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This is to certify that the

dissertation entitled

Multifragment Emission in Central Collisions of

36Ar+197Au at E/A=50, 80 and 110 MeV

presented by

Larry William Phair

has been accepted towards fulfillment
of the requirements for

Ph .0. degree in Physics

 

 

 

 

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Date 9/9/93

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MULTIFRAGMENT EMISSION
IN CENTRAL COLLISIONS
OF 36Ar+197Au AT E/ A = 50, 80 AND 110 MeV

By

Larry William Phair

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1993

ABSTRACT

MULTIFRAGMENT EMISSION IN CENTRAL COLLISIONS OF
35Ar+197Au AT E/A = 50, 80 AND 110 MeV

By
Larry William Phair

Multifragment disintegrations of highly excited nuclear systems may
carry information about the equation of state and the liquid-gas phase
transition of low density nuclear matter. The mechanism causing
multifragment decays is, however, not yet understood. To study this
phenomenon we constructed a low threshold 4n: charged-particle detector, the
Miniball, and studied the reaction of 36Ar+197Au at E / A=50, 80 and 110 MeV.

For central collisions, the chance for equilibration of the system is
highest. We therefore determined the most efficient method of selecting
central collisions by comparing the ability of several global observables to
select events with suppressed projectile-like fragment emission and small
anisotr0pic azimuthal emission patterns. Similar event selection was
obtained by using the following global observables: the charged particle
multiplicity NC, the total transverse energy E,, and summed charge emitted
at midrapidity 2,.

An average multiplicity of 4 intermediate mass fragments (IMF) was
observed for the most central collisions at E/A=110 MeV. These IMF

multiplicities are consistent with predictions from a statistical model for

evaporation from an expanding compound nucleus. The statistical decay
model predictions are sensitive to the low-density nuclear equation of state.

The IMF yields are also consistent with predictions from a standard
percolation model for the 36Ar+197Au system. Assuming a spherical decay
geometry, percolation fails, however, to predict the measured fragment
multiplicities for the heavier system 129Xe+197Au at E/A=50 MeV
investigated by our group. For noncompact decay configurations (bubbles and
toroids) the percolation model can reproduce the large fragment

multiplicities observed for this reaction.

to Kendra, Nathan and Jordan

iv

ACKNOWLEDGMENTS

I would first like to thank my advisor Konrad Gelbke for his time,
patience, teachings and leadership. His ability to cut to the core of a problem
has made my work at the laboratory fruitful and enlightening. Bill Lynch,
Betty Tsang, Scott Pratt and Hartmut Schulz have given their friendship as
well as invaluable instruction during the time of this analysis. Wolfgang
Bauer suggested and oversaw several theoretical projects, as well as sitting on
my guidance committee. Maris Abolins, Aaron Galonsky and SD. Mahanti
gave a careful reading of this work and also served on my guidance
committee.

Three postdocs gave me my first exposure to experimental work here at
the lab. Dave Bowman, Nelson Carlin and Romualdo de Souza taught by
example the virtue of hard work and took the time to share their knowledge
of experimental nuclear physics. As they left, other capable scientists took
their places. Nicola Colonna, Graham Peaslee, Carsten Schwarz, Thomas
Glasmacher, Bill Llope and Carlos Montoya have added their friendship and
guidance to my research.

My fellow graduate students have offered much support. Mike Lisa
never minded being a sounding board for my ideas (some of which were not
so bright, but he was always polite telling me so). He also had many helpful
suggestions for my analysis. Erik Hendrickson, Paul (Rooster) McConville,
Raman Pfaff, Don Sackett, George Jeffers, Mike Wilson and others helped me

survive my course work during the first two years of graduate school. I am

V

especially indebted to two graduate students, Cathy Mader and Paul
McConnville, without whom I am pretty sure I would not have passed the
comprehensive exam the first time. Observing their approach to problem
solving was very instructive. The graduate students in our group: Yeongduk
Kim, Hongming Xu, Fan Zhu, Tapan N ayak, Wen Guang Gong, Damian
Handzy, Wen Chien Hsi, Min-Jui Huang, James Dinius and Cornelius
Williams have all spent long hours building and maintaining the Miniball
and/ or running my thesis experiment. Easwar Ramakrishnan, Shigeru
Yokoyama, Afshin Azhari, Mathias Steiner, Phil Zecher, Tong Li, Dietrich
Klakow, Qiubao Pan, Brian Young, Jim Bailey, Michael Fauerbach, Stephan
Hannuschke, John Kelley, Eugene Gualtieri and others have also helped to
make my experience enjoyable and instructive.

All of the staff deserve my thanks, but I would like to single out Jack
Ottarson and Jim Vincent. I interrupted both of them countless times with
different projects, requests, problems, etc., but every time they were always
willing to help (quickly). This helped to smooth out what could have been
rocky times during the construction of the Miniball.

None of my education would have been possible without my parents
and the selfless hours they gave in teaching me everything (from music to
Scouting). They nurtured in me a desire to learn for which I am eternally
indebted.

And finally I am indebted to my wife, Kendra, for her love and
patience and to our two boys, Nathan and Jordan. When I came home late at
night there was something about a sleepy two-year-old saying "I love you to

come home, Dad" that just helped keep things in perspective.

vi

Table of Contents

List of Tables .................................................................................................................. ix
‘ List of Figures ................................................................................................................ x
Chapter 1 Introduction ............................................................................................... 1
Chapter 2 MSU Miniball ............................................................................................ 4
2.1 Mechanical construction .......................................................................... 4

2.2 Detector design ........................................................................................... 11
Phoswich construction ....................................................................... 11

Uniform scintillation response of Csl(Tl) ...................................... 15

Scintillator foils .................................................................................... 21

Light pulser system .............................................................................. 25

Chapter 3 Experimental Setup and Data Reduction ............................................. 29

3.1 Experimental setup .................................................................................... 29

3.2 Data acquisition electronics ..................................................................... 31

3.3 Particle identification ................................................................................ 34

Bicron crystals ....................................................................................... 43

3.4 Energy calibration ...................................................................................... 47

Chapter 4 Impact Parameter Selection.............., ..................................................... 53

4.1 Motivation .................................................................................................. 53

4.2 Definition of filters .................................................................................... 55

4.3 Comparison of relative scales ................................................................. 62

vii

viii

4.4 Suppression of projectile-like fragments .............................................. 71
4.5 Suppression of azimuthal correlations ................................................. 84
Azimuthal correlation functions ..................................................... 85
Comparison of relative scales ........................................................... 89
4.6 Directivity .................................................................................................... 103
4.7. Summary .................................................................................................... 107
Chapter 5 General Reaction Characteristics Selected by Impact
Parameter ....................................................................................................................... 110
5.1 Multiplicity distributions ......................................................................... 110
5.2 Element distributions ............................................................................... 121
5.3 Angular distributions ............................................................................... 127
Chapter 6 Model comparisons .................................................................................. 132
6.1 Expanding emitting source model ......................................................... 133
6.2 Percolation ................................................................................................... 137
Toroids and bubbles ............................................................................. 147
Chapter 7 Fluctuations in multifragment emission ............................................ 157
Chapter 8 Summary .................................................................................................... 167
Appendix A Physics tape format .............................................................................. 170
Appendix B Temperature estimate from percolation theory ............................ 172

List of References ......................................................................................................... 173

List of Tables

Table 2.1 Coverage in solid angle, polar angles, azimuthal angles and

distance to the target (d) for individual detectors of the Miniball. ................... 9
Table 3.1 Fit parameters of Equation 3.8 for Figure 3.14 ...................................... 50
Table 3.2. Calculated energy for which the listed particle punch through

2.0 cm of CsI ................................................................................................................... 50
Table 4.1 Moving source parameters used to fit the energy spectra in

Figures 4.10-12 ............................................................................................................... 76
Table A.1 Physics tape format for particle identification ..................................... 171

ix

List of Figures

Figure 2.1 Artist’ 5 perspective of the assembly structure of the Miniball

41: fragment detection array. For clarity, electrical connections, the light
pulsing system, and the cooling system have been omitted .............................. 5

Figure 2.2 Half-plane section of the Miniball array. Individual detector

rings are labeled 1 through 11. Numbers of detectors per ring are given

in parentheses. The polar angles for the centers of the rings are

indicated. The dashed horizontal line indicates the beam axis. ........................ 6

Figure 2.3 Front views of different detector shapes. The detectors are
labeled by their ring number. Numbers of detectors per ring are given
in parentheses ............................................................................................................... 8

Figure 2.4 Isometric view of the target insertion mechanism. .......................... 10

Figure 2.5 Schematic of phoswich assembly of individual detector

elements. The u-metal shield covering the photomultiplier is not
included. ........................................................................................................................ 12

Figure 2.6 Photograph of 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 housing the voltage
divider. The can has been removed to expose the voltage divider. ................. 14

Figure 2.7 Schematic of the active voltage divider used for the Miniball
detectors. ........................................................................................................................ 16

Figure 2.8 Relative variation of scintillation efficiency measured for

two parallel surfaces of a CsI(Tl) crystal by using a collimated on-source.
The axes of the coordinate system are parallel to the sides of the
scintillator. The coordinates are fixed with respect to the scintillator. ............ 19

Figure 2.9 Variations of scintillation efficiency measured with a charge
integrating ADC for two different time gates selecting the fast and slow
components of scintillation for CsI(Tl). .................................................................. 20

xi
Figure 2.10 Variations of scintillation efficiency detected with
collimated a-particles of 8.785 MeV energy (solid points) and

collimated y—rays of 662 keV energy (open points). The left-hand panel

shows the measurement for a detector which was rejected. The right-

hand panel shows the measurement for a detector which was

incorporated in the Miniball. ................................................................................... 22

Figure 2.11 Relation between scintillator foil thickness and rotational
frequency of spinning measured for solutions of Betapaint of different

viscosity. The lines show fits with the power law to: v'“. ................................. 24
Figure 2.12 Schematics of light pulser assembly and LED trigger circuit ......... 26

Figure 2.13 Open points - gain variations of a CsI(Tl) photomultiplier
assembly determined by measuring the detector response to 8.785 MeV

(1 particles. Solid points - same data corrected for gain shifts in the off-

line analysis by using information from the light pulser system. .................... 28
Figure 3.1 Schematic diagram of the data acquisition of the electronics

of the Miniball. ............................................................................................................. 32
Figure 3.2 Timing and widths of the fast, slow and tail gates ............................ 33
Figure 3.3 Fast versus slow spectrum for detector 3—6 (ring 3, position

6). ..................................................................................................................................... 35
Figure 3.4 Tail versus slow spectrum for detector 3-6 (ring 3, position

6). ..................................................................................................................................... 36
Figure 3.5 Slow versus fast’ for detector 3-6. The solid line is used to

separate light charged particles from intermediate mass fragments ................ 37
Figure 3.6 Schematic tail versus slow describing variables used in the
construction of the PID function. ............................................................................. 39
Figure 3.7 PID versus slow for detector 3-6 ............................................................. 40

Figure 3.8 Z resolution of detector 3-6. Yield of Z (counts) as a function
of Z. ................................................................................................................................. 41

Figure 3.9 Fast versus time for particles that stop in the fast plastic of
detector 3-6. Time t=0 is arbitrary. ............................................................................ 42

Figure 3.10 Fast versus slow for a detector with a CsI(Tl) crystal from
Bicron, detector 3-16. ................................................................................................... 44

xii
Figure 3.11 Fast versus tail for a detector with a CsI(Tl) crystal from
Bicron, detector 3-16 .................................................................................................... 45

Figure 3.12 Tail versus fast’ for a detector with a CsI(Tl) crystal from
Bicron, detector 3-16. ................................................................................................... 46

Figure 3.13 CsI calibration of detector 3-6. The light output response

(slow) as a function of energy deposited in the C51 (Egg). The circles
represent all previous calibration data for this detector while the

squares represent calibration taken during the present experiment. The

solid lines are fits using Equation (3.8) for the elements listed. ........................ 48

Figure 3.14 CsI calibration of Miniball rings 2-4. The average light

output response (slow) as a function of energy deposited in the CsI

(ECsl)- The solid lines are fits using Equation (3.8) for the elements

listed. The fit parameters are given in Table 3.1. .................................................. 49

Figure 3.15 The circles represent all previous calibration He calibration

data for detector 3-6 while the squares represent calibration taken

during the present experiment. Solid line - from Equation 3.8 using

the fit parameters for detector 3-6. Dashed line — spline fit of a punch
through point to low energy calibration ................................................................. 52

Figure 4.1 Correlations between charged-particle multiplicity NC,
transverse energy 15,. intermediate rapidity charge Z, and identified

hydrogen multiplicity N, observed for 36Ar+197Au collisions at E/A=50
MeV. Adjacent contours of different color differ by factors of 5 ........................ 57

Figure 4.2 Correlations between charged-particle multiplicity NC,
transverse energy E“ intermediate rapidity charge Z,, and identified

hydrogen multiplicity N, observed for 36Ar+197Au collisions at E/A=80
MeV. Adjacent contours of different color differ by factors of 5 ........................ 58

Figure 4.3 Correlations between charged-particle multiplicity NC,
transverse energy E}, intermediate rapidity charge Z,, and identified
hydrogen multiplicity N, observed for 36Ar+197Au collisions at

E/A=110 MeV. Adjacent contours of different color differ by factors of
5 ........................................................................................................................................ 59

Figure 4.4 Reduced impact parameter scales 5(X) extracted from the
measured quantities X = NC, E" Zyand N, . Each panel shows the

relation extracted for the indicated observable ...................................................... 61

Figure 4.5 Conditional impact parameter distributions extracted for
Y = 2, (top panels), Y = N, (center panels), and Y = E, (bottom panels)

xiii
for the 35Ar+197Au reaction at E/A=110 MeV. Left and right hand
panels show distributions selected by impact parameter cuts 13(X) =0.05-

0.1 and 5(X)=0.35-0.45 on the indicated observables (X = N,, E,, and Z, ).

All impact parameter scales were constructed according to Equation
(4.4). ................................................................................................................................. 63

Figure 4.6 Conditional impact parameter distributions extracted for

Y = Z, (top panels), Y=Nc (center panels), and Y = E, (bottom panels)
for the 36Ar+197Au reaction at E/A=50 MeV. Left- and right-hand
panels show distributions selected by impact parameter cuts 5(X)=0.05-

0.1 and 5(X)=0.35-0.45 on the indicated observables (X =NC, E, and Z,).

All impact parameter scales were constructed according to equation
(4.4). ................................................................................................................................. 64

Figure 4.7 Conditional impact parameter distributions extracted for
Y = Z, (top panels), Y = NC (center panels), and Y = E, (bottom panels)

for the 36Ar+197Au reaction at E/A=80 MeV. Left- and right-hand
panels show distributions selected by impact parameter cuts 5(X)=0.05-

0.1 and 5(X)=0.35-0.45 on the indicated observables (X =Nc, E,, and Z, ).

All impact parameter scales were constructed according to equation
(4.4). ................................................................................................................................. 65

Figure 4.8 Conditional impact parameter distributions extracted for

Y = Z, (top panels), Y = NC (center panels), and Y = E, (bottom panels)
for the 35Ar+197Au reaction at E/A=110 MeV. Left- and right-hand
panels show distributions selected by impact parameter cuts 5(X)=0.05-

0.1 and 5(X)=0.35-0.45 on the indicated observables (X =NC, E, and Z,).

All impact parameter scales were constructed according to equation
(4.4). ................................................................................................................................. 66

Figure 4.9 Centroids and widths at half maximum of conditional
impact parameter distributions selected by cuts on impact parameters
constructed from other observables X indicated in the individual
panels. For ease of presentation, the open points have been displaced
from the centers of the gates on 13(X). Top, center and bottom panels
show the values for impact parameter scales constructed from NC, E,
and 2,, respectively. Left, center and right hand columns show data for

the 35Ar+197Au reaction at E/A=50, 80, and 110 MeV. ........................................ 69

Figure 4.10 Energy spectra of beryllium (left-hand panels) and carbon
(right-hand panels) nuclei emitted in peripheral (5>0.6, top panels) and

xiv
central (5<0.3, bottom panels) 35Ar+197Au collisions at E / A=50 MeV.

The exact cuts on the measured charged-particle multiplicity are given
in the figure. The solid curves are fits with equation (4.5). The

parameters are listed in Table 4.1. ............................................................................ 72
Figure 4.11 As Figure 4.10, for E/A=80 MeV. ......................................................... 73
Figure 4.12 As Figure 4.10, for E/A=110 MeV. ....................................................... 74

Figure 4.13 Relative contribution of projectile-like source extracted
from the energy spectra of He, Li, Be and C nuclei selected by different
cuts on charged-particle multiplicity for 36Ar+197Au collisions at

E/A=110 MeV. A scale of the reduced impact parameter 5(Nc) and a

pictorial illustration of the geometric overlap between projectile and
target nuclei are included ........................................................................................... 78

Figure 4.14 Fast-particle fractions (fractions of particles detected at 0=9°-
23° with velocities greater than half the beam velocity) for 35Ar+197Au
collisions at E/A=50 MeV. Solid points, open squares and open circles
depict values determined for narrow cuts on the reduced impact
parameters determined from NC, E, and Z,. Star-shaped points

represent simultaneous cuts on 13(Nc), 5(E,) and 5(Z,). .................................... 81
Figure 4.15 As Figure 4.14, for E/A=80 MeV. ......................................................... 82
Figure 4.16 As Figure 4.14, for E/A=110 MeV83

Figure 4.17 Azimuthal correlation functions constructed from particle
pairs of protons, deuterons, tritons and He nuclei emitted in peripheral

35Ar+197Au collisions (l;>0.75, NC =2-9) at E/A=50 MeV. The
correlation functions were constructed for particles emitted at polar

angles of 913b=31°-50° using an energy threshold Eth/A=12 MeV .................... 88

Figure 4.18 Azimuthal correlation functions for He nuclei emitted in
35Ar+197Au collisions at E/ A = 35 MeV. Panels from left to right show

data selected by cuts on reduced impact parameters 5 = 0.8, 0.6, 0.4, and

<02, respectively .......................................................................................................... 90
Figure 4.19 As Figure 4.18, for E/ A = 50 MeV. ....................................................... 91
Figure 4.20 As Figure 4.18, for E/ A = 80 MeV. ....................................................... 92

Figure 4.21 As Figure 4.18, for E/ A = 110 MeV. ..................................................... 93

XV

Figure 4.22 Azimuthal correlation functions for He nuclei emitted in
125’Xe4-197Au collisions at E/ A = 50 MeV. Panels from left to right show

data selected by cuts on reduced impact parameters 5 = 0.8, 0.6, 0.4, and
<02, respectively .......................................................................................................... 94

Figure 4.23 Reduced-impact-parameter dependence of the coefficients

2.1 and 2.2 used to fit the measured azimuthal correlation functions for
emitted He nuclei emitted in 35Ar+197Au collisions at E/ A = 35, 50, 80
and 110 MeV. ................................................................................................................ 96

Figure 4.24 Reduced-impact-parameter dependence of the coefficients

l1 and 12 used to fit the measured azimuthal correlation functions of

protons, deuterons, tritons or He-nuclei emitted in 36Ar+197Au
collisions at E/ A = 50 MeV. ....................................................................................... 98-

Figure 4.25 Reduced-impact—parameter dependence of the coefficients

l1 and 2.2 used to fit the measured azimuthal correlation functions of

protons, deuterons, tritons or He—nuclei emitted in 129Xe+197Au
collisions at E/ A = 50 MeV. ....................................................................................... 99

Figure 4.26 Reduced-impact-parameter dependence of the coefficients

1.1 and 1.2 used to fit the measured azimuthal correlation functions of

He nuclei emitted in 35Ar+197Au collisions at E/ A = 50 MeV. Results

from different impact parameters filters are shown by the different

symbols indicated in the figure ................................................................................. 101

Figure 4.27 Reduced-impact-parameter dependence of the coefficients

l1 and It: used to fit the measured azimuthal correlation functions of

He nuclei emitted in 129Xe-t-197Au collisions at E/ A = 50 MeV. Results

from different impact parameters filters are shown by the different

symbols indicated in the figure ................................................................................. 102

Figure 4.28 Conditional impact parameter distributions,
dP[5(E,)]/ d5(E,), for 35Ar+197Au collisions at E/ A = 110 MeV selected

by cuts on 5(NC) = 0.05 - 0.1 (dashed curve), directivity D S 0.2 (dotted-

dashed curve), and for the simultaneous cuts 5(Nc) = 0.05 - 0.1 and D s
0.2 (solid curve). ........................................................................................................... 104

Figure 4.29 Distribution of transverse momentum directivity D for.

35Ar+197Au collisions at E/ A = 110 MeV selected by cuts on 5(Nc) <0.2
(dashed curve). The solid curve shows the distribution of the
transverse momentum directivity obtained after randomizing the

xvi

azimuthal distribution of the emitted particles according to an isotropic
distribution (solid curve). .......................................................................................... 106

Figure 5.1 Upper panel — NC distributions. Lower panel - NM,
distributions. Both from the reaction 36Ar+197Au at E/A=35, 50, 80 and
110 MeV. ................................................... , ..................................................................... 1 11

Figure 5.2 Measured relation between charged particle multiplicity, N,,,

and total charge of identified particles (2525) for the reaction

35Ar+197Au at the indicated energies. Different colors represent

contours that change by factors of 3. ........................................................................ 112

Figure 5.3 Percentage of detected charge Z", appearing in clusters (A>1)
and intermediate mass fragments (IMF) as a function of reduced impact
parameter for the reaction 35Ar+197Au at E/A=50, 80 and 110 MeV ................ 114

Figure 5.4 Measured IMF multiplicity distributions for the indicated
gates on charged particle multiplicity N,,. Panels are labeled by incident
energy. ........................... 116

Figure 5.5 First and second moments of IMF multiplicity distributions
as a function of charged particle multiplicity, NC. Different symbols
represent results for the indicated beam energies. ............................................... 117

Figure 5.6 First and second moments of IMF multiplicity distributions
as a function of 5(Nc). The two circles show the approximate overlap
between target and projectile for 13:02, 0.4, 0.6 and 0.8. ...................................... 119

Figure 5.7 First and second moments of IMF multiplicity distributions
as a function of 13(E,). The two circles show the approximate overlap
between target and projectile for 5:02, 0.4, 0.6 and 0.8. ...................................... 120

Figure 5.8 Elemental distributions at E/A=50 (circles), 80 (stars) and 110
MeV (diamonds) for I3=0.75 (top), 0.5 (middle) and 0.25 (bottom). Solid

lines are exponential fits using Equation (5.1) ....................................................... 122
Figure 5.9 a as a function of 5(NC) for E/A=50, 80 and 100 MeV. ..................... 123
Figure 5.10 a as a function of 5(E,) for E/A=50, 80 and 100 MeV ...................... 124

Figure 5.11 Mass yields for a percolation simulation at fixed excitation
energy. The open (solid) symbols correspond to distributions
constructed for multiplicity N=67(77). The curves are power law fits .............. 125

xvii
Figure 5.12 Evolution of the fit parameter I. as a function of
multiplicity N for a percolation simulation at fixed excitation energy ............ 126

Figure 5.13 Angular distribution of elements Z=1-6 from the reaction
35Ar+197Au at E/A=50 MeV. Solid symbols - central cut 5(NC)<0.25.

Open symbols - peripheral cut 5(NC)>0.75. ............................................................ 129
Figure 5.14 Same as Figure 5.13 for E/A=80 MeV. ................................................ 130
Figure 5.15 Same as Figure 5.13 for E/A=110 MeV. .............................................. 131

Figure 6.1 First and second moments of IMF multiplicity distributions

as a function of charged-particle multiplicity, NC. Different symbols

represent results for indicated beam energies. The solid, dashed, dotted-
dashed and dashed-dotted-dotted curves show results calculated for the
statistical decay of expanding compound nuclei of finite-nucleus
compressibility K=144, 200, 288 and co, respectively. The dotted curves
represent the calculations for K=200, filtered by the detector response. .......... 135

Figure 6.2 Elemental multiplicity distributions detected in 36Ari-197Au
collisions at E/A=50, 80, 110 MeV (top panel) and in 129Xe+197Au

collisions at E/A=50 MeV (bottom panel). The curves represent

calculations with the bond percolation model (described in the text) for

the indicated bond-breaking probabilities p. All calculations are filtered

by the response of the experimental apparatus except for the dot-dashed
curve. .............................................................................................................................. 139

Figure 6.3 Angular multiplicity distributions of light particles, Z=1,2

(circles), and intermediate mass fragments of Z=3-5 (squares) and Z=6-

12 (diamonds) detected in 35Ar+197Au collisions at E/A=50, 80, 110

MeV (top panel) and in 129Xe+197Au collisions at E/A=50 MeV (bottom
panel). The curves represent calculations with the bond percolation

model for a bond-breaking probability of p=0.7. The calculations have

been normalized to the data at 0=45°. Dashed and solid curves show

raw calculations and calculations filtered by the response of the

experimental apparatus. ............................................................................................. 142

Figure 6.4 Relation between average IMF and charged-particle

multiplicities detected in 35Ar+197Au collisions at E/A=50, 80, 110 MeV

(open diamonds, open squares, and open circles, respectively) and in
129Xe+197Au collisions at E/A=50 MeV (solid circles). Thick and thin

curves show the results of filtered and unfiltered percolation

calculations, respectively. Details are given in text. ............................................. 144

Figure 6.5 IMF admixture as a function of 1, according to Equation (6.2). ....... 145

xvni
Figure 6.6 Relation between average IMF and charged-particle
multiplicities. Solid points represent values measured for 12‘e’Xe-i-197Au
at E/A=50 MeV. Thin (thick) solid line shows the raw (efficiency
corrected) percolation calculation for a solid sphere. The hatched area
shows the range of average IMF and average charged-particle
multiplicities predicted by percolation calculations for toroidal breakup
configurations ............................................................................................................... 149

Figure 6.7 Relation between average IMF and charged-particle

multiplicities. Solid points represent values measured for 129Xe+197Au

at E/A=50 MeV. Thin (thick) solid line shows the raw (efficiency

corrected) percolation calculation for a solid sphere. The hatched area

shows the range of average IMF and average charged particle

multiplicities predicted by percolation calculations for bubble-shaped

breakup configurations ............................................................................................... 150

Figure 6.8 Extracted power-law exponents 1 fit to mass distributions
predicted by the bond percolation model for the break up of toroidal
systems as a function of Rt and p. ............................................................................ 153

Figure 6.9 Extracted power-law exponents A fit to mass distributions
predicted by the bond percolation model for the break up of bubble
shaped systems as a function of Rb and p. .............................................................. 154

Figure 6.10 It as function of p for a solid sphere (circles), a toroid of

radius Rt = 3.0xa (diamonds), and bubble with inner radius Rb = 3.0xa
(squares). ........................................................................................................................ 155

Figure 7.1 Upper part - Measured relation between transverse energy

E, and total charged particle multiplicity for 36Ar+197Au reactions at
E/A=110 MeV. Lower part - Charged particle multiplicity distributions

for the cuts on E, indicated in the top panel. ........................................................ 160

Figure 7.2 Bottom, center and top panels show the mean values (NC),

variances a: , and ratios 0'2 / (NC) of the charged particle multiplicity

distributions for 36Ar+197Au reactions at E/A=110 MeV. These
quantities were selected by various cuts on the transverse energy ................... 161

Figure 7.3 Relation between mean charged particle multiplicity (NC)

and the ratios 0% / (NC). Solid circular points - experimental values

extracted from near-central 35Ar+197Au reactions at E/A=35, 50, 80 and
110 MeV. Open symbols are explained in the text. ............................................... 163

xix
Figure 7.4 Scaled factorial moments as a function of binning resolution.
Solid points show experimental results for central 36Ar+197Au
collisions at E/A=110 MeV. Open points show results from percolation

calculations using p=0.7. Open squares depict calculations in which the
total number of broken bonds is allowed to fluctuate ......................................... 165

 

Si

11‘.

Chapter 1 Introduction

Weakly-excited nuclei decay primarily by fission and light particle
evaporation. For excitation energies much higher than the binding energy,
explosive disintegration into light particles (232) takes place. Between these
two extremes, there is a region in which copious production of intermediate
mass fragments (IMF: 3.<_Z$20) is observed [Ogil 91, Bowm 91, Wadd 85]. The
mechanism causing multifragment decays is not yet understood and is a
subject of current debate.

Many mechanisms have been proposed [Bert 83, Cser 86, Lync 87, Schl
87, Gros 90, Frie 90, More 75, Frie 83a, Frie 83b, Baue 87, Peil 89, Boal 88, Boal
90, Baue 85, Baue 86, Biro 86, Cerr 88] for the production of the fragments. For
example, the action of thermal pressure may force hot nuclear systems to
expand to low density where the exponential growth of density fluctuations
may lead to a complete disintegration of the nuclear system [Bert 83, Schl 87],
in analogy to a liquid-gas phase transition in infinite nuclear matter.
Unfortunately, detailed comparisons of various fragment production models
to experimental data have been lacking since few experiments [Doss 87, Boug
88, Troc 89, Boug 89, Kim 89, Blum 91, Ogil 91, Bowm 91] performed to date
provided sufficient phase space coverage to allow the extraction of exclusive

quantities.

2
Exclusive experiments have become necessary as different models,

based on very different approximations, often give very similar results for
inclusive data. Inclusive measurements suffer from the implicit averaging
over impact parameter which makes it difficult to separate and understand
statistical and dynamical effects. For studies addressing the thermodynamic
properties of nuclear matter, the selection of central collisions is of particular
interest since reaction zones formed in central collisions promise to reach the
largest degree of equilibration. To address the question of multifragment
decay of highly-excited nuclear systems and to provide exclusive (i.e. impact
parameter selected) measurements for constraints on various theoretical
models, we have measured fragmentation in the reaction 36Ar-i-197Au at
E/A=50, 80 and 110 MeV.

After addressing the technique of impact parameter selection we select
central collisions and study the IMF yields making comparisons with two
statistical models. The measured IMF multiplicities from central collisions
are compared with predictions from the expanding emitting source model, a
model in which an equilibrated source loses mass by evaporation as it
expands. The fragment measurements are also compared to predictions from
percolation theory. Simple percolation theory is of interest since it allows the
study of finite system effects in a well defined model which exhibits a phase
transition in the limit of infinite systems. Statistical models are of interest
since microscopic transport calculations [Boal 88, Boal 90] capable of treating
nonequilibrium-fragment emission predict fewer fragments than observed
experimentally [Bowm 91]. Dynamics, as treated by these models, may not
play the decisive role in fragment production. In this context, it is interesting

to neglect dynamical effects and explore to what extent fragment formation

3
could be dominated by the geometric considerations contained in the

percolation ansatz.

The thesis is organized as follows: the experimental details are
described in Chapters 2 and 3; Chapter 4 provides a detailed study of impact
parameter selection for the 3‘5Ar+197Au reaction at intermediate energies; in
Chapter 5 several reaction characteristics are evaluated as a function of
bombarding energy and impact parameter; in Chapter 6 comparisons are
made to predictions from the emitting expanding source model and
percolation theory, and in Chapter 7 we look for intermittency signals in

central collisions; a summary and conclusions are given in Chapter 8.

Chapter 2 MSU Miniball

2.1 Mechanical construction

The Miniball phoswich detector array is designed to operate in a
vacuum vessel. An artist’s perspective of the three-dimensional geometrical
assembly is shown in Figure 2.1. The array consists of 11 independent rings
coaxial about the beam axis. For ease of assembly, as well as servicing, the
individual rings are mounted on separate base plates which slide on two
precision rails. The rings and detector mounts are made of aluminum. Good
thermal conductivity between detectors and the mounting structure allows
the conduction of heat generated by the photomultiplier voltage divider
network into the array superstructure. This heat is removed from the
Miniball by cooling the base plates to 15°C. By this means, constant operating
temperature in vacuum is achieved after a brief equilibration time. The
individual detector mounts are designed to allow the removal of any detector
without interfering with the alignment of neighboring detectors. The entire
assembly is placed on an adjustable mounting structure which allows for the
alignment of the apparatus with respect to the beam axis.

Figure 2.2 shows a half-plane section of the array in the vertical plane

which contains the beam axis. Individual rings are labeled by the ring

VSU-QO-O“

 

 

 

 

 

 

Figure 2.1 Artist’ 5 perspective of the assembly structure of the Miniball 41:
fragment detection array. For clarity, electrical connections, the light pulsing

system, and the cooling system have been omitted.

MSU-90-046

 

 

 

 

 

 

 

 

 

Figure 2.2 Half-plane section of the Miniball array. Individual detector rings
are labeled 1 through 11. Numbers of detectors per ring are given in

parentheses. The polar angles for the centers of the rings are indicated. The
dashed horizontal line indicates the beam axis.

7
numbers 1-11 which increase from forward to backward angles. For each ring,

the number of detectors is given in parentheses. For a given ring, the
detectors are identical in shape and have the same polar angle coordinates
with respect to the beam axis. These angles are indicated in Figure 2.2.

Since the angular distributions of the emitted particles are strongly
forward peaked, the solid angle subtended by forward detectors is smaller
than for backward detectors. Variations in solid angle were achieved largely
by placing detectors at different distances from the target while keeping their
size approximately constant. The front face geometries of the individual
CsI(Tl) crystals are shown in Figure 2.3. Different detector shapes are labeled
by the respective ring numbers with the number of detectors per ring given in
parentheses (see Figure 2.2 for the definition of the ring numbers). The
crystals are tapered such that the front and back surfaces subtend the same
solid angle with respect to the target location. In order to reduce cost of
fabrication, the curved surfaces were approximated by planar surfaces. The
resulting loss in solid angle coverage is on the order of 2%, comparable in
magnitude to the loss in solid angle coverage resulting from gaps between
individual detectors (which must be provided to allow for mechanical
tolerances and optical isolation between neighboring crystals). A listing of the
detector solid angles is given in Table 2.1.

An isometric drawing of the target insertion mechanism is shown in
Figure 2.4. The targets are mounted on frames made of flat shim stock 0.2 mm
thick. Each target frame is attached to an insertion rod. The insertion rods are
mounted on a tray which can be moved parallel to the beam axis. An
electromagnetic clutch provides the coupling to the insertion and retraction
drive once a target rod is in the appropriate position. A third drive allows

rotation of an inserted target about the axis of the insertion rod. This

 

 

 

 

 

 

 

 

 

 

MSU-90-045
l 2 3 4
(:2) (IS) (20) (24) I
5 6 7 8
(24) (20) (20) (18)
9 IO ll
(l4) (:2) (8) IL. 1 i i 1'
5 cm

 

 

 

 

 

 

 

 

Figure 2.3 Front views of different detector shapes. The detectors are labeled
by their ring number. Numbers of detectors per ring are given in parentheses.

Table 2.1 Coverage in solid angle, polar angle, azimuthal angle and distance to
the target (d) for individual detectors of the Miniball.

 

 

 

Ring Detectors AQ (msr) 0 (°) A0 (°) A4) (°) d (m)
1 12 12.3 12.5 7 30.0 260
2 16 14.7 19.5 7 22.5 220
3 20 18.5 27.0 8 18.0 180
4 24 22.9 35.5 9 15.0 160
5 24 30.8 45.0 10 15.0 140
6 20 64.8 57.5 15 18.0 90
7 20 74.0 72.5 15 18.0 90
8 18 (-1) 113.3 90.0 20 20.0 70
9 14 135.1 110.0 20 25.7 70
10 12 128.3 130.0 20 30.0 70
11 8 125.7 150.0 20 45.0 70

 

 

10

 

 

Figure 2.4 Isometric view of the target insertion mechanism.

 

Pht

5C5:

l l
rotation of the target is useful for the determination of the shadowing a

detector experiences when it is located in the plane of the target frame. Also,
sources can be mounted in the target position and rotated to point to different
areas of the Miniball for debugging and calibration purposes.

In its present configuration, the detector array covers a solid angle
corresponding to about 89% of 41:. The loss in solid angle can be decomposed
into the following contributions:

(i) beam entrance and exit holes (4% 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:);

(iv) removal of one detector (ring 8 position 6) at 0=90° to provide space for

target insertion mechanism (1% of 41:).

2.2 Detector design

Phoswich construction

All phoswich detectors of the array are composed of a thin plastic
scintillator foil, spun from Bicron BC-498X scintillator solution, and a 2 cm
CsI(Tl) scintillator crystal. The foils in Rings 2-11 have an average thickness of
4 mg/cm2 or 40 um while Ring 1 foils have an average thickness of 5 mg/cm2
or 50 um. A schematic of the detector design is given in Figure 2.5. In order to
retain flexibility in the choice of scintillator foil thickness, the scintillator foil
is placed on the front face of the CsI(Tl) crystal without bonding material. The
back face of the CsI(Tl) scintillator is glued with optical cement (Bicron BC600)

to a flat light guide made of UVT Plexiglas. This light guide is glued to a

12

MSU-90-03l

Fast scintillator L‘ght gunde

 

PMT

 

 

WA

 

 

 

 

 

 

 

W

\”

 

 

 

Al-mylar foil \

Optical cement

Figure 2.5 Schematic of phoswich assembly of individual detector elements.
The u-metal shield covering the photomultiplier is not included.

13
second cylindrical piece of UVT Plexiglas (9.5 mm thick and 25 mm in

diameter) which, in turn, is glued to the front window of the photomultiplier
tube (Burle Industries model C83062). The photomultiplier tube and the
cylindrical light guide are surrounded by a cylindrical u-metal shield (not
shown in the figure). Front and back faces of the CsI(Tl) crystals are polished;
the tapered sides are sanded and wrapped with white Teflon tape. The front
face of the phoswich assembly is covered by an alurninized mylar foil (0.15
mg/cm2 mylar and 0.02 mg/cm2 aluminum).

The primary scintillation of the plastic scintillator used has its
maximum intensity at 370 nm. In bulk material of the scintillator, the
intensity maximum is shifted to 420 nm by the addition of a wavelength
shifter. Our scintillator foils are, however, too thin for an effective
wavelength shift and maximum emission remains in the far blue region of
the spectrum. The absorption of this light in CsI(Tl) places a constraint on the
maximum useful thickness of the CsI(Tl) crystals. Additional absorption in
the light guides can be minimized by using UVT Plexiglas rather than
standard Plexiglas light guides. Such considerations become particularly
important for phoswich detectors utilizing thin scintillator foils in efforts to
reduce particle detection thresholds.

Figure 2.6 shows a photograph of the basic photomultiplier assembly
used for all detectors. The phoswich and matching first light guide have not
yet been attached. A precision-machined aluminum ring is glued to the [.1-
metal shield surrounding the photomultiplier and the cylindrical light guide.
This ring provides the alignment for a precision-machined aluminum can
which houses the voltage divider and which defines the detector alignment
when bolted to the rings of the array support structure. In order to expose the

voltage divider chain, this aluminum can has been removed and placed next

5'ng
5"“ In.
unifies

3991‘, f6]

l4

 

Figure 2.6 Photograph of 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 housing the voltage divider. The can has
been removed to expose the voltage divider.

15
to the photomultiplier. The active voltage divider chain is soldered to the

flying leads of the phototube. To prevent destruction of the FETs by sparking
during operation in poor vacuum, the entire divider chain, including the
leads to the photomultiplier tube, is encapsulated in silicone rubber (Dow
Chemical Sylgard 184). Vacuum accidents occurring with fully biased
detectors do not lead to divider chain failures. In fact, the detectors can
survive a full pumping cycle from atmospheric pressure to vacuum with bias
applied to them.

The 10-stage Burle Industries model C83062E photomultiplier tube was
chosen because of its good timing characteristics (tR=2.3 ns), its large nominal
gain (=107) and its good linearity for fast signals. Since the apparatus is
designed to operate in vacuum, active divider chains were chosen to
minimize the generation of heat. A schematic of the active divider chain is
given in Figure 2.7. The final stages of the divider chain are of the “booster"
type which provide improved linearity for high peak currents generated by

large signals of the fast scintillator.

Uniform scintillation response of CsI(Tl)

Previous experience with CsI(Tl) crystals used for the detection of
energetic particles had revealed difficulties with the production of
scintillators with uniform scintillation response [Gong 88, Gong 90].
Therefore, considerable attention was paid to select CsI(Tl) crystals of uniform
scintillation efficiency. In previous tests of large cylindrical crystals [Gong 88,
Gong 90], nonuniformities of the scintillation efficiency were detected by
measuring the response to collimated y-rays. Such measurements are
relatively easy to perform since they can be done in air. However, they are less

suitable for small volume crystals, since collimated y-rays sample a relatively

16

MSU-SO-OSZ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AhfiDE 50a COAXIAL CABLE
”(0%: l 7,-
D 10% a, ‘n'uE-T 411,: Km“
4722i
”II-HE F1EIOMQ
09F NHvAr I" T :
W. t— 473+:31582M9.
08% \rw ,
, 47a VN2406L
on '“F *— 4“ moist-'57:]: | E33 mo
firvv .
mess
W. l— 479 i, 820 110 :: 3.9m:
05% MP F- 5; 680110.
I— eeoko
[Mr '"F
03% ”F 1—1 680m
02% W: I— um
01% “1,, 1— 680m
PHoro- lnF *— "8 ”a SHv CABLE -HV
E \I v 3‘—
CATHODE , tom .1.
lnF ' 7
J.-

Figure 2.7 Schematic of the active voltage divider used for the Miniball

detectors.

17
large volume of the crystal. Small scale fluctuations of the scintillation

efficiency may remain undetected. In addition, measurements for small
noncylindrical crystals are less precise, since the shape of the Compton
background depends on the position of the collimated y-ray source. Such
dependencies lead to additional uncertainties in the extraction of the
photopeak position.

It was determined, however, that nonuniformities of the scintillation
efficiency can be detected very sensitively by scanning the CsI(Tl) crystals with
a collimated a-source in vacuum. All crystals ordered from various
manufacturers were rectangular in shape with dimensions of 2 in.x1.5 in.x 1
in. They were polished at the front and back faces (with dimensions of 2
in. x1.5 in.) and sanded at the sides. The back face was optically coupled to a
clear acrylic light guide with the same dimensions as the crystal. This light
guide, in turn, was optically connected to a photomultiplier tube of 1 in.
diameter. The sides of the CsI(Tl) crystal and of the light guide were wrapped
with white Teflon tape. By covering the front face of the CsI(Tl) scintillator
with an aluminized mylar foil, a uniform light collection efficiency was
achieved. (Without a reflective entrance foil, the light collection efficiency
decreased by about 5% from the center of the front face to its sides.) The front
face of the crystal was scanned in vacuum and the peak location of the 8.785
MeV a-line from a collimated 228Th a-source was monitored. In order to
avoid edge effects, regions within about 2mm of the side boundaries of the
crystal were not scanned. Most tests were performed with a simple
multichannel analyzer equipped with a peak sensing ADC; in those instances
the anode signal of the photomultiplier was shaped and amplified with

standard electronics, using integration and differentiation times of 1 us.

18
Figure 2.8 shows the results of a scan for a crystal exhibiting a large

gradient of the scintillation efficiency. The horizontal axis of the plot shows
the location of the collimated a-source with respect to the center, along the
short symmetry axis of the front face. Different surface treatments of the front
face of the scintillator did not affect the measured variation of the
scintillation efficiency. In order to demonstrate that such variations were
related to the bulk material of the scintillator, we exchanged the role of front
and back faces of this scintillator and performed an equivalent scan of the
parallel surface (i.e. the previous back face). The results of the two scans are
compared by the solid and open points in Figure 2.8 (The coordinate system
was kept fixed with respect to the CsI(Tl) crystal.) Nearly identical variations
of the scintillation efficiency are observed across the two parallel scintillator
surfaces, indicating that the measured large gradient of the scintillation
efficiency persists through the bulk material of the sample.

The measurements shown in Figure 2.9 were performed by integrating
the anode current of the photomultiplier with a charge integrating ADC
using time gates of At=0.1-0.5 us and At=1.1-4.1 us, which select the fast and
slow scintillation components of CsI(Tl). The fast component exhibits a larger
variation of the scintillation efficiency than the slow component. Since the
relative intensity of fast and slow scintillation components depends strongly
on the T1 concentration [Birk 64, Mana 62], the observed variations of
scintillation efficiency are most likely due to gradients in the T1
concentration.

Crystals incorporated into the Miniball were preselected by scanning
the 1.5 in. x 2.0 in. rectangular surface of original crystals along two
perpendicular axes and requiring a uniformity of scintillation response better

than 3%. The preselected crystals were then milled into their final shapes and

19

10 MSU-90-036
IIrIrrTTIITIIIIIIIrIIIIIIITITII

 

— front side ' é
--- back side i”

 

lllllllllllllllllllllllllllllll

 

 

Relative Light Output (70)
N
rlilijl'l'IIUIUIIIIIWUIIIUIIIII

 

6LllLLiLLlllJllllLLillLLlL4lll11

-15 -10 -5 O 5 - 10 15
X(mm)

Figure 2.8 Relative variation of scintillation efficiency measured for two
parallel surfaces of a CsI(Tl) crystal by using a collimated a-source. The axes of
the coordinate system are parallel to the sides of the scintillator. The
coordinates are fixed with respect to the scintillator.

20

 

MSU-90-035
I I T I T I I I I I I T I I I I I I I r I j

—— ADC gate : 0.1-0.5 ps
u" ADC gate: 1.1-4.1 [LS

NOOIFU‘O)

Relative Light Output (%)

 

 

 

 

Figure 2.9 Variations of scintillation efficiency measured with a charge
integrating ADC for two different time gates selecting the fast and slow
components of scintillation for CsI(Tl).

 

Here,

21
scanned a second time, requiring uniformity of response within 2.5%. The

preselection process avoided expensive machining of poor quality crystals; it
was about 90% efficient for the selection of crystals of the desired quality.
Figure 2.10 compares variations of scintillation efficiency detected with
collimated a-particles of 8.785 MeV energy (source: 228Th) and y-rays of 662
keV energy (source: 137Cs). The left- and right-hand panels give examples for
a rejected and an accepted crystal, respectively. The enhanced sensitivity of
the a-particle scan is obvious. It is probably caused by the fact that a-particles
sample a much smaller volume than y—rays and that the two kinds of
radiation exhibit different sensitivities to the T1 concentration [Birk 64, Mana

62].

Scintillator foils

Scintillator foils were spun [Meye 78, Norb 87] from Betapaint, Bicron
BC-498X plastic scintillator dissolved in xylene. The original solution was
ordered with a 40% weight ratio of solute to solvent. It was then diluted by
adding xylene until the solution had the desired viscosity of 20-30 P.

The viscosity was determined by measuring the terminal speed v of a
steel ball sinking in a glass tube filled with a sample of Betapaint. Correcting
Stoke's Law for the finite diameter of the glass tube gives the following

expression for the viscosity [Dins 62]:

2 3 5
n= 23’ (”0 ’p)[1-2.104(L)+2.09(-'—) -o.95(-’-) ]. (2.1)
9v R R R

 

Here, n denotes the viscosity, r and R are the diameters of the steel ball and
the glass tube, g is the gravitational acceleration and p0 and p are the
densities of the steel ball and the Betapaint, respectively.

For the fabrication of scintillator foils, a glass plate of 23 cm diameter

was mounted horizontally on a small platform connected to the drive of an

22

MSU-90° 034

 

4 IITIIIIIrTTIIIrII’TIIIIIIr IIIIIIITIIIIIIfll’TIIII’IrT

3— Rejected -— Accepted -

 

Relative Light Output (7.?)
O
I

 

 

 

 

-4 LllllllLlllllllLlllllllllzL IIllLllLllllllllllllllllll

-10 -5 O 5 10 -10 -5 O 5 10 15
X(mm)

 

Figure 2.10 Variations of scintillation efficiency detected with collimated 0:-
particles of 8.785 MeV energy (solid points) and collimated y-rays of 662 keV
energy (open points). The left-hand panel shows the measurement for a
detector which was rejected. The right-hand panel shows the measurement
for a detector which was incorporated in the Miniball.

ho

am
PEI
ii‘jc

63c]

23
electrical motor which allowed the plate to spin about its center at a

preselected speed. To facilitate the removal of spun foils, the glass plate was
covered successively with metasilicate solution and Teepol 610 and then
wiped to leave only a thin film of the releasing agents on the glass substrate.
An appropriate amount of Betapaint was poured on the center of a glass plate.
In order to provide rapid spreading of the initial solution, the plate was spun
at an enhanced speed for the first few seconds until the entire plate was
covered with Betapaint. Following this rapid startup, the glass plate was spun
at the preset rotational frequency for approximately 4 minutes until a solid
foil had formed. After spinning, the glass plate was stored in a flow of dry
nitrogen for about eight hours. The foil was then peeled from the glass plate,
mounted on a frame, and placed in a dry nitrogen atmosphere for another 24
hours to allow further evaporation of residual xylene.

We obtained good and reproducible results by using dilute solutions
and spinning at low rotational frequencies. A number of measurements were
performed to determine the relation between rotational frequency and foil
thickness. The results of these measurements are shown in Figure 2.11. For
each foil, thickness and homogeneity were determined by scanning the foil in
vacuum with a collimated 228Th a-source and measuring the energy of the
transmitted (it-particles in a calibrated silicon detector. The energy loss in the
foil was then converted to an areal density according to reference [Lift 80]. The
spun foils were uniform in thickness to within typically 1-2% over an area of
7x7 cmz. Scintillator foils used for instrumenting the Miniball in its present

configuration were selected to have a thickness of 4.0 1:012 mg/cmz.

24

 

  
 

 

 

 

MSU-90-037
B I I I I I I r I’ T I I I I I I I I I I I I
7 — 1) = 41 poise _
5 '" 1" x t cc V" —
1‘ I
618‘ 5 -— \ \ \a=0.67 .—
o ‘ ‘ ~ ‘5
>0 4 — s ' ——
E 17 = 25 poise
v
a.)
3 - _
2 p l L l L 1 L I l l l l l l l L I. l l L L
200 300 400 500 600
v (rpm)

Figure 2.11 Relation between scintillator foil thickness and rotational
frequency of spinning measured for solutions of Betapaint of different
viscosity. The lines show fits with the power law toe v'“.

25
Light pulser system

Gain drifts of the photomultiplier tubes are monitored by a simple and
compact light pulser system which operates in vacuum. In order to preserve
the modularity of the device and avoid unnecessary removal of optical fibers
during transport, each detector ring is provided with its own light pulser
system. Figure 2.12 shows schematics of the mechanical assembly of the light
pulser system and of the driving circuit for the light emitting diodes (LEDs)
which is triggered by an external NIM logic signal. During experiments, the
light pulser is triggered at a rate of about 1 Hz. Light is generated by
simultaneously pulsing an array of eight LEDs (Hewlett Packard HLMP-3950)
which generate light at wavelengths around 565 nm. The emitted light is
diffused by reflection from an inclined Teflon surface. Light fibers which view
only the scattered light transport the light to the individual photomultiplier
tubes.

Because of temperature fluctuations and aging effects, operation of
light emitting diodes is not stable over long periods of time. Therefore, the
intensity of each light pulse is monitored by two PIN diodes (Hamamatsu
5123) read out by standard solid state detector electronics. The ratio of the
signals of the two PIN diodes can be used to monitor their stability. The ratio
of PIN diode and photomultiplier signals can then be used to monitor the

gain of the individual photomultiplier tubes according to the relation

Ch’:Chx(_1:_Afl)_).
1+APMT

(2.2)
Here, Ch denotes the ADC conversion measured for a given event, Ch’ is the
conversion corrected for gain shifts and APD and APMT are percentage
changes (measured with respect to some arbitrary time t=0) of the average of
the two PIN diodes and the individual photomultiplier signals for LED

generated light pulser events. Better than 1% gain stabilization is achieved if

26

 

  

 

 

 

 

 

 

 

 

 

 

 

MSU-90-049
Fiber optics bundle
LED drivercircuit
+24V
560
353335333233333332323333:3}» HLMP
rift-3:3: ----- 395°
........ 2N
2N2222
2700 2700
e .mv
1.5us

 

 

Figure 2.12 Schematics of light pulser assembly and LED trigger circuit.

 

27
the temperature of the CsI(Tl) crystals is kept constant. (Variations of the

scintillation efficiency of CsI(Tl) caused by temperature fluctuations cannot be
detected with the light pulser.) It was verified, however, that active cooling of
the base plate ensures rapid achievement of a stable operating temperature for
the Miniball.

Figure 2.13 illustrates the gain stabilization achieved with the light
pulser system. The gain variations of a photomultiplier (enhanced by
variations of the supply voltage) were directly measured by irradiating a
CsI(Tl) crystal with a-particles emitted from a collimated 228Th source and
monitoring the peak location of the 8.785 MeV a-line; they are shown by the
open points in the figure. The solid points in the figure show the peak
positions obtained in the off-line analysis after correcting the gain variations
according to information obtained by the light pulser system. Gain stability to

better than 1% was achieved.

r- .- flu. C

C>.~.~ ~w-mvv~

28

 

 

 

 

MSU-90-048
14: I I I F I l I I | l -
- 1
. o o .
1.3 _- O uncorrected '1
O corrected 3
1.2 —- —‘
.5 : 0 u :
(U : .
U 1.1 _— ;
Q) h :
.2. : ‘
*3 1.0_— --°- ~0- 0- ..-._ --4---o--‘°'~-O —:
5:6 : :
0.9 C.— ":
- 0 0 :
0.8 E— —'
.. u 9 :
0.7 - I I l l l l I l l L d
O 1 2 3 4 5 6 '7 8 9 10 11

Run #

Figure 2.13 Open points - gain variations of a CsI(Tl) photomultiplier
assembly determined by measuring the detector response to 8.785 MeV 0:
particles. Solid points — same data corrected for gain shifts in the off-line
analysis by using information from the light pulser system.

Chapter 3 Experimental Setup

and Data Reduction

Details of the 36Ar+197Au and 129Xe+197Au experiments along with
techniques used to extract particle identification and energies are described in
the following sections. As the thrust of this thesis work is the study of the
reaction 36Ar+197Au at E/A=50, 80 and 110 MeV, the focus of this chapter will
be on details of this experiment. Only sparse details will be given with respect
to the 129Xe+197Au at 50 MeV/ nucleon and 3'6Ar+197Au at 35 MeV/ nucleon

experiments.

3.1 Experimental setup

The experiments were performed with 36Ar and 129Xe beams extracted
from the K500 and K1200 cyclotrons at the National Superconducting
Cyclotron Laboratory of Michigan State University. The argon beam energies
were E/A=35, 50, 80 and 110 MeV, and extracted intensities were typically 108
particles per second. The xenon beam energy was E/A=50 MeV with
intensities of about 107 particles per second. The areal density of the gold
targets was approximately 1 mg/cmz.

Light particles and complex fragments were detected with the MSU

Miniball phoswich detector array. For the argon beam energy of E/A=35 MeV

29

30
the array covered scattering angles of Olab=16°-160° (rings 2-11) and a solid

angle corresponding to 87% of 41:. At beam energies of E/A=50, 80 and 110
MeV, the array covered scattering angles of Olab=9°-160° (rings 1-11) and a
solid angle corresponding to 89% of 41:. Details about the detector geometry
are given in Chapter 2.

For the 129Xe+197Au experiment the Miniball consisted of 171 detectors
(rings 2-11 with 4 detectors missing from ring 2) with a solid angle coverage of
approximately 87% of 41:. At very forward angles 2°-16°, fragments of charge
Z=1-54 were detected with high resolution using a 16-element Si (300 um)-
Si(Li) (5 mm) - plastic (7.6 cm) array [Keho 92] with a geometrical efficiency of
64%. Where counting statistics allowed, individual atomic numbers were
resolved for Z=1-54. Representative detection thresholds for fragments of 2:2,
8, 20 and 54 fragments were approximately 6, 13, 21 and 27 MeV/ nucleon.
Energy calibrations were obtained by directing 18 different beams ranging
from 2:1 to 54 into each of the 16 detector elements [McMa 86]. The energy
calibration of each of these detectors is accurate to better than 1%, and position
resolutions of :15 mm are obtained. The complete detector system subtended
angles from 2°-160° with respect to the beam axis and had a geometric
acceptance of 88% of 41:.

The detector array was actively cooled and temperature stabilized. Gain
drifts of the photomultiplier tubes were monitored by a light pulser system
(see Chapter 2). All events in which at least two detectors fired were recorded
on magnetic tape. Random coincidences were negligible due to the low beam
intensity.

Each Miniball phoswich detector consisted of a 40 um (4 mg/cmz) thick
plastic scintillator foil backed by a 2 cm thick CsI(Tl) crystal. All detectors had

aluminized mylar foils (0.15 mg/cm2 mylar and 0.02 mg/cm2 aluminum)

31
placed in front of the plastic scintillator foils. As a precaution against

secondary electrons, the detectors of ring 11 (Olab=140°-160°) were covered by
Pb-Sn foils of 5.05 mg/cm2 areal density for the higher energy argon
experiments (E/A=50, 80 and 110 MeV) and rings 2 and 3 were covered by
aluminum foils of 0.81 mg/cm2 for the lowest bombarding argon energy. In
the xenon induced reaction, rings 9-11 of the Miniball were covered with the
Pb-Sn foils. Particles punching through the 4 mg/crn2 plastic scintillator foils
were identified by atomic number up to Z=18. Hydrogen and helium were
identified by isotope as well. Approximate energy thresholds are Eth/Asz
MeV for Z=3, Eth/AE3 MeV for Z=10 and Eth/AE4 MeV for Z=18 fragments.
Low energy particles stopped in the scintillator foils were recorded but could

not be identified by atomic number.

3.2 Data acquisition electronics

Figure 3.1 shows a block diagram of the data acquisition electronics for
the 36Ar+197Au experiments at E/A=50, 80 and 100 MeV where the Miniball
ran as an independent detector. A fast clear circuit (not shown) was added to
the acquisition electronics for the 129Xe+197Au experiment where the
Miniball ran as a slave to the forward array. In events where the Miniball
detected particles but nothing was detected in the forward array, the fast clear
vetoed the gate to the tail FERAs, cleared the fast, slow, tail and time FERA,
and cleared the bit register.

Figure 3.2 shows the gate widths for the fast, slow and tail and their
relative timing along with a typical signal from a Miniball detector. During
the CAMAC readout of the FERAs, integer‘2 words for fast, slow, tail and
time are written to tape for detectors that have signals above threshold. The

dynamic range of the FERA is 2048 channels.

32

 
 
   
   

x12 ECL x12

  

COMO“ SW!

150 as

Figure 3.1 Schematic diagram of the data acquisition of the electronics of the
Miniball.

33

 

 

 

 

 

F
0 1500: 15m
|_J l l |
"fast" "slow" "tail"
33m 390m 15'“

Figure 3.2 Timing and widths of the fast, slow and tail gates

3.3 Particle identification

Typical spectra used for on-line analysis are shown in Figures 3.3 and
3.4. These spectra were generated for detector 3-6 using one run of the
36Ar+197Au reaction at 110 MeV/ nucleon (about 4 hours of beam time, =107
events in the Miniball). By plotting the fast versus slow components of the
signal (here slow=slow FERA word/ 4 and fast=fast FERA word/ 4), clear
element identification is obtained. Using the tail (tail=tail FERA word/4)
versus slow, isotope resolution for H and He is obtained. The lines in Figure
3.4 are used later to make isotope identification easier.

To facilitate the gate-setting procedure, we have constructed some

simple particle identification functions. For element resolution we define
fast - e(slow)
2

 

fast’ = (3.1)

where fast is the fast FERA word, slow is the slow FERA word and e is a
detector dependent constant which ranges from O to 1. A spectrum of fast’
versus slow (slow word/ 4 + 15 channels) for all runs (36Ar+197Au, E/A=50, 80
and 110 MeV) is shown in Figure 3.5. Of particular note is the separation of
the double 0: line from Li. This separation allows a rather clean measurement
of the number of intermediate mass fragments (particles which fall to the
right of the red line with 223) from the light charged particles on an event by
event basis.

In order to facilitate setting the gates for isotope identification we
constructed the PID function shown schematically in Figure 3.6. Using the
upper and lower lines shown in Figures 3.4 and 3.6 we construct the

following variables:

tailr = tail + 6 (3.2)

08f

35

 

 

 

 

 

500 —I I I I I I I I I I I I I I I l I T I I I I l ‘I—

400 o” _.

300 .3

.1.) -

(I) -

c0 _
‘H .

200 % n‘;v -4

100‘ -—

l 1

l .

l -

O l. J l l l L I l 1 J. J l l L l l l l l l l l 1— 1 l
0 100 200 300 400 500
slovv

Figure 3.3 Fast versus slow spectrum for detector 3-6 (ring 3, position 6).

36

 

500
400
r

300

tail

200
j-

100

 

 

 

 

Figure 3.4 Tail versus slow spectrum for detector 36 (ring 3, position 6).

 

 

slow

37

 

500 —I I l l T I I I I I I I l I l I I I I I l I I ‘1—

400

300

slow

200

100

 

 

 

Figure 3.5 Slow versus fast’ for detector 3-6. The solid line is used to separate
light Chal‘ged particles from intermediate mass fragments.

Wher

from

whet

Figui

and

lsoto
slow
gates
Itjecl
the V

idem

38

where 6 is a random number between -0.5 and 0.5 (to remove digitization
from the displayed spectra);
tail, = line, (slow) (3.3)
tail2 = line2(slow) (3.4)
where line, and line2 are shown by the upper and lower lines respectively in

Figure 3.4 and Figure 3.6 and are parameterized as a function of slow;

 

Arm = tail, -tai12; (3.5)
A = tail, - tailr (3.6)
and
A
PID=512XA . (3.7)

[DIX

Isotope resolution when using this PID function is shown in Figure 3.7 (note:
slow = slow word/ 4 +15, the extra 15 channels just to make it easier to set the
gates). In this figure are plotted only the particles that fall to the left of the
rejection line in Figure 3.5 (i.e. just light charged particles). Gates were set in
the valleys between different isotopes of Figure 3.7 to determine particle
identification.

On the other hand, for element identification, rather than set gates in
the valleys, the gates were set along the ridges in Figure 3.5. A linear
interpolation between ridges was then used to determine a ”real” (as opposed
to ”integer”) Z value for the point (fast’,slow) in question. The resulting
identification of elements for detector 3-6 is histogrammed in Figure 3.8 for
the 110 MeV per nucleon runs.

For particles which stop in the fast plastic (the horizontal line at the
bottom of Figure 3.5) we cannot extract the Z value. However, it is possible to
make some classification of the particles using time of flight. Figure 3.9 shows

the fast signal versus time (time=tdetector-tRF, the time difference between the

39

 

   
  
    

 

 

500”"I""I l“"lr‘r'1
400:.— (slow, tail,) i
300— A _‘
08 :
4.)
200—-

  

I I I I

100

 

 

 

 

Figure 3.6 Schematic tail versus slow describing variables used in the
construction of the PID function.

4O

 

500 '—

400

300

PID

200 m

 

100 —

 

l

 

O 1 l l l l l l l l L J l l L l
O 100 200 300

slow

Figure 3.7 PID versus slow for detector 3-6.

 

39

 

I I I I I I I I I

500

   

 

I I I I I

400

 

(slow, tail,)

 

300 _.

 

tail

200

100

 

 

T I I r

I.... l . . . .l
O 100 200 300 400 500

 

 

Figure 3.6 Schematic tail versus slow describing variables used in the
construction of the PID function.

40

 

 

 

 

 

500 :I I I I I I I I I I I I I I I I I I I I I L

I .4

400 —

300 —

Q .
I—t

Q" i

200 ~

100 — —

O F- l L l L l l I 1 l J l l l l l I I l l I L [d

0 100 200 300 400 500

slow

Figure 3.7 PID versus slow for detector 3-6.

41

 

 

 

 

 

 

2000 I I I I I I I I I I I I I I
: Det 3-6 :
1500 —' '—
c _
\N, 1000 w
>.. _
j H
500 —-
0 _L1 1

 

 

 

 

Figure 3.8 Z resolution of detector 3-6. Yield of Z (counts) as a function of Z.

42

 

500 *-

400 —

300 *—

fast

200

100

 

 

 

tjnle (us)

I J l L l

1 L1 1 14

l

or ill

1

 

Figure 3.9 Fast versus time for particles that stop in the fast plastic of detector

3-6. Time t=0 is arbitrary.

43
rf time from the cyclotron and the time that a detector fires) for particles that

stop in the plastic. Two branches are clearly seen — a prompt branch (on the
left) and a slower branch (on the right), presumably due to fission of the
heavy target. In this figure the time t=0 is arbitrary - only the relative timing

between the two branches can be extracted.

Bicron crystals

The identification techniques described above work well for more than
two-thirds of the detectors in the Miniball. The remaining one-third have
been classified as ”Bicron-like” detectors and require special attention. Figure
3.10 shows a typical fast versus slow spectrum for detector 3-16 which has a
CsI(Tl) crystal from Bicron. The characteristics of a detector which we call
”Bicron-like” are two-fold: a large Z separation in fast for small slow (which
quickly disappears for large slow) and poor separation of Li, He, and H
particles. Improved resolution (shown in Figure 3.11) can be obtained by
plotting the fast versus tail components of the PMT signal. It is now easier to
separate He from Li fragments. Figure 3.12 shows fast’ (replacing slow with
tail in Equation 3.1) versus tail for several runs. Again a rejection line can be
set to separate the Li fragments from the light charged particles. Light charged

particles fall below the rejection line and are further identified by isotope

using PID versus slow.

 

 

 

 

300 400 500
slovv

200

100

Figure 3.10 Fast versus slow for a detector with a CsI(Tl) crystal from Bicron,

detector 3-16.

45

 

500 t__'_‘ I I' I I 1T T I I I I I I I I I I I I I I I I I L
. 1
400 —' °. . - "J

300

fast

200

100

 

 

 

 

Figure 3.11 Fast versus tail for a detector with a CsI(Tl) crystal from Bicron,
detector 3-16

46

500 I I I IfTI T—III I I I

 

400

300

tail

200

 

100 7

       

 

 

1 J_L L JL I; ;L LJ 1 l I L l I
0 100 200 300 400 500
fast’

Figure 3.12 Tail versus fast’ for a detector with a CsI(Tl) crystal from Bicron,
detector 3-16.

47

3.4 Energy calibration

Energy calibrations of forward Miniball detectors were obtained by
measuring the elastic scattering of 4He, 6Li, 10B, 12C, 160, 20Ne and 35Cl beams
from a 197Au target at incident energies of E(4He)/A=4.5, 9.4, 12.9, 16 and 20
MeV; E(6Li)/A=8.9 MeV; E(10B)/A=15 MeV; E(12C)/A=6, 8, 13 and 20 MeV;
E(16O)/A=16 and 20 MeV; E(20Ne)/A=1O.6, 11.3, 13.3, 15.0 and 19.8 MeV; and
E(35Cl)/A=8.8, 12.3 and 15 MeV. Figure 3.13 shows the response of the light
output as a function of energy deposited in the C51 crystal of detector 3-6 for
the listed elements. For calibrations we assumed a functional form for the
light output (slow channel) as

slow = yEC, + mew-E“ -1). (3.8)

yis the slope of the light output in the region linear with energy. The values
of the parameter [3 come from the extrapolation of the asymptote (yEcsl-fi) to
the slow axis. The values of a come from the data points in the low energy
curved region of the response where quenching effects make the response
nonlinear. For detectors in rings 1-4, these calibrations are estimated to be
accurate within 5%. Calibrations of more backward detectors were obtained by
using the energies of light particles punching through the CsI(Tl) crystals to
normalize to extrapolations of the average response of detectors at more
forward angles. The average response of rings 2-4 is shown in Figure 3.14. For
elements where the light output as function of ECsI has not been measured, a
linear interpolation between measured curves is used. Table 3.1 contains the
fit parameters for the curves in Figure 3.14. The resulting uncertainties in
energy calibration at more backward angles are considerably larger, typically of
the order of 10% and for some detectors as large as 20%.

In the 35Ar+197Au at E/A=110 MeV we had for the first time He and Li
punching through the 2.0 cm CsI crystals in the forward detectors. The

~m.v:~.~.$ -nv

\SOTfl

Detector 3—6
600 I I I I I I I T I I I I l I I

 

lllJl

500 z=2 3 5 6 a 10 17
400

300

200

slow Channel

100

IIIIIIIIIIjIIIIIIIIIWIIIIIII

 

llllllJlllllllJJlllLLll

 

O
H
O
O
N
O
O
03
O
O

ECsI (MeV)

Figure 3.13 CsI calibration of detector 3-6. The light output response (slow) as a
function of energy deposited in the CsI (ECsI). The circles represent all
previous calibration data for this detector while the squares represent
calibration taken during the present experiment. The solid lines are fits using
Equation (3.8) for the elements listed.

49

Energy Calibration

 

 

 

 

 

1200 .- I T I I I I I r I r I I I II, r I I/ ,«r 1 ‘
E I l/ :
L Ring 2.3.4 I ,’
1000 — Z=2 3 I 5 6 / 8 --
I- I I .
: I, / I 10 :
800 :- 2 I I, ' ..
0/ . ..
E / II o .4
=3: - /I . I, : I
g 600 — / :/< 1 “j
0 '_'_ ’ y . J .
D I I
L . ,’ ‘ J
400 :- I’ .
200 —- '1
O - 1 I l L l 1 l l l ' I [A l L d
0 200 300 400 500
EN(MeV)

Figure 3.14 CsI calibration of Miniball rings 2-4. The average light output
response (slow) as a function of energy deposited in the CsI (ECsI)- The solid
lines are fits using Equation (3.8) for the elements listed. The fit parameters
are given in Table 3.1.

50

Table 3.1 Fit parameters of Equation 3.8 for Figure 3.14

 

 

 

Z y 6 a

2 5.00842 84.1273 0.0293086
3 4.32194 86.1841 0.023502
5 3.9500 199.942 0.0138215
6 3.46959 189.468 0.0164807
8 2.71884 126.610 0.0194922
10 2.4808 178.975 0.016974
17 2.01865 261.832 0.00867138

 

Table 3.2. Calculated energy for which the listed particle punch through 2.0

cm of C31.

 

 

 

Z A E (MeV)
1 1 75.2

1 2 100.6

1 3 119.4

2 3 266.7

2 4 300.8

3 6 576.0

3 7 615.2

 

 

 

51
calculated punch through energies are listed in Table 3.2. These punch

through points can be used to extend the calibration to higher energies. In
doing so, it was discovered that the punch through points lie below the
extrapolated He and Li curves of Figure 3.14 by typically 20%. Clearly there is a
problem region (above 80 MeV for He) for which we have only the calculated
punch through energy with which to calibrate. Our crude solution to this
problem consisted of using the measured calibration curves out to the highest
calibration point and then performing a spline fit out to the punch through

point that connects the two regions (see Figure 3.15).

52

 

 

 

 

 

1500 I I I I I I I I I I I I I I
3 l l -
- low energy fit p :
1250 _—— - including punch through / —:
_ /

-—-I " / :
cu - ’ -
Ci 1000 :— / / —..
_ / _
g .. , _
4: 750 I / _
_. / _
U — / _
— / —
3 E / "
,9. 500 — / / —~
U) _'_ / -
_ / Z
250 — —:
p _
O _ l L l l I L l l L I l I J L I I I-4

0 100 200 300

ECsI (MeV)

Figure 3.15 The circles represent all previous calibration He calibration data
for detector 36 while the squares represent calibration taken during the
present experiment. Solid line - from Equation 3.8 using the fit parameters
for detector 3-6. Dashed line — spline fit of on punch through point to low
energy calibration.

Chapter 4 Impact Parameter Selection

4.1 Motivation

Still today, it is not yet clear how experimental observables from
nuclear collision experiments can provide quantitative information about
phase transitions in nuclear matter [Saye 76, Jaqa 83, Jaqa 84, Curt 83, Rose 84,
C011 75, Morl 79, Shur 80, McLe 81, Kuti 81, Snep 88, Schl 87, Cser 86, Lync 87] -
whether it is a liquid-gas phase transition at moderate temperatures and low
densities, or a transition between a nucleon gas and a quark-gluon plasma at
high densities and temperatures. Intermediate energy nucleus-nucleus
collisions may produce finite nuclear systems at temperatures and densities
commensurate with a liquid-gas phase transition in infinite nuclear matter.
However, interpretations of inclusive measurements are complicated by the
implicit average over impact parameter which makes it difficult to unravel
the complex interplay between statistical and dynamical effects. Comparisons
between experiment and theory are expected to become more tractable and
more sensitive to unknown model parameters as research becomes more
focused upon exclusive experiments in which specific reaction filters are

employed to select narrow ranges of impact parameter.

53

54

In most experiments, information about the impact parameter is
extracted from quantities which relate to the collision geometry via simple
intuitive pictures. Many impact parameter filters represent some measure of
the "violence" of the reaction which, in turn, is assumed to be related to the
collision geometry. Common impact parameter filters are based upon the
measured multiplicity of charged particles [Tsan 89a, Stoc 86, Stéc 86, Cava 90],
the transverse energy [Ritt 88], or the summed charge of particles emitted at
intermediate rapidity [Ogil 89]. For collisions with incident energies of a few
hundred MeV per nucleon, the summed charge, Zbow, of particles with
atomic number Z22 [Hube 91] has also been used. This quantity is the
complement of the combined p, d, and t multiplicity. At lower energies,
E/A=20-50 MeV, comparable information on impact parameter has been
extracted from measurements of the velocities of fusion-like residues [Cali 85,
Awes 81, Chen 87], charged-particle multiplicities [Chen 87, Tsan 89b], or
neutron multiplicities [Gali 85, Morj 88]. A recent analysis of data with solid
angle coverage restricted to forward angles (elabgm) suggests that improved
selectivity for central collisions could be achieved by introducing a new
observable, the transverse momentum directivity (defined in Section 4.6) and
by simultaneous cuts on large charged-particle multiplicities and small
transverse-momentum directivities [Alar 92].

A priori it is unclear to what extent the various techniques select
similar or equivalent impact parameters, and whether one technique
provides superior resolution to another. At low energies, cross calibrations
have been performed between the linear momentum transfer techniques and
the emitted charged [Tsan 89b] or neutral particle multiplicities [Gali 85]. In
this chapter, we investigate 36Ar+197Au collisions at incident energies of

E/A=50, 80, and 110 MeV and 129Xe+197Au collisions at E/A=50 MeV and

IIkIet
pafidex
waged]
hydrogex
uefiflna

Th
wiousi
duhtd]
mrdadc
fitsls:
ganth
be ‘mpaa
I?pts
Pflflnme

0ndu§o

.l.
I
.ml.

“-lPie 1
RIan
3.7;: by at

‘1‘!
033

EctDISH

55
explore the relation between impact parameter filters based upon the charged-

particle multiplicity, NC, the total transverse kinetic energy of detected
charged particles, E,, the mid-rapidity charge, Z), and the multiplicity of
hydrogen nuclei, N1 (the complement of med). We also examine the
usefulness of a directivity cut in selecting central collisions for these systems.
The chapter is organized as follows: in section 4.2, we define the
various impact parameter filters; in section 4.3, we compare the relative scales
derived from the different observables and investigate their cross-
correlations; in section 4.4, the efficiency of the individual impact parameter
filters is quantified in terms of their ability to suppress contributions from
projectile fragments and fast particles emitted at forward angles; in section 4.5,
the impact parameter filters are studied again, in terms of their ability to
suppress azimuthal correlations; in section 4.6, we study the effects of impact
parameter selection using directivity cuts; in section 4.7, a summary and

conclusions are given.

4.2 Definition of filters

Throughout this work, we will use the following quantities to extract
information on the magnitude of the impact parameter:

(1) The charged-particle multiplicity, NC. This quantity includes all
charged particles detected by the Miniball (and the forward array for the xenon
induced experiment), even if they are not identified. For example, heavy
fragments stopped in the scintillator foils are included in the definition of NC.
Multiple hits in a single detector module are counted as single hits, even if
they can be clearly identified as double hits (as is generally the case for double
hits by (at-particles). The number NC is therefore equal to the number of

detectors in which at least one charged particle is detected in a given event.

[1560'

Here,
angle

the M

charge

OI, eq'.

Here, -
iOIAI S:
referer

Sj‘sterr

ddlneg

Where

56
(2) The total transverse kinetic energy of identified particles, 15,, defined

[Tsan91]as
E: ZEsin25=9 2— “52mm (4.1)

Here, 5,, pi, and 9,. denote the kinetic energy, momentum and emission
angle of particle i with respect to the beam axis. E, was calculated using only
the Miniball detectors.
(3) The mid-rapidity charge, Z,, defined [Ogil 89a] as the summed

charge of all identified particles of rapidity y with

0.25),“ S y S 075ij + 0.25))” (4.2a)
or, equivalently,

0.75%, S y’ S 0.75y;mj (4.2b)

Here, primed quantities are defined in the center-of—mass rest frame of the
total system and unprimed quantities are defined in the laboratory frame of
reference; yam, y,” and ym. denote the rapidities of the total center-of-mass
system and of target and projectile, respectively. The rapidity, y, of a particle is
defined [Gold 78] as

2 2
y=%m[\’m +P +pcos9]=mnh—1(ficosg), (4.3)

\[mz +p2 —pcosG
where m, ,8, and p denote the particle's mass, velocity and momentum,
respectively.

(4) The identified hydrogen multiplicity, N,. This quantity is defined as
the number of detectors in which a Z=1 particle is identified. The definition
includes hydrogen nuclei which punch through the CsI(Tl) crystals and
double hits by p, d, or t Nl is the complement of Zbow, [Hube 91], the summed
charge of particles with atomic number of 222.

Figures 4.1-4.3 show the measured correlations between the quantities
NC, 5,, Z, and N,. At all incident energies, the four quantities are strongly

57

E/A=50MeV Ar + Au

 

 

 

I

 

 

 

 

0 200 400 500 800
c Et(MeV)

Figure 4.1 Correlations between charged-particle multiplicity NC, transverse
energy E,. intermediate rapidity charge Z, and identified hydrogen

multiplicity Nl observed for 36Ar+197Au collisions at E/A=50 MeV. Adjacent
contours of different color differ by factors of 5.

30

20

60
N 40

20

F
‘Blile 42
9.] . ,
9'3" 5,.
muifiplicn

 

58

E/A=80MeV Ar + Au
30 I l I I I I 40

 

130
-20 oz

‘10

 

0
'60

—40 4“

 

'20

 

 

 

 

0
0 10 20 30 O 300 600 9001200

NC Et(MeV)

Figure 4.2 Correlations between charged-particle multiplicity NC, transverse
energy E,, intermediate rapidity charge 2,, and identified hydrogen
multiplicity Nl observed for 36Ar+197Au collisions at E/A=8O MeV. Adjacent
Contours of different color differ by factors of 5.

3C

2C

1C

BC

N4C

2C

59

E/A=110MeV Ar + Au
30 I I I 45

 

    

20

 

 

‘40

 

‘20

 

 

 

O
0 15 3O 0 500 1000 1500
E*(MeV)

Figure 4.3 Correlations between charged-particle multiplicity NC, transverse
energy 8,, intermediate rapidity charge Z}, and identified hydrogen
multiplicity NI observed for 36Ar+197Au collisions at E/A=110 MeV. Adjacent
Contours of different color differ by factors of 5.

60
correlated. In general, an increase in the value of one observable is

accompanied by increases in the values of the other three observables. From
this observation one may already conclude that all four quantities (NC, E,, Z,
and N,) are suitable for impact parameter selection — or none of them are.
However, the correlation between E, and NC gives evidence for a slight
saturation of NC at high E,, indicating that the transverse energy might
provide a better central collision trigger than the charged-particle multiplicity.
In order to construct an approximate scale for the impact parameter, we
adopt the geometrical prescription proposed by Cavata [Cava 90]. For each of

the quantities NC, E,, Zy and N,, we assume a monotonic relationship to the

impact parameter and define the reduced impact parameter scale via

:00 =b(X)= {jde—L-dx’f ) —dX’}/ (4.4)

 

where X =Nc, E,, Z, and N,. dP(X) / dX is the normalized probability
distribution for the measured quantity X, and bu,“ is the maximum impact
parameter for which particles were detected in the Miniball (NC 22). In the
following, we will use the reduced impact parameter scale 5 which ranges
from [3:1 for glancing collisions to 5:0 for head-on collisions.

The quantitative relation between the reduced impact parameters 5(X)
and the measured observables X (X =NC, E,, Z, and N,) is shown in Figure

4.4. Individual panels present the relationships between the reduced impact
parameters 5(X) and the observables X =NC, E,, Z, and Nl shown on the

abscissae, and the different curves show the relationships extracted for the
three different incident energies.

While geometric prescriptions implicit in equation (4.4) may provide
reasonable scales for the average relationship between charged-particle

multiplicity and impact parameter, it is not clear, a priori, whether the scales

61

 

   
 
  

    

 

NC E,(MeV)
1 O0 10 20 30 40 0 250 500 750 1000 1250
. IIIIIIIIIIIII IIIIIVIII:~L: FIIIIIIIIIITfIIIITIIIIIQg
°°Ar+‘"Au. E/Aa :: E
0.8 110MeV TT 9’
aouev 3: g

 

 

.........

IIII

1111

 

1111

IIII

 

 

 

IIII IIrI

.4

.__4

1

...

.4
11111111

111111111 LLLII1111I111111111I1

1111'1111'1111I1L11I

IIIIleleIIIIIIIIII

\

._. \
1 1111IllllIlLllIlLlJIl.PLh

5 10 15 20 25 30
ZY' DI1

 

 

 

111111111111111

 

O
N
O
4:.
O
O}
O
0

Figure 4.4 Reduced impact parameter scales 13(X) extracted from the measured
quantities X =NC, E,, Z, and N,. Each panel shows the relation extracted for

the indicated observable; different curves represent the relations extracted at
the three incident energies.

62
extracted from the various quantities are commensurate. Furthermore, for

collisions at fixed impact parameter, these quantities exhibit fluctuations of
unknown magnitude. Therefore, reaction filters constructed from the various
observables could have different resolutions. This question will be addressed

in the next sections.

4.3 Comparison of relative scales

In order to investigate the relationship between impact parameter
scales extracted via equation (4.4) from the various measured observables, we
have set narrow gates on impact parameters 5(X), defined by means of an
observable X, and determined the conditional distributions of impact
parameters 5(1’), constructed from different observables Y (Y at X and X ,Y =
NC, E,, 2,, N,).

Conditional impact parameter distributions are presented in Figures
4.5-4.8. Individual panels of these figures show conditional impact parameter

dP(13(Y))
distributions T determined from the indicated observables Y. Left and

right hand panels show distributions extracted for the cuts 13(X)=0.05-0.1 and
5(X)=0.35-0.45, respectively. Dashed and dotted curves show results obtained
by cuts placed on one observable, and solid curves show results obtained by
simultaneous cuts placed on two observables. The observables X used for
these cuts are indicated in the individual left-hand panels; the conventions
for left and right panels are identical. For better comparison, all conditional
impact parameter distributions are normalized to unit area.

For the present reaction, impact parameter filters based upon N1 (the
complement of 2,0“) are considerably less selective than reaction filters based

upon NC, E, and 2b,“. This effect is clear from Figure 4.5, which compares

63

36Ar + 1971.11, E/A=110MeV

 

   
    

 

 

 

 

 

 

  
 
 

    

 

   
    

 

 

 

6 : I I I I I I I I I I I I I I I I I I I I I I I::I I I I I I I I II I I I I I I I II I I I I I g
.. .1.- m
.7.— “ A ._‘.‘Z_ A A .—'1 C:

5 E b(z,) ; b(X) = 0.05-0.1 3E b(z,) : 1100 = 0.35-0.45 3 3°

4 E- """ X a N‘ iii—- _-:1 ('0

3:—// \.... “""E‘ -—::—- -—i§
E I .'\ x = E" N1 55 E

2 r- -' \ '- 1:— '1
El . \ 3E -, :

1 : \ ‘71:?— _5

1'1111111111 11-111111111::'11111111111 "2111111:

0 :I I I I I I I I I I I I I I I I I I I I I I I I::I I I I I I I I I I I I I I | I I I I I II I I:

5 g— S(N,) : Soc) - 0.05—0.1 "51-: $01,) :B(X) - 0.35-0.45“;

4 :_- """ X 8 Zr :: —§
E ‘- ‘- x a E :1: :

3 :— ‘ -::- -—

25 — X =- 31.2, 55 j

1 E— ‘Z’é‘ “S

' 1" I 1 1 L121: 1 I 1 1 1 =

0 I I I I I I I I I I::I I l I I I I I l I I I I:

5 Bus.) : 1300 = 0.05-0.1 1:? 1:03,) 801) = 0.35-0.45 "5

4 ----- x - zY -E':'- -E

3 — - X " N! ._:.E_ _:

2 — x " "v z, 55 3

EE ".\ E

1 .' \ -—ZE— , / ‘- —5

O{ .'-11~11111111112E 11111111111'31L111L:

0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0

Figure 4.5 Conditional impact parameter distributions extracted for Y = Z, (top
panels), Y = N, (center panels), and Y = E, (bottom panels) for the 36Ar+197Au
reaction at E/A=110 MeV. Left and right hand panels show distributions

selected by impact parameter cuts 5(X)=0.05-0.1 and 5(X)=0.35-0.45 on the
indicated observables (X =N,, E,, and Z). All impact parameter scales were

Constructed according to Equation (4.4).

dP(b)/db

64

36Ar + 19713.11, E/A=5OMeV

 

 
 

 
 

 

 

 

 

 

      

  

 

 

GE IIIIWIIIII IIIIIII I I I IIIEErII IIIFIIIFITI I I I I II—FTII: 5
:_ - - _::_ - - _: c:
5 : b(z,) : b(X) = 0.05-0.1 :: b(z,) : b(X) = 0.35-0.45 : 4,
IL— ..... ._":_ _: N
4 : x " N0 :: :1 I
3 E. " - X " Br .55. .3 2
E ,—."' _X=EvN EE 5°
2 - ' \ c l-‘_ d
- -' . —::_' _:
1 :— <-- 55—- —;
”.1"111|1 1|1111"'1_IL111:: 1 llllllllll. 41:
0 : I II I I I I I I I I I I I I I I I lira-1:1 I I I I I I I I I I I I I I I I I I I rrI:
5 E’ b(Nc) : b(X) - 0.05-0.1 ‘3: b(Nc) : b(X) - 0.35-0.45 3
4 E— ..... x - z, -—:E— -2
d— -1
E - — x = E :: :
3 _ £ —_ ——
25 -.._—X=Evzr ii i
E ' ' ii i
1 : .- 75— ":
"1.111|1111|1111 ‘~11|1111:: LL:
0 :I I I I I I I I I I I II I I I I I I I I I I I::I I I:
L - 3: .3
5 ~_- 5(3.) : b(X) = 0.05-0.1 55 :
4 E- ''''' x = z, —§§— —:
3 E— "' " X 3 Ne __::_ __'-‘
E (~--\ ——x-Nc.z HE E
2 - A“ \. I _:.-__ .4
E 1" V. 35 E
1 ' ‘ \". ._‘:_ . '- ._"‘
. V._ :I- ' \I. _ :1
0 I l l l I l l l l I l l l "II-44 l l l LLL: LI 1 l l l I l l I l I 1 L ' 4.1 L l:
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0
A

Figure 4.6 Conditional impact parameter distributions extracted for Y = Z, (top

panels), Y = NC (center panels), and Y = E, (bottom panels) for the 36Ar+197Au
reaction at E/A=50 MeV. Left- and right-hand panels show distributions

selected by impact parameter cuts 5(X)=0.05-0.1 and 13(X)=O.35—0.45 on the
indicated observables (X =NC, E, and Z). All impact parameter scales were

constructed according to equation (4.4).

65

36Ar + 19713.11, E/A=80MeV

 

 

 

 

 

  

 

 

 
    

 

 

0")

6 I—I I I I I I I I I I I I I II I I I I Can-I I If I I I I IT I I I I I I I I I
5 I A I A I I a; I A I A I I 5
5 E“ b(z.,) : b(X) = 0.05-0.1 3E— b(z,) : b(X) = 0.35-0.45“:
4 E— """ X 3 NC i}. -_:
3 a. , .- - - x = E. _fia. __=
I / -' '. __ - "b :1
2 = / x - 3.. N. 3%.. _a
I: .- \ .. :: d
1 = _- \a, .55. _3
\2 35 ‘ a
'D IILIIIIILII .' _LIILIIIII‘” ' LIILILIIIILI' IJII"
0 —I I I I I I I I I II I I I I I I I I I—IEI I I I I I I I I I I I I I I I I I I I-1
a 5 l A I A l I a. I I A I I 5
> 5 E— b(Nc) : b(X) - 0.05-0.1 _EE— awe) : b(X) - 0.35-0 45";
.0 4 E._ ..... x - zY _g_ _i
IE: 3 3.. / -'-’XI= Et ._3E_. :
p- / ..' -.. a _—
a _ _. me; E
.- ' db— .4
: .' :t: z
1 I .' -. —‘E— .5
0 ‘i111 IILII 11 - 111 LIL41.§E ILI :
:Fl I I I I I I I I I III I I—II II I II I3:I I I II I:
5 g" 5(3‘) : Soc) = 0.05-0.1 —§E— S(E,) : b(X) = 0.35-0.45 "5
4' :'—- """ X 3 z? '—::—- —:
3 E " ’ x ‘ "c €5— ——f
2 E _ x ‘ Nc- 2? SE 5
a '\. EE :3
1 ’ ‘- ‘73;— *5
~$L4I I I I L I I I I F: I l I I I I I I II " I I LL I I:
0
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.

IIIO-ZB-GSW

Figure 4.7 Conditional impact parameter distributions extracted for Y = 2, (top

panels), Y = NC (center panels), and Y = E, (bottom panels) for the 35Ar+197Au
reaction at E/A=80 MeV. Left- and right-hand panels show distributions

selected by impact parameter cuts 5(X)=0.05-0.1 and 13(X)=0.35-0.45 on the
indicated observables (X =NC, E,, and 2,). All impact parameter scales were

constructed according to equation (4.4).

66

36Ar + 197Au, E/A=110MeV

 

ZPO-ZG-OSK

 

 

 

 

 

 
   

 

 

 

 

 

 

6 -I I II I I II III I I I II I I I I—II—I I II II I f I I II I I I I II fl
5 I AI A I I 55 I AI A I I i
5 :_ b(z,) : 1300 = 0.05-0.1 "E: b(Z.,) : bot) = 035-045“;
4 ;— , ----- x - a. -:;—- —-3
3%.—Is" ——X'E‘ -—f§- -—f
E /-' \'2 — x 3 Ev Ne 55 E
2 : \°. —::— '1‘
: _' \'- EE '3
1 .' \'-. “:5— _z
oIIIIIILII \' 1111'1111:: ~' 1 1'111 :1
: I I I II I I I I I I II I I I I I I I IE:I I I I II I I I I I I II :1
5 E" Sm) : Sm - 0.05-0.1 ‘gg‘ Ewe) : Bot) - 0.35-0.45?
4;.— ..... x-zr __;§_ —;
3 FE.— /4 - _ - x " E‘ -—fE—- —5.‘
: I". \'-A — x - E. z, :: :
I .' \'. ‘I—- -:
2 5 ’-' \'-.' I; . - a
1' .- \n. ‘55-— - "E
OIILII111L1'1IL111I111F: ”1'1111 IIII..'IIIIIL:
—II II I I I I I I II I I I I I I I I—u—I I II I I I I I I I I I I I I III I-
E. I A I A I I 35 I A I A I l g
5 : baa.) : b(X) = 0.05-0.1 “5.?" has.) : b(X) = 0.35-0.45 ‘5
: ' I: :
4E- '\ “““ x-z, 15— -=
3 3 ' " "‘ X ' N0 .35.. .5
2% _ x ‘ NI» 2, ii i
1“ -—;;— -:
0L '141I1L11L111F” 11Lfi
0.0 0.2 0.4 0.6 0.8 0. 0.8 1.0
A
b

Figure 4.8 Conditional impact parameter distributions extracted for Y = Z, (top
panels), Y = NC (center panels), and Y = E, (bottom panels) for the 36Ar+197Au
reaction at E/A=110 MeV. Left- and right-hand panels show distributions

selected by impact parameter cuts 5(X)=0.05-O.1 and 13(X )=0.35—0.45 on the
indicated observables (X =NC, E, and 2,). All impact parameter scales were

constructed according to equation (4.4).

67
conditional distributions based upon N,, E, and Z, for collisions at E/A=110

MeV. The reduced resolution of impact parameter filters based upon N1 is
most likely of statistical origin. For collisions at fixed impact parameter, the
relative magnitude of statistical fluctuations is enhanced for the observable
Nl because it contains, by definition, only a subset of the emitted particles.
Figures 4.6, 4.7 and 4.8 present conditional impact parameter
distributions at different energies. In general, the conditional impact
parameter distributions become narrower with increasing particle energy.
Qualitatively, this can be understood in terms of statistical fluctuations: at
fixed impact parameter, statistical fluctuations lead to distributions in X =Nc,
E,, and Z, of finite widths AX. For collisions at fixed impact parameter, the
mean values of all quantities (X =NC, E, and 2,) increase with increasing
projectile energy and the relative fluctuations, AX / X, decrease. Hence, impact
parameter determinations should become more accurate at higher energies.

At a given incident energy, rather similar conditional impact
parameter distributions are extracted from the observables NC, E, and 2,,
With little sensitivity to the applied cut on other observables. While some
differences exist, they are generally small.

Cuts on small impact parameters, 5(X)=0.05—0.1, generally produce
conditional distributions that peak at larger impact parameters, 13(Y) =0.2.
Conditional impact parameter distributions extracted for simultaneous cuts
on Small impact parameters, 5(X,)=5(X2)=O.OS-O.1, are slightly narrower and
the)’ peak at lower impact parameters, 13(Y)=O.12-0.16, than those obtained
fr om cuts on a single observable. Hence, somewhat improved selection of

cen trail collisions can be obtained from multi-dimensional cuts.

One-dimensional cuts on 13(E,)=0.05-O.1 produce narrower distributions

m bCNC) and 5(2,) than the alternative one-dimensional cuts on 5(2,) and

68
5(NC), respectively. Furthermore, two-dimensional cuts on small impact
parameters produce narrower distributions for 13(E,) than for 5(NC) and 5(2)).
These observations suggest that filters based on E, may be more effective in
selecting central collisions than filters based on the other observables
investigated in this work.

For cuts on intermediate impact parameters, 13(X)=0.35-O.45, the
conditional distributions are peaked at values close to 5(Y) =O.4, i.e. close to
the cut on 5(X). This correspondence between the impact parameter scales
derived from different observables improves at higher beam energies,
indicating that all impact parameter filters have improved resolution at
higher energies. Again, small improvements in resolution are obtained by
the application of two-dimensional cuts. However, these improvements are
less pronounced than those seen for very small impact parameters.

The similarities of the various conditional impact parameter
distributions shown in Figures 4.6-8 indicate that the three observables (NC,

5, and 2,) have similar selectivity on impact parameter. Figure 4.9

summarizes and corroborates these findings in a compact form. The
individual panels of the figure show the centroids (points) and widths at half
maXimum (vertical bars) of conditional impact parameter distributions
selected by narrow cuts on impact parameters, centered at 5(X)=0.1...0.9 and
extracted from other observables X . Left, center and right columns show data
at E/ A=50, 80, and 110 MeV, respectively.

Overall, the relation between the individual impact parameter scales is
fairly linear with significant deviations occurring only for small and large
l.mIDact parameters. Such deviations must be expected at the edges of the
l.mlDact parameter scales whenever the two-dimensional correlations shown

i - . . .
n Izlgtues 4.1-3 have finite Widths and some curvature.

f

P‘- \ a .

HIKE»

1.0

0.8
A

’chei
2

<5 0.4
0.2
0.0
0.8

A

’2 0.6

51

(.3 0.4-
0.2
0.0
0.8

A

’2 0.6

S

('3 0.4
0.2
0.0

0.0 0.2

E/A=50MeV

69

“Ar-t 197Au
E/A=80MeV

E/A=110MeV

 

__ I III - III
{III EIIIIIW ,EIIIIIIIII

 

IV'VTI'ITI Iv

 

 

VIITIITT' 'UT'IITII

TV

 

:II

P 1 I l I

a,l

 

l I l

l

 

EIIII

L I, r l 1 I . l

 

 

Bot)

Figure 4.9 Centroids and widths at half maximum of conditional impact
p‘h‘alneter distributions selected by cuts on impact parameters constructed
from other observables X indicated in the individual panels. For ease of
pr eSentation, the open points have been displaced from the centers of the
gates on 13(X). Top, center and bottom panels show the values for impact
Parameter scales constructed from NC, E, and 2,, respectively. Left, center and

right hand columns show data for the 36Ar+197Au reaction at E/A=50, 30, and
110 MeV.

CfO‘ZB‘flSN

0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0

(Jfl‘

._
Cr
(C)

‘ré
Pm.

51v

Parr

70
Furthermore, the widths of the various conditional distributions are

rather similar. Some modest improvement in impact parameter selection can
be obtained by using multi-dimensional cuts (solid circular points). These
improvements are difficult to quantify since none of the conditional
distributions can be narrower than their natural widths at a given sharp
impact parameter. Since the widths from one— and two-dimensional cuts are
rather similar, one may surmise that the widths extracted from the two-
dimensional cuts are close to the intrinsic resolution of the respective impact
parameter filters.

Improved impact parameter selection could be expected from
observables (or combinations of observables) which are less subject to
statistical fluctuations. For example, one may argue that central collisions
could be reconstructed by an accurate determination of the energy deposited
into internal degrees of freedom. For fixed excitation energy, the number of
emitted charged particles will exhibit considerable fluctuations due to
statistical partitions of the de—excitation energy among neutrons, protons and
complex particles, all of which are emitted over a broad energy spectrum.
Improved reconstructions of the initial excitation energy could be expected
from simultaneous measurements of neutral and charged-particle
multiplicities, but some fluctuations would remain due to the finite energy
distributions of the emitted particles (and varying separation energies). One
should expect statistical fluctuations to be smaller for the total transverse

energy of the emitted charged particles since this quantity suffers only from
the random partition of the excitation energy into charged and neutral
particles and into longitudinal and transverse velocities. Close inspection of
Figures 4.5-8 reveals that the double-gated 5(E,) distributions are slightly

narrower than the other impact parameter distributions, possibly indicating

71
that the transverse energy exhibits a particularly good selectivity for central

collisions.

At higher energies and for noncentral collisions, geometric
considerations may lead to the expectation that the intermediate rapidity
charge might become competitive or even more appropriate since it could
provide a better measure of the participant zone. Unfortunately, microscopic
reaction models do not yet describe realistic descriptions of complex particle
emission processes, and fluctuations in Z, due to fragment formation at the
interfaces of the participant and spectator zones are difficult to assess. Since a
substantial portion of the emitted nuclear matter emerges in the form of
bound clusters, it is not clear which observables (or combinations of
observables) provide optimal impact parameter selectivity. In the next
section, we explore this question further by investigating an alternative
measure for the selectivity of various impact parameter filters which is based
upon their ability to suppress the emission of fast fragments at forward

angles.

4.4 Suppression of projectile-like fragments

In this section, we explore the efficiency of various impact parameter
filters in terms of their ability to suppress projectile-like fragments emitted
with near-beam velocity at forward angles.

For illustration, Figures 4.10-12 show the energy spectra of beryllium
(left-hand panels) and carbon (right-hand panels) nuclei detected in Rings 1-
10 in the Miniball for incident energies of E/A=50, 80, and 110 MeV,
respectively. Top and bottom panels show the energy spectra gated by cuts on
the charged-particle multiplicity corresponding to 5(NC)>O.6 and 5(NC)<0.3,

respectively. At all incident energies, the energy spectra gated by large impact

72

36Ar+197Au, E/A=5OMeV

 

 

 

 

 

 

 

 

IIII FlrrrIIIIIIIIT IIIIIIIrjIIIIIIIrlrg
10-3 80. zsncsia 7 c. 2511,3513 I"
_ .0. .. §
_ 71:}. '00-.‘ ‘. .‘.".,l... O
10 4 J _.-... ..... , ...... . g
. 0.. o. . i.‘ .. 0.... ...
A ‘. “ .l . ' .~ ‘. .
> ‘ 0
g \! , .' cg
°o
7 '
a l l I l l l l l l l l L L1 1 1+1 l L 1 J_L
; I l I I I T I Al I I I I T r FYI I I I I I
12.9 ' 4w, mucus
a 19.5' 5’3. ~.
3 27° - ' '
35.9 ‘ , .
No. . ‘5, _ , . .' a 1!!
‘6 57.5. \s , '. on
7:29 p - . ‘ ’.
90° .. V ,_
110° - ! .
130°
10—7_ILLJ[LLIIILI;LL111 LllLlLL.Ll|llllLlLll
O 200 400 600 800 200 400 600 800

E (MeV)

Figure 4.10 Energy spectra of beryllium (left-hand panels) and carbon (right-
hand panels) nuclei emitted in peripheral (1350.6, top panels) and central
(5<0.3, bottom panels) 36Ar+197Au collisions at E/A=50 MeV. The exact cuts
on the measured charged-particle multiplicity are given in the figure. The
solid curves are fits with equation (4.5). The parameters are listed in Table 4.1.

dzP/dfldE (sr"1 MeV“)

73

36Ar+197Au, E/A=80MeV

 

 

 

 

I I I I I T I I I I I I I I I—T I I I r1 I I I Ii I I I I fiT I I I I I
38.25Nc516 - C.2$N§SIB
’1‘? H .‘....
-- - ...... 4:31... .-
. --.. 3" .~ ito,‘ a...
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E; .0 . ..I .o 3‘ .’ . .0
_, o 3‘ .
‘3‘- 00 I. a \‘ , II.... DO
\ o I ‘9 9!!
-. 1 . a if i
J 1 [J l L1 1 1 [I l l l I l I l 1 L l l l L L L l l l l l l l l l l 1
Fl I I I I I r I I I II I I I I I I I TI I I I I Fl I I I I I I I I I I
ix
- jab-k3 Be. N :2: 125° 1-13.1. c was
,, . .. 6 J4 .
'. ‘. 19.5- . '. «x1
‘ m. °. 35.9 ‘ " . '. ‘
'l‘. , - 45' - 0 lg
'l‘ ' 57.5- - _' 'o- ’ Q
: ~. '. i m- ,- fig
' f 90. ° if _-
' «. ‘ 110'
i 130‘ §
LllLngLllllLlLLLll lllllllLllLlllllllL
O 200 400 600 800 200 400 600 800

E (MeV)

Figure 4.11 As Figure 4.10, for E/A=80 MeV.

QVO-ZB-HSN

LIV

.32: :u\Ln 3

(IN
1

f“Ed-re 4.

74

36Ar+197Au, E/A=110MeV

 

 

 

 

 

 

10-3 IIIIIIIIIIIIIIIrrIfIIIIIIjTIIII—IIIIIIT
_4 Be. 25N¢$18 «"00... 12.5- c. 2311:5111
10 , a; ' 19.5' - ,3“ .0.
i. - - . 27° ' "“ ‘1‘ .0‘
- . ........ .V\{‘ o
i . ' . . Do 35.5. _ .‘ . ‘ 3» o. . ..-... I
A 3‘ . O ... . ‘5. ‘ . . o ..
T . ' . 57.5' \i .. '- °
3’ O O 725' I! o '0 '
§ °°° 90' 21-"- ”9 ! 1!
9 110° " g
7 130° ‘ '
"‘ | l l
U! 1 1 1_L 1 1 1 1 1 1 1 1 1 1 1 1 L
V I I I FT I I I I I I I I TfIf I I I
C23 - .-_.
,O ’14:“ . ‘ Bo. mac
9.? ° "“11
\ 0 ‘ ‘ .
0. ° ' '
N o
11 'o
'4" ! .0 f
10“? lllllllllllLllllllJ—lllllllllllllllllll
O 200 400 600 800 200 400 600 800

E (MeV)

Figure 4.12 As Figure 4.10, for E/A=110 MeV.

BIO-ZO-OSN

75
parameters exhibit pronounced maxima at forward angles which correspond

to fragment velocities close to the beam velocity. Such projectile-like
contributions are strongly suppressed in the energy spectra selected by high-
multiplicity cuts corresponding to 13(NC)<0.3, as qualitatively expected from
simple geometrical considerations within the framework of a participant-
spectator approximation. These qualitative findings are consistent with
observations at lower energy, E/A=3S MeV [Kim 92]. They indicate that
impact parameter filters based on charged-particle multiplicity are rather
effective in selecting collisions with large geometric overlap between target
and projectile, even in situations where simple participant-spectator models
are not expected to be accurate in detail.

In order to provide a more quantitative basis for discussing the effects
of impact parameter filtering on the measured energy spectra, we fit the
energy spectra with a simple three-source parameterization corresponding to
the superposition of three Maxwellian distributions centered at velocities v,

and characterized by temperature parameters Ti:

 

2 3 3 —E—V +Ei—21/E,(E-V )cosB
fig: 20‘.(E,9) = XNiJE-vc exp ( C T C ) ,(4.5)
i=1 i=1 1'
where
E. = §mvi2 (4-6)

is the energy of a particle at rest in source 2'. The parameter VC is introduced to
roughly account for Coulomb repulsion from a heavy charge assumed, for
simplicity, at rest in the laboratory system [Awes 81, Chit 86]. Fits obtained
with this parameterization are shown as solid curves in Figures 4.10-12. The
corresponding parameters are listed in Table 4.1.

A word of caution is necessary. Since our energy calibrations at

backward angles have considerable uncertainties, the source parameters listed

76

Table 4.1 Moving source parameters used to fit the energy spectra in Figures
4.10—121

 

 

IP/A 2 NC N1 Bl T1 N2 [32 T2 N3 B3 T3
0 4 7—13 106 0.273 9.2 40.2 0.180 17.0 55.2 0.079 12.6 1
0 6 2-13 179 0.275 9.7 24.1 0.177 17.4 39.6 0.071 14.1
0 4 219 51 0.196 14.5 231 0.098 14.3 171 0.036 8.6
0 6 219 2.07 0.221 16.3 69.3 0.141 14.9 211 0067 14.6
0 4 2-16 107 0.303 11.1 16.3 0.190 20.2 44.1 0.073 12.9
0 6 2-16 54.7 0.313 12.5 5.46 0.169 20.0 34.6 0.66 12.4
0 4 225 28.1 0.218 21.8 152 0.106 21.9 212 0.046 12.0
80 6 225 1.08 0.257 24.5 31.5 0.142 24.7 238 0.071 16.9
110 4 2-18 26.1 0.326 17.5 6.88 0.165 24.3 34.6 0.060 12.6
110 6 2-18 4.37 0.333 19.4 1.67 0.155 22.3 25.4 0.057 12.4
110 4 230 9.48 0.248 31.9 91.8 0.123 29.9 251 0.055 15.8
110 6 230 12.1 0.406 24.3 17.4 0.142 34.8 218 0.068 19.4

 

 

 

in Table 4.1 may have large systematic uncertainties. In particular, the
parameters of the slow target-like sources (i=3) must be viewed with caution,
and the temperature parameters T3 should not be misconstrued as accurate
temperature measurements for target—like residues.

Nevertheless, the fits allow estimates of the relative contributions
from fast projectile-like sources and intermediate-velocity ”nonequilibrium”
sources. At all energies, peripheral collisions (5>0.6) are characterized by
'strong contributions from projectile-like sources. Such contributions are
strongly suppressed for more ”central” collisions (5<0.3). However, central
collisions have significant contributions from intermediate-velocity
”nonequilibrium” sources representing emission during the early,
nonequilibrated stages of the reaction. Our present findings are consistent
with previous observations at a lower energy [Kim 92]. Clearly, accurate

descriptions of energy spectra and angular distributions will require

 

lFor beryllium and carbon nuclei, the Coulomb parameters were Vc=32.9 and 47.2 MeV, respectively. The
normalization constants Ni are given in units of 10’6/(8nMeV3’2); units for EIA and Ti are in MeV. At the
energies E/A=50, 80. 110 MeV. the velocities of the projectile are: Bc.m.=0.051, 0.065 and 0.07,
respectively.

77
theoretical treatments which allow the incorporation of emission from the

earlier non-equilibrated phases of the reaction, as well as from the later more
equilibrated stages.

In order to provide a more quantitative measure of the selectivity of
the impact parameter filter based upon the charged-particle multiplicity, we
have fitted the multiplicity selected energy spectra of representative complex

particles (He, Li, Be, C) with equations (4.5-6) and determined the relative

contribution, om}

/ 0,0,, of the projectile-like source to the total particle yield.
This contribution was evaluated by integrating the respective sources over all
angles and energies:

0W. 2 H (II (E, 9)dEdQ , (4.7)

0,0, = 23“” 0,05, mama , (4.8)

We did not analyze the energy spectra of hydrogen nuclei, because of
the restricted dynamic range of the Miniball (75 MeV protons punch the C51
crystals). Moreover, light particles (especially nucleons) are less suitable for
such an analysis, because of their comparatively large mean free path and
because ”thermal” smearing of the energy spectra is more serious for light
particles than for intermediate mass fragments. As a consequence, collective
source velocity components are more difficult to unravel from the energy
spectra of light particles.

Figure 4.13 shows ratios 0,,0, / 0,0, extracted from the energy spectra of
He, Li, Be and C nuclei selected by different cuts on charged-particle
multiplicity for the 36Ar+197Au reactions at E/A=110 MeV. For illustration, a
scale of the reduced impact parameter 5(NC) and a pictorial illustration of the
geometric overlap between projectile and target nuclei are included in the

figure. While such a simplistic graphic visualization must not be taken too

78

36Ar +197Au, E/A=110MeV

 

 

 

 

 

1 I I Tfir I I I I fl I I I I I I I I I I I I I I I I fl T
I I l 1 l I z
- A ..
- b (3' 9 -§
- .9 .8 .7 .6 .5 .4 .3 .2 r g
0.8 — l r l l l l 1 l — g
— . _.
- CJ He ~
§ 06 "" I O C U '7
Q : . 0 Be :
'§ _ ' ’ I C .
b“ 0.4 -— . _..
- C] O -
_ C] C] S C] -
. [:1 a
0.2 —— ' 0 —
_ D ,
_ I ‘ 7
O P L 1 l 1 I. l l L l I l l 1 1 l l l l l L #1 ! JLI_L L! l l l 1 9 l q
0 5 10 15 20 25 30 35
NC

Figure 4.13 Relative contribution of projectile-like source extracted from the
energy spectra of He, Li, Be and C nuclei selected by different cuts on charged-
particle multiplicity for 36Ar+197Au collisions at E/A=110 MeV. A scale of the
reduced impact parameter 13(NC) and a pictorial illustration of the geometric

overlap between projectile and target nuclei are included.

79
seriously, it nevertheless illustrates that complete overlap between projectile

and target nuclei is only achieved for relatively small impact parameters,
representing less than 10% of the total reaction cross section. Even in such a
naive geometric picture, some emission from projectile-like sources must be
expected down to impact parameters of 5:0.4. These simple expectations are
fulfilled rather nicely for the emission of intermediate mass fragments for
which projectile-like contributions are strongly suppressed at large
multiplicities (small impact parameters). The suppression of projectile-like
contributions is more effective for heavier (e.g. carbon nuclei) than for lighter
particles (e.g. a-particles). In fact, the emission of fast a-particles does not
follow the simple trends expected from simple geometric arguments. Possibly
a-particle emission already sets in at the early contact phase of the reaction
and can, therefore, not be described by a simple participant-spectator picture.
In order to compare of the effects of different impact parameter filters
on the shapes of the energy spectra at forward angles, we have analyzed the
energy spectra of particles detected in rings 1 and 2 (&=9°-23°) and determined
the ”fast-particle fraction” 0(v > vap)/ 0,0,, defined as the fraction of particles
detected in rings 1 and 2 with velocities larger than half the projectile
velocity. This simple quantity provides qualitatively similar insight as the
quantity om]. / 0,0, obtained from the moving source decomposition, without
necessitating cumbersome multi-parameter fits with equation (4.5). The fast-
particle fraction is well defined and it can be established with good statistical
accuracy even for narrow cuts on impact parameter. Furthermore, this
quantity is insensitive to source-parameter ambiguities associated with fits to
energy spectra which have poor statistical accuracy due to narrow cuts on

impact parameter.

80
Fast-particle fractions extracted for various cuts on impact parameter

are shown Figures 4.14-16. Different figures show results for the three
incident energies. Individual panels show the fast-particle fractions for He, Li,
Be and C nuclei, and different symbols depict results obtained by different
impact parameter filters.

At all energies, the fast-particle fractions are monotonic functions of
the reduced impact parameter. The suppression of fast particles for cuts on
small impact parameters is particularly effective for beryllium and carbon
nuclei. The suppression of fast particles is less effective for 131-particles. These
qualitative observations are consistent with the results obtained with the
moving source decomposition shown in Figure 4.14. For impact parameters
5<0.6, all three impact parameter filters produce rather consistent fast-particle
fractions. For intermediate and small impact parameters (5<0.6), filters
constructed from NC, E, and Zy appear to provide comparable resolution.
Slightly better suppressions of the fast-particle fractions can be obtained by
employing triple cuts on values of the impact parameters reconstructed from
NC, E, and 2, (see star-shaped points). Consistent with our previous findings,
these improvements are relatively inconspicuous.

For larger impact parameters, however, the differences between the
various techniques become more significant, particularly at the higher
energies. Such differences may be caused by statistical fluctuations of the

quantities NC, E, and 2,. The relative magnitude of these fluctuations should

be largest for peripheral collisions which are characterized by small mean

values <NC >, < E,>, and <Zy>. As a consequence, selections of large impact

parameters may be associated with larger uncertainties than selections of

smaller impact parameters.

U(v>vP/2)/atot

81

36Ar+ 1971111, E/A=50MeV

 

 

 

 

 

 

 

 

1.0 .- I I I I I I I I I I r I T I Id-I I I I I I t I I I I I I I I I I . .q :
: He I I ; U I U 1: Li a H 3 I: 3 (I?
0.8 :- g fl . _‘__ O i i n ‘1‘ ,3
_ 9 g _1. O t .1 g
1- x .0. O __l a
067 77“ .
0.4 :- ’ 1‘”"1: __‘I" '—_:
: D X=Zy .tt -4
"' O xag, __:_ _“
0-2 Z— x X=N¢. 2,. E. j- ~
:I 1 LJ 1 L L I l 1 1 I 1 I I I I l 1"! L I I l I I I 1 I I I ' 1 1‘
0.0 '- I I I I I I I I I I I I I I I I I:: I I I I I I I I 1 I I I I I I 4
1.0 :- __ E n _1
: Ha I I: C :8! 5 5 :
0.8 :— 5 3 8 01:- g -_j
: I 2: 5 § 2
0.6 _— 9 O i ‘17 a 1 ‘1
fl ._ 8
: a n «l- 9 n 1
0.4 :- . “__T . X -—-1
1— x .1... x d
0.2 E- __ —:
O O :l l L I L l I L l l l L l I L I l l I]- l I I I l l I I l_ l l I l L L I 1 1 1d
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0

1301)

Figure 4.14 Fast-particle fractions (fractions of particles detected at 9=9°-23°
with velocities greater than half the beam velocity) for 3‘5Ar+197Au collisions

at E/A=50 MeV. Solid points, open squares and open circles depict values
determined for narrow cuts on the reduced impact parameters determined
from NC, E, and 2,. Star-shaped points represent simultaneous cuts on

13(Nc), 1305,) and 5(z,).

82

36Ar+197Au, E/A=80MeV

 

 

 

 

 

 

 

 

 

1.0 _ VT I I I I I r I I T I I 1 I4 I I I r r I 1 1 1 I I I T I 1 r _ g
: He 1 l I l a It L, l I V a 2 5%
0.8 — x 1‘ —— g 3 9 e -—_ e
I . 3 1: a .1 é
o 6 ’ I " ' —3 °°
' : 9 g 2: o 9 :
"' x '1- O I d
0.4 :— ' X'Nc ‘1" Q —.
.8 _ DX=ZY __ _,
e ' ”=8. .2; _:
g 0.2 : )1 xauc, 2,, 3,1: 2
a '- l I l I 1 L1 I l L I I I l I I l l Id-l I I I I L1 I I I I IT I I L d
\g 0.0 - r I I I I I I I I I I I I I I I I r___‘1 I I I I I I i 9 1
> " Be _.,.. C a 1
A - ' ’ )1 ll ‘1' _
3, 0~8 .— 3 a 6'2:- x , . _
b : u I :t a 8 ° C :
0'6 '— a i: 9 i
.. .1.- 8 .1
0.4 :- 8 E f:- ° -:
.. 5 x -- a 3 J
02 :— )1 ‘1- a 2 1
.. - i .1
O 0 I. l l I I l I I I I l I L l l l I I l 4‘; L L I I I 1 LI I l 1 I 1 L1 I I I i-
'0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.5 0.8 1.0
b(X)

Figure 4.15 As Figure 4.14, for E/A=80 MeV.

U(v>vp/2)/Utot

83

36Ar+ ‘97Au, E/A=110MeV

 

1.0

 

I I I rI I I I I I I I I I I I I I fiI I I I T I I r I I r

- I I I -- I I d =

: He I g E‘ :: Li I I x E : .9

L. E .. 3 ' I c':

- I 4- d g
0.6 _— 1 f;— 3 -:
: q 5 :: fl :
0.4 — X ' X=Nc ‘Tf' g ‘—
I o x=z, _ 9 I
0 2 :_ 0 Hz 3:. x _:
. - XX=N0 Zr.E‘ :: -4
”I l I I L L L I L LJLI l I L I I 1 1.0-1 1 I I l I I LLL ILLI l I L! I 1 -‘
0.0 .- I I fl, I I I I I I I I T I I I I I I—-- r I I I I r I I I I I I I I I I I i d
: Be x 2: C :
0.8 :- xz __.— —_
.. _. X ..
o 6 ' g g a g " x ‘
__. __.... x _.
. __ g .- g . _
.. -- B o 9 ‘
I— 8 ‘P ' 8 _
0.4 :— 0 -::- l O _:
- g R d- . c:
— -- -I
0-2 3'3 ‘5' a 9 ‘:
I.1 I I I l L 1 I 1 1 L I L l l I L l I‘I-J L l g L 1 I Ii 1 I I L L L I L L L‘

 

 

 

 

0.0
0.0

0.2

0.4 0.6

0.8 0.0 0.2 0.4 0.6

Soc)

Figure 4.16 As Figure 4.14, for E/A=110 MeV.

0.8

1.0

84

4.5 Suppression of azimuthal correlations

In the previous section we have shown that impact parameter
selection techniques based on NC, E, and Z, provide rather similar event
selection and that, indeed, cuts on small, reduced impact parameters strongly
suppress contributions from projectile-like fragments as expected from
qualitative arguments. A recent analysis of data with solid angle coverage
restricted to forward angles (9lab S 30°) suggests that improved selectivity for
central collisions could be achieved by introducing a new observable, the
transverse momentum directivity (defined in equation (4.11), below) and by
simultaneous cuts on large charged-particle multiplicities and small
transverse-momentum directivities [Alar 92].

In the next two sections, we address two questions: (i) can previously
used impact parameters filters (NC, E, and Zy ) be used to select central
collisions, i.e. collisions with small angular momenta?, and (ii) do cuts on the
directivity significantly improve central event selection when used with
detectors providing 41: coverage? In order to quantify the selection of
collisions with small impact parameters, we explore the azimuthal
correlations between emitted light particles [Tsan 84a, Chit 86, Fiel 86, Tsan 90,
Ardo 90, Elma 91, Wang 91]. For truly central collisions, a reaction plane is
undefined and the azimuthal distribution of emitted particles must be
symmetric about the beam axis. If the azimuthal correlations between two
emitted particles reflect their single-particle emission patterns, the azimuthal
correlation function must become flat for central collisions. (Deviations from
strict azimuthal isotropy may arise from final state interactions such as the
sequential decay of primary reaction products produced in particle unbound
states [Poch 87] or, for small systems, from momentum conservation effects

[Chit 86, Lync 87].) For peripheral collisions on the other hand, transverse

85
flow effects or other ordered motion in the reaction plane [Tsan 84a, Chit 86,

Fiel 86, Tsan 90, Ardo 90, Elma 91, Wang 91, Tsan 84b, Tsan 86, Tsan 88, Wils
90, Tsan 91] can cause large anisotropies in the azimuthal correlations. We
apply this analysis to our previously measured [Phai 92a, Tsan 88, Bowm 91,
deSo 91, Kim 92, Bowm 92] data of 36Ar+197Au collisions at E/A=3S, 50, 80 and
110 MeV, and 129Xe+197Au collisions at E/A=50 MeV.

In this section we will define the azimuthal correlations and use them
to test the effectiveness of the different impact parameter filters. In section 4.6
we explore the usefulness of a directivity cut in selecting central collisions. A
summary and conclusions are given in section 4.7.

In the previous section it was demonstrated that impact parameter
filters based upon NC, E, and Z, were similarly effective in suppressing
particles emitted with near-projectile velocities when cuts on small reduced
impact parameters were applied. While the suppression of beam velocity
particles is qualitatively expected for collisions between a relatively small
projectile and a relatively large target nucleus, a suppression of beam velocity
fragments may not necessarily be the best indicator for event centrality.
Therefore, in the following sections we evaluate an alternative observable
which depends less on a participant-spectator picture and which is more
closely related to the angular momentum effects as evidenced by an ordered

motion of the emitted particles in the entrance channel reaction plane.

Azimuthal correlation functions
Ordered motion of the emitted particles can be detected by

measurements of azimuthal correlation functions [Tsan 84a, Chit 86, Fiel 86,

Tsan 90, Wang 91] defined by the ratio

86

Y(A¢)
Y'(A¢) 9,;

Here, Y(A¢) is the coincidence yield of two (identical) particles emitted with

= c[1+ R(A¢)]|9j. (4.9)

relative azimuthal angle A4) at a polar laboratory angle 6 and in collisions
selected by a specified cut on reduced impact parameter 5; Y ’(A¢) is the
background yield constructed by mixing particle yields from different
coincidence events, but selected by identical cuts on the reduced impact
parameter; C is a normalization constant such that the average value of the
correlation I+R(A¢) is one. All azimuthal correlation functions presented in
this paper were constructed from particles detected at 9 = 31° - 50°.

Azimuthal correlation functions may provide an additional diagnostic
tool with regard to the selection of central collisions which is complementary
to the suppression of beam velocity particles investigated in the previous
section. For example, if projectile and target nuclei fuse at finite impact
parameters, the emission of beam velocity particles will be suppressed for all
impact parameters below a certain value. However, off-center fusion
reactions will induce a collective rotation of the residue. The angular velocity
of this collective rotation depends on impact parameter. Particle emission
from a rotating compound nucleus is focused in a plane perpendicular to the
angular momentum vector, i.e. the emission will be enhanced in the reaction
plane [Chit 86]. The degree of this enhancement increases with the angular
velocity of rotation, and it decreases as a function of temperature. The effect
becomes more pronounced for heavier emitted particles [Chit 86]. For the case
of equilibrium emission from a long-lived rotating system, the emission
becomes left-right symmetric and the azimuthal correlation functions exhibit
a characteristic V-shape [Chit 86]. In the limit of 5-9 0, the collective rotation

ceases and the azimuthal correlation function becomes flat, i.e. R(A¢) -> 0.

87
Nonvanishing azimuthal correlation functions have been observed in

a large number of intermediate-energy heavy ion collision experiments. In
many instances, slightly distorted ”V”-shapes were observed [Tsan 84a, Chit
86, Fiel 86, Tsan 90, Elma 91] which could be understood in terms of a
collective rotational motion in the reaction plane caused by the attractive
mean nuclear field [Tsan 84a, Tsan 86, Tsan 88, Wils 90]. A number of other
physical effects can influence the shape of the azimuthal correlation functions
and lead to deviations from symmetric V-shapes. An important example is
the directed transverse flow caused by the interplay of mean field deflection
and pressure due to nuclear compression [Tsan 89a, Moli 85, Ogil 90, Wils 90,
Sull 90]. Additional distortions may arise from phase space constraints
imposed on finite systems by momentum conservation [Lync 82, Ogil 89b] or
final state interactions [Elma 91].

Figure 4.17 shows azimuthal correlation functions of protons,
deuterons, tritons and He nuclei (3He and 4He combined) detected in
peripheral (13> 0.75) 35Ar + 197Au collisions at E / A=50 MeV. (For brevity of
notation, all emitted He nuclei are denoted by the symbol a in the figures;
contributions from 3He are smaller than those from 4He by a factor of about
5). A uniform software energy threshold of E/A=12 MeV was applied. In
addition, the energies of protons, deuterons and tritons were required to be
smaller than 75, 100 and 119 MeV respectively (the punch-through energies
for the 2 cm thick CsI crystals). For He nuclei, no upper energy threshold was
imposed since punch-through He nuclei could still be cleanly identified since
no distinction is being made between 3He and 4He. Consistent with previous
observations for slightly different systems [Tsan 84a, Chit 86,]ie1 86, Tsan 90,
Ardo 90, Elma 91], the azimuthal correlation functions exhibit (slightly

distorted) V-shaped patterns with a clear minimum at A¢ a 90°, reflecting the

1 +12(A¢xx)

88

°°Ar + 197Au, E/A=50 MeV

 

2.0 l I I I I I I I I I I r I T7 I I r f 5
” ‘ ‘F
T 8 x: “E
” 0 p . i ‘ 55
_ 0 d 91.b=31°-50° -
1.5 —- o t Nc=2-9 i —
_ o a 8 _

1+R(A¢XX)

 

 

 

’ i Q o
o
_ § § _
0.5_ 0 o ‘—
l—- o —1
L l l i I l l L I l l l l l l l l l L
0 50 100 150

A¢xx

Figure 4.17 Azimuthal correlation functions constructed from particle pairs of
protons, deuterons, tritons and He nuclei emitted in peripheral 36Ar+197Au
collisions (5>0.75, NC =2—9) at E/A=50 MeV. The correlation functions were
constructed for particles emitted at polar angles of 91ab=31°'50° using an
energy threshold Eth/A=12 MeV.

para"
1.

Com

89

known preferential emission of nonequilibrium particles in the entrance
channel reaction plane [Tsan 84a, Chit 86, Fiel 86, Tsan 90, Ardo 90, Elma 91,
Tsan 84b, Tsan 86, Tsan 88, Wils 90, Tsan 91]. Again consistent with previous
observations [Tsan 84a, Chit 86, Fiel 86, Tsan 90, Ardo 90, Elma 91, Tsan 84b,
Tsan 86, Tsan 88, Wils 90, Tsan 91], the azimuthal anisotropies become
stronger with increasing mass of the particle pair. Since the effect is
particularly pronounced for He nuclei and since He nuclei are emitted in
great abundance, we utilize the He azimuthal correlation function as a
diagnostic tool for assessing whether cuts on small reduced impact

parameters do, indeed, select central collisions.

Comparison of relative scales

Figures 4.18-22 show azimuthal correlation functions for He nuclei
emitted in 36Ar + 197Au collisions at E/ A = 35, 50, 80 and 110 MeV and for
129Xe +197Au collisions at E/ A = 50 MeV. A software threshold of E/A=8 MeV
was used in selecting the He nuclei. Different panels of the figures show
results for different cuts on the charged-particle multiplicity NC. For each
panel, the overlapping circles present a simple geometric picture of the
collision geometry deduced by means of equation (4.4). At large impact
parameters, i.e. low values of NC, the correlation functions show a strong
preference of emission at relative azimuthal angles of A4) = 0° and 180°,
characteristic of preferential emission in the reaction plane. At larger values
of NC, the azimuthal correlation functions become more isotropic. For a
given cut on reduced impact parameter, the azimuthal correlation functions
become increasingly damped as the beam energy is increased. The effect may
be related to the disappearance of flow predicted and observed [Moli 86, Ogil

90, Wils 90, Sull 90] in symmetric projectile-target collisions at comparable

I ~f“ I?(A¢aa)

2.0

0.0
O 50 100150 0 50 100150 0 50 100150 0 50100150

90

36Ar + 19725.11, E/A=35 MeV

MSU—QG-OIG

 

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A¢aa(degrees)

Figure 4.18 Azimuthal correlation functions for He nuclei emitted in
36Ar+197Au collisions at E/ A = 35 MeV. Panels from left to right show data
selected by cuts on reduced impact parameters 5: 0.8, 0.6, 0.4, and <02,
respectively; the actual cuts on charged-particle multiplicity NC are indicated
in the figure. The circles show the approximate geometric overlap between
target and projectile for the different cuts in NC.

2.0

91

°°Ar + 197Au, E/A=50 MeV

 

 

 

 

 

 

 

 

 

 

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Figure 4.19 As Figure 4.18, for E/ A = 50 MeV.

92

36Ar + 197.111, E/A=80 MeV

 

 

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Figure 4.20 As Figure 4.18, for E/ A = 80 MeV.

93

°°Ar + 1971111, E/A;110 MeV
1180-93-016

 

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Figure 4.21 As Figure 4.18, for E/ A = 110 MeV.

 

94

12°Xe + 1""1111, E/A=50 MeV
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0.5
Nc=12 -‘ Nc=22 "' Nc=3o “'- Nc235
0.0 1111IIIIIIIIIIIIIII.LIIIIIIIIIIJLLILLll-bLlLL llllIIIIIIIIII-V-IIIIIIIIIIIIIIIIIII
0 50 100150 0 50 100150 0 50 100150 0 50100150
A¢aa(degrees)

Figure 4.22 Azimuthal correlation functions for He nuclei emitted in
129Xe+197Au collisions at E/ A = 50 MeV. Panels from left to right show data
selected by cuts on reduced impact parameters 13: 0.8, 0.6, 0.4, and <02,
respectively; the actual cuts on charged-particle multiplicity NC are indicated
in the figure. The circles show the approximate geometric overlap between
target and projectile for the different cuts in NC.

95
energies, possibly reflecting an increased balancing of attractive and repulsive

forces from the mean nuclear field and from pressure, respectively.

In order to allow a more compact presentation of the main features of
the observed azimuthal correlation functions, we have fitted them by
functions of the form

1+ 1.1 cos(A¢) + 2.2 cos(2A¢) , (4.10)
where 1.1 and 12 are treated as free parameters. Large values of 22 may be
associated with collective motion resembling a rotation [Lace 93]. Positive
(negative) values of M indicate preferential emission of the particle pair to
the same side (opposite sides) of the beam. Positive values of M can come
from large final state interactions (e.g. the decay 8Be —-) 20:) or, alternatively,
from directed sideward flow. Negative values of M (preferred emission on
opposite sides of the beam) may reflect phase space constraints for small
systems due to momentum conservation [Lync 82, Ogil 89b]. For isotropic
distributions l1=l2=0.

Figure 4.23 presents the values of the parameters 11 and 12, extracted as
a function of reduced impact parameter 5(NC) for 36Ar + 197Au collisions at
the different bombarding energies. At all bombarding energies, 12 decreases as
a function of decreasing reduced impact parameter. Indeed, for E/ A 2 50 MeV,
12 -) 0 as 5(Nc) —) 0 with rather good accuracy, indicating that small reduced
impact parameters do, indeed, select near-central collisions for which R(A¢) z
0 by necessity. At E/A=35 MeV, the selection of central collisions appears to be
of slightly reduced quality since 12 2 0.1 even for very small values of 5(NC).
To some extent, reduced selectivity for very central collisions at the lower
energy may result from a loss of statistical resolution as ANC [NC becomes
larger at lower incident energy. It is more likely, however, that this loss of

resolution is related to the breakdown of the participant-spectator picture

96

 

  
 

 

 

 

 
 

 

 

 

1150—93-01:;

:I I II I I I I III I I I7 I I I l I I I I:

0.6 — 36Ar+197Au 0‘: Q":

C E/A(MeV) = gt: §§§ j

“ E] 35 Q Q -
0.4:... 50 D .I Qi

02 Z 0 30 D D .' 099 ~
’< P I 110 D ' go i Z
I D ° 000 _

02 :— C] D . j i _

_ 0 CI D . . . 00 I.- -4

_ . I ‘

00 _. iIO 909099?" __‘
0.2:HHHHHHiHHiHH:

-’- 3

0.1 — ..

; o D D D I: D I: 99 g I

r<H Do—éé I’éi. -:
I _ .0590'95bohp :
—0.1 E— ——:
_O.2 -I L I 1 I I I I I I I I I I I I I I LI I I I I—
0 O 0.2 0.4 0.6 0.8 1.0

80%)

Figure 4.23 Reduced-impact-parameter dependence of the coefficients M and
12 used to fit the measured azimuthal correlation functions for emitted He
nuclei emitted in 35Ar+197Au collisions at E/ A = 35, 50, 80 and 110 MeV.

97
which underlies impact parameter filters measuring the ”violence” of the

collision.

As expected from the qualitative trends apparent in Figures 4.18-22, the
values of 12 at fixed 5(NC) are largest for the lowest bombarding energy where
rotational deflection of the emitted particles by the mean nuclear fields is
most pronounced. At E/A=50, 80 and 110 MeV, M is nearly zero for reduced
impact parameters 5(NC) < 0.4. For larger reduced impact parameters, as well
as for E/A=35 MeV, M assumes small but positive values. Positive values of
M may indicate a weak sideward directed flow, but they can also arise from
the sequential decay of particle unstable nuclei, in this case 8Be.

In order to discern between these two possibilities, we show in Figures
4.24 and 4.25 the parameters M and 12 which characterize the azimuthal
correlations of protons, deuterons, tritons and He nuclei emitted in 35Ar +
197Au and 129Xe + 197Au collisions, respectively, at E/ A = 50 MeV. The clear
monotonic dependence of the extracted values of 12 on the mass of the
detected particle pair (top panels) illustrates the well-known fact [Chit 86] that
collective effects are most readily discerned in the emission patterns of heavy
particles. In contrast, the parameters M do not exhibit a monotonic mass
dependence. At large impact parameters, the extracted values of M are
consistent with zero for protons, they assume small negative values for
deuterons and tritons and small positive values for He nuclei.

Most likely, the small negative and positive values of M reflect
distortions due to final state interactions (possibly modified by additional
momentum conservation effects [Lync 82]). Indeed, the changes in sign of M
follow the qualitative differences of the final state interactions which
determine the shapes of small angle correlation functions of these particles,

(see e.g. ref. [Poch 87]). For pairs of deuterons and tritons, correlations at small

98

 

 

 

 

 

 

MSU-93-019

:I I If I I I I I I I I I I I I F I I I I I:

0.6 "_— 3aAr-i-IWAu _:
— E/A = 50 MeV fl -

0 4t 0 p .3; :

- _. d . —_‘
(<N : O t I.- Q§§§§ :
‘ ' °‘ ' 00° ii —‘-

0.2 r". I I OO §§ -
: . . g 3 0° .IQQEDDDD :

I O D ..

00 fig% la 8 g '3 I IBIBDDD l _q
LnI I I I I I I I I I I I I I I I I I I I-J

0.2 '- Fl I I I I I I I I I I I I I I I I I I I I I I:
C _

0.1 E— i —:
’< 0'0 $98; i i #9 .EEDDDQ¥QD —_
:§§I §§ ’8.§S§g§§ :

r- . —
-O.1 '_—“ —:
_O.2 -L I I I I I 1 I I I I I I I I LI I 1 I I I I I-

0.0 0.2 0.3: 0.6 0.8 1.0
b(Nc)

Figure 4.24 Reduced-impact-parameter dependence of the coefficients M and
12 used to fit the measured azimuthal correlation functions of protons,
deuterons, tritons or He-nuclei emitted in 36Ar+197Au collisions at E/ A = 50
MeV.

99

 

 

 

 

 

 

MSU-93—020
:I I I rI I I I I I I I I I I I I I I I I I :
0.6 __ 129Xe+197Au if __..
I E/A = 50 MeV i _
: U P .i _
0.4 f0 d . §§% ‘j
N - O t OQ :
f< : I 01 I QQ é; -
0.2 — I O I '—_
: g I a O 9.: §DQ _
_ 0.0 DOD :
0.0%“35590530000 _-
'-I I I I I I I I I I I I I 1 I I I I I I I I I I“
0.2: I I I I I I I I I I I I I I I I I I I I :
0.1 _— 1
K i -—_
-O.1 :— —:
_O.2 -I I I I I I I I I I 1 I I I I I I I 1 I I I I I
0.0 0.2 0.4 0.6 0.8 1.0
b(Nc)

Figure 4.25 Reduced-impact-parameter dependence of the coefficients M and
12 used to fit the measured azimuthal correlation functions of protons,
deuterons, tritons or He-nuclei emitted in 129Xe+197Au collisions at E/ A = 50
MeV.

100
relative momenta are suppressed [Poch 87]. For these particles, the final state

interaction is dominated by the repulsive Coulomb interaction, since there
are no contributions from low-lying resonances. For a-particles, on the other
hand, correlations at small relative momenta are strongly enhanced due to
large contributions from the decay of 8Be [Poch 87]. The case of two-proton
correlations lies in between: here the correlation function exhibits a
minimum at very small relative momenta followed by a broad maximum at
relative momentum 20 MeV/c [Poch 87]. For the present data, a quantitative
interpretation of the parameter 11 solely in terms of collective flow effects
appears inappropriate, and final state interactions or other many-body
correlations may have to be considered. Since these effects are of little interest
in the context of the present work, we do not pursue this issue further.
Figures 4.26 and 4.27 show that impact parameter filters based upon NC,
E, and Z’ select, as expected (Section 4.3), classes of events characterized by
very similar azimuthal distributions. The figures show the reduced-impact-
parameter dependence of the parameters 11 and 12 which characterize the
azimuthal correlations functions of He nuclei emitted in 36Ar +197Au and
129Xe +197Au reactions at E/ A = 50 MeV. The different symbols in the figures
indicate the use of different impact parameter filters, 5(Nc) (solid circles),
1305,) (open circles), and 5(2),) (solid diamonds). As determined from the
shape of azimuthal correlation functions, the three impact parameter scales
provide equivalent event selection. Only for the most peripheral reactions,
some slight differences exist. Small discrepancies between different impact
parameter filters for peripheral collisions are not surprising since the
determination of large impact parameters must be associated with relatively

large statistical and systematic uncertainties.

101

36Ar+197Au, E/A=50 MeV

 

 

0.8: T I I I I I I I I I I I I I I I I I I I I I I I i;

” ~ I

0.6 :— X= g?

E ' Nc i§ j§
m 0.4 1— 0 Et 438%0 —3
’< : ’ Zr f I
0.2 E— 50 -—~:
W60“? :
OB‘IHIIIIIHIIIIIIHIItItHT
' 5 I | | I | 3
0.2 E— —;
0.1 :— éégfi —Z
r< 0.0 wOOOQQQQW —;
—0.1 f.— —;
—0.2 E- —_—:_
_O 3 I I I I I L I l I I I I I I I I LI I I l I I I I ;

 

 

 

0.0 0.2 0.4 0.6 0.8 1.0
Boo

Figure 4.26 Reduced-impact-parameter dependence of the coefficients l1 and
12 used to fit the measured azimuthal correlation functions of He nuclei
emitted in 36Ar+197Au collisions at E/ A = 50 MeV. Results from different
impact parameters filters are shown by the different symbols indicated in the
figure.

102

129Xe+197Au, E/A=5O MeV

 

 

0.8 L__rI I I I I I II II I I I I I I I I I I I II I15
I :c:
0.63- X‘ g? €13
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r< I 9 Z ..
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0-2 :— 9 —:
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0.1

IIIIII

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O
%
n—W
n—u—u
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IIIIIIIIIIIIII

IIIII

IIII

 

 

-IIIL

IIIIIIlLIIIIIIIIIIILILIII
0.0 0.2 0.4 0.6 0.8 1.0

Boo

 

Figure 4.27 Reduced—impact-parameter dependence of the coefficients 11 and
12 used to fit the measured azimuthal correlation functions of He nuclei
emitted in 129Xe+197Au collisions at E/ A = 50 MeV. Results from different
impact parameters filters are shown by the different symbols indicated in the
figure.

103

4.6 Directivity

For intermediate energy nucleus-nucleus collisions, the highest degree
of equilibration is expected to occur in central collisions. Investigations of the
thermodynamic properties of nuclear matter should therefore be performed
with reaction filters optimized to provide high selectivity of very small
impact parameters. Recently an improved method for the selection of central
collisions was suggested [Alar 92] which employed simultaneous cuts on large
charged-particle multiplicities and small transverse-momentum directivities.
The experimental evidence for improved selectivity of small impact
parameters was based upon data at significantly higher incident energy and
with experimental apparatus of restricted solid angle coverage (Blah < 30°). In
this section we explore whether similar improvements in small-impact-
parameter selectivity can be obtained for the present data taken at lower beam
energies and with more complete solid angle coverage.

The transverse-momentum directivity D is defined as

 

 

> . (4.11)

 

, y.2y..
In equation (4.11), the sum includes all identified particles with rapidities
larger than the center-of—mass rapidity, and pii denotes the transverse
momentum of detected particle i. The momenta are calculated assuming
A=ZZ for 332318.

Following the procedure outlined in Section 4.3, we explored the
conditional distribution of impact parameters 505,) selected by the cuts

5(NC) = 0.05 —O.1 and D s 0.2. The results are shown in Figure 4.28. The

104

36Ar+197Au, E/A=110 Me

5 . MSU—93—023
I I I I

 

— — — - b(Nc) = 0.05—0.1
/ -‘-‘-' D§0.2
Simultaneous cut

 

A

dP(B)/db

a
. .-.-C-O‘O-o_0-
‘
.

 

 

 

Figure 4.28 Conditional impact parameter distributions, dP[5(E,)]/ d5(E,), for

36Ar+197Au collisions at E/ A = 110 MeV selected by cuts on 5(NC) = 0.05 - 0.1
(dashed curve), directivity D S 0.2 (dotted-dashed curve), and for the
Simultaneous cuts 5(NC) = 0.05 - 0.1 and D S 0.2 (solid curve).

105
dashed and dotted-dashed lines show the results for cuts on 13(NC) and D
only, and the solid line shows the results for a simultaneous cut on 5(NC)
and D. A single cut on D alone provides little selectivity of central collisions,
and no improvement in the 5(E,)-distribution is observed if an additional cut
on D<O.2 is imposed beyond the cut 5(NC) = 0.05 — 0.1. This insensitivity to
additional cuts on D should be compared to that observed for other impact
parameter filters, based for example on the mid-rapidity charge Z]: narrower

5(E,) distributions were obtained when double cuts on NC and Z, were

employed (Figures 4.6-8). We conclude that cuts on directivity provide little
additional selectively, at least for the present set of data taken at lower energy
and with full tin-coverage.

Cuts on small values of the transverse-momentum directivity
suppress collisions which exhibit significant transverse flow. Hence, such cuts
are ineffective in systems for which transverse flow is small or negligible.
Indeed, for the reactions investigated here, small values of M were extracted
from the azimuthal distributions, and these small values could not be
associated with transverse flow effects. The dashed curve in Figure 4.29 shows
the distribution in transverse momentum directivity for 36Ar+197Au
collisions at E/A=110 MeV selected by a cut on small reduced impact
parameters, 13(NC) S 0.2. For comparison, the solid curve shows the
distribution for the same set of events after randomization of the azimuthal
emission angles, i.e. after an artificial elimination of any existing flow effects.
The true experimental distribution and the randomized distribution are very
similar. In particular, the experimental distribution does not exhibit an
excessive tail toward larger values of D which could be eliminated by cuts on
small values of D. For the present reactions, additional cuts on transverse

directivity do not enhance the selectivity of central collisions beyond that

106

36AT+197AU» E/A:1 10 MeV MSU-93—024

 

4 '- I I I I I I I I I f IiI I I I f I I I I I I I _‘
'- ’- \ A -4
_ ,’ \\ —-- b(NC)§O.2 «
- l . . . ~
\ —— Isotroplc Slmulatlon

 

Yield
N

 

 

Figure 4.29 Distribution of transverse momentum directivity D for 35Ar +
197Au collisions at E/ A = 110 MeV selected by cuts on 5(NC) <O.2 (dashed
curve). The solid curve shows the distribution of the transverse momentum
directivity obtained after randomizing the azimuthal distribution of the
emitted particles according to an isotropic distribution (solid curve).

107
achieved by tight cuts on small reduced impact parameters. The success of

applying cuts on the transverse directivity at higher energies [Alar 92] may be
due to the presence of larger transverse flow effects at these energies, and
partly due to a relatively poor impact parameter selection by multiplicity

filters which only cover part of the full solid angle [Tsan 89a, Tsan 89b].

4.7. Summary
In this chapter, we explored the selection of impact parameters via

impact parameter filters based upon the detected charged-particle multiplicity

NC, the transverse energy E,, the intermediate rapidity charge Z), and the
identified hydrogen multiplicity Nl (which is the complement of Zhu,, the
summed charge of clusters with 222 [Hube 91]). For the reactions studied in
this work, all of these quantities were shown to be correlated with each other
and all display some sensitivity to the impact parameter. By applying a simple
geometric prescription for the construction of a reduced impact parameter
scale, quantitative comparisons of the various impact parameter scales were
performed.

Average impact parameters deduced from these quantities were found
to be mutually consistent, with small deviations occurring at the boundaries
of the reduced impact parameter scale, [3:0 and 13:1. In order to assess the
relative resolutions of the various impact parameter filters, we explored
distributions of impact parameters 5(Y) for events selected by narrow cuts on

alternative impact parameter scales 13(X), where X, Y = NC, E,, 2,, and N1.

As an alternative measure, we compared the suppression of fast fragments

emitted at forward angles by corresponding cuts on 13(NC), 13(E,) and 5(2)).

Based upon these criteria, impact parameter selections derived from

the total charged-particle multiplicity NC, the transverse energy E, and the

108
intermediate rapidity charge Z, were found to be very similar, with some

differences occurring at large impact parameters where fluctuations appear to
become a limiting factor. Somewhat worse resolution was obtained by
employing the hydrogen multiplicity N1 (the complement of 2M). This
effect is related to the fact that Nl S NC and, hence, ANl / Nl 2 ANC INC. For a
given impact parameter, the larger fractional fluctuations of ANl INl therefore
lead to larger fluctuations A5(Nl)/ 5(Nl)in the deduced impact parameter.

Slightly improved selection of central collisions can be obtained by the
use of multi-dimensional gates placed on different observables. In this regard,
the future use of instruments capable of detecting neutral and charged-
particle observables in 41: geometry appears particularly promising.

We also explored the effect of impact parameter filters (based on NC, E ,

I

and 2,) on the azimuthal correlation function between emitted light

particles. Azimuthal correlation functions are sensitive to the presence of
collective velocity components in the reaction plane. In many reaction
scenarios, the ordered motion in the reaction plane depends on the angular
momentum of the emitting system and, hence, upon impact parameter. This
impact parameter dependence is complementary to quantities which provide
a measure of the geometric overlap of projectile and target nuclei, for
example, the relative intensity of emission from a projectile-like source
explored in Section 4.4.

As was done in Section 4.2, we employed a simple geometric
prescription for the construction of reduced impact parameter scales based on
the observables NC, E, and Z’ to provide a quantitative basis for the
comparison of these different impact parameter filters. For the reactions
studied in this chapter, these reduced impact parameter scales were found to

select classes of events with very similar azimuthal correlation functions. At

109

large reduced impact parameters, the azimuthal correlation functions exhibit
strong anisotropies indicating preferred emission in the reaction plane. At
small reduced impact parameters, the correlations are nearly flat indicating
isotropic emission. This evolution of the azimuthal correlation functions as a
function of reduced impact parameter is consistent with the expected
dependence on the true impact parameter of the reaction. For the reactions
investigated in this chapter, cuts on small reduced impact parameters were
found to be effective in selecting central or near-central collisions. No
additional selectivity for central collisions was found by imposing additional

cuts on transverse momentum directivity.

Chapter 5 General Reaction Characteristics
Selected by Impact Parameter

5.1 Multiplicity distributions

Intermediate-energy heavy-ion reactions exhibit broad multiplicity
distributions shown in Figure 5.1. The upper panel shows that with
increasing bombarding energy, the charged particle multiplicity (NC)
distribution extends to larger NC values, reaching as many as 40 charged
particles at the highest bombarding energy. A simple model (see Chapter 4)
can be used to relate the charged particle multiplicity to a ”reduced” impact
parameter I; that ranges from zero (most central collisions) to one (most
peripheral collisions).

The lower panel shows the inclusive measurement of the number of
intermediate mass fragments (NW) for the four different bombarding
energies. At all bombarding energies the inclusive measurement is peaked at
zero IMFs with the tail of the distribution pushing out to larger NM. for
increasing bombarding energy. At the highest bombarding energy the tail
extends to nearly 10 IMFs.

A qualitative perspective of the evolution of the reaction with
increasing bombarding energy is provided by Figure 5.2. In this figure, the
correlation between the total detected charge, 2”, of all particles (ZSZS) and

110

111

 

   

 

 

 

 

 

 

NC
0 o 10 20 30 4o
10 I I I I I I I I I I I I I I I I I I I I I I:
1 3°Ar+197Au :
10’ = "a
E -------- \°"‘ 5
A - ‘—_'__..—.—\ .—
0 — . . _
\Z/ 10‘3 ?— E/A(MeV)= ‘. —§
0.. E - -35 ‘.
3 — ..... 50 \\ _
10 E_—__80 .\ '_§
E ——110 '\ E
_ \ _
_I I III I I I L I I LIr LI'II I I IIII I T I
E \V: E/A(MeV)= 3
10"1 :—- \’K + 35 I
g - u 50
A - \
g : 3 O 80
o—c -—2 __ \‘ .110
z 10 "E
E: E “ °
10-3 -=-—' o
: fl
10-4'1.,..I. L.
0 5 10
NIMF

Figure 5.1 Upper panel - NC distributions. Lower panel - N,Mr distributions.
Both from the reaction 36Ar+197Au at E/A=35, 50, 80 and 110 MeV.

112

 

IOO

80 _ 36Al’ + l9TAU

60-

40?

  

  

 

  

  

 

 

 

 

20-
: E/A=35MeV E/A=50MeV
a I I l I I L n 1
N
80-
60-
40-
20v-
E/A=80MeV E/A=IIOMeV
o l l I l l l l I
I0 20 3O 40 IO 20 30 40
NC

Figure 5.2 Measured relation between charged particle multiplicity, NC, and
total charge of identified particles (ZS25) for the reaction 36Ar+197Au at the
indicated energies. Different colors represent contours that change by factors
of 3.

 

113
the total charged particle multiplicity, NC, is presented. At each bombarding

energy, the total detected charge increases with increasing multiplicity as the
collisions evolve from peripheral to central. At the lowest energy, E/A=35
MeV, the charged particle multiplicities are relatively small and only a small
fraction of the nuclear system is observed as light particles and intermediate
mass fragments. This observation is consistent with the survival of heavy
reaction residues which are not identified in the present experiment and
therefore not included in 2”. As the bombarding energy per nucleon is
increased to 50, 80 and finally 110 MeV, one observes a correlated increase in
both the total charged particle multiplicity and the total detected charge. The
multiplicity associated with central collisions increases from NC =11 at E/A=35
MeV to NC =32 at E/A=110 MeV. The average total detected charge for central
collisions increases from 22% of the total charge of the system at E/A=35 MeV
to 57% at E/A=110 MeV, where a number of events were observed for which
more than 80% of the total charge of the system was detected. These latter
events correspond to a nearly complete disintegration of the system into light
particles and intermediate mass fragments.

Figure 5.3 shows the contributions of charged particles bound in
clusters (A>1) and bound in intermediate mass fragments to the detected total
charge, 2”. For this measurement all hydrogen nuclei punching through the
C51 crystals were treated as mass number A=1. The percentage of the detected
charge Zm observed in clusters (open symbols) and in intermediate mass
fragments (solid symbols) is plotted as a function of 5(NC) for the three
bombarding energies E/A=50, 80 and 110 MeV. At all bombarding energies
nearly 30% of the detected charge appears in the form of intermediate mass
fragments for central collisions. The charge bound in clusters accounts for

about 80% of the detected charge, independent of beam energy.

114

36 197
Ar+ Au
100 I I I I I l I I I I I I I I I I I I I I I

 

so M83886? €510<>CI<>OGO<> 00001:] [100000 _—

0
000 0000 D D
000

— 0 Do _
/\ ‘ <><><> U -
E 60 — IMF A>1 E/A(MeV) 00 o _—
5’, ' I D 50 00 ° ‘1
N : . o 80 O o :
qES - 0 O 110 0 o;
N 40 — O —
v - O-
y'rfrfin-u {,3 I n ._

  

20!.

 

 

 

B<NC>

Figure 5. 3 Percentage of detected chargeZ M appearing in clusters (A>1) and
intermediate mass fragments (IMF) as a function of reduced impact parameter
for the reaction 36Ar+197Au at E/A=50, 80 and 110 MeV.

115
To better characterize the various classes of multifragment events, we

have extracted the probability distributions P(N,MF) of detecting NM,
intermediate mass fragments in a single collision for different gates on the
charged particle multiplicity NC. The four panels in Figure 5.4 depict these
distributions. At each energy, the IMF multiplicity distributions become wider
and shift toward higher average multiplicities as the charged-particle
multiplicity increases. Peripheral reactions, selected by NC <7, exhibit narrow
IMF multiplicity distributions peaked at NW =0. For these collisions, IMF
emission is an unlikely process. On the other hand, for central collisions
selected by large values of NC, IMF emission is a common process for which
the average IMF multiplicity increases from (N,MF)=1 at E/A=35 MeV to
(N,MF)=4 at E/A=110 MeV. At the highest incident energy, events are observed
in which as many as 10 intermediate mass fragments are detected in the exit
channeL

To allow quantitative comparisons of IMF multiplicity distributions,
we have determined their first and second moments, (NW) and of”. The
dependence of these moments on the total charged particle multiplicity NC is
shown in Figure 5.5. Values extracted at different incident energies are shown
by different symbols as indicated by the key in the figure. At all energies,
(MW) and 0er exhibit an initial, approximately linear increase as a function
of NC. The slope of this increase is rather similar for the four energies
investigated. For large values of NC, corresponding to the extreme tails of the
respective NC distributions, both (NW) and 0er increase only marginally as a
function of NC. This saturation arises from working in the high multiplicity
tail of the NC distribution where the excitation energy is roughly constant. For
lower NC , the average violence of the collision is smaller when changing the

gate on NC. But making a higher NC gate does not increase the average

116

 

 

 

 

100 I I I I
E/A = 50MeV
_1 Ne
10 C] 2—9
I 10-16
0 17-20
10"2 O 221
10-3
A
(a.
E§
Z I I I : I
\_/ l
0.. E/A = 110MeV
10‘1 "°
I3 2-9
I 10-23
0 24-31
10'”2 O 232
10-3
10—4 . .

 

 

 

Figure 5.4 Measured IMF multiplicity distributions for the indicated gates on

charged particle multiplicity NC. Panels are labeled by incident energy.

8931-88-08"

117

 

III
.0
O

36
Ar+197Au

4~fiTI I I I I I I I l l I I I rTI III I-

. .. ‘

I .0. Z

3“ 00 —

- 0'0 -

/\ ~ 0. -
Ln _ o -
E 2:. O . -
z " IOI'I ~
v : I'O ° ~
” .580. :

1_ CF? 0 —_

 

O
I
-
.1—
——P
~0-
~1—
—r—
—1-—
_—
Ill-
~—
qr—
~0-
_—
_-
.1}-
_—
-II-
_—
*I‘
.—

 

 

 

3}- —‘
_ o ' "
m h '0';0 . :
e2- 00"” —'
._. I 00 E/A(MeV) j
b ” 1"..- 035 "'
1:. lb”. .50 _:
60.0.. 251320
Ofll....l....l....l.f

O 10 20 3O 40

Figure 5.5 First and second moments of IMF multiplicity distributions as a
function of charged particle multiplicity, NC. Different symbols represent
results for the indicated beam energies.

118
excitation energy of your event selection, thus giving a saturated value of

(NW)-
This same information can be plotted as a function of 5 instead of NC

and this is done in Figure 5.6. This allows an easier comparison of the
different reactions as a function of impact parameter. Most striking is the
saturation of (NW) at E/A=50 MeV. This could come from a breakdown in
the assumption of the participant spectator picture upon which 5 is based.
Another possibility is that NC is not the best measure of centrality at this
bombarding energy. As an alternative scale we plot the (MM) and of“, as a
function of 5(E,) (Figure 5.7). For this impact parameter scale the saturation
in (MW) and 0qu is much less pronounced. In addition, a central cut (5<0.3)
using the impact parameter scale 13(E,) selects events with larger numbers of
intermediate mass fragments than that from the corresponding cut using
5(NC). The (NW) measured for the two impact parameter scales (from NC
and 5,) are similar down to 52.03 for the highest bombarding energies (80
and 110 MeV/ nucleon). For the 50 MeV/ nucleon reaction the measurements
diverge for 5=0.4.

A saturation in (NW) as a function of 13(E,) is not observed because E,
is perhaps a better measure of the violence of the collision than NC. E, is also

“Ct a bounded quantity. At very high bombarding energies, NC is limited by
For very large NC an auto-correlation between NC and

the faCt that NC S Zsystem °
where by

(NIMF) is introduced. The obvious extreme example is NC=Z,y,,,,,,,

CorlStraint N W =0.

119

 

36 197
Ar+ Au

4 I r. I I l I I r I I I I I I T I I r I I I r -

.~ ..

2 ° 0 . I

3 _QDO o o T

. O ..

Ah. - o. -
E. 2 L O. __i
Z 2" ' l I .I.) I
V : e :
1 __.. 0c, _

_ Col _

 

O
_—
.—
nu—
HF
qr—
‘—
up—

.-
__.

_—
‘-
.‘F-
-n--
.0—
—I—
.m—
r-
l 1

C9 C9 GD OD

 

 

 

3_ _‘
me. 3:5. 0 . E/A(MeV) I
. O . -
tag 27 O O 0' :23 T
;_ "a. __'
1_ (’- _
I 0; :
O'....l...,l....l....QQILQ_]
0.0 0.2 0.4 0.6 0.8 1.0
b(Nc)

Figure 5.6 First and second moments of IMF multiplicity distributions as a
f“fiction of 13(NC). The two circles show the approximate overlap between
t"=ll'get and projectile for 5:0.2, 0.4, 0.6 and 0.8.

120

 

 

 

 

36 197
Ar+ Au
5 it I 1 I I I r I I I I I I I I r I T I I I I I :
4 '0. ‘:
I O .. :
- O O -
Ann. 3 _ Oo .0 —_
2 o 00. q
"' - I oo 3
z 2 :_ I I I I C’s __
V I l I a I
: "I :
1 :— ‘3'9' . —.‘.
I 0.0. I
O .- 1 1 1 1 l 1 I I Fl I I I I l 1 1 1.L 1m:
_ I I I I I I I I I r I I I I -
I © (9 CD 03 I
3:‘ i
l
N _Cbo~.. E/A (MeV) I
an. _ '0 -
I
: I I I I I E. I 110 :
1 7 “W. ‘i
.. 0:). _
- . -
O b 1 1 L l 1 1 1 1 l I 1 14 l 1 1 1 1¢ m

 

(10 (I2 04. (16 013 ‘L0
b(Et)

Figure 5.7 First and second moments of IMF multiplicity distributions as a
fuIlction of 5(5). The two circles show the approximate overlap between
tal‘get and projectile for 13:02, 0.4, 0.6 and 0.8.

121
5.2 Element distributions
Element distributions for three different impact parameter cuts are
shown in Figure 5.8 for the three bombarding energies. The solid lines are fits
for Z=3-20 assuming a functional form for the average yield of
N(Z) o. e‘” (5.1)
The different distributions are normalized such that N(Z) is the average
multiplicity of Z for a given cut on 5. The steeper distributions come from the
highest bombarding energy. This can be seen more clearly in Figure 5.9. The
fit parameter a is plotted as a function of 5 (constructed from NC) for the
three bombarding energies. With this impact parameter scale the flatness of
the element distributions (as measured by (1) seems to saturate at about
5(NC)=O.5. For 5(NC)<O.3, (1 begins to rise again. If we instead extract a from
element distributions constructed with cuts on 13(E,) (see Figure 5.10), the
saturation in a is less pronounced. There is a change in slope near 5(E,)z0.5,
but or always decreases for smaller 5(E, ).

The observed saturation and subsequent rise of a at small 5(NC) in
Figure 5.9 can be caused by autocorrelations. For a fixed excitation energy, the
NC distribution has a finite width. Gating in the extreme tails of the
distribution may not select more central collisions or higher excitation

energies. If we make cuts on large NC in the tail of the distribution (where the
eXcitation energy does not increase) we may force the IMF spectrum to
become more steep due to the selection of events with more light particles
el'Ilitted. We simulated such a scenario with a standard bond percolation
IIlodel (see Chapter 6) and the results are shown in Figures 5.11 and 5.12. In
this percolation simulation, each nucleon in the compound system is
represented by a lattice site. The nucleons are bonded to their six nearest

neighbors in a simple cubic lattice structure. The simulation consists of

 

122

 

 

 

 

    

 

 

 

101 r5 I v I l v I I I I Y T v I I l I I T 1
s 0 ~ 5
100 :- b(Nc)aIO.75 1
g 10—1 r 1
z 10“2 i i
10-3 I i
10‘4 E.: ‘
101 [8
100 E
5 10 I-
2 10-2 I
10‘3 f
10-4 I ..1 .1. 1 . 1
101 J; I l A I E;
100 r b(Nc)~O.25 .3
A —1 " :
El 10 E- ”AH-m'Au 1
2 10-2 I- E/A(MeV) 1
10-3 if 50 ‘
_ so “I
10‘4.°.1.1?.L..1....1....1‘
0 5 10 15 20

Figure 5.8 Elemental distributions at E/A=50 (circles), 80 (stars) and 110 MeV
(diamonds) for 13=o.75 (top), 0.5 (middle) and 0.25 (bottom). Solid lines are
EXponential fits using Equation (5.1).

123

 

 

 

 

)_ l I l I I l I l l I I I I r l I I I I I I!I _
' 36 97 § _
1

_ E/A (MeV) i Q —

— I 50 Q; Q I —

_ 80 E O _

O {500 '

I o 110 O ‘
0.4 —— .99 O ' —
... I O . _

_. .. O ..
a __ .. .. O . _
_ .'O can.” 00 . _
0.3 —— 00O 0 C)0 . _
_. I _

I . I
_ . . . ' I -
0.2 —— ——
_J l l l l l l l l l l l L l l l l L l l l l l _l
o 0 o 2 04,. o 6 o 8
b(No)

I:igure 5.9 a as a function of 5(NC) for E/A=50, 80 and 100 MeV.

124

 

 

 

 

_ I I I I I I I I I I I I I I I I I I I I I r I r_
_ 36 197 _
_ E/A (MeV) i 4
- I 50 Q Q -
i! 0 d
r o 80 i O
_ Q g _
o 110 o _
1.0

Figure 5.10 a as a function of 5(E,) for E/A=50, 80 and 100 MeV.

 

H

125

 

  
     
 

 

 

 

105 .— I I f‘I I I l I l I I I I I r _.
5 7 l I ' g
' -1
104 :— 1
A -
:5 103 __.— _E
>" I :
102 :— o N=67 —:
: o N=77 1
- s>
- :; Y(A)«A"‘ i 4
' 1.
101 l l l l I l l l l I l l l l I l l l l
o 5 1o 15 20

Figure 5.1] Mass yields for a percolation simulation at fixed excitation energy.
The open (solid) symbols correspond distributions constructed for
multiplicity N=67(77). The curves are power law fits.

126

 

 

 

 

 

”I I I I I I I I I I I I I I I q
t. d r1
2.4 — _
2.3 _ g i _
,< - D .
. o _
- o .
2.2 — o ._
L. 0 q
. O a
_ O a
I- Q .
I”I l 1 l l l l l l 4 I l 1 l l IJ
65 7O 75 80

Figure 5.12 Evolution of the fit parameter A as a function of multiplicity N for
a percolation simulation at fixed excitation energy.

127
assigning a fixed number of bonds to be broken (representing a fixed

excitation energy) and then randomly breaking the bonds between nucleons
in the compound system. Clusters with their bonds intact are identified by
mass in each event. Each event is also identified by the multiplicity N , the
total number of clusters emitted. In Figure 5.11 is plotted the mass
distributions for breakup of an A=233 system at a fixed excitation energy (i.e.,
fixed number of bonds broken). The two distributions plotted are selected for
multiplicities N of 67 (open circles) and 76 (solid circles). The solid lines are a
fit with a power law function Y(A) cc A"? The higher multiplicity cut gives a
steeper mass distribution even though the two distributions correspond to
events with the same excitation energy. Figure 5.12 shows the auto
correlation more clearly. The fit parameter A is plotted as a function of
multiplicity. Again we observe larger values of 2. (steeper mass distributions)
for increasing N. We therefore conclude that the rise of a for very small 13(NC)
likely an artifact produced by an autocorrelation of the particular impact
parameter filter. This autocorrelation is not apparent for the impact

parameter filter based upon the observable E,.

5.3 Angular distributions

Angular distributions as a function of bombarding energy and event
Cer'ttrality for elements with Z=1-6 are shown in Figures 5.13-15. The
distributions are normalized such that the integrated yield (over solid angle
aCceptance) is equal to the average multiplicity of a given element for the

giVen impact parameter cut. The open circles are the angular distributions for

a peripheral cut, 5>0.75, and the solid circles are for a central cut, l3<0.25. The
yield for lithium is not plotted for the last two rings. Poor separation of
1ithium and helium in these detectors prevented an accurate measurement of

the lithium yield at these backward angles (see discussion in Section 3.3). The

 

128
fragment yield is suppressed at 9=90° because of target shadowing at this

angle. The fragment yield is also suppressed at the most backward angle
(9=150°) because of the Pb-Sn foils used to suppress electron detection (see
Section 3.1). The slopes of the angular distributions are steeper for peripheral
collisions than for central collisions which indicates more equilibration and
smaller contributions from projectile-like sources for events with large
multiplicities (see Section 4.4). In addition, for central collisions the

distributions become less steep as one goes up in bombarding energy.

 

129

E/A=50 MeV

 

 

 

 

 

 

 

.- I I I I rfle I I I I I I I I I I I I I ITI I I I I I III -1
10’2 r "a
330. E
. o O .
10’3 g- 0 .' 0 . ‘2
: ° . O o 3
. o o . u
_4 I- o o O . I. ..
10 r O o O O . . '1:
E o 0 o o o 5
_5 _ A 00 O o q
10 5‘ . 33(Nc)<o.25 o 1
"‘ : o b(Nc)>O.75 :
'L. 10—6 5‘ ° ° ‘21
—7 I . L L I . . 1 . l 1 I . . l . I
\ ’ 10-2 _I I I rI IfTII I I I I I I II I I II I I I I I I I I r II I I I I f -1
c: 1° :— z=4 z=6 I
"O E. 5
10_4 g9... .- E
E 0 ' .0 5
_5 I- O . . 0 I -1
10 E— 0 . . . O 1
E O I I o . :1
: ° 0 . ° . I
10_6 E" O o 0 -§
5 ° ° 0 o o ' 3
10-7 .- 14LIIILIJLIAITII 1 i llljijbll 1I141?1?1?lIl1-‘
O 50 100 150 0 50 100 150 0 50 100 150

9 (degrees)

1‘I‘igure 5.13 Angular distribution of elements Z=1-6 from the reaction

36Ar+197Au at E/A=50 MeV. Solid symbols - central cut 13(NC)<0.25. Open

Symbols - peripheral cut 5(NC)>O.75.

dN/dQ OnsnI)

10-5

10-2

36 197
Alri-

130

Au, E/AzBO MeV

 

   

 

 

 

 

 

 

 

9 (degrees)

Figure 5.14 Same as Figure 5.13 for E/A=80 MeV.

I I III fI I I I I III I I I I I er1 I II III I I I I I I I II I '4
: O .I 8.. i
o . I .I
E— o . 0 ° . o _5
5 o . ° . o o 3
,_ o O O I . O. 4
5' ° ° 0 0° 0 ° “3
E I O 0 O O o . o E
I' o "
E‘ . b(N¢)<0.25 0° 1:
: ° b(Nc)>O.75 o 3
F ° 1
5 z=1 2:2 2:3 .3
stale. :1: fits Mtltt .llh.‘ rtlH :1: 'I':‘
h I I I I I I ' I I I ' I ' ~
f 2:4 2:5 2:6 E
5' ‘I
:I. .. . :
o o 0
EU I.. ... . o I.. 13
I . . ' I
.. o .
o ' o ° ’
o o o o 1
° 0 o o ' 2:
° ' 0 ° 3
o o . 0
° 0 ° o ‘3
° ° 0 o :
o .
l l I l l LLLHLATI 1 7 LII l l l 1 L11 ll Lié L1 L111 1?]?lelll l
O 50 100 150 0 50 100 150 0 50 100 150

10“2
10-3
1o-4
10-5
A
I ._
h 10 6
.U)
33, 10’7
c: 10-2
3 3
0-
2: 1
"o
10“4
10“5
1o—6
10-7

131

36Ar+197Au, E/A= 110 MeV

 

 

 

r fl I rI I I I I I I I I I I I I I I I I I I I I IT I I I rI' r7 I I I T I d
. ' ‘5
o. :
0 . 0 g... :
o
O o . I . 1
oo ' 0 ° 0 . '0. i
o oo o . o '. . .
° 0 0 ° 0 o 0 -—§
0 O O O O o I . i
A o I
o b(Nc)<O.25 0° “5'
° b(Nc)>O.75 o o o :
0

2:1 z=2 2:3 I
Z=4 2:5 2:6 I
‘5
. o 1
I... . I... . Co... 1:
o . o . 5
o . . o . . . a
O . O . . .1:
O o o o I 2
o o °° o °° ‘
o O o 0 1
O O o O O O O O i
l 11 l l l I l l l 1 IT]. 1 l l l l l l 1 l l I l l L 1 ¢ L l l I l l 191? l 9 I; l L‘

50 100 150 O 50 100 150 O 50 100 50

9 (degrees)

Figure 5.15 Same as Figure 5.13 for E/A=110 MeV.

Chapter 6 Model comparisons

Over the broad range of incident energies in this study of 36Ar+197Au,
the mean values and variances of the intermediate mass fragment
multiplicity distributions exhibit an approximate scaling with the total
charged-particle multiplicity (see Figure 5.5). The relationship between the
IMF and total charged-particle multiplicities can be qualitatively understood
by assuming that the charged-particle multiplicity is strongly correlated with
energy deposition and that the production of intermediate mass fragments
depends primarily upon this energy deposition. The nearly universal increase
of (NW) as a function of NC can thus be viewed as due to the selection of
interactions involving a progressively increasing amount of internal energy
deposition. Deviations from a universal relationship between (NW) and NC
can arise from different amounts of preequilibrium emission and/ or angular
momentum transfers to the equilibrated nuclear systems at the different
incident energies. In addition, different impact parameters can be expected to
involve decaying systems of different mass. The measured multiplicities of
light particles and intermediate mass fragments are compared in this chapter
with both a model involving statistical decay of an expanding compound

nucleus and with a model involving percolation theory.

132

133
6.1 Expanding emitting source model

We have compared the experimentally measured IMF multiplicity
distributions with the predictions of the expanding emitting source model of
Friedman [Frie 90]. This schematic model couples the phase-space features of
statistical decay with the dynamical features of expansion driven by thermal
pressure. It has the capacity for predicting multiplicity distributions for
different species of particles (IMFs in particular). The model has been
successfully employed [Frie 88] to interpret the low emission temperatures
deduced from the relative population of states [Chen 87], as well as trends in
IMF multiplicity with excitation energy [Troc 89] where the IMF multiplicity
was found and predicted to be less than one [Frie 90] (lower than predicted by
the instantaneous breakup model of [Bond 85b]). The model also predicted
[Frie 90] a sharp rise in multiplicity for excitation energies on the order of 8
MeV/ nucleon.

The expanding emitting source (EES) model characterizes an ensemble
of emitting sources by a time-varying average density p(t) to which the level
densities of the source are calculated via the Fermi-gas approximation. For
the purpose of calculating the mean collective expansion energy, the model
assumes that the density of the source is uniform. This assumption requires
that the collective radial velocity increase linearly with radius.

In the EES model, the dynamical response of the source is governed by
the interplay of thermal pressure and the nuclear binding forces, which tend
to return the density to its equilibrium value p0. The binding effects are
parameterized in terms of the finite-nucleus compressibility. For simplicity,
the binding energy per nucleon at densities different from p0 is assumed to

deviate from liquid-drop values by a quadratic function of density,

134

A A 18 p0 '

where the first term on the right represents the liquid drop binding energy
values (~-8 MeV) and K is the finite-nucleus compressibility coefficient. This
compressibility includes surface and Coulomb effects.

The calculations require an assumption of initial source mass, charge
and thermal excitation energy when the system is at normal density. For
simplicity we assumed the decaying source to be the full composite system
(A=233, 2:97), and we further assumed that the system expanded with no
initial expansion velocity from normal density. For different compressibilities
we calculated the correlation between the predicted total charged-particle and
IMF multiplicity distributions. The results are shown by the curves in Figure
6.1 (the data were described previously in Chapter 5, Figure 5.5).

For orientation, the upper panel of Figure 6.1 includes an approximate
scale of the relation between the excitation energy of the emitting system and
the mean charged-particle multiplicity calculated from the BBB model. In the
extreme tails of the NC distributions, the correlation between internal energy
and NC becomes dominated by fluctuations of the charged-particle
multiplicity. Hence, very large values of NC become ineffective in selecting
nuclei of increasing internal energy thus causing the observed saturation of
(MW) and of”, at large values of NC (see discussion in Section 5.1).

The calculations are sensitive to the nuclear compressibility at low
density. The solid, dashed and dashed-dotted curves in Figure 6.1 show
predictions for the relationship between the (Nm)and (NC) for finite-nucleus

compressibilities of K=144, 200 and 288 MeV, respectively [Blai 80]. IMF

 

 

  

 

 

 

 

6 .- I I I I I I I I I I I I I r I I I 7 l’ I :‘

: 5 1. 1.5 2 2.5 d g

5 :— l l l l . f j In

: E. (08V) ..-'// j ('0
.. / .,
A : 3“... :
In . ., .
a 3 :— 0‘? ’9" 1
6 2: .O' .............. 1
1 .— fl
m 1 14 I 1 1 1 1 l l 1 1 L l i L:
O l. I I I I I I I I I I I l r I I I lfi I‘
- 1
3 ‘_ _.
NEH " 'l
2 2 r- 1
6" . -<
1 _— T
O ‘ L

0

 

Figure 6.1 First and second moments of IMF multiplicity distributions as a
function of charged-particle multiplicity, NC. Different symbols represent
results for indicated beam energies. The solid, dashed, dotted-dashed and
dashed-dotted-dotted curves show results calculated for the statistical decay of
expanding compound nuclei of finite-nucleus compressibility K=144, 200, 288
and co, respectively. The dotted curves represent the calculations for K=200,
filtered by the detector response.

136
multiplicities for a non-expanding compound nucleus, corresponding to the

limit K—->oo, are shown by the dashed-dotted-dotted curves. To illustrate
instrumental distortions, the dotted curves show the calculations for K=200
MeV, filtered by the response of the experimental apparatus. At low
multiplicities, corresponding to low excitation in this model, the nucleus
does not expand and all calculations predict IMF multiplicities consistent
with the measured values. For multiplicities larger than NC =20, however,
expansion strongly influences the predicted number of clusters in the final
state. Large observed IMF multiplicities, comparable to the measured values,
are only predicted for an equation of state that is sufficiently soft to allow the
nucleus to expand in response to thermal pressure. Very stiff low-density
equations of state hinder the expansion of the system leading to a suppression
of the production of multifragment final states and a consequent
underprediction of the observed mean IMF multiplicities by nearly a factor of
two.

The model ignores angular momentum, fluctuations in source size
and excitation energy, and the lack of thermal equilibrium. While the
inclusion of such effects might affect the final observations, we nonetheless
find that the essential features of the observed data are included in the
predictions of the schematic model shown by the curves in Figure 6.1. The
sensitivity to the source size was explored by changing the initial mass and
charge by 20%. The qualitative conclusions remain valid despite this change.

The calculations with the schematic model suggest that multifragment
decays of highly-excited nuclear systems may exhibit considerable sensitivity
to the low-density nuclear equation of state. They indicate that the expansion
dynamics, which is governed by the compressibility, may be intimately

connected to the production of IMFs. A more quantitative exploration of

137
these properties, however, will require a more complete model which also

incorporates non-equilibrium effects.

6.2 Percolation

In the search for a signature of the liquid-gas phase transition in low
density nuclear matter, percolation models are attractive since they exhibit a
well-defined phase transition for infinite systems and since they allow
straightforward generalizations to finite systems and the incorporation of
important geometrical ingredients [Biro 86] for multifragmenting systems. In
rather general terms [Siem 83, Hirs 84], the fluctuations at the critical point of
the nuclear matter phase diagram are expected to lead to mass distributions
which follow a power law, o(A)ocA"‘, with a critical exponent of the order of
1:22-23. A number of theoretical investigations of phase transitions in finite
nuclear systems have been based on percolation models [Baue 85, Baue 86,
Baue 88, Camp 86, Camp 88, N g6 90, Biro 86, Jaqa 90].

Most percolation models are governed by a single bond-breaking or
site-vacancy parameter and cannot be expected to reproduce the two-
dimensional phase diagram of nuclear matter in the temperature vs. density
plane. Despite this limitation, they have been rather successful [Baue 85, Baue
86, Baue 88] in describing the observed [Hirs 84] power-law behavior of
measured fragment mass distributions and in developing techniques to
extract critical exponents from exclusive fragmentation data [Camp 88, N g6
90]. In this section, we provide a test of the bond percolation model of
references [Baue 85, Baue 86, Baue 88] and compare its predictions to fragment
yields measured for the reactions 129Xe+197Au at E/A=50 MeV and

36Ar+197Au at E/A=50, 80 and 110 MeV.

138
Calculations were performed with the bond percolation model of ref.

[Baue 85, Baue 86, Baue 88]. In this model, the nucleus is considered to be a

cubic lattice, the sites of which are randomly occupied by protons and
neutrons within a spherical volume of radius R ——= (:3; A)M a, where a is the
lattice parameter (a = p?” z 1.8 fm). Initially, all nucleons are connected in
one cluster. The bonds between the sites are randomly broken with a
probability p. Each nucleon is assigned a random momentum consistent with
the momentum distribution of a Fermi-gas of temperature T estimated [Baue
88] from the bond-breaking probability as T=11.7JE MeV (see Appendix B).
Emitted fragments are defined in terms of connected clusters. Initial fragment
energies are calculated from the total momenta of the clusters. Final kinetic
energies are calculated by incorporating the final state Coulomb repulsion
between the fragments. For this purpose a given fragment partition is
translated into a spatial distribution of clusters characterized by an average
freeze-out density p=0.2p0 , where p0 =0.17 fm'3 is the density of normal
nuclear matter. (Different choices of the freeze-out density lead to slightly
different shapes of the low-energy portion of the energy spectrum due to
changes in the Coulomb repulsion between the fragments. These Coulomb
barrier fluctuations are of minor importance in the present context.) The final
momenta of the emitted fragments are boosted by the velocity of the center-
of-mass of projectile and target. In order to allow meaningful comparisons
with our data, we have filtered the theoretical distributions with the response
of the experimental apparatus.

Elemental multiplicity distributions measured for the reactions
36Ar+197Au and 129Xe+197Au are shown (as points) in the top and bottom
panels of Figure 6.2. These distributions were obtained by integrating all

identified fragments over all detectors. In order to select central collisions,

139

MSU-92-019

 

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

”A: + "'Au. E/A =
o 50MeV, Mezzo

 

 

 

 

A 100 \ “I.” o aouev, Ncgze
g 1 \\ o 110MeV, chao
_. \ ‘
10 \ ‘ - ~.__
\ ~~~~~~~~ -
\
10—2 \ \ \
\ \ ...... e a
10_3 : % = I if i : : I : l l [_— h l l l 1"
I I I I I l I I I r
101 ‘9. mice + 1""111. E/A = 50MeV
\ kn, o Nc233
A 100 \ \OKI, .
v \ ‘°"\., . . .
z 10‘1 \ "\"r.°°
‘,~ ~.-. .
- -P=0.8 \\ 'O-g- -
_2 P=O.8 \ \ .
1° —P=0.7 \ g "
_ . -p=o.7 (unfiltered) \
10—3 I I I I L I I L I I I I I LL I I I 1 l g
0 5 10 15 20
Z

Figure 6.2 Elemental multiplicity distributions detected in 36Ar+197Au
collisions at E / A=50, 80, 110 MeV (top panel) and in 129Xe+197Au collisions at
E/A=50 MeV (bottom panel). The curves represent calculations with the bond
percolation model (described in the text) for the indicated bond-breaking
probabilities p. All calculations are filtered by the response-of the
experimental apparatus except for the dot-dashed curve.

140
these Z-distributions were selected by the total charged-particle multiplicity

cuts indicated in the figure. When one adopts a strictly geometric
interpretation of the measured charged-particle multiplicity [Cava 90, chapter
4], these cuts were chosen to represent the range of impact parameters of
550.3 . The elemental distributions observed for the various reactions exhibit
rather similar shapes. For the 36Ar+197Au system (top panel), the elemental
multiplicity distributions become slightly steeper with increasing bombarding
energy. The multiplicities for heavier IMFs are significantly larger for the
129Xe+197Au system (bottom panel) than for the 36Ar+197Au system (top
panel). Part of this difference in the observed yields of heavier fragments may
be due to an increased detection efficiency in 129Xe+197Au reactions resulting
from the larger center-of—mass velocity.

The curves in Figure 6.2 depict elemental multiplicities predicted by
the standard bond percolation model and filtered by the detection efficiency of
the experimental apparatus. Calculations are shown for representative bond-
breaking probabilities above and below the near-critical bond-breaking
parameter of p=0.7. (In the percolation model, the critical point marks a
second order phase transition: for p>0.7, the percolation cluster disappears and
the system breaks up completely.) Calculations performed for bond-breaking
probabilities much larger or smaller than the critical value predict Z-
distributions which are too steep. For the 36Ar+197Au reactions, the overall
magnitudes and shapes of the experimental Z-distributions are in reasonable
agreement with predictions of the percolation model when the bond-breaking
parameter is taken close to the critical value. However, for the 129Xe+197Au
reaction, the percolation model underpredicts the yield of heavier fragments

(Z=6-20) for any choice of bond-breaking parameter.

141
Representative angular distributions of emitted particles are shown in

Figure 6.3. The top and bottom panels show results for the 36Ar+197Au
reaction at E/A=110 MeV and for the 129Xe+197Au reaction at E/A=50 MeV,
respectively. The angular distributions are shown for three different ranges of
element numbers, Z=1-2 (circles), Z=3-S (squares), and Z=6—12 (diamonds). For
both systems, the angular distributions become more forward peaked with
increasing fragment charge. Angular distributions for the 129Xe+197Au system
are more forward peaked than those for the 36Ar+197Au system. These effects
are largely due to kinematics. Differences between the angular distributions of
the two reactions arise primarily from the larger velocity of the emitting
source of the 129Xe‘I-197Au system.

The curves in Figure 6.3 show results of percolation calculations. To
facilitate a better comparison of shapes between observed and predicted
angular distributions, the calculated angular distributions were normalized to
the experimental yields at 6=45°. To display the effects of the detector
response, filtered and unfiltered calculations are shown by solid and dashed
curves, respectively. In view of the fact that the percolation model does not
include dynamical preequilibrium effects, the shapes of the experimental
angular distributions are reasonably well reproduced by the calculations.
Discrepancies between theoretical and experimental angular distributions due
to preequilibrium emission are most pronounced for lighter fragments
emitted at forward angles. For such fragments the measured angular
distributions are slightly more forward peaked in the laboratory than the
calculated angular distributions. Similar discrepancies exist for other
treatments in which statistical equilibrium is assumed. For this purpose, the

calculated angular distributions may be sufficiently realistic to assess effects of

142

1 HSU-QZ-OO 1
10-. “ l I I FT] I I I I I T’I I I I I II

 

3'Ar + mAu

10"2 '0
10‘3

10’4

 

.. 12.}(6 + "7A1:
E/A = souev. chss

   

dN/do (msr’l)

10"4 .

   
  

I 3-5
0 6-12
10"5 - - P-0.7

— P-0.7 (filtered)

6 lab

Figure 6.3 Angular multiplicity distributions of light particles, Z=1,2 (circles),
and intermediate mass fragments of Z=3-5 (squares) and Z=6—12 (diamonds)
detected in 35Ar+197Au collisions at E / A=50, 80, 110 MeV (top panel) and in ‘
129Xe+197Au collisions at E/A=50 MeV (bottom panel). The curves represent
calculations with the bond percolation model for a bond-breaking probability
of p=0.7. The calculations have been normalized to the data at 9=45°. Dashed
and solid curves show raw calculations and calculations filtered by the
response of the experimental apparatus.

143
instrumental distortions on the energy- and angle-integrated particle

distributions presented in Figures 6.2 and 6.3.

In order to display more clearly the fraction of IMFs among the emitted
charged particles, Figure 6.4 presents the average IMF multiplicity, (Mm), as a
function of charged-particle multiplicity, NC [Bowm 91] (see Figure 5.5). Points
in the figure show the average IMF multiplicity (MW) as a function of
charged-particle multiplicity NC. For a given charged-particle multiplicity,
more intermediate mass fragments are observed for the 129Xe-I-197Au reaction
than for the 36Ar+197Au reactions. At large charged-particle multiplicities (NC
230), IMF admixtures of (NW) /NC =0.18 and 0.1 are observed for the systems
129Xe+197Au and 36Ar+197Au, respectively.

If the elemental distributions strictly followed a power law distribution,

0(2) = 0'02", the IMF-admixture would be determined by the exponent 1::
20
002"
”mp = 2:3

N Zw
C 2 0.02-1
2:1

For power law distributions, the (raw detected) ratios of (N,,,F) / NC =0.18 and

(6.2)

0.10 correspond to exponents of 122.15 and 2.6 (see Figure 6.5). If we use the
shapes of the energy and angular distributions predicted by the percolation
model to correct for the detector efficiency, we obtain efficiency corrected
values of (N,MF)/NC =0.19 and 0.116, corresponding to 1:--2.12 and 2.5. These
values are slightly below and above the value, “rm-32.2, at the critical point.
According to Figure 6.2, the percolation model underpredicts the yield
of IMFs with Z>5 for the 129Xe+197Au reaction. This failure is displayed more
clearly by the curves in Figure 6.4 which show maximum IMF admixtures
predicted by the percolation model. Thick and thin curves represent results of

filtered and unfiltered percolation calculations, respectively, using the near-

‘Y
I

Figu
dElEI
Open
E/A.
and l

144

 

 

 

 

 

MSU-92—055
I I I I I I I I I W I I I -‘
l. o ‘”Xe+""Au, E/A = sonev ,' .
8 __. o “A:+‘°"Au. E/A = 110MeV ,.’ fl
_ o ”Ar+‘"Au. E/A = aouev l.’ -
. o ”Ar+‘°"Au. E/A = souev q .r' .
6 — _.
A r- _
In - ..
a I
z «-
v 4 ~— —
2 — _
- 197+129 .
. Cb .
0 l L l l l 1 1 1 1 1 1 J_ l
0 20 40 60

Figure 6.4 Relation between average IMF and charged-particle multiplicities
detected in 35Ar+197Au collisions at E/A=50, 80, 110 MeV (open diamonds,
open squares, and open circles, respectively) and in 129Xe+197Au collisions at
E/A=50 MeV (solid circles). Thick and thin curves show the results of filtered
and unfiltered percolation calculations, respectively. Details are given in text.

145

 

0.25

0.20

0.15

0.10

NIMF/NC

 

|llllllJlLLLlllLllll

0.05

IlIfilTIIIIjIIIIIIIII

llll

 

 

0.00 llllllllllllllLlLJ
2.00 2.25 2.50 2.75 3.00

T

)—

 

Figure 6.5 IMF admixture as a function of 1, according to Equation (6.2).

146
critical bond-breaking parameter, p=0.7. (For the percolation model, these

calculations give upper bounds for the admixture of IMFs among the emitted
charged particles. Smaller IMF admixtures can be obtained by using larger or
smaller bond-breaking parameters.) The dashed curves in Figure 6.4 show
percolation calculations for the combined 36Ar+197Au system. For this
system, filtered and unfiltered IMF admixtures are very similar, and the
relative abundance of Ms observed in central collisions can be reproduced
by the model calculations. The solid curves represent percolation calculations
for the combined 129Xe+197Ausystem. Here, the filtered calculations represent
slightly higher IMF admixtures than the unfiltered calculations. This effect is
largely due to an increased IMF detection efficiency resulting from the larger
center-of-mass velocity of the 129Xe+197Au center-of—mass system. However,
both filtered and unfiltered calculations predict IMF admixtures that are too
small. This failure is most dramatic for central 129Xe-I-197Au collisions for
which the comparison with an equilibrium model is most meaningful.

As an alternative scenario, we have also performed percolation model
calculations for the separate multifragment decay of excited projectile and
target nuclei. The unfiltered calculations, shown by the thin dot-dashed
curve, predict slightly higher IMF admixtures. The thick dot-dashed curve
illustrates the effect of filtering for an extreme two-source scenario in which
the relative velocity of projectile and target was reduced by only 50% from the
initial value, taking total momentum conservation into account. Even in
such an extreme scenario, the major discrepancy remains.

High-resolution coincidence experiments indicate that a significant
portion of primary fragments can be expected to be produced in highly excited,
particle unbound states which decay by light particle emission [N aya 92]. Such

sequential decay processes will result in secondary fragment yields which are

147
smaller and secondary light particle yields which are larger than the

corresponding primary yields. The portion of primary fragments may
therefore be even larger than the portion of particle-stable secondary
fragments. This aggravates the failure of the percolation model to predict the
large proportion of intermediate mass fragments among the particles emitted
in the 129Xe+197Au reaction.

The inability of the bond-percolation model to reproduce the large
intermediate mass fragment multiplicities observed for the 129Xe+197Au
system is unexpected and represents, to our knowledge, the first significant
failure of the percolation model. This model is only one representation of a
large number of phase transition models which all belong to the same
universality class and should therefore show similar deficiencies. At present
it is not clear whether one can rule out all such models or whether one may
be forced to consider dynamical enhancements of fragment yields due to

collective expansion or rotation.

Toroids and bubbles

In the bond-percolation calculations of the previous section, the
disintegrating systems were assumed to have compact spherical
configurations. Recent microscopic transport calculations [Baue 92, More 92]
indicate, however, that multifragment decays may proceed via more complex
toroidal or bubble-shaped decay configurations. In this section, we employ the
bond-percolation model used previously to investigate how multifragment
disintegrations might be affected by the occurrence of ring- and bubble-shaped
decay configurations.

All calculations presented in this section were performed for a system

consisting of A=N+Z=250 nucleons, with Z=102. The decaying system is

a 1
F0
ad

me
sol
per
nea
ma;
3 CC
mu]
thic.
line

eXpE

for t:

Prob;
tOl’OiI
the 5

defim'

148
represented by those points on a simple three-dimensional cubic lattice which

fall within the specified decay volume. Each site represents a nucleon that is
"bonded" to its nearest neighbors as described in the previous section. In the
following discussion of non-compact breakup geometries, instrumental
distortions will be ignored since we wish to outline some general trends
without trying to fit a specific set of data. Nevertheless, it is useful to provide
a reference which allows the reader to gauge the magnitude of various effects.
For this purpose, the solid points in Figures 6.6 and 6.7 show the fragment
admixtures (i.e. the mean number of detected intermediate mass fragments,
(N,,,F), as a function of the detected charged-particle multiplicity, NC)
measured [Bowm 91] for the 129Xe + 197Au reaction at E/ A = 50 MeV. The
solid lines in the figures show previous calculations with the bond
percolation model for a compact spherical breakup configuration using a
near-critical bond-breaking parameter p = 0.7. These calculations represent the
maximum fragment admixtures predicted by the bond-percolation model for
a compact spherical geometry; they underpredicted the measured fragment
multiplicities (see previous section). The difference between the thin and
thick lines illustrates the magnitude of instrumental distortions. The thin
line represents the "raw" calculation (not corrected for the acceptance of the
experimental apparatus), and the thick line represents the calculation filtered
by the acceptance of the experimental apparatus.

The hatched area in Figure 6.6 shows fragment admixtures predicted
for toroidal breakup configurations. In these calculations, the bond-breaking
probabilities were varied between p = 0.5 - 0.8, and the central radii of the
toroids were varied between Rt = (2.0 - 4.5)xa, where a = p01 / 3 ~ 1.8 fm denotes
the spacing between adjacent lattice sites (see insert in the figure for a

definition of the geometry). Because of volume conservation, the thickness d

149

 

 

     
 

 

 

 

MSU-93-039
10 h I I I T‘r I I I T I I I I 77 I I I I I H
: o 12"Xe+""’Au, E/A=50 MeV :
" sphere, A=326, p=0.7 «
8 — fl
: 1
/\ ‘_ _“
k. 6
E ' 1
Z _ <Nc> toroid .
V - A=250 —
4 _— P=0.5-0.8 —_
_ R,=2.o-4.5 .
2 ‘ CH9 ‘
F / a
l l '.l l I l L l l l l l I l L L l W
O I |
0 20 4O 60 80

Figure 6.6 Relation between average IMF and charged-particle multiplicities.
Solid points represent values measured for 129Xe + 197Au at E/A=50 MeV.
Thin (thick) solid line shows the raw (efficiency corrected) percolation
calculation for a solid sphere. The hatched area shows the range of average
IMF and average charged-particle multiplicities predicted by percolation
calculations for toroidal breakup configurations.

150

 

 

 

 

 

 

MSU—93—040

10 '- l I I I l I I I I \ I I I I 4

I o 12")(e+“"’Au. E/A=50 MeV \ Z

- sphere, A=326, \ I

8 _— p=m \ ‘1

_ %-\\u .

- §§ \\ .

A 6 — §~ § 3 —
in P §§5’ E .
2 § §:: E .
z : _5. $3: <Nc> bubble 3‘ ‘
V - .5 §f A=250 ‘3
4 — Q; P-O 5—0 a I
5%; - ' ‘ -

' r§§ Rb=0.0-4.0

2 r—- § > —

L ~§ .

.. § -

. § .

O b L .1 ’ 1 I l g l 1 l 1 1 l L l l l 1 l ‘*
0 20 40 60 80

NC

Figure 6.7 Relation between average IMF and charged-particle multiplicities.
Solid points represent values measured for 129Xe + 197Au at E/A=50 MeV.
Thin (thick) solid line shows the raw (efficiency corrected) percolation
calculation for a solid sphere. The hatched area shows the range of average
IMF and average charged particle multiplicities predicted by percolation
calculations for bubble-shaped breakup configurations.

151
of the toroid is defined by Rt and the nucleon number: A a antdz/ 2. For

A=250, the toroid has a hole in the center only for Rt > 2.5xa. The upper
boundary of the hatched area is determined by the breakup of a toroid of
radius Rt = 4.5xa, and the lower boundary represents the breakup of an oblate
object of Rt = 2.0xa. Over the range of NC z 30 - 50, the percolation model can
produce significantly larger fragment multiplicities for toroidal than for
spherical breakup configurations. Qualitatively such an effect may be expected
since the surface of a toroid is larger than that of a sphere.

Enhanced fragment admixtures can also be obtained for bubble-shaped
breakup configurations. In Figure 6.7 the hatched area shows the range of
average IMF multiplicities predicted for bubble shaped density distributions
using bond-breaking probabilities of p = 0.5 - 0.8 and inner bubble radii of Rb =
(0.0 - 4.0)xa (see insert in the figure for a definition of the geometry). For A =
250 and Rt > 4.0xa, the bubble has a thickness less than 1.0xa, and the
simulation models the breakup of a thin sheet. The upper rising boundary of
the hatched area corresponds to calculations with a fixed bond-breaking
probability of p = 0.5 and varying inner radii, Rb = (0.0 - 4.0)xa. The falling part
of the upper boundary represents calculations for an inner radius of R], = 4.0xa
and varying bond-breaking probabilities p = 0.5 - 0.8. For (NC) 5 70, the lower
boundary is given by the breakup of a sphere with varying bond-breaking
probabilities, p = 0.5 - 0.8.

The results in Figures 6.6 and 6.7 demonstrate that objects with larger
surfaces can produce more fragments than objects with smaller surfaces.
Hence, geometrical considerations may play an important role for
multifragment disintegrations. A relatively cool object (small bond-breaking
probability) with a large surface may decay into more fragments than a hotter
object (larger bond-breaking probability) with a smaller surface. For

152
noncompact breakup geometries, the large fragment multiplicities observed

for the 129Xe + 197Au reaction can be reconciled with predictions of the bond
percolation model.

A number of investigations have aimed at obtaining information of
near-critical behavior from the shape of fragment mass or charge
distributions [Gros 90, Pana 84, Mahi 88, Ogil 91, Li 93]. Near the critical
region, scaling theory of large systems predicts mass distributions of the form:

N409) .. A"f(A"(p - 19.)). (63)
Here, 1: and o are critical exponents and f (A°( p — 19.)) is a scaling function that
modulates the power law behavior near the critical bond-breaking probability,
p=pc, above which the "infinite" percolation cluster ceases to exist. In the
Fisher droplet model f is an exponential so that
NA (P) °¢ A.f exp(const- A"(p - pc )). (6.4)
f has the appropriate limiting behavior of f =1 at p = Pa so that N A cc A"

In practice, the mass or charge distributions are often fit [Pana 84, Mahi
88, Ogil 91, Li 93] by a simple power law,

Y(A) .. A“, (6.5)
where A is treated as a fit parameter, and the critical exponent 'c is identified
with the extracted minimum value of A [Baue 85].

We will now show that this empirical approach can lead to misleading
results if the geometry at breakup is not compact. For this purpose, we
performed power law fits to the mass distributions predicted by the bond
percolation model over a broad range of parameters and geometrical
configurations. The results of these calculations are summarized in Figures
6.8 and 6.9 for toroidal and bubble-shaped geometries, respectively. In both
cases, the best fit-parameter A. exhibits a clear valley as a function the bond-

153

toroid, A=250

 

Figure 6.8 Extracted power—law exponents 7L fit to mass distributions predicted
by the bond percolation model for the break up of toroidal systems as a
function of Rt and p.

 

wo-co-nsw

154

bubble, A=250

 

2.75
2.5

2.25

Figure 6.9 Extracted power-law exponents 7» fit to mass distributions predicted

by the bond percolation model for the break up of bubble shaped systems as a
function of Rb and p.

zvo-co-nsw

155

 

 

 

 

MSU-93-043
3 5 L: I l I I I I I I I I r I I I I I l r _J
- A=250 -
' o sphere . ‘-
3 0 ’_ g o toroid R.=3.0 _:
° _ g g I bubble Rb=3.0 -
_ I .
< 5 . I «
2. — -—
_ I 0 fl
- § § ' . I O .
I- § . s a —I
" I 3 3 ° ‘
2.0 — _ . —
h- . . - - -1
1 5 ln] 1 l l l l l [A l l L L l l l J l 1 Id

0 5 0 6 0.7 0 8

P

Figure 6.10 A as function of p for a solid sphere (circles), a toroid of radius Rt =
3.0xa (diamonds), and bubble with inner radius Rb = 3.0xa (squares).

156
breaking (p) and geometry (Rt and Rb) parameters. In this valley, smaller

values of 1 occur for less compact breakup geometries.

The strong geometry-dependence of the relation between the power-
law exponent A. and the bond-breaking parameter p is depicted more clearly in
Figure 6.10 for three representative breakup configurations, a solid sphere
(solid circles), a toroid of central radius Rt = 3.0xa (open diamonds) and a
bubble with inner radius Rb = 3.0xa (solid squares).

Microscopic transport calculations [Baue 92, More 92] predict a strong
dependence of the breakup geometry upon beam energy, impact parameter
and projectile-target combination. Indeed, under favorable conditions, the
formation of unstable bubbles and rings has been predicted [Baue 92, More 92].
The present model calculations for finite systems indicate a strong
dependence of extracted "critical" parameters (1 and pc) on the geometrical
configuration of the system at breakup. Compilations of power law exponents
1 determined from data for different entrance channels [Pana 84], for different
impact parameters [Ogil 91] or for excitation functions covering broad ranges
of energies [Pana 84, Mahi 88, Li 93] may contain samples representing
sufficiently different geometrical configurations to render a minimum in A
difficult to interpret. For infinite systems, critical exponents govern the
scaling laws near critical points. We suspect that the application of scaling
laws to finite systems of potentially complex breakup geometries is much less

straight forward than originally surmised [Camp 92].

Chapter 7 Fluctuations in multifragment
emission

A recent analysis [P105 90] of fragment-size distributions observed in
reactions induced by gold on emulsion at E/A=1 GeV [W add 85] saw evidence
for intermittency, which might indicate that fragmentation processes are scale
invariant. We take up the problem of intermittency by analyzing the factorial
moments of charge distributions observed in 36Ar+197Au reactions at beam
energies between 35 and 110 MeV per nucleon [deSo 91].

The occurrence of intermittency is deduced from the factorial moments

[Bial 86, Bial 88]
2%
2(N,(N, -1)...(N,. -k + 1))

F.(A)= z.)
201,)"

i=1

(7.1)

 

where 20 is the total charge of the disintegrating nuclear system, A is a binning
parameter, and N,- is the number of fragments with charges in the interval (i-
1)A <2 5 iA where i=1,...,Zo/ A. The ensemble average < > is performed over all
fragmentation events considered. Intermittency is defined by a relation [Bial
86, Bial 88]

F,(A’) = F,(aA) = a‘“’"Fk(A) (7.2)

157

158
between factorial moments F ,(A’) and F ,(A) obtained for two different

binning parameters A and A’ = 0A. Generally, evidence for intermittency has
been obtained by examining the double logarithmic plot of lnF, versus -lnA.
Plots which show lines of positive slope are consistent with nonzero fractal
dimension and considered to display intermittency. By construction, the
factorial moments F k (A) are unity for Poisson distributions. Hence, Poisson
distributions do not exhibit intermittency. The expression for the second
factorial moment can be written as

F2(A) = 1 +2410} — W» (7.3)

2.002
where (N,)and of denote the mean value and the variance of the multiplicity
distribution in the ith bin. Distributions which are narrower (broader) than
Poisson distributions possess moments which are less (greater) than unity.
For A = Z0, the right hand side of Equation (7.3) reduces to
1+(ofi —(NC))/ (NC)2, where (NC) and of; are the mean value and the variance
of the charged particle multiplicity distribution.

Intermittency as an indicator of nontrivial physics is generally sought
for in systems exhibiting larger than Poisson fluctuations. Evidence for such
large fluctuations must be sought in events representing similar initial
conditions. However, constraints from conservation laws may lead to
reduced fluctuations because statistical independence of individual bin
occupations is lost. As the number of bins becomes larger, the individual bin
occupations may become more independent, and the moments may increase.
The corresponding rise in the factorial moments as a function of decreasing
bin size would be of little interest.

In order to explore whether there is a basis for non-trivial intermittent

behavior in a reaction in which multifragment emission has been observed,

159
we analyzed the first and second moments of the charged particle multiplicity

distributions measured (see Section 5.2), with the Miniball [deSo 90] for
36Ar-I-197Au collisions over a broad range of energies, 355 E/ A5110 MeV.
Event selection was performed by cuts on the total transverse energy,
E, = E sin2 0, of the emitted charged particles. For orientation, we also provide
an empirical impact parameter scale by using the geometrical prescription of
Section 4.2 to construct a ”reduced” impact parameter IS. The reduced impact
parameter assumes values of I; zl for peripheral collisions and 5 z 0 for the
most violent collisions characterized by large values of E,.

The top panel in Figure 7.1 shows the measured two-dimensional
correlation between transverse energy E, and charged particle multiplicity NC
for 36Ari-197Au collisions at E/A=110 MeV. The dashed and dot-dashed

curves in the bottom panel of Figure 7.1 depict charged particle distributions
selected by narrow cuts on E,, corresponding to reduced impact parameters
5:101 and 0.6. The dotted curves illustrate the effects of increasing the
widths, A5,, of the cuts as indicated by dotted horizontal lines in the top panel.
For central collisions, the NC distribution is rather insensitive to AE,, but for
more peripheral collisions it suffers considerable broadening as AB, is
increased.

The broadening due to impact parameter averaging is illustrated more
quantitatively in Figure 7.2. Top, center and bottom panels of the figure depict
the quantities 02 /(NC), a: and (NC), respectively, as a function of transverse
energy. Solid circular, open square-shaped and star-shaped points show
values obtained for cuts of widths AE,=20, 180 and 340 MeV, respectively. For
narrow cuts on AE,, the charged particle multiplicity distributions are

inconsistent with F2(A = 20) > 1, since of. /(NC) < 1. For near-central collisions,

5<0.3, the extracted values of of. / (NC) exhibit little dependence on the

160

 

 

 

 

 

 

MSU-92-038
12507....1. "' "' H'flfl
1000' — = = = 0.1
I?) 750 ‘ 0.2
‘2: _ «0.4
m 500 ............... I
b -:::::::::= 06
250 """""" ' """
‘ 0.8
O . l *t t q
0.10 _ €
0.05 " ‘
000 .u-;./l.. . Li". r\n'-°1'uuL~L 1 .l'x-L‘ ‘
0 10 20 30 40 50

Figure 7.1 Upper part - Measured relation between transverse energy E, and

0")

total charged particle multiplicity for 36Ar+197Au reactions at E/A=110 MeV.
Lower part - Charged particle multiplicity distributions for the cuts on E,

indicated in the top panel.

161

 

2.0 I ' . g

" 1 M

I ‘3] I?

A 1.57 \ X fl co

0 .. ' . N

Z : 5’ ‘- c';

\/ 1.07 \ 1 21’

\ : fix.

Nb!) _ \0\71\ d

0.5 I

All

 

IIUU'I'UI

2

C
H
01

II

 

      

All}.a
0 ZOMeV I
0 1801191!
)1 340MeV

I UIVUTUIUU IU'UIVIIUIYIU' I II

<NC>
8
'l

 

 

0'.DLL...L...1..11.AJ
0 200 400 600 800 1000

Et(MeV)

 

Figure 7.2 Bottom, center and top panels show the mean values (NC),
variances oi , and ratios 0% / (NC) of the charged particle multiplicity
distributions for 36Ar+197Au reactions at E/A=110 MeV. These quantities
were selected by various cuts on the transverse energy; the mean values of
these cuts are given by the abscissa and the widths are given in the figure. The
upper scale gives the reduced impact parameter 5(E,) .

162

widths of the applied cuts. However, for larger impact parameters, 13>0.4,
wide cuts on E, cause an artificial broadening of the multiplicity distributions
resulting from the superposition of distributions with different centroids.
Poorly defined ensembles of events may therefore exhibit larger than Poisson
variances, of. /(NC)>1. However, these large variances are an artifact from
impact parameter averaging, and they do not represent intrinsic fluctuations
of the decaying system.

Figure 7.3 depicts the relation between (NC) and 0% / (NC) extracted for
central (13<0.3) 3'5Ar+197Au collisions at the incident energies of E/A=35, 50,
80 and 110 MeV. At all energies, the fluctuations of the charged particle
multiplicity are considerably smaller than expected for Poisson distributions.

In order to explore effects resulting from phase space constraints such
as energy conservation, we performed calculations with the bond-percolation
model of refs. [Baue 85, Baue 86, Baue 88]. For simplicity, we assumed the
decay of the composite system (A=233, Z=97). Calculations with bond-breaking
parameters close to the critical value of p=0.7 have already been shown
(Section 6.2) to reproduce the element distributions measured for the present
reaction. Standard percolation calculations (in which the number of broken
bonds is allowed to fluctuate from event to event) predict fluctuations in NC
which are much larger than observed experimentally. Open circular points in
Figure 7.3 show representative results for p=0.6 and 0.7. The indicated shift
from the open circle to the open triangular point illustrates the magnitude of
instrumental distortions for the case p=0.7. These distortions are too small to
affect our conclusions. If one introduces a constraint analogous to energy
conservation by requiring a fixed number of broken bonds, much narrower
charged-particle distributions are produced (see open diamonds). These

illustrative calculations suggest that the widths of impact-parameter—selected

163

 

I IFIII I I I I II I I [I IIIII I I K

.. a g

1.0— - |

. O 0 .§

b / " C

CD

.- u O.
0.8"“ A -
A . -I

U
z 0.6— -
v r ,
\ . o .
NCO . I
0.4. . .
O

. o o o .
' 'l
0.2- )1 31 .-
. x0 .
. [31:1 .
00 1 lklllLLl [1 L1 1 LLI 111.1 1 1 1

 

 

 

0 10 20 30 40 50
No

Figure 7.3 Relation between mean charged particle multiplicity (NC) and the
ratios of. / (NC). Solid circular points — experimental values extracted from
near-central 36Ar+197Au reactions at E/A=35, 50, 80 and 110 MeV. Open
symbols are explained in the text.

164
charged-particle-multiplicity distributions are strongly affected by phase space

constraints due to energy conservation.

This conclusion is corroborated by more realistic statistical model
calculations which incorporate energy conservation on an event-by-event
basis. The open square and star-shaped points show predictions of the
sequential decay model GEMINI [Char 88] and of the Copenhagen
fragmentation model [Bond 85a, Bond 85b, Barz 86], respectively. (The
individual points represent results obtained for the decay of heavy compound
nuclei at various excitation energies.) Both microcanonical and sequential
decay models predict ratios of. / (NC) somewhat smaller than observed
experimentally, and they do not show intermittency [Ella 92, Barz 92]. These
smaller ratios from theoretical predictions may not indicate an inconsistency
with experiment since we do not select a sharp value of impact parameter
and/ or excitation energy for the data in our event selection. In other words,
the data contain some residual impact parameter averaging and represent an
upper limit on the ratio of. / (NC).

In Figure 7.4, second factorial moments calculated from Equation (7.1)
are presented as a function of binning resolution. The solid points represent
experimental data selected by a narrow cut on central collisions at E/A=110
MeV. A slightly positive slope is observed, but the moments are smaller than
unity. As argued above, this positive slope may be of trivial origin. In order to
corroborate this point, we include the results of percolation calculations
performed for p=0.7. Standard calculations for which the total number of
broken bonds is allowed to fluctuate (open squares) predict large factorial
moments, but no intermittency. When the number of broken bonds is
constrained to be constant (open diamonds), the factorial moments are

strongly reduced in magnitude, and they exhibit a small increase as a function

165

 

 

 

 

 

 

I ‘ I I ‘ I g
9:
’ P=0.7 do
0.000 - - I}:
" %B:E a a
. B .
as E] °’
1- \S \ fl 4
—0.002 ~ _ _ fl—
’1 . . . . 1 L . . . 1 A . . . 1 ‘
I I ’ II I
P=0.7 q
f?) —0.016 _ energy conserved _
Ln. - e
. ..
5 L W090 3 .
-0.018 2' -.
1 . . . . 1 . . . . 1 A . . . 1
O 022 I . r . T r . . . . I r . r . I
- ' _ “Ar + ""Au. E/A=110MeV - ‘
central collisions .
1 . . 1 L . . 1 . 1 1
-3 -2 -1 O

—ln(A)

Figure 7 .4 Scaled factorial moments as a function of binning resolution. Solid
points show experimental results for central 36Ar+ 197Au collisions at
E/A=110 MeV. Open points show results from percolation calculations using
p=0.7. Open squares depict calculations in which the total number of broken
bonds is allowed to fluctuate; open diamonds represent calculations in which
the total number of bonds is kept fixed.

166
of binning resolution, similar to that observed experimentally. This increase

appears to be of little significance.

It has recently been shown [Elat 92] that rate-equation models used to
describe sequential decay processes are incapable of producing intermittency
in fragment mass distributions. The few statistical models that could produce
intermittency signals [DeAn 92, Barz 92] had to resort to mixing
fragmentation events of very different initial excitation energies (effectively
averaging over impact parameter). Our results are consistent with the small
fluctuations predicted from the statistical multifragmentation approach for

initial conditions with a narrow range of excitation energies.

Chapter 8 Summary

In this work we have studied multifragment emission in 35Ar+197Au
reactions at incident energies of E/A=50, 80 and 110 MeV. Of particular
interest are central collisions, where the chance for equilibration of the system
is the highest. To select such collisions we have made a detailed comparison
of impact parameter scales constructed from the following global observables:
the charged particle multiplicity NC, the total transverse energy of an event
E,, the sum of the charge emitted at midrapidity Zy and the hydrogen
multiplicity N, (the complement of 2M, the total charge bound in clusters
of ZZZ). Each of these observables was evaluated in its ability to select events
with suppressed projectile-like fragment emission and azimuthally
anisotropic emission patterns. We found that scales constructed from NC, E,
and Z’ provide comparable measures of impact parameter. Impact parameter
scales based on E, provide slightly better resolution those based on NC or 2,,
while the scale constructed from N1 is the worst. N 0 additional selectivity for
central collisions was found by imposing additional cuts on the transverse
momentum directivity D, a variable which has been applied successfully in
symmetric systems at higher bombarding energies where flow effects are
important.

The mean values and variances of the multiplicity distributions of

intermediate mass fragments were found to increase as a function of the total

167

168
charged-particle multiplicity; this may be interpreted as a rough measure of

the internal energy of the fragment emitting system. An average multiplicity
of 4 intermediate mass fragments is observed for the most central collisions at
E/A=110 MeV. These large IMF multiplicities are consistent with predictions
of a statistical model for evaporation from an expanding compound nucleus.
The statistical decay model predictions are sensitive to the low-density
nuclear equation of state.

In our investigation of fragment admixtures predicted by a standard
bond-percolation model, we found rather good agreement with the values
measured for 36Ar + 197Au collisions. On the other hand, larger fragment
admixtures measured for 129Xe + 197Au collisions were found to be
inconsistent with the model (for compact geometries).

We have also explored the multifragment breakup of different
geometrical configurations (toroids and bubbles) with a bond-percolation
model. Calculations for these finite systems predict enhanced fragment
production for less compact decay configurations. For noncompact breakup
geometries, the large fragment multiplicities observed for the 129Xe + 197Au
reaction can be reconciled with predictions of the bond percolation model.
Power law fits to the predicted mass distributions reveal a strong sensitivity of
the extracted critical exponents to the geometry of the decaying system. Of
course, the existence of such non-compact breakup configurations is not yet
established. This question still remains to be answered.

In this fragmentation study we also include an intermittency analysis
based on factorial moments that should permit the establishment of
deviations from Poissonian fluctuations in the fragmentation process. We
argue that an intermittency signal is meaningful only when it is observed in

data selected by a narrow cut in impact parameter, and only when F,(A)>1.

169
These two conditions are never simultaneously met in our data. A

surprisingly small value of about 0% /(NC) = 0.3 was measured in central
collisions, independent of beam energy. The small widths of the charged
particle distributions may be caused by the constraints imposed by energy

conservation.

170

Appendix A

Physics tape format

The information is stored in buffers 4096 words long (integer‘2). The
first 16 words are the header. The first word past the header is the multiplicity
of the event. This number is negative. The second word is the ringmask (one
bit is set in this word for every ring that is hit). The third word is the RF word.
The fourth is the number of intermediate mass fragments in the event.

Knowing the event multiplicity (number of detectors that fired), we
can calculate the length of the event.

length = (multiplicity‘S) + 4 (AJ)
To get to the next event you need to increment the pointer in the buffer by
this length, i.e. ‘
next event = last event + length. (A2)

The information of the current event is stored and accessed as follows:

pointer = pointer + 4 ! move past the multiplicity, ringmask, RF,
! and NM words
do ii = pointer, next event, 5

detector number = buffer(ii) ! from 1-188

slow = buffer(ii+1) ! slow channel number 1-2048
"real Z": buffer(ii+2) ! 2’25 from the PID maps

PID number = buffer(ii+3) ! two byte word: first byte=ISO, for

! isotope resolution
! second byte=IZ of the particle
energy = buffer(ii+4) ! in 100 keV units
end do

171
Particle identification is accomplished through the two words 150 and

12 as shown in the following table:

Table A.1 Physics tape format for particle identification.

 

firticle type

Hydrogen punch through
Proton

Deuteron

Triton

3He

4He

6He or Be decay to 20:
He punch through
Li punch through

2a punch through

Unidentified particles

Fragments that punch through fast plastic

Fission fragment

Fragments that stopped in the plastic, above the
LCP reject line, in the prompt branch of fast-
time spectrum

Particles that stopped in the plastic, below the LCP 0 11
reject line, in the prompt branch of fast-time

 

NgNO moan NNNHHHHE
H
N

 

 

spectrum
No isotope resolution 1,2 5
Hit detector (ring 1, position 5) 0 0

 

The following detectors did not work properly during the experiment:

 

 

fietector fiProblem fifesult

5 PMT pulsed Hit detector

16 double image no isotope resolution
48 discriminator double fired no Z greater than 2
70 double image no Z greater than 2
169 very small signal no Z greater than 2

 

 

None of the above detectors have energies.

 

172

Appendix B

Temperature estimate from percolation theory

In the standard percolation model the bond breaking probability p is
linearly related to the excitation energy per nucleon E‘ of the compound

system such that
E‘ E'
p z =
Ea Emir

where E, is nuclear matter binding energy per nucleon (16 MeV), E,,“, is the

 

 

(B.1)

energy required to break one bond and z is the number of nearest neighbors
on the lattice (2:6). (By placing the nucleons on a lattice, the volume and
surface effects in the binding energy are well approximated [Baue 88].) The

total excitation energy of the system is simply

E5... = pEaAm- (3.2)
EL, in the Fermi gas model is
5;, = ar2 (13.3)

where a is the level density parameter (a= A,”/ 8.5 MeV) and T is the

temperature. The expression for the temperature is then

E A
T = .”+"’ =117Mev43. (13.4)

For the near-critical bond-breaking probability p=0.7, T=9.8 MeV.

[Alar 92]

[Ardo 90]

[Awes 81]

[Barz 86]

[Barz 92]

[Baue 85]

[Baue 86]

173

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