresponds to scattering angles of about 5°. Typical projectile scattering angles are
on the order of 1° (refer to the lower panel in figure 5.13) and therefore all scattered

particles are detected.

5. 1.1 Proton Detectors

The recoiling protons were observed in a set of four telescopes, each of which consisted

of a silicon strip detector (300 ,um) followed by two silicon PIN detectors (500 pm

81

each). The arrangement of the four telescopes is shown in figure 5.3. The telescopes
were mounted at a distance from the target of about 23 cm. Telescopes 1 and 2
covered angles from 9,0,, = 67° — 79° and telescopes 3 and 4 covered angles from

6,01, = 70° — 82°.

Position Sensitive Strip Detector

The strip detectors were about 300 pm thick (see table 5.1) and had, as all other
detectors in the telescope, an active area of 5 x 5 cm2. One side of the strip detector
was segmented into 16 resistive strips, whose resistance was about 3 k0 over the strip
length of 5 cm. Each of these strips was read out on both ends, and the position of
the incident proton was reconstructed from the two signals assuming charge division.
Thus the detector provided position information in 2 dimensions, where the direction
perpendicular to the strips was given by the strip number itself. Within a strip the
measured position resolution was about 0.5 mm for a 5 MeV signal. This should be
compared to the strip width of about 3.1 mm. The telescopes were mounted so that
lines of constant 0 were roughly perpendicular to the strips. Therefore a much better
angular (0) resolution was achieved as compared to an orientation with the strips along
the lines of constant 0 (better by a factor of 3.1/0.5 at 5 MeV). Angular resolution is
crucial for the distinction of elastic from inelastic scattering. The unsegmented back
side of the detector was used to obtain an energy signal. Figure 5.4 shows a schematic

of the telescope and the working principle of the strip detector.

Silicon PINs

Since protons with an energy of more than about 6 MeV were not stopped in the
300 pm thick strip detector, it is followed by two 500 pm thick silicon PINS. These

three silicon detectors st0pped protons up to an energy of about 14 MeV. The com-

82

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.3: Schematic 3D view of the setup around the secondary target for the (p, p’)
experiment. The beam is going from left to right. The telescopes are labeled from
one through four starting with the top right-hand telescope and counting clockwise.
Telescopes one and two cover the same scattering angles, as do three and four.

83

 

Position Sensitive Si-Strip
Detector with Resistive Readout

 

non resistive O

resistive
: Energy

r

I

0—5—1 9 ' . . .’.—0
Y2 Y1
Y1

Y1+Y2

 

 

 

 

 

 

 

 

 

 

 

Position =

 

 

 

 

 

500 um Si PIN

500nm? SiPIN
30011111 f‘lLllllllLLllLlJ]

 

 

 

 

 

 

 

 

 

Figure 5.4: Position sensitive silicon detector telescopes. The telesc0pe composition
is shown in the bottom part and a schematic of the operating principle of the strip
detector is shown in the top part.

84

 

Telescope Strip PIN 1 PIN 2
1 305 471 475
2 296 467 464
3 299 462 469
4 301 462 476

 

 

 

Table 5.1: Detector thicknesses in um.

bination of the three signals allowed the identification of protons that had a higher
energy and thus were not stopped. This was achieved by comparing the proton energy
deduced from the measured energy loss in the first two detectors (E_from_E2), with
the sum of all detectors (E3). If E_from_E2 was equal to E3 then the proton had been
stopped in the third detector and the total proton energy was given by E3. If, on
the other hand, E3 was not equal to E_from_E2 then the proton had passed through
all silicon detectors and the proton energy was reconstructed from the energy loss in
all three detectors, using the known relation between the proton range in silicon as
a function of energy [48]. In order to reconstruct the proton energy from the energy

loss the detector thicknesses had to be known. They are listed in table 5.1.

Energy and Position Calibration

All proton detectors were calibrated with a 228Th oz-source. Figure 5.5 shows the
a—energy spectrum measured with one of the silicon PINs. Indicated in the plot are
the a-energies and the corresponding parent nuclei. The peak centroids and errors
were obtained by fitting the sum of a Gaussian and a skew Gaussian to the spectra.
The skew Gaussian (an exponential tail convoluted with a Gaussian resolution) fits
the low-energy tail that can be observed in figure 5.5. A sample calibration curve is
shown in figure 5.6.

The position along the strips of the strip detectors was calibrated using a mask

containing a set of 1 mm wide slits separated by a center-to—center distance of 6 mm.

85

 

 

 

 

 

 

 

 

 

 

 

°o
4L9, \ 4% Q
° s" as" s
0;)! 6hr %\ «08)
59 56° bath ,9
{E 1; g (X - Energy (parent nucleus)
00 m N
s, s a,
*2 E a;
E 2 2 ,0 8.78437 MeV(212Po)
m [\ N 00
8 °° ‘>’ 38
m 3 o o.
vi tr} 2 \o
/ 00
[x
O
V)
q
\0
Ln“...

 

 

\ average = 6.06171MeV

(weight=intensity of lines)

Figure 5.5: A 228Th a-source was used for the energy calibration of the silicon detec-
tors. The oz energies and the corresponding parent isotopes are indicated.

86

 

I l T l

9 — Backside of Strip Detector 1 _
a 8 — —
E
>. __ _
g 7
L5

6 — _

E: -0.0038(55) + 0.0067207(58)-channel
5 — l l | I ‘

 

 

 

800 1000 1200 1400
Channel Number
Figure 5.6: Energy calibration of strip detector backside for telescope 1. A linear fit

is shown in addition to the data points. The error bars on the data points are smaller
than the size plotted symbol.

87

 

O
fifflf‘v" (Psi-«16M! «1 I-: 7 a":

 

I
C K
J .
‘
.
I
‘
‘
I
a. .
..
" '
.
J ' i
t
:-
1’
.
. .
. .
.
.
s
‘.
. .
r
. .
.
I
.
n‘ c
a o
(
I
.
.
.0
‘
'
.c
O

Perpendicular Position (mm)

I .
a
I

 

 

 

-10 — g: +-
_20 ._ I __
3 1‘ I ' It 3 5!
-30 l l I l
-20 O 20

Position Along the Strip (mm)

Figure 5.7: Position calibration of the strip detector. The position perpendicular to
the strips was randomized. One strip was not working properly and was excluded
from the analysis. The distance between the 1 mm wide slits in the mask is 6 mm
(center-to-center).

A raw position was calculated from the two measurements on either end of a single
strip as x = yA/(yA + 313), where y“; are the measured signals. The true position
was obtained by fitting the raw positions, measured with the calibration mask, to the

true position using a 3rd order polynomial. A calibrated 2D position spectrum, with

the mask on the detector is shown in figure 5.7.

Proton Identification

Protons with an energy of more than 6 MeV and less than 14 MeV passed through
the strip detector and were stopped in either the first or the second PIN. Figure 5.8
shows the energy loss in the individual detectors as a function of proton energy. This
allows the identification of protons through the AE—E method illustrated in figure 5.9

(lower panels). Proton bands can be seen clearly. Protons with an energy of less than

88

N
O

t—d
M

UI

Proton Energy (MeV)
c:

-I>- O\ 00

[\J

Detector Signal (MeV)

 

  

 

 

 

 

 

 

 

 

 

 

 

l \\
l \x
I— \ \\\ —
Strip \\ JIN 1 (4.92) \g‘ A PIN2(4.69) _
(2.29) \V V ‘V‘ ...... 7
__ \ _
‘ \ — - Strip
\ x g - - PIN 1
— — PIN 2 —
l l l l l I l
O 2 4 6 8 10 12 14 16
Detector Signal (MeV)
I I j I I T I j T I I I I I I T I I I
— — - Strip 4
- - PIN 1 ,’
— PIN 2 ,’
__ /\ ,'
// \\ ’1
/ \II
/ ,\
i— / l \
/ ,’ ‘ \
__ / I
/ I’
/ I
’111Lll’llllllllllll
O 5 10 15 20
Proton Energy (MeV)

Figure 5.8: Signals that are registered by the individual detectors in the telescope as
a function of proton energy. As an example, a 12 MeV proton will lose 2.3 MeV in
the strip detector, 4.9 MeV in the first PIN, and 4.7 MeV in the second PIN.

89

6 MeV were stopped in the strip detector and they could therefore not be identified
by the previous method. However, their time of flight (TOF) to the strip detector
was measured and different isotopes can be identified using a E—TOF matrix. This is
Shown in the top right-hand panel of figure 5.9. The time resolution of 2 us did not
allow a complete separation of the different isotopes, but the band structure can be
observed.

In order to remove all non-proton events from the analysis the identification of the
scattered beam particle in the zero degree detector was required. As can be seen in
the left-hand panel of figure 5.9, this condition alone removes almost all non-proton
background.

This can be understood by considering the possible reactions that lead to parti-
cles other than protons being emitted into laboratory angles 0101, around 80°. One
possibility is the occurrence of violent reactions on the carbon nuclei in the target.
These reactions, however, lead to the breakup of the projectile and hence are rejected
by the requirement of observing the beam particle after the target. N ucleon transfer
reactions such as (p,d),(p,t), and (p,3’4He). can not produce the non-proton back-
ground Since they are very forward focused because of the negative Q-value. (The
total kinetic energy has decreased by lQl, which is on the order of several MeV.) For
instance, the Q-value for the reaction p(3°Ar,35Ar)d is Q = -13 MeV and from equa-
tion C.20 a maximum possible scattering angle for the deuteron in the laboratory of
around 03;?" = 32° is inferred. Hence these particles can not hit the detectors that
are mounted around 010;, = 80°.

The groups of events marked A and B in the left-hand panels in figure 5.9, which
are disconnected from the proton bands, are protons that miss the last PIN (B) or
the first PIN (A). These protons enter the telesc0pe close to the edge of the detector

and not perpendicular to the detector plane. Consequently, it is possible to traverse

90

the strip detector and miss the PIN, which is located about 5 mm behind the strip
detector. The energy for these events has been reconstructed from the measured

energy loss.

5.1.2 Beam Particle Tracking

Beam particle tracking before the target is essential for an accurate determination
of the scattering angle, which in turn is needed to distinguish elastic from inelastic
scattering. The tracking detectors were calibrated using a hole mask with a grid of
2 mm diameter holes separated by a center-to—center distance of 1 cm. At the center
of the mask was an additional fine grid with 1 mm diameter holes separated by 2.5

mm. A sample calibration Spectrum is shown in figure 5.10.

New Tracking Method

Unfortunately the in-beam position resolution of the PPACS was not aS good as
expected. The reason was an (unexplained) strong rate dependence of the position
Signal from one of the detectors. A test with an a-source, prior to the experiment,
gave a resolution of about 2 mm (FWHM). With the beam, at a rate of about 30,000
particles/S, the resolution deteriorated to about 8 mm for PPAC 1 and to about
3.5 mm for PPAC 2. With these resolutions it would have been impossible to separate
the first excited state from the ground state in telescopes 3 and 4, which depend more
on good tracking than telescopes 1 and 2 because of the orientation of the target. The
target was rotated, so that the lowest energy protons could traverse the smallest target
thickness. Thus the normal to the target surface should point as much as possible
towards telescopes 3 and 4 which will see the lowest energy protons, Since these
telescopes are located at larger laboratory angles. This however means that a good

position resolution of the tracking detectors iS crucial, Since a small change in the

91

Identification of incoming No identificaton of

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

beam particle after target isotope after target
30 r I T I I I I I
25 __ E - TOF ..- * ' _,
9
0.) ‘ -.. ...
E5 15 — . ., -- —
g 10 _- _* ‘ -- —
0 -~ -- “ l ‘ . - . . . -. |
0 5 10 15 20 0 5 10 15 20
Time of Flight (ns)
30 I I l I I I I I I I
25 — AE (strip) - E -- 4He .. -
3 ~-:‘
20 — -- H° 3.3., -
15 T -- -' . z. :14
g 10 — -- a
2 5 I- B ‘1‘»: -- n
m annual-r“ “
g 0 l". i 4 + IL l “fi' i 'r i ‘r
E; 25 _ AE (strip+pin A) - E -_ - 1'
Q)
5 20 — -— 3 -I
2 H -
15 '_ J'- l H S. ' . ‘ .—l
H (a...
10 _ -"' . ..‘: *‘.!I'. 'd
5 - 7 ,4} " . " I; ... ' _ ._
0 .1! | I l | l . -L - -.'l l l l

0 5 10 15 20 25 0 5 10 15 20 25 30
Energy (MeV)

Figure 5.9: Particle ID spectra in the telescopes. The identification of the incoming
beam particle after the target, in the zero-degree detector AE—E spectrum, is sufficient
to remove the non-proton background from the analysis. Particle ID spectra without
any out are shown on the right-hand Side and bands from protons, deuterons, tritons,
and even 3"’He can be seen. With the requirement of observing the beam particle after
the target (left-hand Side) only the proton band remains, with very little background.
See text for the groups of counts labeled A and B.

92

 

 

 

 

 

 

 

 

 

 

30 - I I .L t —
E 20 —- II _
g 10 — a —
.9
(L
E -10 — ‘5 —
E 20
> - _ Target Frame _
-30 — I I I I I J

 

~40 -20 0 20 40
Horizontal Position (mm)

Figure 5.10: Calibrated PPAC position spectrum. The shadow of the target frame
is visible. Broken ofl' drill bits can be seen in two of the small 1 mm holes close to
the center of the target frame. The separation of the large (2 mm) holes is 1 cm
and the 1 mm holes in the center are separated by 2.5 mm. The L—shaped pattern
(the lower part is barely visible behind the target frame) served to check the correct
orientation of the reconstructed position. The incident rate was about 600 36Ar/s at
E/A=33.6 MeV. The applied voltage on the detector was 550 V and the gas pressure
was 5 Torr (2,2,4 Trimethylpentane).

93

position on the target results in a large change in angle at which this point is viewed
from the telescopes as illustrated in figure 5.11. On the other hand telescopes 1 and 2
“look” almost parallel onto the target surface and are therefore not so dependent on
the particle tracking, even though they also benefit from good tracking results, Since
the beam direction itself enters into the determination of the scattering angle.

Since the usual method of calculating the beam particle trajectory (i.e. one cal-
culates the trajectory assuming the beam particle crossed the two points that were
measured by the tracking detectors) did not lead to resolved states in telescope 3 and
4 a novel technique was employed to obtain the trajectory, which gave satisfactory
results. This technique, which uses information on the beam itself (i.e. the relation
between the position of the particle and the slope of a particle trajectory), is described

in appendix F.

5.1.3 Electronics

The electronics for the strip detector constituted the largest part of the setup, since
for each strip detector 32 channels (2 - 16 strips) had to be read out. In the electronics
diagram (figure 5.12) the strips are labeled 1A through 16A and 1B through 16B,
where A and B correspond to either side of one of the 16 strips. The signals from
the strips were first amplified by charge sensitive preamps developed at Washington
University and built by PICO systems (Kirkwood, MO). The preamps had a feed-
back capacitance Cf of 1 pf (Von, oc Q/q, where Q is the released charged in the
strip detector), which resulted in a high Signal gain. The preamp Signals were fed
into Shaping amplifiers with a Shaping time of 300 ns. This small shaping time was
necessary because, contrary to capacitive noise, the noise Spectrum of the resistive
strip is high at low frequencies. The shaping time could not be lowered below that

value, Since below that integration time, high-frequency pickup in the shapers be-

94

  

 

/ low energy;
*’ Short path

sag, §
high energy; ‘
long path

 

Figure 5.11: Effect of target orientation and beam spot Size on the angular resolution
a for the different telescopes. Telescopes 1 and 2 have a better resolution, Since they
“look” almost parallel onto the target surface. The target angle was chosen so that
the lowest energy protons have to traverse the smallest target thickness as indicated
in the lower left-hand Side. The protons scattered towards telescopes 1 and 2 have
to traverse a large amount of target material (as shown in the inset), but the proton
energy is sufficiently high. Protons scattered towards telescopes 3 and 4 have a lower
energy, but the target material is small enough to still allow a good energy resolution.
The drawing is not to scale.

95

came the dominant source of noise. (The shapers have a high channel density of 16
channels per single width CAMAC module and therefore the circuits are not built for
high-frequency applications.) This shaping time gave sufficient position resolution of
about 0.5 mm for a 5 MeV signal. The Shaped Signals were digitized by 16—channel
CAMAC peak sensing ADCs (P/S 7164H).

The chosen Shaping time would have resulted in a poor energy resolution (about
200 keV), if the energy had been determined from the sum of the two Signals belonging
to one strip. However, the energy Signal in the strip detector was obtained by feeding
the Signal of the unsegmented back Side (hence no resistive noise) into a preamp
followed by a shaper with a shaping time of about 5 us. The same setup was used
for the PIN detectors. This resulted in a good energy resolution of about 50 keV
(F WHM) for all Silicon detectors.

The fast signals from the shapers for the strip backside and the PINS were input
into constant fraction discriminators (Tennelec TC 455), whose output Signals were
ORed (for each telescope separately). This telescope-OR had a width of 200 ns and if
within that time a beam particle Signal was observed a master trigger was generated.
For each telescope a coincidence bit was set and all detectors belonging to a telescope
that triggered were read out. In addition to particle-proton coincidences ’downscaled’
(1 / 500) beam particle singles (i.e. events where only a beam particle was registered)
were also measured for the normalization of the scattering cross section. The readout

of the zero degree detector is essentially identical to that described in section 3.2.3.

5.1.4 Kinematic Reconstruction

The purpose of the present experiment was to obtain the (p, p’) differential cross
section do/dQ as a function of center of mass scattering angle 06",. In order to

separate the elastic from the inelastic events the excitation energy of the particle, or

96

 

_ _ Eon—.2...
in ._ R: “as E 358.5
>3:

 

   

 

 

 

 

 

9»: 02

5355:35 oma 9:33 H mm:

...HEEtSAQ ...:EE .5350 u EU

E523 omen—.3 5:356 335 E @352 u «Emu
5:355 3&5 E uEF n be...

3583 4.85 552:6 1:35 a. ns.—:2 u UQ<
53550 850 n 00

 

V.M.N._ H.. . IIIIIIIIIIIIIIIIIIIII

s2...

 

 

 

n: in

 

030
8.82

is I'— uah _asm .— 55“ p

 

m2 59;}: ...Su: ................................ _

 

Figure 5.12: Electronics Diagram.

97

equivalently its mass, after the scattering had to be determined. These two quantities
can be obtained from the measured quantities 010,,(1H) and Elab(1H) through a Lorentz

transformation. See appendix C and section 5.3.1 for details.

Inverse Kinematics

For a better understanding of the kinematics it is illustrative to look at the kinematic
lines, which connect lines of constant Q-value in a plot that Shows energy versus
scattering angle in the laboratory system.

The kinematics in this experiment is different from normal kinematics where a
heavy particle is bombarded with lighter one. Here, the light particle constitutes the
target and one speaks of inverse kinematics. The kinematic lines for the p(3°Ar,36 Ar)
reaction at E /A = 33.6 MeV are shown in solid in figure 5.13. The top part Shows
the relation of lab energy to lab angle for the recoiling protons and the bottom part
for the scattered beam particles. The lines of constant 06", for the protons (top part
of figure), Shown dashed, are almost independent of 0,0,, in the range covered by the
telescopes. Therefore a very good angular resolution was obtained in the center of
mass, Since the energy in the laboratory system was measured with high precision.
Figure 5.13 also illustrates the importance of good angular resolution, Since good
energy resolution in the lab system is not sufficient to distinguish the excited state
from the ground state.

The bottom part of figure 5.13 Shows the kinematic lines of the scattered beam
particles. The intrinsic beam energy Spread of the secondary beam, typically on the
order of 1%, together with a high beam emittance make the use of a magnetic spec-
trometer, such as the S800, to detect the scattered beam particles (without measuring
protons) impossible, Since even with particle tracking the nuclear states could not be

resolved. Another drawback is the background on the carbon atoms in the plastic

98

E,ab(36Ar) (MeV)

  
   
 
 
 
 

 

l l l
36Ar(p,p') _.

Telescopes l, 2 EM: = 33-6 MCV
: = (E =1.97 MeV)

C
C
C
"

 

Telescopes 3, 4

 

Blah (1H) (MeV)

U!

 

 

 

 

 

 

 

 

 

50 60 70 80 90 100

 

1210 —

p—s
N

O

O
I

p—A
H

\O

O
I

 

1180 —

 

 

l e =55° "

l | "i l |
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

61ab(36Ar) (deg)

 

Figure 5.13: Kinematics for the reaction 36Ar(p, p’) at E /A = 33.6 MeV.

99

target, which would be indistinguishable from the scattering on protons for small
angles. In addition the recoil energy from the photon that is emitted after excitation

further reduces the resolution.

5.2 Simulation

A simulation program was written in DEC FORTRAN [49]. The program simulates
the full detector response, the geometry of the setup, the beam emittance, as well as
energy loss and angular straggling of the recoil protons in the target. A description

of the program is given in appendix D.

5.2.1 Examples

The simulation can be used to investigate the influence of certain effects, such as
energy loss in the target or detector resolutions, on the final energy and angular
resolution. In the following a few examples are given: Figure 5.14 shows the resulting
Spectra for an ideal case, i.e. all detectors have perfect resolution and there is no
straggling or energy loss in the target. The top two panels Show the kinematic lines,
which are identical to the ones Shown in figure 5.13, except that here the detector
acceptance restricts the scattering angle in the lab. The middle panels show the
(reconstructed) excitation energy over the (reconstructed) center of mass scattering
angle and the projection of these spectra on the energy axis is shown in the bottom
two panels. In figure 5.15 all target effects have been included. The differences
between the spectra for telescope 1 and telescope 3 are due to the target orientation.
Figure 5.16 shows the result of the simulation when all target and detector eflects are
included.

One can see that the resolution after the inclusion of the detector effects is not

much larger than that with only the target effects for telescopes 1 and 2. However,

100

the resolution in telescopes 3 and 4 shows a strong dependence on the resolution of
the detectors, especially the position resolution of the tracking detectors. Figure 5.17

shows measured data and the agreement with figure 5.16 is quite good.

5.2.2 Efficiency

The simulation program can also be used to calculate the proton detection efliciency
(acceptance of the telesc0pes). Figure 5.18 shows the calculated efficiency for tele-
scope 3. In the central region around ch = 25° the efficiency is nearly independent
of the scattering angle 6m, and the value corresponds to the geometrical efficiency of

the setup given by

width of detector
27r(distance to target)
( 13 / 16) - 5 cm

27r - 23 cm
0.028 .

 

 

22

The factor of 13 / 16 was introduced, Since for this telescope 3 out of 16 strips were not
working and hence the active width of the detector was (13/16) - 5 cm. For scattering
angles close to the edges of the detectors the efficiency goes from its maximum to
zero in a manner that can not be obtained through geometrical considerations, Since
here the beam spot size and the angular spread of the beam are important. Most of

the data points are not affected by the correction for the detector efliciency.

5.3 Analysis

5.3.1 Excitation Energy

From the measured laboratory angle 6105 and the measured proton energy EM, the

excitation energy was reconstructed using the equations given in appendix C. For each

101

91,1,(deg) Olab(deg)

 

 

 

 

 

 

 

 

 

 

70 75 8O 85 70 75 80 85
20 I I I I I I I I
9 15 *— -"- '—
i
g 10 — -~ —-
LL]
5 __ -_ .4
0 e l l I I : I 'r
2 '— -"" —- "-
§ 1 - -- n
S’
IV 0 _ -.. .-
LL}
-1 ._ -_ __
_2 I I I I I I J J
o 10 20 30 4o 0 10 20 3o 40 50
9...,(deg) 9cm(deg)
Telescope 1 Telescope 3
12x103 — __ ‘
10 — __ —
%, 0;...
x 8 — __ —
O
: 6 — _ _
4 - 2: -- -
5 2 — 7* n
l .
0 I I I I I I

 

 

 

 

 

-2 -1 0 l 2 -2 -1 O 1 2 3
Excitation Energy (MeV)

Figure 5.14: Simulated Spectra under ideal conditions. The left-hand panels Show the
spectra for telescope 1 and the right-hand panels Shows the spectra for telescope 3.
The top row depicts the spectra measured in the laboratory system. The middle two
spectra Show excitation energy plotted over the center of mass angle and the bottom
panels Show excitation energy spectra (projection of the middle panels on the energy
axis). The t0p row reproduces the kinematic lines shown in figure 5.13

102

6...,(deg) 91,,(deg)
70 75 80 85 70 75 80 85

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-2 l l J l l l l l
0 10 20 30 40 0 10 20 30 40 50
06m(deg) 0cm(deg)
Telescope 1 Telescope 3
250 — --
%200 — _
x
3150 .— __
.2100 — _
G
5 50 — J _.
0 l l -..4L_ ,
-2 -1 0 1 2 3

 

Excitation Energy (MeV)

_Figure 5.15: Same as figure 5.14, except a 2.7 mg/cm2 polypropylene target was
Included (energy loss and angular straggling).

103

e,,.,<deg) e,,.,<deg)

70 75 80 85 7O 75 80 85
= l l W l I l

N
O

 

._.
LII

Em, (MeV)
5

LI!

 

 

 

 

 

 

 

 

0 10 20 30 4O 50
0cm(deg) 0cm(deg)
Telescope 1 Telescope 3

0 10

   

h—l
Ln
0

Counts / 10 keV

-2 -1 0 1 2 -2 -1 0 1 l 2. 3
Excitation Energy (MeV)

Figure 5.16: Same as figure 5.14, but this time the full detector response, in addition
to the effects of the target, have been included. The used input file is given in section

104

elab(deg) 91,1,(deg)
70 75 80 85 7O 75 80 85

 

 

 

 

 

 

 

 

J- _
. -— —l
_2 I I .l 'ii .
O 10 20 30 40 0 50
0cm(deg)
Telescope 1 Telescope 3

Counts / 10 keV

 

 

 

 

-2 -l 0 1 2 -2 -1 0 l 2 3
Excitation Energy (MeV)

Figure 5.17: Actual data. The agreement to figure 5.16 is quite good.

105

b.)
O

 

l l

r?

. w/o correction

I
_
...

_

I

 

 

 

A E- . w o l o w/ correction— a
,3 ; 31; II . {y 8 o
E : -— 20 =3
a ’ ' 8.. m

. {21- H5
g 100 if . [_ ‘ 15 E?
v : is FD.

\ f l —- ~<
g 10 :..— ! -. i c

: IE? :2
E ”4 i _ 5 vb.)
_ TCIiiCOpe 3 E

9cm(deg)

Figure 5.18: Efficiency. The open data points Show the elastic cross section after a
correction for the detection efficiency, whereas the solid points have not been cor-

rected.

event a 4-vector was generated that has the direction 010,, and a fourth component
E10,,(1H) 2 mp + 7105(1H), where mp is the proton mass and Tlab(lH) is the measured
kinetic energy. A Lorentz boost was applied to this vector with velocity 6, which
corresponds to the velocity of the center of mass in the laboratory system. The result
iS the total center of mass energy of the proton E 13$... as the fourth component of the
resulting 4-vector and the center of mass scattering angle, which is given in the first
3 Components. By solving equation C.42 for the mass of particle A’ (using the same

notation as in appendix C)
m3, = Efm + m2B, — 2 EBQmEcm (5.1)
the excitation energy is obtained as

Elf = mA: — mA . (5.2)

106

 

Elab (MeV)

 

 

 

 

61,, (deg) 9C... (deg)
50 —
40 _ E = 1.98(6) Mev[
30° 3 9cm 3 34°

Counts / 150 keV

 

 

E" (MeV)

Figure 5.19: The measured proton energy plotted over the measured lab angle for 36Ar
is Shown in the top left-hand panel. Events belonging to the ground state and the
first excited state can be seen. The top right-hand panel shows the same events, but
the excitation energy and center-of-mass scattering angle have been reconstructed.
The bottom panel shows an excitation energy spectrum with an angular cut of 30° 5
0”,, S 34°. The excitation energy of the first excited state in 36Ar (adopted value
1.97039(5) MeV) is reproduced. The energy resolution is about 800 keV (FWHM).

107

 

Isotope E (MeV) E/A (MeV) # of particles
36Ar 1209 33.6 1, 832 - 106
42Ar 1386 33.0 2, 275 - 10°
44Ar 1461 33.2 2, 417 - 106

 

 

 

Table 5.2: Experimental Parameters: The beam energies and number of incident
particles are listed for each isotope. The (effective) target thickness was 3.118 mg/cm2
in all cases.

Here m 3: is the proton mass, which is equal to m 3 since it is the same before and after
the scattering. EC", is the total center of mass energy given by equation C38. The
results of this procedure are illustrated in figure 5.19. The top left-hand panel Shows
the proton energy as a function of laboratory scattering angle. Two groups of counts
can be clearly distinguished: Most counts are located around a diagonal line that
corresponds to the kinematic line for elastic scattering. The other counts are located
along a line almost parallel to the elastic events and the corresponding kinematics
line is shown explicitly in the figure. The top right-hand panel Shows the same events,
but for each event the center of mass scattering angle and the excitation energy was
calculated according the the previously described method. A projection on the energy
axis of this spectrum with the condition that the center of mass scattering angle be in
the range of 30—34° results in the spectrum shown in the bottom. The nuclear states
are clearly separated and the measured excitation energy corresponds to the ad0pted
value for the first excited state in 36Ar [25].

Figure 5.20 shows the excitation energy spectrum measured for 42"‘4Ar and in each

case the measured energy agrees with the adopted value.

5.3.2 Problem with the Cross Section

1° wide cuts in the center of mass scattering angle were applied to the excitation

energy spectra; the resulting Spectra were fitted with two gauss peaks. The resulting

108

 

    
   

 

 

 

 

l I
60 L * ”Ar —
E = 1.172 (43)
% 50 ‘— (29°<ecm< 37°)
.84
g 40 — —
g 30 — —
g
o 20 — —
10 — —
0 ° “1'1: 0
2
l l 44 l
60 — * Ar _
> E =1.167 (40)
0 (27° < 6cm< 33°)
54
E
g 20 — —
U
' I I O O
O . . 0, I I . L‘ . . ,_,.,;:S.T......

 

 

 

0 1
Excitation Energy (MeV)

Figure 5.20: Shown are excitation energy spectra for 42Ar and 44Ar. The excita-
tion energies are reproduced. The adopted values are 1.2082(3) MeV for 42Ar and
1.144(17) MeV for 44Ar, where the latter was measured in this work.

109

 

31'935

then

 

inehr

obser
surer
rneas
At a
beha
toth
cour
exac
addi
CTOE
tlie

beai

are
the
pro
83M

the

areas, the known target thickness, and total number of incoming beam particles were
then used to calculate the differential cross section (see equation E.5) for elastic and
inelastic scattering.

Figure 5.21 shows the resulting cross sections for the known case of 36Ar. One
observes that the cross section for elastic scattering does not follow the previous mea-
surement by Kozub [46]. Similarly the inelastic cross section, which is compared to a
measurement by Johnson and Griffiths [47], seems to fall below the older data points.
At about 28—29° the measured cross sections drop by about a factor of 2. A Similar
behavior can be seen by comparing the Becchetti-Greenlees optical model calculation
to the 44Ar data which are shown in the lower part. The same is observed for 42Ar. Of
course, one does not expect the Becchetti-Greenlees calculation to reproduce the data
exactly, but especially the height of the maximum might be better reproduced. An
additional reason to suspect a problem is the systematics of the other argon isotopes.
Cross sections for 40Ar [50], 43Ar [51] do not Show such a drop in cross section and
the height of the maximum around 66", z 40° is almost independent of isotope and
beam energy.

In the following different possible explanations for the drop in detection efficiency
are presented: The same angle in the center of mass (for all isotopes) corresponds to
the same proton energy in the laboratory, which gives some credence to an electronics
problem that only occurs if the proton energy is above a certain value (which is around
8 MeV). This value can be obtained from figure 5.22 which shows the relation between
the scattering angle in the center of mass and the laboratory proton energy. Figure
5.22 Shows that the relation is almost independent of excitation energy, which can
also be seen in figure 5.13. Hence the elastic and inelastic cross sections dr0p at the
same angle. However it also follows from figure 5.22 that this energy (8 MeV) does

not correspond to any of the detector thresholds and it is therefore unclear what

110

 

 

 

1000 :_ I o I I I I I _E
6E o Kozub E
4- Ego a Johnson -
_ if I . + '-
A 2__ . o I thlS work (0gs ,2?) _
‘0'] I
B 100 _— o -:
g 6; I i 0 !0I a
CI 4: I i 0 I! f 2
E 2- {Ch I {iii -
b I: in D f
1310:— HEI IIEEID D] Dc] ‘5
6E 36 [:1 D E
4: ElA=£MeV } Iiiiff I
2" l l | l l I ‘

 

15 20 25 30 35 40 45 50

 

N
I I TPITIWT

on
I I’I IIIII

dO'ldQ (mb/sr)

b)
I

 

44
10 Ar f II
BIA = 33.2 MeV

I l l
20 25 30 35 40 45

Gem (deg)

00
Illll
lllll

 

 

 

p—d
(It
LII

0

Figure 5.21: Comparison of the measured cross section to previously published data.
One can see that neither the elastic not the inelastic cross section agrees with the
old data for angles larger than 28°. The lower graph Shows the large disagreement
between the Becchetti-Greenlees calculation and the measured data for 06m > 28° for

44 AI‘

111

 

 

 

 

Figux
scatt
the f
Show
ener;
saw
was

Prol
for 3

Can:

the

[Hi

DO

25

 

 

 

 

 

 

 

 

 

 

I I I I
— 13": 0.0 MeV /
20 - — - E’: 1.97 MeV , ’—
2 15 — , ’ 4
A , / 38°
5 10 _ _
3 33°
Lu
5 — 25° -
0 | | l
O 10 20 3O 4O 50
Gem (deg)

Figure 5.22: Relation between the proton laboratory energy and the center of mass
scattering angle. The energies at which the proton is not stopped in the strip detector,
the first PIN, and the second PIN are indicated. The corresponding angles are also
shown. The insensitivity of the relation between EM, and 06m on the excitation
energy E“ should be noted (the two curves are almost on top of each other). The
same relationship is obtained for 42”MAr, since the beam energy per nucleon E/A
was the same for all three isotopes. This, with the assumption that the electronics
problem depended solely on the proton energy, justifies the use of the scaling function
for all isotopes and all nuclear states.
caused such a discrepancy. For each detector (strip, PIN 1, PIN 2) the time between
the zero degree detector hit and the detector response (TOF) was measured. By
inspecting the Energy-TOP matrix it was confirmed that the drop in cross section
occurs at a proton (lab) energy away from any detector thresholds. (If, for example,
the coincidence gate was not setup properly for the PINs then a problem should occur
as soon as the proton energy is high enough to trigger the first PIN (6 MeV) and one
should observe a drop in cross section at the corresponding angle, i.e. 25°.)

Another explanation could be that at a certain scattering angle the beam particle

misses the zero degree detector and therefore no coincidence could be registered. This

possibility can be ruled out, because the radius of the zero degree detector is much

112

 

 

 

 

 

g0 thI
that I
Whlcl
\\
most
(8 .\I
the (
Out
be p
for -
Cur‘
obt
exp

the

Fat

larger than the maximum scattering angle. In addition, the spread in beam spot size
and the angular spread of the incoming beam are larger than the deflection caused
by the scattering and no sharp transition should be observed.

Could it be that an object obstructs the flight path of the protons? This not
possible, for several reasons. Firstly, one expects a smoothly vanishing cross section
and not a step function with a non-zero cross section after the step. Secondly, any
edge such as the target frame would degrade the resolution, since some protons will
go through just a small part of that object. However the resolutions above and below
that critical angle are the same. Lastly, the object should cover half the telescope,
which is very unlikely to be missed during the setup or take down of the experiment.

What could cause about every second event to be lost if Elab(1H) 2 8 MeV? The
most likely explanation is that no coincidence gate was formed. The threshold energy
(8 MeV) and the rate (about every second event) remain unclear.

In conclusion, several possibilities have been investigated, but none can explain
the observed behavior. Regardless of this, an experimental problem can not be ruled
out and the evidence of the previous measurements and systematic trends can not
be neglected. Therefore we decided to use the previous data on 3°Ar as a calibration
for the detection efficiency. The resulting scaling curve is shown in figure 5.23. The
curve was obtained by using the optical model parameters given by Kozub [46] to
obtain the cross section for elastic scattering for the same angles as were used in this
experiment. The calculation was used instead of the experimental values, because
there are not corresponding experimental data points in Kozub’s work for all points
in this work. The rather surprising result is shown in figure 5.23. Shown there is the

ratio of the current measurement to Kozub’s data. The fitted curve is of the form

Q—IUQ

£1113

 

f(0) = we + 1121 -erf ( ) + 21240 + 111502 , (5.3)

113

 

  

 

 

 

 

 

 

 

 

 

1.2 _ I I I I I I _#
............ — Fit (with error lines)
1.0 ~— .. -I . Data Points a
€0.23 _ ..................
b!
'2 0.6 -
b .......
-c
0.4 —
0.2 — _.
36
00 AI I I I I I I
15 20 25 3O 35 4O 45 50

90m(deg)

Figure 5.23: Ratio of the present data to the ones by Kozub. The fit and the corre-
sponding error lines are explained in the text.

where the w,- are fit parameters and erf() is the error function

erf(:z:) = face—I2 dd: (5.4)

—00
that allows for the smooth step in the fitted function. The error lines (dotted in the

figure) were obtained via

02W» = 2 (1’1) (3) of. , (5.5)

ik dw, dw,c
where 03,, is the covariance matrix obtained through the fitting procedure. Only data
points above 28° are affected by the scaling procedure. Data points below that angle
are not scaled. The previously described scaling function has been applied to all

isotopes (36’42’44Ar) and the resulting differential cross sections are shown in figures

5.24 (3°Ar) and 5.25 (42’44Ar).

114

 

   
 

 

 

I I

1000 — oblate ‘g
g — — prolate i
A —- - B2=0.256 -
‘53 2 -
E 100 E
s 4 , =
B ‘ H _

'° -~.‘:. - r
10 """""" “l- - l “:-
° 36 \. E

4 Ar I L r s _

10 20 3O 4O 50

Gem (deg)

Figure 5.24: Different fits are obtained depending on the chosen shape of the nucleus.
However, the measured data do not allow to distinguish between the two cases, even
though it seems the oblate shape fits the inelastic data points better. Almost no
difference is observed for the elastic channel.

5.3.3 Deformation Parameters and Mn/Mp

The computer code ECIS [39] was used to extract deformation parameters from the
data by fitting the elastic and inelastic cross sections, where V, W, and WD (refer
to equation 4.2), in addition to the quadrupole deformation parameter ,62, have been
allowed to vary during the fit. Other combinations of parameters have also been
tried with essentially the same results. In each case the deformations of all potentials
have been set to the same value and have also been kept fixed to one another during
the fitting procedure. The starting point for all fits was the Becchetti-Greenlees
prediction [40]. The results of the fit are displayed in the figures as solid lines. In the
rotational model prolate and and oblate shapes fit the data equally well, as shown
in figure 5.24. In the vibrational model the quadrupole deformation parameter fig
corresponds to the phonon amplitude and as in the rotational model positive and

negative values were tried and they also fit the data well. The obtained deformation

115

 

 

 

 

  
    

 

 

 

 

 

I I I I I I
1000:— . ——fit 3
E . --BG 5
c I _
e100.— ,. -x __
fl E \ /”! ‘5
V : x/i :
g - I _
B 10:— i _=
‘3 E 5
:42 l I
1 ArI I I I I I
15 20 25 30 35 40 45 50
1000?
$1005—
5 E
Cl I
"U
B 10==—""1 .3
'D E .. E
:44 I
1 ArI I I J I I
15 20 25 30 35 40 45 50
90m(deg)

Figure 5.25: Measured cross sections and ECIS fits. Also shown are the optical model
calculations for the elastic channel using the Becchetti-Greenlees optical potential
parameters. The fit of the elastic data points for 44Ar is not as good as for the
other two isotopes, but it has been verified that the extracted deformation fl; is quite
independent of the actual Optical model parameters.

116

 

 

 

 

Figur
was a
samp
One 5
same-
Parai
defor
thou
meal
Show
agre.
ljelo.
Calm
p011)
SeCti

I.
Were

tab},

0.5

 

 

 

 

        

r l I I I l
-O- electromagnetic
A (p,p')prolate ® (p,p') vibrational (neg)
0 4 § V (p,p') oblate — (p,p') literature
' l O (p,p') vibrational I (p,p') adopted here
.... 03 '— Q I 201 _-
<2“.I B a
0.2 — _
0.1 —— A
0.0 I I I I I ArgonI

 

 

 

36 38 4O 42 44 46
Mass Number

Figure 5.26: Deformation parameters obtained with the various fits: The value that
was adopted in this work is the average of all four values. The assigned error bar is the

sample variance of the data points about the average (a = s := \/ 171—1 201:,- — :E)2).

 

One is not allowed to divide this value by 2 (=\/4), since the four methods use the
same data and are therefore not independent.

parameters are listed in table 5.3 and shown in figure 5.26. One can see that the
deformations obtained using the various methods are grouped close together, even
though their (10) error bars do not overlap. The value that was adopted by us is the
mean of all four methods and is shown as a vertical bar. Previous measurements are
shown as horizontal bars. One can see that the previous measurement for “At [47]
agrees with the present result. ECIS fits have also been done, where only data points
below 0cm 3 28° have been used with very similar results. The Becchetti-Greenlees
calculations are also shown in figure 5.25 as dashed lines and they follow the data
points well, especially for center of mass angles below the first minimum in the cross
section. For larger angles, however, some deviation is apparent.

With the help of equation 4.18 and the measured deformations, the Mn/Mp values
were obtained and are listed together with other properties of the Argon isotopes in

table 5.4. The interaction strengths of protons with protons (bp) and neutrons (bn)

117

 

labI
[node
last I‘

have

inchn

 

PI
for 35.
under
Synnn
not a]
be no
Derfln

ln
Slllglt

A! II

”/1.

rotational vibrational value adopted

Isotope
pos neg pos neg here

 

36AI 0.348(13) 0.406(15) 0.362(12) 0.318(10) 0.359(36)
“Ar 0.290(13) 0.324(17) 0.308(14) 0.283(14) 0.301(18)
“Ar 0.277(13) 0.315(8) 0.297(6) 0.263(6) 0.288(23)

 

 

 

 

 

Table 5.3: Deformation parameters [32 obtained with the various ECIS fits for each
model (rotational and vibrational) positive and negative figs have been used. The
last column gives the adopted value.

have been chosen to be 0.3 and 0.7 respectively [43, 52]. The quoted uncertainty
includes a 1-0 error in bp/bn of 0.3.

Figure 5.27 shows the obtained Mn/Mp values. One notices the very large value
for 36Ar, indicating a large isovector contribution to the excitation. This is not
understood, since the excitation of an N = Z nucleus should be isoscalar by isospin
symmetry. Possibly, the prescription of Bernstein (equation 4.18) to obtain Mn/Mp is
not applicable and, as in the 388 case (section 4.1.3), microscopic calculations might
be necessary for a correct interpretation of the data. Such calculations are being
performed now in collaboration with E. Bauge in France.

In future experiments, measurements of the proton scattering cross section on the
single-closed-shell nuclei 38Ar (N=20) and 4°Ar (N=28) would complete the set of

Mn/Mp values for the neutron-rich argon isotopes.

118

 

 

 

FIgurt
incon1
)leasu
been p

Figure
21150 in

Neutron Number
1 8 20 22 24 26 28

 

 

 

2'0 I I I I I I
1.8 —- " Argon —
1.6 — _
Q 14 -— I _
z - I.
2: 1.2 — ] ] _
2 1.0— .................................. .2
0.6 -— _
0.4 I I L I I I

 

 

36 38 4O 42 44 46
Mass Number

Figure 5.27: A comparison of the extracted Mn/Mp values. The value for 36Ar is
incompatible with an isoscalar excitation that is expected for an N = Z nucleus.
Measurements of 38Ar and 46Ar, which are to single-closed-shell nuclei, have not yet
been performed.

 

 

 

2'0 I I I I I I
1.8 — I” o Argon m
o Sulfur
A 1.6 — _
s " --
a 1.4 — _
E 1.2— § 0 ] _
E 1.0 “" ---------------- - ------------- ...
0.8 L .3
1
0.6 l I I I I I

 

 

 

1 8 20 22 24 26 28
Neutron Number

Figure 5.28: Same as figure 5.27, but here the data points for the sulfur isotopes are
also included for comparison.

119

.mumumafieg cosmEuflmc @3855 23 so 828 9.8%: eon
Ev FE monogflom 3.83 mmmmfi mu: .«0 33%: 8m + 3 326:8 855 =< .mmqouofl :owdfl mo $35963 me @9556 Jam. 23H.

 

.9: I l 5%: 2392: 23” a @333 Es
3; 22:? 29.883 23:42 :39.” 26:5 :53: E:
a: 28%? @383 page 8va A33 @384 E2
m3 $856.0 3 $83

:1 So $38.23 33% 6:3 @6383 E2.
:s I I 8325 as $2 $85 a§§£.~ Es
2: 233?: 2836323

r: 8.0 8:82 336% $8.23 @253; Es
as I I gang Gamma €98 @833 E:
S2. .2212 2% am €533me E: 9625me 836$

 

 

 

 

 

 

 

120

Chapter 6

Summary

Two experiments have been performed to investigate the quadrupole collectivity of
the neutron-rich argon and sulfur isotopes.

In the first experiment the nuclei 38140328 and 44"“iAr were studied by Coulomb
excitation on a 197Au target. A position-sensitive, high-efficiency NaI(Tl) array was
constructed and was used to detect the de-excitation photons. The experiment al-
lowed us to determine the energy of the first excited J’r = 2+ states in three nuclei
(40’428, 44Ar), while a previously observed state in 4‘5.Ar was assigned definite spin. The
excitation energy of the first excited state in 388 was reproduced. In addition, the

38.40.4123,“,46 Ar) were measured

Coulomb excitation cross sections for all 5 isotopes (
and the reduced transition probabilities B(E2) were determined. The measurements
revealed a new region of deformation around 40’428. The reduced transition probabil-
ity for 4°Ar demonstrates that the N =28 shell closure persists at Z =18. The power
of Coulomb excitation in the investigation of heavy sd-shell nuclei can be seen in
figure 6.1. Shown there is the inverse quadrupole deformation parameter l/lfigl (the
inverse is shown solely for illustrative purposes, since otherwise the large values in

the foreground would block the view). The left-hand panel shows the available data

before this Coulomb excitation experiment, while the right-hand panel shows all data

121

 

 

 

Figux
R» U
beflor
sequc
avail;
UHSI
quen‘

lI
Huck
in Ft

(‘0er

|I32I1

  
  
 

Q:
3’

i, C‘lCium
1441-30,,

_ _~~ , ;[ -.-, \__/ Slug-0,,
20 ”we—~4— Inn” / 20 T-“““"‘-—-—~
22 r“ xw— — , 22 ‘\““~ —_./ “4 n,
- 2 “ w ~ -—4 2 ~~ n .
Nguln’ll lgulnbif 28 ’ Neutn)n Igumbzef- 28 sues,"

Figure 6.1: Inverse quadrupole deformation parameter. (The inverse was taken solely
for illustrative purposes.) The left-hand panel shows the data that was available
before this thesis work. The result from this work (cross-hatched columns) and sub-
sequent work (hatched columns) are shown on the right.
available at the time of this writing. The cross hatched columns were measured in
this Coulomb excitation experiment, and the hatched columns were measured subse-
quently at the NSCL.

In the second experiment the proton scattering cross sections on the unstable
nuclei 42"‘4Ar were measured. The results are not completely understood and a group

in France is working on microscopic nuclear matter calculations, which will help to

correctly interpret the measured cross sections.

122

 

 

 

A.1

As SlI
assun
mt’ch:

lS gin

The I

“here

But}

1(3)

Appendix A

Coulomb Excitation

A.1 Semi-Classical Approach and Perturbation

Theory

As shown in section 2.3 Coulomb excitation can be treated semi-classically. One
assumes a classical Rutherford trajectory but treats the excitation process quantum-
mechanically. The excitation cross section from an initial state ]i) to a final state If)

is given by

£13 (10
dfl —

— PH). (A.1)
d9) Ruth

The Rutherford cross section is given by

(3%)...=(%>28m”(3)’

with no given by equation A.15. Treating the electromagnetic interaction potential

V(r(t)) as a time-dependent perturbation, P,.,, can be expressed to first order as
1 °° .
PH, = Ia,_,,|2 with a,_,, = IE / e'“f“(f|V(r(t))|i) dt, (A.3)

where (Ufi = (E f —E,-) /h. For sufficiently high energies or large impact parameters the

Rutherford trajectory can be approximated as a straight line and the electromagnetic

123

 

 

pot

Cha:

wuh_

charg

 

 

 

Ther

and

Thes
hues
“lth

lOCatg
the X

Eben

Xkfie

potentials can be obtained through a Lorentz transformation of the field of a point
charge at rest

4(17) = A(B)A'(1‘-"1(fi)flv')1 (Pt-4)
with A’ (2’) being the 4-vector potential of a electric point charge at the origin

A'(:I:’) = (A’, 26') = (0, 0, 0, z’ e—TZ) (A.5)

11(5) is the Lorentz transformation matrix that takes the stationary charge into a

charge moving with velocity ,6 (see appendix C)

1 0 O 0
0 1 0 O
0 0 2'76 7

The resulting fields are the following Lienard-Wiechert expressions:

(”1' = Zpe'y A.7
d) (,t) \/(b—:c)2+y2+’)r2(:1:—'ut)2 ( )

 

and
v
A(”)(r,t) = —C-¢(”)(r, t) . (A8)
The superscript (p) indicates that this is the field generated by the projectile nucleus
’measured’ at position 1‘ and time t. The coordinate system that is used is oriented
with the z-axis in the direction of motion of the projectile. The target nucleus is

located at the origin and the projectile passes with impact parameter b (displaced in

the x direction; i.e. 33,, = b, yp = 0, 2p 2 at ). The electric and magnetic fields are

given by
Ez—la—A- — V05
° 6‘ . (A.9)
B=V x A
More explicitly:
eZp'yb __ 7E0

 

(EJ. :)E:r :

NICO

(b2 + AWN”)g (1 + (:92)

124

 

 

 

Witl‘.

follox

 

with

for e:

for n

for 1

and

Th.

E _ _ eZp'yvt __ 7E0 3
. 02+72v2t2fi (new

(Bi =)B = fiEx=fi ‘32” = — ME“
” (b2 + 72v2t2fi (1+(:)2)%
3,, = 0

 

 

 

 

with T 2 7i” and E0 = 93253. 7' defines the collision time.

After expanding ¢ and A in multipole moments Winther and Alder [5] obtain the

following excitation amplitudes

 

 

Z 2 I M A— AM,-
aH, = —z' hf; 26.1,.(5) (—)" K,(g(b))\/2/\TkA< f ”M“; “l > ,
(A.10)

7041

with (for p 2 0)

0°" (5) Z ”"4253” (Ii-1:35)? ((5) - 1)?
((A+1)(/\+#)P,i (c)_)1(/\—mu+1) u (c))

__ p _
2A+1 H 2A+1 “1 v

 

 

for electric excitations (7r 2: E) and

GM“ (5) = wamlfiw (84:35) ((5) — 1)—%”P£(g) (All)

for magnetic excitations (7r 2 M). Pf(:1:) are associated Legendre functions evaluated

 

for a: > 1. For n < 0 the following relations can be used
GEA—p = (—)”GE,\p and GMA—p = —(—)”GM,\” . (A.12)

Even though [1 is listed as a summation index it is actually fixed by the nuclear matrix

element to be

and can be identified with the angular momentum transfer along the beam direction.
The quantity 5 (b) is the adiabaticity parameter given by

7rao

£(b) = ‘13—; (6+ 27) (A14)

125

where a0 is the half-distance of closest approach in head-on collisions, assuming the

nuclei are point-like and if non-relativistic kinematics is used

2
7710112

 

00

(11.15)

mo is the reduced mass of the two nuclei. The second term in the parentheses in
equation (A.14) is a correction term that takes the recoil of the target nucleus into
account. This expression is obtained by comparing the relativistic result with the
low-energy Coulomb excitation result when large impact parameters are considered,

therefore justifying the use of straight line trajectories [5].

A.1.1 Cross Sections

The excitation cross section is obtained by integrating the product of the Rutherford
cross section and the excitation probability. A sum over final and an average over
initial magnetic substates must be performed, since they are not observed in the

experiment.

00

1
i = 2 f d E i 2

 

min
2

.9# (€(bmin)) -

 

 

 

a... (S)

 

2
he 62

«Au

The function g“ is defined as

9u(§(b)) : 27’ (%)270Pdle#(€(/’))l2
b
= QWflKu($)]2fEd$
E

= «£2 [IK..+1<0I2—IK.(£)I?—2—,“KI+1(€IK..(€) .

The K p are modified Bessel functions.
The above expressions have been programmed using MATHEMATICA [53]. Given
below is the MATHEMATICA code for the symbolic definitions (one can use them

126

 

 

 

for r

9qu

 

 

 

A.1

The

them

The
at [I

SF‘SU

and

Iran.

for symbolic calculations as well as for numerical ones) of the functions GEM, (5) and

9,,(5). The code is very compact, illustrating the power of MATHEMATICA.

GEElam-.mu_,invbeta_]:= (-1)“mu * GE[1am,-mu,invbeta] /; (mu<0)
GEElam-,mu_,invbeta_]:=(
I‘(1am+mu) #
Sqrt[16 PiJ/(lam*(2 1am +1)!!) *
Sqrt[(1am-mu)!/(1am+mu)I] e
SqrtEinvbeta‘2 - 1]‘-1 *
(
(1am+1)(1am+mu)/(2 1am + 1 ) # LogondroPElam-l,mu,invbeta, LogondreTypo -> Complex 1 -
1am * (lam-mu+1)/(2 1am + 1 ) t LegendrePEIam+1,mu,invbeta, LegendreTypo->Complox ]
)
I; ( (invbota > 1)&& (mu<=1am) it (mu>=0) ) )
gAlEmu-.xi-]:8 Hodulo[{mupos},
mupossAbs[mu];
Pi xi‘2 ( Abs[BesselK[mupoa+1,xi]]‘2 - AbsEBesselK[mupos,xi]]‘2 -
2 mupos/xi BessolKEmupos+1.xi] BesselKEmupos.xi]) J

A.1.2 Relation Between Impact Parameter and Deflection
Angle

The impact parameter can be related to the scattering angle by calculating the mo-

mentum transfer. In lowest order the momentum transfer to the target nucleus is

t

(00;) = f — (9%)rzo Zte dt. (A.16)

—00
The subscript r = 0 means that we evaluate the field (generated by the projectile)
at the original position of the target nucleus, which is the origin of the coordinate

system. Evaluation of the integral results in

__:Z£Zj£:p€2

((1:0(m) = b’U ’

(A.17)

where Zt,p are the proton numbers of target and projectile, b is the impact parameter,
and v is the projectile velocity. Since the total momentum is conserved the momentum

transfer to the projectile is the same (but opposite in sign)

__.:2£Zt2:p62

by (A.18)

pi”’(00) = —p§3)(00)

127

 

 

 

Hen

whet

 

moth

of 1111

The I

C46.

The

Mult

The momentum in the z-direction is almost unchanged and given by the product of
the relativistic factor 7, the (initial) velocity of the projectile v, and the projectile
mass mp,

pz 2 7mpv . (A.19)

Hence, the deflection angle in the laboratory is given by

x 2Z Z 2
z p

where small angles have been assumed. Alternatively one can use the equation of
motion in the center of mass system and calculate the scattering angle as a function

of impact parameter assuming a relativistic Coulomb trajectory [54]

mot}? '

00

b = —cot(%06m) with an (A21)

The relation between the laboratory and the center of mass angle is given by equation

C.46. Both methods give identical results (within the approximations).

A.2 Some Formulas

In the following a list of useful equations is given. The relations will not be derived.

A.2.1 Relations between 52, Q0, B(E2) . . .
The nuclear radius for an axially symmetric nucleus is given by
12(0) = R0(1+ 52Y20(9, 4)) . (A22)
Multipole moments are defined as
QLM = i/p(r)rLYLM(6, 05) dr , (A.23)
Q,,- = /p(r)(3:1:,-:I:j — 7260') dr ,and for L=0 and M22 (A24)

128

 

 

 

whr
R0 i
the

The

The

Em

“"11

all

 

167I' 3

Q20 = —5—4—7r_Z€ROfl2 )

(A.25)

where 62 is the quadrupole deformation parameter, R0 is the nuclear radius (e.g.

R0 = 1.2 fmA%), and Z is the nuclear proton number. In first order approximation,

the deformation 52 is related to the B(E2) through

 

 

 

 

 

47r 1
= —- 2' + 2+ . .2
32 3(/B(E .0 —> )ZeRg (A 6)
Electromagnetic transition probabilities are given by
87r(A +1) k2’\+1 . p E,
,- = B A; ,- k = — = — .
W(7rA,J -—> Jf) A((2A+1)!l)2 h (7r J —) Jf) w1th it he
(A27)
The transition probability is related to the lifetime T and half-lifetime T%
T w-1 Tl A 23
_ _ ln2 ° ( ' )
The B(7rA) values are defined as
B(7I’A,0 —) A) = Z [(Jfo]M(7I’Au)]JiM,->]2
#11411
_ 2
— 2, + 1|-<J)||M(7r/\)IIJ>I
Energy-weighted electromagnetic sum-rules from [5]
14.8yA—Z (32fm2 - MeV for A =
ZB( 7r(A, i—>f) (Ef—E ,-) =
5.0 A(2A + 1) 21‘2”“2 e2fm2 - MeV for A 2 2,
(A29)
with R z 1.2A3fm.
Single particle estimates (Weisskopf units) are defined as [9]
Bu, (EA): -—1— E—H 2 —)2 R2’\e2 (A 30)
A + 3 ’ '
and
2
Bw(MA) = E (L) 1229-9332,, . (A.31)
7r

A+3

129

A.2.2

Constants

he 2 197 MeV fm

_ 1
‘137

e2 = ahc = 1.44 MeV fm

(I

_ ehc
_ 2M,,c2

 

MN

130

(A32)
(A.33)
(A34)

(A35)

Appendix B

y-Ray Angular Distribution
Following Relativistic Coulomb

Excitation

The excitation process does not populate the magnetic substates of the excited state
evenly and hence the angular distribution of emitted photons will be anisotropic. In
order to determine the detection efficiency the angular distribution of photons must
be calculated and included in equation (2.33). In the following the 'y-ray angular

distribution for intermediate energy Coulomb excitation is derived.

131

Per

 

and

 

by

and

“her
with

give:

 

)Vhei

 

Slatf

Iff) :

B.1 General Structure

Perturbation theory is applied to the process

 

All l1f>
H7

 

IIfkaI

 

 

 

III)

Here HCE and H7 represent the interaction Hamiltonians for Coulomb excitation
and radiative 'y-decay, respectively. The amplitudes for the two processes are given
by

at»; = (IfolHCElIiMi) (B-1)

and

ar—w = (IffofkalelIfo) , (132)
where the final state consists not only of a nuclear state but also of a photon state
with wave-vector k and polarization 0. The amplitude for the whole process is then
given by

“Hf! == “Hf—MU = 2 am; ar—w
M!

= ZaianHMflkalellfo)
M!

where the sum goes over all possible magnetic quantum numbers of the intermediate

state.1

 

1One could view this process also as one where the state | f) = EM, a,-_,f|Ifo) decays to state
lff) = [I ff M fka’), from which follows that

ai—w = (ffIHvaI = Zai—U (IffofkalelIfo) (B3)
MI

132

 

 

 

 

Fr)
absolt
ing or
which
be drt
initial
of the

0H);

B.i

B.2

1; I

wh,

¢

tex

From the transition amplitude one gets the transition probability by taking the
absolute square, summing over final magnetic substates and polarizations and averag-
ing over the initial magnetic substates. Since the final result is an angular distribution
which will be normalized to unity at the end, constant factors in the equations will
be dropped without special notice; e.g. the factor 5171:; from the averaging over the
initial magnetic quantum number (M,) is not shown in B4. The angular dependence

of the transition probability is due to the presence of the vector k in the amplitude

a,_,ff. 0 and cp are the spherical coordinates of the vector k.

WWW) = : [Oi—4M2
M,’,Mff,0
2

= Z :aHfofofkalelIfo) (B-4)
M5,M”,a' M;

= Z ”Midi—1;!(IffMflkaleIIfo)
Mi’Mff'a
M].M}

>< (IffofkalelIfMj‘y

B.2 Detailed Derivation

B.2.1 What is (IffokalH,]Ifo)?

H, is a scalar operator, a spherical tensor operator of rank zero. This is clear from
rotational symmetry.

Usually H, is expanded into a nuclear part and a (external) field part
nuc ield
H, = 203,, >056, ’ , (3.5)
A.1:

(nuc)

A“ is a spherical tensor of rank A, which usually appears in

where the operator O
textbooks to derive transition probabilities. The difference is that O('"‘°) operates

between two nuclear states and 0(fz'eld) operates between a one photon and a vacuum

133

state. The product of the two amplitudes can be related to the transition probabil-
ity for one particular multipolarity A. This expansion of H, is not needed for this
derivation. The only information needed is that H, is a scalar operator.

In order to use this property H, needs to be sandwiched between angular momen-
tum eigenstates. The nuclear part is already fine, but the photon part is given in plane
wave eigenstates. Therefore we insert a complete set of photon angular momentum

states:

(IffofkalH7lIfol
= Z(IffofkalIffofLMl(IffofLMlelIfol
L,M

: Z(kalLA/Il(IffA/IffLMlH'rlIfo)
L,M

= Z (kalLM)(IffofLMlL’M')(L’M’IH,|Ifo)
L,M
L’,M'

1
= ZIRUILMWHMHLMIIIMII

L,M —’_——21, + 1<IfllH1llIf>°
From the second to the third line the nuclear part in the first bracket was dropped.
In the fourth line another complete set of states was inserted to express | ff) in terms
of the total angular momentum. To get to the last line the Wigner-Eckart-theorem
was used.
(kalLM) is the overlap between a photon-state with wavevector k and polarization
0 with another photon state with sharp angular momentum L and projection M. We

can rewrite |k0‘) in terms of [20) by using the rotation operator
[k0) = B(z -> k)|z0). (B.6)
Inserting a complete set of states and multiplying by (LMI from the left gives

(LMlka) = Z (LM|R(z —> k)|L’M’)(L’M’|z0). (13.7)
L’M'

The rotation operator R(z —+ k) connects only states of the same angular momentum

134

 

and the 8111'

 

 

where the 1
momentum

the angular

 

has to expaI

Was d0ne e.1

Here 603 IS

Combining

Hence the

and the sum reduces to a sum over M’ only.

(LMlka) = :(LM]R(Z —> k)]LM')(LM'|z0) (B8)
= §(D1T4M’(z ‘2 k))*<LM’IZU), (B.9)

where the DIIQM, are the matrix elements of the rotation operator in the angular
momentum basis (see equation B.20). To find an expression for (LMlza), which is
the angular momentum representation of a photon traveling in the z direction, one
has to expand the photon field in terms of angular momentum eigenfunctions which

was done e.g. in [55]. The result is2

M6,, : 7r 2 — L
(zalLM'Ir) = 2 M ( ) (B-10)
Eli—1 0’ 60M 2 71' = (—)(L+1)

Here 605 is the usual Kronecker Delta. The 0 in the second line is very important.

Combining equations B8 and B.10 it follows that

2L+1
2

 

(kolLM) = Dfmz —> k)[6EL + adML]. (B.11)

Hence the following expression for the matrix-elements3 of H, are obtained

(IffofkalelIfo> = 2V2L+1UIJMIILMIIIMf>
L,M

fo40(z —+ k) [6EL + 06ML],

where the 6,1, are factors which are given explicitly in e.g. [56], but this information is
not needed in this derivation. The 6,,L’s can be related to the mixing ratio for mixed

multipole transitions.

 

2At this point we also have to introduce parity, which was not needed so far; in the previous
formulas parity was implicitly contained in the summation over L since for gamma decay 7r and L
are not independent as soon as the nuclear spins are given.

3See also [56] page 329

135

B.2.2 The Angular Distribution

Putting the matrix elements of H, into the expression for the angular distribution

 

B.4 one gets
W(0,(p) = Z a,_,fa:_,f,\/(2L+1)(2L’+1) (B.12)
”li'Mff’a
M,.M’f
L,M.L',M’
X If! L I! If! I" 1! (_)(—L+Mf)+(—L’+M})
MflM—M, M,,M'—M;

X D1160 DU); [dEL + UdML] [6EL’ + UdMUr,

where the Clebsch-Gordan Coefficients have been replaced by expressions involving

3j-symbols (B.28). ngh D5,; can be evaluated as follows":

L LI f _ M'—0’ L LI
DMUDM’U — (—) DMaD—M’

= (—)M"" Z (2j+1)(,L4,’,,_IjJ) (Lj ”)19’23’

am’—0

= (-)M'“’Z(2J' + 1) (,1; #34,) (£31) Diho*(a,fl,7)

, L ' L' L ' L'
(_)M —UZ\/2j+1(M7:l-M’) (03_0_) }/jm‘(fiva)
],m

Here, use has been made of the relations B23, B25, and B27.

Using B.35 one can evaluate the sum over M ff which only occurs in the first two
3j-symbols in B.12

23(11): L’ 1; )(Iff L If)

I LI L k
: _ 2L+Iff+M —Mf _ k 2k
() .23 )( +1)(M,_Mn)

x I, I, k L’Lk
M}—MfK. IfIfIff ,

°if no arguments are given we always assume the same Euler angles

 

136

 

 

with the fol

II'(6. I

Here. 6) staI

 

 

Slimming 0“

TO gm this

Stile-Chou 7‘11. I

Th‘i‘refore

with the following result:

 

W(0,tp) = Z aiafa:_,f:\/(2L+1)(2L'+l) (2k+1)

Mi,a,k,n
I
Mf'Mf
L.1W,L’,M’

X L L’ k I, I, k LL’k
M-M'K Mf—M}I€ IfIfJ
x(_)(M;-L)+(M}—L’)+(2L+M'—M,)+k
,_ . L j L’ L j L,
_ 1W (7 2 1 at ’ .
XI) 2; 3+ )(Mm_M,) (00%) xmahah
Here, 5A stands for [5B, + adMA]. The sum over M and M’ can now be performed:

2, L L’ k Lj L’
ZHM (M—M’n) (Mm—M’)

M,M’

_ Z L L’ k Lj L’
’ M—M’Is: Mm—M’

M,M’

L L, k L L, j (_)L+L’+j
, M —M’n M —M’m

_ L+L’+k

= —2k_+—1—6kj6m" -

II
:M

A
v

Summing over 0 yields:

Lj L’ 2 (f3 5;) (”fly I jeven
2 6L 6L. =
a={1.—1) 0 : jodd

To get this result the symmetry properties of the 3j—symbol (B30) and the parity

selection rules for EA and MA transitions were used.5

 

(—)L : for EA transitions
”1”? = (3.13)
(—)L+1 : for MA transitions
Therefore
W(0, e) = )3 aisfaggm/(zL +1)(2L’+1)(2k +1)
$23512

 

L = even 6 = 6
5e.g.: for mm = + and => L EL
L 2 Odd 6L 2 ngL

X I! If k LkL’ LL’k
Mf—M}I€ 10—1 IfIfIff
x (—)(Ml‘L)+(M}"L')+(2L-M;)+(L+L’+k)+lc

X thc 6L 6U

 

Z aiafa:_,f;\/(2L + 1)(2L’+1)(2k +1)

I: even,n.M,’

MI,M}.L.L’

X I, I, k LkL’ LL’k
Mf—M}I€ 10-1 IfIfIff
X (-)M} Y1; 5L 5U

2 a. a, I, I, k
H’ Hf’ Mf—M}rs:

k even,x.M'-,
MI,M}.L,L’

X (—)M}Fk(L, L’, I”, If) Y’s!“ 6;, 6L’

If particles are detected symmetrically around the z-axis one can integrate over

‘Pparticle which is equivalent to integrating, or averaging, over (p7. Using B.36 yields

the final result:

we)

2, a,- a? ' (—) ’
keven,M'-, —)f _)f (Mf —M}0
M,,M’,,L,L’

X Fk(L, LI, Iff,1f) V2k +1 Pk(COS(0)) 6L (5y

:1 If If k M]
Z alfifa’i—{f (M! —Mf O) ( )

k even,M,-.

M,,L,L’

X Fk(L, LI, Iff,1f) V2k +1 Pk(COS(0)) 6L 6U

I I k
. 2 f f _ M,

k even,Mi.

Mf,L,L’

X Fk(L,L’, Iff,If) V216 +1 Pk(COS(9)) 6L 6y

I I k
kegml fl M,—M,0 () ( )
Mf,L,L’

X Fk(L, L], I”, If) V2k +1 Pk(COS(0)) (51461}

138

B.3 Angular Distribution Using the Winther and

Alder Excitation Amplitudes

Putting the excitation amplitudes from [5] into the expression for the angular distri-

bution B. 14 yields6

W) = Z:

k even,M‘-.
&![.L,L’

x (If If k) (—)Mka(L, L',1,,,1,) \/2k+1Pk(cos(0)) 5,, 5”.

The absolute square of the excitation amplitude can be written as follows

u If /\ Ii 2
- K 6“.) (_)MI (—Mf —/1Mi)
— C [1 If A Ii
— EGAM(;)KII(€)(—) (—Mf "H Mi)

C * 1.. p' If A Ii
X GAp'(;) KM“) (_) (_Mf —#’Mi)

I A I- I A 1'
_ 2 f t f ‘
_ SIGMA: )QLIIK (5 )l (—M;-/1Mi) (-M, —#M.')

The last step could be performed since u and ,u’ are fixed by the 3j-symbols. Hence

 

c u M! [f A I,- 2
gGAu(;)Ku(€)(—) (—) (—Mf "#Mi)

 

 

         

I, A 1,)(1, A 1,)
W0 = G K
() 2M 2| M— 5m (52» (am—m, -M,—uM.-
MfHLL’
If If k _M : /—_
X ( ) fP‘k(L,L,Iff,1f) 2k+1Pk(COS(9)) 61,61}.
M,-M,0

Using 835 one can evaluate the sum over M, which only occurs in the first two

Z< I, A 1,)(1, A 1,)
Mi —Mf—/.I.Mi "Mf—pMi

_ 2I,+I,—M,—p _ k’ r If If I")
(> a ) (2k+1)(_Mfo,c

k’ ,n

x A A k’ IfIfk’
p-pn: AA],-

6See [5] or appendix A for a definition of G “(5), K “(E) and £.

33' —symbols

139

W(9) = Z IGAL(; 62)| IKIIEU -)_Mf "XX- (2k'+1)

k evenm
A1f,L.L’

x If If k' A A k' IfIfk’ If If k
—MfoIt p—pn A AI,- Mf—Mfo

X (—)Mka(L, LI, Iff,1f) V2119 + 1 Pk(COS(0)) 6L 61,!

Summing over the M f in the first and the last 3j-symbol (B.29) gives a 6H6“.

Therefore
c A A k
W 9 = G — 2K 2 — “ -
III :I AIIUIII II(€)|( ) (M10)
L.L’
I I k
X{111.}Fk(L,L,,Iff,If)\/2k+1Pk(COS(0))6L6LI .

Here, we also indicated the impact parameter b in the formula, which was im-
plicitly assumed before. The remaining step is to integrate this formula over impact

parameters with a minimum of bmin, which is e.g. determined by 0mm from the ex-

periment
W(9) = / mm) b db. (3.15)
bmin
As in appendix A the integral over the square of the modified Bessel function
/ IKIIIoIZ’bdb (8.16)
bmin

can be expressed using the Winther and Alder function 9,,(5)

9,460)» = 27%} ) [pdle(€p())l

= 27r/lKu(a:)|2:rd:c

= «£2 [IKu+1(€)l2 — lKu(€)l- 2§KI+II mm

140

Thus

W) = *2: IGAII(-:-)l2gu(€)(-)“(:_Au3) (8.17)

L,LI

X {111/{f } Fk(L, L’,Iff,If) V2k +1Pk(COS(6))6L 6L! . (13.18)

Usually the angular distribution is expressed as

we) = Z akPk(cos(0)) (13.19)

I: even

The following the MATHEMATICA code calculates the coefficients ak in front of the

Legendre polynomials:

141

(s i gets excited to f and decays to ff
StatesI 8 {{Ji,Pi},{Jf,Pf}.{Jff,Pff},delta}
I9)
SRules I {
JiS[x-] > x[[1.1]]. (* initial spin *)
PiSfx-] -> X[[1.2]]. (’ initial parity 1")
JfS[x_] :> x[[2.1]], (* excited spin *)
PfS[x_] '> x[[2,2]], (# excited parity t)
JffS[x-] > x[[3,1]], (t final spin *)
PffSEx-) :> x[[3,2]], (* final parity *)
DeltaMixSEx-] :> x[[4]] (* mixing for simultaneous E2 and M1 decay *)
}

AngDis[K_,P_,T_,S_]:=

Module[{LamHin,LamMax.ParityChange,kHax,tab.PureElamTransition},
CachinEK,P,T]; (* first do the kinematic calculation *)

(a check for the involved transitions *)
Lamflin 8 AbsEJfSESJ-JffSES]]S
LamMin = RationalizeELamMin]; (t make sure we have an integer t)
If[(Lamflin===0)&l(JfS[S]+JffS[S]>0),LamMin=1,,Print["siw"] ];
(t there is no H0 or 80 transition t)

(e find out what multipolarities for deexcitation are possible t)
If[PfS[S]=!=PffS[S],ParityChange=True,ParityChangeaFalse,Print["siw.ParityChange"]];
If[!IntegerQ[LamMin],Print["LamHin is no integer; LamHin 8 ",LamHin];Return[]];

(* we need to consider Elam and H(lam-1) transitions if the following is FALSE t)
PureElamTransition 8 (EParityChange it EvenOELamMinJ) ll (ParityChange It OddQ[LamHin]);
If[ PureElamTransition,

LamMax=LamMin. (* Pure ELamMin transition t)

LamHax=HinELamHin+1.JfSES]+JffS[S]], (t for Hlam transition consider next
higher E transition if at all possilble 0)

Print["siw AngularDistrib"]];

Print["LamMax ",LamMax];

Print["LamMin ",LamMin];

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

(* first we determine kMax #)
kHax=HinE2JfSESJ,2LamMax];
tabSTable[Sum[

(* Print["11 ",11];

Print["12 ",12];
Print["mu ",mu]; f)
If[11===LamHin,1,DeltaMixSES].Print["siw AngularDistrib 2"]] #
If[12===LamHin.1,DeltaMixSES].Print[“siw AngularDistrib 3"]] t
AbsEGEELamNEP].mu,1/BetaK[K]]]‘2 gAlEmu.XiHoK[K]] (-1)‘mu #
ThreeJSymbol[{LamN[P].mu}.{LamN[P],-mu},{k,0}] #
SixJSymbol[{JfS[S],JfSES].k}.{LamN[P].LamNEP].JiS[S]}]
F[k,JffS[S].11,l2,JfS[S]] Sqrtf2k+1].
{11,LamHin,LamMax},{12,LamMin,LamMax}.
(t sum over the possible deexcitation multipolarities *)
{mu,-LamN[P],LamN[P]} (* sum over magnetic ON of excitation *)
] (* close Sum *)
,{k,0,kMax.2}]; (* close Table-loop *)
Do[
Print["a",k," 8 ", N[ tab[[k/2+1]] / tab[[1]] ] J .{k,0,kMax,2}];
tab/tab[[1]] // Together
]//.SRules//.NKRules

(* set the necessary attributes for AngDis *)
SetAttributes[AngDis,HoldFirst]

H[theta_,dis-List,K_:1]:=
Module[{Hcm,Hcontracted,Uboost},
Hcm[t_]:= 1/(4Pi) PlusOO (Table[LegendreP[2k,Cos[t1],{k,0,Length[dis]-1}]'Times’dis);
If[!List0[K], ReturnEHcmEthetaJJ J;
Hcontracted[t_]:= HcmEt] t DomCmDomLab[t,BetaX[K]] //.Kfiu1es;
Hboost[t_]:= Hcontracted[ThetaGm[t,BetaK[K]] J //.KRu1es;
Hboost[theta]

142

 

BA Angular Momentum Re-coupling Coefficients,

Rotation Matrices and Related Formulas

The phase convention used for the Clebsch-Gordan coefficients is that of Condon and

Shortly [57].

B.4.1 Angular Momentum Representation of the Rotation

Matrix

Definition

DLmAmflfl) == (J'm'lR‘1(a,fi,7)ljm)= (J'mllem'>

= <jm'le””e

infleiJza

 

jm)

: (jmlleim’yeinBeima Um)

eim'vdzgmmeima

dj

Symmetry Properties

(DkIa(av 16a 7)): :

Dllll’la(aa 16,7)

Special Values of Indices

Dll'n0(a116a7) :-

D:IW(_77 —:Ba —a)

(_)M—UD£M—a(aa :8) 7)

(‘)M—0DaLM(’YI 3a a)

47r
2l+1

 

143

mmrw) == (jm'leimljm)

Mm(fla 0)

(3.20)

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

03.3.2.7) = (ea/$342.7) (B26)

Product of two rotation matrices of the same Eulerian angles 0,-

B.4.2

Dj‘ ((2)1912 (9.)

mlal "1202

= 2),...(23' + 1) (3:, 3.33,) (3.13.3.2) Dz...‘(0.) (B27)

Some Properties of the Clebsch - Gordan - Coefficients

and 3 j-Symbols

Relation between Clebsch-Gordan-Coefficients and 3j-symbols

()1j2m1m2IJM)=(—)J‘1-J‘2+W21+1(’1 32 J ) (B28)

m1 m2 —M

Orthogonality of the 3 j-symbols

JJ'j1)(JJ’j2) 1
= ...—_5.1.25m1m 3.29
L§41(M.M’m1 MIMI";2 231+111 2 ( )
(J1 J2 J3) : (_)J1+J2+J3( J1 J2 J3 ) (3.30)

m1m2 m3 —m1 —m2 —m3
: (J2 J3 J1) (3.31)
m2 m3 m1
(_)J1+J2+J3 (J2 J1 J3 ) (8.32)
7722 m1m3

B.4.3 Some Properties of the 6j-symbols

Symmetries of the 6 j-symbols

I1 32 J3 )2 I1 )3
: BI
{J1J2J3} {J2J1J3} ( 33)
J1J2j3}
. . B.34
{31 32 J3 ( )

144

Relation between 3 j- and 6j-symbols

2(Ij1j2)(1J1J2)
l lmlmg [Mle

: (_)211+I+m1+Mzz(_)k(2k+1)(jl J1 k) (B35)

k,x m1 —M1 H
X )2 J2 k j. J. 1:
—m2 M2 h) J2 jg I
B.4.4 Y)",

Relation between Yzm and Pf"

 

2l+1(l—m)!

4w (l+m)! Bm<C°S<9>>e""‘” (B36)

 

Mm(67 ¢) : \J

B.4.5 7 -— 7 Correlation Function

Definition

 

Fk(L,L’,Il,12) = (—)’1+’2‘1\/(2k+1)(212+1)(2L+1)(2L’+1)

Lflk LUk
X (1—10) {1212 11} (3'37)

145

Appendix C

Relativistic Kinematics

This chapter develops some formulas that are needed for computer calculations, which
are used in chapter 5 to obtain the excitation energy and center of mass scattering
angle from the measured particle energy and laboratory scattering angle. We will
not have a single formula as a final result, but rather instructions on how to combine
certain formulas to obtain what is desired.

The formulas are derived and in the last section a summary is presented.

C.1 Notation and Preliminaries

We will use the Minkowski notation with x4 = iso and c = 1. To obtain the equations

with the correct power of c at the right place, just follow these replacement rules:

p——>cp , m—>mc2 ,and flag (C.l)

Repeated indices are summed implicitly.
If one knows a certain 4-vector in one system, one applies a Lorentz transformation
to obtain this quantity in a different system. For practical purposes consider only

transformations in the z-direction since one can obtain others by simply combining

146

Lorentz transformations and rotations. Assuming:

(3 = ( 2.12, ) (02)

then 12’ is obtained through

p; = Aijpj , (summing over j is implicit) (C3)
with A given by:
1 0 0 0
0 1 0 O
A— 0 O 7 473 (0.4)
0 0 ivfl ”Y
with fl = g and
1
7 = —— (05)

(/(l -fi2) '

Consider the 4-momentum of a particle initially at rest

p:(i(7)n):(z'1§ ). (0.6)

The momentum after a boost in z—direction with velocity fl is given by

0
, 0
Wm
iym
p’ can also be written as1
0
, 0
p = I (0.8)
p
iE'

Comparison of the two results yields the following very useful formulas:

E = 7m and p = yflm = ,BE. (C.9)

 

1We will use p’ to label both the 4—vector and the magnitude of the total momentum. It should
be clear what is meant from the context.

147

Taking the scalar product of p with itself yields p - p 2 —m2. Since 19 - p is a Lorentz

scalar it has to be equal to p’ - p’ = p2 — E2. One obtains the well known result
E2 = p2 + m2 (0.10)
which can alternatively be obtained through the following

E2 : 72777.2 2 m2 + (72 _1)m2
: m2 +72fi2m2
= m2 +p .

The first line is an identity and going to the second line the definition of 7 (equation
C.5) has been used.
We see that A Operates on the particle states and not on the frame of reference.

A boost by 3 is equivalent to choosing a new frame which moves with velocity —fi.

C.2 Collision of Two Particles

Consider a collision B (A, A’ )B’ , which means that B is the stationary target and A
is the projectile. After the reaction we have the ejectile A’ and the recoil B’ . Figure
Cl and table C.1 show the quantities involved: E denotes the total energy, m the
rest mass, T is the kinetic energy defined as E — m, and p is the momentum. The
subscripts cm and )0), refer to the center of mass and laboratory coordinate systems,
respectively.

To obtain an expression for the total energy in the center of mass system, consider
the contraction of the total 4-momentum. This is a Lorentz scalar, meaning that this

quantity does not change when going from one frame to another (pcm 19m 2 pic), 1010),)
pcm ' pcm : _Egm

_ _ 2 2
— plab ' plab “‘ plab _ Elab

148

Laboratory System Center of Mass System
B(A,A')B' ,

A 3 fGIab A > . A 9cm 3
2/
<9.

Figure C.1: Schematic drawing of a collision of two particles in the laboratory system
and center of mass system. The arrows symbolize the momenta of the particles.

 

 

 

 

 

 

 

 

. Coordinate system
Quantlty
laboratory center of mass
0 0
0 0
PA
pAlab pAcm
ZEAM, iEAcm
0 O 0
0 0 0
p3 =
0 chm —pAcm
2777.3 21?ch ZEch
PA; , s1 (92); ) m; sm(0Ar )
PA! pAiab = 0 PA;,,, = O
iEAiab p4“) cos(6A;ab) z'EAém PAgm cos(0A;m)
iEAiab iEAgm
pBIab Sin(gBIab)
Pegab _ 0 —PA;,,,
[’3' . — .
”38;... PHI... COSWBI...) ZEBém
z.E‘Bfab
0 0
, 0 0 0
P = P = .
pAlab pAlab ZEcm
“EA,” + m3) z.(E‘lab)

 

 

 

 

Table C.1: Momenta of particles before and after scattering in the center of mass
frame and the laboratory frame (see also figure CI). p is the total momentum of
particles A and B combined.

149

= pl2ab - (EX... + 21321:.me + mfg)

= _(mil + ZEAlame + m2B) a
where in the last step E2 = p2 + m2 was used. Therefore
E02," 2 2&4“me + mg, + "123 . (C.11)

In order to obtain an expression for the energies of the individual particles in the
center of mass frame, we use the fact that in the center of mass frame the linear

momenta of the two particles add up to zero (PAC... + chm = 0), therefore:

2 _ 2
pAcm _ chm

2 2 _ 2 _ 2
Ecm_mA — Ecm 7723.

Using ECm = E Am + E30," to replace e.g. Eficm yields

 

E3, = 213ch EC", — mfg + mi, (0.12)
01‘
Efm + m2 — m2
EB“, _—. 2E3 A . ((3.13)

From E ch one can obtain flgm through:

’73 = Ech
cm m8
_ —2
fich _ 1 — 78cm

Since particle B is at rest in the laboratory system, the velocity of B in the center of

mass )6ch is equal to the velocity of the center of mass in the laboratory 5ch

Ban = fig... . (0.14)

Other ways to obtain ficm are

a E
ficm : Pl b _____ pAlab : 5AM. Alab (0.15)
Elab EA“), + m3 Ema), + m3

 

 

150

01‘
c _ _ _ __ _, ac 0.16
7 Ecm ( )

C.2.1 Relation Between 60m and 61a),

From table C.1

pAga, sin(9A;,,) PAL... Sin(9A2m)
0 0
pA/ : and pAém : (018)
’a" 19%» cos(6A; ) pAIm cos(0A: )
213% z'EAr

p Aiab is also given by

PAL”, 2 Acm—>lab PAgm
1 0 0 0 . PAL... Shim/1'...)
_ 0 1 0 0 0
— 0 0 7 475 pAngOS(0A;m)
0 0 2'75 7 iEA’cm
psin(6)
_ 0
_ 7(pCOS(9)+flE)
i7(fiPCOS(9)+E) A,
lab
Three equations result:
PAgabSinWAgabl = PAgm Sin(9A;m)

pAlab cos(0A;ab) = 7(pArcm cos(0A;m) + IBEA'cm)
EAL“, = 7(52922," cos(0,,,m) + EAém)

Dividing the first by the second equation yields
sin(0,4rcm)
[3E I
7(cos(6Arcm) + p—j‘m)
sin(9A2m)

: . (C.19)
7 (COS(9A;m) + [321' )

151

tan(6 A105)

 

 

If [3 > 3.4;," then there is a maximum scattering angle, which can be found by setting

the derivative 00,120“ 80,427” equal to zero. The result is

 

fiA’
t max 2 cm . 0.20
an( Alab) WWW) ( )

Unfortunately there is no closed expression for 0A2...) because there is no closed
expression for E A?“ as a function of 0A2“. To solve numerically one uses the previous
formula iteratively until agreement is reached. But the formulas are useful for the

Special case where one particle is a photon.

sin (0,211”)

7 (cos(6A;ab) — 3232—)

lab

 

tan(0Arcm)

EAém : f)l(—.'BIDA;ab COS(0AIab) + EAIab) .

Photons

For photons the previous formulas can be simplified, since the photon (rest) mass is

zero and it follows that

 

E21) and 3:1. (C.21)
Therefore:
sin(l9)ab)
t 60m
...( ) 7(008(01ab)-fl)
tan(91ab) : sm(0m)

 

7(cos(06m) + 5)
Ecm : 7Elab(1 — fiCOS(glab))

Elab = 7Ecm(1 + IBCOS(ocm))

C.2.2 Solid Angle Relation

To find the relation between the solid angle subtended in the laboratory and the

center of mass system use
data), _ d(COS(0[ab))

do... ‘ d(cos(9m)) ' (0'22)

 

152

 

 

Figure C2: Relation between 2:,y, and 0.

From figure C2 it follows that

tan(6) = g (0.23)

and comparison to equation (3.19 shows that :r and y are given as

 

x = 7(COS(0A'°m)+fl,Zm)

y = sin(0A:cm).

In order to find cos(Hlab) we use

513

 

 

0 = —— . .24
cos( ) mi: +31 (C )
Therefore
'7 (cos(QAICm) + flj )
COS(6(ab) = cm (0.25)

 

2 .
\/Sln2(9Alcm) + ’72 (COS(0Alcm) + 32%;)
With the abbreviations of (sin(0A'cm) —> s), (cos(OAtcm) —> c), and (fi—f— —> fl) it
follows that

———:E:::Ezlab;; = 7(72(c + (1)2 + 82W - V(C + fl)(72(c + H)? + 32)“%(72(c + 3) _ c)
7(72(c + m2 + s?) — 70720: + I?)2 — c2 — cfl)
(72(6 + fl)2 + 82)%
7(1 + Cfl) .
(72(6 + fl)2 + 82)%

After reversing the substitutions

szab _ 7 (1 + cos(OAlcme, )

 

 

 

((3.26)

 

Photons

For photons the last formula simplifies to

dumb __ 7(1— CO5(9cm )5 )

(mam (720mm MW) +sin<o mV)-3-

 

 

((3.27)

The denominator can further be simplified to (using the same notation as before)

32 + 72(6 + fl)? = 32 + 72c2 + 272cfl + ff
2 1—c2+72c2+272cfl+72—1
= (72 —1)62 + 27ch + 72
= 72(fl202 + 2cfl + 1)

= 72(1 + 502,
where 72/32 2 72 — 1 and cos2 + sin2 2 1 was used. Hence

dQlab _ 1 _ (Em)2
dflcm — ”72(1+,BCOS(0 ))2 _ Elab . (C28)

 

 

 

C.3 Decay of A into B»;{¢=1,2,...,N}

The 4-vector of particle A is in the center of momentum is given by

O 0 Z 0

(al)

and in the lab
P30)

19.4101, = P3,“ = Z ( iE "5 ) (C30)

(2')
Blob

pic," has to be equal to pita!) and it follows that

mi = -(};p3,.b)2
(219322.) (223:3)?-

154

m A calculated in this way is known as the invariant mass. The excitation energy of

particle A is obtained as
E"; = mA — mfg) , (C.31)
and the breakup energy is

EST = mA — 2mm. ((3.32)

mg?) is the mass of particle A in its ground state.

CA Summary of Formulas

 

C.4.1 General

 

7: lifl ,fl= 1-7‘2 ; 7232: 2-1 (033)
E = 7m (C.34)

p = flE ((3.35)

E2 = P2 + m2 ((136)

g—; = M (0.37)

0.4.2 Total Energy in the Center of Mass System Ecm

Efm = 2&1“,me + mg + mg with EA,” = TAM + mA (C.38)

C.4.3 Individual Energies in the Center of Mass System

2 2 2
_ Ecm+mA_mB

 

 

 

 

EAM _ 2E0... ((3.39)
wwmi—ma (c...)
EA)“ = E22“ 37;:- m23’ (041)
Egg," = E32" +2";:_ mg" (0.42)

C.4.4 Velocity of the Center of Mass in the Laboratory ficm

Equation numbers are given above the arrows.

 

 

EA)... (“ii Ecm 23 7.3m = g": 9'3 mm ((3.43)

or
EAlab fl Ecm 9.23 Ech 9‘34 73cm "all 53.... = [gem (CA4)

or
EA)” C—3§ ”Men. ——> 5A.". 2‘33 PAM, = plab C43? 16cm = 2:: (C45)

The first possibility is the easiest, but one might choose a different one depending on

what other quantities are needed later on.

C.4.5 Relation Between 60m and 610),

 

tan ( 9A1“) = Sln(0Agm) , (C. 46)
'7 (cos(OArcm) + L)

3A2...

where 3 is the velocity of the center of mass system in the laboratory system.

 

 

Photons
Sln(01ab)
tanwan) 7(C03(01ab) - fl)
Sln(0cm)
tan(9zab) 7(COS(0cm) + )8)

Ecm = 7Elab(1 _ flCOS(61ab))

Elab : 7Ecm(l + BCOS(gcm))

C.4.6 Solid Angle Relation

_ __5_
M- 2(1 COW/2%)

dQcm — 2 . _
(72 (COW/“rm + mi...) + SmwA’mV):

 

((3.47)

 

156

Photons

thab 1
dQcm _ 72(1+flc08(0¢m))2

Ecm 2
_ (Elab)

2 72(1— ,BCOS(0[ab))2

 

 

 

C.4.7 Invariant Mass

2 2
"'32 = (21333)) ‘ (Z 23:22,) - (C48)

The excitation energy of particle A is obtained as
E“ - m — (0) C
A — A mA a ( '49)

and the breakup energy is
Eff = mA — 2mg,“ (0.50)

157

Appendix D

Simulation of the Proton

Scattering Experiment

A simulation program was written in FORTRAN that simulates the full detector
response, the geometry of the setup, the beam emittance, as well as energy loss and
angular straggling of the recoil protons in the target.

The code uses the N SCL histogrammer SMAUG for the generation of the spectra
and since it is also linked to the standard analysis routines, actual data can be used
as a starting point for the simulation. In this case actual tracking data are used in

order to study the effect of the beam emittance on the resolution.

D.1 Structure of Computer Code

0 Event Generation

— scatter two particles in center of mass system; option to use cross section

data or flat cross section
— transformation to the lab system

— depending on settings rotate coordinate system and/or move origin to

match the incoming particle; the incoming particle is defined by actual
measured PPAC data

158

— calculate and subtract the energy loss [48] in the target from the proton en-
ergy; the scattering occurs at a random depth in the target; add straggling
[58] to the proton direction; see figure 5.15;

0 Measurement

— check which detectors are hit (that includes the particle and proton detec-
tors)

— add detector response (energy resolution, position resolution...) to “mea-

sured” signals
0 Analysis

— same (or similar) as analysis of real data

159

D.2 Description of Input File

The following is a sample input file. It was used to create the spectra in figure 5.16.

m_a : 33533.7840
m_b : 938.2723
m_apO : 33533.7840
m-bp0 : 938.2723
beam_energy : 1211.0400
de : 0.0200
use_cs : F

ex_en_ap : 0.0000
target_thickness : 2.7000
axis_rotate : T
target_intercept_ca1c : T
do_angl_strag1 : T
strag1_max_iter : 10
e_noise : 0.1000
pos_res_10mev : 1.0000
strip_x_randomize : T
ppac1_resolution : 8.0000
ppac2_resolution : 3.0000
tof_distance : 40.0000
tof_resolution : 0.0000
hit_check_zero : T
hit_check_te1 : T
zero_radius : 38.1000
t_wethinkitis : 1211.0400
tracking_pos ° T
tracking_angle T
corr_for_target_th T
tof_correction : F
file_cs_gs_data : SYS$GAMMA3z[96035.SIMUL.36AR]cs_gs.data
file_cs_ex_data : SYS$GAMMA3z[96035.SIMUL.36AR]cs_ex.data
tot_cs_max : 300.000

All parameters are described in table D.1. If use_cs=T then two states are in-
cluded, one with excitation energy (=Q-value) of zero and the other with excitation
energy ex_en_ap. The cross section data files file_cs_gs_data and file_cd_ex_data
are used for both states, respectively. If use_cs=F then only one state is used with

a flat cross section and excitation energy of ex_en_ap.

160

 

 

Item Description

m_a projectile mass (MeV)

m_b target mass (MeV)

m_apO ejectile mass (MeV)

m_pr recoil mass (MeV)

beam_energy beam energy (MeV)

de beam energy spread

use_cs if T then use cross section given in the files f ile_cs_gs_data
for the elastic scattering and file_cs_ex_data for the ex-
cited state

ex_en_ap excitation energy of excited state; see text

target_thickness
axis_rotate

target_intercept_calc

do_angl_stragl
stragl_max_iter

e_noise
pos_res_10mev
strip_x_randomize
ppac1_resolution
ppac2_resolution
tof_distance
tof_resolution
hit_check-zero
hit_check_tel
zero_radius
t_wethinkitis

tracking_pos
tracking_ang1e

corr_for_target_th
tof_correction
file-cs_gs_data

file_cs_ex_data
tot_cs_max

 

target thickness (mg/cmz)

if F then use z-axis as incoming direction; if T use direction
supplied by PPAC data

if F projectile hits target in the center (a: = 0, y = 0), if T see
axis-rotate

if T include angular straggling in target

take a maximum of stragl_max_iter steps for the proton to
travel out of the target

noise in energy detectors (Silicon) (MeV) (FWHM)

position resolution of strip detectors (mm) (FWHM)

if T randomize position perpendicular to strip

resolution of first PPAC (mm) (FWHM)

resolution of second PPAC (mm) (FWHM)

distance between time of flight detectors (m)

resolution of time of flight detectors (ns)

if T check if ejectile hits zero degree detector

if T check if proton hit telescopes

radius of zero degree detector (mm)

beam energy used for analysis; can be different from
beam_energy in order to include systematic errors

if T use tracking to calculate position on target

if T use tracking to calculate incoming beam direction; if F
use z-axis

if T correct measured proton energy for half of the target
thickness

if T use time of flight information to correct for beam energy
spread; not implemented yet

path to file containing the elastic cross section; use if use_cs
kiT

same as previous for inelastic scattering

maximum possible cross section for correct calculation; if the
cross section (from the previous two entries is larger than
tot_cs_max then tot_cs_max will be used

Table D1: Explanation of input file for simulation.

161

Appendix E

Differential Cross Section

The number of reactions N, observed in a detector is proportional to the number of
incoming beam particles Nb and the number of target nuclei per unit area M. For
a particle detector that has a perfect intrinsic detection efficiency (like the silicon
detectors used in the proton scattering experiment, but unlike the photon detectors
used in the Coulomb excitation experiment) the factor of proportionality a := N—C’fi;

is given by pure geometrical considerations as
d .
0 = /—31n(6)d¢d0, (13.1)

where 3% is the differential cross section of the reaction to be observed. The integral

goes over 0, which corresponds to the solid angle covered by the detector. Assuming

a constant differential cross section within 9

d0

35 = a (/ sin(6) d¢ do) . (B.2)
Q

The integral over (25 can be replaced in the following way

fsin(6) dab d9 = fsin(0) 2[1(8) qu d0
{2 o 0
= 27r [sin(0)e(0) .

0

162

6(0) is the detection efficiency as a function of angle 0, defined as the fraction of
particles (all with scattering angle 0) that are actually detected. This efficiency can
be obtained from geometrical considerations of the angular coverage or from a Monte
Carlo simulation of the experimental setup.

The differential cross section is obtained from the number or observed reactions
N, in the following way: For a sufficiently small 0 integration range one can assume

6(0) (and also sin(0)) to be constant and obtains

 

 

 

do _ a
0R- _ 27r 6(0) (cos(0zow) — cos(0high))
_ 1 1
‘ " 27r 6(9) sin(0) A0
_ N, 1 1
— Nth 27r 6(0) sin(0) A0 ’
with
N, : Number of reactions observed
Nb : Total number of beam particles
N, : Total number of target particles (1H per cm?) (B.3)
6(0) : efficiency for particle detection at angle 0
010w, 0mg,“ 0 : lower, higher limit and centroid of 0 bin

In the proton scattering experiment N, is given by:

Nt = NIH ‘1}; dt , (13.4)

where N13 is the number of 1H atoms per molecule, (1, is the target thickness in units
of mass per area, N A is the Avogadro number, and M is the molar mass (mass of N A

molecules). For the polypropylene target (CH2) used in the (p, 19’) experiments

NIH-:2

NA = 6.022 - 1023 mol—1

 

M =14i
mol
NA 23 1 1
N — = 2. .22-10 —
“M 60 111011442—

mol

163

 

= 8.603 - 1022 g-1

Therefore
fl_ 1.850~106 .IYL 1 1
d9 — dt/(mgcm‘2) Nb 6(0) sin(0) A0

 

 

mb . (E.5)

Equation 13.5 gives the differential cross section in 1:1} given the target thickness in

mg cm”. 1 mb = 10‘27cm2 was used.

164

Appendix F

A New Method for Particle

Tracking

F. 1 Traditional Tracking

From the geometry of the setup (see figure F.1) it follows that the position on the
target 3:, (the target is located at z, = 0) is given by

xt = M, (F.1)

22-21

and the slope 9% is given by
d1} _ $2 — 5131

(F2)

 

d—Z — 22 — 21 ,
where 151(2) is the (measured) x-coordinate of the particle at z 2 21(2), which is the

z-coordinate of detector 1(2). The error of the target position 0(a) is

 

 

02(xt) = ( Z2 0($1))2+( Z1 00102))2 , (F.3)

22—21 22—21

where 0(x1(2)) is the uncertainty in the measurement of the position 331(2). The error

in the SIOpe of the particle trajectory (IQ?) is

a2 (2%) = ( 1 )2(02(231) +0202». (F.4)

21—22

 

165

 

 

x Tracking 1 Tracking 2 Target
ll 1
l .

xl . .__>§_ Particle Uajectogy
X2 1 21
x! i m dz

I
I
O beam axis 1

 

 

 

 

 

 

i

Z1 22 21:0 2

Figure F.1: Setup for particle tracking. Two tracking detectors determine the position
of the beam particle 31,2 at two points along the beam line, from which the position
on the target and the slope of the trajectory can be determined. The coordinates of
the tracking detectors are 21 (detector 1) and z2 (detector 2). The target is located
at zt == 0

F.2 New Method

In the following a new algorithm for the determination of the target intercept and the
slope of the trajectory is presented. The new method is superior to the traditional

method, especially if the resolution of the tracking detectors is poor.

F.2.1 Some Beam Physics

The slope of a particle trajectory is not independent of the (transverse) position of
the particle. This is illustrated in figure F.2. Shown is the phase-space—ellipse (PSE),
i.e. the slope of the beam trajectory is plotted as a function of transverse position.
In an ideal case the width (= 2 - W) is zero and the slope of the trajectory is a linear
function of the particle position. Hence we can infer from one position measurement
alone the trajectory of the particle. A prerequisite is, of course, a knowledge of the
orientation of the PSE (i.e. the slope S). The width of the PSE introduces a large
uncertainty, however.

Assuming that the main beam direction is along the z-axis of the coordinate

system, it follows that the slope of the trajectory if as a function of position a: is

166

 

 

 

 

X

 

Figure F.2: Phase-space—ellipse (PSE) of the beam can be approximated by a straight

line. The error that is introduced is W, the half-width of the PSE. The slope S is
defined as S = £1393.

167

obtained through

g:- = 3:1: . (F.5)
It is important to not confuse the slope of the trajectory j—: with the slope of the PSE
S. The latter changes as one goes along the beam-direction (S is negative before the
focus, infinity at the focus, and positive after the focus of the beam). It is the same
for all particles and a prOperty of the beam as a whole. In contrast fix; is different for
each particle. For the same particle, however, it is independent of the position along

the beam-line z.

The position on the target is given by

da: da:
$t=$1+(Zt—Zl)d—z=$1—21'&;. (F.6)
The uncertainty of the particle slope is
2 d5” 2 2 2
0' (IE) 2 S 0 (2:1) +W , (F.7)

where the first term is due to the resolution of the detector and the second term is
the half-width of the PSE (see figure F.2). The two contributions are independent
and are therefore added in quadrature. The uncertainty in the position on the target

is
02(1),) = 02(1‘1) + sz202(a:1)+ sz2

= (1 + 2? S2)2a2(a:1) + 2% W2.

If the beam is focused at the target position the term 1 + z? 52 = O and the position
resolution is equal to 2% W2, which yields essentially the beam spot size. The approx-
imation of the PSE as a straight line results in a constant calculated position at the
focus, and hence, the position resolution is equal to the half-width of the beam-spot.
If the resolution of the tracking detectors results in a resolution at the target (using

standard tracking), that is larger than the beam-spot size, then it is of course better

168

to assume a constant position on the target. But what happens if the beam-spot is
on the order of the resolution? Then one should combine the three different methods
(the usual one and the new method applied to both tracking detectors) to determine

the target point 12¢ and the slope 4‘5. The new method relies on the independence

dz

of all three procedures, which is only almost true. For instance the error due to the

width of the PSE is the same for both tracking detectors and should therefore be
added linearly.

Another advantage of the new method is that it can be applied, if the beam is

not focused at the target, but instead slightly before or behind. The assumption of a

constant target point would result in a much worse resolution.

F.2.2 Determination of Position and Slope

Assuming independence of the three measurements one finds the target position as

the weighted average:

 

1,: (2%)“12 ”39:0 , (F.8)

i 02W:
and the slope of the trajectory as
‘1 dz“)
_(277) 2—2) (F...)
with uncertainties given by
1 —1
02(xt) 2 (E; m) (F.10)

and 1

0264:): Z—(‘i-x—(T) . (Ru)

2' 0'2
The slope S and the half-width W of the PSE in terms of the beam emittance e, the

beam-spot size at the focus D, the position of the focal point f at any position 2

169

 

l

    
 
 

new method '—

 

 

 

 

 

0 1 l L l
O 2 4 6 8 10
60(2) (mm)

Figure F .3: Comparison of old and new method. Shown is the resolution of the target
position o(:z:t) as a function of the resolution of the second tracking detector 0(x2).
The positions of the tracking detector with respect to the target 21,2 are roughly 1 m
and 2 m. The beam-spot size D is 10 mm and the emittance e is 0.1 mm. One can
see that if the resolution of the second detector is larger than 2 mm the new method
is better, and even for a resolution of 1 mm the weighted average of both methods
results in an improvement.

along the beam-line are [59]:

 

(F.12)

W = ((2%): (gig—”)2)—% . (F.13)

Figure F .3 shows an example. The new resolution is considerably better as soon

and

as the resolution of the tracking detector is larger than 2 mm. In experiments where
tracking is crucial not only the target position but also the slope of the trajectory
is of importance and the effect of the different tracking methods can not easily be

estimated.

170

F.2.3 Practical Method

For practical purposes it is hard to determine the lepe of the PSE S for a given
detector position and beam profile. A better procedure is to consider it a param-
eter and optimize it until the best resolution is reached. This procedure has been
applied here, and figure F.4 shows the effect on the energy resolution for the re-
action 36Ar(p,p’)36Ar*. The resolutions of the tracking detectors are 8 mm and 3
mm (FWHM) and the new method results in a better resolution by almost a factor
of 2. The position resolution of one detector was so poor because of a strong rate

dependence. See section 5.1.2.

171

200

 

150 — _.

100—
T

 

50— -
m

H

s: 0 1 1 1

g 300

U

250

200

150

100

50

 

 

 

 

30 40 so 60 7o 30 9o
Excitation energy (arb. units)

Figure F .4: Measured excitation energy with angular cut in 66", = 25° — 30°. The
top panel shows the spectrum obtained if normal tracking is used, whereas the bot-
tom panel shows the same spectrum but the new tracking method is applied. The
resolution improves by almost a factor of 2.

172

 

Appendix G

Fitting of Spectra

In order to fit a spectrum one needs to assign proper errors a,- to each point 2:.-
(z' = 0,1 .. .). Since in each channel the number of points is “counted”, Poisson

statistics applies and an estimate is to assign

This is only an estimate, since the real error has to be taken from the parent distri-
bution, which is not known. The estimate from equation G] is especially bad if the
statistics is low. As a general rule, the number of counts per channel should be more
than 10 [22].

For lower statistics it is necessary to take the error from the fitted curve (i.e.
of = f(:1:,-), where f (2:) was fitted to the data points y,- as f (27,) = 3),), which should
follow the parent distribution much closer. However, there is another reason to use
the errors obtained from the fitted curve, which is explained in the following.

Consider a set of data points {23, y,} and a function f (11:) is fitted to these points.

The X2 of the fit is given through

x2 = Z [511. — may] . (G2)

173

 

Writing f (x) as b . f (51:) with parameter b yields

X2 = Z [313(11- — b - fun-M2] - (G-3)

Since x2 has a minimum for the best fit it follows that

—6—X—2 — _—2Z[‘1—0_2(yi bf($z))'f “330] = 0' (GA)

X2 can be written as

 

1% = Xjfam—bM( »—b“3%m—wfla»].

i i

The last term is proportional toQ2L 2'and is thus equal to zero. It follows that

6b
x — —Z;§ y‘( (gr—bf (an). (6.5)

Setting a? = y,- yields
X2 = 2 y,- — Z bf(:1:,-) = Area(DATA) — Area(FIT) ((3.6)

Thus the fit underestimates the area of the data by X2, which can be quite consider-
able.
However if one sets of = f (xi), i.e. the error is deduced from the fitted curve,

then it follows from equation G.4 that

f3__x

0: b:—2Z( y«—bf(a:,-)) , ((3.7)

and thus

23y.- — b f(x,-)) = Area(DATA) — Area(FIT) = 0 . (G.8)

Therefore the fitted function has the same area as the data.
In practice, one starts out with errors derived from the data (0,2 = y.) and when
the fit looks good one uses the errors derived from the fitted function (0-2 = f (11.3))

and fits again.

174

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