6 h.
.01.. 5‘; .3.
{1.7.2 . 2371....

1 V .25 .

‘vgn; I":"
\.x’v
In ..3‘

..
s
a)?“
an .i it
. i .v

...-.11
i. .

:9.
: .... ...

i

. ..1 ,3;

x . : 3.-.- ...);
a... ‘ z. .

323...,
11.1.1? .
..

13:51: I
c :5, 5| it.
.3 tilt»...

 

! ....
.32)..“ Xi)

v‘ {...v: ... r . .A
..29...~.arv|.»r3

\. u: .11».
29...... 1.3 .

v1.5... x . .
3.24.162 t. A

.15... ‘3 ...3. .1
1n 3: f

 

 

 

 

 

 

 

5.1..l1 . ...:

 

 

l’llfl

 

wcmem s

ll/I/Il/I/lllzl’l ii “777" 7%

I/ ll llfl"'I/lI//I/l//l

H l H! /
115939

I]: Ill

meats ’ 3 1 93 014

This is to certify that the

dissertation entitled

The Onset of Vaporization in 197Au + 197Au
Collisions

presented by

Wen-Chien Hsi

has been accepted towards fulfillment
of the requirements for

PH. D degree in Phys i cs

Major professor

Date z/Zfl I,

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

 

LlBfififiY
Michigan State
1 University

 

 

 

PLACE It RETURN BOX to man» chockmnftoin your record.

TO AVOID Flues Mum on or baton an. duo.

r————-_———._—_'=—_——n_-‘
DATE DUE DATE DUE DATE DUE

MSU In An Affirmative AdlaVEqual Oppommlty Initiation
Wan-39.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The Onset of Vaporization in 197Au + 197An Collisions
by

Wen-Chien Hsi

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

1995

ABSTRACT

197 197

The Onset of Vaporization in Au + Au Collisions
by
Wen Chien Hsi

Two bulk phase transitions exist in nuclear matter. At temperatures of the order of
a few MeV up to a critical temperature of about 17 MeV, infinite neutral nuclear matter
supports a mixed phase consisting of Fermi liquid droplets in coexistence with a nucleonic
gas. At significantly higher temperatures of the order of 150 MeV, calculations within the
standard model predict a deconfinement phase transition from hadronic matter to a quark-
gluon plasma.

One of the four major current thrusts in nuclear physics is the extraction of
information about these basic phase transitions from nuclear collisions. Temperatures and
densities relevant to the liquid gas phase transition can be momentarily attained in nuclear
collisions at incident energies of E/A = 35 - 400 MeV. To search for the liquid-gas phase
transition in a large system where signals of phase transitions are expected to be sharper,
multifragment disintegrations in 197Au + 197Au collisions were investigated in this
dissertation at incident energies of E/A = 100, 250 and 400 MeV. These measurements
clearly indicate that the yields of intermediate mass fragments (IMF’s; 3 3 Z s 30)
decrease significantly with the incident energy in central collisions, consistent with the
onset of nuclear vaporization.

These measurements were performed with an experimental array of considerable
complexity, capable of providing considerable information about the dynamics of the
collision. Measurements performed in this dissertation indicate the presence of a
collective radial flow for central collisions that contain of the order of a third to a half of

the total incident kinetic energy, thereby decreasing the energy available for thermal

excitation. Measurements of the kinetic energy spectra also provide information about the
mechanisms of energy deposition in peripheral collisions. Both fragment yields and energy
spectra were compared to molecular dynamics calculations developed to predict these
observations. These comparisons reveal significant shortcomings in these dynamical

models.

ACKNOWLEDGMENTS

I would like to express my thanks to my advisor, Professor William Lynch, for his
invaluable supervision over my research. His ability to cut to the core of a problem has
made my work fruitfill and enlightening. His advise at each time step of my research
period has been always critical and appropriate. Betty Tsang, Graham Peaslee, Thomas
Glasmacher and Korand Gelbke have given their friendship as well as invaluable
instructing during the data analysis of my thesis. Professor Betty Tsang has especially
contributed a lot to this thesis during the data analysis and revising my thesis, as well as
sitting on my guidance committee. S.D. Mahanti, Scatt Pratt, Wayne Repko and Michael
Thoennessen gave a careful reading of this work and also served on my guidance
committee.

When I began my education in experimental nuclear physics, Dave Bowman,
Graham Peaslee and Remualdo de Souza took the time to share their knowledge with me.
It was very pleasant to work with Dr. Graham Peaslee. I greatly appreciate his help and
suggestions on preparing the two experiments performed in Europe, and the fruitfiJl
discussions in the following data analysis. I also have greatly benefited from Dr Dave
Bowman and Dr Remualdo de Souza teaching me about data acquisition and experimental
skills. I also appreciated the fiiendship and the contributions from Thomas Glasmacher,
Carlos Montoya, Carsten Schwarz and Betty Tsang during the data analysis, I would like
to thank G. Peilert and WA. Frideman for their valuable simulations of the IMF
emissions.

My fellow graduate students have offered much support. I appreciate the help
from Mike Lisa during my thesis experiment at GSI, Germany. I also had many valuable
discussions with Larry Phair about techniques applied in data analysis. The other graduate
students in my research our group have helped me very much. Mei-Jui Haung, Yeongduk
Kim, Fan Zhu, Damian Handzy, James Dinius and Cornelius Willams have shared their
knowledge and fiiendship with me, making my work much easier than it would have been
without them. I would also like to acknowledge Esawar Ramakrishnan for the enjoyable

experience of studying nuclear physics with him during the summer school of 1993. Afshin

iv

Azhari, Shigeru Yokoyama, John Kelly, Eugener Gualtiteri and others have also helped to
make my experience enjoyable and instructive.

It has been a great pleasure working with the excellent NSCL staff. Jack Ottarson
was extremely helpful for the upgraded Miniball/Miniwall design. John Yurkon, Dennies
Swan and Jim Vincent have solved countless hardware related problems in upgrading the
Miniball/Miniwall project. I also appreciate our machine shop for doing all the
manufacturing required for the upgraded project. I am also thankful for other people not
mentioned here who have helped me.

I sincerely thank my family in Taiwan, R.O.C.; my parents, bothers and sisters for
their encouragement and support. I also thank my in-laws in Chicago for sharing
happiness and helping to build my self-confidence to live in a new environment. Finally, I
am indebted to my wife, Li Yung, for her love and patience during my graduate studies.

I appreciate my daughter Jane for bringing more joy and happiness to my family.

Table of Contents

List of Tables ................................................................................................................ viii
List of Figures ................................................................................................................ ix
Chapter 1 Introduction ..................................................................................................... 1
1.1 Overview .......................................................................................................... 1
1.2 Theoretical Models for Multifragmentation ....................................................... 2
1.3 Experimental Status Prior to the Dissertation Measurement ............................... 8
1.4 Outline of This Thesis ..................................................................................... 10
Chapter 2 Experimental Set-up ..................................................................................... 11
2.1 Original MSU Miniball Array .......................................................................... 13
2.2 Upgraded Miniball/Miniwall Array .................................................................. 19
2.3 Catania Si-CsI Hodoscope Array ..................................................................... 27
2.4 Aladin Spectrometer ........................................................................................ 27
2.5 Data Acquisition Electronics .......................................................................... 30
Chapter 3 Data Analysis ................................................................................................. 33
3.1 Particle Identification of l\/Iiniball/I\I1iniwall ...................................................... 33
3.2 Energy Calibration of Miniball ......................................................................... 41
3.3 Particle Identification of Catania Hodoscope ................................................... 50
3.4 Particle Identification of the Aladin ToF Wall .................................................. 53
Chapter 4 Experimental Results and Model Comparisons ............................................... 55
4.1 Reaction Filter: Impact Parameter Selection .................................................... 55
4.2 Onset of Nuclear vaporization at 197Au + 19“'Au Collisions ............................... 65
4.3 Detection Efficiency Corrections at E/A = 400 MeV ....................................... 73

vi

4.4 Reaction Dynamics ......................................................................................... 88

4.5 Collective Expansion of Central Collisions at E/A = 100 and 250 MeV .......... 105
Chapter 5 Summary ..................................................................................................... 122
List of References ....................................................................................................... 124

vii

List of Tables

Table 2.1 Coverage in solid angle, polar angles, azimuthal angles and the distance from
the front face of CsI crystal to the target ((1) for individual detectors of the upgraded
Miniball/Miniwall array. The first three rings of the original Miniball array are also

listed in the parenthesis .................................................................................................. 14

Table 2.2 The centroid of the polar angles, azimuthal angles of Catania detector are

listed .............................................................................................................................. 29

Table 3.1 The values of a, b7 and c7 in the set of functions (Eq. 3.13) for energy

calibrations .................................................................................................................... 48

Table 4.4.1 Parameters of three-source fits for the Lithium and Boron fragments at
E/A=100 MeV and five-source fits for the Lithium and Beryllium fragments at
E/A=400 MeV for different centralities listed in this table .............................................. 95

Table 4.4.2 Parameters of moving-source fits to the triple differential cross sections for
Lithium, Beryllium and Boron fragments at E/A= 400 and Lithium fragments at 100
MeV collisions ............................................................................................................. 100

Table 4.5.1 Parameters of three-moving-source fits for fi'agments with Z = 2 to 6 in

central collisions at E/A=100 MeV. ............................................................................. 107

Table 4.5.2 The parameter of fits for the central collisions of em, = 90° energy spectra
at E/A=250 MeV ......................................................................................................... 119

List of Figures

Figure 2.1 The schematic drawing of the experimental set-up. ........................................ 12
Figure 2.2 Half-plane section of the original Miniball array ............................................. 15
Figure 2.3 Schematic diagram of individual detector elements. ....................................... 16
Figure 2.4 Schematic diagram of the active voltage divider ........................................... 18
Figure 2.5 Timing and widths of the fast, slow and tail gates .......................................... 20
Figure 2.6 Half-plane section of the upgraded Miniball/Miniwall array. .......................... 22

Figure 2.7 The set-up for testing the counting capability of the PM tube with different

voltage dividers is shown. .............................................................................................. 25

Figure 2.8 The relate magnitudes of the y-ray peaks are plotted as a function of the

counting rate for the two different voltage dividers. ....................................................... 26

Figure 2.9 Three dimensional geometric diagram of the Catania hodoscope array .......... 28

Figure 2.10 Schematic electronics diagram of the data acquisition of the electronics for

the Miniball array alone ................................................................................................. 31

ix

Figure 2.11 Schematic electronics diagram of the data acquisition of the electronics for

the Miniwall array alone ................................................................................................. 32

Figure 3.1 Fast versus slow spectrum with high statistics for detector 3 ’-5 (ring 3’,

position 5) obtained for Kr + An collisions at E/A = 35 MeV. ........................................ 35
Figure 3.2a The linearized particle-identification (PID) spectrum of Fig 3.1 .................... 36

Figure 3.2b Linearized two-dimensional PID-slow spectrum obtained in the low

statistics Au + Au measurement ..................................................................................... 37

Figure 3.3 The tail versus slow spectrum for detector 3’-5 generated by combining all
the runs of Au + Au collisions at E/A= 100 MeV/A and 250 MeV obtained in this

experiment. .................................................................................................................... 39

Figure 3.4 Schematic tail versus slow plot describes variables used in the construction

of the PDT function. ...................................................................................................... 40
Figure 3.5 The plot shows the PDT versus slow spectrum of detector 3’-5 ..................... 42

Figure 3.6 Fractional deviations of the light output of Miniball detectors as defined in

equation 3.9, as a function of the normalized QDC channel ............................................ 45

Figure 3.7 Calibration curve for 4He and 12C fragments (solid points) including the

saturation corrections .................................................................................................... 47

Figure 3.8 Calibration data for 4He (solid circles), 6Li (open fancy crosses), 10B
(crosses), 12C (open circles), l4N (solid squares), 160 (open diamonds) and 20Ne (open
squares). The lines are fits to the data (solid lines for odd charge elements and dashed

line for even charge elements). ...................................................................................... 49

Figure 3.9 DE (signal of Si) versus E (signal of C31) for the Catania hodoscope is
plotted for the detector 30 at E/A = 100 MeV collisions. ............................................... 51

Figure 3.10 DE (signal of Si) versus E (signal of C51) for the Catania hodoscope is
plotted for the detector 30 at E/A = 400 MeV collisions. ............................................... 52

Figure 3.11 The flight time is plotted versus the light output of particles detected in the
front 100 scintillators of the TOP wall which are used for the particle identification. ....... 54

Fig. 4.1.] Normalized probability distributions for the charged particle multiplicity Nc (left
side), and the participant proton multiplicity Np (right side) for E/A=100, 250 and 400
MeV. ............................................................................................................................. 56

Fig. 4.1.2 Normalized probability distributions for the transverse kinetic energy E (left
side) and the sum of bound charges memd (right side) for E/A=100, 250 and 400 MeV .. 58

Fig. 4.1.3 Reduced impact parameters 5 extracted from the measured observables X =

N, Np, E. and Zbound by using Eq. 4.1.2-3. ...................................................................... 60

Fig. 4.1.4 Correlation between the reduced impact parameters reconstructed from the
charged particle multiplicity Nc , transverse kinetic energy E, participant proton
multiplicity Np, and the sum charge of spectator nucleon Zbomd for E/A=1OO and 400 MeV.
...................................................................................................................................... 63

Fig 4.1.5 Mean charge particle multiplicities <Nc> obtained from the QMD model and

filtered through the experimental acceptance at E/A = 100 and 400 MeV are shown in the

top panel. The mean deduced impact parameter <bded> are plotted in bottom panel ........ 64

xi

Fig. 4.2.1 Correlation between <NM> , the observed mean fragment multiplicity, and

Nc , the observed multiplicity of charged particles detected in the Miniball/wall. ............. 66

Fig. 4.2.2 The measured impact parameter dependence of the mean fragment
multiplicity is shown by the solid points. The open circles and the open squares depict
the unfiltered predictions of the QPD and QMD models, respectively. The dash-dotted
and dashed lines depict the QPD and QMD calculations, filtered through the

experimental acceptance. ............................................................................................... 67

Fig. 4.2.3 The mean total charge of all detected particles as a function of the reduced
impact parameter for E/A=100 and 400 MeV ................................................................. 69

Fig. 4.2.4 Comparisons with hybrid model calculations at E/A=100 and 400 MeV. ........ 72

Figure 4.3.1 .The y-Pt scattered distributions for Li ions at near mid-central collisions

are plotted as a function of normalized rapidity y/y.mm .................................................... 74

Figure 4.3.2 The Gaussian sources obtained from the Box-Muller method with
different variances in x and y directions are plotted as open circles in the two bottom

panels ............................................................................................................................ 77

Figure 4.3.3 The distribution of Li fragments as a firnction of y and pt are plotted for

events with impact parameters of .67<b 5.7 8. The experimental cross sections are plotted
in left panel. The simulation with the detection acceptance correction is plotted in middle

panel. The simulation without the experimental acceptance cut is shown in the right panel.

...................................................................................................................................... I E

Figure 4.3.4 The transverse momentum cross sections of Li fragments are plotted as

solid circles for the data at the impact parameter of .67 <13 S .78. The dashed lines are

xii

the simulations filtered through the acceptance cut. The solid lines are the results

without the acceptance cut. ............................................................................................ 81

Figure 4.3.5 The extracted probability distribution was plotted as a function of

normalized rapidity for Li fragments at impact parameters .67 <13 S .78. ........................ 82

Figure 4.3.6 The transverse momentum distribution for Li fragments are plotted as the

solid circles for the experimental data at impact parameters Es .34 ................................ 84

Figure 4.3.7 The measured mean MS as a function of the reduced impact parameter
with the various device acceptances at E/A = 400 MeV. The vertical lines indicate the
impact parameter ranges for the y-P. simulations. The simulations filtered through the

devices are plotted in bottom panel. ............................................................................... 85

Figure 4.3.8 The mean number of Ms was plotted as a firnction of the reduced
impact parameters. The solid circles are the experimental measurements. The bars are

the range of values determined by the efficient calculations. ........................................... 87

Figure 4.4.1 Mean transverse energies plotted as a function of the reduced impact

parameters for or, Lithium, and Carbon fragments at 25°: 0 7.7, $160° .............................. 89

Figure 4.4.2 Mean transverse energies are plotted as a function of the fragment charge

for 1350.34 (top panels) and .45£b<.75 (bottom panels) at E/A =100, 250 and 400
MeV incident energies. The QMD and QMD + SMM calculations, filtered through the

experimental acceptance, are plotted as solid and dashed lines, respectively .................... 91

Figure 4.4.3 The energy spectra of Lithium and Boron emitted in the Au + Au
collisions at E/A=100 MeV were plotted for 91.7, = 28°, 355°, 45°, 575°, 725°, 90°

xiii

and 110° for the peripheral (top panel), mid-central (middle panel) and central (bottom

panel) collisions. ............................................................................................................ 94

Figure 4.4.4 The energy spectra of Lithium and Beryllium fragments emitted in Au +
Au collisions at E/A = 400 MeV. ................................................................................... 97

Figure 4.4.5 The triple differential cross sections of Lithium fragments emitted with

the impact parameters .42<5 s .74 for Au + Au collisions at E/A = 400 MeV ................ 99

Figure 4.4.5b The transverse velocities of spectator source is obtained by fitting the

triple differential cross sections of fragments emitted with the impact parameters

.42<I; s .74 for Au + Au collisions at E/A = 400 MeV. The temperature of the
spectators is plotted at bottom panel with the temperature of spectator extracted for

the heavier fragments ( Z=7—20) from the Au + Pb collisions at E/A = 600 MeV .......... 101

Figure 4.4.6 The triple differential cross sections of Lithium emitted at central collision
of Au on Au collisions at E/A = 100 MeV .................................................................... 102

Figure 4.4.7 The temperature TSpc and the longitudinal velocity [32 of the fit for the
triple cross sections of Lithium and Boron fragments at the central E/A = 100 MeV

collisions, shown as a firnction of the transverse velocity Bx. ........................................ 104

Figure 4.5.1 Comparisons of the energy spectra for Boron fragments emitted to 91,7, =
280°, 355°, 450°, 575°, 725°, 90° and 110° (solid and open points), with

corresponding moving source fits. ................................................................................ 106

Figure 4.5.2 Comparisons of the energy spectra for Z= 2 to 6 fragments ...................... 108

xiv

Figure 4.5.3 The relation between the chi square x3, fiom fitting the energy spectra

and the temperature of the participant source is plotted at the bottom panel for
Lithium and Boron fi'agments. The corresponding relationship between Tmid and the

radial expansion velocity of the participant source is shown in the top panel. ................ 111

Figure 4.5.4 Best fit values for the radial velocity are plotted at the bottom panel as a
fimction of the assumed breakup density of the participant source ................................ 112

Figure 4.5.5 Lefi panel: The open points correspond to the best fit values for the radial
expansion velocities as a function of the fragment charge. The solid points are the
corresponding values obtained when Tmid is constrained to be 15 MeV. Right panel:
The solid points depict the dependence on the fragment mass of the mean radial

collective energy <Er> extracted fiom the fits upon the fragment mass ......................... 115

Figure 4.5.6 Energy spectra in the center of mass fiame for various fragments

detected at 80°s 6mg 10° in the central collisions at E/A = 100 MeV ......................... 116

Figure 4.5.7 Energy spectra in the center of mass frame for various fragments

detected at 80°s 6cm3110° in the central collisions at E/A = 250 MeV ......................... 117

Figure 4.5.8 Measured mean energies are plotted as a fiJnction of the fragment mass at

80°S 6ms110° of the incident energies E/A=100 and 250 MeV ................................... 118

Figure 4.5.9 Mean radial energies per nucleon for Lithium and Boron fi'agments are

plotted as a function of incident energy. ....................................................................... 121

Chapter 1

Introduction

1.1 Overview

The nucleon-nucleon interaction is strongly repulsive at distances less than 0.4 fm,
attractive at distances of about 1 fin, and vanishes exponentially at larger distances. While
the distance scales of the nucleon-nucleon interaction are considerably smaller than those
of molecular interactions, these characteristics, short-range repulsion and mid-range
attraction, are also found in the molecular Lennard-J ones potential [Lenn 31, Slat 75].
Thus it comes as no surprise that infinite nuclear matter exhibits a liquid-gas phase
transition.

Infinite nuclear matter is a theoretical construct that finds its closest physical
realization within a neutron star. Nearly all experimental constraints upon the properties
of nuclear matter, however, have been derived from experimental measurements of
binding energies [Waps 77, Myer 82], radii [Vrie 87] and compressibilities [Siem 79], etc.
of finite nuclei. Temperatures and densities relevant to the liquid-gas phase transition can
be attained momentarily in central nucleus-nucleus collisions at energies of E/A = 35-400
MeV. The investigation of such collisions offers the best opportunities for experimentally
determining the properties of this bulk phase transition.

Within the region of liquid-gas coexistence, thermodynamical properties of nuclear
matter are strongly reflected in the relative abundance of fragments [Lync 87]; therefore,
the study of fragment observables in multifragment disintegrations is essential to
investigations of the liquid-gas phase transition. Initial investigations of fragment
observables consisted of inclusive measurements [Finn 82, Hirs 84, Buaj 85, Cser 86,
Lync 87, Troc 89] and were unable to address many of the important issues. Since the
latter part of the last decade, a number of new multifragment detection arrays that are
capable of exclusive measurements have come into operation [Bade 82, West 85, deSo 90,

Gobb 93], and the information about multifragmentation has grown enormously [Boug 87 ,

Doss 87, Boug 89, Blum 91, deSo 91, Kim 89, Bowm 91, Phai 93, Sang 92, Alar 92, Ogil
91, Hube 92].

This dissertation work began at a stage of great discovery in this field.
Measurements [Boug 87, Doss 87] immediately prior to those discussed here had shown
for the first time that multifragmentation was feasible for central collisions of heavy ions at
intermediate energies [deSo 91, Kim 89, Bowm 91, Phai 93] and for peripheral collisions
at considerably higher energies [Ogil 91, Hube 92]. It became a high priority to determine
the range of experimental conditions for which this new phenomenon was manifested.
Excitation functions for the Kr + Au [Peas 94] and Au + Au [Tsan 93] systems were
initiated spanning a wide range of incident energies and requiring experiments at three
difi‘erent facilities for their completion. This dissertation work consists of studies of Au +
Au collisions performed at E/A = 100, 250, and 400 MeV using heavy ion (Schwerionen)
Synchrotron (SIS) facility of the Gesellschaft fiir Schwerionenforschung (GSI) at

Darmstadt Germany.

1.2 Theoretical Models for Multifragmentation

As many of the experimental observations place significant constraints upon
theoretical interpretations of multifragmentation, it is relevant to provide some brief
descriptions of current theoretical models [Frie 90, Peil 92, Baue 85, Dani 88, Stoc 86] of
this phenomenon. These models can be categorized in a variety of ways. Here, we
choose two broad categories: 1) dynamical models that follow the non-equilibrium
transport of energy, mass and momentum throughout the nuclear collision, and 2)
statistical models which calculate the probabilities of various experimental observations

using some form of statistical weight.

Dynamical models
While considerable effort is being directed towards the development of quantum
transport theory, the description of fluctuations leading to fragment production is an

unsolved problem. Here we only describe some of the presently used approximate

techniques. For the description of phenomena less sensitive to fluctuations at incident
energies of E/A 2 20 MeV, considerable success has been achieved in descriptions based
upon the Boltzmann Uehling Uhlenbeck (BUU) equation [Nord 28, Uehl 33] which
describes the time evolution of the Wigner transform of single particle density matrix

f ('f, p. t) as follows:
if. r» if

at
at “1.6? 55—1 (1.1)

 

d3r'd3r d3" A _ .. . . _‘ g _. _.
:l pugs p1'GIV'Vrllflf(1-f1)(1—f)“f1f(1—f1)(1—f)]5(P+P1‘P ‘91)

Here, U is the nuclear mean field potential which may depend upon both the position 'r‘

and momentum p of the particle. For momentum independent potentials, local Skyrme

interactions with two- and three-body components are commonly used. The right-side

collision integral of a binary collision depends upon the relative velocityv‘ — v1 between the

two nucleons, and includes a cross section 0' for nucleon-nucleon scattering via the
residual interaction (which may differ from the corresponding free space value), as well as
the occupancy and Pauli blocking factors, f and (l-f), respectively.

To solve the VUU/BUU equation, nucleons are represented as a sum of point-like

test particles
RA

f('fi,131,t)= 25% ‘90)5051-910), (1-2)
i=1

where A is the number of nucleons and I? is the number of test particles per nucleon. Each
test particle is propagated throughout the nucleus-nucleus collision using Hamilton’s
equations of motion [Mari 70]. Solutions of Eq. 1.1 provide a description of the time
dependence of the single particle phase space density [Huan 63]. While fluctuations may
be produced by collisions described by the R.H.S. of Eq. 1.1 in some simulations and may
result in the production of some fragments, such fluctuations only reflect numerical
instabilities which should be negligible for accurate solutions of Eq. 1.1. Efforts at
extending Eq. 1.1 to higher order in the BBGKY hierarchy [Bogo 62] by including an
additional fluctuating collision integral I on the R.H.S. of Eq. 1.1 [Dane 91, Batk 92] are

still in their infancy. Thus Eq. 1.1 is mainly used to establish initial conditions such as the
excitation energy, density or collective expansion velocity. It cannot describe fiagment
production.

Because the time evolution of the density matrix approaches that of a classical A-
body system at high temperature, molecular dynamics has been used to describe A-body
fluctuations and correlations during nuclear collisions [Peil 92, Aich 88]. Some difficulties
occur due to the explicit lack of quantum mechanical effects, such as the Pauli exclusion
principle in such models [Boal 88]. In an attempt to address such limitations, some models
represent each nucleon by a Gaussian “wave packet”. In the Quantum Molecular

Dynamics (QMD) model of ref. [Peil 89 & 92], this takes the form

.. .. 2 2 2 - - 2 2
w, it) 1. [in 3e-lp-Poi(t)l L #7 e—[r-rman IL (13)
11:

where the 130, and to, are the centroids of wave packet of the ith particles and L is a width

parameter of the wave packet. A Pauli potential, which acts as a repulsive potential in

phase space, is used to simulate the Pauli exclusion principle [Dors 88].

2 ,3. p2.
1.] 1]

ex - - 5 . .5 . . 1.4

] p qu 213(2) 1,1] 01ch ( )

 

 

1 0
U = —v
Pan 2 Pau[qopo

Here the 07 and 1:. are the third components of the spin and isospin of nucleon i. The
inclusion of a Pauli potential in such molecular dynamics approaches allows for well-
defined fermionic ground states [Peil 92] and thereby permits accurate estimation of the
excitation energies of the emitted fragments. The nuclear system is then described by a

Hamiltonian of the form:
1,?
Hi2? :Z‘an'l'upau +U0 (15)
i

where U0 is a potential which may include two- and three-body Skyrme potentials as well
as Coulomb, Yukawa and symmetry potential related terms. The time evolution of the
system is obtained by solution of Hamilton’s equations. A similar approach is also taken

within the Quasi-Particle Dynamics (QPD) model of ref. [Boal 88 a&b].

Promising approaches, such as F emion Molecular Dynamics [Feld 90] and
Antisymmetrized Molecular Dynamics [Hori 91, Bauh 85] are now being developed which
like time-dependent Hartree-Fock, and satisfy the Pauli principle by considering the time
evolution of Slater determinant wave functions. Unfortunately, calculations of systems
containing of the order of 400 nucleons are not currently feasible in these approaches and

therefore will not be considered further in this dissertation.

Statistical Models

In a nuclear collisions after the initial pre-equilibrium cascade, one is frequently left
with large highly excited fragments and reaction residues. In a complete dynamics theory,
the decay of these residues should be adequately described, but might require inordinate
amounts of computer time. Presently, none of the available dynamics models can
reproduce the decay rates predicted by statistical models. In the case of the BUU
equation, this is understandable as the fluctuations relevant to cluster production are not
included in the theory. In the case of the QMD and QPD molecular dynamics models, the
inclusion of the Pauli potential does not alter the fact that specific heats of the systems
described by these calculations are essentially classical and therefore exceed the specific
heat of an equivalent quantum mechanical system composed of Fermions [Bere 92]. Thus,
for fixed total excitation energy, the kinetic energy per degree of freedom is less for these
molecular dynamics models than for real nuclear systems and consequently, fragments do
not so frequently surmount the Coulomb barrier.

There is, therefore, a need for a better description of the longer time scale decays
of excited fragments and heavy residues. A variety of statistical fragmentation models have
been developed. One may classify such models according to whether the final states result
from the solution of a statistical rate equation, as is the case of the Hauser-Feshbach [Mekj
77, Gran 86, Boal 83] or fission [Morr 78, More 75] theories of the compound nucleus, or
by assuming static equilibrium of the fragmenting system. Models which assume static
equilibrium can be further distinguished as to whether the statistical weights are calculated
according to phase space assumptions, as in the case of multiparticle phase space models

[Gros 82, Bond 85, Fai 82] and liquid-gas equilibrium models [Frie 83, Cser 86, Lync 87],

or from purely geometric considerations, as in the case of site or bond percolation theory
[Baur 85]. In this dissertation, two of these statistical models were used [Tsan 93]; the
Expanding Evaporating Source (EES) model of Ref. [Frie 90] and the Statistical
Multifragmentation Model (SMM) of Ref. [Bond 85 a&b] and [Botv 87]; these will be
described in the following.

The SMM model [Botv 87] has the advantage of being an efficient Monte Carlo
event generator that produces multifragment final states with statistical weights that are
similar to the weights incorporated in the Copenhagen [Bond 85 a&b] or Berlin [Gros 90]
multiparticle phase space models. All three approaches have the central goal of producing
a set of possible multifragment final states that conserve the total energy, momentum,
particle number, and charge. The relative probability of each specific decay configuration
is assumed to be proportional to its corresponding phase space volume. When the
predictions of these models are calculated solely as a function of the excitation energy,
there are some differences in the assumed freeze out configurations. In the original
approach of the Copenhagen model, for example, the density of the breakup configuration
is allowed to vary with excitation energy [Bond 85 a&b], while in the SMM approach, the
density is held fixed at po/6 [Botv 90, Bond 94], similar to the philosophy of the Berlin
multifragmentation model [Gros 90].

In addition to the phase space corresponding to the transitional motion of the
various particles, the internal phase space of the excited nuclear fragments is calculated via
empirical level density formulae [Myer 82]. The multiparticle phase space is sampled in the
SMM model by an algorithm described in ref. [Rand 81, Bond 82] and a breakup
configuration, consisting of free nucleons plus stable and excited fragments, is Monte
Carlo chosen for each event. These excited fragments are then allowed to decay until all
fiagments are particle stable.

The underlying assumptions of such multiparticle phase space models are: 1) that
local equilibrium is maintained during the expansion of the system until a very low freeze-
out density is reached, and 2) that the breakup is sufficiently rapid that the system can not
globally re-equilibrate between decays. The first assumption is problematic and its validity

has yet to be adequately tested; the neglect of nuclear interactions between the various

particles at freeze-out in these models has the consequence that none of them can be used
to explore critical phenomena. The validity of the second timescale assumption about
multifragment decays can be clearly tested via measurements of fragment-fragment
correlation functions [Kim 92]. It is clear that there are cases where this assumption is
invalid, e. g., at low excitation energies where compound nuclei have been extensively
studied [Fox 93]. There, hot nuclei decay sequentially, emitting one particle after another,
a process that has been successfirlly described by statistical rate equation approaches like
the Weisskopf or Hauser-Feshbach theories [Weis 37].

The Weisskopf and Hauser—Feshbach compound nuclear rate equation techniques
have been generalized to describe the emission of complex fragments from hot nuclei [Frie
83]. The basic rates incorporated in such models may be trivially calculated by assuming a
detailed balance between the emission of a particular particle species and its hypothetical
re-absorption which would occur if the hot nucleus was in thermal equilibrium with hot
gas of the emitted species in which it was embedded [Frie 89]. Hot and highly charged
nuclei are not hydrodynamically stable at temperatures in excess of 5 MeV, however.
Therefore, in the EES model of ref. Frie 90, simultaneous expansion and fiagment
emission are considered.

These statistical models have been tested in a variety of contexts. Both
multiparticle phase space models, such as the SMM and the evaporative EES model,
predict a rapid rise in the fragment multiplicities when the systems expand to sub-nuclear
densities [Frei 90]. In the case of the SMM model, calculations display the rapid onset of
multifragmentation as the temperature of the system rises above 5-6 MeV; this rise is
attributed to the onset of a “cracking” phase transition [Boal 85b, Bovt 90]. A rapid rise
in fragment multiplicities and fragmentation at temperatures of about 5 MeV is also
predicted by the EES model when the system expands to a density less that 0.4po [Frie
90]. The occurrence of this common feature in both models is a reflection of the
underlying thermodynamic instability of a homogeneous thermally excited system at a sub-
nuclear density which gives rise to the liquid-gas phase transition [Siem 83]. There are

detailed differences between the predictions of these models that are most pronounced

when the predicted energies, momenta and correlations of emitted fragments for the

various models are compared [Bowm 93].

1.3 Experimental Status Prior to the Dissertation Measurement

Multifragment emission has now been explored in a variety of nuclear systems
[deSo 93, Phai 93, Tsan 93, Peal 94]. Large fragment multiplicities are observed in
central collisions of complex nuclei which exceed the predictions of statistical compound
nuclear decay at normal density. The multiplicities, however, may be described by SMM
and EES models which assume the disintegration occurs at subnormal density. The
fragment multiplicities also exceed those predicted by the molecular dynamics models.

Estimates of the time scales for fragmentation in 36Ar + 197Au collisions [Fox 93],
Kr + Au collisions [Baug 93], 197Au + 197Au collisions [Kamp 93], and 12S'Xe + l9fl’Au
collisions [Bowm 93], have been obtained by analyzing the Coulomb final state
interactions between fragments. These extracted timescale decrease with incident energy.
Time scales less than 100 fm/c = 3.3 x 10'22 seconds are typically observed in such
experiments, consistent with a bulk disintegration and much less than the time required for
the system to decay stepwise, equilibrating between each successive step. This and the
observation of large fragment multiplicities are necessary conditions for the existence of a
liquid-gas phase transition, but considerably more detailed information is needed. It is
especially important to explore how these nuclear systems evolve from liquid to gas with
excitation energy. Variations in excitation energy can be achieved either by varying the
incident energy for central collisions or by varying the impact parameter for peripheral
collisions at high incident energies.

Prior to this dissertation work, there existed only one excitation fimction for
multifragmentation in the literature [deSo 91]. This previous study was performed for the
36Ar + 197Au system at E/A = 35-110 MeV [deSo 91]. It revealed that the fragment
multiplicities in central collisions increase with incident energy over the range of energies
measured. At significantly higher incident energies E/A = 600 MeV, other measurements
were performed [Ogil 91, Hube 91] with a 197Au beam which suggested that the fragment

multiplicities are smaller in central than in peripheral collisions. These latter high incident

energy measurements were performed with a device with a relatively small efficiency for
the detection of particles emitted in central collisions [Hube 91].

As indicated previously, comparisons of fragment multiplicities to QMD and QPD
molecular dynamics calculations [deSo 91, Bowm 91] indicated that such calculations
generally underpredicte the observed fragment multiplicities. Molecular dynamics predict
that these fragment multiplicities should increase strongly with incident energy and reach a
maximum at incident energies of the order of 100 A MeV. At energies of E/A = 200 MeV,
a successful description of the fragment observables was in fact achieved with the QMD
model for central Au + Au collisions [Peil 89]. Beyond E/AleO MeV, however,
decreasing fragment multiplicities are expected on the general grounds that the fragments
which are produced will likely be so highly excited that they will disintegrate into nucleons
[Ogil 91]. This general consideration is also reflected in various calculations [Frie 90, Peil
92]; specific predictions for this decrease are provided by molecular dynamics models [Peil
92]. There is considerable interest in determining the incident energy dependence of the
onset of nuclear vaporization and whether this energy dependence can be reproduced by
molecular dynamics models.

Complimentary information may be obtained by examining the impact parameter
dependence of multifragmentation at relatively high incident energies [Hube 92, Ogil 91,
Kean 94, Wang 95]. Here, the onset of vaporization occurs as a function of impact
parameter reflecting the impact parameter dependent excitation energy deposition into the
reaction residues [Tsan 93, Ogil 91]. Both the sizes and the excitation energies of the
residues vary with impact parameter complicating the interpretation of such data,
however.

From previous studies [Phai 92], a quantitative understanding of the impact
parameter and incident energy dependence of multifragmentation is now achieved. Despite
these promising indications, information about the time evolution of the system in the
(p,T) plane for bulk disintegration is necessary to proceed with the accurate extraction of
thermodynamic quantities from such collisions, and to discern non-equilibrium and
dynamical effects. With respect to the latter issue, it is particularly important to determine

the relative importance of collective expansion to thermal excitation for the

10

multifiagmentation process. This issue was also carefully investigated in this dissertation
work. Estimates of the energy contained in collective motion were obtained [J eon 94],
and the role of this collective expansion upon the fragmentation process was explored [Hsi

94, Kund 95].

1.4 Outline of This Thesis

This thesis is organized as follows: Details of the experimentally designed
procedures are described in Chapter 2. The experimental analysis is described in Chapter
3. The resulting experimental data, including the energy dependence of the fragment
emission and the extraction of the energy of collective expansion, are described in Chapter

4. A summary is given in Chapter 5.

Chapter 2
Experimental Set-Up

The experiment was performed at the heavy ion Synchrotron (SIS) facility of
Gesellschaft fiir Schwerionenforschung (GSI) at Darmstadt. Au projectiles with incident
energies of E/A = 100, 250 and 400 MeV bombarded Au targets of various thicknesses.
The experimental set-up included the Miniball/Miniwall phoswich array, Catania Si-CsI
hodoscope array and Aladin spectrometer as shown in Fig. 2.1. Together, these detector
arrays provided a coverage of more than 95% of 41: for detecting fragments with Z 2 2.
At polar angles of 145° S 01,7, S 160°, charged particles were detected in 215 plastic-
scintillator-CsI(Tl) phoswich detectors of the IVIiniball/Miniwall arrays. Fragments (2 S Z
S 79) moving with velocities around the beam rapidity were detected by the Aladin
spectrometer [Aldi 89]. This spectrometer covered lam] S 10° in the horizontal (bend)
plane and |013b| S 5° in the vertical plane. Due to the dynamic range of the magnetic field
used in this experiment, hydrogen particles were partially deflected out of the Time of
Flight (ToF) wall acceptance. Since the detection efficiency of Z = 1 particle is not well
defined in this spectrometer, Z = 1 particles emitted forward with 07.7, S 10° were not
included in the data analysis. Charged particles emitted to angles between the Aladin
spectrometer and the Miniball/Miniwall array were detected by the 84 elements Si-CsI(Tl)
hodoscope array.

The main focus of this thesis is on the data obtained with the Miniball/Miniwall
array which covered more than one half of the total solid angles in the center of mass
frame. Therefore, only a brief description of the techniques relevant to the Aladin
spectrometer and the Catania hodoscope will be described in this thesis. Further details of
the Aladin spectrometer can be found in Refs. [Aldi 89] and [Kund 94]. The original
Miniball array is described first in this chapter. Then, the upgrades to Miniball/Miniwall
array and the Catania hodoscope and Aladin spectrometer are described. Finally, details of

the data acquisition electronics of the Miniball/Miniwall are given in the final section.

11

12

ALAD i N

Magnet
TP-MUSIC Ill

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1 The schematic drawing of the experimental set-up. This experimental
apparatus includes the IVIiniball/Miniwall array, the Catania Si-CsI hodoscope array and
the Aladin spectrometer. The TP-MUSIC chamber and TOF wall are two main

components of the Aladin spectrometer.

13

2.1 Original MSU Miniball Array

The original configuration of the Miniball is listed in Table 2.1 and a half-plane
section of upgraded Miniball is plotted in Figure 2.2. This array consists of 11 independent
rings coaxial about the beam axis. For ease of assembly and servicing, the individual rings
are mounted on separate base plates which slide on two precision rails. For a given ring,
the detectors are identical in shape and have the same polar angle with respect to the beam
axis. Each phoswich detector of this array is composed of a thin 4 mg/cm2 (or 40 um)
plastic scintillator foil, prepared from Bicron BC-498X scintillator solution, and a 2 cm
CsI(T1) scintillator crystal.

A schematic plot of the detector design is given in Figure 2.3. The front and back
faces of the CsI(Tl) crystal were polished and the crystal was glued with optical cement
(Bicron BC 600) to a flat light guide made of ultra-violet transparent (UVT) Plexiglass.
The light guide was glued to a second cylindrical piece of UVT Plexiglas which, in turn,
was glued to the front window of the photo-multiplier (PM) tube. The PM tube and the
cylindrical light-guide were enclosed inside by a cylindrical u-metal shield (not shown in
Figure 2.3). The front face of the detector was covered by an aluminized mylar foil (0.15
mg/cm2 mylar and 0.12 mg/cm2 aluminum) to keep the detection assembly light-tight and
also to suppress low energy secondary electrons form hitting the plastic scintillator. The
fast scintillator of foil was used for charge particle identification; the thickness of foils (40
um) were minimized to reduce particle detection thresholds. These low detection
thresholds are particularly important for detecting intermediate mass fragments.

Since the angular distribution of emitted particles in heavy-ion collisions is strongly
forward peaked [Kim 92], the solid angles of the forward detectors needed to be smaller
than those of the backward detectors. Variations in solid angle were achieved largely by
placing detectors at different distances from the target while keeping their sizes
approximately constant. The crystals of Miniball detectors were tapered so that the front
and back surfaces have the same solid angle with respect to the center of the target. The
curved surfaces corresponding to the constant polar angle were approximated by planar
surfaces. The absorption of light from the plastic scintillator by the CsI(Tl) crystal

constrained the maximum useful thickness of the crystal. Under this constraint, a 2 cm

14

Table 2.1 Coverage in solid angle, polar angles, azimuthal angles and the distance from the
front face of CsI crystal to the target (d) for individual detectors of the upgraded
Miniball/Miniwall array. The Miniwall rings are labeled 1-2. The 3’—11 are the rings of
upgraded Miniball. The first three rings of the original Miniball array are also listed in the

parenthesis.

 

 

Ring Detector A!) 0 (degree) A0 Ad) (1
(msr) (degree) (degree) (mm)
1 24 5.30 16.6 4.14 15.0 305.2
(12) (12.3) (12.5) (7.0) (30.0) (260)
2 24 10.36 21.9 6.15 15.0 312.5
(16) (14.7) (19.5) (7.0) (22.5) (220)
3’ 28 11.02 28.0 6.0 12.86 280
(24) (18.5) (27.0) (8.0) (18.0) (180)
4 24 22.9 35.5 9.0 15.0 160
5 24 30.8 45.0 10. 15.0 140
6 20 64.8 57.5 15. 18.0 90
7 20 74.0 72.5 15, 18.0 90
8 18(-l) 113.3 90.0 20. 20.0 70
9 14 135.1 110.0 20. 25.7 70
10 12 128.3 130.0 20. 30.0 70
11 8 125.7 150.0 20. 45.0 70

 

 

15

 

 

 

 

 

 

 

 

usu-so-m
7‘2“” Bus)
5 (24) 3120) 9 (14)
4‘2" 10am
3 (2m
2 (16’ n (8)
1(12)
27'
12.9 \\ 1'
a.
i v r
20 cm

 

 

 

Figure 2.2 Half-plane section of the original Miniball array. Number of detectors per ring
are given in parentheses. The polar angles for the centers of detectors are indicated. The

dashed horizontal line indicates the beam axis.

Fast scintillator L131“ gurde

\

 

PMT

 

 

 

11

Al-mylar foil

 

 

 

 

 

Optical cement

Figure 2.3 Schematic diagram of individual detector elements.

l7

thicknesses of the crystals were chosen to obtain a dynamic range suitable for intermediate
energy heavy-ion collisions and also to limit the effect of light absorption.

The lO-stage PM tube (Burle industries model C83 0622B) was chosen for its good
timing characteristics (tR~2.3 ns), its large nominal gain (~107), and its good linearity for
large signals (Imz 30 mA). Since the detection array was operated in vacuum, an active
divider was designed to minimize the heat generation. A schematic diagram of the active
divider is given in Figure 2.4. The values of the resistors in the original divider are listed '
in parenthesis. The voltage steps are shaped in order to maintain large voltage drops on
the final amplification stages, because that minimizes the influence of space charge
dependent gain shifts that occur for large peak currents of order Ipcak z 30 mA. The choice
of an active base allowed for a higher degree of gain stability under count rate
fluctuations. The current supporting the dynode voltages runs through the resistors for the
first seven voltage stages and then principally through the transistors on the last four
voltage stages when the tube is quiescent. This allows for large currents to resupply the
dynodes without the large heat dissipation one would experience with a passive base. The
resistors on the last four stages provide reference voltages about which the voltages on
D7-D10 are stabilized, The large capacitors at the final stages provided additional stability
against sagging under large pulses in the phototube.

The CsI(Tl) crystals used in the Miniball array were selected to have good
uniformity of scintillation response. In a test of large cylindrical crystals [Gong 88, Gong
90], possible non-uniformities of the scintillator due to non-uniformities in the thallium
doping of the CsI(Tl) crystal were measured with a collimated 661 KeV y-ray source.
Since the range of electrons [Knol 7 9] excited by the photoelectric absorption of these y-
rays is comparable to the sizes of these small volume Miniball crystals, the y-ray method is
not well suited to testing Miniball detector uniforrnities. Instead, the non-uniformity of the
scintillation efficiency was measured by scanning the crystals in vacuum with a collimated
8.75 MeV 228Th a—source. This procedure takes advantage of the fact that or particles are
more sensitive to Thallium doping and that the short range of the alpha particles allows for
tests of local variations in the doping concentration [Birk 64, Mana 62]. To avoid edge

effects, regions within about 2 mm of the sides of the crystals were not scanned. For

18

Anode 50 Q Coaxial cable

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

C {l >
' .1.
1K9; 'L “II—7 $5.1(10)Mo '-
D10.—__L_\J——‘V\A/1“F:II: 7:;
“IF II— 4701 F'— I'Jl_l $5.1(10)MQ
p.
D 9 .—'—\/W\/ A"!
_H_ 47 Q 4 I3 '— 4.3(8.2) MQ
lnF JVW luF J I “l
D8 I : VN2406 ' I_ 7
lnF—H 47“ MOSFET é 2.0(3.9)Mr2
D 7 o—x/wx, :,
”FL"— 4752 g 430(820)KQ IN965 % 2.0(3.9)MQ
D6 0 l ”\J 7
lnF é 360(680)KQ
D5 0
I Y
D4 . 1 nF—ll 360(680)KQ
'— \/
lnF "— 360(680)KQ
énF _”_ é .510(1.0)MQ
D2 .
1_ fi’
lnF ” g 360 (680) K!)
D 1 e l v
— — .910 1.8 M!)
photo_ 1 n1: % ( )n SHV Cable\ HV
’ L \J‘ W 11 1' "
Cathode ”— 10m =
1 nF __

 

Figure 2.4 Schematic diagram of the upgraded active voltage divider used for Rings 9, 10
and 11 of the Miniball array. The value of resistors for the original divider are listed in

parenthesis.

l9

acceptance in the array, a uniformity of scintillation response better than 3% to the or
particles was required. The pre-selected crystals were then machined into their final
shapes and scanned a second time, requiring uniformity of response within 2.5%. More
details of the quality control tests can be found in Ref. [deSo 90].

The detection principle of the Miniball detectors is based on the plastic-CsI(Tl)
phoswich technique. The anode signal from the PM tube has the shape shown
schematically in Fig. 2.5. It reflects signal components from the scintillation of the plastic
foil (fast) and the CsI crystal (slow and tail). (The amplitude of the signal can be
individually adjusted for each detector by a computerized adjustment of the voltage on the
PM tube.) These signals were integrated with the three time gates indicated at the bottom
of Figure 2.5, corresponding to the various scintillation components. The fast and slow
signals are combined to determine the charge of the detected particles. The slow and tail
signals are combined to determine the mass of the hydrogen and helium isotopes. The
energy of the detected particles was extracted from the amplitude of the signal within the
slow gate. Further details of these particle identification and energy calibration techniques

are discussed in the next chapter.

2.2 Upgraded Miniball/Miniwall Arrays

The geometry of the original Miniball array was designed to address the problems
of accurately measuring multiplicity and angular distributions in asymmetric 36Ar + 197Au
collisions [Kim 92, deSo 91, Phai 92]. The original Miniball array contained 187 phoswich
detectors to cover the angular domain 9° S 01,7, S 160°. However, heavy symmetrical
systems, such as Au+Au, result in higher multiplicities and problems with double hits. For
such heavy systems, modifications in the Miniball at 01.1, S 30° were required to address
these problems. Thus, 48 detectors, covering 9° S 0..., S 31° of the first three rings in the
original Miniball array were replaced in this dissertation work by 48 detectors of the
Miniwall array at 145° S 01.7, S 25°, 28 detectors of an improved third ring of the Miniball
array at 25°S 01.7, S 31°, and by 84-elements of the Catania hodoscope and the Aladin

spectrometer at 07,7, S 145°. It would have been usefirl to upgrade Rings 4

20

 

 

 

 

0 151!- 1.5::

L! I I I

"fut" ”slow“ "an”
33m 3901- 1518

Figure 2.5 Timing and widths of the fast, slow and tail gates.

21

and 5 of the Miniball, but the double hit probabilities were not so significant, and there

were insufficient resources to replace the detectors in these rings.

Upgrade of Miniball Array

The final configuration of the Mniball/Miniwall array is listed in Table 2.1 and a
half-plane section of the upgraded Miniball/Miniwall is plotted in Figure 2.6. Rings 1 and
2 of the upgraded array are part of the Miniwall array built at Washington University (St.
Louis), described in detail in the next section. To improve the detection efficiency at the
forward angles and to match ring 4 of the Miniball to the Miniwall, a new forward ring 3’
of the Miniball, containing 28 detectors, was built. These detectors were located at a
longer distance (280 mm) from the target, significantly reducing the solid angle and double
hit probability relative to that for the original array. For comparison, the configuration for
the first three original Miniball rings is also listed in parenthesis in Table 2.1.

Some technical improvements were employed in constructing these new detectors.
In the original Miniball detectors, one observed occasional sparking between the external
high voltage field shaping electrode and the u-metal magnetic shield which surrounded the
entire PM tube assembly and was held at ground potential. To suppress such problems, the
gap between these two surfaces was filled with epoxy. Unfortunately, there were voids in
this epoxy layer that became depleted of air after several days of operating in vacuum and
subsequently became a weak point in the insulation where sparking could occur. To
reduce this problem, the epoxy layer was degassed under vacuum while it was curing.

This and some additional refinements in the gluing procedure resulted in a completely
stable operation under vacuum.

A second improvement was made to the Rings 9, 10 and 11 of the Miniball
detectors to handle higher counting rates. In heavy ion bombardment, secondary electrons,
X-rays and to a lesser extent y-rays are emitted more copiously at all angles than in the
case of light ion bombardment. Since solid angles of the backward detectors are larger
than those at forward angles by a factor of 10-20, the electron, X-ray and y-ray counting
rates in these backward detectors are far bigger than the rates in the forward detectors. At

high beam intensities, these high counting rates can cause the bases of the PM tubes for

22

 

 

 

 

 

 

9
5 (24) 5am \ (14)
3125) \,
4(24 \ \\ \\ ”I 10 (12’
\‘ ‘\ \
2 (2 ‘) \“ \\‘ \\ \\ “ 0, ’
~ \ ‘ \ \ I I
\ s \ l I
t (24) ‘N ‘ ~ ‘~ ‘ \ . I .
\x“ x.“ x. .\ \ \ '71" ’ ”I 11(3)
~“‘ ~‘5 \\ \ \ \ I
~“‘ \ \s ‘ \ ‘ "'
K.‘ ‘ \\‘ \ 'I' "t //
s \ \s\ ‘ 9' A ’I
\ uw‘ ‘ ' \‘ ‘ ’0
\‘ ‘\ “ \ \ P "'
K‘:‘\‘ \\‘\.c‘.‘ a.
‘-‘ .\\~ \1 I "

 

Figure 2.6 Half-plane section of the upgraded Miniball/Miniwall array. The rings of the
Miniwall are labeled with 1-2 and rings 3’-ll label the individual rings for Miniball. The
number of detectors per ring are given in parentheses. The polar angles for the centers of

detectors are indicated. The dashed horizontal line indicates the beam axis.

23

these detectors to become over-loaded. The overloaded detectors will have a worse
charge particle resolution, and the bases can shut off when the currents in the PM tubes
are sufficiently large. Some reduction in sensitivity to the electrons can be achieved by
covering the detectors with absorber foils (typically of an Pb-Sn alloy with 5 mg/cm2 aerial
density). Even with covered foils, at times the tube currents at the back angles can be too
high, requiring reductions in the beam intensity and a resulting loss of data. This can
prevent investigations of issues for which high statistics are required. One example of such
an investigation can be found in Ref. [Zhu 92].

To increase the count rate capability of these backward detectors, new bases were
constructed for the detectors in rings 9-11 to handle higher count rates. The original
bases did not always precisely display the same failure mode. Nevertheless, an increased
count rate stability will result when the DC current passing through the resistor network
of the bases considerably exceeds any fluctuations in the load caused by a fluctuating
count rate [Gupt 67]. Thus the tube stabilities were increased by decreasing the resistance
in the passive divider networks by about a factor of two, as shown in Fig. 2.4.

To test these modifications, the variation of the pulse height [Mich 65] of a typical
Miniball detector was studied as a firnction of the counting rate of the detected y-rays
from a 27Co"0 source. A drawing of the setup and the associated electronics used for this
purpose is shown in the Fig. 2.7. The detector was put in a light-tight chamber. The C060
source could slide towards and away from the detector, thus allowing changes in the
counting rate due to changes in they-ray flux at the detector. The counting rates were
read out by a scalar. The signal of the PM tube was amplified and read out by a
multichannel-analyzer (MCA). Two 1.174 and 1.332 MeV y—rays are emitted in a
(4+—)2+—> 0+) cascade in the excited ”Ni“, which is the product of the B decay of C060.
The y-ray energy spectra reveal two photopeaks at these two energies and a Compton
scattering continuum. The shifts of both peaks were examined to reveal any count-rate
dependencies of the PM tube gain.

To simplify the comparison between the modified and original Miniball voltage
dividers, the two dividers used the same PM tube coupled to the same CsI crystal. The

same high voltage was also applied to the PM tube for each divider (base). Under this

24

condition, the performances of the dividers could be compared by considering the count
rate dependent shifts in the peaks in the y-ray spectra, reflecting a gain change in the PM
tube due to alterations in the voltages supplied by its base (divider). The y-ray photopeak
pulse height divided by a reference pulse heights obtained at 25 counts/sec are plotted in
Fig. 2.8 as a firnction of the counting rate for the two bases. The corresponding distances
between the source and detector were labeled at the top of Fig. 2.8. Large shifts in the
peaks measured with the original divider occurred when the counting rate was greater than
300 counts/sec. Large shifts in the peaks measured with the modified divider did not occur
until the count rate exceeded 900 counts/sec, indicating a significantly higher count rate

capability with the new base.

Miniwall Detection Array

The Miniwall detector array was built by LG. Sobotka et. al. at Washington
University in St. Louis. The array consists of 6 rings, a total of 128 detectors, spanning
from 97.7, = 33° to 250°. In this experiment, only the two backward rings covering angles
ranging from 0..., = 166° to 250° were used. The rings are labeled as rings 1 and 2 in Fig.
2.6. Similar to the Miniball, each detector contained a fast plastic foil (~8. mg/cmz) and a
3 cm long CsI(Tl) crystal. The crystal was coupled by a light guide to a PM tube
(Hamamatsu model R647). For these tubes, a passive voltage divider was used, requiring
significant cooling of the array. The PM tubes with similar gains were used for
neighboring detectors so that the same high voltage was applied in parallel. This method
reduced the number of high voltage channels, but had the disadvantage that the gains of
individual detectors could not be individually adjusted by applying different voltages.
Similar to the Miniball, three different time gates; the fast, slow and tail, were applied on
the signal out of the PM tube. The mechanical configuration of the Miniwall was very
compact with all the detectors attached to the same plate. A cooling line was mounted
around the edge of the plate to keep the temperature of the PM tubes constant during the

experiment.

25

Setup for Testing Counting Capability

 

I] 25 50 75 cm

 

 
    
 
   

'l Detector
r PMT

r-ray M
Cf'ml Sourice

# q I‘ .Em,

Electronic schematic diagram

 

auo n§|_l"

Figure 2.7 The set-up for testing the counting capability of the PM tube with different
voltage dividers is shown. The electronic diagram also is drawn in here. The Co‘50
was moved to vary the intensity of y-rays hitting the detector and to allow one to
monitor the shifts of the peaks with a multi-channel-analyzer (MCA). The counting
rate was read out by a scalar as a fiinction of source intensity and the distance between
the source and the detector. The down scalar was inserted to ensure negligible MCA

dead time.

26

 

1-21 1 1 1 1

~ a Modified Base —
0 Original Base —

 

 

 

 

 

“5
L. _.
(D 0.8 *— \ _
D—r * S
\ T \ _.
x . 1 _
cu _ 7 _
c1)
D—t 0.6 _ Q —
. \ _
\ —t
- b :
0.4 — —
i—l l l l l i 1 l l l l l l l I i l l l l l l l 1 I‘ll
o 250 500 750 1000
Counts/Sec

Figure 2.8 The relate magnitudes of the y—ray peaks are plotted as a firnction of the
counting rate for the two different voltage dividers. The distance between the source and

detector is indicated at the top of this Figure.

27

2.3 Catania Si-CsI Hodoscope

The Catania Si-CsI hodoscope array contains 84 Si-CsI telescopes, arranged as
shown in Fig. 2.9. The polar and azimuthal angles for the centers of these detectors are
listed in Table 2.2. Each detector telescope consists of a 300 pm thick Si detector backed
by a 6 cm thick CsI(Tl) crystal. The signal of each CsI(Tl) detector was read out by a pin-
diode. The size of the fiont face is approximately 30 x 30 m2. Each Si detector had a
dead region 1 mm wide along the edges of the detectors. At polar angles greater than
145°, the telescopes were blocked by the Miniwall array, as discussed in the previous
section. Software gates were set to avoid the double counting of particles that punched
through the Miniwall into the partially covered Cantania array. Signals from the array were
amplified by preamplifiers inside the scattering chamber and firrther amplified by computer
controlled shaping amplifiers before being digitized by a Fast Bus charge integrating ADC.
Information from the Catania array for each event was written on tape along with the data
from the Miniball/Miniwall and Aladin spectrometers. Further details of the mechanical
and detector designs can be found in refs. [Kund 94, Lind 93].

2.4 Aladin Spectrometer

The Aladin spectrometer consists of three main components: a larger bore dipole
magnet, a multiple-sampling ionization chamber (TP-MUSIC), and the time-of-flight wall
(TOP). In the data discussed in this dissertation, the beam intensity was chosen to be too
large for the TP-MUSIC to function, and so data was obtained only with the TOP Wall.
The ToF wall is a layered structure consisting of two arrays of 96 plastic scintillators
arranged with one array in front of the other. Each plastic scintillator is 1100 x 25 mm2 in
its frontal area with a thickness of 10 mm. Signals from each scintillator were readout by
two PM tubes, one at each end of the 1.1 m long scintillator. The signal from each PM
tube was discriminated to obtain a timing signal, and the time and amplitude of each pulse
were digitized in Fast Bus modules and written on tape along with the rest of the
information from each event. Further details of the design were described in Refs. [Aldi

s9, Lind 93].

28

 

Figure 2.9 Three dimensional geometric diagram of the Catania hodoscope array.

29

Table 2.2 The centroid of the polar and azimuthal angles of the Catania detector are listed.

The polar angles of these detectors:

 

 

 

 

 

17.8 16.2 15.1 14.8 15.4 16.1 17.6
17.4 15.1 13.4 12.3 11.8 12.2 13.3 14.9 17.2
15.2 12.8 10.8 9.3 8.9 9.4 10.9 12.6 15.2
16.6 13.5 10.8 8.3 6.4 5.6 6.4 8.3 ’ 10.6 13.3 16.1
13.5 10.9 11.1 13.9
13.3 10.9 10.8 13.6
13.5 10.9 11.1 13.9
16.2 13.2 10.6 8.1 6.3 5.5 6.2 8.1 10.4 12.9 15.7
14.8 12.5 10.5 9.1 8.6 9.1 10.6 12.3 14.8
17.4 14.8 13.1 12.0 11.5 12.0 13.0 14.5 16.8
17.3 15.8 14.8 14.5 15.0 15.7 17.1
The azimuthal angles of these detectors:
122.0 112.8 101.9 90.7 79.3 68.1 59.0
135.4 127.8 118.1 105.6 91.3 76.6 62.9 52.8 44.3
143.1 135.3 124.6 109.5 90.6 71.2 55.6 47.7 36.9
158.4 153.8 147.8 136.3 118.0 90.9 63.8 44.1 32.3 25.9 21.6
169.0 166.3 13.7 11.0
180.0 180.0 0.0 0.0
191.0 193.7 346.3 348.3
201.6 206.2 212.2 223.7 242.0 269.1 296.2 315.9 327.7 334.1 338.4
216.9 224.7 235.4 250.5 269.4 288.8 304.4 315.3 323.1
224.6 232.6 241.9 254.4 268.7 283.4 297.1 307.2 315.7
238.0 247.2 258.1 269.3 280.7 291.9 301.0

 

 

 

30

2.5 Data Acquisition Electronics

Fig. 2.10 shows a block diagram of the electronic set-up used for the Miniball
array in this experiment. The shape of the anode current from the PM tube, shown in Fig.
2.5, was split via passive splitters into “fast”, “ slow”, “tail” and “trigger” branches with
relative amplitudes If... : 1,70“, : 1m. : It“.g z 0.82: 0.04: 0.04: 0.10 for the currents. The gate
widths for the fast, slow and tail are also shown in the bottom of Fig. 2.5. During the
readout of FERA’s (Fast Encoding and Readout ADC), integer*2 words for fast, slow,
tail, and time are written to the tape for the detectors that have trigger signals above the
threshold. Since high energy fragments were emitted abundantly into the forward angles in
Au+Au reactions, the splitter ratios for slow and tail signals of rings 3’- 5 were reduced by
a factor of two to keep the signals within the dynamic range of the FERA, ADC.

Signals from the Miniwall detectors were processed by different electronics. Fig.
2.11 shows the electronics block diagram for the Miniwall array [Stra 90]. The anode
signal from each PM tube was amplified by a fast variable gain amplifier before the signal
was split into the “fast”, “ slow” and “tail” parts. The logic signal came from the second
output of the amplifier. Similar to the Miniball detectors, the fast, slow, tail and time

signals were written to tape for all detectors that had trigger signals above the

discriminator thresholds.

    

 

Dlsc premaster
Disc veto
Master
veto
logic
FI/FO
tail trig. tail dclay L

L. U. instant busy

veto

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.10 Schematic electronics diagram of the data acquisition of the electronics for
the Miniball array alone. The Disc, Split., DGG, Fera, Amp, L.U. and FI/FO present
Discrirninator, splitter, delay and gate generator, Fera ADC, amplifier, logical unit and the

fan in/fan out models respectively.

32

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.11 Schematic electronics diagram of the data acquisition of the electronics for the

Miniwall array alone. The Phis QDC is the Phillips QDC model.

Chapter 3
Data Analysis

In general, each device in this experiment uses a different detection principle to
resolve the charge, mass and energy of the detected particle. To extract this information
from the raw data, different particle identification procedures and energy calibrations were
used for each device. Details of the particle identification and the energy calibration for
the Mniball/Miniwall will be described first in following two subsections. The energy
calibration of the Catania hodoscope was recently done by the Catania group [Raci 95].
Since the energies of the particles detected in the Catania hodoscope were not used in the
following data analyses, only the particle identification principle of this device is briefly
described in this thesis. This is followed by a description of the particle identification, and
the energy and angular determination techniques used for the TOP wall of the Aladin

SpCCtI'OI'IlCteI'.

3.] Particle Identification of the Miniball/Miniwall

For each detector of the Miniball/Miniwall, the “fast”, “slow”, “tail” and “time”
signals were recorded in the raw data, as described in Chapter 2. By plotting the fast
versus slow components of the signal, clear element identification can be obtained [deSo
90]. By using the tail versus slow spectra, the isotopes of hydrogen and helium can be
separated [deSo 90, Kim 91]. A total of 300,000 events were collected in the Au + Au
collisions at E/A = 100, 250 and 400 MeV. Due to a lack of statistics, the drawing of the
charge particle gates became very difficult, especially for the higher charges. The
response of each detector was therefore mapped from scintillation response firnctions for
the same detectors obtained in high statistics measurements at lower energies. To provide
guidance for this procedure, we chose two experiments, the Satume experiment and the
NSCL-91014 experiment. Both experiments were designed to study Kr + Au collisions
from E/A = 35 to 400 MeV, and included the same set of Miniball/Miniwall detectors used

33

34

in the present setup. NSCL-91014 collected a total of 60,000,000 events, while 5,000,000

events were collected in the Satume experiment.

Charge Identification

Figure 3 .1 shows the fast versus slow spectrum for the fifth detector of ring 3 ’
(detector 3’-5) in the NSCL-91014 experiment. To facilitate drawing the charged
particle gates, these charge lines were linearized as shown in Figure 3.2a. The final charge
gates were drawn on these linearized spectra. To reduce the number of charge gates set in
the fast versus slow spectra, the detectors used in the experiment were classified into
several groups with similar detector response. Within each group, the detector with the
best charge resolution was chosen as the reference. The two-dimensional fast versus slow
spectrum of each detector within a group is scaled to match the spectrum of its reference
detector. The proton-punch-through point of the “slow” signal of each detector was first
scaled to the one on the reference spectrum to provide the scaling factor for the slow
component. Since the “fast” signal is very sensitive to the temperature dependent
scintillation efficiency of the plastic foil and the thallium doping of the CsI crystal, the
alpha particle line of each detector was separately fitted to the reference alpha line with
individual “slope” and “off-set” parameters for the fast component.

After the two dimensional fits were obtained, the “fast” versus “slow” spectra of
individual detectors had shapes similar to the reference spectra. Ideally, the particle
identification lines of all the charged particles would sit in the same positions as those in
the reference spectra and could be optimally linearized with the same parameters. In
practice, this proved to not be so simple. Nevertheless, charge gates could be easily set on
the linearized spectra for the high statistics measurements, and mapped to the low
statistics spectra of the Au + Au measurements presented here. Fig. 3.2b shows the
corresponding low statistics linearized spectra for the G81 measurement. With the
procedure described above and guided by the spectra with high statistics, charge gates up
to Z = 6 were set on the low statistics GSI data. Beyond Z = 6, extrapolations could be

used to obtain the charge gates, but in general, the quality of the energy spectra for these

35

 

.350

2262

Fast
:1
or

87

 

   

 

 

Kr + Au, E/A = '7
35 MeV
Dot 3'-5
l I l l
O 35 70 105 140 175

Shaw

Figure 3.1 Fast versus slow spectrum with high statistics for detector 3 ’-5 (ring 3’,

position 5) obtained for Kr + Au collisions at E/A = 35 MeV.

 

 
   

 

 

 

400 I l
Kr + Au, E/A =
_ 35 MeV
300 7‘— o.+3--5 —
o
a 200 _
1oo '. a _
- Be
u
, - Ho
0 J l I
O 20 4O 60 80 100

Figure 3.2a The linearized particle-identification (PID) spectrum of Fig 3.1. The charge
gates were set such high statistics spectra during the data analysis and mapped to the low

statistics data of figure 3.2b.

37

 

 

    

 

 

400 .. I I I I
in. . Au+Au
300 E- . ' Dew-5 -
o ," ..
a" 200 ?._..
M “'53 "' ~27 ‘ 7"”??- i
100 ' ' '-
1
o
0 4o 80 120 160 200

Figure 3.2b Linearized two-dimensional PID-slow spectrum obtained in the low statistics
Au + Au measurement. Due to the lack of statistics, charge gates were set on the high

statistics spectra shown in Fig.3 .2a and mapped into the low statistics data shown here.

38

heavier particles was quite poor. Therefore, the energy information with Z > 6 was not
used in this analyses which are described later.

Most of the Miniball detectors had the particle identification lines drawn by the
above procedure. This procedure was also used for Miniwall detectors which had crystals
with similar doping as the bulk of the Miniball detectors. However, for CsI crystals in the
Miniball that were produced by Bicron Corporation, the doping concentration was
different. Such crystals were generally used in rings beyond 01,7, = 90°. Charge resolution
for these Bicron-like detectors cannot be achieved via gates on the fast versus slow
spectra, but require gates on the fast versus tail spectra. In general only charge gates up
to Z = 4 were set for these detectors. Due to a mistake in the electronics setup, charge
resolution was achieved only up to Z = 4 in the Miniwall at E/A = 400 MeV; this problem

was corrected for the measurements at E/A = 250 and 100 MeV.

Isotope Separation of Hydrogen and Helium

The pulse shape discrimination of CsI(Tl) crystals allows the isotope separation of
hydrogen and helium particles. The procedure described here was first developed by Y.D.
Kim [Kim 91]. Due to the high abundances of hydrogen and helium particles, spectra in
the present experiment were sufficient to set the gates. Figure 3.3 shows the tail versus
slow spectra for detector 3’-5. From this spectra, p, d and t isotopes of Z = 1 particles
and 3 He and alpha isotopes of Z = 2 particles were quite difficult to separate. Using the
upper and lower limit lines which encompass the area of the spectrum (see Fig. 3.4) with
counts, we construct the following parameters:
Tailr=Tail+5 (3.1)
where 5 is a random number between -.5 and .5 which is added 0 remove digitization from
the displayed spectra. We also define
Taill = Line] (slow) ( 3.2)
Tail; = Line; (slow) ( 3.3)
where Linel and Linez are used to designate to the upper and lower lines respectively
which are parameterized as a firnction of the slow signal;

A... = Taill - Tail; (3 .4)

39

 

‘75 1 I I I ,
Au + Au, E/A =

1008c250MeV .47

131 " Dow-5 _.;'-:. :7

.I It.

Ta”
or
'u

l

43"

 

 

 

l l l

 

0 35 70 105 140 175
Sflow

Figure 3.3 The tail versus slow spectrum for detector 3’-5 generated by combining all
the runs of Au + Au collisions at E/A= 100 MeV/A and 250 MeV obtained in this

experiment

40

 

500

400

300

Tail

200

100
t

 

 

(slow, tailz) S

 

 

 

 

Figure 3 .4 Schematic tail versus slow plot describes variables used in the construction of

the PDT fimction.

41

A = Tail - Tail] (3.5)
and a new parameter is defined as

PDT = 512 * (A/Amx) (3.6)
Figure 3 .4 illustrates the procedure used to maximize the dynamic range in the spectra in
order to achieve the optimal isotope separation. The new slow channel is equal to the raw
slow channel /4 + 15. An extra 15 channels were added to ensure that all particles lie
within the range of the new spectra. As shown in Fig. 3.5, there is a clear separation
between the Z = 3 and the light charged particles at high energies. At low energies, all
particles merge in the tail versus slow spectra and must be identified through the charge
identification method using the linearized PID versus slow spectra described above.

All isotopes of the hydrogen and helium particles were separated up to the punch
through points. At higher energies, the punch through lines from the helium particles
merge with the hydrogen punch through lines on the left side of the figure. The electronic
discriminator levels were set high during the experiment to suppress low energy electrons.
These discriminator settings also strongly suppress the low energy hydrogen particles
leading to a sharp cutoff of the p, d, and t particle identification lines as shown in Figure

3.5. These isotope separation procedures were also applied to the Miniwall detectors.

3.2 Energy Calibration for Miniball Detectors

Since the response firnction of CsI(Tl) scintillation depends strongly on the atomic
number, its energy and to a less extent on the mass of detected ions, several calibration
measurements [Stor 58, Quin 59, Alar 86, Gong 88, Souz 90, Kim 91, Colo 92] have been
performed by directly bombarding CsI(Tl) crystals with various light and heavy charged
particles of known energies. These measurements indicate that the non-linearity of the
light-output of the CsI(Tl) crystals becomes greater at lower energies and with higher
atomic numbers of the detected particles. Above 10 MeV/A, the light output of
scintillation increases more or less linearly with the energy deposition. Previous
calibrations the Miniball detectors [Kim 91, Phai 93] using the elastically scattered

particles have parameterized the scintillation light output L as

42

 

    
   
   

 

  

 

 

‘39 l I 4 ,it 7 1‘: ,
3 ' '1'; .. ’t
I... 5* . ' '.‘
97 1" ' "g.
:5 " ‘7: '-:
3 H3 p.I. f ' .5 if _
.9 65 — £1 'I-I.
(f) .- . r...
I,'-. fa
t -’.' :11
d t. p
p .
32 —Au + Au, E/A =
100 8: 250 MeV
0.1 3'-5 .
l
O l I
O 51 102 153 204 255

Figure 3.5 The plot shows the PDT versus slow spectrum of detector 3 ’-5. Good isotope

separation of proton, deuteron, triton, 3He and alpha is achieved.

43

L(E,Z)=a(Z)E+B(Z)[C'Y(Z)E -1.] (3.7)

where or, B and y are charge dependent, adjustable parameters. Here, E denotes the
energy deposited in the CsI crystal [Kim 91]. This functional form is consistent with
previous measurements [Quin 59, Colo 92, Valt 90]. It reproduces most of the nonlinear
behavior of the CsI(Tl) light output in the low energy region and the roughly linear
behavior at higher energies up to E/A ~ 25 MeV.

Additional calibration points can be obtained from charged particles that punch
through the 2 cm long CsI(Tl) crystal, since the energy loss for known fragments in the
crystal can be calculated. A close examination of these additional high energy data points
obtained at GSI and Satume revealed that they could not be described by equation 3.7.
Indeed, Eq. 3.7 overpredicts the light output by 20 i 5% for or-particles that punch
through the CsI crystal due to a saturation effect described below. Detailed examination
reveals that this discrepancy correlates with the PM tube gain of the individual detectors.
As there was no active amplifier in the electronics for the slow signal, the proton punch-
through channel provides a measure of the PM tube gain. Correction for this gain
dependent effect required a readjustment of the energy calibration procedure described

below.

Correction for Saturation Effects

The nonlinear “saturation” of the energy-light relation at high energies (E/A > 30
MeV) reflects a saturation of the PM tube gain. As mentioned in Section 2.2, several
factors affect the tube gain for large signals. First, the electron distribution from a large
signal can significantly reduce the electron field between the last 2 dynodes, thereby
reducing the amplification on D10 and the subsequent anode signal. Moreover, the charge
on the capacitors in the last dynode stages can be depleted by the pulse train, thereby
reducing the tube gain by lowering the voltage drops between the amplification stages,
leading to reduced signal amplitude. In order to calibrate this nonlinear behavior of a large

signal, we bombarded low intensity 6Li, 12C and 18O beams directly into 10 different

44

Miniball detectors at incident energies ranging from E/A = 22 to 80 MeV. Figure 3.6
shows the fractional deviation of the observed light output from the light output predicted
by Eq. 3.7 using calibration constants determined from lower energy signals. Here, the
light output is expressed as a function of the measured QDC (charge-digit-converter)

channel. The fractional deviation is defined as

#norm- L(E, Z)
(E, Z)

 

f(ch#) = Ch (3.8)

here ch# is the measured QDC channel for a particle with charge Z and the energy E in a
particular detector. The Ch#,,mm is the same data point renorrnalized by a linear
transformation consistent with the measured pedestal, and one that puts the proton punch
through point in channel 240.37 so that it can be compared to a light output firnction
L(E,Z) which has been renorrnalized in the same fashion. In order to convert CH# into the
tube gain, the measured QDC channel was corrected to take into account any differences
in the splitter ratios discussed in Section 2.5. For 12C fragments, the saturation effect was
parameterized by the straight line shown in the upper panel of Figure 3.6. The same fit
also approximates the 6Li data (shown in the bottom panel of Fig. 3.6) By combining
calibrations obtained in previous Miniball experiments [Kim 91, Peas 94], a similar
description of the saturation effects for 4He, 6Li, 10B, 12C, 12N, 16O and 20Ne ions was
achieved.

Since the fractional deviation varies roughly linearly with the measured QDC
channel, i.e., to the amount of light passing through the PM tube, one can obtain the

model-dependent corrected channel by rewriting Eq. 3.8 into a different form:

f(ch# ) ___ Ch#norm— Ch#corrected . (3 '9)
Ch#corrected

Then, one can write the corrected channel as

ch#
h# =——n°£“—. 3.10
C ”mime“ 1+ f(ch#) ( )

Since f (ch#) = fC 0 ch# and fc is assumed to be independent of the charge and the energy

of the detected particles, Eq. 3.10 can be written as

45

 

      

 

 

 

 

:TFII IIIRIIIII[IIIIIIIIIIIIIf:
0.0_ _:
E 12C 3
I: _ _
.9 : :
> C i
'8 —06i'"71711117111111.1171111171"
(___, . :1 7 l I IIFIII:
cs 00_ § __
E 3 iii GM 3
3 —0.2:— Hi} j
3‘; E 9 H 3
: if :
_O.651111LL11111111l111111111111n1

0 500 1000 1500 2000 2500 3000

QDC channel

Figure 3.6 Fractional deviations of the light output of Miniball detectors as defined in
equation 3.9, as a function of the normalized QDC channel. The upper (lower) panel
shows the results for 12C( 6Li) fragments, respectively.

46

ch#norm

_ 3.11
l+fcoch# ( )

Chllicor'llected =

The spreading of the data points from 10 different Miniball detectors is of the
order 10-15 % as indicated in Fig. 3.6. The origin of this spread is unknown, but may be
related to the accuracy of some of the electronic components used in the PM voltage
divider, or the differences in the doping of individual crystals, or the internal tolerances of
some of the PM tubes. It represents a systematic error associated with using a universal
correction without specific modifications for the unique characteristics of each Miniball
detector.

To demonstrate this correction effect, we compare the corrected and uncorrected
light outputs of a specific detector for ‘He and 12C fragments. As seen from Fig. 3.7, this
correction becomes more important where there is a higher amount of light passing
through the PM tube, i.e., for larger QDC channels. While the 250 MeV 12c data points
are only corrected by about 10 %, the correction for the 300 MeV ‘He data points is on
the order of 30-40 %.

The parameters a, B and y in Eq. 3.7 are charge dependan Following the method
proposed by Colonna et a1. [Colo 92], we parameterize this charge dependence as follows:

(1(2) = a1 +32 0 C-a3z ,

B(z) = b1 - b2 . e‘b3z, and (3.12)

7(2) = 01 +02 'C-C’Z

The fractional saturation constant f c in Eq. 3.11-was also treated as a fit

parameter in the overall calibration. The a, [3 and yparameters for light particles (Z S 2)
cannotbe fitted well by the smooth function and are treatedas three additional 5
parameters. A total of 13 parameters have been fitted by a x2 mininrization procedure
using all of the existing Miniball calibration data. All data points have been appropriately
normalized to correct for differences in the splitter ratios. The values of at, b; and or are
listed in Table 3.1 for fragments with Z/A = 0.5.

47

 

12C corrected —
12C uncorrected ~
4He corrected -
4He uncorrected —

1500 —

(>90.

 

 

 

m t—
3; 1000-— i -—
S : <> i
CD
,3 - _
O
_ (D _
500 — O -—
_ ' .1
O— l I 11 111i 1 1 11 i l l~
o 100 200 300

energy [MeV]

Figure 3.7 Calibration curve for 4He and 12C fragments (solid points) including the

saturation corrections. The open symbols show the uncorrected data points.

48

Table 3.1 The values of a, b7 and c7 in the set of functions (Eq. 3.12) for energy
calibrations

 

 

 

 

 

 

 

 

xZ/n = 3.27 i=1 i=2 i=3
ai 0.4142 4.995 7.824E-2
h 299.7 50393-3 1.967E-4
ci 1.339E-2 59255-3 3.846

 

 

In order to compare this data to previous calibration curves, the curves and data were
normalized to place the 200 MeV 12C point at channel 511. The fit to the saturation
correction factor in Eq. 3.11 for the Miniball detectors yielmd rc = 1.7x10“.

An additional constraint is applied to the fit that keeps the calibration curves for
different charges from crossing. For clarity, multiple data points for the same energy and
fragment charge were averaged to create Fig. 3.7; the spread of the calibration data is
indicated by the error bars. For energies larger than 30 MeV per nucleon, the calibration
is accurate to 10-15% and this accuracy improved to 5- 10% at lower energies. The insert
of Fig. 3.8 shows the fit for various fragments more clearly at lower energies. While most
of the fragments follow the predicted systematic trends, both ‘Li and 1"B particles (the
crosses) seem to deviate from the calculated curves. As this data has been taken from
various experiments and analyzed by different people at different times, a more
comprehensive set of data for these two elements, taken under more controlled
. circumstances, would be desirable to understand the observed deviation and to establish a
systematic trend.

Even though the Miniball array contains CsI crystals from different manufacturers,
all CsI detectors have a similar light output. The final energy calibration for a specific
detector has a 10-15% uncertainty for particles with energies above E/A = 30 MeV. The

49

0 5 10 15
‘OOOIIIIIIIII'TIIHI/ / I

 

 

I I I I

3000

' |

 

2000

corrected channels

1000

 

 

 

Figure 3.8 Calibration data for 4He (solid circles), 6Li (open fancy crosses), loB (crosses),
12C (open circles), l‘N (solid squares), 160 (open diamonds) and 20Ne (open squares). The
lines are fits to the data (solid lines for odd charge elements and dashed lines for even

charge elements).

50

eflect of the fluctuations between the calibrations of different detectors are reduced when

the data for the different detectors in a ring are combined to obtain the energy spectra.

3.3 Particle Identification of Catania Hodoscope

Each telescope in the hodoscope consists of a 300 um Silicon detector backed by a
6 cm long CsI(Tl) crystal. The signal of each Silicon detector is proportional to the energy
loss in that detector and is designated as “DE”. The signal from the CsI(Tl) is
correspondingly designated as “E”. The charge lines in the DE-E two-dimensional spectra
are approximate hyperbolae, as shown in Fig. 3.9, in accordance with the energy loss
predictions of the Bethe-Bloch Equation [Goul 69, Knol 7 9]. While the operation of
Silicon CsI(Tl) energy loss telescopes is in principle straightforward, non-linearities in the
electronics and some difficulties in calibration prevented the full utilization of this device.
Therefore, it was simply used to determine the number of intermediate mass fragments
emitted into the angular range subtended by the Catania hodoscope.

Since the punch though energies in the 6 cm CsI crystals of the Catania
Hodoscope are around 200 to 250 MeV/A for intermediate mass fragments, a lot of IMFs
emitted at incident energy of E/A = 400 MeV have energies higher than 200 MeV/A. Such
energetic [MFs can punch through the detectors and be indistinguishable from lighter
charged particles that also punch through these detectors. This is illustrated in Fig. 3.10.
Therefore, sufficiently energetic punch-through Lithium and Beryllium fragments could
not be separated from punch-through Helium particles, leading to an additional upper
threshold for detecting these MS. The cut used in the analysis for suppressing light
particles is shown as the line in Fig. 3.10. It removes punch through particles and also
suppress double hits by the or particles. The effective upper thresholds per nucleon for the
MS introduced by this procedure is approximately Emu/A ~180. + 80.x(Z-3 )2 Mev where
the Z is the charge of detected particle. To simulate the number of emitted IMFs via
theoretical calculation, these punch-through energy cutofl‘s were also included in the filter

routines for the experimental acceptance.

51

 

256 , f.. I . .. .II'" I -'.‘ T . a.

 

- U f. I
.‘ ' J .
~ - ' "-' I 3' -
‘I .$' . . o-
. I ‘
C. ’-
.I o
. 0"”:E;\ - .. . -
I- " 3.. .' .
. . ' . a .
uol. .. ’. ....
. . no. ... f
' . I I g
. ‘ , - r
: .I';Iu :- . .9 I
‘- - no '. f .’ ' .. . .0 .
in ‘I .5 . .P o
I - ‘ I .l: I, ' ’ .’ .-
.. . 0 ... I.
o I o '-
so' 1? " . , '
I ‘ .fi. .P. . g -
I uni-J a .h t‘ . I
. I

”7A =‘i1oo MeV.
{fir-1'5 Dat SO :
‘ J

204 256

 

 

Figure 3.9 DE (signal of Si) versus E (signal of C31) for the Catania hodoscope is plotted
for detector 30 at E/A = 100 MeV collisions.

52

 

200 |'- r I I

Au+Au

 

E/_A =4oo MeV -

  

Dot 30

8100

 

 

 

 

0 4O 80 120 160 200

Figure 3.10 DE (signal of Si) versus E (signal of C31) for the Catania hodosc0pe is plotted
for detector 30 at E/A = 400 MeV collisions.

53

3.4 Particle Identification of the Aladin ToF Wall

Each of the plastic scintillators of the TOP wall were read out by two PM tubes at
the ends of the scintillator. The sum of the two signals from the PM tubes is proportional
to the light output of the scintillators; the light output is proportional to the square of the
charge by the Bethe Bloch equation and roughly inversely proportional to the square of
the velocities of the detected particles. The flight time of these particles could also be
measured with a resolution of 160 ps if a start scintillator is put into the beam line.
Unfortunately, such a start detector could not be used in this experiment because it would
be thicker than the target. Therefore, the time of arrival of Hydrogen and Helium particles
in the Miniball was used to generate a start signal, given an overall of resolution of 500 ps.
In Fig. 3.11, the resulting flight time is plotted versus the light output of particles detected
in the front 100 scintillators of the ToF wall. Particle identification was achieved in this 2-
D spectrum by gating on the indicated charge lines. Similar methods were applied to both
the front and back ToF walls. A tracking procedure was applied to avoid double counting
the particles which were detected in more than one scintillator.

The ratio of the two PM signals from each scintillator gave the vertical position of
detected particles with a position resolution of about 1 cm. Despite knowing the positions
of particles detected in the TOP wall, the polar and azimuthal angles of the detected
particle could not be accurately determined without knowing the trajectories. Therefore,
the possible trajectories were reconstructed using the flight time, the x and y positions, and
the charge of detected particles, assuming masses consistent with simulated isotope
distributions which reproduced the isotope distribution measured in previous experiments
with the Aladin spectrometer. This allowed approximate polar and azimuthal angle
reconstruction. The acceptance of the Aladin spectrometer was modeled via GEANT

simulations, and this acceptance was incorporated into the filter routine.

54

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o ’ .5... .. ."I.
o ... 2.3. ' 'U .50. .0 p. 0.: n.
I... . ' .0 . . .I ' ‘
Al/IU 4 1 r 1 r L

o 25 56 75 100 125 150 175 200

 

 

 

TUVSM

Figure 3.11 The flight time is plotted versus the light output of particles detected in the
front 100 scintillators of the TOP wall which are used for the particle identification. The

charge lines are indicated by the straight lines.

Chapter 4

Experimental Results and Model Comparisons

In this chapter, the principal experimental results of the dissertation are presented.
Since little can be learned about the thermodynamics of a nuclear collision without
knowing something about the impact parameter of the collision, this chapter begins with a
description of techniques used for impact parameter selection. This is followed by
experimental evidence for the onset of nuclear vaporization in central collisions. At the
highest incident energy, the complimentary nature of the experimental setup allows one to
extrapolate to a “threshold-less” setup. Such an extrapolation is presented in Section 4.3.
The final sections of this chapter addresses the experimental information regarding the

dynamics of the multifragment breakup.

4.1 Reaction filters: Impact Parameter Selection

Information about the impact parameter has been obtained via measurements of
the total charge particle multiplicity N, [Kim 91, Phai 92], the total transverse kinetic
energy E [Wils 90, Tsan 91, Phai 93], the total charge Zbound contained in bound remnants
of the projectile [Hube 91], and the participant proton multiplicity [Doss 85, Gust 88].

In the present work, the total charged particle multiplicity N, includes all the
charged particles detected by the Miniball/Miniwall array, even if they were not identified.
Heavy fragments stopped in the scintillator foils are included in NC. Multiple hits in a
single detector module are counted as single hits, even if they can be clearly identified as
double hits (such as double hits by two or-particles which can give rise to a single particle
identification line). The left panel of Fig. 4.1.1 shows the normalized charged particle
multiplicity distribution detected in the Miniball/wall array for E/A = 100, 250 and 400
MeV. Similar to other measurements [Phai 92], the normalized probability distributions
exhibit a rather structureless plateau and a near-exponential fall-off at the highest
multiplicities. The probability where one observes the exponential fall-off increases from

Nc = 55 to 75 as the beam energy is increased from E/A=100 to 400 MeV.

55

56

 

 

 

 

 

 

 

 

 

 

 

 

1 r06 0.6101102] I I I T013‘ 076 l0101.2I I I I T
10‘1 r~cf~1 ‘ ‘ dent/x ilPeriI ‘ dent
.."'ll
10 — 2 ... I... '.'".'I'III'IIIO . b -
E/A : .O
10“8 1 O -
100 MeV O “ OO
10“4 . o
10—5 1 11 1 l 1 9311 _ 11 11 I1 11 1I 1
1 0.8 0.6, 0.4 0.2 — 0.8 0.6] 0.4 0.2 l E
_1 I 1 1 1 1 1 l l I ~
10 3 I -§
_ I _ .‘w'l'. I
10'" O -- —a
D“ 250 MeV o g
10_4 OO _ 0 ~
1 11 1 l 1 1 11 11 11 I1 11 1 l1
1 0.8 0d 0.4 0.2 ‘a 0.8 0.15 0.4 02'
10_1 1 1 T 1 1 _ 1 1 1 1
1 O _ 2 .‘"_l '.'.'.l"".l'l.l ‘I'I'I'I'IU'l-IIIOII'Q'...
3 o
10—
400 MeV
10—4
l l l l J l l l
O 50
N. fl

Fig. 4.1.1 Normalized probability distributions for the charged particle multiplicity Nc (left
side), and the participant proton multiplicity Np (right side) for E/A=100, 250 and 400

MeV. The reduced impact parameters 6 are indicated at the top of each panel.

57

The participant proton multiplicity N, is defined as the sum of the true proton
multiplicity and the number of participant protons bound in light Hydrogen and Helium
isotopes. In the present work, the participant proton multiplicity is calculated from the
yields of these particles detected at Elm/A > 15 MeV in the Miniball/Miniwall. Normalized
probability distributions for Np, shown in the right hand panel of Fig. 4.1.1, are very
similar to the distributions for NC.

Zbound is defined in this experiment as the sum of the charges of fragments with ZZZ
detected in the forward Aladin spectrometer [Hube 91]; most of these particles are emitted
fiom the projectile spectators. Zbound depends on the disassembly of the projectile
spectator which will be strongly influenced by the energy deposited in the projectile
spectator nucleus during the initial stage of the reaction [Ogil 91, Hube 91]. The
normalized probability distributions for Zbound at the three energies are plotted in the right
hand panel of Figure 4.1.2. The probability distribution is nearly constant before dropping
off exponentially above med > 70. For reactions at E/A = 400 MeV, the Aladin detection
efficiency for fragments emitted within its angular acceptance and with Z > 3 and y > 0.5
ybm, , is nearly 100%. For E/A=100 MeV, on the other hand, the detection efficiency in
the Aladin spectrometer for heavy ions of Z > 40 rapidly vanishes, because such fragments
may stop in the 50 mg/cm2 Kevlar entrance foils of the MUSIC chamber. Thus, Zbound data
at E/A = 100 MeV does not provide useful impact parameter information. The statistics of
the measurements at E/A = 250 MeV are insufficient to determine the suitability of the
med data for impact parameter determination.

Most of the detected particles move at much less than the speed of light; therefore,
the total transverse kinetic energy E is calculated nonrelativistically [Phai 93].

EI=ZEisin20i=ZM (41.1)
i 1 2mi

Here m,, E,, p; and 0, denote the rest mass, kinetic energy, momentum, and polar angle of

the i"‘ emitted particle in the laboratory frame. For particles that punch-through the CsI

crystals, there are no energy calibrations. Thus B, does not include any of the punch

through particles. The normalized probability distributions for E at E/A = 100, 250 and

58

 

 

 

 

 

 

 

 

 

10“1
10—2
10—8
10“4
10—5 l 11 1 1
I 0.8 1
10‘1 p'°
61"]
a ' n . '
A 1 O — 2 mung-“m” . 00.9-11.1“...‘u'tu. gut-9.61130
>< _ ‘0' ‘ . O
E: 1 O — 3 "“39
GE)
10‘4 50 MeV o
10-5 1111i1111|1111l1 1 1 11 i 1 1 11
2 4 0.6 0.8 1
_ l I l I l
10 1 g _
‘25; . '9 . -
1 O _ 2 '01"ng , .— e'uvulu-r-um-renI'I-I'W'“...
a
10‘3 '%5 O
'5.
10‘4 :00 MeV cm
10_5 lllllllgLJllllllglgLLl l llgl
0 2500 5000 75000 50 100
EJMEV) Zbound

Fig. 4.1.2 Normalized probability distributions for the transverse kinetic energy E (left
side) and the sum of bound charges Zbound (right side) for E/A=100, 250 and 400 MeV.

The reduced impact parameters 6 are indicated at the top of each panel when they can be

reasonably assigned.

59

400 MeV are shown in the left hand panels of Figure 4.1.2. Since light particles punching
through the CsI can not be included in E., the suitability of the current detection
configuration for reconstructing the impact parameter using E at E/A = 250 and 400 MeV
is questionable.

To establish an impact parameter scale at these incident energies, the geometrical
prescription proposed by Cavata [Cava 90] has been adopted. For each quantity, N,, E,

med and ND , we assume a monatomic dependence upon the impact parameter and

define the reduced impact parameter scale Via

00 I 1 1/2 00 1 1 1/2
b(X)__ _b_(X) _Uxmx )dx] {jxwmflmx )dx] (4.1.2)
where X = N, E and Np
, , 1/2 , , 1/2
b(X)=— b—(—X) {jinx )dX] {I fwm“)1>(x )dX ] (4.1.3)

For impact parameter selection using med, P(X’) is the normalized probability
distribution for the measured quantity and bm is a maximum impact parameter at which

the detected charged particles multiplicity equals the minimum bias threshold. For this set

of experiment b = bm at <Nc>=4. The corresponding 5 scales are shown near the top of

each normalized probability distribution plotted in Figure 4.1.1-2. Neglecting fluctuations,

the reduced impact parameter scale ranges fromb = 1 for glancing collisions to b = 0 for
head-on collisions.

The quantitative relationship between the reduced impact parameters 5(x) and the
measured observable X where X= NC, Et, Zbound and Np, is shown in Figure 4.1.3 for three
incident energies, E/A = 100 (dotted lines), 250 (dashed lines), and 400 MeV (solid lines).
As punch-through particles were not included in B, only data for E/A=100 MeV is shown
in the lower left panel. Due to a low detection efficiency in the TOP wall at E/A = 100
MeV and problems with statistics at E/A=250 MeV, only the data for E/A=400 MeV is
shown in the lower right panel. Based on the acceptance of the device, one expects N,, N,

and Zbound to provide reasonable impact parameter information at E/A = 400 MeV, and NC,

60

 

 

 

 

 

 

 

 

NC Np

0 20 40 60 80 100 0 25 50 75 100 125
1.0FTTTITTITIIIT!ITTYT TTTT TIL—IIIIIIIITTrFTIIIITIIIII‘le—J
E . ”"Au + ‘° Au, E}A= 1E . i
0.8 — —— 400 MeV 4:- J
: — —- 250 MeV :_ :
_ ------- 100 MeV __ 1
0.6 E- _:_— i
0.4 :— €:— id
0.2 % —EZ— —E
(E) 0.0 :ITLTliT lllLlIli:ll 11111 1 1111:
0.8 :_°._ —3 __:
1 2 2
0.4 ~_— 1 i
0.2 _L —:_ __:
0.0 :1 l l l l l 1 L1 141 l 11 l..l.'l l 1;:11 l l l l l l l L|_l l 1J1 l l 1 Lil:

0 2000 4000 6000 8000 20 40 60 130

Et(MeV) Zbound

Fig. 4.1.3 Reduced impact parameters 6 extracted from the measured observables X =

N,, Np, E and Zbomd by using Eq. 4.1.2-3. Each panel shows the relationship between

the extracted variable X and b; the dotted, dashed and solid curves represent the
relations extracted at E/A=100, 250 and 400 MeV respectively.

61

N, and E to provided reasonable information at E/A = 100 MeV. For E/A = 250 MeV,
N, and N1) are the experimental observables available for constructing the impact
parameter.

Figure 4.1.4 shows the measured correlation between the impact parameters
deduced from those observable at the two extreme incident energies, E/A = 100 and 400
MeV, where adequate statistics were available. As expected, N, is strongly correlated to
Np for both incident energies. N, is correlated to Efor E/A = 100 MeV. It is a general
fact that N, is anti-correlated with Zbound for EA = 400 MeV collisions; N, is correlated to
the multiplicity of particles emitted from either the target spectator or the participant
region, while Zbound reflects survive bound protons of the projectile spectator residue.
Previous comparisons of these diflerent observables by Phair et a1. [Phai 92, Phai 93]
suggest a close relationship between the various impact parameter scales with significant
deviations occurring mainly at small and large impact parameters. A significant loss in
sensitivity is also expected whenever the eventwise fluctuations at fixed impact parameters
exceed the variations of the mean value of the relevant observables with the impact
parameter. For simplicity, we chose the reduced impact parameter constructed from N, as
the primary source of the impact parameter selection for the current data analyses. A two-
dimensional cut on two different impact parameter gates has not been used to place more
restrictive gates on the event selection, even though that might lead to greater precision in
the impact parameter selection.

Using theoretical calculations and folding in the experimental acceptance,
the normalized probability distributions for the charge particle multiplicity can be
used to test the reduced impact parameters that were experimentally assigned. For
this purpose, we analyzed QMD molecular dynamics model calculations as if they
were data, and compared the impact parameters deduced from the charged particle
multiplicity to the actual impact parameters where calculations were performed.

The upper panel in Figure 4.1.5 shows the impact parameter dependence of the
mean charged particle multiplicity. The lower panel compares the deduced and
actual impact parameters. Typical FWHM deviations of about 1 fin between the

actual and deduced impact parameters are observed, indicating that reasonably

62

accurate impact parameter selector can be achieved using N, as the relevant

observable.

1.0
0.8
’106
(_Q 0.4
0.2

0.0
1.0

0.8

’1 0.6
v

(,0 0.4

0.2

0.0

63

 

 

  
  
    

 

 

 

 

 

 

l1111l1111l1111l111—1l11 11111111111111111
" c
— L
_ Au + Au __
; E/A=100 MeV /_::_
: ,. , ::
— 1 . ——
p—‘f- ‘—
: a‘ .‘ ::
*1“ l l l
1111 1 1 1111 1111 LL11
I1111|Tr11|1111|11111111J
h—- ’l—i—
— .' -—0— ad
._ O __. _,
— . k—w— _-
P— " . -0— —1
I— .« —0— —1
1 __.... _
hi 0.. . . —1I-— —
— ' - 9.. dr— —1
h- . O ‘ .. .11— _4
. ‘ o 0..
'— I . o O -'_ _1
9° . r _ 4
p— ‘ . —l- —
— . .~O' .‘ —u— —
._ 0" '(-' '0 ...... ..
11111 1111111111111111114+|111|1L1J11111111111111111

 

 

 

 

 

0.0 0.2 0.4 0.6 0.8

1.(D.O 0.2 0.4 0.6 0.8

1.0

MM.)

Fig. 4.1.4 Correlation between the reduced impact parameters reconstructed from the

charged particle multiplicity N, , transverse kinetic energy E, participant proton

multiplicity Np, and the sum charge of spectator nucleon med for E/A=100 and 400 MeV.

Adjacent contours difl‘er by a factors of 5.

64

 

150 1 1 1 1 1 1 1 1 I I

100— 1 —

1.0; i1M :
_ ., _

<NC>
e<>~4

w

1—0—4

 

_ MD _
10——Q 111/1—

<bded>(fm)
K
1'0

 

 

 

Fig 4.1.5 Mean charge particle multiplicities <N,> obtained from the QMD model and
filtered through the experimental acceptance at E/A = 100 and 400 MeV are shown in the
top panel. The mean deduced impact parameter <bd,d> are plotted in bottom panel. The

bow is the impact parameter of which the QIVfl) calculations were performed.

4.2 Onset of Nuclear Vaporization at 197Au + 197Au collisions

The incident energy dependence of multifragmentation in central collisions was
explored for 36Ar + 197Au collisions between E/A = 35 and 110 MeV [deSo 91]. These
investigations reveal fragment multiplicities which increase monotonically with incident
energy. Over a broader range of incident energies, however, models such as the Quantum
Molecular Dynamics model (QMD) predict a maximum in the fragment multiplicity for
central collisions with symmetric system at energies of the order of 100 A MeV [Peil 89],
and decreasing multiplicity thereafter, consistent with the onset of nuclear vaporization
[Cser 86, Bert 83]. Until recently [Peas 94, Tsan 93], the availability of data to test such
predictions was very limited. Measurements of central Au + Au collisions at E/A = 150
and 200 MeV [Doss 87, Alar 92] exhibited multifragmentation, but did not measure the
incident energy dependence of the phenomenon. Measurements of multifragmentation in
peripheral collisions at higher incident energies [Ogil 91, Hube 92] suggested declining
fragment multiplicities with increasing impact parameters, but the measurement lacked
suficient phase space coverage to draw definitive conclusions about the central collisions.

Figure 4.2.1 shows the correlation between <N1MF> the mean IMF multiplicity
measured in the combined array and N, the total charge particle multiplicity detected in the
Miniball/Miniwall array at the three incident energies. The observed dependence of
<N1MF> upon Nc reflects the dependence of both quantities upon the impact parameter.

Figure 4.2.2 shows the mean IMF multiplicities as a function of the reduced impact

parameter 5, obtained using Eq. 4.1.2 for X = N,. At E/A = 100 MeV, <NM> is the
largest for small impact parameters, consistent with increased multifragmentation for
collisions with increased compression and excitation energy. With increasing incident
energy, the multiplicities observed in central collisions decrease, however, reflecting the
onset of nuclear vaporization.

To demonstrate that the decreasing IMF multiplicities observed in central
collisions at E/A = 400 MeV did not arise from detection inefficiencies of the experimental

apparatus, the mean total charge , <Ztot>a is shown in Fig. 4.2.3 for the two extreme

incident energies. At both incident energies, the measured mean total charge is a

monotonic firnction of the charged particle multiplicity. The windows of the MUSIC ion

65

66

 

 

 

 

12 1 1 1 I 1 1 1 I 1 1 1 I 1 1 1 1 1 1
197 197
10 — Au + Au —
3111111
, §§ E/A=100 MeV -
§
_ §§ if i _
A i
Z _ i M.‘ i 250 z
\/ '. I i
4 — o‘o. .0 ii __
0°. ... 1
2 .1" '-.. ‘
_ 1' "'°-. _
O (T 1 1 l 1 Li J 4 1 1 A 1 l 1 l l l
O 20 4O 60 80 100
. No
Figure 1

Fig. 4.2.1 Correlation between <N1Mp> the observed mean fragment multiplicity and N, the
observed multiplicity of charged particles detected in the Miniball/wall. The measured
quantities have not been corrected for the energy and angle dependent detection efficiency

of the experimental apparatus.

67

 

 

 

197 1 7 K
Au + 9 Au 8
E/A= 100 Mev E/A= 250 MeV E/A= 400 no g
I I I l I I 1 I I I I I I
' 2.5 5.0 7.5 10. " 2.6 5.0 7.5 10. "' 2.5 5_o 7.5 10. Ԥ
b (rm) —_ 0 Data —
_, .. --o- QPD .
__--- w/filter __
-a- QMD

 

-- A .- ...— w/filter .

 

 

 

 

 

 

 

 

-- .0 e «- n .
_ 9 O o __
71" . °

C
.. 6Q. . . -
o 0' . ' —

37?? ° . .0
. d9' ' . .'. 0. .
1 . °.° 1 +9” 1 . '1
0.0 0.33 0.67 1.0 0.33 2.67 1.0 0.33 0.67 1.0

b

Fig. 4.2.2 The measured impact parameter dependence of the mean fragment multiplicity
is shown by the solid points. The open circles and open squares depict the unfiltered
predictions of the QPD and QMD models, respectively. The dash-dotted and dashed lines
depict the QPD and QMD calculations, filtered through the experimental acceptance.

68

chamber in the Aladin spectrometer cause a loss in the detection efliciency at E/A = 100

MeV for beam velocity particles emitted to 0lab < 10°. A loss of efficiency also occurs for

heavy target-like residues which do not penetrate the scintillator foils of the phoswich
Miniball/Miniwall detectors and therefore are not identified. Both effects combine to
cause a significantly reduced detection efficiency for peripheral collisions at E/A = 100
Mev. In contrast, the Aladin spectrometer is quite efficient in detecting the beam velocity
particles at E/A = 400 MeV, and the mean total detected charge <Z.,,> is nearly constant
as a firnction of N, at about 102 (z65% of the total charge). More details of this detection
inefficiency will be described in Section 4.3.

Since <Zm> is nearly independent of the impact parameters, the decrease in IMF
production, observed in central collisions at E/A = 400 MeV, is not principally caused by
detection inefficiency. Thus, for 197Au + mAu collisions at incident energies significantly
above E/A = 100 MeV, the fragment multiplicities in the central collisions decrease with
incident energy, consistent with the onset of vaporization in systems that are too highly
excited to produce significant numbers of fragments. Multifragmentation is especially
suppressed for the overheated systems produced in central collisions at E/A = 400 MeV.

For the more weakly excited systems produced in more peripheral collisions,

multifragmentation persists, and large fragment multiplicities, e. g., <N1Mp> z 5-6 for bz
0.67 at E/A= 400 MeV, are observed.

Over much of the incident energy domain spanned in this data analysis, both
multifragmentation and collective flow have been successfirlly modeled for central
collisions via microscopic molecular dynamics models [Peil 89, Boal 88, Peil 92]. It is
interesting to explore whether such models can also describe the decline of
multifragmentation in central collisions. The open squares in Fig. 4.2.2 are the IMF
multiplicities predicted by the quantum molecular dynamics (QMD) model of Ref. [Peil
92]. The open circles in Fig. 4.2.2 are the IMF multiplicities predicted by the quasi-
particle dynamics (QPD) models in Ref. [Boal 88]. Both calculations were plotted as a

firnction of the reduced impact parameter b: b/bm, where bmax was determined from

QPD calculations according to the requirement <Nc(bm)>,b,,w,d = 4. This yielded bmax =

69

 

150 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 I 1

125

100 *—

<Ztot>

 

I 100 MeV

25— , _

 

 

 

Fig. 4.2.3 The mean total charge of all detected particles as a function of the reduced
impact parameter for E/A=100 and 400 MeV. The reduced impact parameter is
reconstructed from the charge particle multiplicity using Eq.4. 1.2.

70

11.1, 11.7 and 11.9 fin at E/A= 100, 250, and 400 MeV, respectively. The actual impact
parameters used in the calculations are given at the top of Figure 4.2.2. Both models
predict enhanced fragment multiplicities for central collisions at E/A = 100 MeV.
However, the models underpredict the measured peak IMF multiplicities, and
underestimate the shift in the peak fragment multiplicity to large impact parameters with
incident energy. These discrepancies are even larger when the calculations are corrected
(filtered) for the detection efficiency; filtered calculations are indicated by the dash and
dash-dotted lines in Figure 4.2.2.

Previously observed failures of the QMD and QPD calculations to reproduce
larger IMF multiplicity observed at lower incident energy [deSo 91, Bowm 91, Tsan 93]
and large impact parameters [Ogil 91] have been attributed to an inadequate treatment of
statistical fluctuations that lead to the decay of highly excited reaction residues [Lync 92].
Such residues are produced at b 2 4 frn in the present QMD and QPD simulations, but are
predicted to decay primarily by nucleon emission [Peil 92], not by fragment emission as
predicted by statistical models [Frie 90, Cser 86, Lync 87, Bond 85, Botv 87, Gros 86].
The suppression of statistical fragment emission in QMD and QPD calculations is not fully
understood, but it may be related to the classical heat capacities [Peil 92, Lync 92, Boal
89], the suppression of Fermi motion [Peil 92], the insuflicient collective motions [Bond
94], or the neglect of quantum fluctuations within the hot residual nuclei [Peil 91], as
modeled therein.

To illustrate such statistical decay effects, we have taken the mass and excitation
energies of fragments produced in the QMD and QPD calculations as the initial conditions
for statistical model calculations, using two different statistical models which both predict
a multifragment decay of sufficiently hot residues at the low density [Frie 90, Bond 85,
Botv 87]. For the QMD model, the decays of all fragments with A > 4 were calculated via
the statistical multifragmentation model (SMM) of Ref. [Bond 85, Botv 87], which
contains a “cracking” phase transition at low density. Input excitation energies and
masses for the SMM calculations were taken from the QMD calculations at an elapsed
reaction time of 200 fm/c. For the QPD model, the decays of bound fragments with A >

20 were calculated via the expanding evaporative source (EES) model of Ref [Frie 90],

71

which describes the evaporative decay of a hot residue expanding self-consistently under
its own thermal pressure. Here, the residue properties are evaluated within 10 fm/c after
the separation of the hot projectile and target-like residues.

The open squares and the circles in Figure 4.2.4(b) are the predictions from the
hybrid QMD-SMM and QPD-EES models, respectively, without correction for the
detection efficiency of the experimental apparatus. The dashed curve shows the QMD-
SMM predictions after the efliciency corrections for the experimental apparatus were
applied. Including the statistical decay of heavy residues increases the peak values for
<N1Mp> in both models to <N1MF> z7-9 for E/A = 400 MeV, and moves the peak to larger
impact parameters, consistent with experimental observations. Both hybrid models
underpredict the IMF multiplicity at the small impact parameters. This reduction is even
more evident in the QMD-SMM model predictions at E/A=100 MeV (see Fig. 4.2.4(a)).
For such collisions, IMF’s are either produced by the QMD model in insufficient
quantities, or are too highly excited to survive the SMM statistical decay in numbers

consistent with the experimental observations.

72

 

197Au + 197Au E
E/A= 100 Mev E/A= 400 nov '
1 1 1 1 1 1 1 1 .1 ¥
10 :_ 2.5 5.0 7.5 10. _-_~_ 2.5 5.0 7.5 10. _« g
- 1111‘; -- ' .
.- . :

LLLLI

   

 

 

 

 

, 0 TE—
. Data ... _
I "0° QPD + EES . I :
2 :- o __.— j
. -B—qrm + 31m 0 .1 .
: —- w/filtor .. I {§.: . :
o__llllllllllJlllLllllllllllTl I_llllll l lllllllllllllllllll—l
0.0 0.2 0,4 0,6 03 1.0 0.0 0.2 0.4 0.6 0.6 1.0

A

b

Fig. 4.2.4 Comparisons with hybrid model calculations at E/A=100 and 400 MeV. The
solid points depict the data. The Open circles and open squares depict the unfiltered
predictions of the QPD-EES and QMD-SMM models, respectively. The dashed lines
depict the QMD-SMM hybrid model calculations, filtered through the experimental

acceptance. The impact parameter scales are identical to those given in Fig.4.2.1.

4.3 Detection Efficiency Correction at E/A = 400 MeV

Fragments emitted from the participant region in collisions at E/A = 250 and 400
MeV can have energies far higher than those energies where fragments penetrate through
the CsI(Tl) crystals of the Miniball/wall array. Such fragments and light particles which
“punch” through the Miniball/wall detectors are often indistinguishable. Similar problems
occur for particles which “punch” through the CsI(Tl) crystals of the Catania hodoscope.
The resulting inefficiency for detection of Ms makes it difficult to precisely assess the
fiagment yields in the regime of nuclear vaporization attained in central collisions at E/A =
400 MeV. For this reaction, efficiency corrections to the fragment yields may be
performed using the mass symmetry of the Au + Au system and the wide acceptance of
the detection array used in this experiment.

Figure 4.3.1 shows the measured rapidity-transverse momentum distribution

 

d d for Lithium fragments emitted at E/A = 400 MeV and reduced impact parameters
Y Pt

in the range 0.43 <6 3 0.75, where the maximum multiplicity of Ms is measured.
Experimental data is shown for the Miniball and ToF wall, where both the energies and the
yields of Lithium ions could be measured with a significant and well understood efliciency.
For reference, the boundaries of the angular and energy acceptances of the Miniball,
Miniwall, Catania hodoscope, and TOP wall are indicated by the individual lines in Figure
4.3.1. Note that the boundary between the Catania hodoscope and Aladin spectrometer,
represented by the cross hatched region in the figure, does not correspond to a fixed value
of 01.1,. The acceptance of low energy ions in the Miniball is strongly influenced near the
ynm= 0 by energy loss in the target. The acceptance of low energy ions at the ynm< 0.3 in
the Aladin spectrometer is also low because such ions can be stopped in the front window
of the TP-MUSIC detector or bent away from the TOP wall by the Aladin dipole magnet.
These detection inefficiencies can be carefiilly examined to obtain efficiency

corrected IMF yields. Since there is mass symmetry in the entrance channel, values for

 

d d at a rapidity y in the center of momentum (c.m.) system is a reflection of the
Y Pt

“mirror” distribution at -y. For example, the distribution for fragments detected but not

73

74

 

   
   
     

400 _ ,

"Au + An; L1
*1/5-400 IN

'0.“ < ’1‘: s 0.75

Miniball
"" / Minilall

I Hodoscope

300 —

 

I
I Hodo 8: TOP

A
Q
>
§ - 4
v 200 — —
i - - .
02' : “°' ' '_ ' ,1 i ,/0 nu :
100 — f7 . ‘ ... ... _
- ~ . ' Q .0 . -

 

 

Figure 4.3.1 The y-P, scattered distributions for Li ions at near mid-central collisions are
plotted as a function of normalized rapidity y/ylmm (yum). The distributions at the polar
angles 5°S IBM, | s 25°are not included where energy calibrations were lacking. The
distributions are also strongly shielded by the target at the target-rapidities. Also, the

Ms with small P: and ynom<.5 will not be detected by the TOP wall.

75

calibrated at positive rapidities in the Catania array can be determined from the distribution
measured by the Miniball array. Low energy fragments stopped in the target can be
determined from the distribution measured with the TOP wall. Near the cm. rapidity,
however, there is a gap in the acceptance of the array. Emission into this region was

estimated by methods described below.

Reconstruction of the Emission Pattern

Corrections for the inefficiency of the experimental array were assessed by
simulating the fragment emission patterns and projecting them upon the experimental
acceptance. In the limit of a “two step” model for peripheral collisions, where fragments
are emitted via the statistical decay of a hot prefragment, the p. distributions of emitted
fi'agments are approximately independent of rapidity. This suggests a factorization ansatz

i = P1(pt)-P2(y), where P1 can be determined at rapidity values where the

dydp.
experimental acceptance is relatively complete, and P2 must be determined from the
measured dependence upon rapidity, as described below.

Assuming the central limit theorem is applicable, P1 was calculated by assuming
Gaussian distributions in px and py. The Box-Muller method [Pres 86] was used to
generate Gaussian distributed values for p, and py. It works as follows: If x1 and x2 are
random numbers with a joint probability distribution P(x1, x2), and y; and y; are each

functions of XI and x;, then the joint probability distribution for y1 and Y2 is

 

 

x , x
P01,Y2)dY1dY2 = P(X1,X2)MdY1dY2 (4-3-1)
6(Y1, Y2)
where (XXI’ x2) is the Jacobian determinant for the coordinate transformation

 

 

5(Y 1, Y 2 )
x1, x2 :> yl, Y2. To obtain any arbitrary probability distribution P(y1, yz) starting from a
given probability distribution P(x1, x2), it suffices to find the appropriate coordinate

transformation that gives the correct Jacobian determinant. In this case, one wants to

76

obtain the two-dimension Gaussian (normal) distribution

2 2 2 2
e'Yl ”"1 e'Y2 ”“2 dyldy2 (4.3.2)

 

P01,Y2)dY1dY2 =
21:0102

starting from the unit random number distribution,
P(x1, x2) = l for 0 < xis 1; i= 1, 2 (4.3.3)
= 0 otherwise.

By direction substitution, it can be shown that the transformation,

yl /01 = ,/—21nx1 cos(21rx2) (4.3.4)
yz /02 = ./—21nxlsin(21cx2) (43,5)

satisfies Eq. 4.3.1. Physically, 27tx2 corresponds to the angle between yl and y; (px and p3,
in our case), and x1 comes from a mapping of the angle integrated Gaussian distribution
for p, onto a unit probability distribution in X}, where x]: 0 corresponds to pt= 00, and
x1= 1 corresponds to pt: 0 . In cases where speed is of essential importance, it is
sometimes useful to define additional random variables v1 and v; as the coordinate and

abscissa of the random points inside the unit circle. Then, the sum of their squares, R E

v.2 + v: can be used for x1, and sin(21tx2 ) and cos(21tx2 ) are given by vl / JR and
v2 / x/E , respectively.

The choice of independent width parameters 0,, and CY allows for a two component
fit to the angle integrated pt distribution, as well as the ability to describe possible
anisotropies in the experimental azimuthal distributions about the beam axis. The result of
a simulation with two different width parameters ox and cry is shown in Figure 4.3.2. The
individual px and py distributions are shown in the two bottom panels in the figure. The
total pt distribution is shown in the top panel. Even with 105 events, these simulations
closely follow the corresponding Gaussian distributions calculated analytically (solid lines).
Thus, the Box-Muller transformation method provides for efficient and accurate Gaussian

random number generation.

77

 

[Lilillllilllllllllillll

   

 

 

 

 

 

 

 

 

Figure 4.3.2 The Gaussian sources obtained from the Box-Muller method with different
variances in x and y directions are plotted as open circles in the two bottom panels. The
vertically dashed lines indicate the width of the OS. The total transverse moment
distribution is plotted as the open circles in the top panel with the vertical dashed lines
indicating the as in the x and y directions. The solid lines in all the panels are calculated

analytically.

78

Experimental y-P. Cross Sections

Experimental rapidity-transverse momentum distributions obtained by using only
the data from the well calibrated Miniball and TOP devices. Angles between 5° to 10° in
the TOP wall were not used, because one could not obtain the full azimuthal distribution at
these polar angles. Smooth rapidity and transverse momentum distributions were created
by randomly reassigning the emission angles of particles detected in specific detector of
the Miniball to spread them uniformly over the solid angle of that detector. Furthermore, a
correction factor is applied for each polar angle in the Miniball, to compensate for the loss
of efficiency that occurs whenever there are detectors in the Miniball that malfunction.

The spectra at the projectile and target rapidity were combined to compensate for
the deficiencies of each and thereby obtain a uniform and wide dynamic range pt spectrum.
The distributions in the Miniball at low pt were replaced by distributions measured in the
TOP wall at yum > 0.5 and 01.1, < 5° [Kund 94], to avoid complications due to the
nonlinear response in the CsI(Tl) detectors in the Miniball at low energies. This provided
continuous p. distribution at target rapidities ranging from pt z 0 up to the “punch through”
points where particles penetrate through the CsI(Tl) detectors of the Miniball. This
distribution can be reflected about the cm. rapidity to obtain the corresponding
distributions near the projectile rapidity. The combined distribution for Lithium particles
emitted at E/A = 400 MeV and peripheral impact parameters .67<b S .78, is shown in the
left panel in Figure 4.3.3. It should be noted that this distribution indicates only those
portions of the experimental acceptance where accurate energy determinations could be
made. The experimental fragment multiplicity distributions in Fig. 4.2.1 also included the
yields of many fragments at mid-rapidity, ynorm z 0.5, for which accurate energy
calibrations were not available.

At both peripheral and mid-central collisions, projectile- and target-like spectator

sources dominate the observed IMF distributions. The measured distributions were

 

modeled using the factorization ansatz = P1(pt)-P2(y), where

t

79

 

I'll'llII'_IIII‘IIII.IIIII lll'll'l'l'llll'lll'llll lllI'lllllillI'llvvrvlll

- . .'., . Au -l_-'..Au; Li .. I
_ y '_ I. z/A __.;400 my " '
- "r. " 0.07 <3: 5 0.7.3 l. .

.L'i.‘
fail".- -.';'-. 3'." ‘
-g'. '.':~ .

.'i

     

Pt/A (MeV/c)

 

 

 

 

Figure 4.3.3 The distribution of Li fragments as a function of y and p, are plotted for
events with impact parameters of .67<b 3.78. The experimental cross sections are plotted
in left panel. The simulation with the detection acceptance correction is plotted in middle

panel. The simulation without the experimental acceptance cut is shown in the right panel.

80

2 2 2 2
1 e-(Px/2ox-l-Py/2oy)

P:(pt) = Px(px) -Py(py) = (4.3.6)

oxoy
and 6,. and cry are determined at the values of .19 Synms .29 where the coverage in p is
nearly complete from threshold to the punch-through points. The resulting distributions
are somewhat wider than those observed at the ynoun s .19, but that was irrelevant because
the coverage was complete at smaller ynomrl values. Measured transverse momentum
distributions of those modeled using Eq. 4.3.6 and corrected for the experimental
acceptance, are plotted for different rapidity bins as the solid points and dashed lines in
Figure 4.3.4. Here, P2(y) was determined by requiring the same number of simulated
counts in each rapidity bin, as observed in the experimental data. Applying this procedure
at all measured rapidities, one obtains the values for P2(y) shown in Figure 4.3.5. The
correction for the experimental acceptance can be obtained by comparing those
simulations with the acceptance corrections to those without the corrections. Calculations
without the acceptance correction are shown by the solid lines in Figure 4.3.4. Similarly,

Figure 4.3.3 compares the simulations with (center panel) and without (right panel)

 

acceptance corrections to the measured distribution for peripheral collisions at .67

dydp.
<6 3.78 and E/A = 400 MeV.

Contributions to the measured yield at central rapidities ynmz 0.5 at large impact
parameters were determined by comparing simulated to measured distributions at smaller
pt. This provided a lower limit for the fragment yields. The low acceptance for the
experimental array at mid-rapidities has the consequence of large uncertainties in the
deduced contributions at mid-rapidities, however. At large impact parameters, the yields
at mid-rapidity are comparatively small and the interpolation leads to relatively small
uncertainties as shown in Fig.4.3.4. At small impact parameters, this method can be
applied. Its accuracy is more uncertain because it inadequately constrains the interpolation
to mid-rapidities of the yields measured accurately at both larger and smaller rapidities,
and because these mid-rapidity yields are such a large fraction of the total yield for these

central collisions.

81

 

  
    
   

     
  

  
    

        

 

   
     
 

 

 

 

 

 

.‘ 50 I I I I LII I I I I I IT TI II_0_I I I I IITII I I I I I I I I I_‘I ITTTI I Fr I I I I I TIT
+ .13§yn°m§—.0 T 035%,”,st -_ .135): msza 3
>— -0— —‘P- -I
__ ___— /\ ___—— —
40: AU + AU 2- ’ .67 < b g .78 a -
: E/A = 400 MeV :: 0 Li 3: 1
Z I: 0 I: I
A 20 E— —— —— —
<1: ' 0 ‘3 1” ‘
P —4b— r— -‘
b— —0— —o— —:
\ .— . —0— .4»— _4
7’ 10 — y if’ —:
DH -<»- . —u- —4
v .... . — _
U10 I—ilAifipiiiLliiirilp 41:11:! 7
>3 ,— I Ij I I I_”_I I I I I I I VI I I I I I I I I___I I I I l .4
E : .195ynm529 :: .29synms.39 :: .assymsAs I
8 _— __— _———__ _.
fl. ” *" ‘* ‘
N — -1- ~- 1
>— —-v— -I
L— — —o-— —
g I: 3: Z
r- —4r- —‘>- -
4 ___— —I_ T
4. .... -
2 ___ ___:— *t
_ 4— _
_ J ._ _
_ I I _ ‘ ”I o 0 115 I ii I _

l l l l l l l l l l 1 L1 L1 1 l l l l l

 

o
0—5
0
N
.0
w
o
A

.O 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0
Pt/A (GeV—C)

00

Figure 4.3.4 The transverse momentum cross sections of Li fragments are plotted as

solid circles for the data at the impact parameter of .67 <5 3 .7 8. The dashed lines are
the simulations filtered through the acceptance cut. The solid lines are the results

without the acceptance cut.

82

 

 

 

 

0.04 I I I I I I I I I I I liI jj I I I I
E/A = 400 MeV _
” Li; .6? < b g .78 r
003 ~— —
L _
”E L 2
{I
3) 002 — ——-
CL.
0.01 -— —
0.00 l L l i i L l i L L l l l l l L
—0.5 0.0 0.5 1.0 1.5

y/yb(Lab)

Figure 4.3.5 The extracted probability distribution was plotted as a function of normalized

rapidity for Li fragments at impact parameters .67 <5 3 .7 8. This probability distribution

was folded into the final simulations of the transverse momentum-rapidity distribution.

83

For efficiency determination at smaller impact parameters, an alternative
extrapolation to mid-rapidity was obtained by using the parameterization of a single
participant source. In this procedure, the contributions from the spectators are obtained at
y“mm less than .19 and contributions at ymm. greater than .19 are determined fi'om the decay
of a single participant source centered at mid-rapidity given by

M = const. P(pt) e'mg /2mT) El, (4.3.8)

dPtdY dy
Here, T represents the temperature of the source emitting the fragment. P03.) is the

probability distribution given by Eq. 4.3.6 where the width 6,, and the ratio of 0,, /try are

 

determined by fitting the distribution at laboratory values of the rapidity yum z

dydp.

0.19-0.29 where the contributions are dominated by the participant source. By assuming
that the distribution along P2 to be governed by an effective temperature determined from
the P. distribution of yum ~0.19-0.29, a temperature T = 210 MeV was assigned and the
yields evaluated according Eq. 4.3.8.

The largest yields extrapolated from this thermal source with T = 150 and 210
MeV were used to provide an upper limit of estimation of the fragment yields. The
Miniball/Miniwall gates are drawn rather tightly about the Z = 3 lines to exclude the a
particles. This combined with nonlinearities in the Miniball/Miniwall energy calibrations
may conspired to reduce the yield in the Miniball at law energies. To provide an upper
limit for potential losses of low energy fragments in the Miniball, an upper limit of 50%
loss in IMF yields at E/A = 7 MeV was assumed. The filtered simulations for this upper
limit are plotted as doted lines in Figures 4.3.6 for central collisions with . The unfiltered
simulations are plotted as the dotdashed lines. Since energy spectra are not available for
the Miniball detectors beyond Z = 10, simulations were only performed up to Z = 10. For
charges fi'om Z = 11 to 30, the correction to the fragment yield is taken from the
correction factor for Z =10.

To see how well this simulation reproduces the fragment yields presented in
section 4.2, the mean numbers of detected MS in the various devices are plotted as a

fianction of the reduced impact parameter in the upper panel of Figure 4.3.7. The vertical

dZP/dydan/A)

Figure 4.3.6 The transverse momentum distribution for Li fragments are plotted as the
solid circles for the experimental data at impact parameters 13 s .34. The dashed lines are
the results of simulation which the filtered yields are normalized with the measured data at
all rapidities. The dotted lines are the results of simulation for which the normalization
was taking from the thermal source. The solid and dotdashed lines are the corresponding

15

10

0

...
0|

...
O

 

IIIIIIIITIFIIIIIIIIITrTIIT

TII

UTII

IIIIIIIITIIIIIIIIIIIIIII

-— -.1asy__s-.oe ‘1— cosy-.543
’ Au + Au "" $5.34
— E/A = 400 Mev ——+ Li
: :: H
L _" i
:5 i" a
C o - ’1
_ o I 9

 

d II-

0 -
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.
Pt/A

 

165264)

—

l l l I L.J l l |.i l l l

—

IIIIITIIUIIIIIIIIIIIIIll'

.1asy_s.aa

llll

I l l l 1 l l l l I I

 

 

 

  

du-
GP

"1111

 

3 04 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6

results without the experimental acceptance cut.

85

 

q

          
   

I T

: I I :
-_ | l l _-
- I g g I I ~
I I i I i I I I
4.— | 5 III I ‘2
: I i I I II I ::
3.— I. I I I. I ‘j
: ’.'-—-‘-.\-. x I I :
2 :- u } '.° 0 .Poo't.\l\ l I f
: q. C I O I I j
11". I I I 1 .m

11

 
 

Ill

<Nmp>
0.2}
_ .. r
9

I

V

    
    

 

 

 

I I I I 3
I I I I -
5 -_ I I I I I 7
I I ."I ' I I I
4 C‘ I _.'l I I —.‘
L" I ' I '-I I 2
3 _— | . I '- I 1
; .I" -. I :
2r J i
_ - ' .\ -. -I
1 _— ‘I'xi, '1
- I I I I I). 4 -
0 I. l l l l i L l I ll 1 L I II I I l L I 1 ll 1 I.”l J
0.0 0.2 0.4 0.8 0.8 1.0

b

Figure 4.3.7 The measured mean IMFs as a function of the reduced impact parameter with
the various device acceptances at E/A = 400 MeV. The solid circle points are the number
of IMFs detected in the Miniball array only. The dot points are added with the Miniwall
array. Adding in the Catania hodoscope is presented by the dashed line. The solid line
further add in the IMFs detected with the polar angle 19...,I 5 5° in the ToF wall. The
highest open circle points are the total number of detected IMFs within the combined
devices. The vertical lines indicate the impact parameter ranges for the y--Pt simulations.

The simulations filtered through the devices are plotted in bottom panel.

86

lines represent boundaries of impact parameter gates for the simulations. Within these
boundaries, the efficiency was determined and interpolated linearly between the centroids
of each gate. The corresponding simulated results are shown in the lower panel of the
figure.

A comparison of the measured fragment multiplicities (solid points) to the
efficiency corrected fragment yields (Bars) is provided in Figure 4.3.8. The bars are
bounded at the lower part by the procedure where the filtered yield are normalized to the
measured data at all rapidities. The upper limits of the bars are the upper limit of yields
extracted as described in previous paragraph. The efficiency corrected multiplicities are
about 50% larger than the ones measured for central collisions, and nearly a factor of two
for mid impact parameter collisions, where they reach their maximum values. These
simulations confirm that the observations of nuclear vaporization were not simply due to

problems with the experimental acceptance at mid-rapidities.

87

 

10 —' 0 Data

 
 
  

'§ W/corr‘ectlon

 

 

 

Figure 4.3.8 The mean number of Ms was plotted as a firnction of the reduced impact
parameters. The solid circles are the experimental measurements. The bars are the range of
values determined by the efficient calculations. These bars reflect the range of efficiency
corrected values that would be allowed considering the uncertainties of extrapolating into

the region at mid-rapidity, where the array provided little energy information.

4.4 Reaction Dynamics

In this section, the mean transverse energies of emitted fragments are compared to
the theoretical predictions. The impact parameter and incident energy dependence of the
energy spectra and triple differential cross sections are compared to moving source
calculations to search for evidence of collective motion, and determine the extent to which
such collective motion influences the fragment observables. The details of the radial

expansion will be discussed in the next section.

4.4.1 Transverse Energies of Emitted Fragments

Even though the mean number of the intermediate mass fiagments (MS) (as a
function of impact parameter) were not well-reproduced by the QMD or QPD calculations
for these Au on Au collisions, it is still relevant to ask whether the collective and thermal
motions predicted by these models are comparable to the experimental observations. Due
to the large amounts of time needed to perform such calculations, it is very difiicult to
generate comprehensive comparisons to measured energy spectra. However, one can

make quantitative comparisons to the experimental mean transverse energies defined by

< Iat >= 2 Eisinzei x P(Ei,0i) / 21>(Ei,ei) (4.4. 1)
i i

where Bi, 0. and P(EI, Or) are the energy, the polar angle and the probability of the
detected fragment. Collective and thermal motions contribute differently to the transverse
energies; to first order, the thermal contributions reflect the local temperature and are
independent of mass, while the collective contributions scale linearly with the fragment
mass. The mean transverse energies for the emitted fragments were evaluated between
25.°.<_ 61.15160.o where the phase space is covered by the Miniball detectors and good

energy calibrations were obtained at all incident energies.

The measured mean transverse energies for or, Li and C fragments are plotted as a
function of the reduced impact parameter in Fig. 4.4.1 for collisions at E/A = 100, 250 and
400 MeV. The error bars reflect a 8% systemic uncertainty in the energy calibration. At

88

89

 

 

 

200 ITIIIITIIIIIIIIIITWIIIII’IIIIIIIWIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
_ E/A = 100 MeV I 250 MeV __ 400 MeV .
_ 25.°§B,,b§160.° «a
_ D C ..
150 . Li
_ O a -_
9 - -.
Q, _ l -_
100 —
/\ _ I l a
La r i T r
V ~ \¥T “"
50 __ T ——

 

 

 

 

 

LillLlJJLLliUELLiLIJZLillllllllliLlllilllllllllillllillullJ llllLLlJLllIlllllllllilllIll

O
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.81.0

l)

 

Figure 4.41 Mean transverse energies plotted as a function of the reduced impact
parameters for or, Lithium, and Carbon fragments at 25°S 0 m, $160°. The three panels
correspond to the collisions at E/A = 100(lefl panel) , 250 (middle panel) and 400 (right
panel) MeV.

90

small impact parameter, these error bars are enlarged to reflect the effects of coincidence
summing which are estimated by adding randomly signals from different events according
the observed single hit probabilities. The observed mean transverse energies for the
emitted fragments generally increase with a decreasing impact parameter. Because many
particles punch through the detectors, the E values for at particles can not increase as
much as those of the heavier fragments with increasing centrality or incident energy or
both. In general, heavier fragments display larger values of E, consistent with the
existence of a significant collective motion in the system at breakup. The mean transverse
energies of Carbon fragments in central collisions at E/A = 400 MeV are less than those
for Lithium fragments suggesting that the heavier fragments do not participate as fully in

the collective motion as the lighter fragments do.

To study the dependence on the mass of fragment in greater detail, the mean
transverse energies are plotted as a function of the fi'agment charge in Fig. 4.4.2. Due to a

lack of statistics from model calculations, comparisons with the QMD and QMD + SMM
models are only shown for two impact parameter regions: 5 $0.34 (central, upper panels)

and 0.45313 <0.75 (mid impact parameters, lower panels). This mid impact parameter
range was chosen to bracket the impact parameter range corresponding to the maximum
number of fragments produced in collisions at E/A = 400 MeV. The experimental data is
plotted as open and closed circles respectively. The solid and dashed lines represent the
QMD and QMD + SMM simulations which have been filtered through the experimental
acceptance. The QMD calculations generally underpredict the mean transverse energies in
these collisions, perhaps reflecting the absence of Fermi motion in the original projectile
and target nuclei as modeled by the QMD model. By including the statistical decay of hot
fragments and residues, as in QMD + SMM simulations, the mean fragment transverse
energies are further reduced. As shown in Fig. 4.4.2, including the statistical decay of the
spectator residues and hot fragments via the SMM model, the observed mean number of
IMF’s can be reproduced for these mid central collisions at E/A = 400 MeV. This might
indicate that the MS were produced from the decay of the colder thermalized spectator

[Kund 95, Moll 95]. However, the underpredictions of the mean transverse energies by

200

150

100

50

<Et> (MeV)

100

50

Figure 4.4.2 Mean transverse energies are plotted as a filnction of the fragment charge for

b 30.34 (top panels) and .4535 <.75 (bottom panels) at E/A =100, 250 and 400 MeV

91

 

 

 

 

 

 

 

h IIIIITITIIIIIIIIII-«I—IITTIITITIIITII[III—r—IIIIITTTIIIIIIIIII d
" E/A = 100 MeV _"_ 250 MeV _j“_ 400 MeV _‘
F 25.°s9,_,g160.° ~e— QMD ~— ~
_ O gs 34 :— QMD+SMM § I (I I
'- _. A “— § ‘9‘ “I
~—— 0 .45Sb<.75 -—_— (I __ {i _
I (i) f § 1: Z
: {5 L : (I :
_ é _ ;_ _
e t? e —— «
I- I— —-I7— ..
: / : :: 9— \ ,. :
_. / __. _Z___ \ / __I
I I I
Fl 1 l l I l lJ_l I l l l l I I l l l ljhl l l l I l l l l I l l l l I l l l 1“
IIIIIIIIIIIIII jIII IIIIIIIIIIIIIIIIIII
_ _p _
_ .- 1 9 i «I :
: g i i ‘1 :: :
T .-_/\Q ib— .\ / ‘ j
,_ --.. —./ —I~— \ v -4
I— —1'— —4
L-1L114Lllllillllll_LJ_l llll—‘bllllllelllLlLllllllq
O 2 4 6 0 2 4 6 B

 

incident energies. The QMD and QMD + SMM calculations, filtered through the

experimental acceptance, are plotted as solid and dashed lines, respectively.

92

the QMD + SMM model indicates that many of the MS are not produced by the slow
thermalized decay processes described by the SMM model. This discrepancy is largest for
Z = 4 and decreases slightly by Z = 6. It has been reported that the energies of Ms with
Z > 10 are consistent with the thermal breakup of a projectile residue at a temperature of
about 20 MeV [Lind 93]. How one reconciles these observations with the transverse

energies in Fig. 4.4.2 is not clear at present.

4.4.2 Three-Moving-Source Model

With the failure of the QMD and QPD models which were specially developed to
explain the dynamics of fragment emissions, it is appropriate to perform more systematic
analyses of the fragment energy spectra and triple differential cross sections. For this
purpose, schematic moving-source models [Chit 86, Phai 92, Bowm 92] are ofien useful.
One often assume three sources: a projectile-like, a target-like and a mid-rapidity source.
For example, three source parameterizations have been successfillly used to describe mass
asymmetric heavy ion collisions at low energies [Phai 92]. Assuming a mass symmetric
entrance channel, and allowing for some transverse deflection of the projectile— and target-

like spectators sources, one can write

 

 

d2P 3 dZP, d2P1 2" 3 d2p. ..
= = + d ——-(l , xV' , 4.4.2
dEdQ Edam dEdQ I ¢ngdQ p V1 1) ( )

where each of these sources is centered at a velocity V, and characterized by a
temperature Ti. E and 'p denote the kinetic energy and momentum of the emitted particle,

Vi denotes the effective Coulomb barrier of the ith source, and (lip denotes the azimuthal
angle of the reaction plane. Source 1 represents the participant and sources 2 and 3

represent the target and projectile-like spectators moving with a velocity vi with both

longitudinal and transverse components. The integration over (bu is appropriate for

analyses in which the reaction plane has not been defined.

93

In Eq. 4.4.2, dPi / dEdQ(p,0, Vi) is defined in the rest frame of the source (vi 2 0) by

dPi

dEdQ 05,0, Vi) = ai®(E - VIXE + 11102 — Vi)

 

 

E-Vi

 

x\/(E + mc2 — Vi)2 — mzc4 x exp(- ), (4-4-3)

i
and dPi / dEdQ(p, Vi, Vi) is obtained from Eq. 4.4.3 by Lorentz transformation. In Eq.

4.43, a,- is a normalization constant, T, is a temperature parameter, and @(E-Vi ) is the
unit step function. In the following analyses, the mass mistaken for simplicity from that
of the most abundant natural isotope, e. g. A =11 for Boron. Because of the mass

symmetry of the entrance channel, V2 =-v3, a3 = a2, and T3 = T; are stringently required

in the CM frame for the spectator sources.

4.4.3 Differential Cross Sections of Fragment

Energy Spectra of Fragments

To illustrate the impact parameter dependencies of the IMF energy spectra, the
spectra of Lithium and Boron fragments were created for three different impact parameter
bins at E/A = 100 MeV, as shown in Figure 4.4.3. The solid lines are the sum of
contributions from the three sources, and the dashed lines are the contributions from the
participant source alone. The corresponding parameters for the fits are listed in table
4.4.1. The cross sections at low energies are sensitive to the decay of the target-like
residues formed in peripheral collisions, as shown in the top panels of each column. As
the impact parameter is decreased, the spectra become flatter, reflecting larger yields at
higher energies. This increased yield is due to a bigger contribution fi'om the participant
source. This source, however, is not well described by the fits at 01,1, S 45 MeV and E/A >
40 MeV. Instead, they display a shoulder that is significantly underpredicted by the
calculations. This shoulder is evidence for a collective expansion of the system which will

be discussed in more detail in section 4.5.

94

 

?
a
_I
-I

I I I Y 7 1' '1?‘ l' 1' Y I I I Y I l' I 1' T I T T V 71 Y I I T
-I>

Au + 197Au :_ E/A = lOOMeV

 

 

 

dEP/deEfinb sr'1 MeV—1)

 

 

l l l l

 

 

200 400 600 800

E (MeV)

Figure 4.4.3 The energy spectra of Lithium and Boron emitted in the Au + Au collisions at
E/A=100 MeV were plotted for 0hb=28°, 355°, 45°, 575°, 725°, 90° and 110° for the
peripheral (top panel), mid-central (middle panel) and central (bottom panel) collisions.
The solid lines are the sum of contributions of three moving sources, and the dashed lines
are the contributions from the participant source alone. The corresponding parameters are

listed in Table 4.4.1.

95

Table 4.4.1 Parameters of three-source fits for the Lithium and Boron fragments at
E/A=100 MeV and five-source fits for the Lithium and Beryllium fragments at E/AMOO

MeV for different centralities listed in this table. (The units for a;, Ti and Vi are MeV'3 sr'l,

MeV and MeV respectively.) Source 1 represents the participant source which moves

with the velocity of the center of mass along the beam direction. Source 2 represents the

decays of target-like and two projectile-like spectators which move with the vx and v2 at

the transverse and longitudinal directions in the center mass frame. Source 3 designates

two sources with velocities midway between the center of mass and the spectator sources.

 

 

E/A Z
100 Li

Li

Li

100 B

400 Li

Li

Li

400 Be

Be

Be

> .067

.34-.67

_<_ .34

> .067

.34-.67

S .34

> .067

.43-.74

S .34

> .067

.43-.74

S .34

SOUTCC

H

WN—iWNI—IWN—‘WNI—‘WN—iwmI—INI—INt—INI—iNF—‘Nh—IN

3i
9.2E-11
3.7E-10
3.5E-10
5.9E-10
6.9E-10
4.5E-10
6.6E-12
5.2E-11
3.6E-11
7.6E-11
9.1E-11
6.5E-11
5.8E-12
1.2E-10
2.5E-11
1.2E-10
3.0E-10
2.3E-11
6.8E-10
2.2E-11
2.2E-11
1.8E—12
3.9E-11
1.7E-14
1.5E-11
1.0E-1O
1.5E-13
7.9E-11
3.1E-11
3.3E-13

Ti
31.9
20.6
48.5
32.7
58.7
29.6
35.6
23.4
61.3
39.6
70.1
35.5
60.0
39.6
60.0
60.0
53.4
60.0
60.0
34.9
60.0
60.0
30.3
60.0
60.0
49.1
60.0
60.0
46.2
60.0

Vx,i/ C

0.
.02
O.
.02
O.
.06
0.
.02
O.
.02
0.
.06
0.
O3

.015

O

.087
.044

.15
.08

.06
.03

.068
.034

.10
.05

VLi/C

0.

.155

0

.136

O

.132

O

.162

0

.134

O

.120

0

.35
.17

O

.35
.17

O

.35
.17

O

.35
.17

0

.35
.17

O

.35
.17

Vi
22.1
9.12
22.1
9.12
22.1
9.12
35.0
13.0
35.0
13.0
35.0
13.0

0.
9.12
9.12

0.
9.12
9.12

0.
9.12
9.12

0.
11.3
11.3

0.
11.3
11.3

0.
11.3
11.3

Bap/c

 

 

96

The energy spectra for Li and Be emitted in the Au + Au collisions at E/A = 400
MeV are shown in Figure 4.4.4 for three different reduced impact parameters. In contrast
to those E/A = 100 MeV, the angular dependence of the energy spectra for peripheral and
mid impact parameter collisions is weak; these spectra would overlap the most forward
angles, 28°, 35.5° and 45°, if the energy spectra were not multiplied by factors of 8, 4, and
2 respectively. These overlapping spectra suggest a strong transverse source velocity.
Such a hypothesis is explored below in conjunction with the triple differential cross section
data. The solid and dashed lines are the best fits obtained with the spectators and
participant component of a five moving model. Here, the source parameter involves a
projectile- and target-like spectators source, a participant source moving with GM.
system and additional sources with velocities halfway between the participant and the
spectator sources and a temperature sets equal to that of the participant source. For the
participant source, a non-zero radial expansion velocity was assumed (see section 4.5), but

it had no effect on the quality of the fits.

Triple Differential Cross Section of Emitted Fragment

To explore azimuthal angle dependence, the triple cross sections must be
extracted. It requires determination of reaction plane for each event. Such reaction planes

can be reconstructed with the transverse moment vector Q. [Dani 85] defined by :

Q = ZwiPh, (4.4.6)
with w; = +1 (-1) for particles with rapidity y > ycm ( y < ya“). Only particles with well-
calibrated energies were included in the reaction plane reconstruction, thus excluding
particles emitted at 01,1, < 25°. Since punch through particles did not have well-defined

energies, they were also not included in the reaction plane reconstruction.

The component of the 6 vector transverse to the beam axis is used to define the

reaction plane, ¢R=O°. Due to a lack of statistics, only three broad azimuthal gates (1b = 0°-
600, 600-1200, 1200-1800) could be used to explore the triple differential cross-sections.

The triple differential cross sections for the Lithium fragments, subject to a gate at the

97

 

I I II I IIfiTT I I I II I l I.‘ IITTIIIIIIIIIIIITITTer

11.11 1 l

 

 

 

 

  

7 I
10 2 t 197AU + ‘9 A E/A = 400MeV
_. a; 5
3 :c&\ L1 :0 A j
10‘ _— o. are" Be; .67 < b S 1. a
E i :3” \ i
/"\
H .—
| 3
>
(D 4
H
I .
L.
m
.0 I
m ;
"U
C3 :
”U 'r.
04
N .
"U
=iIII \
‘hl 1 1 fl 11 1 A l l l l 1411 1‘

 

 

 

 

0 200 400 600 80(1) 200 400 600 800

E (MeV)

Figure 4.4.4 The energy spectra of Lithium and Beryllium fragments emitted in Au + Au
collisions at E/A = 400 MeV. See the caption of Fig. 4.4.3 for details. The cross sections
of 0hb=28°, 35.5° and 45°are multiplied by factor of 8, 4, and 2 respectively. The

corresponding parameters are listed in Table 4.4.1.

98

mid-impact parameters which brackets the maximum in the fragment multiplicity at E/A =
400 MeV, are plotted in Figure 4.4.5. Since the number of Ms is small at GM, > 100° ,
the triple differential cross sections are only plotted for first six rings of the Miniball array.
A strong enhancement of the differential cross-sections at 4) = 1200-1800 is observed. This
enhancement increases with the particle energy and polar angle. This observation is
consistent with a significant transverse flow [Gutb 89 a&b, St0c 82, Bert 87, Kamp 93].

The corresponding parameters for the fits are listed in the table 4.4.2. The lines
present the contributions from the target-like spectator. The triple differential cross
sections for fragments with Z = 3, 4 and 5 can be reasonably well described by assuming
that the target-like spectator has significant transverse velocity 0.08-0.05 and a
temperature around 40-50 MeV (as listed in table 4.4.2). The heavier fragments move
with a smaller transverse velocities, consistent with the trend displayed by the mean
transverse energies as shown in the Fig. 4.4. 1. The trends are shown in Fig. 4.4.5b.
Recent analyses suggest that the spectra of heavier fragments emitted for this impact
parameter range are relatively insensitive to incident energy, consistent with the
assumption of limiting fragmentation. Also, show the temperature parameter for heavier
fragments 7 < Z < 20 extracted for Au + Pb collision at E/A = 600 MeV [Lind 93]. This
comparison indicates that the lighter fragments are indeed more sensitive than the heavier
fragments to the dynamics of the initial energy deposition to the spectator residue.

The corresponding triple differential cross sections for Lithium fragments emitted
in central E/A = 100 MeV collisions are plotted in Fig. 4.4.6. The lines represent the sum
of contributions of the three sources. The emission is relatively isotropic with respect to
the azimuthal direction around the mid rapidity. The anisotropy of the emission patterns
around the projectile and target rapidity at these collisions is much smaller than that
observed at E/A = 400 MeV. The corresponding parameters of these triple differential

cross section fits using the three source model are listed in the table 4.4.2.

Fig. 4.4.7 shows some of sensitivities of the fitting procedure to the transverse
velocity of the spectator sources. The solid points in Fig 4.4.7 are obtained by allowing

the longitudinal velocity and the temperature of spectator sources to be free fit

99

 

_lllllllllllllllllll lllllllTTITllTIllIl llIT TlllIIllIlllTT

p—L

O

I
N

D—i

O

I
(.0

   

 

   

     

10—4
\
10—5 \
—... 72.50 m 90.0
\ E/A = 200 MeV .. 0120_180
.. .42 < b g .74 0 60—120

0- 60
Li

h-I

O

I
(J

   

dZP/dEdmmb MeV—1 sr“)
°.

H
C
|

A

   

p—I

O

l
U‘

0»
i \ ll 1]
lll>lllll lllllll llllLLJllllLlle lIELJlllllll

O 200 400 600 0 200 400 600 0 200 400 600 800

E (MeV)

Figure 4.4.5 The triple differential cross sections of Lithium fragments emitted with the

impact parameters .42<5 s .74 for Au + Au collisions at E/A = 400 MeV. Each panel
represents the triple difl‘erential cross sections at each polar angle as indicated. The lines

are the contributions of the target-like spectator.

100

Table 4.4.2 Parameters of moving-source fits to the triple differential cross sections for
Lithium, Beryllium and Boron fragments at E/A= 400 and Lithium fragments at 100 MeV

collisions. The units for a;, Ti and Vi are Mev'3 sr'l, MeV and MeV respectively.

 

E/A b Z s a; T; va/c vu/c Vi Bap/C
400 42-74 Li 1 7.2E-ll 60. O. O. O. .25
2 6.6E-10 53.4 .087 0.35 9.2 O.
3 5.2E-11 60. .044 0.17 9.2 O

400 .42-.74 Be 1 2.8E-ll 60. 0. 0. 0. .25
2 1.4E-10 49.1 .068 0.35 11. O.
3 4.0E-12 60. .034 0.17 11. 0

400 .42-.74 B l 1.0E-ll 60. 0. O. 0. .25
2 9.4E-ll 44.1 .054 .35 13. 0
3 4.6E-12 60. .027 .17 13. O.

100 .34 Li 1 6.3E-09 14.7 0. O. 0. .199
2 2813-10 46.7 .6 .12 9.2 0.

 

 

 

101

 

0.150

—4
—i
..4
..q
_
.._4
...4
_J
__.
an
_1
-—
_4
—4
—1
q
_
q

0.125

0.100

Q1 0.075

0.050

0.025
60

X
17[llll[lllllllllllllllllll
f—H

lllllLlillllillllillllillll

>—
‘-0—-

>—

...

....

.-

...

.—

.-

._

1-
-v-—

...

.—

.—

—.

)—

 

fib—
_.
.4
—4
_‘
._1
_4
—<
.1
__.
_
_‘
_4
.—
__.
.—
_—
2L
—0—

1'34

50

40

30

 

TSPC (MeV)

20

10

lllIITTIWITTlTllTTTTTlTTIlTTT

llllllllll llllllllillllillll

 

 

 

(\3
A
0')
CD
p—I
O
H
N

Figure 4.4.5b The transverse velocities of spectator source is obtained by fitting the triple

differential cross sections of fragments emitted with the impact parameters .42<5 .<_ .74 for Au +
Au collisions at E/A = 400 MeV. The temperature of the spectators is plotted at bottom panel
with the temperature of spectator extracted for the heavier fragments ( Z=7-20) from the Au + Pb
collisions at E/A = 600 MeV.

102

 

      

 

 

  
 

 

 

 

  
  
     

 

 

 

 

 

5 11 1111 11T1 rTl4J11 1111 1111 11LL. 11 1111 1111 111—1—

- I ...... I -. I I I - I I I :

10_3 : __:l

A 3 i

T : 1

L4 ._ _

U)

_4 ._4

'7 ‘0 gIg 45.0 g

> _.. i

é :I :

ElO—S +1{%HILILHfij:LrLH+LlH}1 IILI:

v : j
55 3

La 10 72.50 ’3? 9o.° ‘3

U E/A — 100 14ve __E

\ “ —~ 0120-180 —

NIL i) 5.34 +\ Q] 915%- 60—120 A

“U 10_4 _5:¢_\ \ O 0— 60 ___-1

_=5‘1’ \ _3.

7” \ 3

‘ IE 1’ -

10—5 llllllllllllllillLLillllll llllllllllL LlLlJiJLlllllllllJ
o 200 400 600 o 200 400 600 o 200 400 600 800
E (MeV)

Figure 4.4.6 The triple difl‘erential cross sections of Lithium emitted at central collision of
Au on Au collisions at E/A = 100 MeV. Each panel represents the triple differential cross
sections at each polar angle as indicated. The lines were the sum of the contributions fi'om

three-moving-source model.

103

parameters. The open circles are obtained by fitting with constant longitudinal velocity and
the open squares with a constant spectator temperature T8,. Here is a clear tradeoff
between T8pc and the transverse velocity 0,, of the spectator source that is shown in the
middle panels of the Figure 4.47. Best fits for the Lithium and Boron fragments in central
collisions at E/A = 100 MeV are obtained by assuming that spectator moves with a
transverse velocity of about 0.06 c. With the range in Bx explored here, there is a very
little change in the parameters of the participant source. Thus the radial expansion of the '
participant source described in Section 4.5 is not influenced by including the transverse

velocity of the spectator sources.

104

 

 

 
  
   

   
 
  

 

  

  

   

 

 

0.200 E1—FL1111 1111 1 111 1111I111 ,_11l1111]1111[1111l111[1I1111_‘
u+A,EA=10MeV ES E

0175 _ /[ i:— / —..

: Li; Nc > 52 E] :: B; No > 52 j

... —«»— y] ...I

0.150 —- —— —

: :: [El—E] :

N 012553—03 e—oeo—ee—oj; B _3
Q’- - : :: Geo a :
0.100 :— —:_L._,{‘3 .3

: 2: 3

0.075 E— ——§;—- ‘3

hHl Lll+ 4'14 L01” T 14 lLLILll Pl 1 J l_Ll L] 11 l l] l l l l I 1 L11 l l 1L1:

80 ;t.__ T1| 1 1 1P1‘re1e fTitT 1 1 I 1 11_—1? I T1 f1 l 1 1 1 1 l 11 1T| 1 1 1 1 [11 1-_1:

1; 0 fixed: B: :: Z

60 _ C] fixed: '1‘.” —::——- _E

8. [G~o :: B—E] a {I EH3 31343 B—a :
— [3—8 8 [HS] B—D ~_ 2

Em 4O :— 77 i
20 :— -——:g— —:

:1 l l l l l l l l l Ll ll 1 l l '1 ill [111171111111 l l l l 1 L1 lLLlJ ll 1:

:11l1111 | F1 11[1111 l11j1[111i:11[1111|111111fl1l1 1&111:

3-75 SEQ ID €11 1

3.50 E- \ J2] —:— —:

C P 2 EE 2

N a 3.25 E— % )9 —:_— f
X : go :: :
3.00 3— —EE— —2

2.75 :— {E— —E

— —< —-4
:lllllllllLllLllLl1111111ll—:llllllllllllllllllllllllJll:

 

 

 

 

2.50
0.025005000750100012501

5x

00250050007501000125

Figure 4.4.7 The temperature Tm and the longitudinal velocity B, of the fit for the triple
cross sections of Lithium and Boron fragments at the central E/A = 100 MeV collisions,

shown as a function of the transverse velocity Bx.

4.5 Collective Expansion in Central Collisions at E/A = 100 and 250 MeV

At E/A 2 100 MeV, dynamical models [Peil 89, Peil 92, Boal 88] predict that
fragment multiplicities are strongly influenced by a rapid expansion from supranormal
densities achieved early in the nuclear collision [Peil 89, Peil 92, Bond 94]. Experimental
evidence now suggests a significant collective "radial" expansion [Jeon 94, Barz 91, Baur
93, deSo 93, Hsi 94]. This collective expansion may persist even to lower incident
energies where transverse directed flow [Gutb 89 a&b, Gutb 90, Leif 93] vanishes [Ogil
90, West 93] due to cancellations between the attractive and repulsive deflections that
result from the mean field attraction and nucleon-nucleon collisions, respectively. Like
measurements of directed transverse flow [Gutb 89 a&b, Gutb 90, Leif 93], measurements
of radial flow [J eon 94, deSo 93, Hsi 94] can provide unique constraints on nuclear
transport properties such as the in-medium nucleon- nucleon cross section [Dani 92].

Radial flow effects should be enhanced for central collisions [Dani 92]; therefore a

gate on the reduced impact parameter, b = b/bmax s 0.33 was imposed in the data analysis
discussed in this section. If the contribution from radial flow are large, one would expect
this to be manifested in the energy spectra of the emitted fragments. The solid lines in the
upper panel in Fig. 4.5.1 indicate the best fits to the energy spectra for B11 fragments at
E/A = 100 MeV, assuming three relativistic Maxwellian distributions [Laud 80], as
described in Section 4.4. Contributions from the participant source (dashed lines)
dominate fits at the forward angles 013., = 35.5° and 45°. The projectile- and target-like
spectator sources contribute strongly at the backward angles and for very high energies at
forward angles, but not at 28°s 0m, 3 575° for 400 MeV 3 EM, 3 700 MeV, where the
shapes of the measured spectra are very poorly described by these thermal source fits.
Similar difficulties are encountered for the energy spectra of other IMFs with 2 3 Z; _<_ 6,
as shown in the left panels of Fig. 4.5.2. The corresponding parameters of these fits
without radial expansion are given in Table 4.5.1 (type I) for all of the fragments. These
temperatures are higher than expected from systematics of asymmetrical systems. From
such systematics, one expects a temperature of about 16 MeV [Jack 83, Chen 88, Gelb

87], consistent with a Fermi gas system with thermal energy E/A z 25 MeV, comparable

105

106

MSU—094—038

 

 

 

   

 

 

 

__ 1 1 1 1 I 1 1 1 1 l 1 1 1 1 1 1 1 1 ITfirl 1 Trfi1 1:
3 ” .. Au + Au ‘
10‘ E— ,.-n==_ _ \0 Do 100 MeV —:
E.” 9. ".W .'. Boron é
: 7 / 0'6 ‘ \ Q \ \DB \ :
4 \O ' \
10_ _—_—' —:
g i :
G) 10’5 ?— —5
2 E 3
\ *- Z
a ~ -
\ 10_6 _1 ‘_
Q i I
E 10‘3 :— "'°°" ‘5
E E '1’." __ ' "‘-"~ _- \H 5
PU ” o 9 \ \.\ \ 4
93 10‘4 — .' \ °‘ —
\ E . \ .‘. i 5
% : \9 ° H :
_ , 0 _
10_5 E_\ \ \ _' 28 1
I 0 o 2
: i \ 0 . 0 35.5 :
~ 725 45 r
_ \ \ _
10'6 2—\ \90." \ 57.5” —:
\110.\° 3
10—7 4 [\l l I l l l l l l l l l J_ LLLIJ l l l L l l l l l
0 200 400 600 800 1000 1200

Elab (Mev)

Figure 4.5.1 Comparisons of the energy spectra for Boron fragments emitted to 01“, =
280°, 355°, 450°, 575°, 725°, 90° and 110° (solid and open points), with
corresponding moving source fits. Upper panel: The solid lines correspond to the fits
obtained with Eq. 4.4.2 and no radial expansion. Lower panel: The solid lines correspond
to fits obtained with Eq. 4.4.2 and 4.4.5, incorporating a radial expansion. The dashed
lines in both panels correspond to the respective contributions fi'om the participant sources

only.

107

Table 4.5.1 Parameters of three-moving-source fits for fragments with Z = 2 to 6 in
central collisions at E/A=100 MeV. (Fits l and 2 are without and with expansion,
respectively, and the units for a;, T; and Vi are MeV3 sr", MeV and MeV respectively.)
The source 1 represents the participant source which moves with the velocity of center
mass along the beam direction. The source 2 represents the target-like or projectile-like
spectator sources which move with velocities VR and v2 in the transverse and longitudinal
directions in the center of mass frame. The Coulomb energies of the emitted fragments
were estimated and added to the fragment energies without changing the direction of the

emitted fragments.

 

Z fit source a; T, va/c VyJ/C Vi Bap/C
He 1 1 1.2E-08 37.4 0. 0. 15.1 0.
2 1.0E-08 12.76 .06 .167 6.6 0

2 1 2.4E-08 23.5 0. 0. 0. .129
2 LIE-08 14.3 .06 .159 6.6 0.
Li 1 1 6.9E-10 58.7 0. 0. 22.1 0.
2 4.5E-10 29.6 .06 .132 9.12 0.

2 1 4.3E-09 17.6 0. 0. 0. .178
2 5.6E-10 27.6 .06 .131 9.2] 0.
Be 1 1 1.1E-10 73.9 0. 0. 28.6 0.
2 1.0E-10 45.8 .06 .107 11.3 0

2 l 1.5E-09 13.9 0. 0. 0. .172
2 8.3E-ll 46.6 .06 .117 11.3 0.
B 1 l 9.1E-11 70.1 0. 0. 35.0 0.
2 6.5E-11 35.5 .06 .120 13.0 0.

2 1 8.9E-10 16.6 0. 0. 0. .153
2 5813-11 35.3 .06 .120 13.0 0.
C 1 1 4.0E-ll 64.6 0. 0. 41.2 0.
2 7.1E-10 39.1 .06 .096 14.1 0.

2 l 5.5E-10 12.8 0. 0. 0. .144
2 6.6E-ll 37.9 .06 .099 14.2 0

 

 

 

108

Au + Au, E/A = 100 MeV
central

10—1 IIITTIIIIlI—TIIrIIIIII rTIIl’IIIITIIIIlITinTrE

 

.-
.u“‘ .‘--u—

h- . wo

 

 

 

«
111

1 11111111 11111111]

dP/dEdQ(mb/sr/MeV)

  

1 1.1111111 1 1111111]. L 11111111 1 1 1‘ 1 lllllul

   

 

 

111%11f11111 114111 I 7114L11

0 25 50 '75 10(1) 25 50 75 100

E/A (MeV)

 

 

Figure 4.5.2 Comparisons of the energy spectra for fragments with Z =2 to 6. (See Fig.
4.5.1.) Lefl panels: The solid lines correspond to fits obtained with Eq. 4.42 and no
radial expansion. Right panels: the solid lines correspond to fits obtained with Eq. 4.4.2
and 4.4.5, incorporating a radial expansion. The dashed lines in both panels correspond to

the respective contributions from the participant sources only.

109

to the total kinetic energy per nucleon in the cm. system. This discrepancy suggests that
there is a significant enhancement in the apparent temperature due to some form of
collective motion.

As discussed in Section 4.4, the triple differential cross sections for the central
collisions at E/A = 100 MeV show that contributions fiom the directed transverse
collective flow are negligible at mid-rapidity. Instead, transport model calculations predict
the existence of a radial expansion at breakup. To explore this idea within a moving source

parameterization, a self similar radial expansion, via) = cBexp'r' / RS , of a spherical

participant source (i = 1 in Eq. 4.4.2) was assumed which attains its maximum velocity
chxp at the surface r = Rs. The velocities of the individual particles were assumed to be

thermally distributed with temperature T; about the local radial expansion velocity.

Coulomb expansion after breakup was modeled in the limit of large Bap, i.e., particles with

charge Zf, emitted fiom a source with charge ZS, were assumed to gain a kinetic energy
_ 2 2 3
AECoul(r) — Zf(ZS — Zf)e r /RS (4.4.4)

without changing direction. In the CM frame, one obtains

dP 3 R, .
dEdIQ = 41:11; [0 rzdrI (10,] dB

 

dp1 g. g“ .
x . ,v r,0 XSE —E+AE r 4.4.5
dE (19(1) ( ) ) ( Cou1( )) ( )

 

The total energy spectrum is obtained by inserting Eq. 4.4.5 into Eq. 4.4.2 as the
participant source.

The best fits, assuming ZS = 118 and RS = 11.1 fm for Boron fragments, are shown
by the solid lines in the lower panel of Fig. 4.5.1; the inclusion of collective expansion
changes the curvatures of the calculations so as to accurately follow the curvatures of the
energy spectra at 28° 3 0..., 3 575°, where the participant source dominates. Similar
curvatures of the energy spectra for fiagments Z = 2 to 6 are shown in the right panels of

Fig. 4.5.2.

110

The extracted value for the expansion velocity Bap depends on the temperature
Tmid, the total charge and size of the participant source. Setting RS = 11.1 fm and ZS =118,
the sensitivity of the expansion velocity to the temperature of the mid-rapidity source
(Tmid) is shown in Fig. 4.5.3. Here the radial expansion velocity and the chi squares ( 7(3)
of the fits are plotted as a fimctions of Tmid for Li and B fragments. A minimum chi square
around Tmid = 15-18 MeV is observed. Throughout this minimum, the radial expansion
velocities of the participant source decrease with increasing ngd, as shown in the top panel
of the Figure 4.5.3. The extracted values of chp are not very sensitive to the temperature
of the participant source, changing by about i 10% for 5 MeV 3 T1: 20 MeV, where
reasonable fits were obtained. For Tmid > 30 MeV for which fits dictate a smaller radial
expansion velocity, the curvatures of the energy spectra were not well described.

Both 0“,, and Coulomb expansion dynamics have a similar influence on the energy
spectra. The source size can be related to the breakup density by pB~poo(7.4 fin/Rs)3,
where p0 is the normal nuclear matter density. Here, 7.4 fin is the radius of the participant
source with ZS = 118 at normal nuclear matter density. In this estimation, a charge of Z81,“
= 20 is assumed to be taken away by each spectator residue. Figure 4.5.4. shows the
dependence of the Tmid and chp on the pB /po. For a fixed Tmid = 15 MeV, the Bap and Tmid

values obtained from the fit are shown by the solid circles in the bottom and middle panels
of Fig. 4.5.4 respectively. The corresponding x3, values are shown in the top panel. Since
smaller contributions from the Coulomb energy to the energy spectra occur at smaller p3
/po, and can be partly compensated by requiring larger contributions from the collective
energy, these is the monotonically decreasing chp dependence on p3 loo as shown in the
bottom panel of Fig. 4.5.4. There is no clear x3, minimum in the density dependence.

Similar conclusions are obtained by allowing both the density parameter and the
temperature of the source in the fits to vary freely, fits as indicated by the open points in
the figures. Over the range 0.1 3 p3 /po 3 0.3, there is a i 5% change in the best value of

Bap. Energy spectra alone do not provide sufficient information to determine the breakup

lll

 

0.25 I

0.20

 

0.05

Bexp

O

;_. .

O
1111I1111I1111I1111I1111

11111

 

0‘90 1 1 1 1 I j 1 1 1 I 1 1 1 1 I

3.5

111111

3.0

XV

2.5

2.0

1111I1111I1171I1111IT711

 

 

 

1.5 ‘

 

0

(MeV)

Figure 4.5.3 The relation between the chi square x3, from fitting the energy spectra and

the temperature of the participant source is plotted at the bottom panel for Lithium and
Boron fragments. The corresponding relationship between Tm“ and the radial expansion

velocity of the participant source is shown in the top panel.

112

 

 

 

 

 

 

2.6».1111 I1 111I1 1 11I1 1 11 1101 I 111—

Z BOI‘OHI 0 Free fit 3

2.5;— . Fixed: Tmid fi

3 : Z
N 2.4— B ‘5 _—
>< Z a —e 70:0/5’; Z
: B’O’ . I

z I
2.2hkliiIit+lIllilIlil%|ilH/dLlHl‘

E /Cg~ a I

17:— /O _

_ Z I

32 : o/ 3
E 16:— / '_.:
E" 2 HO —8/—I—o— .— O —. —o—. :
15E— / ‘5

I— x9 —I
141—IlrljjlllllL14lll 1111111111 111112

Q11I11Tf|fi 111I1111I1111I117 1

0.17}— ‘Cak —:

:— — EC‘; __________

a 0.16:— ——
.. _ \1 :
Q1 : —————— \o- +—‘.— — — — —3
0157 \O\O\\.\' i

.. a _

r 1
0.14—111J Ll1111111L1111111111\T1111«
0.1 0.2 0.3 0.4 0.5 0.6

p/po

Figure 4.5.4 Best fit values for the radial velocity are plotted at the bottom panel as a
function of the assumed breakup density of the participant source. The solid points are the
best fit values for a fixed temperature of the participant source and the open points are the
best fit values allowing the temperature of the participant source vary as a free parameter.

The best values of the radial velocity vary by about 5% as the density values from p0 =
0.3 to p0 = 0.1 of normal density. x3, values from the fits and the participant temperature

Tmid are plotted in the top and middle panels respectively.

113

density. Additional information such as fragment-fragment correlation functions may be
needed [Kim 89, K'amp 93].

When the fitting procedure described above is applied to other fragments, similar
values of Bap for the different Zf's are extracted (left panel, Fig. 4.5.5) for fixed Tm“ = 15
MeV (solid points), as when Tmid is varied freely (open points). However, there is a
systematic decrease of Bap with Zf, suggesting that heavier fragments may not participate
as fully as the lighter fragments in the collective expansion. Such an effect could arise if

heavier fragments originated fiom the more dense central regions of the expanding system.

Values for Bap are not significantly changed by making a more restrictive gate, 5 < 0.16,

on the impact parameter.

Radial Expansion for the Central Collisions at E/A = 100 MeV
Since Bap and the Coulomb expansion have a similar influence on emitted

fragments in the fitting, the mean total radial collective energy, defined by
<13r xii-mafia“,+zf(z,-zf)e2/R,I (4.5.1)

is considerably less sensitive to p13 . The values for < E, > are shown as solid points in the
right panel of Figure 4.5.5. Here, < Er > increases with mass (charge) but not linearly as
expected for a uniform participation of these fragments in the radial expansion. The radial
energies for heavier fragments with Zr 2 5 increase less than one would expect if a linear
dependence were followed.

To further support these conclusions, the energy spectra in the total CM frame at
0m. = 900 are shown in Fig. 4.5.6 for Zf = 2-6. The total fit and participant source
contributions are represented by the solid and dashed lines, respectively. Assuming a
temperature Tmid = 15 MeV, the mean radial collective energy was independently
estimated by integrating these spectra and then subtracting the mean thermal kinetic

energy of a Maxwell gas, i.e.

< E, >=<E>—%Tmid (4.5.2)

114

These estimates for < E,> (open squares, right panel of Fig. 4.5.5) are somewhat smaller
than the fitted values for the participant source (solid points), reflecting additional low
energy spectral contributions from the spectator sources. Differences between the solid
and open points provide indications of the systematic uncertainties in extracting mean
radial kinetic energies for the only participant source.

Energy spectra for Ms produced in central Au on Au collisions at E/A=100
MeV indicate large radial collective expansion velocities at breakup. The radial expansion
energies, EJA = 8.3-13.5 MeV, decrease with the fragment charge, but are relatively
insensitive to assumptions about the density of the system at breakup and the contributions

from the transverse flow or from the breakup of projectile and target spectator matter.

Radial Expansion for the Central Collisions at E/A=250 MeV

Center of mass energy spectra at 0cm, = 90° are shown in Figure 4.5.7. Due to the
lack of statistics in this data set, comprehensive analyses similar to that at E/A = 100 MeV
could not be performed. The three-moving-source model was used to fit these energy
spectra for Z = 2 to 4 (due to the lack of statistics for the higher charges). As shown in
Table 4.5.2, fits to the center of mass energy spectra at 90° do not unambiguously choose
the participant temperature and the values for Tmid (best fit) vary widely with Z. The
extracted Bap are listed in Table 4.5.2 for the best fit and for a fixed participant source
temperature of Tmid = 30 MeV, assuming p/p = 0.3. The values of radial expansion
energies EJA for the fits assuming Tmid = 30 MeV also given.

To better estimate the temperature of participant source, the mean center of mass
energy, <Ecm> is plotted in open squares in Fig. 4.5.8 at 0m = 90° energy spectra for
E/A=250. For comparison, the corresponding data at E/A = 100 MeV is plotted as the
open circles as well. Assuming the thermal energies of the detected fragments are charge
independent and that the radial energies are proportional to the mass of fragments, a linear
fit to the data for Z = 2-4 at E/A = 250 MeV was obtained, consistent with a thermal
energy of 45 MeV, corresponding to a temperature of 30 MeV. The extracted
temperature is close to the systematic of Refs. [Chit 83] and [Gelb 88]. Using Tmid = 30

MeV, the values of EJA have been obtained. These values are given in the last column in

115

MSU—094—040

 

 

 

 

 

I I I I 1 I I I I 1 II I III LI-I I I Fl I I 1 fl I I rfi 150
0.20 :— __:_ __3 125
i 1 III :E 1 11' I
1 i 11 “” I [1% . 3‘
0.15e 1 T14:— T —; 100 Q,
i l 11 fl) 1 2 [f q E
S : I : [I i
Q) ’— ‘T _ 75 A
P ‘1‘ T 4 50 V
: :1 % j
005 _— OT1 = 15 MeV f: . Fitted :
. or, Best Fit “2 D CM spectraf 25
u _: :I
000 1 14QLL11111111 11 1 1 1 1 1 1 1 1 1 1 1 1 1 O
0 2 4 6 0 5 10
Zr A

Figure 4.5.5 Left panel: The open points correspond to the best fit values for the radial
expansion velocities as a function of the fragment charge. The solid points are the
corresponding values obtained when Tmid is constrained to be 15 MeV. Right panel: The
solid points depict the dependence on the fragment mass of the mean radial collective
energy <E,> extracted from the fits upon the fragment mass. The open squares depict the
corresponding values extracted fiom the energy spectra of the center mass fiame at 90°,
assuming Tmid = 15 MeV. Both values are plotted at the mass of the most abundant

natural isotope.

116

 

    
    
    

m> Au + Au
q) - r
2 10_10 __ E/A = 100 MeV—
s o -
m _
\
E 10—12 __
Q4 - -1

L1(><10 )
0° -14
"U 10 — _.
E Be(><10—2) M
"£2 —16
mo 10 B(x10_3) _‘
\ _
F—i

 

 

 

Figure 4.5.6 Energy spectra in the center of mass frame for various fragments detected at
80°s 0m3110° in the central collisions at E/A = 100 MeV. The solid lines correspond to

the three source fit, assuming a radial expansion. The dashed lines depict the participant

source alone.

117

 

~ Au + Au ‘
10‘10 ______ E/A = 250 MeVT

 

Tc (x10‘4)

10—18 _

1/03%dP/d3p(mb/sr/M6V3)

 

 

11 111 ll_l [L4 1 1 111111

0 20 40 60 80
ECM/A (MeV)

 

Figure 4.5.7 Energy spectra are plotted in the center of mass frame for various fragments
detected at 80°: 6cms110° in central E/A = 250 MeV collisions. The solid lines

correspond to the three source fit, assuming a radial expansion. The dashed lines depict

the participant source alone.

118

 

 

 

 

BOO lilfIllll 11T1‘IT111I1111 1111 1
197 1'7 /5
_ Au+ Au / r
c / z
_ / ..
250* / —
p z

/
I I3 250 MeV /T/ —
200 ~— 0 100 MeV T // I? T j
_ By E]
/>\ _ / /C
Q) _ / / _
2 150“- // // fl
.— / —<
v H / / g) 6 _,
/\ — / /(T3 i —
/
Lil ~ g; Ey L —
v 100k / 1 A —I
I— / / —I
/

:/ f :
507— // —
h— / .1
_ / z
_/ 1
ObhlLl 11011111111L1111111111114 I:

0 2 41 6 8 10 12
A

Figure 4.5.8 Measured mean energies are plotted as a function of the fragment mass at
80°s 0cm.<.110° of the incident energies E/A=100 and 250 MeV. The dashed lines indicate

linear fits of the mean energies.

119

Table 4.5.2. Very much same method was applied at E/A = 100 MeV to obtain the open
squares in right panel of Fig. 4.5.5. Here, we fit the three lightest charges to obtained the
thermal energy 22.5 MeV is corresponding to a temperature 15 MeV, and the radial

energies consistent with our previous extraction of radial energies.

Table 4.5.2 The parameter of fits for the central collisions of 0““, = 90° energy spectra at
E/A=250 MeV.

 

Z Tmid(best fit) chp Bap Br Br
MeV (best fit) (Tmid=30 MeV) fits spectrum
(Tmid=30 MeV) (Tmid=3 0MeV)
4He 6.2 .267 .190 59 60
7Li 13.9 .267 .254 153 143
9Be 18.7 .195 .187 126 157

 

 

 

The collective radial energies of Lithium and Boron fragments extracted in the
present work at E/A = 100 and 250 MeV are plotted in Figure 4.5.9. For reference, we
also show data points measured at E/A = 150 MeV [Jeon 94] with the GSI 471: phase I
detectors. In the analysis at E/A = 150 MeV, energy spectra at 0cm, z 45° were measured
and compared to the statistical model, FREESCO, to estimate the thermal, Coulomb and
collective radial energy contributions to the mean fragment kinetic energy. The lower
limits of bars at E/A = 150 MeV are obtained by subtracting Coulomb and thermal
contributions out of the mean kinetic energy. The upper limits of bars include Coulomb
energy by assuming the temperature of system equal 20 MeV.

Recently, multifragment disintegrations have also been measured for central 197Au
+ 197Au collisions at E/A = 35 MeV. These fragments are found to be emitted
predominantly at the low center of mass energies, about E/A ~ 5 MeV, consistent with a
Coulomb dominated breakup of a single source [Dago 93]. When the latter measurements

are placed into global content, it is clear that onset of radial expansion occurs at incident

120

energies somewhere between E/A = 35 and 100 MeV. At E/A _>. 100 MeV, the extracted
collective energy appears to be approximately one third to one half of the incident kinetic
energy per nucleon in the CM frame. This reduces the available excitation energy for

thermal motion.

121

 

 

 

 

 

25 l T 1 1 I 1 1 1 1 I 1 1 1 VI 0‘0 1 1 I 1 1 1 l

— Presen —
’ 197 197 work ‘
* Au+ Au Z=3 9 *
20 — —
A i i
> 7 Present —
CD 7 work - u
2 15 7— xJeong et 611 I#
v ” a 1
<2 ; ‘ 3 3
\ I- E ZZB —
/\ 1Q — _
L. _ _
[11 — _
\/ _ _
_ 0 2
5 7‘ I ———————— c—oureab— i
7 D'Agostino et al 7
e 4

O F l l l 1 1 L 1 l 1 J 1 1 1 1 1 LJ 1 1 1 1 l 1 L1 1

0 50 100 150 200 250
Ebeam/A (Mev)

Figure 4.5.9 Mean radial energies per nucleon for Lithium and Boron fragments are

plotted as a function of incident energy.

Chapter 5
Summary

Multifragmentation has been measured for 197Au + 197Au collisions at E/A = 100,
250 and 400 MeV collisions. The mean fragment multiplicity increases monotonically
with the charged particle multiplicity at E/A = 100 MeV. For central collisions at E/A =
100 MeV, an average number of nearly 10 intermediate mass fragments is detected, about
50 % larger than the largest fragment multiplicities in the Ar + Au and Xe + Au reactions
[Bowm 91, Phai 92]. The mean number of IMF multiplicities are reduced to less than 2
for central collisions at E/A = 400 MeV before any correction for detection efficiency, and
the peak IMF multiplicity is shifted to the larger impact parameters. To perform
quantitative corrections for the experimental detection efficiencies, the experimental
distributions in rapidity-momentum phase space were simulated and the symmetry of the
system was exploited to extrapolate to the poorly measured regions of the phase space.
After correction for the efficiency, the mean number of IMF 3 increases to about 3 for the
central collisions and about 8.5 at the peak IMF multiplicity. This further confirms the
observations of nuclear vaporization in central Au + Au collisions. -

Microscopic dynamical models generally underestimate fragment yields and predict
an incorrect impact parameter dependence for IMF multiplicity at the highest incident
energy. The description of peripheral collisions at E/A = 400 MeV can be improved by
including statistical decay, via the SMM or EES models of the bound residues produced at
b 2 4 frn in the molecular dynamics simulations. At the same time, however, the number
of Ms are significantly reduced in the central collisions at E/A = 100 MeV by the decay
of excited fragments within the SMM model. Thus, fragments are either produced in
insufficient quantities or are too highly excited to survive the statistical decay in quantities

consistent with experimental observations.

122

123

Both the fragment mean transverse energies and the kinetic energy spectra are
strongly dependent on the mass of fragments, indicating a significant collective motion
within the fragmenting system. In central collisions at E/A = 100 MeV, this motion is
largely radial; in mid-central collisions at E/A = 400 MeV, there are significant transverse
collective velocities. To extract a quantitative value for the radial collective expansion
velocity at E/A = 100 MeV, moving-sources analyses were performed. These analyses are
consistent with the radial expansion energies per nucleon of E/A at 8.3-13.5 MeV, about
one-third to one-half of the incident kinetic energy per nucleon. The radial energies of the
emitted fragments decrease with the fragment charge, but are relatively insensitive to the
assumptions about the density of the system at breakup. Similar results are obtained from
the averages of the fragment energy spectra at E/A = 150 [Icon 94] and 250 MeV. At
both E/A = 100 and 250 MeV, the radial expansion energies decrease with the charge of
fragment, suggesting that all fragments do not equally participant in the collective
expansion. The transverse energies per nucleon of the emitted fragments from the
breakup of projectile and target spectator matters also decrease with the fragment charge
in mid-central E/A = 400 MeV collisions. Both these transverse energies and the triple
differential cross sections indicate the heavier fragments leave the system at the later
stages in the multifragment breakup.

Despite the significant findings in this dissertation, many of the principal objectives
of investigating multifragmentation have not been achieved. There is still no definitive
determination of the thermodynamic parameters relevant to the liquid-gas phase transition.
These prospects still lie ahead. The principal achievements of this work are the delineation
of the initial conditions where the mixed phase final states relevant to this bulk phase
transition may be achieved. Many current and future investigations are, and will be,
focused upon the interpretation of multifragment decays in central collisions at incident
energies less than 100A MeV, where this process is fully developed and the prospects for

future investigations are very exciting.

[Aich 88]
[Alar 86]
[Alar 92]
[Aldi 89]

[Bade 82]
[Batk 92]
[Barz 91]
[Baue 85]
[Baue 93]
[Bang 93]
[Bauh 85]
[Bere 92]
[Bert 83]

[Bert 87]

[Birk 64]

[Blum 91]
[Boal 83]
[Boal 88a]
[Boal 88b]
[Boal 90]
[Bogo 62]

[Bond 82]
[Bond 85a]

List of References

J. Aichelin eta1., Phys. Rev. C37, 2451 (1988).

J.Alar_ia et al., Nucl. Instand Meth. A242, 352 (1986).

J. P. Alard et al, Phys. Rev. Lett. 69, 889 (1992).

The ALADIN collaboration, Nachrichten GSI 02-89 of GSI at
Darmstadt Germany 1988 (Unpublished).

A. Baden et al., Nucl. Inst. and Meth. A203, 189 (1982).

G. Batko et al., Nucl. Phys. A536, 786 (1992).

H.W. Barz et al., Nucl. Phys. A531, 453 (1991).

W. Bauer et al., Phys. Lett. 3150, 53 (1985).

W. Bauer eta1., Phys. Rev. C47, R1838 (1993).

E. Bauge et al., Phys. Rev. Lett. 70, 3705 (1993).

W. Bauhoff et al., Phys. Rev. C32, 1915 (1985).

M. Berenguer et al., J. Phys. G18, 655 (1992).

G. Bertsch and P. J. Siemens, Phys. Lett. B126, 9 (1983).

GP. Bertsch et al., Phys. Lettt. B189, 384 (1987).

J .B. Birks, The theory and Practice of Scintillation counting, P.2
(Pergamon Press, New York, 1964).

Y. Blumenfeld et al., Phys. Rev. Lett. 66, 576 (1991).

D.H. Boal, Phys. Rev. C28, 2568 (1983).

D.H. Boal and J. N. Glosli, Phys. Rev C37, 91(1988).

D.H. Boal and 1N. Glosli, Phys. Rev. C38, 1870 (1988).

DH. Boal and James N. Glosli, Phys. Rev C42, R502 (1990).

N. N. Bogoliubov, Studies in Stastical Mechanics, J. de Boer and
GE. Uhlenbeck Eds, Vol. 1 ( North-Holland, Amsterdam 1962).
The BBGKY stands for Bogoliubov-Bom-Green-Kirkwood-Yvon.
J.P. Bondorf et al., Nucl. Phys. A387, 25c (1981).
JP. Bondorf et al., Nucl. Phys. A444, 321 (1985).

124

[Bond 85b]

[Bond 94]
[Bord 93]
[Botv 87]
[Botv 90]
[Botv 93]
[Boug 87]
[Boug 89]

[Bowm 91]
[Bowm 92]
[Bowm 93]

[Buja 85]
[Cava 90]
[Cebr 90]
[Chen 88]
[Chit 83]

[Chit 86]

[C010 92]
[Cser 86]

[Cuss 85]
[Dani 85]
[Dani 88]
[Dani 91]
[Dani 92]
[deSo 90]
[deSo 91]
[deSo 93]
[Dors 88]

125

JP. Bondorf et al., Nucl. Phys. A444, 460 ( 1985).

JP Bondorf et al., Phys. Rev. Lett. 73, 628 (1994).

B. Borderie et al., Phys. Lett. B302, 15 (1993).

A.S. Botvina et al., Nucl. Phys. A475, 663 (1987).

A.S. Botvina et al., Nucl. Phys. A507, 649 (1990).

A.S. Botvina eta1., Z. Phys. A345, 297 (1993).

R. Bougault et al., Phys. Rev. C36, 830 (1987).

R. Bougault et al., Proceeding of the Symposium on Nuclear
Dynamics and Nuclear Disassembly, P. 193 (1989).

DR. Bowman et al., Phys. Rev. Lett. 67, 1527 (1991).

DR. Bowman et al., Phys. Rev. C46, 1834' (1992).

DR. Bowman et al., Phys. Rev. Lett. 70, 3534 (1993).

A. Bujak, et. al., Phys. Rev. C32, 620 (1985).

C. Cavata et al., Phys. Rev. C42, 1760 ( 1990).

D. Cebra et al., Phys. Rev. Lett. 64, 2246, (1990).

Z. Chen and CK. Gelbke, Phys. Rev. C38, 2630 (1988).

C. B. Chitwood et al., Phys. Lett. B131, 289 (1983).

CB. Chitwood eta1., Phys. Rev. C34, 858 (1986).

N. Colonna et al., Nucl. Inst. and Meth. A321, 529 (1992).

L. P. Csemai and Kapusta, Phys. Rep. 131, 223 (1986), and
references therein.

Y. Cusson et al., Phys. Rev. Lett. 55, 2786 (1985).

P. Danielewicz and G. Odynice, Phys. Lett. 8157, 146 (1985).
P. Danielewicz eta1., Phys. Rev. C38, 120 (1988).

P. Danielewicz and GP. Bertsch, Nucl. Phys. A553,712 (1991).
Pawel Danielewicz and Qiubao Pan, Phys. Rev. C46, 2004 (1992).
RT. De Souza et al., Nucl. Instr. and Meth. A295, 109 (1990).
R. T. de Souza et al., Phys. Lett. B268, 6 (1991).

RT. De Souza eta1., Phys. Lett. B300, 29 ( 1993).

C. Dorso and J. Randrup, Phys. Lett. B215, 611 (1988).

[Doss 85]
[Doss 87]
[Fai 82]
[Feld 90]
[Finn 82]
[Fox 93]
[Frie 83a]
[Frie 83b]
[Frie 90]
[Gelb 87]

[Gobb 93]
[Gong 88]
[Goul 69]

[Gros 82]
[Gros 86]
[Gros 90]

[Gutb 89a]
[Gutb 89b]
[Gutb 90]
[Gust 88]

[Haus 52]
[Hirs 84]
[Hori 91]
[Hsi 94]

126

KHGR. Doss et al., Phys. Rev. C32, 116 (1985).

K.G.R. Doss, et al., Phys. Rev. Lett. 59, 2720 (1987).

G. Fai and J. Randrup, J. Nucl. Phys. A381, 557 (1982).

H. Feldmeier et al., Nucl. Phys. A515, 147 (1990).

J.B. Finn, et al., Phys. Rev. Lett. 49,1321 (1982).

D. Fox et al., Phys. Rev. C47, R421 (1993).

WA. Friedman and W.G. Lynch, Phys. Rev. C28, 16 (1983).
WA. Friedman and W.G. Lynch, Phys. Rev. C28, 950 (1983).
WA. Friedman et al., Phys. Rev. C42, 667(1990).

C.K. Gelbke and D. H. Boal, Prog. in Part. and Nucl. Phys 19, 33
(1987).

A. Gobb eta1., Nucl. Inst. and Meth. A324, 156 (1993).

W.G. Gong et al., Nucl. Inst. and Meth. A268, 352 (1988).

RS. Goulding and DA Landis, “Recent Advances in Particle
Identifier at Berkeley,” in Semiconductor Nuclear-Particle Detectors
and Circuits, Publication 1593, National Academy of Sciences,
Washington, DC, P 757 (1969).

D.H.E. Gross et al., Z. Phys. A309, 41 (1982).

D.H.E. Gross et al., Phys. Rev. Lett. 56, 1544 (1986).

D.H.E. Gross et al., Rep. Prog. Phys. 53, 605 (1990), and Refs
therein.

H.H. Gutbrod et al., Rep. Prog. Phys. 52, 1267 (1989).

H.H. Gutbrod et al., Phys. Lett. 8216, 267 (1989).

H.H. Gutbrod eta1., Phys. Rev. C42, 640 (1990).

H. A Gustafsson et al., Modern Physics Letter A, Vol. 3, No. 14,
1323 (1988).

W. Hauseer and H. Feshbech, Phys. Rev. 87, 366 (1952).

A.S. Hirsch, et al., Phys. Rev. C29, 508 (1984).

Hisashi Horiuchi, Nucl. Phys. A522, 257 (1991).

W. C. Hsi et al., Phys. Rev. Lett. 73, 3367 (1994).

[Hube 91]
[Hube 92]
[Huan 63]

[Icon 94]
[K'amp 93]
[Kean 94]
[Kim 89]
[Kim 91]
[Kim 92]
[Knol 79]

[Koch 91]
[Kund 94]
[Kund 95]
[Land 80]

[Leif 93]

[Lenn 31]
[Lind 93]
[Lync 87]

[Lync 92]
[Mari 70]

[Mekj 77]
[More 75]
[Morr 78]
[Mull 95]

127

J. Hubele et al., Z. Phys. A340, 263 (1991).

J. Hubele et al., Phys. Rev. C46, R1577 (1992).

Kerson Hunag, Statistical Mechanics, p69, John \Vrley & Sons Inc,
1963.

S.G. Jeong et al., Phys. Rev Lett. 72, 3468 ( 1994).

B. K'ampfer et al., Phys. Rev. C48, R995 (1993).

D. Keane et al., Preprint LBL-36370, September 1994.

Y.D. Kim et al., Phys. Rev. Lett. 63, 494 (1989).

Y.D. Kim, PhD Thesis, Michigan State University, 1991.

Y.D. Kim et al., Phys. Rev. C45, 338 (1992).

Glenn F. Knoll, Chapter 10, Radiation Detection and Measurement,
John Wiley & Son Inc.(]979).

V. Koch et al., Nucl. Phys. A532, 317 (1991).

GJ. Kunde PhD thesis, Unversity Frankfort 1994.

G]. Kunde et al., Phys. Rev. Lett. 74, 38 (1995).

L.D. Landau and EM. Lifshitz, Statistical Physics (Pegammon Press,
New York, 1980), 3rd ed..

Y. Leifels et al., Phys. Rev. Lett. 71, 963 (1993).

J .E. Lennard-Jones, Proc. Cambridge Phi. Soc. 27, 469 (1931).

V. Lindenstruth, GSI report, GSI-93-18, May, 1993.

W. Lynch, Ann. Rev. Nucl. Part. Sci., 37, 494 (1987), and references
therein.

Williams G. Lynch, Nucl. Phys. A545, 1990 (1992).

Jerry B. Marion, Classical Dynamics, p220 (Academic Press Inc.
1970)

AZ. Mekjian, Phys. Rev. Lett. 38, 640 (1977).

LG. Moretto, Nucl. Phys. A247, 211 (1975).

DJ. Morrissey et al., Phys. Rev. C18, 1267 (1987).

W.F.J Muller, J. Pochodzalla and W. Trautmann for ALADIN/LAND
collaboration, GSI 03-95 (1995).

[Myer 82]

[Nord 28]
[Ogil 91]
[Peas 94]
[Peil 89]
[Peil 92]
[Phai 92]
[Phai 93]
[Pres 86]

[Quin 59]
[Raci 95]

[Rand 81]
[Sang 92]
[Schw 93]
[Siem 79]

[Siem 83]
[Slat 75]

[Stoc 80]

128

W.D. Myers and W.J. Swiatecki, Ann. Rev. Nucl. Part. Sci. 32, 309
(1982)

L.W. Nordheim, Proc. Roy. Soc. A49, 689 (1928).

CA. Ogilvie, et al., Phys. Rev. Lett. 67, 1214 (1991).

GP. Peaslee et al., Phys. Rev. C49, R2271 (1994).

G. Peilert et al., Phys. Rev. C39, 1402 (1989) and refrences therein.
G. Peilert et al., Phys. Rev. C46, 1457 (1992).

L. Phair et al., Nucl. Phys. A548, 489 ( 1992).

L. Phair et al., Nucl. Phys. A564, 453 (1993)

Williams H. Press, Brain P. Flannery, Saul A Teukolsky and \Vrlliam
T. Vttterling; Numerical Recipes The Art of Scientific Computing,
Cambridge University Press 1989.

AR. Quinton et al., Phys. Rev. 115, 886 (1959).

G. Raciti; The firll range of particle identification for the Catania
hodoscope are achieved with the parameters of PID defined as
log(DE) + .l"‘log(E)2 -.65*log(E); DE is measured QDC channel fiom
the Silicon and the E is the measured QDC channel from the CsI. A
2-D spectra with the PID and the log(E) are used to set the charged
particle identification gates. The energy deposited into the Silicon
were calibrated by a polynomial fit. The energy deposited in C51 were
done by a power law approximation fit.

J. Ranfrup and SE. Koonin, Nucl. Phys. A356, 223 (1981).

T. C. Sangster et al., Phys. Rev. C46, 1404 (1991).

C. Schwarz et al., p.248, NSCL/MSU Annual Report 1993.

Philip J. Siemens and Joseph I. Kapusta, Phys. Rev. Lett. 43, 1486
(1979)

R]. Siemens, Nature 305, 410 (1983).

John. C. Slater, Solid-State and Molecular Theory: A Scientific
Biography, P92-95, Wiley-Intersience Publication 1975.

H. Stocker, J. Maruhn and W. Greiner, Phys. Rev. Lett. 44, 725

[Stoc 86]

[Stor 58]
[Troc 89]
[Tsan 91]
[Tsan 93a]
[Tsan 93b]
[Uehl 33]
[Vrie 87]
[West 85]
[Wils 92]
[Wang 95]
[Waps 77]
[Weis 37]
[Zhu 92]

129

(1980).

H. Stocker and W. Greiner, Phys. Rep. 137, 279(1986), and
references therein.

R.S. Storey et al., Proceedings of the Phys. Society 72, 1 (195 8).
R. Trockel et al., Phys. Rev. C39, 729, 1989.

MB. Tsang eta1., Phys. Rev. C44, 2065 (1991).

MB. Tsang et al., Phys. Rev. C47, 2717 (1993).

M. B. Tsang et al., Phys. Rev. Lett. 71, 1502 (1993)

EA. Uehling and GE. Uhlenbeck, Phys. Rev. 43, 552 (1933).
H. de Vries et al., Atomic data nd Nuclear data table 36, 495 (1987).
G. D. Westfall et al., Nucl. Inst. and Meth. A238, 347(1985).
W.K. Wilson et al., Phys. Rev. C45, 738 (1992).

S. Wang et al., Phys. Rev. Lett 74, 2646 (1995).

A. H. Wapstra and K. Bos, Nucl. Data table 19, 175; 20, 1 (1977).
V.F. Weisskopf, Phys. Rev. 52, 295 (1937).

F. Zhu et al., Phys. lett. B282, 299 (1992).

"‘i111111111111“