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This is to certify that the

dissertation entitled

TRANSITIONAL PHENOMENA IN INTERMEDIATE—ENERGY
HEAVY-ION REACTIONS

presented by

Nathan Thomas Boden Stone

has been accepted towards fulfillment
of the requirements for

Ph. D. Natural Science

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TRANSITIONAL PHENOMENA IN INTERMEDIATE-ENERGY
HEAVY-ION REACTIONS

By

Nathan Thomas Boden Stone

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

1996

ABSTRACT

TRANSITIONAL PHENOMENA IN INTERMEDIATE-ENERGY

HEAVY-ION REACTIONS

By

Nathan Thomas Boden Stone

Many studies have been performed pertaining to the subject of nuclear matter
phase transitions and also related transitions in nuclear reaction decay mechanism.
In Chapter 3 we present a study of central heavy-ion collisions in the entrance chan-
nels, 20Ne+27Al, 40Ar-l-“Sc, 87Kr+93N b, and 129Xe+139La, for a large range of beam
energies, for the purpose of identifying transitions in the reaction decay mechanism
for those systems. We have defined quantitative measures of the relative sizes of
the three largest fragments in these central events using new variables derived from
charge Dalitz plots. We then relate the values of these quantities to the predom-
inance of specific reaction decay mechanisms. Transitions from sequential binary
disassembly to multifragmentation were observed at ~50 and ~30 MeV/ nucleon in
the 40Ar+45Sc and 129Xe-l-139La entrance channels, respectively, and trends observed

for the 87Kr+93Nb reactions indicate a similar transition at a beam energy slightly

below 35 MeV/ nucleon.

Recent theoretical calculations based on transport theories have predicted the oc-
currence of exotic break-up geometries at intermediate energies. These geometries

include bubbles and toroids, which are considered non-compact because of their hol-

low centers, as well as disks. Experimental searches to date have not been able to
confirm these predictions. However one group has shown an agreement between their
data, for one system at one beam energy, and emission from a toroidal configuration
as opposed to a single-source ellipsoidal configuration. In Chapter 4 we present a
study of central collisions in the 86Kr+93Nb system at incident energies ranging from
35 to 95 MeV/ nucleon. Our results comprise the first systematic experimental results
which both provide evidence for the decay of nuclear matter from a toroidal geom-
etry and confine this occurrence to a finite range within the beam energies studied
for this system. We use two experimental observables, intermediate mass fragment
multiplicities and power-law exponents of ordered charge distributions, to establish
that the system decays from a non-compact break-up geometry. We then use event
shapes to show that the decaying system is more coplanar in shape, thus implying
that the geometry is toroidal. We find that the toroidal geometries are produced for

beam energies between 60 and 75 MeV/ nucleon.

To my wife Kathy,
to our children both now and yet to come,
and to our LORD.

To you I have pledged my lifelong love and service.

Soli Deo Gloria.

iv

ACKNOWLEDGEMENTS

Here I would like to acknowledge those people who have made substantial contri-

butions to my work, training, and general well-being.

To my advisor, Gary Westfall: I shall be always in your debt for your guidance,
insight, support, and even defense. You have counseled me in everything from what
to wear to an interview, to what to write in a PRL, to how to survive in East
Lansing. Your contribution is not only professional, but personal. Your character
and leadership was as compelling to me as your analysis when I first met you as an
undergraduate. It is largely to your credit (or blame) that I pursued post-graduate

training in physics, and beyond. You have my thanks.

To Skip Vander Molen: You have taught me much of what I know about the art
of data acquisition. You have fanned into flame that ember in me which burns for
the real truth, as opposed to the most convenient interpretation, or straight—line fit.

You have bailed me out on many late nights after many long days.

To Robert Pak: You have been a listening ear and an open mind in many dis-
cussions of methodology, presentation, FORTRAN, and faith. You are almost always
the one to hear my latest ideas first. We have shared many successes as well as trials,
and our commonality has been a strength to our friendship, our families, and our
work. May it continue to always be so. I am grateful to you for sharing not only your

time, but your self with me.

To Bill Llope: You introduced me to PAW and got me started on analysis before
I was even started in classes. You also guided me through what was to become my

first publication. Thanks for helping to show me the ropes.

To my fellow graduate students: Gene, Yee, Omar, Dan (who told me everything

I need to know about Berkeley), Betina, and Justin. Your combined brain power,

V

camaraderie, humor, and character are what has made being in the 471' group one of
the most enjoyable parts of my training. I cannot over-estimate the significance of

being accepted, challenged, and appreciated by my peers.

To Paul Simms: Your influence beginning in my early undergraduate years contin-
ues to impact my life today. You have been my professor, my employer, my mentor,
and my friend. You were also careful not to neglect my character training for my
academic training. It was you who first recommended me to my research position as

an REU, and to several positions since then.

I am also grateful to the coordinators and participants in the REU program,
without whose efforts I would not have seen the side of research which led me to
pursue post-graduate training and physics as a career. I must also thank those at
the NSCL who have provided the environment that has made my work easier, faster,
and better. This includes the computer group in their relentless pursuit of glitch-free
programming (and disk storage), the operations group for the beam which I often
have taken for granted, and the NSCL faculty as a whole whose combined vision has

secured a place for this lab in the future.

A particular thanks must go to Jules Kovacs and Stephanie Holland, for their
administrative work on my behalf. Her Serene Highness has faithfully ensured this
Humble Servant’s enrollment, payment, and otherwise official existence. Your contri-

butions are invaluable.

To my Mother, who could attest from my early years that I am an experimentalist,
and to my Father, Whose curiosity and lack of inhibitions I have inherited; to my
siblings, who forced me at an early age to learn to overcome difficult situations; and
to my dog, who unwillingly participated in some of my earliest experiments. You put

the “form” in my formative years.

vi

To God, for giving me the talents and predisposition fitting to my task, and for
filling my life with “rare and beautiful treasures.” To my wife, one such treasure, who
has given me her faithful love and support, and many beautiful treasures of her own.
Her strength has been my strength, her hopes my hope, and her dreams our dream.
For this, and all of her daily sacrifices along the way, my Ph. D. is her Ph. D. “He
who finds a wife finds a good thing.” I must also thank my wife’s family, now my
family, who have provided not only the obvious contribution, but who have provided

much love and support to us in these past 5 wonderful years.

There are undoubtedly others whose names could fill more pages as their memories
fill my mind. My overwhelming sentiment is both of gratefulness and indebtedness.
I hope that I will give to those who would ask of me as willingly and fully as those

who have given to me.

And last but not least, this work was supported by NSF grants Nos. PHY 89-

13815, PHY 92-14992, and PHY-95-28844 (NSCL/MSU). Thank you, tax-payers all.

vii

Contents

LIST OF TABLES ix
LIST OF FIGURES x
1 Introduction 1
1.1 Phase Transitions ............................. 1
1.2 Decay Mechanisms ............................ 7
1.2.1 Relation to Charge Observables ................. 8

1.3 Non-Compact Geometries ........................ 12

2 Experimental 22
2.1 The 47r Array ............................... 22
2.2 The Zero Degree Detector ........................ 28
2.2.1 Design ............................... 29

2.2.2 Construction ........................... 35

2.2.3 Detector Response ........................ 39

3 Transitions in Decay Mechanism 47
3.1 Dalitz Triangles .............................. 47
3.2 Experimental Measurements ....................... 53
3.2.1 Mean Distances: Dcent and Dcdge ................ 55

3.2.2 Constraints ............................ 57

3.2.3 Acceptance Effects ........................ 60

4 Observation of Non-Compact Geometries 64
4.1 Signatures ................................. 64
4.2 Experimental Measurements ....................... 68
4.2.1 IMF Multiplicity ......................... 68

4.2.2 Ordered Charges ......................... 70

viii

4.2.3 Event Shapes ........................... 76

5 Conclusion 89
A High Voltage Circuit Components 91
B Centrality Selection 93
B.1 Numerical Calculations .......................... 94
B.2 Autocorrelations ............................. 96
C Simulation Parameter Definitions 100
C.1 Temperature ................................ 101
C2 Charge and Multiplicity Distributions .................. 104
LIST OF REFERENCES 108

ix

List of Tables

1.1

2.1

2.2

4.1

Critical exponents extracted from charge moments analysis of 1.0 GeV/
nucleon Au+C listed with values derived from other critical systems

[Gilk94]. .................................

Punchin energies for the ZDD, based on 0.67mm thick scintillating
plastic ....................................

Azimuthal angles for the ZDD. (Note: Since the ZDD chamber is sep—
arate from the 41r ball, if the ball is rotated, these angles will change
accordingly. These angles are reported assuming a 3.7° tilt (clockwise,
looking down the beampipe) of the 41r ball.) ..............

Definitions of event shape variables. The limiting cases shown in the

last columns represent the values obtained for the aspect ratios §:%:%,

%:%:0 and 1:0:0, respectively[Cugn83]. The definition of Eccentricity

depends on the momentum-space configuration. Prolateness (oblate—
ness) is defined via the relation ql — q2 >(<)q2 — q3. .........

A.1 List of Components for PMT socket. ..................
A.2 List of Components for the PMT divider .................

32

39

List of Figures

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

The second conditional moment m2 (see Equation 1.1) plotted ver-
sus the reduced multiplicity n for three-dimensional bond percolation
lattices. Four different system sizes, Z0 = 33(open circles), 53(closed
circles), 93 (crosses), and 503(triangles), are shown[Camp88]. .....

Extracted values of the power-law exponent 1' versus the calculated
temperature for six experimental systems: p+Ag (0.21-4.9 GeV) (cir-
cles); p+U (4.5, 5.5 GeV) (crosses); p+Xe and p+Kr (80, 350 GeV)
(squares); C+Ag and C+Au (180, 360 GeV) (triangles)[Pana84]. The

inset shows a charge distribution at the minimum in r. ........

Extraction of the critical exponents (a) ,8 and (b) 7 from experimental
data [Gilk94] and (c) 7' from a percolation calculation [Elli94]. Ordinate
and abscissa values are the conditional moments defined in Equation
1.1. ....................................

(a) 2-Body and (b) 3-Body asymmetry (defined in the text) plotted
versus the centrality variable Zbound[Kreu93]. Markers are mean val-
ues extracted for 600 MeV/ nucleon collisions of Au on C (circles), Al
(triangles), Cu (squares) and Pb (stars). The dotted, dashed and solid
lines represent values extracted from GEMINI, COPENHAGEN, and

percolation calculations, respectively. ..................

Logarithm of the largest charge versus logarithm of the normalized
second charge moment (defined in the text) for (a) 35 MeV/ nucleon
40Ca+4°Ca data, (b) a multifragmentation model by Sa and Gross, (c)

a sequential decay model by Richert and Wagner, ((1) the GEMINI
sequential decay model[Hage92]. ....................

Surface of constant density (p = 0.3po) resulting from BUU calcula-

tions of 60 MeV/ nucleon 93N b+93N b reactions at t=160 fm/c [Baue92].

Surface of constant density (p = 0.4po) resulting from BL calculations
of a 3 MeV system of 200 nucleons[Guar95]. The four panels show the
reaction at times 0, 40, 80 and 120 fm/c, as labelled. .........

Density of test-particles in a BNV lattice resulting from calculations
of 75 MeV/nucleon 90Mo+9°Mo reactions[MoreQ2]. The two column
groups show the calculation for two different nuclear matter compress-
ibilities, and rows (a)-(d) show the collision at times 20, 60, 120 and
180 fm/c, respectively. ..........................

xi

11

14

15

16

1.9

1.10

2.1

2.2

2.3

2.4

2.5

2.6
2.7

2.8

2.9

2.10

2.11
2.12

2.13

2.14

3.1

Mean distance from the CM-frame origin (Equation 1.8) in the BUU
position lattice versus time for the 86Kr+93N b system at 7 representa-

tive beam energies, as labelled. ..................... 19
Same as Figure 1.9, except using quadratic normalization (Equation
1.9). .................................... 20
Mechanical drawing of the frame of the main 47r ball, revealing the
arrangement of the hexagonal and pentagonal faces. ......... 23
Side view of a hexagonal 47r ball module, revealing three independent
layers of plastic and gas detectors. ................... 24
Arrangement of the forward array phoswiches at the time of data col-
lection for the charge correlations analysis. .............. 26
The High Rate Array, which consists of 45 close-packed phoswich tele-
scopes. .................................. 27
The Maryland Forward Array, with the outline of the High Rate Array
to show positioning. ........................... 28

ZDD High Voltage Circuits. (a) HV divider card. (b) PMT socket card. 31
The fractional energy loss of fragments (2 S Z S 27) in 0.67 mm

scintillating plastic, revealing the “punch-in” energies. ........ 33
Assembly diagram of the ZDD, showing location of PMTs and light
guides. The beam enters from the lower right. ............. 34

Mechanical drawing of the ZDD chamber. Above: Top view, with
beam entering from the top of the page. Below: Rear View, looking
upstream. This also shows the goniometer, used for positioning of the

detector ................................... 36
Energy deposited in the AE plastic versus time-of-flight for fragments
(Z=2-27) in 0.67 mm scintillating plastic, for a 2.74 m flight path. . . 37
Frontal photograph of the Zero Degree Detector. ........... 38
Raw experimental data recorded in one ZDD telescope for 45 and 65
MeV/ nucleon 87Kr incident on 93Nb. .................. 40
Calibration points measured in segment 2 of the ZDD. One atomic
number (Z=17) and isotope ratio (N=Z) are labelled. ........ 43
Response function minimization results, comparing Cebra and Birks

parameterizations. (a) AL Measured vs. Calculated from Cebra pa-
rameterization. (b) AL Measured vs. Calculated from Birks parame-
terization. (c) L Measured vs. Calculated from Cebra parameteriza-
tion. ((1) L Measured vs. Calculated from Birks parameterization. . 45

A Dalitz triangle. For each event, the reduced charges (Z,’) are set
equal to the distances to the three sides of the triangle. Dam, is the
distance from each event entry to the center, and Dcdgc (2 Z3) is the
distance to the nearest edge. Further features are explained in the text. 48

xii

3.2

3.3

3.4

3.5

3.6

3.7

3.8

4.1

4.2

4.3

4.4

Charge Dalitz plots created for the reaction 43 MeV/ nucleon Kr+Au
[Biza93]. The four panels represent four different excitation energy
bins, as labelled. Details of the figure are discussed in the text.

The relative probabilities of populating the three regions in a Dalitz
triangle for 43 MeV/nucleon Kr+Au collisions[Biza93]. Each of the
three lines has been labelled according to the reaction decay mechanism
leading the population of that region in the triangle. .........

Dalitz plots for central collisions of the the 129Xe+139La system. Beam
energies are labelled in each frame. The color contours depict the
logarithm of the counts in the histogram. ...............

The average values of Dcent (with points) and Dcdge (without points)
in the central events versus the projectile energy. The solid lines depict
the results for events selected by the two—dimensional cut on E. and
Zmr, while the dashed (dot-dashed) lines correspond to events selected
by a one-dimensional cut on E¢(Zm,.) ...................

The average values of Dedge versus Zwm in the central events for three
representative beam energies in MeV/ nucleon for each entrance chan-
nel, as labeled. The solid lines indicate the minimum and maximum
values of Dedgc that are possible for each value of Zwm by definition. .

The average values of Dccnt (with points) and Dedge (without points)
versus the beam energy for specific gates on the quantity Zsum: 6S
me S 17 for the 20Ne+27Al and 87Kr+93Nb entrance channels, and
GS Zwm $20 for the 40Ar+458c and 129Xe+l39La entrance channels.

The average values of Zwm (upper left frame) and Dec,“ (upper right
frame) obtained from eight samples of events, each generated at a spe-
cific excitation energy for central 40Ar+45Sc reactions using the Berlin
code. Each sample is boosted from the CM frame to the laboratory,
and then filtered, for beam energies from 25 to 105 MeV / nucleon. Some
points in the upper frames have been offset for clarity by the amounts
shown to the right. The lower frame compares the excitation energy
dependence of the unfiltered Berlin events with the beam energy de-
pendence of the filtered Berlin events, using BUU calculations to relate
the beam and excitation energies for central 40Ar+45Sc reactions.

IMF multiplicity versus the total charged particle multiplicity resulting
from percolation model calculations[Phai93]. The solid lines (both pan-
els) show the values obtained from percolation of a compact spherical
configuration, while the shaded regions show the values obtained from
percolation of initial toroid (panel a) and bubble (panel b) geometries.
The markers (both panels) show a comparison to an experimental data
set. ....................................

Mean intermediate-mass fragment multiplicity versus the incident beam
energy ....................................

Energy dependance of estimated efficiency for IMF detection, extracted
from filtered simulations. ........................

Mean total charge of IMFs versus the incident beam energy. .....

xiii

50

52

54

56

58

61

62

65

69

72

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

B.1

B2

B3

C.l

Ordered Z distributions (markers) for all beam energies, as labelled.
The solid line is a fit to the function (Zo,d(z')) or 2"“. .........

Power-law (a) values extracted from ordered charge distributions plot-
ted versus the incident beam energy. ..................

Center-of-momentum frame momentum profiles of spherical (top row),
disk-like (middle row) and rod-like (bottom row) simulated events.

Calculated behavior of all shape-variables for simulated spherical(solid
lines), disklike (dashed lines) and rodlike (dotted lines) events, plotted
versus IMF multiplicity. .........................

Same as Figure 4.8, except plotted versus total charged particle mul-
tiplicity, allowing extended (asymptotic) multiplicity scale. ......

Mean values of sphericity extracted for spherical (dashed), disklike
(dot-dashed) and rodlike (dotted) filtered simulations compared to
those extracted from the experimental data (solid) for all beam en-
ergies, as labelled. ............................

Measured values of the mean sphericity of IMF emission (markers) ver-
sus the incident beam energy for three representative IMF multiplic-
ities, as labelled. The dotted (dot-dashed) lines in each case indicate
the mean values obtained for filtered spherical (disk-like) simulations.
Small offsets have been added for clarity. ...............

Scaled values of all eight shape-variables (as labelled at right) for IMF
multiplicity 5. This reveals the suppression or enhancement of the all
shape variables at 60 MeV/ nucleon. ..................

Cosine values of the azimuthal angles (termed the “flow angles”) of
the three eigen-vectors of the kinetic energy tensor, plotted versus the
incident beam energy. ..........................

Two centrality variable distributions for (a) E. and (b) Zm, (see text
for definitions). The unshaded histogram shows the inclusive distribu-
tion for System-2 (see text) events, while the horizontally (vertically)
shaded sections show the subsets selected by l0% cuts on Zmr (E1)
alone. The intersection of these two subsets shows the portion of the
inclusive data selected by simultaneous l0% cuts on both of these vari-

ables (roughly 7%). ...........................

RMS widths of the charge distributions of first, second and third largest
fragments in central events in the 40Ar+458c system selected by 5% cuts
on 5 centrality variables (as labelled). .................

Mean charge of the first, second and third largest fragments in central

events in the 40Ar+45Sc system selected by 5% cuts on 5 centrality
variables (as labelled). ..........................

Kinetic energy (per nucleon) spectra (histograms) for 7Li fragments
for each energy, as labelled. Each spectra has been fit with a Maxwell-
Boltzmann distribution (solid line) in order to extract the emission
source temperature. ...........................

xiv

75

78

80

81

83

84

86

88

95

98

99

O2
C3

C4

C.5

Kinetic temperatures extracted from energy spectra for 7Li fragments.

Comparison of charged particle multiplicity distributions (histograms)
generated by the simulation (top panel) and measured for central events
in the experiment (bottom panel). Gaussian distributions have been
fit to the data (solid lines) and fit parameters are shown in the upper
right corner of each panel. ........................

Same as Figure C.3, except for charge distributions, fitted with expo-
nential functions. .............................

Same as Figure C.3, except for IMF multiplicity distributions.

XV

103

Chapter 1

Introduction

1 .1 Phase Transitions

Particular trends in the slopes and functional form of the charge distributions of frag-
ments emitted following central heavy-ion collisions have been noted [Fish67, Baue88,
Camp88, Stau79, Hirs84, F inn82] as strong signals for liquid-gas phase transitions in
excited nuclei [F ish67, Good84, Pana84]. Many studies have been performed with the

intention of providing signatures of these transitions.

One dominant characteristic of phase transitions is the divergence of conditional

charge moments[Camp88]
mk(n) = szN(z,n)/Zo (1.1)

where n is the reduced multiplicity, z is the size of the measured fragments, N (z, n)
is the number of fragments of charge 2 in events with reduced multiplicity n, and 20
is the largest possible charge (i.e. Ztarg + Zproj). One example of these moments is
shown in Figure 1.1. In this figure, the authors have plotted the second conditional
moment m2 versus the reduced multiplicity for three-dimensional bond percolation
lattices of sizes 33, 53, 93 and 503. The second-order phase transition which occurs

in percolation model calculations is clear from this figure. For the largest (effectively

 

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Figure 1.1: The second conditional moment m2 (see Equation 1.1) plotted versus the
reduced multiplicity n for three-dimensional bond percolation lattices. Four different
system sizes, Z0 = 33 (open circles), 53(closed circles), 93(crosses), and 503(triangles),
are shown[Camp88].

infinite) system, the singularity occurs at a reduced multiplicity of n = 0.25. In
smaller (finite) systems, such singularities and other signatures for phase transitions

are less prominent, and may be more difficult to extract.

Another example of a phase transition signature is that a relatively shallow power-
law charge distribution for charges ISZ§20 is expected at the transitional excitation
energy, as compared to steeper exponential distributions at energies both above and
below[Stau79]. It is possible to perform power-law fits (dP/dz cc 2") to experi-
mentally measured charge distributions and search for a minimum in the parameter
r as a signal of the phase transition[Finn82]. This procedure has been followed by
the authors of Reference [Pana84]. They have plotted the “apparent exponent” 7'
versus the extracted source temperature in Figure 1.2 for six experimental systems:
p+Ag (0.21-4.9 GeV) (circles); p+U (4.5, 5.5 GeV) (crosses); p+Xe and p+Kr (80,
350 GeV) (squares); C+Ag and C+Au (180, 360 GeV) (triangles). By calling 7' the
“apparent exponent”, the authors acknowledge that this power-law behavior of the
fragment charge distributions may also result from averaging over impact parameters
or excitation energies [Baue88, Aich88, Li93], imitating the expected signal of a liquid-
gas phase transition. This figure shows a clear minimum in the extracted 7' value,
which is indicative of a phase transition[Stau79, F inn82, Baue88]. The value at this

minimum is also consistent with those values extracted for liquid-gas systems[Stan71].

The authors of References [Elli94] and [Gilk94] have provided another indication
of phase transitions in nuclear matter. Using the equations describing the decay
of systems which display critical behavior [Camp88], they have extracted critical
exponents from experimental data collected for 1.0 GeV/ nucleon Au+C reactions.
By comparing the extracted values to those obtained from actual liquid-gas sys-
tems, the authors have provided perhaps the most persuasive arguments to date

concerning the liquid-gas -like nature of nuclear matter, although their conclusions

:5
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Figure 1.2: Extracted values of the power-law exponent 1' versus the calculated tem-
perature for six experimental systems: p+Ag (0.21-4.9 GeV) (circles); p+U (4.5, 5.5
GeV) (crosses); p+Xe and p+Kr (80, 350 GeV) (squares); C+Ag and C+Au (180,
360 GeV) (triangles)[Pana84]. The inset shows a charge distribution at the minimum
in T.

are contested in other works[Baue95]. The equations describing critical systems are

as follows:[Camp88, Stau79, Elli94]

M2 oc [cl—7 (1.2)
Zmax oc lelfi (1.3)
N(Z) oc Z‘T (for m 2 me) (1.4)
Anatfimr as)
3Scorr — 4
= —-——-—— I.
T Scorr _1 ( 6)

The exponents 7, ,8, and T can all be constrained by the relation

T=2+—3—. (LU

In the above equations, M2 and M3 are the second and third conditional moments
(see Equation 1.1), e is a measure of the distance from the critical multiplicity (Le.
e = m — me), N (Z) is the multiplicity of fragments with charge Z, Sea" is the slope
of the correlation between M3 and M2, and 7, fl and T are three critical exponents.
By plotting combinations of the conditional moments, Zmax and the multiplicity (m)
corresponding to Equations 1.2 thru 1.6 (see Figure 1.3), the authors have extracted
the values for these exponents listed in Table 1.1. The authors conclude by noting
that the values obtained from the experimental data are closest to those observed in

liquid-gas systems.

Discussions of transitions in nuclear disassembly can alternatively be phrased in
terms of the predominant reaction decay mechanism[Gros93], as transitions in reac-
tion decay mechanism have been observed on the same energy scale[Botv87, Hage92,
Cebr90]. The goal of this chapter is to use charge correlations to determine which de-
cay mode is dominant for an experimentally measured set of target-projectile-energy

combinations.

 

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5 21.0 - -
20.5 h c-
20.0 - t = 2.21 :1; 0.02 -
19-5 . . . l . - . . 1 . . . .. I . . . . l L . . . "
14.4 14.8 15.2 15.6 16.0 18.4

In ("2)
c)

Figure 1.3: Extraction of the critical exponents (a) 6 and (b) 7 from experimental
data [Gilk94] and (c) 7' from a percolation calculation [Elli94]. Ordinate and abscissa
values are the conditional moments defined in Equation 1.1.

 

 

 

Liquid-gas
Quantity Experiment Liquid-gas Percolation mean field
mc 26 :i: l
7 1.4:l:0.1 1.23 1.8 1.0
6 0.29 :l: 0.02 0.33 0.41 0.5
T 2.14 :t 0.06 2.21 2.18 2.33

 

 

Table 1.1: Critical exponents extracted from charge moments analysis of 1.0 GeV/
nucleon Au+C listed with values derived from other critical systems [Gilk94].

1 .2 Decay Mechanisms

We will evaluate the predominance of sequential binary (SB)[Barb86, Casi93, Dela86,
Siwe93, Sowi86, Piihl77, More93] and multifragmentation (MF) [More93, Barz86,
Botv87, Gros87] disassembly mechanisms in small impact parameter events for a range
of entrance channel masses and incident beam energies. The observation of a transi-
tion from SB disassembly to MF for increasing beam energies would be suggestive of
the transition from liquid to liquid-gas coexistence phases, i.e. the “cracking” transi-
tion, that has been predicted by several models [Barz86, Botv87, Gr0387, Gros93]. We
classify as sequential binary those disassembly mechanisms involving a cascade of two-
body decay steps, in which each step is independent and may involve (a)symmetric
binary fission [Barb86, Casi93, Dela86], (a)symmetric ternary fission [Siwe93, Sowi86],
or evaporation [Barb86, Piihl77]. Over the course of a purely evaporative SB decay,
the probability of emitting large fragments decreases due to the increasing importance
of the Coulomb and angular momentum barriers. The three largest charges in final
states produced in such SB decays would thus be expected to consist of one larger
fragment and two smaller ones. A binary or ternary fission step in the SB decay
cascade would result in the emission of two larger fragments and one smaller one.
Disassembly leading to three similarly sized largest fragments has been noted as a

signal for the process known as multifragmentation[More93, Barz86, Botv87, Gros87].

1.2.1 Relation to Charge Observables

Previous work has been done to investigate the effects of the decay mechanism upon
the charges of the resulting fragments. In one study[Kreu93], the authors com-
pared results from three different models to their data using quantitative measures of

charge symmetry to determine the sensitivity of charge-based observables to different

methods of populating phase space in nuclear decay processes. Two of the models
compared were hybrid combinations of dynamical and statistical calculations. The
Boltzmann-Uehling-Uhlenbeck model (BUU)[Bert88], a dynamic transport model,
was used to describe the first stage of the collision. The second stage of the col—
lision was described by each of two statistical models. The first was the GEMINI
model[Char88], a sequential binary decay model, and the second was the COPEN-
HAGEN model[Bond85], a simultaneous multifragmentation model. The version of
COPENHAGEN called CRACKER was used, which includes a specialized treatment
of the evaporation stage. The third model compared was a 3-dimensional bond per-

colation model[Desb87, Baue88].

The authors of that study calculated two- and three- body asymmetries for the
largest three fragments on an event-by-event basis. The two-body asymmetry is
defined as (Zmam — Z2)/(Zma:z: + Z2) and measures the asymmetry of the first and

second largest charges. The three-body asymmetry is defined as

 

a. = 7(me — <2»? + (z. — <2»? + (z. — <2>)2/\/6<Z>

where (Z) = %(Zma:c + Z2 + Z3), and measures the asymmetry of the combination
of the largest three charges. The mean values of these asymmetries are plotted in
Figures 1.4(a) and 1.4(b) versus a centrality variable Zbound. Zbound is defined as the
sum of the charges of all bound fragments (Z _>_ 2). With this definition, central

(peripheral) collisions should result in small (large) values of Zbound.

The data shown in this figure were collected for 600 MeV/nucleon collisions of
Au on C (circles), Al (triangles), Cu (squares) and Pb (stars). The mean asymmetry
values found for these systems is similar for all values of Zbound. Superimposed over
this figure are the resulting mean values extracted from the sequential binary decay

model GEMINI (dotted lines), the simultaneous multifragmentation model COPEN-

 

 

   

 

 

 

 

 

 

A ,0, I /
S o 0-8 ' '.
t v : '
° E
E, 0.6 k
S _
(\i Z
'7’ 0.4 ~
x _
o .
E
v 0.2
V
o -' A A A A A A A A A A A
0 20 4O 50
Z bound Z bound

(2%) (b)

Figure 1.4: (a) 2-Body and (b) 3—Body asymmetry (defined in the text) plotted
versus the centrality variable memd[Kreu93]. Markers are mean values extracted for
600 MeV/nucleon collisions of Au on C (circles), Al (triangles), Cu (squares) and Pb
(stars). The dotted, dashed and solid lines represent values extracted from GEMINI,
COPENHAGEN, and percolation calculations, respectively.

10

HAGEN (dashed lines) and the percolation model (solid line). While the percolation
model has good agreement with the data for all Zbound, for mid-peripheral events the
GEMINI values are considerably higher than the data and the COPENHAGEN val-
ues considerably lower. The authors point out that some input parameters to these
models could have been adjusted to improve the overall agreement with the data,
and even that the point at which the BUU calculation was stopped (determining the
input configuration for the statistical models) could have been changed. However, the
authors’ goal was not to show agreement or disagreement between their data and the
models, but to explore the sensitivity of their experimental observables to the fun-
damentally different methods of populating phase space in the three models. These
figures clearly illustrate the quantitative differences in charge symmetry which one
may expect to observe due to the predominance of fundamentally different reaction

decay mechanisms.

Another study also illustrates similar sensitivities of charge observables to the de-
cay mechanism. The authors of this study[Hage92] also perform calculations using
hybrid models. The first stage of these models was described by BUU calculations,
which provided input parameters to each of three statistical models. The statisti-
cal models were: a simultaneous multifragmentation model by Sa and Gross[Sa85,
Zhan87a, Zhan87b], the GEMINI sequential decay model[Char88], and another se-

quential decay model by Richert and Wagner[Rich90].

The experimental data used in the comparison were collected for collisions of 35
MeV/nucleon 40Ca+4°Ca. The charge observables examined here were the largest
detected charge (Zmax) and the normalized second moment of the charge distribution

with the largest charge excluded:

2.3mm...“ ZiM(Zi)

 

11

 

 

 

 

 

 

 

 

EXI'DERINiENI ' I -- siciier't e'md' viag'nf'r «
3.0 — -~® .1, c -
Ag X7: ‘F \ W -.
u 2.0 - q‘ -- \w—"fm ‘
.5 ' ‘" '
1.0 - -— -

o.o...:.::::+::4=:
- Sa and Gross «- GEMIN ~
3 0 JG?“ bi __ d) d
“E ' F ‘—C%:%__ ._ Lyw—‘b ‘7‘? .
0' gr __ \MM’ _

t.“ 2.0 - l,_\\_:;_%/

5 _ _ _
1.0 — -_ '

0.0 l 1 A l I. I A l A I A I A I A I

0.0 0.5 1.0 1.5 2.0 05 1.0 1.5 2.0

LniSzi lniSzi

Figure 1.5: Logarithm of the largest charge versus logarithm of the normalized second
charge moment (defined in the text) for (a) 35 MeV/nucleon 40Ca+4°Ca data, (b) a
multifragmentation model by Sa and Gross, (c) a sequential decay model by Richert
and Wagner, ((1) the GEMINI sequential decay model[Hage92].

where M (Z,) is the multiplicity of fragments with charge equal to Z,». Figure 1.5
shows contour plots of ln(Zmu) versus ln(.5'2) for the experimental data (panel a),
the multifragmentation model by Sa and Gross (panel b), and the sequential decay
models by Richert and Wagner (panel c) and GEMINI (panel (1). Two predominant
peaks appear in the plot for the experimental data. The authors of that work attribute
the population of the first peak (at ln(5'2) z 0.5) to the sequential decay mechanism,
and the population of the second peak (at 111(52) z 1.2) to the multifragmentation

mechanism. By comparing the relative strengths of these two peaks to those yielded

12

from the model calculations, it is clear that the pure multifragmentation model over-
populates the second peak, while the sequential decay models over-populate the first
peak. The authors continue by showing that sequential codes modified to include
expansion[Frie88, Frie90] are better able to reproduce the admixture of charge sym-

metry with charge asymmetry.

For the purposes of this thesis, we concentrate on the commonality between these
two studies, which is the establishment of the fact that purely sequential decay mech-
anisms generally yield asymmetrically charged reaction products while purely mul—
tifragmentation models yield symmetrically charged fragments. Since previous work
has shown good agreement between experimental data and sequential model calcu-
lations for low energy nuclear collisions[Bada93, Barb86], and similar agreement be-
tween experimental data and simultaneous multifragmentation model calculations at
higher energy collisions[Botv87, Hage92, Cebr90], it is reasonable to expect a transi-
tion in the predominant reaction decay mechanism at some intermediate beam energy.
The two studies presented above have further shown a correspondence between the
specific behavior of fragment charge observables and the predominant reaction decay

mechanism.

1.3 Non-Compact Geometries

Recent theoretical calculations based on transport theories have predicted the occur-
rence of exotic break-up geometries at intermediate energies[More92, Baue92, Gr0392,
Xu93, Souz93, Guar95]. These geometries include bubbles and toroids, which are con-
sidered non-compact because of their hollow centers, as well as disks. In Figure 1.6

is shown a surface of constant density (p = 0.3po) resulting from Boltzmann-Uehling-

Uehlenbeck (BUU)[Bert88] model calculations[Baue92]. The system was 93Nb+93Nb

13

at 60 MeV / nucleon incident energy, and this Figure depicts the density configuration
for a single time at 160 fm/c. The toroidal ring in the density profile is quite clear in

this figure.

Figure 1.7 shows another surface of constant density (p = 04%) resulting from
Boltzmann-Langevin (BL)[Ayik90] model calculations[Guar95]. The system was pre-
pared with 200 nucleons at a temperature of 3 MeV. This Figure illustrates the
time-progression of the density configuration at times 0, 40, 80 and 120 fm/c. The
emergence of a bubble occurs in these calculations roughly at 40 fm/c, after which

point the shell decays into 5 large proto-fragments.

Figure 1.8 shows a time progression of a nuclear collision, illustrated by the
density of points, resulting from Boltzmann-Nordheim-Vlasov (BNV)[Bona92] model
calculations[More92]. The system was 9°Mo+9°Mo at 75 MeV/nucleon incident en-
ergy. Rows (a) thru ((1) show density profiles for the times 20, 60, 120 and 180 fm/c,
respectively, and the two column groups show the calculation for two different nuclear
matter compressibilities (K). The profile of a disk can be seen in row (c) for the high
compressibility case, while a toroid can be seen in row ((1) for the low compressibility

case. However, even these distinctions appear somewhat subjective in this figure.

What is common to each of these results is that there does not appear a simple
ellipsoidal structure expanding outward from the origin. There appears in each case
some exotic configuration. In this discussion, we now make one important distinction
among these three examples. While the first two, the toroid and the bubble, have a
hollow center, the disk does not. This quality is the origin of the term “non-compact”
geometry, which we hereby qualitatively define to mean any position-space configu-
ration of nuclear matter with a hollow center. This includes bubbles and toroids but

excludes disks. While experimental predictions and discussions regarding such geome-

14

 

 

i0 '10

Figure 1.6: Surface of constant density (p = 03,00) resulting from BUU calculations
of 60 MeV/ nucleon 93Nb+93Nb reactions at t2160 fm/c [Baue92].

 

 

 

 

 

 

 

Figure 1.7: Surface of constant density (p = 0.4po) resulting from BL calculations of
a 3 MeV system of 200 nucleons[Guar95]. The four panels show the reaction at times
0, 40, 80 and 120 fm/c, as labelled.

K = 200 MeV K = 540 MeV

front view side view front view side view

 

 

   

 

 

 

 

 

 

 

 

 

 

 

Figure 1.8: Density of test-particles in a BNV lattice resulting from calculations of
75 MeV/nucleon 9°Mo+9°Mo reactions[More92]. The two column groups show the
calculation for two different nuclear matter compressibilities, and rows (a)-(d) show
the collision at times 20, 60, 120 and 180 fm/c, respectively.

17

tries flourish, experimental searches to date have not been able to confirm predictions
regarding the existence of non-compact geometries[Gla393, Phai93, More96]. One
group, however, has shown an agreement between their data, for one system at one
beam energy, and emission from a toroidal configuration as opposed to a single-source

ellipsoidal configuration[Dura96].

It is a goal of this thesis (see Chapter 4) to provide experimental evidence which
either confirms or refutes claims regarding the decay of nuclear matter from a non-
compact geometries. In order to do this, we rely upon the predictions of theorists
who have examined model calculations and have provided both qualitative and quan-
titative descriptions of observables sensitive to the formation/ decay of these exotic
geometries. These observables, termed signatures, provide criteria suitable for a con-
firmation of the appearance (and disappearance) of non-compact geometries in the
early stages of nuclear interactions. We have performed a study of central collisions
in the 86Kr+93Nb system at incident energies ranging from 35 to 95 MeV/ nucleon.
The results we present comprise the first systematic experimental results which both
provide evidence for the decay of nuclear matter from a toroidal geometry and confine

this occurrence to a finite range within the measured beam energies for this system.

One author has commented on the role of expansion, specifically a stall in the

expansion of the combined system, in the formation of bubble and toroid structures.

“When a compressed sphere expands radially, a rarefaction wave prop-
agates from outside towards the center, leaving matter at normal den—
sity streaming outward. When the wave reaches the center, the outward
streaming matter depletes the center, sending the density to a low value.
At low energies the system slows down and recompresses the hole, but at

higher energies, ..., the bubble grows.” [Baue92]

18

As an illustrative example of this, we have run BUU model calculations for the
86Kr+93Nb system and have calculated a mean radial distance from the center of the

position-space lattice for the ensemble averaged nuclear density:

2:1,“: P(Z,j, k)T(i,j, k)

R =
< I) Zigflc p(2,j,k)

 

(1.8)

In Figure 1.9, we plot this mean radial distance versus time for 7 beam energies.
From this observable, it appears that nuclear matter streams outward for all energies,
unchecked. In fact, this is the case for very light fragments, many of which are
emitted as a blast of pre-equilibrium light-charged-particles. If, however, we modify
the observable slightly so as to focus on higher density regions (by analogy, larger
fragments), the picture is quite different. We thus define a quadratic mean radial

distance
= Zi,j,k p2(iaja k)7'(i,j, k)
Zi,j,kp2(i$j1 k)

and plot this versus time for the same beam energies (Figure 1.10). In this case,

 

(Re) (19)

we can identify the expansion and subsequent collapse at low energies, stalling at
intermediate energies, and continued growth at higher energies mentioned in the above
quote. Furthermore, the stall in the expansion does not persist even 5 MeV/ nucleon

away from 60 MeV / nucleon.

The above quote, supplemented by the preceeding example, implies an energy
dependance of this phenomenon, as do other authors[More92]. The sensitive depen-
dence upon initial conditions (e.g. initial compression) coupled with a basic depen-
dence upon parameters such as system size and the nuclear equation of state, lead us
to classify the formation of non-compact geometries as a transitional phenomenon.

The results of Chapter 4 will also confirm the transitional nature of this occurrence.

The precise process by which non—compact geometries, whether toroids or bubbles,

are formed and decay is a matter of great interest, but is outside the scope of this

19

 

     

60 _
: — so AMeV
55 _- ~~~~~~ 70 AMeV
50 :_ — — — 65 AMeV
: — 60 AMeV
45 _— ———- 55AMeV
40 50 AMeV

   
 

—— 35 AMeV
35

(R1) (1'!!!)

30

llriIIlllllllllll

25
20

15”

 

 

10 11lllliJlllllllllIlilLLllllmllllllllllmimil
0 20 40 60 80 100 120 140 160 180 200

 

t (fm/c)

Figure 1.9: Mean distance from the CM—frame origin (Equation 1.8) in the BUU po—
sition lattice versus time for the 86Kr+93Nb system at 7 representative beam energies,

as labelled.

20

 

 

 

 

 

 

35 _
i — 80AMeV
30 _ -'-'--- 65AMeV
I —— 60AMeV
27-5 _- ———- 55AMeV _-
g : — 35AMeV //"
i} 22.5 _— ”/./"
20 _ "__—
17.5 :— “
15 :—
12.5 }
10:111l111l111l1111111l111[111llimliiiliii
0 20 40 60 80 100 120 140 160 180 200

t (fm/c)

Figure 1.10: Same as Figure 1.9, except using quadratic normalization (Equation

1.9).

21

thesis. In Chapter 4, we will focus on experimental methods for determining whether
or not these geometries are formed. Using two charge-based signatures, intermediate
mass fragment multiplicities and power-law exponents of ordered charge distributions,
we will establish that the system decays from a non-compact break-up geometry. With
the use of event shape observables, we will then show that the decaying system is more
coplanar in shape, thus implying that the geometry is toroidal. We find that the

toroidal geometries are produced for beam energies between 60 and 75 MeV / nucleon.

Chapter 2

Experimental

2.1 The 47r Array

The 47r Array[West85, Cebr91, West93] is a close-packed detector array that has been
operational for almost 10 years, and has yielded terabytes of data during that time.
In shape, it is a truncated icosahedron, with 20 hexagonal and 12 pentagonal exterior
faces, as shown in Figure 2.1. The 47r Array is an evolving system, so that it is
necessary to delineate the components that were used to collect the data for this
thesis. We shall describe the configuration first for the charge correlation analysis
(Chapter 3), which was performed on an older data set, and then list the differences
in the detector configuration used to collect the data for the non-compact geometries

analysis (Chapter 4).

At the time of the charge correlation analysis, each of the hexagonal and pentago-
nal faces on the main ball, which excludes the entrance and exit pentagons, supported
a module composed of several independent, layered detectors. One such hexagonal
module is shown in Figure 2.2. Each hexagonal (pentagonal) ball module consists
of 6 (5) truncated triangular phoswich detectors[West85] made from fast and slow
scintillating plastic. These phoswich towers are preceeded by a Bragg Curve Counter

(BCC), a single gas volume with a cathode window at the front and a conductive

22

23

 

 

 

Figure 2.1: Mechanical drawing of the frame of the main 471' ball, revealing the ar-
rangement of the hexagonal and pentagonal faces.

24

FAST/SLOW PLASTIC SC I NT I LLATORS

  
 

LOH PRESSURE
MWPC

 

 

 

 

 

 

 

/§//

\ \ BRAGG CURVE
SPECTROMETER
“‘1

Figure 2.2: Side view of a hexagonal 47r ball module, revealing three independent
layers of plastic and gas detectors.

 

\

 

 

 

 

 

 

L.

25

coating which serves as an anode at the back; the anodes are evaporated onto the
front of the thin fast plastic. The five forward hexagonal ball modules which surround
the forward array pentagon have segmented anodes, providing a greater granularity in
those modules while still using a single gas volume. Although low-pressure multi-wire
proportional counters (MWPCs) are also shown in this figure, these were not instru-
mented at the time these data were collected. At a pressure of 125 Torr of C2F6, this
BCC/Phoswich detector combination has a lower energy threshold of approximately

3 MeV/ nucleon for 7Li fragments.

The forward array consisted of 15 square and 30 cylindrical elements, a total of
45 fast and slow plastic phoswiches distributed evenly across the forward pentagon,
covering roughly 5° < 0 < 18°. (see Figure 2.3) Detectors in this array had a lower
energy threshold of approximately 14 MeV/nucleon for 7Li fragments. It was with
this forward array and the phoswiches and BCCs of the main ball that the charge
correlations data for Chapter 3 were taken. It is clear from this figure that there is
uncovered solid angle in the forward direction (0° < 9 < 18°), a significant portion
of the center-of-momentum frame solid angle. The improvements in detector cover-
age which took place between the charge correlation analysis and the non-compact
geometries analysis all took place within this forward pentagon, in order to improve

the forward coverage.

The first major upgrade was the addition of the High Rate Array [Pak92, Pak93,
Pak96], shown in Figure 2.4. It consisted of 45 close-packed fast and slow plastic
phoswich telescopes which covered the lab polar angles 3° < 0 < 18°, thus replacing
the forward array shown in in Figure 2.3. It was built at the NSCL, where the
project was led by Robert Pak. Detectors in this array had a lower energy threshold

of approximately 14 MeV / nucleon for 7Li fragments.

26

   
   
     
     
 
   
        

®@@
©9¢0
QQ @
seas.
@@ Beam ><Pipe 0%

  

159

   

 

Figure 2.3: Arrangement of the forward array phoswiches at the time of data collection
for the charge correlations analysis.

27

 

Figure 2.4: The High Rate Array, which consists of 45 close-packed phoswich tele-
scopes.

28

 

Figure 2.5: The Maryland Forward Array, with the outline of the High Rate Array
to show positioning.

The second upgrade was the addition of the Maryland Forward Array[Llop92],
shown in Figure 2.5. This array consisted of 16 fast and slow plastic phoswiches,
preceeded by a multi-segmented annular silicon detector. It covered the angles 1.5° <
0 < 3°, and was built by Bicron Corporation at the request of collaborators in the
University of Maryland Department of Chemistry. The Maryland Forward Array
is positioned inside the High Rate Array, as indicated in Figure 2.5. Detectors in
this array had a lower energy threshold of approximately 10 MeV/nucleon for 7Li

fragments.

29

2.2 The Zero Degree Detector

Following these upgrades, another detector now called the Zero Degree Detector
(ZDD) was added[Ston93, Ston94]. Such a detector was proposed both to further
improve forward solid angle coverage and to enhance existing methods of experi-
mental determination of the impact parameter of detected events. Simulations were
performed in which the impact parameter was related to observable quantities derived

from fragments detected within the remaining forward solid angle (0° < 0 < 1.5°).

2.2.1 Design

Once the need for the ZDD was established, we performed a series of calculations in
order to optimize the exact configuration of the detector. Although typical rates for
event-by-event fragment population in this detector can be obtained from detailed
simulations, the greatest rate limitations arise from the scattering of non-interacting
beam particles into the detector; such scattering rates can be calculated directly from
the Rutherford scattering equation. One other factor contributing to this direct-
beam rate is the divergence of the beam itself, measured in terms of beam emittance.
The beam emittance is the product of the size of the beam spot and the resulting
angular divergence of the beam, with an additional constant factor of 1r. It has been
empirically measured to be as high as 57r (1071') mm-mr in the x (y) direction. Given
a 1 mm beam spot in the 47r experimental vault (worst case), the divergence of the
beam was estimated to be 10 mr (0.6° ). Based on the typical beam currents used
in previous 47r experiments, divergence of beam particles, and the typical fragment
particle population in this detector, the total count rate was projected to reach as
high as one million particles per second, with eight-fold segmentation and a small hole

in the center, to minimize hits from non-interacting beam particles. With such high

30

rates projected, a concerted effort was mandated to facilitate high speed processing
of the signals emitted from both the AE and E scintillating plastic layers. This effort

resulted in the following detector and signal processing features:

1. Fast scintillating plastic for both AE and E layers
2. Separated read-out of both layers

3. Adiabatic light-guides for the AE readout

4. High-Speed Photo-Multiplier Tubes (PMTs)

5. Negative High Voltage bias for the PMT

6. Specialized high-power voltage dividers to drive the PMTs

First, the ZDD could not be made from fast and slow scintillating plastic, as is
the majority of the 47r array. Typical slow plastic light emission times are on the
order of hundreds of nanoseconds. Both the AE and E layers were made of equally
fast scintillating plastic (NE 104, r = 1.9 nS) to avoid the overlap of light signals
from separate events. Second, once the choice is made to use AE and E layers with
equally fast plastic, the signals can not be resolved if they are read out through the
same PMT, as is done with a phoswich. Thus each layer was given a dedicated PMT,
resulting in 16 PMTs for the eight-fold segmented array. Third, the light emitted
from the AE plastic must be collected from its edge, requiring adiabatic light-guides
to couple the thin plastic to the PMTs. Fourth, fast PMTs (XP226ZB) were chosen
for optimal signal transit time (31 nS) and minimal time spread (3.5 nS). Fifth, the
PMTs were operated in the negative HV bias mode to avoid the capacitive coupling
inherent in positive HV bias designs. And finally, we designed, prototyped, and

constructed specialized high—voltage dividers for these PMTs to allow processing of

31

near 1 volt output signals at a rate of up to 500 kHz. The design goal of lMHz
processing was deemed infeasible given the findings of the prototype testing. The
final circuit designs for both the high voltage dividers and the PMT sockets used are
shown in Figure 2.6. Corresponding lists of components are tabulated in Appendix

A.

We then chose the optimal thickness of the AE and E layers of the detector. Of
primary consideration were the transmission of heavy fragments, signal-to-noise ratio
in the AE layer, and the stopping power of the E layer. We have calculated the
fractional energy loss in a 0.67 mm thick scintillating plastic layer and plotted that
versus the incident energy per nucleon for all fragments with 2 3 Z S 27, the charges
successfully resolved in experimental conditions, in Figure 2.7. From this figure, the
“punch-in” energies, those at which the particle is no longer stopped in the AE layer,
can be clearly seen. At this thickness, all fragments of interest having an incident
energy of roughly 25 MeV/nucleon or more are transmitted to the E layer, and can
be identified via AE vs. E techniques. The actual lower energy thresholds are listed
in table 2.1. Similar calculations were performed which showed that a 30.0 cm plastic

layer would stop all particles with incident energies (KE/ A) < 220 MeV.

The final assembly design of the ZDD is shown in Figure 2.8. From this figure, all
of the relevant mechanical design features can be seen: the eight-fold segmentation,
the small hole in the center, the separated readout of AE and E layers, and the

light-guides used to collect the light from the thin AE layers.

A dedicated chamber was also designed for the ZDD, since there was insufficient
space for an additional detector in the existing 47r ball. This chamber was given the
added feature that the ZDD could be moved inside the chamber without breaking

vacuum, through the use of a goniometer. This was deemed necessary due to the

(a)

 

 

 

 

 

 

 

 

 

 

 

o
1

 

 

 

 

z< is
251 313 ' at
09/
I:\ ,
.1 . .
1/ {F a:
:\ 9
2< .1
léi 3% : a
K a
1% Bil : a-
2< 3
351 i: 12..
K i
8 .
gig a; a i
‘- b‘
2., a"
5':\
a
E
a
E
a
2.1
E\
3?: 3
352 5 3==—
.25 s
31/ I
I\ '

Figure 2.6: ZDD High Voltage Circuits

32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r:\ ”A \9
8 11/ "r I I:
:11\ J3. H \9
a 11/ "fl 3 II:
YH—gL—a”
a a: s':
l
s 3 H
a e. A:
1"—t>
a a ‘3 ‘ >2
5 s H 2
a E
3
n t =
11 11> 39' 3 )5:
=3\ 1%.. H \5
8 B/ "' i ll:
. {ll—o“
21> 3.... >1

 

(b)
. (a) HV divider card. (b) PMT socket card.

33

 

” Stopped in AE
l— m.

     
 

0.8 —

 

0.6 —

AE/Eincirlent

Transmitted
0.4 -

0.2 -

 

 

 

Eincident (AM‘N)

Figure 2.7: The fractional energy loss of fragments (2 3 Z S 27) in 0.67 mm scintil-
lating plastic, revealing the “punch-in” energies.

 

35

 

 

 

 

 

 

 

 

Z Punchin energy Z Punchin energy Z Punchin energy

(MeV) / (AMeV) (MeV) / (AMeV) (MeV) / (AMeV)
1 (p) 7.2 / 7.2 13 504 / 18.7 25 1375 / 25.0
2 29.5 / 7.4 14 555 / 19.8 26 1450 / 25.9
3 57.5 / 8.2 15 625 / 20.2 27 1535 / 26.0
4 90 / 10.0 16 685 / 21.4 28 1590 / 27.4
5 119 / 11.8 17 756 / 21.6 29 1695 / 26.9
6 158 / 13.2 18 845 / 21.1 30 1770 / 27.6
7 201 / 14.4 19 895 / 22.9 31 1880 / 27.2
8 248 / 15.5 20 952 / 23.8 32 1995 / 27.0
9 304 / 16.0 21 1055 / 23.4 33 2075 / 27.7
10 347 / 17.3 22 1135 / 23.6 34 2190 / 27.4
11 396 / 17.2 23 1220 / 23.9 35 2250 / 28.5
12 440 / 18.3 24 1290 / 24.8 36 2360 / 28.1

 

 

Table 2.1: Punchin energies for the ZDD, based on 0.67mm thick scintillating plastic.

potential misalignment of the incident beam, which could strike an individual detec-
tor with a rate surpassing its maximum count rate. The mechanical drawing of this
chamber is shown in Figure 2.9. By placing this chamber sufficiently far from the
target, time—of—flight (TOF) measurements could also be made, allowing the identi-
fication of particles which stop in the AE layer. Given the space constraints of the
47r experimental vault, the maximum separation between the ZDD and the target
was 2.74 m. We performed additional TOF calculations for particles with a range
of incident energies (5 - 100 MeV/ nucleon) and plotted the energy deposited in the
AE layer versus the time of flight for all particles 2 3 Z s 27 in Figure 2.10. From
this figure, we extract the difference in time-of-flight for particles that leave the same
energy in the AE plastic, between 1 and 3 115. This small time difference provided
further incentive to choose high-speed plastics and PMTs. A calibration of the TOF
response of the ZDD is not included in the data shown in the chapters which follow.

However, this feature is available for future use.

36

 

 

 

 

 

 

 

 

Figure 2.9: Mechanical drawing of the ZDD chamber. Above: Top view, with beam
entering from the top of the page. Below: Rear view, looking upstream. This also
shows the goniometer, used for positioning of the detector.

37

 

1600

: Transmitted Stopped
1200

I I I I I I T I I I I I I I I I I I I I I I '

 

 

 

 

TOF (nS)

Figure 2.10: Energy deposited in the AE plastic versus time—of—flight for fragments
(Z=2-27) in 0.67 mm scintillating plastic, for a 2.74 m flight path.

38

 

Figure 2.11: Frontal photograph of the Zero Degree Detector.

2.2.2 Construction

The Zero Degree Detector was built at the NSCL, where the project was led by
Nathan Stone. It was built according to the specifications described in Section 2.2.1.
Construction was completed in November 1994, and the ZDD has been used in many
experiments since that time. Figure 2.11 is a photograph of the ZDD as it appeared
before the evaporation of aluminum onto the face of the E plastic layers for optical
decoupling.

After its installation into the ZDD chamber, it was precisely aligned to the cham-

39

 

Detector 45 Detector <15
1 338.3 5 158.3

 

2 293.3 6 113.3
3 248.3 7 68.3
4 203.3 8 23.3

 

 

 

 

 

 

 

Table 2.2: Azimuthal angles for the ZDD. (Note: Since the ZDD chamber is separate
from the 47r ball, if the ball is rotated, these angles will change accordingly. These
angles are reported assuming a 3.7° tilt (clockwise, looking down the beampipe) of
the 47r ball.)

ber, and the relative orientations of the 47r ball and ZDD chamber were measured.
The geometric centers of each telescope were also measured. Each ZDD telescope has

its geometric center at a laboratory polar angle of 1.1° , and the azimuthal angles are

listed in Table 2.2 (relative to the conventional azimuthal origin of the 411' ball).

2.2.3 Detector Response

Following the installation of the ZDD, we performed a calibration in which we bom-
barded a 1 mg/cm2 93Nb target with a 87Kr beam with incident energies ranging from
35 to 95 MeV / nucleon. The ZDD performed well for this experiment, providing spec-
tra with charge resolution up to Z=27. We plot the measured light output spectrum
(AL versus L) in Figure 2.12. From this spectrum the qualitative response of the
detector (i.e. energy resolution, AL versus L line shape, and detected charges) can
be seen. To extract quantitative information from these AL / L spectra regarding the

actual energies of fragments, we performed a detailed energy calibration for the ZDD.

We derive the light response equation by beginning with the basic relation between

specific energy loss and light production, as reported in a study on the performance

40

: 1F“

‘

A 5‘.

.4...
.3 ..
v

-.~...=1..

L1

 

500

l

.0
..0
l

500

Figure 2.12: Raw experimental data recorded in one ZDD telescope for 45 and 65

MeV/nucleon 87Kr incident on 93N b.

41
of phoswich detector performance:[Past83]

.15 ~Sl%l

dx 1+kB %|

where S' is the absolute scintillation efficiency, It the quenching parameter, and B a

 

constant relating density of ionized and excited molecules to dE/dx. We can then
substitute the formula for specific energy loss, as reported by Birks in his text on
scintillation detectors:[Birk64]

2

dB 47re4Z2 2mev 2 2
'82.?— mm NDZD [In( I )—ln(1—fl)—fl]

 

 

where Z is the atomic number of the incident particle, and N D and Z0 are the density

and atomic number of the atoms in the detecting medium. Now, if we let 6 << 1 then

 

we can substitute v2 2 #733. We also make the following assignments:
a = 27re2NDZD mn
C = kBaAZ2 In (@713)

If we make the additional assumption that the term In (9%) is slowly varying in the

range of interest, then we can write the differential light output as

|
03

 

El“;

DJIQ

1+

This relation can be integrated directly between the incident (E0) and punch-out (E1)

energies, yielding the total light output for a given medium

 

AL = s [AE—Cln (C+E°)]

C+E1

It can be shown that the constant kB itself has an approximately inverse relation-

ship to Z, and that the logarithm which was previously assumed to be constant (or

42

slowly varying) has a slight Z and A dependance as well. So we make the following
replacement:

C = 7A°Zz
in which 7 is a constant, and a and z are the strengths of the A and Z dependencies

contained in the constant C. The final response equations between AL, L, AE, and

 

E are thus
_ Cd! + (AE + E)
AL — 5 [AE Cd: ln( Cd] + E )] (2.1)
and
E
L =s[E—C.1n (1+ —)] (2.2)
C:

where C, = 7A°Zz (:1: 2 (11,1) and AL, L, AE, and E, in these last two equations
now represent the light and energy loss in the transmission and stopping media,

respectively.
These relations can be approximated by the two following equations:[Cebr92]

AL = CdzAEed‘sz’A°d' (2.3)

and
L = GlEe‘Zz'Aa‘ (2.4)
where G; (a: = dl,l) is a simple gain factor, and ex, 23, and a1. are the strengths of

the AE (or E), Z and A dependencies, respectively.

To determine which of these two sets of functions best reproduces the actual light
response of the ZDD, we measured a set of calibration points, each of known energy
and isotopic identity, and compared the calculated light output from each of these
relations to the measured light output, adjusting the free parameters with a X2 mini-
mization routine. The calibration points were drawn from three sources: a secondary
fragmentation beam from the NSCL A1200 reaction particle spectrometer, AE en-

ergy values extracted from punchin points on the raw spectra, and 95 MeV/ nucleon

43

points also extracted from the raw spectra. The raw calibration points from the frag-
mentation beam are shown in Figure 2.13. The isotopic identity of the points were
identified by locating the 8Be vacancy, since 8B6 is particle unstable. One atomic

number (Z=17) and isotope ratio (N=Z) have been labelled, for clarity.

The AL and L values of each of the calibration points drawn from the sources
described above can be calculated from the known AE, E, Z, and A values in each
case via the response equations (Equations 2.1-2.4) derived above. A minimization
was performed for each of these equations to determine the parameters that best
match the data. Below are the final equations (with calculated parameters) which
best fit the data in each case, which compare favorably with our previous work and

standard parameterizations.

 

Cebra:
AL = 0.880AE0'519Z0'279A0'488 (2.5)
L = 0.243E1.118Zo.105A—0.596 (2.6)
Birks:
Cd: '1' (AE + E))(
AL 2 0.959 AE — C l 2.7
[ dl n ( Cdi + E ( )
where
Cdi : 6.679A0.021.444
and
E
L = 0.281 (E — 0.1.. (1 + Ell (2.8)
l
where

C; = 13.563/10'021'559

We have plotted the measured values of AL (panels a and b) and L (panels c

and (1) versus the calculated values from both the Cebra (panels a and c) and Birks

44

 

1400 ,1

1200 ’5

      
     

a
c
C
1 I f1 1'1 1 I 1
.'II'I’III~

I H I
I

3
e

 

l' I I I ll

:1

  

L

Figure 2.13: Calibration points measured in segment 2 of the ZDD. One atomic
number (Z=17) and isotope ratio (N=Z) are labelled.

45

(panels b and d) parameterizations, in Figure 2.14. Because, in the case of a perfect
fit, these lines should be straight lines with slope equal to 1.0, any deviation from
a straight line is an indication of an imperfection in the fit. Thus, one can gain a
qualitative measure of the fit from the line shape. In addition, the X2 per degree of
freedom values are written on each panel as well. From a visual inspection of the fits,
it appears that the Cebra parameterization yields a better fit to the data, and the
X2 values are less than half those found with the Birks parameterization. For this

reason, the Cebra parameterization was used in the final calibration of the detector.

46

 

   

 

 

 

 

 

 

800
i" H I
...~_ 1 ll
1 l 1‘
- (l
: 13+
200 _—
; x2/D0F=0.963 x’mor=2.009
800 1 1 l 1 1 1 l 1 1 1 l n. 1 1 1 1 l 1 1 1 l 1 1 1 l 1 1 1
_ c)
.0. E-
. 4.. : (ill 7 +11“
: 7+ +1 1
. “if H
200 L— L
. 2 _ ~ 2 _
2". . . . . AWE"??? 1’. .. .1 579.1382“?
200 400 600 800 200 400 600 800
fCebra fBirks

Figure 2.14: Response function minimization results, comparing Cebra and Birks
parameterizations. (a) AL Measured vs. Calculated from Cebra parameterization.
(b) AL Measured vs. Calculated from Birks parameterization. (c) L Measured vs.
Calculated from Cebra parameterization. (d) L Measured vs. Calculated from Birks
parameterization.

Chapter 3

Transitions in Decay Mechanism

3.1 Dalitz Triangles

In this chapter, we concentrate on quantitative information extracted from one type
of charge correlation called the charge-Dalitz plot[Kreu93, Biza93, Rous93]. To con-
struct a Dalitz plot, we use only the charges of the three largest fragments in an
event (Z1 > Z2 > Z3 > ---.) We normalize these charges by dividing by their sum
(Z: = Z,-/ 21:13 Zj). A Dalitz plot is then created by setting these three reduced
charges equal to the distances to the three sides of an equilateral triangle of unit
altitude (see Figure 3.1). This method uniquely defines one pair of coordinates for

each event, given by the following equations:

I

—Z’t (30°)+ Z"
x— i an cos(30°)

 

and

y=Z£

where the indices (i,j) are non-duplicating pairs of 1, 2, and 3 (e.g. 1,2 ; 2,3 ; 1,3 ;
2,1 ; .. ). This latter definition brings to light the fact that a Dalitz triangle in fact

contains 6 symmetrized copies of one unique triangle, outlined in Figure 3.1 with the

dashed line.

47

48

  

 

 

 

Figure 3.1: A Dalitz triangle. For each event, the reduced charges (Z) are set equal
to the distances to the three sides of the triangle. Dee,“ is the distance from each event
entry to the center, and D6496 (= Z3) is the distance to the nearest edge. Further
features are explained in the text.

49

As noted, this technique has been used in many other analyses to illustrate qual-
itative trends in experimental data[Rou893, Kreu93], but it has been used to extract
quantitative information as well. The authors of Reference [Biza93] have classified the
relative abundance of three decay mechanisms in their data by recording the relative
population of events in different regions of the Dalitz triangle. The experimental sys-
tem they studied was Kr+Au at 43 MeV/ nucleon. The authors of that study divided
the observed range of excitation energies into four bins, and generated a Dalitz plot

for each corresponding bin (Figure 3.2).

By construction, the region in the middle of each side of the triangle (Figure 3.1,
unshaded) is populated by events which have, as their three largest fragments, two
relatively large fragments and one relatively small fragment. The region at each of
the corners (Figure 3.1, hatched) is populated by events with one large fragment
and two smaller fragments. And the region in the center of the triangle (Figure 3.1,
shaded) is populated by events with three equally sized fragments. Since the triangle
is filled according to the reduced charges, as opposed to the actual magnitudes, the
total charge of the three largest fragments (Zwm = 21:1,3 Z.) is a free parameter
(i.e. events having three largest charges equal to 2:2:2 would fall in the center of the
Dalitz plot, as well as those with charges 20:20:20). We shall examine this degree of
freedom in more detail later in this chapter. It is for this reason that the authors

of Reference [Biza93] have included the mean total charge of events plotted in each
panel ((Zfragl)-

One final detail we note is that the authors have drawn around the data in each
panel the limiting edges of the Dalitz triangle. The outer lines drawn depict the
boundaries of the Dalitz triangle itself. The inner lines, however, show the maximum
extent possible for their experimental data set. This boundary arises from the fact

that their detector apparatus had an inherent efficiency problem limiting the size

50

 

8'<2.5 MeV/u 2.5(5'<3.5 MeV/u
<z.,.,>275 <z.,_>=72

 

 

3.5(5'<4.5 MeV/u £'>4.5 MeV/u
<z,,.,>=68 <z...,><64

     
  

      
 

- [30:05:05]
| (Shoooonofll I :\
flag ”I 1Dr—T10D9 n no ,

 

 

 

 

\

Figure 3.2: Charge Dalitz plots created for the reaction 43 MeV/nucleon Kr+Au
[Biza93]. The four panels represent four different excitation energy bins, as labelled.
Details of the figure are discussed in the text.

 

51

of detectable charges to Z 2 8. Because the distance from the edges of the Dalitz

triangle are based on the relative charges, the nearest distance to any edge is therefore

I
mtn

= 8 / (Ztarg + Zproj). We shall examine this boundary in more detail later in this

chapter.

An examination of the experimental data in Figure 3.2 reveals that, for low exci-
tation energies in this system (6* < 2.5 MeV/nucleon) the regions at the centers of
the sides are predominantly populated. The authors ascribe the occurrence of such
events to the predominance of symmetric fission at these excitation energies, in which
two massive fragments are emitted in conjunction with an IMF. At high excitation
energies (6* > 4.5 MeV/ nucleon) the central region is predominantly populated. The
authors attribute this to the predominance of symmetric ternary fragmentation. They
then proceed to quantify these trends by counting the percentage of events which fall
into the three regions of the Dalitz triangle (shaded, unshaded, and hatched regions
in Figure 3.1). The relative probabilities for populating these three regions has been
plotted in Figure 3.3 versus the excitation energy. This figure shows the decline
in fission processes and subsequent rise in symmetric fragmentation processes with
increasing beam energy. The authors proceed by locating the energy at which the
probability for symmetric fragmentation becomes greater than that for fission and
labelling that as point at which a transition occurs between those two decay mecha-

nisms.

In this chapter, we choose to quantify the predominant population in Dalitz tri-
angles via two distances Dcent and Dedgc. These quantities are the distance from the
coordinates of each event entry to the center of the Dalitz triangle, and the distance
from the coordinates of each entry to the nearest edge, as labelled in Figure 3.1.
Events with three or more nearly equally sized largest fragments will populate the

center of the triangle (Dcent<Dedge), while those with one large fragment and two

52

0.9 O Fission
I Residues
A ng. Frag

9.0.0
mum

Probability
O O
'4: m

0.3
0.2

0.1

         

I‘

O

   

o . 2 :1 4 5
Excitation Energy [Hell/u)

Figure 3.3: The relative probabilities of populating the three regions in a Dalitz
triangle for 43 MeV/nucleon Kr+Au collisions[Biza93]. Each of the three lines has
been labelled according to the reaction decay mechanism leading the population of
that region in the triangle.

53

smaller, or two large fragments and one smaller, will populate the corners or sides
of the triangle, respectively (Dccnt>Dcdge). In this way we can calculate an average
value for this quantity and determine whether the measured events have an overall
charge symmetry or asymmetry. Based on the above studies, we make the associa-
tion of (Dcent)<(Dedge) with multifragmentation events, and of (Dcent)>(Dedge) with

sequential binary or symmetric fission events[Kreu93, C01092].

3.2 Experimental Measurements

The reactions studied in this chapter were 20Ne-l-27Al at beam energies of 55, 75, 95,
105, 115, 125, 135, and 140 MeV/nucleon; 40Ar+45Sc at 15, 25, 35, 45, 65, 75, 85,
95, 105, and 115 MeV/nucleon; 87Kr+93Nb at 35, 45, 55, 65, and 75 MeV/nucleon;
and 129Xe+139La at 25, 30, 35, 40, 45, 50, 55, and 60 MeV/nucleon. The beams
were obtained from the K1200 cyclotron at the National Superconducting Cyclotron
Laboratory, and all of the data were collected using the MSU 47r Array (see Sec.
2.1) with a minimum bias trigger requiring two hits in any detectors of the Array.
The fragment charges were well resolved over the range ISZSI5 in the 20Ne+27Al
and 87Kr+93Nb entrance channels, and ISZ318 in the 40Ar+45Sc and 129Xe+139La
entrance channels. The distortions to the present results that are caused by the
inefficiencies in experimental measurement were studied using events generated by
model codes and a software replica of the detection system. These are discussed
as appropriate below. The total mass and excitation energy of the excited systems
formed for each reaction are constrained via the selection of central collisions, which
is discussed in Appendix B. A method for avoiding autocorrelations in this selection

process is also discussed there.

Sample Dalitz plots are shown for the 129Xe+139La system for all beam energies in

54

Figure 3.4. From this figure, we can identify a qualitative transition with increasing
beam energy. At higher energies (Eproj >40 MeV/ nucleon), population of the trian—
gle is confined mainly to the center, and events having coordinates near the edges
of the triangle are rare; this indicates MF decay. At low energies (25 MeV/ nucleon
S Eproj£35 MeV/nucleon), the Dalitz triangle is populated even at the edges. Fur-
thermore, the events populating the center of the triangle lie mainly along the lines
joining the center of the triangle to the corners (the counts axis is logarithmic). Since
we have attributed population of the corners of Dalitz triangles with sequential bi-
nary decay, our data indicate SB decay at these low energies. This is in contrast
to what we see 6.9. in Figure 3.2, in which the authors examined an asymmetric
entrance channel (Kr+Au). In each of the two cases (Figure 3.4 and Figure 3.2),
multifragment decay is indicated at high energies. However, we have stated that the
symmetric (asymmetric) system indicates sequential binary decay (symmetric fission)

at low energies.

3.2.1 Mean Distances: Dcent and D6619.3

In order to become more quantitative in our description, we will now focus on the
distance observables (Deent and Dedge) defined earlier. The average values of Dcent
(with points) and Dedge (without points) in the selected central events are plotted
in Figure 3.5 for all reactions. The solid lines depict these average distances for the
events selected by the two-dimensional small impact parameter cut on E1 and Zmr,
while the dashed (dot-dashed) lines correspond to the events passing one-dimensional
cuts on Et(Zm,.) alone. For the lowest beam energies in both the 40Ar+45Sc and
129Xe+139La systems, (Deent)>(Dcdgc) in the central events. This implies an over-

all asymmetry in the charges of the largest three fragments in these events, and is

thus consistent with a SB disassembly mechanism. These events are predominantly

55

 

25 30 35

 

 

 

 

 

 

 

 

Figure 3.4: Dalitz plots for central collisions of the the 129Xe+139La system. Beam
energies are labelled in each frame. The color contours depict the logarithm of the
counts in the histogram.

56

 

 

 

 

 

 

 

 

 

 

0.3:-
o.2:- : """" w
0"; Ne+Al
0.3:-
0.2:-

é 0'1? Ar+Sc
0.3:-
0.1%- Kr+Nb
0.3:-
0.2:- .
.15- '
0 1 Xe+La

 

20 40 60 80 100 120 140

Eproj (MeV/nucleon)

Figure 3.5: The average values of Dam, (with points) and Dedge (without points) in the
central events versus the projectile energy. The solid lines depict the results for events
selected by the two-dimensional cut on E1 and Zmr, while the dashed (dot-dashed)
lines correspond to events selected by a one-dimensional cut on Et(Zm,).

57

near the corners of the charge Dalitz triangles, implying that asymmetric fission or
sequential evaporation decays are more common than those involving a symmetric
binary or ternary fission. For higher beam energies in these entrance channels, as
well as for all of the beam energies in the 20Ne-i-T’Al and 87Kr+93Nb entrance chan-
nels, (Dcent)<(Dedge). The largest three fragments in the final state are thus similarly

sized, implying multifragmentation.

A beam energy that is transitional between SB and MF disassembly is defined
at the crossing of the lines interpolated from the points shown in Figure 3.5. The
transitional beam energies extracted from this figure for the central 40Ar+45Sc and
129Xe+139La reactions are ~50 and ~33 MeV/ nucleon, respectively. The results for
the central 87Kr+93Nb reactions are similar to those for the central 129Xe+139La
reactions, implying a transitional beam energy somewhere in the range of 30-35
MeV / nucleon for this system. The transitional beam energies are 3-5 MeV/ nucleon
higher in the samples of events from the two one-dimensional cuts (dashed and dot-
dashed lines) than those obtained following the two-dimensional cut (solid lines). The
values of (Dow) and (Dedge) appear to reach asymptotic values within a few tens of

MeV / nucleon above the transitional beam energies.

3.2.2 Constraints

There are hidden constraints on the present method. For example, events character-
ized by a low value of Zwm cannot, by definition, populate the extreme corners or
sides of the Dalitz triangles. Indeed, there are well-defined maximum and minimum
possible values of Dam, and Dedge for each value of Zsum. These extrema are shown
for Dedge in Figure 3.6 as the solid lines (analogous limits apply for Dem). As Zsum
is decreased, the range of possible values of Dedge decreases, and this range tends

towards larger values of Dedge. It is therefore possible that the crossings noted in

58

 

OEll>
31

 

0Dt>
3.

 

OEll>
a

 

OCII>
a

 

 

 

 

Figure 3.6: The average values of Dedge versus Zwm in the central events for three
representative beam energies in MeV / nucleon for each entrance channel, as labeled.
The solid lines indicate the minimum and maximum values of Dedgc that are possible
for each value of Zwm by definition.

59

Figure 3.5 are simply an artifact of similarly beam energy dependent decreases in
the average values of Zwm. The values of Zwm averaged over beam energies in each
entrance channel are 6.7, 9.9, 15.7, and 19.0, for the 20Ne+27Al through 129Xe+139La
systems, which correspond to 29%, 25%, 20%, and 17% of the total entrance chan-
nel charge. The percentage standard deviations about these average values over the
different beam energies in each entrance channel are 8%, 12%, 3%, and 7%. Thus,
(Zwm) is only weakly dependent on the beam energy in each entrance channel, so
that the transitions noted in Figure 3.5 cannot simply be an artifact of decreases in
(Zwm) for increasing beam energies. The average values of Dedge are also plotted in
this figure for three representative beam energies in each entrance channel. These re-
veal a dependence of 06498 (and Dccnt) on Zsum that generally follows the shape of the
region allowed by definition. For 33231111135, there is only one allowed value of Dedge
(and Dam). While no relative charge information can be extracted for 3SZ3umS5,
central events with relatively small values of Zsum are nonetheless interesting, as they
indicate the presence of events in which no large fragments were observed, implying

the rather complete vaporization of the system.

It is important to consider also the constraints on the present observables that are
imposed by the inefficiencies of the detection system. The most obvious constraint
of this kind results from the maximum charge that could be detected, which was
~15 for the 20Ne+27Al and 87Kr+93Nb reactions, and ~18 for the 40Ar+“5Sc and
129Xe+139La reactions. The effect of this limit is to decrease the sensitivity of the
present observables for large values of Zwm. Specifically, as the values of 21 and Z2
approach the maximum detectable charge, the number of possible permutations of
Z1, Z2, and Z3 having the same measured Z,“m decreases. This leads to a situation
similar to that for 332111111135, where the limited range of possible values for Dam,

and 08.196 reduces their sensitivity. In Figure 3.6, the values of (Badge) manifest fairly

60

abrupt changes at roughly the maximum detected charge plus two, and twice that
maximum plus one, corresponding to the detection of such large charges in coincidence

with protons.

The values of (Dam) and (Badge) for the central events with 6SmeSI7 for the
20Ne+27Al and 87Kr+93Nb entrance channels, and GSZwm _<_20, for the 40Ar+"5Sc and
129Xe+139La entrance channels, are depicted in Figure 3.7. This gate on Zwm allows
those central events for which the present charge observables are not affected by the
limiting maximum detectable charge. These curves indicate the same transitional
behavior as shown in Figure 3.5, although at slightly lower beam energies: ~47 and

~29 MeV / nucleon for the 40Ar+45Sc and 129Xe+139La entrance channels, respectively.

3.2.3 Acceptance Effects

A more complete investigation of the effects of the experimental acceptance involves
the generation of software events, and the comparison of the present relative charge
observables in these events with and without filtering by a software replica of the
apparatus. The filter code includes a complete description of effects such as detector
geometry and kinetic energy thresholds (see Appendix C). Separate samples of events
were generated with the Berlin code [Zhan87b], each at a specified excitation energy, in
a system expected to be common in the present central 40Ar+“5Sc reactions (Z ~31,
A~68). The upper two frames in Figure 3.8 depict (Zsum) and (Dam) after each
of these samples of events is boosted from the center of momentum frame into the
laboratory, and then filtered, for beam energies in the range of 25 to 105 MeV / nucleon.
The events were generated at excitation energies of 2 (solid squares), 4, 6, 8, 10, 12,
16, and 20 (open crosses) MeV/ nucleon. The filtered (Zwm) and (Dam) show only a
very weak dependence on the magnitude of the boost into the laboratory for all of the

samples of generated events. The crossings noted in Figures 3.5 and 3.7 are therefore

61

 

 

 

 

 

 

0.3:-
o.25- *

E w
Mg Ne+All
0.3:-

0.2? K
.15-
é 0 E Ar+Sc
0.34:-
o.2:- C...
0.1:- Kr+Nb
0.3:-
0.2:- <
0.15-

 

 

Xe+La

 

20 40 60 80 100 120 140

Epmj (MeV/nucleon)

Figure 3.7: The average values of Dcent (with points) and Dedge (without points)
versus the beam energy for specific gates on the quantity Zsum: GS Zwm $17 for the
20Ne+27Al and 87Kr+93Nb entrance channels, and 6S Zwm $20 for the 40Ar-i-“Sc
and 129Xe+139La entrance channels.

62

 

 

 

 

 

 

 

 

 

 

 

40Ar+45Sc
18 3i -.--<>---4»-~<>--+10E~~~I
16 Lfi_—({6} <>——O-—-Z— +8 0.6 :— ""' ------ I.- +0 35
L __ _~_ ——A—- -—+ ...... ’
A 14 gg._.g....g__.a._..g..+4 A 0.5 :‘_---A-----A--~~A--—--A-~+0.30
12 :‘O""“"O””..__ .5 : — ——v——v—— —+0.25
8 :-.y.-- 0 O 0 +2 0 0.4 :—
a 10:— “" _____ , o W—an—wzo
§ 85- "" """" V e 0.3 %D-~-C}-~-G---D-~D-‘+0.15
6 E—_‘__‘__‘__ O 2 :_..A ......... A ......... A ......... A .......... A”... +010
5 “—7 ' g-<>--—--o—-r--<>--»--<>--«-~<>—~-+0.05
2W 0 1 W ‘3’ ‘9‘
_J t1
E20 40 6O 80 100 E20 40 6'0 80 100
pr 0 J (MeV/nucleon) Etop j(MeV/nucleon)
Epro (MeV/nucleon)
20 30 ’50 7o 80 90 100
3 I I 40I 6I0 I I I I 3
0.5 I- -I
A 04:“ E
e 0.3%— i
E ““““ E
0.2 — __.
E :
0.15- ”—3

 

 

 

 

8 f0 112 14 116 18 20
BUU E* (MeV/nucleon)

N
.h
0\

Figure 3.8: The average values of Zwm (upper left frame) and Dccnt (upper right
frame) obtained from eight samples of events, each generated at a specific excitation
energy for central 40Ar+“53c reactions using the Berlin code. Each sample is boosted
from the CM frame to the laboratory, and then filtered, for beam energies from 25
to 105 MeV/ nucleon. Some points in the upper frames have been offset for clarity
by the amounts shown to the right. The lower frame compares the excitation en-
ergy dependence of the unfiltered Berlin events with the beam energy dependence of
the filtered Berlin events, using BUU calculations to relate the beam and excitation
energies for central 40Ar+"58c reactions.

63

not the result of a strong beam energy dependence of the experimental acceptance.

The solid curves in the lower frame in Figure 3.8 depict the excitation energy de-
pendence of (Dem) (with points) and (Dedge) (without points) for the unfiltered
Berlin events. The unfiltered Berlin events evolve from the corners to the cen-
ter of the charge Dalitz triangles, and exhibit a crossing excitation energy near 6
MeV/ nucleon. Boltzmann-Uehling-Uehlenbeck (BUU) calculations are one means of
specifying the relationship between the beam energy and the average excitation energy
for central collisions. These calculations were performed as described in Reference
[Made93, Bowm92], and the predicted relationship between the beam and excitation
energies is visible by comparing the upper and lower abscissa in this lower frame. The
dotted lines in this frame depict the values of (Dccm) and (D6498) following the boost-
ing and filtering of the generated events assuming this relationship. The apparent
crossing (beam) energy is only weakly affected by the imposition of the experimen-
tal inefficiencies via the software filter. The major effect is to reduce the apparent
asymmetry of the three largest charges for asymmetric events, i.e. those in the SB

region.

Chapter 4

Observation of Non-Compact
Geometries

4.1 Signatures

In this chapter we will examine experimental observables, termed signatures, pre-
dicted to show sensitivity to the formation of non-compact geometries. Examples of

these signatures include the following observables:

o IMF Multiplicity

Charge Similarity

Fragment Charge Correlation Functions

Event Shapes

Multipolarity Coefficients

Fragment Velocity Correlation Functions

Kinetic Energy Spectra

Each of these observables has been proposed for use in identifying the occurrence of

exotic geometries. We examine them each in the following.

64

65

 

 

 

    
 

   
  
    

  

 

 

 

 

 

 

 

10 10_ . . - I . . v ‘ l . . ' ' l v 4
- 4
i o “'Xu '"Au, E/A=5O uav 1
- Iph-u. A-32G, p-O'l
8. 8.— “
" 6: "1!- 67 i
Z: - <N¢> bubble z : (NC) toroid
v " _ v .
4 :— P-o.b-0.l 4 .. Psos—M ‘
- neon-4.0 j : n,=2.o—4.5
2 1 one
o....I....I....I...‘ o.."'.I........I..1_
0 20 40 60 50 0 20 40 60 80
NC NC
(a) (b)

Figure 4.1: IMF multiplicity versus the total charged particle multiplicity result-
ing from percolation model calculations[PhaiQ3]. The solid lines (both panels) show
the values obtained from percolation of a compact spherical configuration, while the
shaded regions show the values obtained from percolation of initial toroid (panel a)
and bubble (panel b) geometries. The markers (both panels) show a comparison to
an experimental data set.

IMF Multiplicity

An enhancement in the IMF multiplicity (Ngmf) has been proposed by several au-
thors [Baue92, Xu93] as a consequence of the formation of a non-compact geometry.
The authors of Reference [Phai93] have performed percolation model calculations
to quantitatively demonstrate that decay from non-compact geometries (toroids and
bubbles) will result in emission of more IMFs than would be observed in the decay of
a compact spherical geometry (see Figure 4.1). In this figure, IMF multiplicity values
resulting from percolation of compact spherical distributions are depicted by the solid
lines. The shaded regions show the range of multiplicity values that result from a
range of parameters defining a bubble (panel a.) and a toroid (panel b). Specifically,
values near the bottom of the shaded region will result when the interior radius of
the configuration (R4,, R1, defined in the inset) goes to zero. As the radius parameter

is increased, observed multiplicity values move toward the upper limits of the shaded

66

regions. Thus, in both cases, we see that percolation from non-compact geometries
results in IMF multiplicities that are higher than those found from the percolation of

a compact spherical configuration.

Charge Similarity / Charge Correlation Functions

Several authors have predicted that the formation of non-compact geometries will
result in the emission of large fragments with nearly equal sizes[More92, Guar95].
They motivate this effect by comparing toroid and bubble formation to the formation
of a tube or sheet. If such a tube (sheet) forms, then the nuclear matter in that tube
(sheet) will tend to coalesce into fragments with diameters similar to the thickness
of the tube (sheet). If then the tube (sheet) has a uniform thickness, then such a
mechanism would yield fragments with similar diameters (i.e. masses). The authors
of Reference [Xu93] have also quantified the sizes and multiplicities of similarly sized
fragments. The authors of Reference [More96] have proposed looking for this charge
similarity with Charge Correlation Functions, and have demonstrated some success

at quantifying even small admixtures of events having similar charges.

Event Shapes / Multipolarity Coefficients

Event shape observables, so called because of their ability to assign a quantitative
measure to the shape of an event in momentum space, have been used widely used
to characterize the shapes of nucleus-nucleus events[Gyul82, Fai83, Lope89, Cugn83].
It is their sensitivity to the dimensionality of fragment emission which has led to
the anticipation that these variables should also show sensitivity to the breakup
geometry[Xu93]. Specifically, if the combined system were coplanar (e.g. toroid
or disk) as opposed to spherical (e.g bubble or compact sphere), one might expect

emission angles of fragments to lie predominantly within the plane defined by the

67

initial configuration.

The use of multipolarity coefficients was proposed by the authors of Reference
[Sou294] as a means of quantifying angular distributions of fragments. The concept
is essentially one of a fourier transform of the angular distribution in terms of the
spherical harmonics (Yin, (0, 45)). By examining these coefficients, the authors propose
another way of quantifying the shape of the event via angles alone as opposed to
momenta (angles and energy). However, the authors of this study issue a strong
caution due to the sensitivity of this type of analysis to slight non-centrality in event

selection.

Fragment Velocity Correlation Functions / Kinetic Energy
Spectra

The sensitivity of velocity correlation functions to the formation of non-compact
geometries relies upon the suppression of the Coulomb acceleration of emitted frag-
ments. Specifically, fragments distributed within a compact sphere are closer together
than they would be in a non-compact configuration. Since the relative Coulomb ac-
celeration between two particles is inversely proportional to their initial separation,
fragments emitted from non-compact geometries should experience lower Coulomb
boosts. Examining velocity correlation functions was proposed by the authors of
Reference [Glas93], however the observed effect was small, even for their simulation
calculations. Since relative coulomb boosts between fragments are expected to be
suppressed for non-compact configurations, there should also arise a number of frag-
ments with lower kinetic energies than would be otherwise observed. However, this
enhancement at low kinetic energies (less than 1 MeV/ nucleon for large fragments)

is a difficult signature to pursue due to considerations of detector energy thresholds.

The analysis presented in this chapter will focus on a subset from the signa-

68

tures described above. We will present data pertaining to IMF multiplicities, charge
similarity and event shapes, in lieu of the others listed which suffer from distinct

shortcomings, as noted.

4.2 Experimental Measurements

The experimental data for this study were collected for the 86Kr+93N b system for
the beam energies 35, 45, 50, 55, 60, 65, 75, 85, and 95 MeV/nucleon. Our study
includes only central events from this data set because the formation of non-compact
geometries is predicted for central events[Baue92, Xu93]. We selected central events
from our data with the well-established method of imposing cuts on global observ-
ables called centrality variables[Cava90, Phai92] (see Appendix B). In order to avoid
autocorrelations between the centrality variables and the experimental signatures we
present (see Appendix B), we have used two different centrality variables: the total
transverse kinetic energy (E1) and the total charge of detected fragments traveling
at mid—rapidity (Zmr). To ensure the centrality of the data sample we applied cuts
on E, accepting only the 5% most central events when dealing with charge-based

signatures, and similar 5% cuts on Zm, when dealing with the event shape signature.

4.2.1 IMF Multiplicity

The first experimental signature we present is an anomalous increase in the number of
intermediate-mass fragments (IMFs). We have measured the multiplicity of IMFs in
central events selected via cuts on E1 and plotted the mean values versus the incident
beam energy per nucleon in Figure 4.2. Statistical error bars are smaller than the
marker size. Two features are clear in this figure. The first is a general trend in which
this multiplicity increases with increasing beam energy. This trend in IMF emission

has been documented in previous experimental studies[Peas94, Llop95a]. The second

69

 

5..
E _
5

4-

 

 

I l l l l l l l l] l I

30 ' 40 50 60 70 80 90 '100
Beam Energy (MeV/nucleon)

 

Figure 4.2: Mean intermediate-mass fragment multiplicity versus the incident beam
energy.

70

feature is an increase in the mean IMF multiplicity, deviating from the more basic

trend, at a beam energy of 65 MeV/ nucleon.

This multiplicity enhancement is not caused by systematic effects of our detector
acceptance. To support this claim, we have performed detailed simulations in which
we generated realistic events which we passed through a software replica of our detec-
tor (see Appendix C). In so doing, we are able to extract the efficiency for detecting
IMFs as a function of the incident beam energy. The extracted efficiencies are shown
in Figure 4.3. Clearly, the IMF detection efficiency increases smoothly over the range
of energies shown in this Chapter. The enhancement in IMF production observed in
the experimental data is consistent with the increase in IMF emission predicted to

accompany the formation of non-compact geometries[Baue92, Xu93, Phai93].

To further elucidate the character of these events, we have also calculated the total
charge of detected IMFs (ngf) in these same events. The mean values of the this
quantity are shown in Figure 4.4. From this figure, it is apparent that not only does
the number of IMF 5 increase at 65 MeV/ nucleon, but the amount of the combined

system being converted to IMFs also increases.

4.2.2 Ordered Charges

The second signature we present is an enhanced similarity in the charges of larger
fragments. Some theoretical models have quantitatively predicted that the formation
of non-compact geometries will also result in increased cross sections for the emission
of fragments with nearly equal masses[Xu93]. To quantify this for our data, we have
calculated the power-law exponents for ordered Z distributions. This calculation
begins with the ordering of the charges of the detected fragments, from largest to
smallest, on an event—by-event basis. In this way, each fragment is assigned an index

from 1 to NC, the total number of charged particles. We then calculate the mean

71

 

 

 

0.7 -
a 0.65 -
fl
.2
9
*3
g 0.6 -
§
9 1.
"U
E 0.55 '-
0.5 -
n I l l n J 1 l I l n

 

Beam Energy (MeV/nucleon)

Figure 4.3: Energy dependance of estimated efficiency for IMF detection, extracted
from filtered simulations.

72

 

22L

18-

 

 

l n l n I n I n l n l

i 40 50 60 70 80 90
Beam Energy (MeV/nucleon)

 

Figure 4.4: Mean total charge of IMF 3 versus the incident beam energy.

73

for each of these indexed charges over all events in our sample. The ordered charge
distributions measured for central events are shown in Figure 4.5 (markers) for all
beam energies. We have observed empirically that these ordered Z distributions are
best reproduced by a power-law (i.e. (Zord(z')) o< 2""), although the qualitative
features of the signature we present below are relatively insensitive to the fit type.
Sample power—law fits are also shown in Figure 4.5 (lines). It follows that the fit
parameter a for these distributions would be large for events which contain very
differently charged fragments and small for events which contain similarly charged
fragments. Thus, the presence of a non—compact break-up geometry, giving rise to
more similarly charged large fragments, should be accompanied by a suppressed value

of this exponent.

We have extracted the power-law exponent a for central events selected via cuts
on E, and plotted them versus the beam energy in Figure 4.6. Statistical errors are
smaller than the markers, and systematic fitting errors are plotted. We observe an
overall smooth trend in a, decreasing nearly linearly with increasing beam energy.
It is clear that this observable, like the IMF multiplicity, also undergoes a departure
from the overall smooth trend, and at similar beam energies. The extracted a values
at 65 and 75 MeV/ nucleon are suppressed so that they lie below the values consis-
tent with a smooth trend. This suppression in 01 is an indication of more equally
sized large fragments and is predicted to accompany the formation of non-compact

geometries [Xu93].

Based on the aforementioned quantitative and qualitative predictions, the two
charge-based observables shown above demonstrate trends that are well explained by
the decay of non-compact geometries. However, to differentiate between the possi-
bilities of toroid and bubble formation an additional signature is required. The use

of event shape observables to make precisely this distinction was proposed by the

74

 

 

 

 

 

(Zm(1)>
G N 5 ON co 6 N :5 a W G N «B a co

 

 

 

 

i (index)

Figure 4.5: Ordered Z distributions (markers) for all beam energies, as labelled. The
solid line is a fit to the function (Zord(z')) cc 2"“.

75

 

0.9 -

0.8 -

0.7 '-

 

 

l n I n I n l n I n I 1

30 40 50 60 70 80 90 100

 

Beam Energy (MeV/nucleon)

Figure 4.6: Power-law (a) values extracted from ordered charge distributions plotted
versus the incident beam energy.

76

authors of Reference [Xu93].

4.2.3 Event Shapes

The third signature we present is a suppressed sphericity in the emission of heavy
fragments. It is derived from the sphericity of particle emission in momentum space,
and it ascribes a quantitative measure to the dimensionality of the break-up geometry.
Such an observable includes all of the information measured for each fragment (i.e.
mass, energy, emission angles), and thus provides a good balance to observables based
on charges alone. Sphericity is defined[Fai83] by first generating the kinetic energy

tensor[Gyul82] such that

Nim! p- p-
T1" = M 4.1
.7 ”2:31 2m“ ( )

where Nimf is the number of IMF s in each event, and pm and mu are the 2"" compo-
nent of the momentum and the mass of each IMF, respectively. Next, the eigenvalues
(Ag) of this tensor are calculated, which correspond to the sizes of the axes of the
ellipsoid in momentum space. These eigenvalues are then ordered (A1 > A; > A3)
and normalized (q,- = x\,-/ 22:1 x\,). All of the event shape variables that we will show
are defined in terms of these normalized eigenvalues. Their definitions, complete with
limiting values, are listed in Table 4.1. We have defined all of the shape variables
that we have found discussed in the literature[Fai83, Lope89, Cugn83], although we
will highlight the Sphericity later in this chapter. Given these definitions, events
with isotropic emission of IMFs will have a high value of sphericity, while those with
coplanar or otherwise non-spherical emission will have lower values. The values of the
event shape variables are affected not only by the overall shape of the event, but also
by the multiplicity of particles (Ngmf) in the tensor sum. The range of allowed values
of these variables is a limitation we must treat explicitly. To account for this we will

compare events having the same IMF multiplicities, as in Reference [Llop95b].

77

 

 

 

 

 

 

 

Limiting Values

q Name Symbol Definition Sphere Disk Rod
Sphericity 5 $012 + (13) 1 § 0
Coplanarity C 325(q2 — q3) 0 3g 0
Jettiness j (11 — (12 0 0 1
Prolateness d) H Undef. 1 0
Eccentricity 6 q, — %(q2 + q.) 0 -1 1

q, = (11, qr = <13 (prolate)

q, = q3, q, = q] (oblate)

 

 

 

 

 

 

 

 

 

Table 4.1: Definitions of event shape variables. The limiting cases shown in the last

columns represent the values obtained for the aspect ratios %:§:%, %:%:0 and 1:0:0,

respectively[Cugn83]. The definition of Eccentricity depends on the momentum-space
configuration. Prolateness (oblateness) is defined via the relation ql — (12 >(<)q2 — q3.

We can illustrate the multiplicity distortions to the shape variables through the
use of an event generator. We have used the event generator EGSIM[Gual95] to
create events with well—defined momentum-space fragment emission patterns. The
aspect ratio of the momentum ellipsoid is one of the primary input parameters to
this generator. Details regarding the other input parameters necessary to run this
generator are discussed in Appendix C. Thus, we can generate events with con-
strained momentum-space aspect ratios. Since the values of event shape variables are

directly related to this aspect ratio, EGSIM provides an excellent vehicle for probing

multiplicity distortions and also acceptance effects.

In order to first demonstrate that EGSIM is capable of generating distinctly spher-
ical, disklike and rodlike events, we have run simulations with input aspect ratios of
1:1:1, 2.5:2.5:1 and 1:1:2.5, respectively. The resulting center-of-momentum frame
momentum distributions are shown in Figure 4.7. The disk-like (rod-like) emission of

fragments can be clearly seen by the flattening (elongation) along the Z-axis in these

P; vs. P; and P; vs. P; histograms.

 

 

 

 

 

 

Px’(AMeV)

 

 

 

Px’(AMeV)

 

 

 

 

Py’(AMeV)

  

78

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A 200
>
0
E 0 Sphere
. . ./ 3:;
0 200 20(1200 0 200
Pz’(AMeV) Pz'(AMeV)
200 200
I \‘ g
o E 0 Disk
I Eh" I
20(£200 0 200 200-200 0 200
Pz’(AMeV) Pz’(AMeV)
7? Q 0
E
0 s
V f
_200 M in
-200 o 200
Pz'(AMeV) Pz’(AMeV)

Figure 4.7: Center-of—momentum frame momentum profiles of spherical (top row),
disk-like (middle row) and rod-like (bottom row) simulated events.

79

We have calculated mean values for each of the 5 shape variables defined in Table
4.1, and have plotted them versus the IMF multiplicity in Figure 4.8. In this case,
only IMF s are included in the tensor sum (Equation 4.1). Initially, all variables ap-
pear to be well-behaved. Each variable shows some sensitivity to the three different
configurations, most efficiently for higher multiplicities. However, a close examina-
tion of the plots for Coplanarity (C), Prolateness(¢), Eccentricity(e) and the second
reduced eigenvalue (Q2) reveals crossing points between the sphere and disk lines.
These crossing points represent severe multiplicity distortions to the shape variables
which display them. For example, it is clear from the Coplanarity plot that a sphere
has a higher Coplanarity than a disk, if it is created with roughly 5 particles or less.
This effect is not only counter-intuitive, but compromises its ability to make a correct

event shape identification.

To further clarify this distortion, we have plotted the mean values of the 5 event
shape variables and 3 reduced eigenvalues versus the total charged particle multi-
plicity (see Figure 4.9). In this case, all particles, not only IMFs, are included in
the tensor sum (Equation 4.1). We should expect to see the same trends in these
plots, since the kinetic energy tensor is coalescence invariant. This method allows us
to plot on an expanded scale, which provides an indication of the asymptotic lim-
its of the shape variables. Even with the expanded scale, agreement between the
IMF multiplicity plots of Figure 4.8 and these plots can be verified (i.e. coalescence
invariance). From Figures 4.8 and 4.9 we can identify that there are actually two
types of multiplicity distortions. The first is the narrowing of the range of observed
values at low multiplicities (all three curves join at Ngmf = NC = 1), but the second
is the reversal of the expected (asymptotic) trends at low multiplicities, evidenced
by the crossing of the sphere and disk lines. For the remainder of this chapter, we

will confine our discussion of event shapes to the sphericity, because this variable is

80

 

 

 

(S)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l.
3 Z r
0 - .
- 1 j M 1 0 I. 1 n 1 1
0 NM 15 0 Nimf 15
’ —— Sphere
3' - - - Disk
...... Rod

 

 

 

 

 

 

   

Figure 4.8: Calculated behavior of all shape-variables for simulated spherical(solid
lines), disklike (dashed lines) and rodlike (dotted lines) events, plotted versus IMF
multiplicity.

81

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.9: Same as Figure 4.8, except plotted versus total charged particle multi-
plicity, allowing extended (asymptotic) multiplicity scale.

82

both well-known and well-behaved (meaning that it does not show the second type

of multiplicity distortion).

We have calculated the mean sphericity of central events in the experimental data,
sorting them by IMF multiplicity, and compared the mean values to those obtained
from the simulations in Figure 4.10. If we focus momentarily on the mean sphericity
values obtained for IMF multiplicity 5, which is the mean multiplicity for these events
(see Figure 4.2), then we can see an interesting energy progression. At low energies,
the experimental data (solid lines) lie between the spherical (dashed) and disk (dot-
dashed) values. This is also the case at high energies. However, at an intermediate
energy of 60 MeV/ nucleon, the experimental values are nearer to the disk values.

This same basic trend is furthermore observed for all multiplicities Nimf Z 4.

To clarify this trend, we have calculated the mean values of the sphericity of
IMF emission and plotted these (markers) versus the incident beam energy per nuc-
leon for three representative IMF multiplicities (Nimf = 4,5, 7) in Figure 4.11. We
have also plotted the mean values resulting from spherical (dotted lines) and disk-
like (dot-dashed lines) emission patterns above and below each of these three mul-
tiplicity curves. Statistical errors on the mean values are smaller than the marker
size and systematic errors are shown. The suppression of the sphericity value at 60
MeV / nucleon is now clear, and occurs for all IMF multiplicities shown, and those not

shown (Nimf Z 4) as well.

It is worthwhile to note two features of the filtered simulation data. First, the
sphericity values resulting from the disk-like simulation are lower than those result-
ing from the spherical simulation. These lower values illustrate the effectiveness of
sphericity in quantitatively distinguishing disk-like emission from spherical emission.

Second, an examination of the excitation function of the sphericity values yielded by

0.6

0.4

0.2

0.6

0.4

(S)

0.2

0.6

0.4

0.2

83

 

TilllillrrTlI

IIIIIITIIIIII

Illlllll'f'l'

 

 

 

IIIIIIIIIITITrITIIIIIIIIlI

 

 

I I I I I I I I

I O
WjijllllllI'lIIIIIIIITTTII

 

 

J I I I I I I I

O
IIIrIITIIIIITlll'lITrIIIII

 

 

 

I I I I I I I I

 

5

- - Sphere
---- Disk

10

5

Nimf

l0

5 10

Rod
—-- Data

Figure 4.10: Mean values of sphericity extracted for spherical (dashed), disklike (dot-
dashed) and rodlike (dotted) filtered simulations compared to those extracted from
the experimental data (solid) for all beam energies, as labelled.

84

 

0.6 -

0.5- MN. .....

 

(S)

 

 

0‘

1.11 = 7 (+0.06) 1 -~ ................... '

 

 

I 1 A 1 I 1 I 1 I 1 I 1 I 1 L
30 40 50 60 70 80 90 100
Beam Energy (MeV/nucleon)

Figure 4.11: Measured values of the mean sphericity of IMF emission (markers) versus
the incident beam energy for three representative IMF multiplicities, as labelled. The
dotted (dot-dashed) lines in each case indicate the mean values obtained for filtered
spherical (disk-like) simulations. Small offsets have been added for clarity.

 

85

the filtered simulation reveals that changes in the acceptance effects upon (S) with
increasing beam energy are smooth, so that any significant change in the experimen—
tal data that does not follow this smooth trend cannot be attributed to systematic

changes in acceptance.

Over the range of beam energies measured, the experimental values of sphericity
fall between the limiting values obtained for the spherical and disk-like simulations,
and the shape of the curves for each multiplicity reflect the same basic trends which
we associate with acceptance effects. However, at a beam energy of 60 MeV / nucleon,
the experimental value of (S) is suppressed, approaching the disk-like values. This
suppression cannot be attributed to experimental acceptance and is beyond statis-
tical fluctuations, and indicates a more disk-like, or coplanar, IMF emission at this
energy. To be more quantitative, the value of sphericity achieved at 60 MeV / nucleon
is consistent with a disk of aspect ratio 2:2:1. For completeness, we have calculated
the mean values for each of the other shape variables and have scaled the values to
allow placing them on the same plot. The resulting scaled values, extracted for an
IMF multiplicity of 5, are shown in Figure 4.12. Despite the multiplicity distortions
shown in Figures 4.8 and 4.9, the trends shown in the other event shape observables

are quite similar to those observed for the sphericity.

Early studies involving the flow tensor[Gyul82j quantified the collective flow of
the system in terms of “flow angles,” the azimuthal angles of the three principle
axes of the ellipsoid in momentum space. Thus, we can not only gain information
from the eigenvalues, which we have done in the preceeding pages, we can also gain
information from the eigenvectors. If the formation of a toroid were the cause of the
increased coplanar emission of IMFs indicated by the event shapes analysis, then the
one might expect to see a shift in the flow angles as well. (The toroidal and disk-like

configurations shown in Figures 1.6 and 1.8 were formed perpendicular to the beam

 

86

 

Scaled Shape Variable Value

IIII]IIIIIIIIIIjj—IfiIIj—FIITIIIIIIIIIIIIIIIIII’TTI‘IT
m

‘11

‘12

‘13

 

 

 

IlIIlIlIlIIIll IIIIIILIIIIIJAMJII

30 40 50 60 70 80 90 100

 

Beam Energy (MeV/nucleon)

Figure 4.12: Scaled values of all eight shape-variables (as labelled at right) for IMF
multiplicity 5. This reveals the suppression or enhancement of the all shape variables
at 60 MeV/nucleon.

87

axis.) So we can examine the flow angles for the experimental data to determine
whether there is a shift in the alignment of the major axis of the flow tensor. We
have calculated the mean values of the cosines of the three flow angles (01, 02, 03)
corresponding to the three reduced eigenvalues (q1, q2, q3) and plotted them versus
the beam energy in Figure 4.13. The largest eigenvector also has the largest cosine of
its flow angle, thus the momentum ellipsoid is pointing mostly forward. However, at
65 and 75 MeV/ nucleon, the major axis shifts slightly toward smaller cosine values,
indicating a shift toward a more perpendicular alignment. The shift is small, but it

does move in a direction consistent with the scenario discussed above.

The suppression in the mean sphericity value observed for all IMF multiplicities is
indicative of a transition to more two-dimensional emission of fragments. Increased
coplanar emission of large fragments has been suggested as an additional consequence
of toroid formation specifically[Xu93]. Having established the non-compact nature of
the breakup geometry through the use of the previously shown charge signatures, and
the two-dimensional nature of the fragment emission, toroidal break-up is a current
scenario which can explain the anomalous behavior shown in all three observables
presented. The fact that all three of these signatures appear and then vanish rather
abruptly supports related predictions that the occurrence of non-compact geometries

has a sensitive dependence upon the initial conditions of formation[More92, Baue92].

88

 

 

 

 

3 cos(Bl)
A C
a: .
a: .
v l—
m I.
8 : cos(Gz)
V
7 ....
1T .. . ....... . ....... .0...
: . 1.1. . . ’.
: ~..‘_*, ..- _., cos(93)
i-IIIJLIIIIIIIIIIIIIIIIIIIIIIIIIIIII
30 40 50 60 70 80 90 100

Beam Energy (MeV/nucleon)

Figure 4.13: Cosine values of the azimuthal angles (termed the “flow angles”) of the
three eigen-vectors of the kinetic energy tensor, plotted versus the incident beam
energy.

 

Chapter 5

Conclusion

In Chapter 3 we described a method of evaluating the importance of the specific disas-
sembly mechanisms in a comprehensive set of central heavy-ion reactions using charge
Dalitz plots. Small impact parameter collisions were selected using two-dimensional
cuts on centrality variables that do not autocorrelate with the relative charge distribu-
tions. The observables Dam, and Dedge were introduced for the purpose of quantifying
the distribution of events in the charge Dalitz triangles for each reaction. The con-
straints imposed on these observables, by definition and by the inefficiencies in the
detection system, were shown not to affect our conclusions. Transitions from sequen-
tial binary disassembly to multifragmentation were observed in the central 40Ar+"5Sc
and 129Xe+139La entrance channels at beam energies of ~50 and ~30 MeV/ nucleon,
respectively. The results for the central 20Ne+27Al (87Kr+93N b) reactions are consis-
tent with the trends noted for the central 40Ar+458c(129Xe+139La) reactions over a

more limited range of available beam energies.

In Chapter 4 we presented a systematic study of three global experimental ob-
servables for the 86Kr+93Nb system for incident energies ranging from 35 to 95
MeV / nucleon. Our study reveals enhanced emission of intermediate-mass fragments

(IMF s) and a suppression in the power«law exponent a of ordered Z distributions,

89

90

which together establish the non-compact nature of the break-up geometry. We also
observe a suppression in the mean value of the sphericity of IMF emission, which in-
dicates toroidal, as opposed to bubble-like, geometries. All three of these signatures
occur at energies between 60 and 75 MeV/nucleon. These signatures are predicted
by many of the same theoretical models that have provided us with the most recent
descriptions of non-compact geometries, and should be interpreted in that context as

experimental evidence for the existence of toroidal break-up geometries.

We have established two basic changes in the reaction mechanisms for the systems
studied. The first is the transition from sequential binary decay to multifragmenta-
tion in the incident beam energy range of 30 to 50 MeV/nucleon, for the systems
studied. Second, we have shown that with increasing energy, the system evolves from
a state in which a brief expansion is followed by a collapse, through an intermediate
state in which the initial expansion stalls, to a state in which the expansion continues
unabated. The stall in the expansion of the system studied has been related to the
creation of exotic geometries. Previous theoretical work has already indicated that
these exotic geometries will be subject to different decay modes. Further theoreti-
cal work concerning these distinct effects will advance the understanding of nuclear

matter and its equation of state.

APPENDICES

 

 

Appendix A

High Voltage Circuit Components

Following are lists of the components used in the high voltage circuits shown in Figure

 

 

 

 

 

 

 

 

 

2.6.

Label Part Quantity Vendor (part #)

Rel, Re2 20MQ, 1/2 W res. 2 Mouser (29SJ500-20M)

Rbl, Rb2 10k”, 1/2 W res. 2 Mouser (29SJ500-10K)

Ral - Ra8 1000, 1/4 W res. 8 Mouser (29SJ250-100)

Cal - Ca8 1nF, 3kV cap. 8 Newark (46F 5277)

Ccl 10nF, 3kV cap. 1 Newark (95F4048)

P1 14 pin IDC socket 1

P2 PMT Socket 1

P3 SHV Connector 1

P4 BNC Connector 1

Hardware Components
PC Board 1 Hughes
1” Spacer (4/40 thread) 3 DigiKey (8440GK—ND)
5/16” Screw (4/40 thread) 6 1 DigiKey (H143-ND)
PMT Base Cover 1 ‘ Classic Precision

 

 

 

 

 

Table A.1: List of Components for PMT socket.

91

 

 

92

 

 

 

 

 

Label Part Quantity Vendor (part #)

Rdl - Rd3 1M9, 1/2 W res. 3 Mouser (29SJ500—1M)

R1 - R3 43kQ, 1 W res. 3 Mouser (261~43K)

R4 - R6 82kQ, 1 W res. 3 Mouser (261-82K)

R7, R8 100kf2, 1 W res. 2 Mouser (261-100K)

R9 130kQ, 1 W res. 1 Mouser (261-130K)

R10 180kQ, 1 W res. 1 Mouser (261-180K)

R11 150kfl, 1 W res. 1 Mouser (261-150K)

R12 220M), 1 W res. 1 Mouser (261-220K)

R13 200kf2, 1 W res. 1 Mouser (261-200K)

Ral - Ra8 1009, 1/4 W res. 8 Mouser (29SJ250-100)

Z1, Z2 100V Transorb 2 DigiKey (P6KE100CACT~ND)
Z3 150V Transorb 1 DigiKey (P6KE15OCACT-ND)
Zal - Za8 15V Zener Diode 8 DigiKey (IN5245B)

Val - Va8 360V Varistor 8 DigiKey (P7243-ND)

Qal — Qa8 E—HexFET 8 DigiKey ( IRF820-ND)

P1 14 pin IDC docket 1 DigiKey (MHD14G-ND)

 

Hardware Components

 

 

PC Board

Pomona Box

 

Hughes
Newark (35F3520)

 

Table A2: List of Components for the PMT divider.

 

Appendix B

Centrality Selection

The importance of centrality selection in experimental nuclear physics has been well
established in previous studies[Phai92, Ogil89, Cava90] and merits careful considera-
tion in this study as well. Of primary importance is the effective selection of central
collisions from among the inclusive data in such a way as not to bias the sample of
events. Several global observables have been shown in the cited studies to correlate
to the impact parameter (b) in nuclear collisions. We have chosen to consider the

following five variables from among the list of known centrality variables:

0 The total number of detected charged particles (NC).
0 The total number of detected protons (NP).
0 The total charge of light charged particles (Zlcp).

o The total transverse kinetic energy (E. = Z,_._1,NC E, sin(0,-)), where E,- and 0,-

are the kinetic energy and polar angle of each detected fragment.

0 The total charge of fragments travelling at mid-rapidity

(Z1... = 21.0.75)”

targ

I

mm]- are the

I
<Y’ .<0.75Y;r01 Zi). where Y , and

I
frag” targ’ fragfi

center-of-momentum frame rapidities of the target, emitted fragments, and pro-

jectile, respectively.

93

94

In defining this subset, we have attempted to choose well known, reliable and simple

observables, in order to minimize the number of unknown effects in the analysis.

B.1 Numerical Calculations

The application of a centrality observable is based on the simple geometric concept
that the scalar value of the impact parameter is monotonically related to the value

of a centrality variable by the relation[Cava90]

27rbdb

max

 

where f(q) is the probability of measuring an event with centrality variable value q.
This can then be integrated to yield the relation between the value of the reduced im-
pact parameter (b/bmax) and the integrated probability as a function of the centrality

variable

b/bmax : 1— F(Q) (B°1)

where F (Q) :2 fgcema, f (q)dq, and bmax is the maximum impact parameter leading
to a triggered event. In this strict geometrical picture, bmax = (Rproj + Rwy), where
Rpm,- (Rtarg) is the radius of the projectile (target) and can be approximated by
R = roA1/3[Cava90].

For example, in Figure B.1 we show the probability distributions of two cen-
trality variables 15', and Zn, (unshaded histograms) for the 40Ar+45Sc system at 95
MeV / nucleon incident energy. The horizontally (vertically) shaded sections show the
subsets selected by 10% cuts on Zm, (Et) alone. From the relation given above, a

10% cut corresponds to a reduced E, value of a reduced impact parameter of

b/bm = 1/1 — 0. = 0.31

95

 

10
1
0 500 1000
E. (MeV)
10 6

 

0 10 20 30 40
Z

mr

Figure B.1: Two centrality variable distributions for (a) E, and (b) Zm, (see text for
definitions). The unshaded histogram shows the inclusive distribution for System-2
(see text) events, while the horizontally (vertically) shaded sections show the subsets
selected by 10% cuts on Zm, (Et) alone. The intersection of these two subsets shows
the portion of the inclusive data selected by simultaneous 10% cuts on both of these
variables (roughly 7%).

96

so that events above this 10% threshold have a value of the impact parameter

b g 0.31(1.2fm)(40’/3 + 451/3) 3 2.5fm .

B.2 Autocorrelations

Efficiency is not the only consideration when choosing a centrality variable. We need
to verify also that the centrality variable we choose is not autocorrelated to the observ-
ables in our study. An autocorrelation is a correlation between the centrality variable
and the experimental observable which is inherent in the definition of the observable.
A trivial example of an autocorrelation is between the number of charged particles
(NC) and itself. Other similar, but more subtle, autocorrelations are between N, and
the number of intermediate-mass fragments (IMFs), the number of light charged par-
ticles, or the number of protons detected, the common thread between all of these

latter observables being a number of particles.

Autocorrelations are not always so obvious as those stated in the above examples,
and they must be thoroughly investigated before any serious analysis is performed. In
order to identify autocorrelations, we rely on the fact that the width of the distribution
of values for an experimental observable will be narrower when selected by an auto-
correlated centrality variable. Consider our trivial example again; if we make a very
narrow cut on the centrality variable NC, then the distribution of NC (or Nimf, etc.)
values will be correspondingly narrow. Intuitively, one should expect that observ—
able distributions selected via centrality cuts will become more narrow, as compared
to inclusive distributions, by virtue of the selection of a well—characterized subset of
events. In this light, the separation between autocorrelated and non-autocorrelated
centrality variables based on the observed widths, as described, becomes one of a

relative suppression of the widths.

 

97

Since the analysis in Chapter 3 is an analysis of the three largest charges in each
event, we have examined the effects of cuts on the five centrality variables listed
above on the sizes and RMS widths of the distributions of these charges (Z1, Z2, Z3).
We have plotted the RMS widths for these three charges versus the incident beam
energy for 5% cuts in the 40Ar-I-"58c system (Fig. B2). The corresponding mean
charge values are also plotted in Fig. B.3, for completeness. For each of the three
charge widths measured, the widths resulting from cuts on chp are the most strongly
suppressed for nearly all beam energies. The two centrality variables which cause
the least suppression in the charge distributions are E. and Zmr. Similar suppression
of widths was observed for Dam and Dedge, although not as pronounced. For this
reason, centrality cuts for the study in Chapter 3 will be performed on these two

variables.

In general, the correlation between any given centrality variable and the impact
parameter (b) is not a perfect correlation, i.e. it has some non-zero width due to
fluctuations, as mentioned above. To minimize the effects of these fluctuations upon
the determination of centrality, we will perform two—dimensional cuts to select the
central events we will study in Chapter 3. Specifically, we will require that events
have values of both E, and Zm, which fall in the top 10% of the respective distributions
simultaneously. This is illustrated in Figure B.1 by the intersection of the vertically
and horizontally shaded areas (which represent the one-dimensional cuts) into the
hatched area in each panel. This region includes roughly the top 4-8% of the total

inclusive data.

A similar analysis was performed to select E, as the centrality variable for the
charge-based observables shown in Chapter 4 and Zm, as the centrality variable for

the energy / momentum based observables in that chapter.

 

o(Zl)

0(Z2)

0.5

0.8

0.6

o(Z3)

0.4

0.2

98

 

 

 

 

 

 

 

I.
. O Zlcp
I.
I.

1 I I 1 1 J I 144 I 1 1 I I 1 I 1 I l 1 l I V Et
; A Zmr
1’.

1 1 I 1 j I I I l I I 1 I 1 In 1 I J I 1 I L I Np
. 0 No
:1 1 I 1 I L] I l I I 1 n n I n n 1 I I 1 1 I

 

20 40 60 80 100 120
Beam Energy (MeV/nucleon)

Figure B.2: RMS widths of the charge distributions of first, second and third largest
fragments in central events in the 40Ar+“5Sc system selected by 5% cuts on 5 centrality

variables (as labelled).

 

99

 

 

   

 

 

6:-

A‘ 1 OZ]

5 4_M cp
2~11I1I1I11IIIIIIIIIIIIIJ 'Et
3'.

3' . AZmr
1EIIIIIIIIIIIIIIIIIIIIIIJ 'NP
2?—

1.75-
a, ONc
V1.5;-
1.253—
'IIIIIIIIIIIIIIIIIIIIIII

 

 

 

20 40 60 80 100 120
Beam Energy (MeV/nucleon)

Figure B.3: Mean charge of the first, second and third largest fragments in central
events in the 40Ar+“5Sc system selected by 5% cuts on 5 centrality variables (as

labelled).

Appendix C

Simulation Parameter Definitions

We have generated simulated events having well-defined momentum-space distribu-
tions (see Figure 4.7). The purpose of generating events in this thesis is twofold: (1)
to compare the overall magnitudes of the extracted event shape observables to the
experimentally measured values and (2) to elucidate how measured values are affected
by the detector apparatus itself. In order to achieve either of these goals, it is neces-
sary to filter the generated events. All simulation data shown in this thesis has been
generated by the code EGSIM and subsequently passed through a software replica of
the 47r detector. This event filter accounted for particle loss due to effects such as
upper and lower energy thresholds, uncovered solid angle, malfunctioning detectors,

target shadowing and multiple particle hits in the same detector.

Before such a comparison can be made, particular care must be taken to ensure
a meaningful event structure. In this Appendix, we shall detail the process by which
we selected a set of input parameters for the event generator EGSIM[Gua195]. The
workings of this event generator were based on a model described by the authors of
Reference [Bond90]. The basic concept consists of the selection of momenta from
a three-dimensional gaussian distribution, to fix the direction, and the selection of

kinetic energies from a Maxwellian distribution, to fix the energy normalization.

100

 

101

Of particular value to the analysis of Chapter 4 is the ability that this generator
provides the user to define the relative sizes of the axes of the momentum ellipsoid.
It is these relative sizes, extracted from the kinetic energy tensor[Gyul82], that are
used in an event shape analysis to define observables like Sphericity, Coplanarity,
Prolateness, Eccentricity[Cugn83, Fai83, L6pe89], etc. In this way, one can define
a particular event shape in momentum space and examine the effects of detector

acceptance on the measurement of event shapes variables.

The four available input parameters are the relative sizes of the principle axes,
the emission source temperature (or temperatures, if multiple sources are used), and
the charge and charged particle multiplicity distributions. The aspect ratio used, as
discussed, is the parameter which will be varied in the discussions of Chapter 4. A

discussion of the determination of the others follows below.

C.1 Temperature

We base the selection of an emission source temperature on inverse-slope parameters
extracted from fits of experimental data to Maxwell-Boltzmann distributions. The
functional form is given by:

dP __ P101, (-KECM>
dE —- NPCMl/KECMeXp T

 

, where N is an overall normalization, Bab (PCM) is the lab- (CM-) frame momentum,
K ECM is the CM-frame fragment kinetic energy, and T is the source temperature.
Since the analysis of Chapter 4 involves intermediate-mass fragments (IMF s) only, we
have plotted kinetic energy (per nucleon) distributions for 7Li, the smallest IMF, in
Figure CI. The histograms show the measured kinetic energy spectra, while the solid
line superimposed on each frame is the x2 fit of the Maxwell-Boltzmann distribution

to each histogram. The extracted kinetic temperatures are shown in Figure C.2. It

102

 

 

 

 

 

 

 

 

. 45
1 I
A g 60
M _
ll :
N I
% ‘
I I]
104. 75 i 85
103' i
102 ‘
10
1 1 1 1 I 1 1 1 ..1 1 1 1 I 1 1 1
0 50 0 50 0

Fragment Energy (MeV/nucleon)

Figure C.1: Kinetic energy (per nucleon) spectra (histograms) for 7Li fragments for
each energy, as labelled. Each spectra has been fit with a Maxwell-Boltzmann distri-
bution (solid line) in order to extract the emission source temperature.

103

 

35-

 

30-

25-

Kinetic Temperature (MeV)

20-

 

 

_-
fiI1I1I1I1I1I1l1I1I1I1I1I1lu

35 40 45 50 55 60 65 70 75 80 85 90 95
Beam Energy (MeV/nucleon)

Figure C.2: Kinetic temperatures extracted from energy spectra for 7Li fragments.

104

is these values which were used as the emission source temperatures for EGSIM.

C.2 Charge and Multiplicity Distributions

In order to reproduce the observed charge and multiplicity distributions, we adjusted
the input parameters to the event generator and compared the filtered distributions
those obtained by experiment. A distribution of the total charged particle multiplicity
is shown in Figure C.3 for both the filtered simulation (top panel) and the experiment
(bottom panel). Experimental distributions shown in this Appendix were taken from
central collisions (see Appendix B) in a 65 MeV/ nucleon “Kr+Nb sample. Each of the
distributions has been fit with a gaussian function, and the resulting fit parameters
are shown in the upper right corner of each frame. For the input parameters chosen,
the filtered simulation well-reproduces the multiplicities observed in the experiment.
In a similar fashion, we show simulated and measured charge distributions for the
same data sample in Figure CA. The filtered simulation well-reproduces the observed

charged distributions as well.

With these two sets of parameters defined, it is of interest to examine the resulting
simulated IMF multiplicity spectrum. These are shown in Figure C.5 with the ex-
perimental distributions. These two have been fitted with gaussian distributions and
the extracted parameters are shown in the upper right corner of each frame. Here,
for the first time, the filtered simulation data do not quite reproduce the observed
experimental distribution. The mean IMF multiplicity for the simulation is shifted
with respect to the experimental value by roughly 3 units. Even with this enhanced
IMF multiplicity in the simulation, we can compare the event shapes yielded in these
events to those from experiment if we only compare events having the same IMF

multiplicity. This constraint is already required in event shape analyses by virtue of

105

 

 

 

 

 

 

 

 

iConstant 0.1289E+05 i 50.21
10 5 Mean 19.93 i; 0.9852E-02
7:? Sigma 3.089 3: 0.6922E-02
.3 4
.3 10
E 103
m
V 2
5° 10
\
% 10
1 IIIIl IIIIIIIIIIIIIIIIIIIIIIJII lIIIlIIIlIIIII
Constant 0.2040E+05 :I: 65.14
10 5 Mean 20.03 i: 0.7574E-02
:3 Sigma 2.904 :1 0.533SE-02
O
E 10"
§- 103
g" 102
\
% 10
l IIIIIILIIIILIIIIIIIIIIIIIIIIIIIIII IIIlIIIIlIIII
0 5 10 15 20 25 30 35 40 45 50
N

C

Figure C.3: Comparison of charged particle multiplicity distributions (histograms)
generated by the simulation (top panel) and measured for central events in the exper-
iment (bottom panel). Gaussian distributions have been fit to the data (solid lines)
and fit parameters are shown in the upper right corner of each panel.

106

 

 

 

 

 

10 6 Constant 14.56 :I: 0.333lE-02
Slope -0.5202 :1: 0.6315E-03
i§ 105
i3 4
E 10
g; 103
N 2
g 10
10
1 I I l I L I I I I I I I I L I I I I I I I I_ I I I I
10 6 Constant 14.12 :1: 0.5393E-02
Slope 41.4568 :1: 0.9522E-03

 

 

dP/dZ (Experiment)
1-1
a
U

 

 

 

 

102
10
l 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 L
0 5 10 15 20 25 30
.2

Figure C.4: Same as Figure C.3, except for charge distributions, fitted with exponen-
tial functions.

107

 

 

     
 
      

 

 

 

 

Constant 0.1695E+05 :t 63.61
10 5 5 Mean 8.941 :t axons-02
E? 5? Sigma 2321 j: 0.4514E-02
° :
g 10 4 5
a 5
3 _
in 10 5
V :
z3 10 2 5
E :
% 10 z
1 EIIIIIIIIIIIIIJIIIIIIIIIILIIIIIIIILIIII
Constant 0.3705E+05 i 119.7
A 5 3 Mean 5.583 :t 0.4276E-02
*5 10 Sigma 1.590 2!: 0.3008E-02
9
Z
S
10 E
ILIIIIIIIIIIIIILIIIIIIIIIIIII IIIIIIILI

 

 

0 2 4 6 8 10 12 l4 16 18 20

Nlmt’

Figure C.5: Same as Figure C.3, except for IMF multiplicity distributions.

108

the low-multiplicity distortions to event shape observables[Fai83, Llop95b].

By the above methods, we have constrained the model input parameters relating
to the emission source temperature and the charge and multiplicity distributions. In
this way, we are now able to examine both the effects of varying the momentum space
aspect ratio upon filtered simulation, and the detector acceptance effects upon the

experimentally measured values.

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