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MICHIGAN STATE UNIVERSITY LIBRARIES
lllll|||ll|llllllllHillllllllllll||||l|||||llllllllllllllll w
7 3 1293 01563 5844 ,
4,

This is to certify that the

dissertation entitled

Multifragmentation in 84Kr + 197Au Collisions at

Beam Energies of
E/A = 35, 55, 70, 100, 200, and 400 MeV

presented by
Cornelius Fitzgerald Williams
has been accepted towards fulfillment

of the requirements for

PhD degree in PhYSiCS

 

 

72/4; 4 47/1,,

Major professor

 

Date //5/ێ
/ /

MSUis an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

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m

MSU Is An Affirmative Action/Equal Opportunity Institution
abimmpmMJ

 

 

 

 

 

Multifragmentation in 84Kr + 197Au Collisions at
Beam Energies of

E/A = 35, 55, 70, 100, 200, and 400 MeV
by

Cornelius Fitzgerald Williams

A DISSERTATION

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

DOCTOR OF PI-HLOSOPHY

Department of Physics and Astronomy

1996

ABSTRACT

Multifragmentation in 84Kr + 197Au Collisions at Beam Energies
of E/A = 35, 55, 70, 100, 200, and 400 MeV

by

Cornelius Fitzgerald Williams

This dissertation concerns investigations of multifragmentation processes induced
by “Kr + 197Au collisions at beam energies of E/A = 35, 55, 70, 100, 200, and 400 MeV.
The experimental data in this dissertation reveal the fragment multiplicities to increase
with incident energy to a maximum at about E/A=100 MeV and to decrease thereafter
with the onset of nuclear vaporization. The measured fragment charge distributions have
been characterized by power law distributions in the fragment charge. The extracted
exponents increase monotonically, reflecting an evolution towards steeply falling charge
distributions with increasing incident energy. The measured fragment transverse energy
distributions are characterized by mean values that increase with incident energy,
consistent with the onset of a collective outward expansion. This latter expansion can be
placed within a systematics that includes data obtained from collisions of identical
projectiles and targets.

These experimental data were compared to several dynamical and statistical
multifragmentation models. The Quantum Molecular Dynamics model (QMD) was
generally unable to describe the experimental trends, 3 failure that may be related to the

classical heat capacities of the computational nuclei. The Statistical Multifragmentation

Model (SMM), on the other hand, could successfully describe the average fragment
multiplicities, and charge distributions, but failed to describe the dependence of the
fragment charge distribution on the fragment multiplicity. This latter failure indicates an
internal inconsistency within the SMM approach whose origin will require additional

theoretical study.

This dissertation is dedicated to my family for all the support that they have given me,
and to all of the administrative assistants and secretaries who are always around when you
need them.

iii

ACKNOWLEDGMENTS

To finish many things in life one needs the help of others in some form or another.
When one has finished these things it is often good to take a look back and thank all of
those who have helped reach one’s goal.

The first person I like to and must give thanks to is my Lord and Savior, Jesus
Christ. He has allowed my to finish this dissertation and he has given me everything I
needed to finish it, including the time and the right people to help. And I pray that he will
continue to use me as I am needed.

I would also like to say thank you to my family, who have encouraged and
supported me in every way possible.

To my advisor, Dr. William Lynch, I say thank you for all of your help and
support in working with me to complete this dissertation.

I would also like to say thanks to my advisor committee for their time, Dr.
Wolfgang Bauer, Dr. Raymond Brock, Dr. Thomas Glasmacher, and Dr. Wayne Repko.

Before I go any farther, I would like to give my personal thanks to all of the
members of the “Miniball” group at MSU from 1990—1996.

To all the collaborators, I must give great thanks to you, since your work was of

course a major part of me finishing: Michigan State University, D.R. Bowman, M.

Chartier, J. Dinius, C.K. Gelbke, T. Glasmacher, D.O. Handzy, W.C. Hsi, M.J. Huang,
M.A. Lisa, G.F. Peaslee, L. Phair, C. Schwarz, M.B. Tsang, W. Bauer, C.M. Mader;
Washington University, G. Van Buren, R.J. Charity, L.G. Sobotka; Indiana University, D.
Fox, R.T. deSouza, T.M. Hamilton; Laboratoire National SATURNE, M-C. Lemaire,
S.R. Souza; Gesellschaft fiir Schwerionenforschung, G.J. Kunde, U. Lynen, J.
Pochodzalla, H. Sann, W. Trautmann; Instituto de Fisica, Universidade de 850 Paulo, N.
Carlin; Lawrence Livermore National Laboratory, G. Peilert; University of Wisconsin,
W.A. Friedman; Institute for Nuclear Research, A. Botvina.

To collect the data for this dissertation we had to use two facilities, the National
Superconducting Cyclotron Laboratory at Michigan State University and Laboratoire
National SATURNE in Saclay France, and I would like to take the time to thank the staff
at both locations for their assistance. Since one of these facilities is in a different country,
I would like to thank all of those who made it possible to ship the needed equipment to
France and back, especially Graham Peaslee.

Next I would like to give special thanks to the following people; Fan Zhu, Wen-
Chien Hsi, James Dinius, Christine Hampton, Njema Frazier, Steve Russell, John “Larry”
Bendler, Sergio Souza, Min-Jui Huang, and Sally Gaff, all graduate students whom had to
deal with me.

A personal thanks to all those at NSCL that have supported me over the years with
words of wisdom and a good push every now and then, especially, Jim Vincent, Allyn
McCartney, John Marcotte, John Bonofiglio, Timothy Hoepfner, Jill Franke, Renan
Fontus, Jack Ottarson, Greg Humenik, Gary Horner, Andrew Thulin, Jackie Bartlett, and

Michal Garbek.

Also I like to give a personal thanks to everyone on the faculty and staff of the
Physics Department at Michigan State University, especially Stephanie Holland, Victoria
Simon and Jean Strachan.

To Dr. H. Alan Schwettman, Dr. John C. Hiebert, and Dr. Robert A. Kenefick, I
say thank you helping me get started in experimental research.

To all fellow graduate students, present, passed, and still to come, I wish you all
luck and I will be praying for you.

I realize that I have just given thanks to many people who have help with me
finishing this dissertation in one way or another, still there are many that have been left
out and to those that I have not mentioned I say thanks and God bless you.

Supported by the National Science Foundation under Grant numbers PHY-90-
15255 and PHY-92-14992 and the US. Department of Energy under Contract number

DE-FG02-87ER-403 1 6.

vi

Table of Contents

 

List of Tables ............................................................................................................... ix
List of Figures ............................................................................................................. X
Chapter 1 Introduction .......................................................................................... 1
1.1 Background and Motivation ................................................................................ 1
1.2 Theoretical Background ...................................................................................... 3
1.2.1 Molecular Dynamics Models .................................................................... 4

1.2.2 Statistical Models of Fragment Emission ................................................. 7

1.3 Thesis Organization ............................................................................................. 11
Chapter 2 Experimental Design ........................................................................... 13
2.1 Miniball/Miniwall Design ................................................................................... 13
2.2 Detectors .............................................................................................................. 20
2.2.1 Miniball .................................................................................................... 20

2.2.2 Miniwall .................................................................................................... 22

2.2.3 Ion Chambers ........................................................................................... 22

2.3 Target ................................................................................................................... 27
2.4 Solid Angle Coverage of the Miniball/Miniwall Array ....................................... 27
2.5 Electronic Readout of the Miniball/Miniwall Array ........................................... 30
2.5.1 Photomultiplier Tubes .............................................................................. 30

2.5.2 Electronic Setup ........................................................................................ 32
Chapter 3 Data Analyses ........................................................................................ 38
3.1 Particle Identification .......................................................................................... 38
3.1.1 PID Spectrum ........................................................................................... 38

3.1.2 Problem Detectors .................................................................................... 46

3.1.3 PDT Spectrum ........................................................................................... 46

3.2 Energy Calibration ............................................................................................... 52
3.2.1 Light Output Equation .............................................................................. 52

3.2.2 Saturation Effect ....................................................................................... 53

3.2.3 Determining the Parameters .................................................................... 56

vii

Chapter 4 Data and Calculations ........................................................................ 61

4.1 Impact Parameter Selection ................................................................................. 61
‘1.12.1 .litzitczticarz ................................................................................................... (51
‘1Lil.£? fllatzzl (Cliclrygez ............................................................................................. (555

4.2 Intermediate Mass Fragment Multiplicities ......................................................... 65
4.2.1 Experimental Data .................................................................................... 69
4.2.2 QMD Calculations .................................................................................... 73
4.2.3 QMD+SMM Calculations ........................................................................ 74
4.2.4 SMM Calculations .................................................................................... 77

4.3 Charge Distributions ............................................................................................ 80
4.3.1 Experimental Data .................................................................................... 82
4.3.2 Comparisons with QMD Calculations ..................................................... 87
4.3.3 Comparisons with QMD+SMM Calculations .......................................... 87
4.3.4 Comparisons with SMM Calculations ...................................................... 88

4.4 Multiplicity Dependence of the Charge Distribution .......................................... 91

4.5 Collective Flow .................................................................................................... 99

4.6 Collective Expansion ........................................................................................... 101

Chapter 5 Conclusion .............................................................................................. 106

5.1 Main Results ........................................................................................................ 107
5.1.1 Fragment Yields ........................................................................................ 107
5.1.2 Charge Distributions ................................................................................ 108
5.1.3 Collective Expansion ................................................................................ 110

5.2 Future Outlook ..................................................................................................... 111

Bibliography ................................................................................................................ 1 13

viii

List of Tables

Table 2.1 Information on the number of detectors in a ring, the solid angle of a
detector in that ring, the polar angle of a ring, and the range in the polar angle and
azimuthal angle that each detector covers. The letter W by the first six rings indicates
that these are Miniwall rings ............................................................................................ 24

Table 3.1 Values for the parameters a,-, b,-, and c,- of the functions 0L(Z), B(Z) and 'y(Z)
given in equation 3.9a, 3.9b, and 3.9c, where Z/A = 0.5 ................................................. 59

Table 4.1 Parameters of SMM calculations chosen so that calculations will match the
experimental data ............................................................................................................. 78

ix

List of Figures

Figure 1.1 The upper panel shows the temperature, T, versus the excitation energy
per nucleon, E‘lA, for 1/3 nuclear matter density, the solid line, and 1/6 nuclear
matter density, the dashed line. The bottom panel shows corresponding values for

(NW ..............................................................................................................................

Figure 2.1 Photograph of the Miniball array when mounted in the 92” scattering
Chamber at MSU. The picture shows is the main configuration (rings 3’ through 11)
used in these experiments ................................................................................................

Figure 2.2 Shown here are rings 2-6 of the Miniwall detection array, ring 1 is also
attached, but is recessed so it is not visible in the photo. The CsI(Tl) crystal are
facing the camera. They are about 3 cm in thickness. The photomultiplier tubes are
contained in the u-metal tubes shown here ......................................................................

Figure 2.3 A half-plane schematic diagram of the Miniball/Miniwall array. The
individual detector rings are labeled, 3’ to 11 for the Miniball and 1 to 6 for the
Miniwall. While the polar angle for each ring is indicated correctly, the drawing has
been simplified and the exact position of each the Miniwall detectors may be slightly
different than the actual experimental setup position. The dashed horizontal lines
shows where the beam axis is located ..............................................................................

Figure 2.4 These are the front views of the detectors in the Miniball/Miniwall array.
They are labeled according to the ring number and by the a letter (B for the Miniball
and W for the Miniwall) telling to which array they belong. The number in
parentheses tells the number of detectors in that ring ......................................................

Figure 2.5 Schematic diagram of individual detector elements [deSo 90] .....................

Figure 2.6 Photograph of the photomultiplier assembly. The scintillator and the first
matching light guide are removed. The ring glued to the u—metal shield defines the
alignment of the can that houses the voltage divider; this can has been removed to
expose the voltage divider ...............................................................................................

Figure 2.7 A schematic diagram of the Ion Chamber Telescope ....................................

Figure 2.8 Isometric view of the target insertion mechanism mounted on ring
8 ........................................................................................................................................

Figure 2.9 Schematic of the upgraded active voltage divider used for the Miniball
detectors [Hsi 95] .............................................................................................................
x

12

14

16

18

19

21

23

26

28

31

Figure 2.10 Schematic electronics diagram of the electronics for the Miniball array
alone. The Disc, Split., DGG, Fera, Amp., L.U. and FI/FO labels indicate the
locations of discriminators, splitters, delay/ gate generators, Fera ADC’s, amplifiers,
logical units and the fan in/fan out models respectively .................................................. 33

Figure 2.11 This is schematic drawing of the analog signal of the phoswich detectors.
The fast, slow, and tail gates of the Miniball are shown. The top numbers show when
the gates open after the first gate has opened. The time intervals shown are the time
durations for the gates ...................................................................................................... 34

Figure 2.12 Schematic diagram of the data acquisition electronics for the Miniwall
array alone ........................................................................................................................ 36

Figure 3.1 Fast versus Slow spectrum of detector 4—18 of the Miniball (ring 4 and
position 18) for 84Kr + 197Au collisions at E/A = 55 MeV .............................................. 39

Figure 3.2 Fast versus Slow spectrum of detector 6-8 of the Miniwall (ring 6 and
position 8) for 84Kr + 197Au collisions at E/A = 55 MeV ................................................ 40

Figure 3.3 Tail versus Slow spectrum of detector 3’-20 of the Miniball (ring 3’ and
position 20 ) for 84Kr + 197Au collisions at E/A = 55 MeV ............................................. 41

Figure 3.4 Tail versus Slow spectrum of detector 6-8 of the Miniwall (ring 6’ and
position 8) for 84Kr + 197Au collisions at E/A = 55 MeV ................................................ 42

Figure 3.5 This is a plot of a Slow versus PID spectrum of detector 3’-20 of the
Miniball. Visible are the different charge lines making it easier to draw the gates for
particle identification when analyzing the data ................................................................ 44

Figure 3.6 This is a plot of a Slow versus PID spectrum of detector 6-8 of the
Miniwall. Visible are the different charge lines making it easier to draw the gates for
particle identification when analyzing the data ................................................................ 45

Figure 3.7 Tail versus Fast spectrum of detector 8-5 of the Miniball (ring 8 and
position 5) from the 84Kr + 197Au collisions at E/A = 55 MeV ....................................... 47

Figure 3.8 This is a plot of the Tail versus Slow and the variables that are used to
make the PDT function. In the plot you can clearly see the relationships between
T ail], Tail; and Tailr ......................................................................................................... 49

Figure 3.9 This is a plot of a PDT versus Slow spectrum of detector 3’-20 of the
Miniball. Visible are the Hydrogen isotopes, protons, deuterons, and tritons, as well
as the Helium isotopes, 3He and He ............................................................................... 50

Figure 3.10 This is a plot of a PDT versus Slow spectrum of detector 6-8 of the
Miniwall. Visible are the Hydrogen isotopes, protons, deuterons, and tritons, as well
as the Helium isotopes, 3He and 4He .............................................................................. 51

xi

Figure 3.11 The upper panel shows the results obtain for the fractional deviation
when using 12C as the reference particle. The line is linear fit to the 12C data and
defines the function f(Ch#), the fractional deviations. The light output for the
Miniball and Miniwall was less then the output predicted by equation 3.8. The lower
panel shows how the result for f(Ch#) obtained with 12C compares to the deviation
observed from the 6Li data ...............................................................................................

Figure 3.12 This is the calibration curve with the saturation correction for the 4He
andthe 12C particles. The top line is the 4He energy calibration curve and the bottom
line is the ”C energy calibration curve using equation 3.8. The filled in symbols are
the energy calibration data points using the saturation correction. The open symbols

are the energy calibration data points without the saturation correction ..........................

Figure 3.13 This plot shows the calibration curves for the detected particles with Z/A
= 0.5. Also plotted are the calibration data for: 4He, (solid circles), 6Li, (open fancy
crosses), 10B, (crosses), 12C, (open circles), 14N, (solid squares), 160, (open
diamonds), and 20Ne, (open squares) particles .................................................................

Figure 4.1 The normalized probability distributions for the charged particle
multiplicity NC for E/A = 35, 55, 70, 100, 200, and 400 MeV. The reduced impact

parameter, b , is shown at the top of the panels. The open circle indicates the middle
impact parameters, 0.3 < b < 0.66 ................................. . ................................................

A

Figure 4.2 The dependence of reduced impact parameter, b , upon the charged
particle multiplicity, NC, for E/A = 35, 55, 70, 100, 200, and 400 MeV .........................

Figure 4.3 The dependence of the total charge, <Z[o[>, on the total charged particle
multiplicity for E/A = 35, 55, 70, 100, 200, and 400 MeV ..............................................

Figure 4.4 The dependence of the total charge, <Ztot>, upon the reduced impact
parameter, b, for E/A = 35, 55, 70, 100, 200, and 400 MeV ..........................................

Figure 4.5 The dependence of the mean intermediate mass fragment multiplicity,
<N1Mp>, upon the charged particle multiplicity for E/A = 35, 55, 70, 100, 200, and
400 MeV .........................................................................................................................

Figure 4.6 The dependence of the mean intermediate fragment multiplicity, (NIMF>,

upon the reduced impact parameter, b, for E/A = 35, 55, 70, 100, 200, and 400
MeV .................................................................................................................................

xii

55

57

6O

63

66

67

70

71

Figure 4.7 The upper panel shows a comparison between the measured mean charge
particle multiplicity, <Nc>, and QMD calculations for central collisions as a function
of incident energy. The lower panel shows the corresponding values for the mean
IMF multiplicity, <N1MF>. The solid dots are the experimental data points. The
corresponding statistical uncertainties are smaller than the data points. The dashed
lines are from unfiltered QMD calculations. The solid and dotted lines are QMD
calculations filtered by the experimental acceptence. The solid square is the
prediction of QMD calculations without the Pauli potential ...........................................

Figure 4.8 The upper panel shows a comparison between the measured mean charge
particle multiplicity, <Nc>, and QMD+SMM calculations for central collisions as a
function of incident energy. The lower panel shows the corresponding values for the
mean IMF multiplicity, <NM=>. The solid dots are the experimental data points. The
corresponding statistical uncertainties are smaller than the data points. The dashed
lines are from unfiltered QMD+SMM calculations. The solid and dotted lines are
from QMD+SMM calculations filtered by the experimental acceptance ........................

Figure 4.9 The upper panel shows a comparison between the measured mean charge
particle multiplicity, <Nc>, and SMM calculations for central collisions as a function
of incident energy. The lower panel shows the corresponding values for the mean
IMF multiplicity, <N1Mp>. The solid dots are the experimental data points. The
corresponding statistical uncertainties are smaller than the data points. The dashed
and dot-dashed lines describe unfiltered SMM calculations which include radial and
rotational flow, respectively. The dotted and solid lines are the corresponding filtered
SMM calculations ............................................................................................................

Figure 4.10 The probability of emitting a fragment of charge, Z, per collision in
center collision at all six incident energies. The solid dots are the experimental data
points. The statistical uncertainties of the data are smaller than the data points. The
solid line is the power law fit ...........................................................................................

Figure 4.11 Comparison of the energy dependence of the charge distribution
exponent T from the data to QMD and QMD+SMM calculations for central 84Kr +
197Au collisions. The solid line describes the ”C values from the QMD calculations
using the Pauli potential. The dashed line describes the 17 values from QMD+SMM
calculations. The solid square is the 1: value at E/A = 50 MeV from a QMD

calculation that does not use the Pauli potential. The solid points are values for 17
extracted from the experimental data ...............................................................................

Figure 4.12 The probability of emitting a charge, Z, per mid-impact parameter
collision at the six incident energies. The solid dots are the experimental data points.
The statistical uncertainties of the data are smaller than the data points. The solid
lines are the power law fits ..............................................................................................

xiii

72

75

79

83

84

85

Figure 4.13 Comparison of the energy dependence of the charge distribution
exponent T from the data to QMD and QMD+SMM calculations for mid-impact
parameter 84Kr + 197Au collisions. The solid line describes the T values from the
QMD calculations using the Pauli potential. The dashed line describes the T values
from QMD+SMM calculations. The solid square is a 1? value at E/A = 50 MeV from a

QMD calculation that does not use the Pauli potential. The solid dots are values for I
extracted from the experimental data ...............................................................................

Figme 4.14 Comparison of the measured energy dependence of the charge
distribution exponent 1: from the data to SMM calculations for central 84Kr + 197Au
collisions. The dashed and dashed-dotted lines describes the T values from SMM
calculations assuming radial and rotational flow, respectively, before secondary decay
is allowed to occur. The dotted and solid lines describes the corresponding 1 values
for SMM calculations after secondary decay of the excited fragments. The solid dots
are values for I extracted from the experimental data .....................................................

Figure 4.15 Charge conservation parameter c as a function of transverse energy E for
the 84Kr + 197Au system at incident energies of E/A = 35 - 400 MeV .............................

Figure 4.16 IMF multiplicity gated OL-OL correlation functions for the 84Kr + 197Au
SYStem at E/A = 35 MeV and Et < 100 MeV ...................................................................

Figure 4.17 Left hand panel: c plotted as a function of incident energy and central
cOllisions for data (solid points) and SMM predictions (open points). Right hand
Panel: c plotted as a function of source size used in the SMM calculations for E/A =

200 MeV. The doubled dashed lines bound the value for 0 consistent with the
experimental data .............................................................................................................

Figure 4.18 Comparison of the measured mean transverse energy, <Et>, (solid points)
to SMM calculations with radial (dashed lines) and rotational (solid lines) flow as a

function of fragment charge at E/A = 35, 55, 70, 100, 200, and 400 MeV and 13 <

86

89

93

95

97

0.25 ................................................................................................................................... 100

Figure 4.19 Comparison of the measured mean transverse energy, <Et>, (solid points)
to QMD calculations without the Pauli potential (solid squares) at E/A = 50 MeV,
QMD and QMD+SMM calculations with the Pauli potential (solid and dashed lines,
respectively), as a function of fragment charge at E/A = 35, 55, 70, 100, 200, and 400

MeV and b < 0.25. Note, charged particles are not produced with Z > 4 within the

QMD model for 35 MeV at this reduced impact parameter ............................................ 102

xiv

Figure 4.20 The mean collective velocity. <B>, versus the excitation energy per
nucleon, EI/A of the participant source. The solid squares are from the analysis of the
841(1' +197Au system. The other data are described in the text ......................................... 104

XV

Chapter 1

Introduction

1- 1 Background and Motivation.

The experimental exploration of the properties of bulk nuclear matter at extrem
densities and temperatures began in earnest over a decade ago. A series of experiments 2
the Bevalac at Lawrence Berkeley Laboratory (LBL) [Gutb 89a, Stile 86] initiated th
prospects for a creation of nuclear shock wave in supersonic nucleus-nucleus collision:
and related investigations are now being carried out at places like the Nation:
Superconducting Cyclotron Laboratory at Michigan State University (NSCL), th
Gesellschaft fiir Schwerionenforschung, (G81) and the Alternating Gradient Synchrotro
(AGS) at Brookhaven National Laboratory. These experiments have succeeded i
PlaCing constraints upon the compressibility of nuclear matter at high densities. Aroun
the Same time, an interest in the properties of nuclear matter at subnuclear densities wa
arouSed by a series of pioneering measurements at Fermilab, N SCL and the 88” cyclotro
at LBL [Hirs 84, Chit 83, Sobo 83]. Discussion of these latter measurements in th
Context of a low density phase transition of liquid-gas character [Curt 83, Saue 76, Bon
85a: Gros 82, Fai 82] played an important role in increasing the interest in this area and

c - . . . . . . .
ontlnues to motivate work, 1ncludlng the present dlssertatlon, 1n thls area.

The emission of intermediate mass fragments (IMF’s; 3 S Z S 20) is the mat

0b . . . . . .
Ser\’able that IS presumed to be connected With a tranSltlon between llqu1d and gaseou

phases of nuclear matter [Curt 83, Hirs 84, Gros 82, Bond 85a]. However, the notion of i
liquid-gas phase transition is already present in evaporation theories of nuclear decaj

wherein nucleons and light clusters (d, t, on) are emitted from the surfaces of a hot nucleu

[W eis 37]. The emission of heavier IMF’s at low emission rates is a prediction 0
evaporation theories [Frie 83a, Lync 87]. The observation of very high IM]
multiplicities, however, lies outside of the realm of prediction of evaporation theorie
[Bowm 91, Frie 83a]. Such an observation suggests a different mode of decay via a bull
disintegration [Bond 85a, Frie 83a, Gros 82] which, if it occurs over an appropriatt
timescale, could allow the existence of a phase equilibrium between nuclear liquid drop
(W’s) and vapor (nucleons and light particles) within the breakup volume. T hi
possibility, if this occurs, differs significantly from the phase changes which occur durin;
evaporation since a hot evaporating nucleus can never be in equilibrium with tht

Surrounding vacuum.

After initial inclusive fragmentation experiments, there was a waiting period ii
the latter part of the last decade during which various multifragment detection arrays wen
ConStructed. At the NSCL, the Miniball array was completed and began its firs
experiments in 1990. Soon after that, it was determined that bulk multifragmen
diSiIItegrations were observed in collisions between heavy nuclei [Bowm 91, deSo 91]. 1
then becomes important to determine the experimental domain in incident energy ant
projectile and target masses for the optimal experimental exploration of this new decaj
Ineel'lamism. Therefore, the Miniball group, in conjunction with groups at WashingtOl

U -
r11Varsity, Saclay and GSI agreed to carry out a set of experiments in Europe using th

Miniball/Miniwall array. Two sets of measurements were done, the first one was i
of measurements of 197Au + 197Au collisions at the Schwerionen Synchrotron
accelerator located at GSI, and the other was a series of measurements of 84Kr -l
collisions at the Laboratoire National Saturne (LN S), synchrotron accelerator loc
Sacl ay. These experiments allowed us to map out the fragmentation process as a fl

of incident energy and impact parameter.

This dissertation concerns itself with the measurement of 84Kr + 197Au re
at SATURNE and with a follow up experiment at the lower energies using bean
the NSCL. Altogether reactions at six different incident energies, E/A = 35, 55, 7
200, and 400 MeV were measured, giving a good systematic information over a 111
broad dynamic range. These measurements reveal that the beams provided by the
are close to ideal for the examination of multifragmentation processes. In 0
Central collisions at significantly higher energies produce breakup configurations 1

tOO highly excited and expand too violently for any possibility of phase equilibriun

1.2 Theoretical Background

Theoretical approaches can be subdivided into models which try to calcr
equilibrated multifragment decay and others which try to model the time evolutiOl
System via dynamical or rate equation approaches. Among models which
equilibrium one can count multiparticle phase space models like the St:
Multifragmentation Model (SMM) [Bond 85a, Botv 87], Berlin [Gros 82], or
[Bert 83, Fai 82] approaches. Time dependent approaches include the oval

a
p p r 0aches like the Expanding Evaporation Source (EES), model [Frie 88], or mt

dynamical models such the Quantum Molecular Dynamics (QMD) model [Peil 89] or
Quasiparticle Dynamical models (QPD) [B031 83]. Clearly, an accurate fully dynamical
approach is preferable to any other, since it should also relax and describe an equilibrated

decay configuration given sufficient time.

1 -2- 1 Molecular Dynamics models _

Prior to this dissertation, considerable credibility was given to the results of
molecular dynamics calculations [Peil 91, Peil 92] based upon numerical techniques
similar to those employed in the solution of the semi-classical Boltzman - Uehling -
Uhlenbeck (BUU), or Vlasov - Uehling - Uhlenbeck (VUU) equation [Bert 83, Pan 95].
Probably the most frequently employed models of this type are the QMD models

deve10ped by the Frankfurt—Name collaboration [Aich 86, Peil 89].

The initial implementation of this approach differs little conceptually from the
BUU, VUU, or Landau-Vlasov theories [Greg 87]. The nucleons are represented by
Wigner transforms of the form [Aich 86, Peil 89]:

2
pt): (in )36xp -[pO --i5(t)]2L4 -[rO -'r'(t)]2 h (1.1)

h2L2

Solutions within the QMD model are attained by classically propagating nucleons

accoI‘ciing to Hamilton’s equations of motion and the Hamiltonian:

+—Zjdr,drdp. dp {v§“°"8+v,.f°"' +v;""}f(i,,p,,t)f(rj,p,,t) (1.2)

21¢}

Here, 150, is the average of the Guassian wave packet of the ith particle, m is the mass 0
the particle, Vijc°“1 is the two body Coulomb potential and Vi].SK is the two body stron;

potential, approximated here by an effective interaction of the Skyrme type. The tWt

body Skyrme potential is of the form:
VSK=t1-5(f-—‘f-)+t2-5('fi—fj)~6('fi—'fk) (1.3)
and ‘1‘“ is a local Yukawa term of the form:

Vyuk = Viiuk 'eXPH-fl -f2l/7Yuk) . (14)
El -f2l/7Yuk

 

In addition to these two body effective potentials described by the Skyrme interaction
scattering via the residual interaction is implemented by a classical collision integra
Which incorporates phase-space “Pauli” blocking factors in the final state. This Pauli
bIOCking correction to the collision integral is the essential ingredient that distinguishe
this model from classical molecular dynamics. Unfortunately, this distinction is mor
forn'lal than real, since it is practically impossible to completely enforce the Pau‘
exclusion principle with this approach. Calculations reveal that both ground state an
Weakly excited nuclei are unstable with reSpect to decay via pathways that al
inconsistent with the Pauli exclusion principle. This failure renders the calculatior
unreliable when extended over long time periods. It therefore prevents getting a
accurate description of the multifragment disassembly in collisions at lower incidel

e - . . .
nergles where the decay timescales are characteristlcally long.

To address this deficiency, theorists tried to improve the implementation of Pauli
principle by introducing a two-body “Pauli” potential which acts in phase space [Peil 89,
Boal 83]. In the case of the “QMD with Pauli potential” model, the Hamiltonian is
written in the following manner:

..2.
Hag-3%)

iii drainer,- {vim + Vii-3°” + V? "k }f (iii-WV (TD-91’1")

iaej

 

 

~2
1 o 2 if pi].
+— d‘f.df.d”.d‘. v ex .. ' __d d
2;! i J pl p1 Pau {lo-P50 P 2a: 2§2 t,r/. 0,ch1
0
Xf(?r,f>l.t)f(fj,l3,~,t) (1.5)

Values for the parameters in equation 1.1 and equation 1.2 are given in ref. [Peil 91]. The
inclusion of the Pauli potential makes it possible to create stable computational ground
State nuclei. This, in turn, permits the extension of numerical simulations to long breakup

tin'leSc ales.

While permitting a numerically stable technique for calculating dynamical
multifragmentation, the “QMD with Pauli potential” model displays features that are not
consistent with quantum mechanics. For example, the repulsive Pauli potential scatters
p atItiCles to prevent the overpopulation of phase space. Such scattering corresponds to a
Change of state in quantum mechanics. True Pauli blocking, on the other hand, prevents a
Challge of state whenever the collisional rearrangement would result in a phase space

0
VeI‘LDOpulation. Thus, true Pauli blocking lengthens the mean free path of particles near

the Fermi surface while the Pauli potential actually shortens the mean free path [Prat 95,
Wile 91]. A second shortcoming is that the computational nuclei simulated by the “QMD
with Pauli potential” model and other similar models with “Pauli” potentials, all exhibit

classical heat capacities [Peil 91, Lync 87].

This latter limitation is particularly vexing because it prevents the realistic
modeling of the later stages of multifragmentation whenever such stages occur over
timescales commensurate with statistical emission timescales [Lync 87]. Unphysically
low temperatures occur in “QMD with Pauli Potential” model calculations in which large
classical heat capacities replace the smaller quanta] heat capacities extrapolated from
compound nuclear decay measurements [Lync 87]. When this happens, statistical
emission processes are greatly inhibited due to the resulting unphysically low

temperatures.

These limitations were not apparent at the initiation of this dissertation but
became very apparent after comparisons between our data and other similar data to the
reStilts of simulations using the QMD model with the Pauli potential. These comparisons

are Presented in Chapter IV of this dissertation.

1-2.2 Statistical Models of Fragment Emission

Fragment yields can be calculated within statistical models in the limit of global
equilibrium [Bond 85a, Botv 87, Gros 82, Fai 82, Pan 85] or via equations which describe
the eInission rates [Frie 83a, Frie 88, Boal 83]. Investigations using either approach can

reproduce the available multifragmentation data provided that fragmentation occurs at

Subrlllclear density. Within global equilibrium approaches, the requirement that roughly

sphelical fragments do not overlap dictates multifragment decay configurations with
densities less than a third of normal nuclear matter density. Within the Expanding
Evaporation Source (EES) rate equation approach of Friedman [Frie 88], the relative
fragment multiplicities are only enriched to the level that is experimental observed,
provided the system expands to less that 0.4 normal nuclear matter density [Bowm 91,
Frie 88]. Thus, it is a common prediction of either approach that multifragmentation
principally occurs after the system expands to subnuclear density.

In this dissertation, comparisons have been primarily made to the Statistical
Multifragmentation Model (SMM) of Botvina, Bondorf and collaborators [Bond 85, Botv
87] - In this model, fragments and light particles are consider as Boltzman particles

moving in a reduced free volume:
V=xVo (1.6)

II) this equation Vo = 41tro3Atm/3 with r0 = 1.17 fm, Amt is the total nucleon number, and x
is a model parameter, described below; V is the total volume of the system minus the

V01ume occupied by the fragments.

Accurately sampling the multiparticle phase space is difficult. This is achieved in

the SIVIM approach using the following steps [Bond 85, Botv 87]:

(i) The number of ways to break a prefragment of mass A, into fragments with a
multiplicity, M, is computed in a combinatorial approach [Bond 85, Snep 88].

These partitions {NA,z}are then sampled according to their combinatorial weights.

(ii)

(iii)

Each partition {NA,z} has a well defined set of fragments and nucleons. An
additional multiplicative weight coming from the phase space for each fragment is

assigned to each partition {NA,z}and is written as follows:

W({NA,z}) = 6XP[S({NA,2},T,V)]- (17)

Here the total entropy is written as:

S( { NA,Z} 9T9V) =ZA,Z NA,Z SA,z (T)- (1-8)

In this expression, NA,Z is the multiplicity of fragments of A and Z, SA,Z (T) is the
“entropy” of individual fragments of A and Z, and T is the “temperature” of the
decay configuration. In defining the “entropy” and “temperature” an implicit
ensemble averaging over the various ways of sharing energy within a partition was

taken.

Both entropies and excitation energies for individual fragments were calculated

from an empirical liquid drop expression for the free energy:

 

 

v A T T2
FA,Z (T) = —T1n[gA’Z 21(3) ]+ NA 2 lnN A,Z! + [W0 — E—JA
s 0

A—ZZ2 322 2 1
y( ) + e {1— 1/3 . (1.9)
A 5RA,zL (1+1)

 

+fl(T)A2/3 +

Here, g AZ is the degeneracy factor which used for light fragments and is set
equal to 1 for A > 4, W0 = -16 MeV is the bulk energy parameter, 8,, = 16 MeV,
)(Vo = V, is the free volume that the fragment moves in, and the asymmetry

coefficient 7 = 25 MeV. The nucleon thermal wavelength, 7t, is given by:

(W)

10

2 %
i=(27‘fir) , (1.10)
m

 

where m is the mass of a nucleon. The surface free energy parameter [3(T) is zero

for T 2 Tc, but for T < Tc has the following form:

_T2 5/4
fl(T)'-= flo[:§ —:—T2] , (1.11)

where Bo is a constant with the value of 18 MeV. The ratio of the saturation
density over the freezeout density is given by the parameter x. In the model x is

related to the multiplicity M of the emitted particles by:
d 3
_ 1/3 _ _
Z—[HRo (M 1)] 1, (1.12)

where R0 = 1.17 A g 3, and 2d is the crack width between fragments at freezeout.

The “temperature” of the system is calculated from the total free energy by
requiring a fixed energy in the event. The “entropy” of an individual fragment
SA; (T) can be calculated from the empirical liquid drop formula for the free
energy, FA,Z (T), given in equation 1.9. Then S({NA,Z},T,V) is obtained by
summing SA; (T) over fragments and a multiplicative weight W({NA,Z}) can be

assigned for each partition.

The secondary decay of the various excited fragments at “temperature” T is then

calculated.

11

(v) The experimental observables are obtained by averaging over simulated events
generated in a Monte Carlo approach. The calculations do consider the various
weights of each partition and the various decay pathways for the excited

fragments.

The top panel in figure 1.1 shows the relationship between temperature and total
excitation energy per nucleon predicted within this approach, where the total system is

Z10, = 158 and Amt = 394. For 3 MeV S E*/A S 8 MeV, the temperature is predicted to

increase only gradually with fragment mass. Proponents of this model believe this
behavior to be an indicative of the enhanced heat capacity characteristic of a phase
transition from a Fermi liquid to a nucleonic gas. The bottom panel of figure 1.1 shows
the corresponding relationship between the fragment multiplicity and the total excitation
energy per nucleon. In the domain where the temperature remains roughly constant, the
fragment multiplicity grows rapidly. At higher values of the excitation, energy where the
temperature begins to climb rapidly, the fragment multiplicity declines sharply.
Advocates of this model interpret this decline as a consequence of the onset of

vaporization.

1.3 Thesis Organization

Chapter 2 describes the experimental setup, and Chapter 3 describes the spectra
and analysis procedures. Chapter 4 describes the final physics results and provides
comparisons to various fragmentation models. Chapter 5 provides a summary and
discussion of the physics conclusions of this dissertation. An outlook toward future

investigations is also discussed.

 

ATOT=394’ ZTOT: 158
20 I I I I I I I I I I I I I I I I I I I I

— p/po=1/3 /
-—' p/po=1/6

 

I I I I I I l I I I T I I I I I I I I

I I I l I I I l I I I l I l I I I I I

 

<NIMF>

IIIIIIIIIIIIIIIIIIIIIIIII

CD
IIIIIIIIIIIIIIIIIIIIIIIIIIIII

 

 

 

 

O
O1

10
EVA (MeV)

Figure 1.1 The upper panel shows the temperature, T, versus the excitation
energy per nucleon, E*/A, for 1/3 nuclear matter density, the solid line, and
1/6 nuclear matter density, the dashed line. The bottom panel shows
corresponding values for <N1Mp>.

Chapter 2

Experimental Design

To cover the wide range of incident energies used by this thesis, it was necessary
to conduct experiments at two different facilities. The measurements for the lower
incident energies of EA = 35 , 55 and 70 MeV were performed at the National
Superconducting Cyclotron Laboratory (NSCL) located at Michigan State University
(MSU) in East Lansing, Michigan. The measurements at the higher incident energies of
E/A = 100, 200 and 400 MeV were performed at the Laboratoire National SATURNE
(LNS) located in Saclay, France. The beams at LNS had lower intensities than the beams

at NSCL which made the analysis more difficult to carry out.

This chapter describes the two experimental setups, including the detectors, the
electronics and the geometrical arrangements in the scattering chamber. While there are
differences that exist between the two experimental setups, these differences are relatively
minor. For this reason the Miniball/W all array is described first in general terms, and the
changes of the components that relate specifically to the SATURNE or NSCL

experiments are explained when describing those components.

2.1 Miniball/Miniwall Array

The Miniball phoswich detector array is designed to operate in a vacuum vessel.

The layout of the Miniball/Miniwall array is described in this section and a description of

13

 

Figure 2.1 Photograph of the Miniball array when mounted in the 92”
scattering Chamber at MSU. The picture shows is the main configuration,
(rings 3’ through 11), used in these experiments.

15

the individual detectors is given in section 2.2. Figure 2.1 is a photo of the Miniball
taken while it was in the 92 inch chamber at NSCL. The array consists of 11 independent
coaxial rings centered the beam axis. For easy assembly and servicing, the individual
rings are mounted on separate base plates that slide along two precision rails. The
individual detectors can be removed or inserted without interfering with the alignment of
neighboring detectors. The entire assembly is placed on an adjustable mounting structure

which allows the array to be aligned to the beam axis.

The Miniwall array shown in figure 2.2 is also a phoswich detector array designed
to work in a vacuum. It is constructed with six azimuthally symmetric rings where all the
detectors in a ring are at the same polar angle. These rings are not mounted on individual
supports like those in the Miniball; instead they are mounted on one plate. This plate is
designed to fit on the same precision rails and use the same cooling system as the

Miniball.

The rings and detector mounts are made out of aluminum, which provides good
thermal conductivity between detectors and mounting structure. This allows the heat
generated by the photomultiplier voltage divider network to conduct into the array
superstructure where the heat is removed from the Miniball/Miniwall array by a cooling
bar attached to the base plates. These bases are cooled by alcohol or other refrigerants
which flow through an attached cooling line. By regulating the temperature of the

cooling fluid, the temperature of the base plates are held at 15° C, leaving the individual

detectors at approximately room temperature. This ensures that the array will reach an

equilibrium temperature quickly when the phototubes are at full bias.

16

I

 

Figure 2.2 Shown here are rings 2-6 of the Miniwall detection array, ring 1
is also attached, but is recessed so it is not visible in the photo. The CsI(Tl)
crystals are facing the camera. They are about 3 cm in thickness. The
photomultiplier tubes are contained in the u—metal tubes shown here.

17

Figure 2.3 shows a half-plane section of the array in the vertical plane containing
the beam axis. The individual rings are labeled by ring numbers 1-6 for the Miniwall and
3’-ll for the Miniball, which increase in numbers from the forward to backward angles.
The number of detectors in each ring is given in the parentheses next to the ring number.
All detectors in a given ring are identical in shape and are located at the same polar angle.
These polar angles are defined in the coordinate system shown in figure 2.3 where the

target is the origin and the beam axis is the polar axis.

The angular distributions of the emitted particles are strongly forward peaked
[Kim 92]. To equalize the counting rates between detectors the solid angles of the
detectors at forward angles are smaller than those at backward angles. This variation in
solid angles is achieved mainly by placing detectors at different distances from the target,
allowing the size of the detectors to remain fairly constant. Still the shape of the detectors
changed from ring to ring. The geometries of the front faces of the detectors (which are
made from CsI(Tl) crystals) are shown in Figure 2.4. The front faces of each detector is
labeled according to the ring number. In order to reduce the cost of fabrication, the edges
of the front faces of the crystals were machined flat instead of curved, which could have
allowed the polar angle corresponding to the edges of the detectors to be constant. This
resulted in a loss of solid angle coverage on the order of 2%. These flat edges contribute
to a loss in the solid angle that is comparable in magnitude to the loss in solid angle
resulting from the gaps between individual detectors (which is needed to make
allowances for mechanical tolerances and to allow optical isolation between neighboring

crystals).

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W l (16) W 2 (16) W 3 (22) W 4 (26)
W 5 (24) W 6 (24) B 3’ (28) B 4 (24)
B 5 (24) B 6 (20) B 7 (20) B 8 (18)
5 cm
B 9 (14) B 10 (12) B 11 (8)
Figure 2.4 These are the front views of the detectors in the

Miniball/Miniwall array. They are labeled according to the ring number and
by the a letter, (B for the Miniball and W for the Miniwall) telling to which
array they belong. The number in parentheses tells the number of detectors

in that ring.

 

20

The configuration of the arrays in the N SCL experiment differed from that at
SATURNE by the inclusion of ring 1 of the Miniwall, and by the replacement of the
phoswich detectors directly over the beam axis in rings 3’ - 11 with Ion Chamber, Silicon,
CsI(Tl) telescopes. These Ion Chamber telescopes were constructed at Indiana
University, and were included to provide information about the fragment energy spectra

at energies below the Miniball particle identification thresholds.

2.2 Detectors
2.2.1 Miniball

Each phoswich detector of the array is composed of a 40 um (4 mg/cmz) thick

 

plastic scintillator and a 2 cm thick CsI(Tl) scintillator crystal. The plastic scintillator foil
was spun from a Bicron BC-498X scintillator solution [deSo 90]. Figure 2.5 is a
schematic of the detector design. To retain flexibility in the choice of scintillator foil
thickness, the foil was placed on the front face of the CsI(Tl) crystal without any
adhesive. The shape of the CsI(Tl) is dependent on which ring the detector is in. The
back face of the CsI(Tl) scintillator is glued to a flat ultraviolet transmitting lightguide
(UVT) Plexiglas with optical cement (Bicron BC600). This light guide is 12 mm thick
and matches the geometrical shape of the back face of the CsI(Tl) crystal. This light
guide is glued to a second cylindrical light guide, a piece of UVT Plexiglas 9.5 mm thick
and 25 mm in diameter. The cylindrical guide is then glued to the front window of the
photomultiplier tube, a Burle Industries model C83062E. Both the front and back faces of
the CsI(Tl) crystals were polished so as to allow the transmission of light through them.
The sides are sanded and wrapped with white Teflon tape to help reflect internal light and

to prevent cross talk between neighboring detectors. To farther isolate the detectors from

-LI'I.I' *

 

Fast scintillator L13ht guide

\

 

 

k

 

 

 

 

 

 

 

 

 

 

\\\\\\\\\\\\\

 

Z pm
1:

.. «I
Al—mylar foil \/

Optical cement

 

 

 

Figuer 2.5 Schematic diagram of individual detector elements [deSo 90]

22

outside light, the front face of the phoswich assembly is covered by an aluminized mylar

foil (0.15 mg/cm2 mylar and 0.02 mg/cm2 aluminum).

Figure 2.6 is a photograph of the basic photomultiplier assembly used in all of the
Miniball detectors. In this photograph the phoswich detector and the matching
trapezoidal light guide have not yet been attached. The photomultiplier tube and the
cylindrical light guide are surrounded by a cylindrical u-metal shield -- not shown in the
figure 2.6. A precision machined aluminum ring is glued to the u—metal shield
surrounding the photomultiplier and the cylindrical light guide. This ring also helps to
align the detection with respect to a precision-machined aluminum can that houses the
voltage divider. It is this can that defines the detector’s alignment when it is bolted to the

rings of the array support structure.

2.2.2 Miniwall

This Miniwall array is made of six rings of phoswich detectors constructed in a
manner very similar to those of the Miniball. The Miniwall detectors were made with 3
cm thick CsI(Tl) scintillator crystals and 8 mg/cm2 thick scintillator foils spun from
Bicron BC-498X scintillator solution. It should be noted that like the Miniball, the
boundaries of the polar angles between the rings were approximated by machining the flat
surfaces on the CsI(Tl) crystals very close to the boundaries corresponding to the ideal
surfaces of constant polar angle. The resulting polar angles and the solid angles for

detectors in each ring of the Miniball/Miniwall array are given in Table 2.1.

1 2.2.3 Ion Chamber
The Ion Chamber telescopes used in the experiment at MSU were built by RT. de

Souza at Indiana University. These telescopes where designed to have a low threshold

L..______—____——_______

23

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24

Table 2.1 Information on the number of detectors in a ring, the solid angle
of a detector in that ring, the polar angle of a ring, and the range in the polar
angle and azimuthal angle that each detector covers. The letter W by the
first six rings indicates that these are Miniwall rings.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ring Detector AQ(msr) 0(°) A0(°) Acp(°)
1(W) 16 1.11 4.375 2.15 22.50
2(W) 16 2.57 6.950 3.10 22.50
3(W) 22 2.59 10.000 3.00 16.36
4(W) 26 2.85 13.000 3.00 13.84
5(W) 24 5.56 16.625 4.25 15.00
6(W) 24 10.64 21.875 6.25 15.00
3’ 28 11.02 28.000 6.00 12.86
4 24 22.90 35.500 9.00 15.00
5 24 30.80 45.000 10.00 15.00
6 20 64.80 57.500 15.00 18.00
7 20 74.00 72.500 15.00 18.00
8 18(-1) 113.30 90.000 20.00 20.00
9 14 135.10 110.000 20.00 25.70
10 12 128.30 130.000 20.00 30.00
1 1 8 125.70 150.000 20.00 45.00

 

 

 

 

 

 

 

 

 

25

energy and a high resolution. As shown in figure 2.7 the telescopes have three main
active parts, an ion chamber, a silicon detector, and a CsI(Tl) Crystal scintillator. For

more detailed information on the Ion Chamber telescopes see ref. [Fox 96].

The ion chamber is the first active component of the telescope, it has an electric
field aligned along path of the ion, it is 55 mm long and uses seven square copper rings to
shape its field. These rings are 2 mm thick with 4 mm of spacing between them with the

central ring serving as the anode. The gas used in the ion chambers was CF4 and was

operated at pressures between 18 to 30 Torr.

The next active component in the telescope is a 500 pm thick silicon detector.
The silicon detector was made by Micron Semiconductor and is a ion implanted, Si02

passivated silicon. Its active area is 30 mm by 30 mm, mounted on a 37 mm by 40 mm

printed circuit board 1.28 mm thick.

The third component is a CsI(Tl) crystal, with a photodiode readout. These
Cl’ystals are 30 mm thick and 37 mm square and were wrapped with Teflon tape and
a1uIIJinized mylar to maximize the amount of light available for the photodiode. The

Photodiode used was a 2 cm by 2 cm Hamanatsu photodiode.

While the array can in principle be accurate calibrated, mistakes were made by the
IUCF group in calibrating the ion chambers. These mistakes unfortunately reduced the
utility of the ion chambers for this the experiment. For this reason, data from the ion

chambers were not included in this dissertation.

26

PIN PHOTO DIODE

_ LIGHT GUIDE
. / 051 (TI)
'IT /
I I

H

 

 

 

 

:7/4 WINDOW
ION CHAMBER ’ _ —
|

 

 

 

 

/A///‘:

 

 

 

 

\ PHOTO DIODE FIELD SHAPING RINGS
PRE - AMP (COPPER)
Si DETECTOR
3cm X 3cm
5001.:

Figure 2.7 A schematic diagram of the Ion Chamber Telescope

 

27

2.3 Target

Figure 2.8 is a drawing of the target insertion mechanism. The targets are
mounted on frames made of flat shim stock 0.2 mm thick, which are attached to insertion
rods with screws. These rods are mounted on a tray which can be moved parallel to the
beam axis with the aid of small motor. An electromagnetic clutch is used to insert and
retract the target rod when it is located in the appropriate position. A third drive rotates
the target about the axis of the insertion rod. Being able to rotate the target can be helpful
in determining the shadowing a detector experiences when it is located in the plane of the

target frame.

2.4 Solid Angle Coverage of the Miniball/Miniwall
Array

The original 11 ring configuration of the Miniball detection array is calculated to

cover a solid angle corresponding to about 89% of 41: [deSo 90]. The 11 % loss in solid

angle can be calculated in the following manner:

(i) beam entrance and exit holes (4% of 4n);

(ii) approximation of the curved surfaces corresponding to constant polar angle by

planar surfaces (2% of 4n);
(iii) optical isolation of detectors and allowance for mechanical tolerances (4% of 411:);

(iv) removal of one detector to provide space for target insertion (1% of 41:).

 

28

 

 

 

Figure 2.8 Isometric view of the target insertion mechanism mounted on
ring 8.

29

The Miniball was designed to handle the charged particle multiplicities produced
in reactions like 40Ar + 197Au at E/A = 60 - 100 MeV [deSo 91, Kim 92, Phai 92].
Heavier projectiles have larger multiplicities, making the granularity of the Miniball
insufficient to prevent double hits in some of the detectors of the Miniball array. To
reduce this problem, original rings 1, 2, and 3 of the Miniball (with 12, 16, and 20
detectors covering 9°S 6 S 31°) were replaced by rings 3, 4, 5, and 6 of the Miniwall
array (with 22, 26, 24, and 24 detectors covering 8.5°S O S 25°) constructed by
Washington University, and a new Miniball ring 3’ (with 28 detectors covering 25°S 9 S
31°). This new forward array doubled the granularity of Miniball rings 1 - 3. Additional
Miniwall rings 1 and 2 (with 16 and 16 detectors covering 5.4°S 0 S 85° and 3.225°S 0
S 5.4°) were also constructed by Washington University to extend the coverage to even
more forward angles. The configuration of the Miniball/Miniwall array contained 112
detectors in the Miniwall and 167 detectors in the Miniball in the experiment at Saturne.

In the experiment at NSCL 128 Miniwall detectors and 167 Miniball detectors were used.

The new configuration of the Miniball/Miniwall array with its complete
compliment of detectors is calculated to cover a solid angle corresponding to about 90%

of 41:. The 10% loss in solid angles can be broken down in the following manner:
(i) beam entrance and exit holes (3% of 41:);

(ii) approximation of the curved surfaces corresponding to constant polar angle by

planar surfaces (2% of 41:);

(iii) optical isolation of detectors and allowance for mechanical tolerances (4% of 41:);

30

(iv) removal of one detector to provide space for target insertion (1% of 41:).

At MSU when the Ion Chamber telescopes are added the coverage drops to about

87% of 41:. This is mostly because the telescopes are square and don’t fit into the

Miniball array as well as the detectors that they replace.

2.5 Electronic Readout of the Miniball/Miniwall Array
2.5.1 Photomultiplier Tube '
The primary light produced by the plastic scintillator used in the phoswich
detectors in this array has its maximum intensity at 370 nm. Now by adding a
wavelength shifter to the solution the intensity maximum is shifted to 420 nm. However
the scintillator foils are too thin for a fully effective wavelength shift, so the maximum
emission remains in the far blue region of the spectrum. The CsI(Tl) crystals absorb this
wavelength of light, this places a constraint on the maximum useful thickness of the
CsI(Tl) crystals at about 3 cm. To minimize the absorption of light from the plastic
scintillators by the light guides of the Miniball/W all detectors, the light guides were made

from UVT Plexiglas rather than ordinary Plexiglas.

The lO-stage Burle Industries model C83062E photomultiplier tube was chosen

because its good timing characteristics (TR ~ 2.3 ns), large nominal gain (~107) [deSo

90], and linearity for fast signals. Each Miniball detector was soldered to its base to
ensure a good electrical contact at all times. The Miniball was designed to operate in
Vacuum, therefore active bases (active divider chain) for the photomultiplier tubes were
Chosen to minimize the amount of heat generated by the detectors. A schematic of the

aetive divider chain is given in figure 2.9. The final stages of the divider chain are of the

 

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Cathode
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Figure 2.9 Schematic of the upgraded active voltage divider used for the
Miniball detectors [Hsi 95].

32

"booster" type which helps improve the linearity of the high peak current generated b;
large signals from the fast scintillator. A 1.1-metal shield was placed around thi
photomultiplier and cylindrical light guide to protect the photomultiplier currents from
stray magnetic fields. To prevent sparking when the detectors are running in a poo
vacuum, the entire divider chain, including the leads to the photomultiplier tube, wa
encapsulated in silicone rubber (Dow Chemical Sylgard 184), thus protecting the FET’

from destruction.

2.5.2 Electronic Setup

Figure 2.10 shows a block diagram of the electronics used to gather the data fo
the Miniball. The typical anode current signal from the photomultipliers is shown i:
figure 2.11. In the figure one can see that a fast signal from the plastic scintillator i
superimposed on a signal from the CsI(Tl) crystal. The fast CsI(Tl) component of th
signal is of the order of several hundred nanoseconds and the slow component of th
signal is of the order of several microseconds, and the size of these components depem
on the mass and charge of the detected particle. This analog signal is separated by

,9 ‘6

passive splitter into several branches called the “fast”, “slow , tail”, and “trigger” wher
the typical relative amplitudes Ifast:Islow:Ita“:Img are 0.82:0.04:0.04:O.1. The slow and tai

branches of the splitter are directly connected to their respective fast encoding reado:

analog-to-digital converters (FERAs).

From figure 2.11 one can see that the gates for the “slow” and “tail” FERAs ar
390 ns and 1.5 its wide. For the fast branch, a linear gate is inserted between the passiv
splitter and the “fast” FERA. This linear gate allows each of the “fast” channels to b

individually gated by the discriminator signal for that detector, something that cannot b

33

x1

1
NM

sat delay 13‘“ premaster
L.G. 150 ns .
DISC

Fera TFC Ogle
time veto
gat stop
cle start

DGG start

tail trig. tail delay
L. U. DGG instant busy
VCIO DGG

DOG busy logic
FI/FO

 

Figure 2.10 Schematic electronics diagram of the electronics fc
Miniball array alone. The Disc, Split., DGG, Fera, Amp., L.U. and l
labels indicate the locations of discriminators, splitters, delaj
generators, Fera ADC’s, amplifiers, logical units and the fan in/fa
models respectively.

34

Photomultiplier Tube Signal

 

 

 

 

"faSt" "slow" iftailit
33ns 390ns 1.5115

Figure 2.11 This is schematic drawing of the analog signal of the phoswich
detectors. The fast, slow, and tail gates of the Miniball are shown. The top
numbers show when the gates open after the first gate has opened. The time
intervals shown are the time durations for the gates.

 

35

done with the common gate FERAs. The linear gates are opened z 20 ns prior to the

leading edge of the linear signal and for a duration of 33 us as shown in figure 2.11. Then

the fast FERA gate is set to precede the gated fast signal and has a width of 150 ns. The
trigger branch, Im-g is reamplified by a fast amplifier and fed into a leading edge

discriminator module, the output of which provides the stop signal for the Time-to-FERA

Converter (TFC) and opens the linear gate for the fast channel.

Each discriminator module has an output called the sum output, which is the
linear sum of attenuated output signals for all of the 16 channels. The amplitude of the
sum signal is N . 50 mV where N is the number of channels with a signal greater then the
discriminator level. The sum outputs of each discriminator are used as the inputs of the
“Mult Box”, which is another linear summer that generates an output that is proportional
to the number of channels that are triggered in the event. By placing a discriminator level

on the output of the “Mult Box” one gets a simple multiplicity trigger.

Since the size of the analog signal from the detectors is reduced as the signal
travels through the cables, it is advantageous to place the data acquisition electronics as
close to the detectors as possible. To accomplish this, it is necessary to place the
electronics in the vault near the detectors and to control the discriminator thresholds and
the photomultiplier gains remotely with the computer. Then the signals from each of the
detectors can be checked remotely. This is done looking at the sum output of the linear

gate modules while selectively masking the discriminator and linear gate outputs.

The electronic setup of the Miniwall was a little different than that of the Miniball
see figure 2.12. The anode signal first passes through a fast amplifier and is then split

into the fast, slow, and tail components. The fast signal is processed through a Philips

36

90 ns

ECL out

Fast
Split.

 

Figure 2.12 Schematic diagram of the data acquisition electronics for the
Miniwall array alone.

37

(QDC). The slow and tail are still processed through FERA ADC’s. The time signal is
taken from the discriminator and passes through about 290 ns of delay before going to

into a TFC and then is sent to a FERA.

Chapter 3

Data Analysis

The detectors in the Miniball/Miniwall array are plastic scintillator, CsI(Tl)
phoswich devices. The response of these detectors is non-linear and can also present
problems for precise particle identification. The methods for dealing with these
difficulties are presented in this chapter. The chapter begins with a description of the

particle identification procedure followed with a description of the energy calibrations.

3.1 Particle Identification

There are two basic types of spectra that are used to identify the particles; the
choice of spectra depends on the particle to be identified. They are the Slow vs Tail and
the Fast vs Slow spectra [deSo 90, Kim 91]. Figures 3.1 through 3.4 give examples of
these two spectra for both the Miniball and the Miniwall. While both of these spectra are
good for identifying particles, it turns out that by applying a few transformations to these
spectra, particle identification becomes much easier. The PID spectrum (Particle
Identification), and the PDT spectrum (for the Slow vs Tail spectrum), are used for

particle identification in this thesis.

3.1.1 PID Spectrum

The PID spectra are constructed by performing a transformation on the Fast and

Slow components of the signal. These are constructed so that particles of the same

38

39

   
 

 

 

400 _-
Be
A3oo 4'45? __i'j'
6 L1
:1
8
.2
8
g. 200 He
LL
84Kr+197Au
E/A=55MeV
100 H Miniball
Det4-l8
O " " 1 1 L
O 50 100 150

Slow (channel)

Figure 3.1 Fast versus Slow spectrum of detector 4—18 of the Miniball (ring 4
and position 18) from the 84Kr + 197Au collisions at E/A = 55 MeV.

40

 

 

 

400 —§‘
350 —:" _s4K'H197Au
_. ' .E/A=55 MeV

€300 3 Miniwall
c: a ' .-_ _ '
s: 250 9.65.88;
«3 __ = _
vzoo
'5
CU
IL 150 = _

100

5 0 1.; »

0 0

5 0 1 0 0 1 5 0 2 0 0
Slow (channel)

Figure 3.2 Fast versus Slow spectrum of detector 6-8 of the Miniwall (ring 6
and position 8) from the 84Kr + 197Au collisions at E/A = 55 MeV.

41

    
 

 

 

400-—
E300—
_ d
'3200—
H p
34Kr+197Au
,, . . E/A=55MeV
__ 9.: Miniball
100 - Det3’-20
0 l l I | l 1
o 50 100 150 200 250 300

Slow (channel)

Figure 3.3 Tail versus Slow spectrum of detector 3’-20 of the Miniball (ring
3’ and 20 position ) from the 84Kr + 197Au collisions at E/A = 55 MeV.

42

       

 

350 ——g
300 —§
A250 —.;
T) .
£200 —f.
4:: .' ..
2 3 p
'E 150 —;-
H i 84KI+197Au
100 —";".. E/A=55MeV
' Miniwall
:' I~ .; Det 6‘8
50 '—’:'
O I I I I I I I

 

O 50 100 150 200 250 300 350
Slow(channel)

Figure 3.4 Tail versus Slow spectrum of detector 6-8 of the Miniwall (ring 6’
and position 8) from the 84Kr + 197Au collisions at E/A = 55 MeV.

43

charge appear on the same horizontal line, see figures 3.5 and 3.6. This makes it easier to
draw the charge gates, and to extrapolate gates to higher charge states that may be harder
to see. The success of this procedure relies on the uniformity of the Miniball/Miniwall
crystals, i.e. the crystals all have the same doping factor and thickness. This allows one to

develop a transformation procedure for a detector and use it on the other detectors.

These transformations requires normalizing the Fast vs Slow spectra to a common
reference spectrum. The normalization is done by matching the punch-through energies
of the hydrogen and helium isotopes of the detectors to those of the reference detector.
The punch through energies are defined as the minimum amount of energy a particle must
have to completely pass through the crystal. Once the Fast vs Slow spectra are
normalized to the reference spectrum, the transformation used to produce the PID
spectrum for the reference spectrum is then used on the spectra for the other detectors to

produce their PID spectra.

To construct the PID spectra the normal Fast vs Slow is changed so that the
charge particle lines form horizontal lines and have equal separation between each of the
lines. The first step in transforming the spectrum is to draw a line on the maxima of all
the charge lines in the Fast vs Slow spectra of the reference spectrum. The Fast signals
that lie on these charge lines are assigned values according to Y = Z . 20, where Z is the
charge of the fragment. A new two-dimensional plot is then constructed with this Y
value and with an X value is equal to the Slow component. To transform the points

between the charge lines the following equations are used:

X = Slow (3.1)

 
 
 
    

 

   

 

500
_— _ 84Kr+l97Au
400 ' - BIA: 55 MeV
Minlball
Q Det4—18
o
g 300
£1
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V
2 200
Q-I
_C—- —
100 3__
_—B_e.‘_‘_
, —_Li -
'- , - _ _ He
0 O 20 40 60 80 100 120

Slow (channel)

Figure 3.5 This is a plot of a Slow versus PID spectrum of detector 4-18 of
the Miniball. Visible are the different charge lines making it easier to draw
the gates for particle identification when analyzing the data.

500

400

PID (channel)
‘8

N
O
O

100

 
 
 
  

 

45

h: - -- 841(1- + 197Au

' ‘ E/A = 55 MeV
2, _; MMww
Det 6—8

    

 

 

 

'50- '100 150 200
Slow (channel)

Figure 3.6 This is a plot of a Slow versus PID spectrum of detector 6-8 of the
Miniwall. Visible are the different charge lines making it easier to draw the
gates for particle identification when analyzing the data.

46

 

Y = 20. 2mm + Fa“ _ FaStWE— . (3.2)
FaSt Upper — FaSt Lower

Note, that Slow is the slow channel of the detected particle, Fast is the fast channel of the
detected particle, ZLower is the charge of the charge line immediately below the fast value
for the detected particle, Fastbwe, is the fast channel of that ZINC, charge line, and
Fastuppe, is the fast channel of the Z charge line right above the detected particle. All of
the fast channels in this expression have the same slow channel associated with them.

The X and Y are the new channels in the PID spectra.

Using this method the PID spectrum have nearly horizontal charge lines with
carbon expected at Yz 120 for example. The gates for charge particle identification are
then drawn on this new PID spectra. Identification of the charge of the detected particles

was done by this method for both the Miniball and the Miniwall.

3.1.2 Problem Detectors

Gates drawn on PID spectra were used to identify the charged particles in most of
the detectors. Some detectors with a different doping mixture in the CsI(Tl) crystals,
displayed poor Fast vs Slow spectra, and this method of particle identification could not
be used. Instead Fast vs Tail spectra were used to identify the charges. Figure 3.7 shows

how the charge lines can be separated in these spectra.

3.1.3 PDT Spectrum

Figure 3.5 shows that while the PID spectrum is useful for identifying the charge
it is not sufficient to separate hydrogen and helium isotopes. Therefore a different “PDT”

spectrum is used to separate the isotopes. The PDT spectra are made from the Slow and

47

2 5 0 ,_. 84Kr+ 197Au
lyA=55NRV
Det 8—5

200 ;:

   
   

150

 

Fast (channel)

. l l |
0 50 100 150 200 250 300
Tail (channel)

Figure 3.7 Tail versus Fast spectrum of detector 8-5 of the Miniball (ring 8
and position 5) from the 84Kr + 197Au collisions at E/A = 55 MeV.

48

Tail components of the detector signal by expanding the region where most of the data is
located [Kim 91]. This is done by first drawing two gates lines which contain the

detected data between them; figures 3.8 shows how these lines were drawn.

Taill = Line1(slow) (3.3)
Tailz = Linez (slow) (3.4)
Amax = Taill — Tailz (3.5)
A = Tailr — Tailz (3.6)
PDT =512*(l—-A/Am) (3.7)

Taill and Tailz are functions that define the lines used in the transformation to
obtain the PDT spectrum. The difference between two points at a given slow value is

defined as Amax, and A is defined as the difference between Tailz and the Tailr value of

the data point. Multiplying the ratio of A to Amax by the number of displayable channels,

512, gives the quantity “PDT”. The new PDT component becomes the x-coordinate and
the Slow component becomes the y-coordinate of the PDT spectra. Figures 3.9 and 3.10

show the PDT spectra with all of the isotopes marked.

In the PDT spectra one can draw the gates for the hydrogen and helium isotopes
fairly well up to the punch through energies. Particle identification beyond this point can

be problematic and determining the energies of such particles is usually not possible.

49

84KI + 197 Au {if 1 ._ i h
__ E/A = 55 'MeV

4 0 0 Miniball .

Det 3’-20

 
 
 
  
  
  

(slow, tail!)

     

9 3 0 0 —
8
5
a !
o _ _-_3 ‘.I I.
: (slo , tail). 5" I...
.8 00 _ . : ....) . rig...
H 2 . 3‘.

l 0 0 :7 jg»; . .-

(slow. 121112) A

O I I I I 1 I
O 50 100 150 200 250 300
Slow (channel)

 

Figure 3.8 This is a plot of the Tail versus Slow and the variables that are
used to make the PDT function. In the plot you can clearly see the
relationships between Taill, Tailg and Tail,. '

50

  
    

 

   

 

500 —
34Kr+197Au _
E/A=55 MeV Li;
__ Miniball ‘ ' ~ _
400 Det3’—20 “a
a
0
E300 —
J:
2
3
3200 —
m _
100 e . __
HJpgt. — _ |-‘
0 o 50 100 150 200
PDT (channel)

Figure 3.9 This is a plot of a PDT versus Slow spectrum of detector 3’-20 of
the Miniball. Visible are the different Hydrogen isotopes, protons, deutrons,
and tritons, as well as the Helium isotopes, 3He and 4He.

51

  
 
  
   

 

 

500 — _
84Kr+197Au
E/A=55MeV 6 '
Miniwall
400 — Det6—8
2
E300 —
J:
8
E, 200 fi
V)
.P. '-
100 ’—
jH-p2t_._'
0 . . ‘ i... - I.
o 50 100 150 200

PDT (channel)

Figure 3.10 This is a plot of a PDT versus Slow spectrum of detector 6-8 of
the Miniwall. Visible are the different Hydrogen isotopes, protons, deutrons,
and tritons, as well as the Helium isotopes, 3He and 4H6.

 

 

 

52

3.2 Energy Calibration
3.2.1 Light Output Equation

To find the energy of a detected particle one looks at the light output of the
CsI(Tl) crystal. This light output is dependent on both the mass and the energy of the
particle. This relationship was found by bombarding the CsI(Tl) crystals with different
types of charged particles of known energies. The dependence of the light output can be
written as a function that is dependent on Z, the charge of a particle, and E, the energy
absorbed by the crystal from the particle [Stor 58, Quin 59, Alar 86, Gong 88, Souz 90,
Kim 91, C010 92]. The following is the light output equation used for the Miniball

detectors [Kim 91, Schw 94]:
L(E,Z)= 0t(z)E + [3(2) [e'KZ’E-l] (3.8)
E is the energy of the particle and Z is the charge of the particle. The parameters 0t, [3,

and 'y are also functions of the charge of the particle. Equation 3.8 predicts a light output

that is not linear, but the value of the actual parameters are such that once the energy goes

above 10 MeV/A the dependence of L on E is nearly linear.

The three parameters used in equation 3.8 are all functions dependent on the
charge of the particle. It has been proposed by Colonna et a1. [C010 92] to use the

following method for defining these calibration functions.
a(Z) = a1 + 6126—032 (3.9a)
3(2) = b1+b2e—b32 (3%)

7(2) = Cl + 626—632 (3.9c)

53
The new parameters 61,-, b,, and C,- are all constants that are fitted to the calibration data.
Using this method, the light output from the light particles (Z _<_ 2) can not be fitted that

well, so different on, B, and “y constants are found for the light particle data.

Since the thickness of the CsI(Tl) crystals of the array are known, 2 cm for the
Miniball and 3 cm for the Miniwall, the punch through points can also be used to help
calibrate the detectors. Comparing the punch through energies with those generated from
equation 3.8, it is found that the light output from the detector should be about 20 i 5%

greater than what was observed for the punch through of a—particles. This difference was

determined to be caused by a saturation effect within the PM tube.

3.2.2 Saturation Effect

This saturation was observed at large signals and was found to be related to the
gain of the photo tubes. The capacitors on the final dynodes of the tubes were not large

enough to prevent reduction in the inter dynode voltages for these signals.

T o evaluate the saturation effect, 10 different detectors were directly bombarded
with low intensity beams of 6Li, 12C, and 18O, with known energies ranging from E/A =

22 - 80 MeV. Since the electronic circuit transmitting the slow signal to the ADC has a
passive current and a well defined attenuator, the QCD Channel, Ch#, of the slow signal
is proportional to the signal from the photomultiplier tube. By comparing the observed
photomultiplier signal with that predicted for the light output by equation 3.8 a fractional

deviation is obtained. The fractional deviation can be defined by the following equation:

f(Ch#) = Ch#norm - Ch#L(E,Z) (3.10)
Ch#L(E,Z)

 

54

The fractional deviation, f(Ch#), is the fractional amount that the data lies below the
calibration curve and is presumed to be a function of Ch#. Ch#norm is the observed QDC

channels normalized so that the proton punch through of the detector is in channel

240.37. This normalization takes advantage of the fact that proton signals are small and

have little saturation. The value of Ch#L(E,Z) is the channel calculated from equation 3.8

given all the known energies of the particles, where it has been normalized in the same
way as the QDC channel. It was found that f(Ch#) depends on the pulse height from the

photomultiplier tube.

The fractional deviation for 12C data was found and is plotted in the upper panel
of figure 3.11. The data shows that, for large QDC channels, the observed pulse height is
less than what is expected from the light output calculated in equation 3.8. The 12C data
was then fitted with the straight line, shown in both panels of the figure. The lower panel
of figure 3.11 shows that this line also describes the saturation effect for the 6Li data. The
fit of the 12C data is also in agreement with the saturation effect observed for 4He, 6Li,
10B, 12C, 14N, 16O, and 20Ne particles in previous experiments with the Miniball [Kim 91,
Peas 94].

By fitting the data in figure 3.11 a linear relationship between the QDC channel

and the fractional deviation is obtained;

f(Ch#): mfd * Chaim", (3.11)

where mm is the slope of the line. When this relationship is combined with equation 3.10

the following equation is obtained:

55

 

| l
.0 .0 .0
I-P N O

lllLlLllllllll

Cb

 

lllllllIllllllll

     

|
.09
00>

fractional deviation
I
.0
m

|
.0
4s.

lllillllllllll

 

 

IIIIIIIIIIIIII

[JIIIILIJJLIIIIILJLJllIIlllll

500 1000 1500 2000 2500 3000
QDC channel

 

l
.0
on
c

Figure 3.11 The upper panel shows the results obtain for the fractional
deviation when using 12C as the reference particle. The line is linear fit to
the 12C data and defines the function f(Ch#), the fractional deviations. The
light output for the Miniball and Miniwall was less then the output
predicted by equation 3.8. The lower panel shows how the result for f(Ch#)
obtained from the 12C data compares to the deviation observed for the 6Li
data.

56

Ch#nm.m
1 ‘1' mm * Ch#norm

 

Ch# L(E, Z) = (3.12)

Equation 3.12 is used to find the QDC channel corresponding to the light output equation
after the saturation effect is taken into account. This light output equation can then be

inverted to find the energy. Figure 3.12 shows the corrected (closed points), and
uncorrected (open points), calibration points for 4He and 12C. The graph shows how well
the corrected data fits the energy calibration curve, obtained from equation 3.8, at the
higher energies as opposed to the uncorrected data. The correction of the 12C data at 250

MeV is about 10% and the correction for the He data at 300 MeV is about 30—40%.

The errors in finding the correct QDC channel with the fractional deviation can
first be traced to the data points in figure 3.11. The fractional deviation values in figure
3.11 vary at fixed channel numbers by about 10—15%, which may reflect differences in the
electronic components used to construct the PM tube. This fluctuation presumably could
be removed by performing detailed calibrations of each detector. It is also expected that
averaging over many detectors with the same polar angle will reduce the sensitivity to this

systematic calibration uncertainty.
3.2.3 Determining the Parameters
There are now 13 different parameters to be fitted, including mfd. These

parameter are fitted by minimizing the x2 associated with the difference between the

measured and calculated light output curves, using the constraint that none of the

calibration lines cross each other. Using this method mfd is found to be -1.7x10’4 for

57

 

 

 

 

— I I I I I I I I I l I I I I I I l
1500 — 0 12C corrected —
- O 12C uncorrected -
- 0 4He corrected -
- 0 4He uncorrected ~
to — .
a 1000 —-— $ -—
1:: _ _
c: _ ‘i’ -
to
.5 _ _
.— c) _
500 P “ —
O .__ l l l l l l l I l l l l I l 1‘
O 100 200 300

energy [MeV]

Figure 3.12 This is the calibration curve with the saturation correction for
the 4He and the 12C particles. The top line is the 4He energy calibration
curve and the bottom line is the 12C energy calibration curve using equation
3.8. The filled in symbols are the energy calibration data points using the
saturation correction. The open symbols are the energy calibration data
points without the saturation correction.

58

the Miniball and -1.860x10'4 for the Miniwall, and the values for a), 17,-, and c,- listed in

Table 3.1 are obtained for Z/A = 0.5.

Figure 3.13 shows the calibration curves and calibration data after all the

corrections have been made and all the necessary constraints have been met. The insert in
figure 3.13 shows how well the data fits at the lower energies. The 6Li and 10B points
tend to fluctuate about their respective curves. This may reflect the fact that not all the
calibration data were taken in the same experiment. For the energies larger than 30
MeV/A the calibrations are within 10-15% of the data points. At the lower energies the

uncertainties are of the order 5-10% [Schw 94, Hsi 95].

 

59

Table 3.1 Values for the parameters a,, bi, and c, for the functions 0t(Z),
[3(2) and y(Z) given in equation 3.9a, 3.9b, and 3.9c, where Z/A = 0.5.

 

 

 

 

 

 

 

 

xz/n=3.27 i=1 i=2 i=3
a, 0.4142 4.995 7.824E-2
b, 229.7 5.039E-3 1.967E-4
ci 1.339E-2 5.925E-3 3.846

60

 

40000

3000

 

2000

corrected channels

1000

 

 

 

o 20 4o 60 so 100
E /A [MeV]

Figure 3.13 This plot shows the calibration curves for the detected particles
with Z/A = 0.5. Also plotted are the calibration data for: 4He, (solid
circles), 6Li, (open fancy crosses), 10B, (crosses), 12C, (open circles), 14N,
(solid squares), 160, (open diamonds), and 20Ne, (open squares) particles.

Chapter 4

Data and Calculations

This chapter will discuss the results of the data analysis and compare them to
various nuclear models. These models will be compared to the measured charge
distributions, multiplicities and transverse energies at the various incident energies of the
84Kr + 197Au system. In these comparisons, both the data and the models must be gated

on the impact parameter and have the same energy thresholds.

4.1 Impact Parameter
The impact parameter of a collision [Gold 80] can be obtained by looking at

several variables that one can obtain from the data. Examples of such variables are the

total charge particle multiplicity, NC, the total proton participant multiplicity, Np, or the
total detected charge, Zwt, since they are all sensitive to the impact parameter [Phai 92].
In this dissertation the charged particle multiplicity, NC, is used to select the impact
parameter.

4.1.1 Equation

To construct an impact parameter filter based on the charged particle multiplicity,

we presume that the charged particle multiplicity decreases monotonically as the impact

61

62

parameter increases. Under this relationship the following general geometrical equation

can be used to assign the impact parameter for a specific event:

13(NC) = béNc) = U; P(X')dX']m {[1:de P(X')dx']m. (4.1)

max

 

Here P(X) a P(Nc) is the experimental charged particle multiplicity distribution, NC is the
charged particle multiplicity, and Ndbmax) = 4 is the minimum bias charged particle
multiplicity corresponding to bmax. The “reduced” impact parameter, b , is the ratio of the
impact parameter for the particular event to the maximum impact parameter bmax
corresponding to the minimum bias of NC. According to this definition b ranges over the

interval 0 5 b g 1. At E/A = 200 MeV, bmax was determined to be 10 fm for Nc(bmax) =

4 using direct beam counting, and should be roughly the same at the other energies.

The charged particle multiplicity, NC, consists of all charge particles detected in

the Miniball/Minimal array. This includes the unidentifiable particles, like heavy

fragments that stop in the fast plastic and light particles that punch through the CsI(Tl).
Multiple hits in a detector increment NC by one, even when double hit events can be
identified. Double oc’s hits in one detector are an example of a clear double hit that is that
is counted as a single particle. Figure 4.1 shows the normalized probability distribution

P(NC) for the six incident energies. The reduced” impact parameter is indicated at the top

of each panel in the figure. Figure 4.2 shows the relationship between NC and b for the

six energies.

63

 

W 35 MeV .1;

E/A = 100 MeV _

 

 

 

 

 

 

. _
_ .. . _
0
_ .... . .3
10-3 WW—l - I :-
___1 .8 .8 .4 .2 .8 .6 .4 .2
10 ' 1- ..
A _. 55 MeV __. 200 MeV _
0 ...-mi 0
z I— —I— —
v 10"2 _ .. “Em” : 1‘. _
D-I _ . _,_ . ..
. o
— o -— o 4
10-3 W11111ll;_. . . 1 I ll. 1 ll 1 .1 , I . .1.
10_1 _ .a .a .4 .2 __ . .6 .4 .2 J
_. 70 MeV __ 400 MeV _.
_ .kqu‘ “.0 _
10-2 P __ “w _
.. 3 _ _
‘ o

.- . — . _
0-3 “HI“..IH..I....I...l.... .. . .l....i;.. . I ...4‘. .

0 10 20 30 40 50 O 20 4O 60 80

Ne

Figure 4.1 Normalized probability distributions for the charged particle
multiplicity NC for E/A = 35, 55, 70, 100, 200, and 400 MeV. The reduced
impact parameter, b, is shown at the top of the panels. The open circles
indicate the middle impact parameters, 0.3 < b < 0.66.

   
  
  
  
   

 

  

 

 

 

1.0 llllll'I'HH'"""""'|"""||||IIII_

84Kr+197Au :

0.8— E/A(MeV) —

_ <>85 .100 -

— ’55 [3200 -

‘ 070 .400 _

0.6— _

(FD ' _

0.4L _

0.2— _

— <> .Q%x _

0.0 llllllllllllil%ll$11lLllllllllllll Ill,
0 2o 40 60 80

Figure 4.2 The dependence of reduced impact parameter, b, upon the
charged particle multiplicity, NC, for E/A = 35, 55, 70, 100, 200, and 400
MeV.

65

4.1.2 Total Charge

If the Miniball/Miniwall was a perfect device, all the charged particles would be
detected. Unfortunately, particles stop in the target or in the external foil of the
Miniball/Miniwall detectors, or escape through the beam line hole in the forward array
and are not detected. At larger impact parameters most of the charge remains in a
projectile residue that passes down the beam tube, or it is in a target-like residue which

stops in the target or the external foils of the detectors. Figure 4.3 show the relationship

between the average total detected charge, <Ztot>, and NC for the six incident energies.

This figure shows that as NC increases so does the efficiency of the array.

Since the relationship between NC and b has been established for all six energies
we can plot <Zt0t> vs. b and to look at the efficiency of the array in terms of b. Figure

4.4 shows that as the b approaches zero (the largest value of NC), the efficiency of the

device improves for all of the incident energies. At the three highest incident energies

and bz O, the ratio of the detected charge over the total charge approaches the

geometrical efficiency (90%) of the experimental array.

4.2 Intermediate Mass Fragments

Using this impact parameter selection, examination of the properties of the
Intermediate Mass Fragments, IMF ’s, charged particles with 3 i Z 5 20, begins. General
phase space and barrier penetrability arguments [More 75, Frie 83, Lync 87, Gros 90,
Sobo 83] lead to the prediction that the fragment emission probabilities should exhibit a

strong initial rise as a function of temperature. However, at very high temperatures, the

66

 

 

MSU—94—008
120 r I I I I I I I I I I I I I I I T
- 84Kr+197Au -
100 - ——
80 — —
/\
*5 6O — —
4.)
N
V - —.
E/A (MeV)
40 —

0 35 .100
9 55 C1200 ‘

O I
20 70 400

 

 

Figure 4.3 The dependence of the total charge, <Ztot>, on the total charged
particle multiplicity for E/A = 35 , 55, 70, 100, 200, and 400 MeV.

67

 

120...]...IT..I..fiI

100 -

80%?th

    

 

 

 

4o- E/A (MeV)
035 0100
.55 0200
20" 070-400
0 I- ..
OO 02 0.4/\06 08 10
b

Figure 4.4 The dependence of the total charge, <Z,O,>, upon the reduced
impact parameter, b, for E/A = 35, 55, 70, 100, 200, and 400 MeV.

68

entropy of the system becomes so high that fragment production is suppressed. These
two conditions together cause the fragment multiplicities to exhibit a maximum at some
intermediate temperature, which may depend on the total charge of the fragmenting

system.

The dependence of the fragment multiplicity on impact parameter was measured
for projectile fragmentation of Au nuclei at E/A = 600 MeV. These measurements two
showed the fragment multiplicity to increase with decreasing impact parameter to a
maximum multiplicity, and then it decreases as the impact parameter is reduced further.
This suggest that there is an excitation energy where fragment multiplicity reveals a
maximum, and that the multiplicity decreases at higher excitation energy, as the system
begins to vaporize [Ogil 91, Hube 91]. Looking at central collision of 36Ar + 197Au for
incident energies of E/A = 35 - 110 MeV, the IMF multiplicity was found to increase with
the incident energies [deSo 91], but a maximum in the fragment multiplicity with
excitation energy was not observed. A decrease in the IMF multiplicity as the incident
energy increases [Tsan 93, Wien 93, Kuhn 93] was observed for central Au + Au
collisions at incident energies of E/A = 100 - 400 MeV; but the domain at lower energies
where the fragment multiplicities increase with incident energy was not observed. One
was lacking measurements of both the rise and the decline of multifragmentation at fixed
impact parameter for a single system, so that the incident energy corresponding to the
location of the maximum in the fragment multiplicity could be determined. Here such

measurements are presented.

 

 

 

69

4.2.1 Experimental Data

Figure 4.5 shows the observed mean IMF multiplicity, <NIMF>. as a function of

detected charged particle multiplicity, NC and figure 4.6 shows <N1MF> as a function

of b For E/A = 35 - 100 MeV, the dependence of <N1Mp> upon NC follows a common

trend roughly independent of incident energy: a similar effect was observed for 36Ar +
197Au collisions at incident energies ranging over E/A = 35 - 110 MeV [deSo 91]. Some
fragments from the statistical decay of projectile-like residues, are lost because they are
emitted to angles smaller than 5.4°. This loss is most important at the two highest
incident energies, and leads to an unknown reduction in the fragment multiplicities at the

medium to low values of N C. This problem has little effect on the data for central
collisions characterized by large values of NC. At E/A = 35 - 200 MeV, the peak IMF

multiplicity is observed at the most central collision characterized by the largest values of

NC. But at E/A = 400 MeV, the maximum <N1Mp> occurs at NC = 60 corresponding to

b = 0.4, and <N1MF> decreases for more central collision. This decline in (NIMF> is not

a consequence of charge conservation or due to a loss in detection efficiency in the

experimental setup, since the observed values of <Ztot> (shown in figure 4.3) increase

with NC and approach yields consistent with the geometrical efficiency.

The lower panel in figure 4.7 shows the relationship between the incident energy
and <N1Mp> for central collisions with b< 0.25. For E/A<100 MeV, the increase of

<N1Mp> is likely due to increases in the thermal excitation energy and in the collective

expansion velocity. Both drive the system towards densities less than 0.4p0 where

70

 

 

 

 

Figure 4.5 The dependence of the mean intermediate mass fragment
multiplicity, <N1MF>, upon the charged particle multiplicity for E/A = 35,
55, 70, 100, 200, and 400 MeV.

71

 

 

 

 

 

 

Figure 4.6 The dependence of the mean intermediate fragment multiplicity,

<NIMF>, upon the reduced impact parameter, b, for E/A = 35 , 55, 70, 100,
200, and 400 MeV.

72

84Kr+197Au, E<o.25

 

 

 

 

 

 

 

125W I I I I I I I I IIIIIIIIIIIIIIW
' — — unfiltered A MD ‘
100 _ filtered :l'b Q / / _
------ filtered Nc / ’
~ 0 Data / / -
I
75 _ no P. pot. / ............ _
A .. -
0
Z 50 — _
v _ -
25 — —
- -l
O
- 4
8 — —
/\ " .
E 6 h / . \ \ _
z - o / \ -
V . / \K
4 — / \ -—
l- . l \ -+
/ ~.\
2 —' / . "~.\. "
/ ..' .
.. / ,v' ..
‘ I
'. l l 1 1 I l Llllllllllllllllll
30 50 70 100 200 300 500

E / A (MeV)

Figure 4.7 The upper panel shows a comparison between the measured

mean charge particle multiplicity, <NC>, and QMD calculations for central
collisions as a function of incident energy. The lower panel shows the

corresponding values for the mean IMF multiplicity, <NIMF>. The solid
dots are the experimental data points. The corresponding statistical
uncertainties are smaller than the data points. The dashed lines are from
unfiltered QMD calculations. The solid and dotted lines are QMD
calculations filtered by the experimental acceptence. The solid square is the
prediction of QMD calculations without the Pauli potential.

73

fragment production is enhanced [Frie 90, Bowm 92, Bond 95, Gros 90] and generally
increases with temperature for O < T < 10 MeV [Frie 90, Bond 95, Gros 90]. For E/A >
100 MeV, the decrease of the multiplicities with incident energy very likely reflects the
suppression of fragment production at high temperature, T > 10 MeV and higher entropy
[Tsan 93, Bond 95, Gros 90]. (Note that bound fragments comprise a small portion of the
total phase space at excitation energies that significantly exceed the total binding energy
of the system.) In this domain, collective expansion can also suppress fragment emission
because the spatial variation in the collective velocity field makes for a small overlap
with the wave functions of bound fragments [Kund 95]. Figure 4.7 is consistent with
these expected trends and provides the first approximate determination of the energy at

which the decrease in fragment production commences.

4.2.2 QMD Calculations

A similar maximum at E/A = 100 MeV was predicted by microscopic molecular
dynamics models [Peil 89] for Nb + Nb collisions. It is therefore interesting to see
whether such models can describe the present data. The lower panel of figure 4.7 also

contains the result of QMD model calculations for central 8‘j'Kr + 197Au collisions for

<N1MF> versus the incident energy and the upper panel of figure 4.7 contains the result

for <NC> verses the incident energy. The dashed lines in the figure are the unfiltered

results from the QMD model including the Pauli potential as described in refs. [Peil 91,

Peil 92] and Chapter 1, where bmax = 10 fm is presumed. The solid lines show the

filtered QMD calculations after correcting for the experimental acceptance. Some of the

general energy dependent trends of the data are reproduced with the calculation, i.e. a

 

74

maximum in <N1MF> at E/A = 100 MeV and monotonically increasing values for <NC>-

However, the calculations greatly underestimate the values of NC and NIMF when E/A <

100 MeV.

4.2.3 QMD+SMM Calculations

The failure of QMD calculations to reproduce the large IMF multiplicities
observed at low incident energies has been attributed to an inadequate treatment of the
decay of highly excited heavy reaction residues produced in the QMD calculations. As
shown in refs. [Peil 92, Sang 92], the description of the statistical decay of a thermally
excited nucleus is not adequate with the present QMD model. The exact cause of this
deficiency of the QMD model is not clear, but could be attributed to the classical heat
capacities of the computation QMD nuclei that result in low residue temperatures that are
physically unreasonable. To correct for this deficiency, the decays of all fragments with
A 2 4 were calculated via the SMM [Bond 85, Botv 87]. The input excitation energies
and masses for the SMM calculations were taken from the QMD calculations; after an
elapsed reaction time of 200 fm/c. The basic parameters of the source, its mass,
excitation energy, and the resulting yields, do not vary drastically with the reaction time
[Peil 93]. Typical excitation energies of 5 - 8 MeV/A, were determined by subtracting
the ground state energy of each fragment from the total energy in the fragment in the rest

frame. For details about the determination of the ground state energies, see refs. [Peil 91,

Peil 92].

The results of these two stage calculations, QMD + SMM, are shown by in figure

4.8. The lower panel of this figure shows that the calculations over predict the <N1Mp>

75

84Kr+197Au, S<o.25

 

125 WW I I I I I I I I IIIIIIIIIIIIIIII

‘ — — unfiltered A -

100 __ —— filtered :l'b / , .. ., ’ .5
"""" filtered Nc /

- 0 Data / / ’ -

 

 

 

 

 

 

I I I I I I I I I I I I II IIIIIIIII
0 MW
30 50 70 100 200 300 500
E/A (MeV)

Figure 4.8 The upper panel shows a comparison between the measured
mean charge particle multiplicity, <NC>, and QMD+SMM calculations for
central collisions as a function of incident energy. The lower panel shows
the corresponding values for the mean IMF multiplicity, <N1MF>. The solid
dots are the experimental data points. The corresponding statistical
uncertainties are smaller than the data points. The dashed lines are from
unfiltered QMD+SMM calculations. The solid and dotted lines are from
QMD+SMM calculations filtered by the experimental acceptance.

 

 

 

76
for EIA < 100 MeV/A; this additional yield comes from the contributions from the decay

of the heavy residue in SMM stage. These calculations also under predict the <N1Mp> at

higher energies because many fragments produced by the QMD stage are evaporated

away in the SMM stage. When these calculation are filtered through the experimental
acceptance (solid lines), the values for <N1Mp> are very close to the data at E/A < 100

MeV. Note, that the IMF detection efficiency is significantly reduced when E/A < 100
MeV, reflecting the fact that the QMD+SMM calculations are peaked at the low kinetic
energies. This leaves the measured spectra and many of the predicted fragments below

the experimental thresholds.

The upper panel of figure 4.8 shows a comparison between the <NC> values for

the filtered and unfiltered QMD+SMM calculations and the data. In general, the filtered
charged particle multiplicities follow qualitatively the experimental trends. Yet there is a
tendency for the calculations to overpredict the production of charged particles at the

higher energies; most of the extra yield is in the form of light particles.

In the previous comparisons, the impact parameters for the data were determined
using equation 4.1, while the impact parameters for the calculations were among the input
parameters to the calculations. To check whether the discrepancies between the
calculations and data are caused by the impact parameter filter, calculations were
performed in which the QMD and QMD+SMM calculations were impact parameter

selected as if they were experimental data. The dotted lines in both figure 4.7 and figure

4.8 are calculations in which the impact parameter was defined by the calculated NC

77

values. The latter calculations are essentially the same as the calculations previously

shown for a fixed theoretical impact parameter.

4.2.4 SMM Calculations

It is interesting to investigate whether the discrepancies between the
measurements and the QMD+SMM calculations reflect a fundamental limitation of the
SM model or whether a satisfactory agreement might be obtained with input parameters
that are chosen independently of the predictions of the QMD model. To explore this
issue, SMM calculations were performed, varying the excitation energy, density, and
collective flow of the fragmenting system. Collective flow was calculated in the limits of
a purely rotational flow and a self-similar radial flow. The collective velocity fields for
) and radial ("v

the rotational (V )flow were given by V"), = E) <8? and the

rat rad

, respectively, where 0) is the angular velocity and Bexp is the radial

expansion velocity. Since two-fragment correlation provide evidence that the limit of a

single freezeout time is not attained [Corn 95], only a fraction fA of the system containing
a fraction f}; of the total excitation energy per nucleon is presumed to be equilibrated.
The remaining excitation energy and mass of the system is presumed to be contained in a
preequilibrium source that is too hot to emit fragments [Bond 95]. The parameters fA, f5,

and rotational or radial flow were adjusted to reproduce the fragment multiplicities,
charge distributions, and transverse energies, respectively. The resulting parameter

values are given in table 4.1.

Figure 4.9 compares the SMM calculations for <N1MF> to the data. Unfiltered and

filtered rotational flow calculations are depicted by dot-dashed and solid lines,

Table 4.1 Parameters of SMM calculations chosen so that calculations

will match the experimental data.

78

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FEED/A (MeV) Set fA fE <Emt/A> (MeV) <E,/A> (MeV)
35 1 0.650 0.890 1.2 0.0
35 2 0.650 0.850 0.0 0.9
55 1 0.650 0.760 2.5 0.0
55 2 0.595 0.710 0.0 1.9
70 1 0.625 0.740 4.0 0.0
70 2 0.563 0.670 0.0 3.0
100 1 0.568 0.760 8.0 0.0
100 2 0.509 0.665 0.0 6.0
200 1 0.420 0.585 15.0 0.0
200 2 0.378 0.490 0.0 11.0
400 1 0.248 0.440 25.0 0.0
400 2 0.207 0.360 0.0 19.0

 

 

79

 

“Kr+197Au, ’B<o.25
. . ,..,.

125 5-1-...I....I....I....,....,........., f . l....,.m,....-
—---- ROTATIONAL ;
100_ _ _ RADIAL UNFILTERED ,. .
T — ROTATIONAL Arm" ‘.

. ...... RADIAL FILTERE3/4, . ,

0 DATA /, f"

 

 

 

 

 

 

8 _- ._

g :

A. -
I 1

Z l- .
V 4 r 1
2 :- _

0 P. . l . . . .l....I....I....l....l....l....| , , , , 1 . P, .LLL. ...,”1”, ‘

3O 5O 70 100 200 300 500

E / A (MeV)

Figure 4.9 The upper panel shows a comparison between the measured

mean charge particle multiplicity, <NC>, and SMM calculations for central
collisions as a function of incident energy. The lower panel shows the

corresponding values for the mean IMF multiplicity, <N1MF>. The solid
dots are the experimental data points. The corresponding statistical
uncertainties are smaller than the data points. The dashed and dot-dashed
lines describe unfiltered SMM calculations which include radial and
rotational flow, respectively. The dotted and solid lines are the
corresponding filtered SMM calculations.

80

respectively. Unfiltered and filtered radial flow calculations are depicted by dashed and

dotted lines, respectively. The conclusions are basically the same for radial flow as for
rotational flow. The <N1Mp> values for the unfiltered calculations exceed the

experimental data. When corrections for the experimental acceptance are made, the

filtered calculations essentially reproduce the experimental fragment multiplicities.

4.3 Charge Distributions
One of the earliest foci of the experimental investigations of fragmentation and
multifragmentation was the excitation energy dependence of the fragment charge

distributions. In many investigations the charge distributions were fitted by power law

distributions Y(Z) ~ A't. Such fits were motivated by arguments based upon the Fisher

liquid drop model phenomenology [Finn 92]. In calculations from these models,
fragments were presumed to be associated with clustering near the critical point of the

liquid-gas diagram, where relationships between thermodynamic parameters are largely

governed by a set of critical exponents [Fish 67, Wils 71]. In this picture, the exponent ’C

is derived from the power law fit of the charge distribution, and is expected to be largest

for reaction trajectories that pass through the critical point or critical region.

There are many issues that complicate this simple picture. One such issue is that
the Coulomb potential energy does not scale linearly with the volume; long range
Coulomb interactions complicate the extrapolation of increasingly heavy laboratory
systems towards the thermodynamic limit characteristic of critical phenomena [Fish 67,

Wils 71]. Aspects of such Coulomb effects are manifested by classical molecular

 

81

dynamics calculations of highly charged systems. Such calculations predict a monotonic

evolution of the fragment charge distributions from flat charge distributions at low

excitation energies, with ”E z 1, to steeply falling distributions at high excitation energies

[Pan 95, Kund 96]. Such a trend is inconsistent with the presumptions of critical

exponent analyses [Lato 95, Camp 88, Gilk 94, Mahi 88, Li 93] which presume a

minimum value of ”C will be observed at intermediate excitation energies where reaction

trajectories pass through a “critical region”. Local chemical equilibrium may also not be
achieved during the collision [Prat 95]. A loss of chemical equilibrium will alter
relationships between the observed fragment distributions and those characteristic of an
equilibrated system at the same temperature and density [Prat 95]. Where local
equilibrium is attained, experimental observables can differ from the equilibrium values
because post-breakup secondary decays of excited fragments [Fiel 87] change the
fragment charge distributions from those of an equilibrated system at breakup. All the
above effects were presumed to be negligible in previous analyses [Gilk 94, Mahi 88, Li

93].

Extremely flat charge distribution have been observed in central Au + Au
collisions at E/A = 35 MeV [D’Ago 95]. While this observation offers support for the
predictions [Pan 95, Kund 96] for highly charged systems to have a monotonic decreasing
slope of the fragment charge distributions with increasing incident energy, an
experimental demonstration of such an energy dependence was not provided until this

dissertation.

 

82

4.3.1 Experimental Data
The charge distributions for the central collision, b< 0.25, of the six incident

energies are shown in figure 4.10. These distributions decrease monotonically with

fragment charge and can be described by a power law of the form Y(Z) = C . Z". Fits

using this functional form, shown by the solid lines, are very close to the experimental

yields.
In figure 4.11 the extracted values of 12 are shown as the solid points. The

observed trend of monotonically increasing values of ”C is consistent with classical
molecular dynamics calculations for highly charged systems [Pan 95, Kund 96]. This
trend is contrary to the presumptions in critical exponent analyses of minimum T values at

intermediate energies. The lowest value obtained for T (T = 1.4 at E/A=35 MeV) is

somewhat more than half the value I = 2.3, which characterizes the mass distribution of a
system at the critical point in the liquid—gas phase diagram [Fish 67, Wils 71].

The charge distributions and the power law fits for the data measured at mid-
impact parameter collisions, 0.35 < b < 0.45, are also shown in figure 4.12. Figure 4.13
show the corresponding values for the extracted slope parameter 1? as a function of the

incident energy. The values taken from these measurements follow the same trends as the

central collisions.

 

83

84Kr+197Au, 3<o.25
. , .

 

 

 

 

' l ' l I l I I
E/A 0 Data
0 _ 35 MeV —Z" _.
10 (1100)
o
<55 ...),
. x10- .
o
70 MezV
_2 _ x _ ..
10 ( 10 ) .
100 MeV
. (no-3) .
A o
5 2(00 MeV
a. x10"
_4 __ _.
10 .
400 MeV °
. (:10_5) .
10"6 ~— —
10—8 . I . I . 1 . l . I . I
o 2 4 6 8 10 12
Z

Figure 4.10 The probability of emitting a fragment of charge, Z, per central
collision at the six incident energies. The solid dots are the experimental
data points. The statistical uncertainties of the data are smaller than the data
points. The solid line is the power law fit.

84

' 84Kr+197Au, 8<o.25

 

 

 

 

 

 

6W I I I I‘IWfiIIIIIIIIIIIIIIIII‘
- 0 Data -

' ' QMD no P. pot. :
5_ QMD + P. pot. __
. - — QMD+SMM E g -

- / ~ 4 .

- / _

.. / -
4 _ / 'r '—
- / 2 -

.. 1- 1

- /I/ 3 -

I. ..
1~a e —
[- .
2 — ...
1 — —.
Own“ 1 1 1 1 I 1111'1111'1111'1111
30 5O 70 100 200 300 500

E/A (MeV)

Figure 4.11 Comparison of the energy dependences of the charge
distribution exponent T from the data to QMD and QMD+SMM calculations

for central 84Kr + 197Au collisions. The solid line describes the T values from
the QMD calculations using the Pauli potential. The dashed line describes
the ”C values from QMD+SMM calculations. The solid square is the 17 value
at E/A = 50 MeV from a QMD calculation that does not use the Pauli
potential. The solid points are values for ”l? extracted from the experimental
data.

 

85

84Kr+197Au, o.35<’13<o.45
. , . , .

 

 

 

 

' l l ' l ' l
E/A 0 Data
0 _ 35 MeV —Z" _
10 (1100)
o
_ 55 Mgv .
(.10 ) .
_2 '70 MgV
10 —(x10 ) . -
100 MeV
. (rm-3) .
A 0
EN: 200 MeV
10—4 _ (‘10-4) __
4 0 MeV
. .10-5) .
10”6 - ..
10—8 . 1 . 1 . 1 . 1 . 1 . 1
o 2 4 6 8 1o 12

Figure 4.12 The probability of emitting a charge, Z, per mid-impact
parameter collision at the six incident energies. The solid dots are the
experimental data points. The statistical uncertainties of the data are smaller
than the data points. The solid lines are the power law fits.

86

 

 

 

 

 

 

84Kr+197Au, o.35<’b‘<o.45
6W I I I I I I IITIIIIIIIIIIIIII;
- 0 Data -
5 _ I QMD no P. pot. _'
- QMD + P. pot. _
- — - QMD+SMM -
4 — _
l~ 3 — —
2 —- _
1 - _
0 WW I l J 1 I I l I l I llllIllllIllll-
30 50 70 100 200 300 500
E/A (MeV)
Figure 4.13 Comparison of the energy dependences of the charge

distribution exponent 'c from the data to QMD and QMD+SMM calculations

for mid-impact parameter 84Kr + 197Au

collisions. The solid line describes

the ”C values from the QMD calculations using the Pauli potential. The
dashed line describes the ”C values from QMD+SMM calculations. The solid
square is a 17 value at E/A = 50 MeV from a QMD calculation that does not

use the Pauli potential. The solid dots
experimental data.

are values for ”C extracted from the

87

4.3.2 Comparisons with QMD Calculations

Classical molecular dynamics simulations like those of reference [Pan 95], can

reproduce the monotonically increasing values for ”C. Because nuclei are quanta] systems,

it is relevant to know whether similar agreement can be obtained with molecular
dynamics calculations which attempt to respect the Pauli exclusion principal. As
mentioned in section 4.2.2 and 4.2.3, there are difficulties with the present semi-classical
molecular dynamics calculations. To further illustrate these problems, we performed
analogous fits of the charge distribution predicted by the Quantum Molecular Dynamics
model, QMD, of reference [Peil 92, Peil 93] in which the Pauli principle is modeled by a

repulsive “Pauli potential” which acts in phase space.

Fits of these calculations for central collisions are depicted by the solid lines

figure 4.11. These calculation show a much more gradual dependence of 13 upon incident

energy than which is experimentally observed. The solid square at E/A = 50 MeV in
figure 4.11 shows the corresponding calculations without the Pauli potential as described
in reference [Peil 89]. Calculations for the charge distribution using the code of reference
[Goss 95], which also does not include a Pauli potential are not available. Figure 4.13
shows the corresponding calculations for the mid-impact parameter collisions. The trends
of the experimental data and the QMD calculations with the Pauli potential are similar to

those observed in the central collisions.

4.3.3 Comparisons with QMD+SMM Calculations

Discrepancies between theoretical and experimental fragment multiplicities and

charge distributions at low incident energies may be traced to the production of hot

 

88

residues within the QMD calculations. These residue do not decay by fragment emission
due to their high classical heat capacities [Lync 92]. As mentioned previously, this
problem can be relieved by allowing these residues to decay via statistical models [Bowm
92, Sang 92, Kono 93]. The input excitation energies and masses for the SMM
calculations were taken from the QMD calculations at a elapsed reaction time of 200

fm/c.

The dashed lines in figure 4.11 are the results obtained for the central collisions
when the decays of all fragments with A 2 4 were calculated using the SMM [Peas 94,

Botv 87, Bond 95], which contains a “cracking” phase transition at low density. The

larger values for ‘t which characterize the QMD+SMM calculations indicate that the

corresponding fragment charge distributions are much steeper than those experimentally
observed, even for incident energies E/A = 55 and 70 MeV where the fragment
multiplicities are well reproduced [Peas 94]. The dashed lines in figure 4.13 depict the
results for the intermediate impact parameters. The trends initially observed in figure

4.11 for the central collisions are also observed at intermediate impact parameters.
4.3.4 Comparisons with SMM Calculations
Figure 4.14 shows a comparison between the values of I obtained from the

experimental data and those obtained from the SM calculation presuming rotational
(solid lines) and radial (dotted lines) flow. The excitation energies per nucleon, EVA, of
the SMM calculations were essentially constrained by the slope parameters, T, of the

fragment charge distribution. These EVA values are smaller than the total available

excitation energy per nucleon. This fact is reflected by the small values for fE given in

be

89

84Kr+197Au, B<o.25
'T—ITT'T'I'I‘I'TTI'I'I'I'I'I'I'I'I'ITTfifIfifl'I'flTI'I I I I rIIIIIIIIIIIIIIIIIIII

 

0 Data

SMM WITH -
_ ROTATIONAL FLOW _

— ------ SMM WITH ~
- RADIAL FLOW -

' -'-" SMM BEFORE DECAY '
' WITH ROTATIONAL FLOW

— — SMM BEFORE DECAY
WITH RADIAL FLOW

 

-1 ,.......__i

I
I

I I I I I I I

I I j I I I I

 

 

 

I I I I I ILIIIIIIIIIIIIIIIII
30 5O '70 100 200 300 500
E/A (MeV)

 

Figure 4.14 Comparison of the measured energy dependences of the charge
distribution exponent”: from the data to SMM calculations for central 84Kr +

197Au collisions. The dashed and dashed-dotted lines describes the T values

from SMM calculations assuming radial and rotational flow, respectively,
before secondary decay is allowed to occur.

decay of the excited fragments. The solid dots are values for T extracted
from the experimental data.

The dotted and solid lines
describes the corresponding 1? values for SMM calculations after secondary

90
table 4.1, which denote the ratios of the excitation energies per nucleon used in the SMM
calculations to the total available excitation energies per nucleon at the various incident
energies. The local thermal environment at breakup is much cooler than would be
expected if the total available energy were thermalized.

Neither nature nor the SMM model have any difficulty producing fragmentation

events with a slope parameter I” that is far less than the minimum value, I = 2.3,

characteristic of a macroscopic (and therefore neutral) liquid-gas phase mixture at the
critical point. These low values for ”C become even more difficult to explain via critical
exponent analyses [Camp 88, Mahi 88, Li 93, Gilk 94] when one considers the influence
of secondary decay on the fragment charge distributions. Present critical exponent
analyses require knowledge of the fragment charge distributions before secondary decay.
Experimental measurements provide the charge distributions after secondary decay. The

charge distributions after secondary decay are predicted to be steeper than the ones before

decay.

To illustrate the magnitude of such effects, secondary decay was calculated with

the SMM model and the charge distributions before and after the decay were compared.
The dot-dashed and dashed lines in figure 4.14 depict the values for I obtained from the
calculated charge distributions before secondary decay for rotational and radial flow,
respectively. Note that the differences between the ”C values before decay and values

given by the corresponding solid and dashed lines after decay are very large. Indeed the

91
predicted T values (I < 1) before secondary decay cannot be understood using the Fisher

liquid drOp phenomenology advocated by proponents of critical exponent analyses.

4.4 Multiplicity Dependence of the Charge Distribution
The behavior of fragment charge distributions may be significantly altered by the
breakup geometry [Phai 93], and finite size effects [Phai 93] for the nuclear systems that
can be produced and studied in the laboratory. Some of these alterations arise from the
constraints imposed by conservation laws, and others from the time dependence imposed

by the expansion and disintegration of the system into the surrounding vacuum.

The influence of charge conservation constraints upon the intermediate mass
fragment (IMF) dependent charge distributions was studied in references [Phai 95, Ferr
96, More 95]. There, the fragment charge distributions were found to follow a simple

exponential relationship:
PNIMF (Z) 0‘: exp[—(a0 + C ° NIMF)Z] (4.2)

Here NM: is the multiplicity of intermediate mass fragments, and Z is the charge of a
specific element. The variables or" and c are constants dependent on the system and its

excitation energy. Studies with the percolation model [More 95] suggest that 0 should be

inversely proportional to the charge of the prefragment (0 oc —Z-l— ).
o

The dependence of c upon the total transverse energy of the system E. has been

investigated [Phai 95 , Ferr 96, More 95]. The total transverse energy , E, is given by:

92

E, = 2E,- sin26i. (4.3)
i

Here, E, is the kinetic energy, 9,- is the scattering angle of the i-th particle and the

summation runs over the detected particles in an event. These investigations suggest that
c vanishes at small E. and increases with 13, until reaching a saturation value. The c = 0

(liquid-gas coexistence) region was interpreted [More 95] as being due to the presence of
a large residue with a mixed phase in the final state, which weakens the charge
conservation constraints on the fragment distributions. The larger values of c were
interpreted as reflecting constraints imposed by charge conservation for a final state

consisting of a gas of nucleons, and fragments without a large residue [More 95].

The relationship between excitation energy deposition and charge conservation
constraints can be explored in two ways. One is by varying the impact parameter at fixed
bombarding energy and the other is by varying the bombarding energy at fixed impact
parameter. Both techniques have their disadvantages. Collision geometries and source
sizes vary significantly with impact parameter [Phai 92] while flow and reaction time
scales vary significantly with incident energy [More 93, Bowm 93, Kim 91, Lace 93,

Corn 95]. Figure 4.15 shows the extracted values of c as a function of the total transverse
energy for the six incident energies. The values for c increases rapidly with E. from the
negative values observed in peripheral collisions at smallest 13,, except at the higher

energies. It passes through zero and then increases slowly and comes to a maximum

value between 0.02 to 0.03.

0.03
0.02
0.01
0.00
-0.01

-0.02
0.03

0.02

O 0.01
0.00
-0.01

-0.02
0.03

0.02
0.01
0.00
-0.01
—0.02

MeV.

93

84Kr+ 197.3.11, 2

=8

 

I I I I I I I I I I ITT I I

III

9

IIIIIIIIIIIIIITII
q.:1!|E!:I

IIIIIIIIIIIIIIIIIIIIIIIIIIIII

IIII
'0

-- I I I I I I I I I I I I I I
P d
b q
— q
_ —
— d
p -
h -
- . Ill
_ ............................... —
- -
:0 :
- -
_

 

IIII
NB

IIIIII

IIIII

I

; E/A= 35 MeV :—- 55 MeV :
E::::':H.L':::::11H'1111'1:Hf
5 l I ' I | z
E ‘ilflflhI‘UHUI'liulfl.

IIIIIIIIIIIIIII IIIIIIII

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIII
0-0-0
NH
qr u.
ul- .
__. I
P I
I
I
'-‘ o
O 0
—CD I
II
E% .
<: no

 

--
-
_—
-
-

.Q
——o
3
co
<:

NM
NM
F.*

IIIII
IIIII

7-

 

HI
m
1.4
10
to-
“‘01.:

 

MN
1111I1111
IIIIITIII

 

IIIIIIIIIIIIIIIIIIIIIIIIII

I 200 MeV

I I I I I I I I I I I I I I

 

400 MeV

I44 I I I IIJ I I I I I I

 

[III 1111
III III!

 

O

500 1000

0 500 1000 1500

Et(MeV)

Figure 4.15 Charge conservation parameter c as a function of transverse
energy E. for the 84Kr + 197Au system at incident energies of E/A = 35 - 400

94

The observation of c < 0 indicates that the fragment distributions become flatter as
fragment multiplicity increases. Such a trend cannot be explained as being caused by
charge conservation. Small E corresponds to peripheral collisions where the excitation
energy is primarily carried away by undetectable neutrons and not by detectable charged
particles. Thus the constraint of small E, may be rather ineffective for constraining the
excitation energy deposition at large impact parameters. Shifts in the mean impact
parameter can occur at fixed B. when additional gates are placed on the IMF multiplicity,
because the IMF multiplicity is strongly impact parameter dependent at large impact

parameters. This may be the cause of the negative c values.

To explore the magnitude of these cross gating effects, a-oc azimuthal correlations

were analyzed at a fixed E. as functions of Nmp. Correlations contain information about
the angular momentum and directed transverse flow. Previous analyses have shown that
these correlations are more anisotropic at larger impact parameters [Phai 92] consistent

with larger angular momentum or “rotational flow” effects. Figure 4.16 shows the OL-Ot

correlations of 84Kr + 197Au reaction at E/A = 35 MeV for events with different values of
the IMF multiplicity and for E, < 100 MeV. These anisotropies decrease with N IMF,
consistent with smaller average impact parameter and larger average excitation energy
deposition when the events have a large IMF multiplicity. This suggests that the negative
values of c may be due to the impact parameter bias, and to the flatter charge distributions
obtained from more central collisions [Kim 92]. The vanishing values for c in figure 4.15
and ref. [Phai 95, Ferr 96, More 95] only reflect a mixture between collisions with poorly

defined impact parameter having a negative c and more central collisions having a

95

84Kr+197Au, E/A=35MeV, Et<100MeV

 

 

 

 

1.2-IIITTIIIIIIIIIIIIdI-IIIIIIII IIIIIIIIq
E9 N _1 35 N -2 i
1.1— IMF—— {a IMF" _n
r9 9 :: 0 0.;
1.0? C Q —__— . O —:
1— —u- . —1

: 0 :: o :
0.9? ’0. 1T .. —_‘
08:":14411'1'"11+l:'1111111'1':1:1":

. :II I IIIIIII::I j I II:

_ —1— .J
11:— NMF=3 _"_ NIMF=4 _"
A ° - -- -
g 2'. 001:. ..2
8.10;. . LL.. . _'
<1" ° ° :: o. :
E: E '0’ :: °° :
0.9.— ": —_
OBZHHHHIlHHlHI'HHHtI‘t'Hl'i:
o: I ::r II I I:

.. * .J
1.1; Nnm=5 _‘"_ NILE—6 J
:1} :: §
1.0;? géééfl'III +++_L
2 i999 :: III :
0.95- —IZ—- —i

 

 

 

 

08 IIIIIIII IIIIIIII IIIIIIIIILIIIIII
O

90 0 90 180
495... (deg)

C

Figure 4.16 IMF multiplicity gated oc-oc correlation functions for the 84Kr +
197Au system at E/A = 35 MeV and E < 100 MeV.

96

positive c. Therefore 0 = 0 does not provide an definite signature for the co-existence

region [More 95].

Figure 4.15 suggests that it may be easier to investigate c in central collisions
where the charge distributions may be less sensitive to impact parameter and the

experimental impact parameter selection may be sufficient. Thus we obtain values of c

for a central collision gate of b< 0.25 as before. The solid point in left panel of figure
4.17 are the values of c that come from fitting the multiplicity dependent charge

distributions for fragments with Z 2 3. These points are gradually increasing as the

incident energy increases. This is consistent with the requirements imposed by charge
conservation accompanied by the general increase in the charge particle multiplicity with

incident energy and excitation energy deposition.

Such considerations are generally embodied in statistical multifragmentation
models [Char 88, Frie 90, Gros 90, D’Ago 96]. To show how measurements of
multiplicity dependent fragment charge distributions can provide meaningful constraints
on statistical models, c was evaluated via SMM calculations. The SMM model
pressumes the bulk fragmentation of a thermalized prefragment [Botv 87, Bond 95]. A
more detailed description of this method was previously given in section 4.2.4 where
predicted values for the fragment multiplicities and charged distributions are presented.
Corresponding values for c from SMM calculations are plotted as the open points in the
left panel of figure 4.17. (All theoretical calculations have been filtered through the

experimental detection arrays.) The solid lines are drawn to guide the eye. In general, the

97

84Kr+197Au, ’B<o.25

 

0.12

 

 

 

 

 

 

WW I I I ITIIII-- I I I I I I I I I I I I I I d
I Zmin=3 I l :
0.10 ; . data __--__ _‘
' U SMM calc. -- -
a - -- _
5 3 ii :
N 0.08 — __ _
v - -- _
> - -- -
z 0 06 :— —::— _-
E: 3 I j
5 0.04 —- __ _
'1’ _ 1.4L”
° 3 IL" """"" “ ;‘
0.02 - —— _
I :ZE/A=200 MeV i

0.00 _LULIJLLIIJJJIIIIIIIIIIIIIIIJIJJ I I l 1I1111 1 1 1 1 I 14 1 1 J J 1 1 1

100 0 100 200 300
E/A(MeV) A0

Figure 4.17 Left hand panel: c plotted as a function of incident energy and
central collisions for data (solid points) and SMM predictions (open points).
Right hand panel: c plotted as a function of source size used in the SMM
calculations for E/A = 200 MeV. The dashed lines bound the value for 6

consistent with the experimental data.

98

predicted c is much higher than the experimental values indicating a stronger charge and

energy conservation constraint within these calculations than is experimentally observed.

The source sizes, A() , used in the SMM calculations were smaller than that of the

total system. For the total system of 84Kr + 197Au, the source size is A0 = 281, but in the
calculation A0 = fA . 281, where fA is given in table 4.1 for the different energies. To

show how much larger the calculated system should be in order to reproduce 0, AC was

increased in the calculations for 84Kr + 197Au at E/A = 200 MeV. While increasing A0:

1) The excitation energies (8.5 MeV per nucleon) and therefore the charge

distributions are held constant.

2) The expansion energy (11 MeV per nucleon) and therefore the mean transverse

energies are held constant.

The right panel of figure 4.17 shows the resulting calculated dependence of c upon A0. In
order for the calculated values of c to agree with the experimental measurements (bound
by the two dashed lines), the source size must be increased from 118 to 240.
Unfortunately, an increase like this in source size would result in the SMM calculations
predicting the mean number of IMF to be a factor of two larger than the experimental
multiplicities. This analysis suggests that testing the charge conservation constraints
contained in the multiplicity dependent fragment charge distributions provides unique and

useful information about the size of the thermalized prefragment.

99

4.5 Collective Flow

The low values of f5 in table 4.1 suggest that a sizable fraction of the total energy

is not thermalized. Experiments indicate that this energy is probably lost to collective
motion. The energy spectra and consequently the detection efficiency for fragments in
the experimental array are influenced by collective motion. Collective flow is not a
feature predicted by the SMM model, and must be incorporated into the kinematics of the

model by hand.

Previous investigations indicate that at low incident energies the attractive nuclear
mean field can support a form of “rotational” flow [Tsan 86, Botv 95]. At higher incident
energies the rotational flow decreases and flow becomes primarily outward or “radial” in
direction [Jeon 94, Hsi 94]. This “radial” flow is caused by pressure from nucleon-
nucleon collisions and from the repulsive high density nuclear equation of state in the

overlapping central region.

The dependence of the mean transverse energy, <Et>, upon the fragment charge
has been used [deSo 91] to experimentally constrain the collective flow. The mean
transverse energy , <E>, is given by:

2 E, sinze,
(15,) = i N (4.4)

 

Here E is the kinetic energy, and 91 is the scattering angle of the i-th fragment, and N is

the total number of fragments of a specific charge. Experimental values of <Et> for

central collisions, b< 0.25, are shown as the solid points in figure 4.18 for the different

100

84 197
100 “Kr+ Au, b<0. 25

E/A = 35 MeV 33 55 Mev 5

 

 

 

 

 

SMMWITH

 

25}

 

 

 

 

: 200 MeV 1 400 MeV l
0 .._L.I....I...-I ........ I ............ I..LLL....I ........ I..-.i. #-
3 4 5 6 7 8 3 4 5 6 7 8
Z

Figure 4.18 Comparison of the measured mean transverse energy, <Et>,
(solid points) to SMM calculations with radial (dashed lines) and rotational
(solid lines) flow as a function of fragment charge at E/A = 35, 55, 70, 100,

200, and 400 MeV and 13 < 0.25.

101

incident energies. As previously mentioned, SMM model calculations were performed
with excitation energies per nucleon, equilibrium source sizes, and collective flow
velocities chosen to optimize the agreement with the experimental fragment charge
distributions, total fragment multiplicities, and transverse energies. Table 4.1 contains
the values of the parameters usedin these calculations. The dashed and solid lines in
figure 4.18 show the mean transverse energies from SM calculations with collective

radial and rotational flow velocities respectively. In the calculations, the mean rotational
flow energy per nucleon <Emt/A> and the mean radial flow energy per nucleon <E,/A>

were chosen to provide the same transverse energy. This can be achieved by setting

2<Er>/ 3 = <Emt>/2.

Figure 4.19 shows values for <Et> calculated by the QMD model with (solid

lines) and without (solid squares) the Pauli potential. Hybrid QMD+SMM calculations
are shown by the dashed lines. In these calculations, the collective flow is a prediction of
the molecular dynamics solutions of the A-body equations of motion. Clearly the model
underestimates the transverse energies at E/A > 55 MeV. This shows that the
experimental data displays a transverse expansion that the fragments produced by QMD
and QMD+SMM models do not. This failure is probably related to the inability of the

models to reproduce the observed preequilibrium fragment emission.

4.6 Collective Expansion
It is impossible to reproduce the large fragment multiplicities shown in Figure 4.9
from statistical models without presuming that the system has expanded to subnuclear

density. The SMM model uses a typical breakup density of about 1/6 of normal nuclear

102

84Kr+197Au, b<O. 25

 

 

 

 

 

 

 

 

 

IOOP'HI'H'I'H'I """""" I """""""""" l""l""l""'
D
: E/A = 35 MeV _ 55 MeV
b 4L
I I: I
50." if . o O 9 1
I . /.‘.— . ’ . . I
’ o l ’ . ..—- -
p
25: I i.” ...... —— I 1
p u . I l d
. 1/1 a .
F ‘F 1
0 ::::I::::{::::}::::§::::§:::::::;: :::::::::::::::::::§::::::::::::::
F ‘F 1
F 1F 1
757 1.’ 1
50." o 9 “.7 . / 1
.... ' O , ‘r / "" / '
> 25L ,- " — .I'. J .‘
a) L .
2 , 70 MeV ’ 100 MeV
v o fit‘flltnflccnhnflnn:;;::%:fi:0:;::}:;::§:::;}:::::::::::::::::::
/\ . I .
b {p .
§1757 0 Data 1,- -.
F ‘F I
V I QMD no P. pot
150 " ‘
_ QMD + P. pot. .;
F (I
r — — ,
QMD+SMM
D 4»
F ‘F
F ‘F
100: -_- § § § § -
F F
p r
F F / A
I
50:- -- \ -\ -
F
I 0 \
25p .0- -
F {F
L l200 MeV I 400 MeV
o’.-..i....1....1 ........ 1 ........ ’....i....1....4 ........ i ........
3 4 5 6 7 6 3 4 5 6 7 6

Figure 4.19 Comparison of the measured mean transverse energy, <Et>,
(solid points) to QMD calculations without the Pauli potential (solid
squares) at E/A = SOMeV, QMD and QMD+SMM calculations with the
Pauli potential (solid and dashed lines, respectively), as a function of

fragment charge at E/A = 35, 55, 70,- 100, 200, and 400 MeV and b < 0.25.
Note, charged particles are not produced with Z > 4 within the QMD model
for 35 MeV at this reduced impact parameter.

103

matter density. Break up densities as large as 1/3 of normal nuclear matter density can be
used by the SMM model. Larger densities are not compatible with the SMM model,

because SMM and similar models require that the fragments do not overlap at breakup.

Slightly lower densities are possible, but presuming an equilibrium decay at very
low densities is not credible. However, the actual lower limit on the density is not
known. For a systems to undergo a bulk fragmentation at a reasonable value of the
density, it is necessary for the systems to expand over a short timescale. This reasoning

leads to the idea that there is some collective expansion in the reaction.

Direct evidence for a collective expansion has been obtained for mass symmetric
197Au + 197Au collisions by fitting the fragment energy spectra. This was done by fitting
with a source pararneterization that presumes the superposition of a self-similar radial
expansion and random (thermal) motion [Hsi 94]. Alternatively, similar information has
been obtained by comparing the mass dependence of the mean fragment energies in the
center of mass system with the expectations of models that superimpose a thermal
breakup upon a self similar radial expansion [Icon 94]. In either analysis, roughly 1/3 to
1/2 of the total energy per nucleon in the center of mass system is found to be carried
away by the collective expansion. The open diamonds in figure 4.20 represent the mean
collective radial expansion velocities extracted for the 197Au + 197Au system at E/A = 100
and 250 MeV by Wen Chien Hsi [Hsi 95]. The horizontal coordinate of the plot is the
energy per nucleon of a participant source calculated from the geometrical overlap of
projectile and target nuclei. For symmetric systems, it coincides with the total incident

kinetic energy per nucleon in the center of mass system. The open circles depict mean

104

 

o.4~~l . '-.~~I I
O 197Au+197Au, Jeong 8c Lisa
0 197Au+197Au, Hsi

I 84Kr+197Au

 

 

 

 

 

a 40Ar+4580, Pak 00
0.3 ' o _
T
O
F I O
/\ o 2 - ; _
Q .l.
V I
o T

 

 

 

 

0.0 ....l l . Ll....l I l .
5 10 20 50 100 200 500

E*/A (MeV)

Figure 4.20 The mean collective velocity. <B>, versus the excitation energy
per nucleon, EVA of the participant source. The solid squares are from the
analysis of the 84Kr + 197Au system. The other data are described in the text.

105

collective radial velocities extracted from light particle spectra measured in central

Au+Au collisions at even higher incident energies [Lisa 95].

In mass asymmetric systems, like the one studied in this dissertation, moving
source fits require more parameters than for symmetric systems, since they lack
symmetry (on the average) between the emission in the projectile and target spectator
domains. The results for collective expansion derived from thermal approaches, like the
SMM, may provide a more straightforward estimate of the collective expansion velocity.

The solid squares in the figure show the mean radial expansion velocity for the

84Kr+197Au system calculated according to (B) = 1.37 - [(Er)/(mNc2)]% , where mN is the

nucleon mass, as a function of the excitation energy per nucleon of the participant source.
Also shown are the flow values obtained for 197Au + 197Au collisions by Jeong et al.
[J eon 94] (open circles) and for 40Ar+45 Sc collisions by Pak et a1 [Pak 96] (open squares).
These systematics suggests that radial flow is essential determined early on in the
collision by the conditions that prevail in the region of initial overlap of projectile and

target nuclei.

 

Chapter 5

Conclusion

At the outset of this dissertation work, little was known about the impact
parameter or energy dependence of multifragmentation. While both dynamical and
statistical models for the fragmentation process had been developed, there was little
experimental data to constrain them. The existence of multifragmentation had already
been demonstrated in central collisions of Au + Au at E/A = 200 MeV. The results of
dynamical QMD calculations were roughly consistent with the Au + Au measurements.
Comparisons between the various models and various inclusive experimental
fragmentation data were also made. It was found that models which predicted a
multifragmentation decay and those which predicted the evaporative emission of one or
two fragments were both successful in reproducing much of the inclusive experimental
data. Information concerning the actual multiplicities of fragments could have

distinguished these models, but was unavailable.

By the summer preceding the dissertation measurements, information confirming
the existence of multifragmentation in central Xe + Au and semi-central Au + C, Al, and
Cu collisions became available. Questions concerning the optimal incident energy for
multifragmentation studies in central collisions became highly relevant. The present

study addressed many of these questions.

106

107

5.1 Main results

Chapter Four contains many of the important conclusions of this dissertation. The
most important result concerns the dependence of the fragment multiplicities for central
collisions on the incident energy. There was a maximum at higher energies in the
fragment multiplicity when the incident energy is about 100 MeV and a subsequent
decrease with incident energy, consistent with the onset of nuclear vaporization. This
observation clearly shows that most of the interesting energy range needed for nuclear

multifragmentation measurements is accessible with beams at the NSCL.

5.1.1 Fragment Yields

This experimental data allows comparisons with theoretical models. Of such
models, the most logical candidates for comparison were molecular dynamics models
such as the Quantum Molecular Dynamics (QMD) model of the Frankfurt-Nantes
collaboration [Aich 86, Peil 89]. When this dissertation was initiated, the Frankfurt
group had abandoned the original formulation of the QMD model in favor of a model
which included a Pauli potential which repelled nucleons from each other in phase space.
Due to problems in describing the statistical decays of hot reaction residues that may be
related to the classical heat copacities of the QMD nuclei. The Frankfurt group later
added a secondary decay stage to the QMD model creating the dynamical-staticistical
hybrid QMD+SMM model. Comparisons of this hybrid model to the measured fragment
multiplicities are given in figure 4.8. Reasonable agreement between experimental and
theoretical multiplicities are observed at low incident energies where the fragments are
predicted to originate principally from the multifragment decay from a single large

equilibrated composite residue. Within these hybrid simulations, the decay of this residue

108

is calculated via the Statistical Multifragmentation Model (SMM) model of the
Copenhagen-Moscow group [Bond 85 , Botv 87] wherein the yields of the various emitted
species are produced within an expanded and equilibrated system undergoing a “cracking
phase transition”. At higher incident energies, fragments are produced via the QMD
model during a rapid non-equilibrium expansion. Many of these fragments are too highly
excited to survive the secondary decay stages, however, and the predicted fragment yields
fall far below the experimental values. The predicted fragment charge distributions fall

more steeply with fragment charge than the measured ones as well.

Direct comparisons with the results of the SMM model were made to assess
whether the discrepancies with the QMD+SMM model calculations were essentially due
to an incorrect prediction by the QMD model for the excitation of the system during the
non-equilibrium initial stages of the collision. Excellent agreement with the fragment
charge distributions for central collisions were obtained by presuming an excited
prefragments with total charges, masses, expansion velocities and excitation energies as
free parameters. Excellent agreement with the measured fragment multiplicities, charge
distributions and mean transverse energies were thereby obtained. The presumed values
of the total mass and excitation energy of the equilibrated systems, however, were
significantly less than half of that available raising questions about the presumption in the
model that the early non-equilibrium stages produced a negligible fraction of the observed

fragments.

5.1.2 Charge Distributions

The dependence of the fragment charge distributions upon the fragment

multiplicity was used to further test the SMM. This observable is sensitive to constraints

 

109

imposed upon the multifragment decay configuration by charge conservation. Using a
method developed by the LBL group [More 75], where the charge conservation
constraints are related to a parameter c, the multiplicity dependence of the fragment
charge distributions was analyzed for the experimental data and for the SMM model
calculations, (which reproduced the average experimental charge distributions). The
values for c from the SMM model greatly exceeded the values for c obtained from the
experimental data. This result is consistent with the conclusion that SMM model

presumes an equilibrated prefragment that is much too small.

Some consideration was given to the incident energy dependence of the measured
fragment charge distributions. To discuss such trends compactly, these charge

distributions were parameterized by power law distributions of the form Y(z) = C . Z". In

contrast to measurements for smaller total systems and the naive expectations of

advocates of the Fisher liquid drop phenomenology, the measured values for I increase

monotonically with incident energy. Exploration of this effect within Lattice Gas Model
(LGM) calculations [Pan 95] reveals that this is a likely consequence of the destabilizing
Coulomb interaction, an feature that has been also observed in studies within the SMM
model [Mish 96]. If true, this raises serious questions about interpretations within models
like the Fisher liquid drop model, which attribute the abundant production of fragments to
clustering near the critical point in the liquid-gas phase diagram. Clearly, long range
interactions like the Coulomb force make contributions to the excitation energy which do
not scale as intensive quantities. To the extent that the multifragment decay
configurations reflect the Coulomb interaction, the result density fluctuations will not be

scale invariant as expected near the critical point.

 

110

Detailed examination of SMM calculations reveal other significant shortcomings
of critical exponent analyses when using the Fisher liquid drop phenomenology. These
analyses require that the observed fragment distributions are an accurate reproduction of
the fragment distributions at breakup. But between breakup and final detection, the
highly excited fragments predicted in thermal approaches will decay. Investigations of
these secondary decay corrections with SMM calculations shown in section 4.2.4 reveal
that the charge distributions before the secondary decay are flatter than those after
secondary decay. Thus, “critical exponents” extracted by fitting the observed fragment
distributions have nothing to do with those that characterize the fragment distributions at
breakup. Therefore they cannot be compared directly to the predictions of Fisher liquid

drop theory or to percolation theory without extensive corrections for secondary decay.

5.1.3 Collective Expansion

Indirect evidence for expansion in the data can be found in the comparisons of
statistical predictions to experimental data for the fragment multiplicities and charge
distributions. More direct indications of expansion can be obtained from the examination
of the fragment momentum distributions or the energy and angular distributions. Such
evidence was obtained by examining the energy spectra obtained for symmetric Au + Au
collisions in 1994 [Hsi 95]. Alternatively, evidence for the role of collective expansion
can be obtained by comparing the measured mean fragment energies to statistical
calculations with and without the superposition of a collective flow velocity [Icon 94,

deSo 93].

In this dissertation the mean transverse energies of fragments were so compared to

SMM calculation while superimposed thermal motions upon a collective radial

 

111

expansion. The values for this collective radial expansion velocity were then extracted.
Comparison of these values to those obtained for central collisions between symmetric
197Au + 197Au collisions indicates that a common systematics can be found if all the data
are plotted as a function of the excitation energy of a putative participant source. This
indicates that the collective expansion is determined in the early stage of the collision by

the dynamics in the region where nucleons from the projectile and target are overlapping.

5.2 Future Outlook

Putting into perspective the systematics obtained in this dissertation, a persuasive
case can be made for the exploration of 84Kr + 197Au collisions and other similar systems

at E/A < 100 MeV, i.e. at energies less than that corresponding to the maximum in
fragment production. At higher energies, our measurements indicate that systems expand
rapidly and there may be difficulties in determining the extent to which the reduction in
fragment multiplicities can be better understood as being due to a purely thermal
vaporization or as being due to low cluster production rates within a region of low phase
space density [Kund 95]. At the lower energies, on the other hand, there is considerably
more elapsed time between the initial overlap of projectile and target at the final breakup,

making it more likely that local thermal equilibrium can be achieved.

In this lower energy domain, it is very important to determine whether simple
freezeout approximations like those employed by the SMM model are sufficient or should
be replaced by rate equation approaches like that employed in conventional compound
nuclear decay or the Expanding Evaporating Source (EES) model [Frie 83a]. If a region

of validity or either approach can be determined, it becomes important to place

112

constraints upon the excitation energy, and in the case of the bulk multifragmentation
models like the SMM approach, information about the density at breakup. Such
investigations will greatly further and hopefully conclude this quest to determine the

thermal properties of hot and low density nuclear matter and its phase transitions.

 

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