fié MN!“WWINWINL’IWWHIIWlllHlllWlfllll

   

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This is to certify that the
thesis entitled
Intermediate Mass Fragment. Multiplicities in

Heavy-Ion ColiiSIOns

presented by

Joy Ann Conrad

has been accepted towards fulfillment
of the requirements for

M. S. degree in Phy51 cs

 

 

ajor professor
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0.7639

 

 

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MSU I: An Affirmative Action/Equal Opportunity institution
encircmpms-pd

 

 

INTERMEDIATE MASS FRAGMEN T
MULTIPLICITIES IN HEAVY-ION
COLLISIONS

By

Joy Ann Conrad

A THESIS

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

MASTER OF SCIENCE

Department of Physics and Astronomy

1993

ABSTRACT

INTERMEDIATE MASS FRAGMENT MULTIPLICITIES IN HEAVY-ION
COLLISIONS

By

Joy Ann Conrad

The average multiplicities of intermediate mass fragments (IMF’s) for the most
central of nucleus-nucleus collisions in the (nearly) symmetric systems: 20Ne+WAl,
40Ar+"SSc, 84Kr-i-93Nb, and 129Xe+139La, each at a wide range of intermediate beam
energies, are studied. Cuts allowing the top 10% of the spectra of observables
assumed to be correlated with the impact parameter are utilized to select small
impact parameters. An investigation is conducted to determine the extent to which
the measurements of the average multiplicities of IMF’S are biased by an auto-
correlation between the number of IMF’S and the centrality condition. The two
observables least auto-correlated with the number of intermediate mass fragments
are identified, then used to select the most central events. The system mass and
beam energy dependance of IMF multiplicities measured in a nearly 47r detector is

then presented.

ACKNOWLEDGEMENTS

I thank Bill Llope for all of his patience in helping me with this analysis; Gary
Westfall, my advisor, for coordinating the whole project; and the entire 47r group
for collecting the data: Darren Craig, Gene Gualtieri, Stephan Hannuschke, Tong Li,

Robert Pak, Nathan Stone, Skip Vander Molen, Jaeyong Yee, and Sherry Yennello.

iii

Contents

LIST OF TABLES V
LIST OF FIGURES vi
1 Introduction 1
2 Experimental Details 5
3 Selection of Central Collisions 7

3.1 Centrality Variables ............................ 7

3.2 Auto—Correlations ............................. ‘29
4 Most Central Collisions 35
5 Discussion 43
6 Summary and Conclusions 47

LIST OF REFERENCES 49

List of Tables

3.1 The most efficient centrality variables for these data taken from the
event shape analysis of Reference [1]. .................. 9

List of Figures

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

The entrance channel mass dependance of the average multiplicity of
intermediate mass fragments versus the number of charged particles
for specified beam energies ........................

The entrance channel mass dependance of the average multiplicity
of intermediate mass fragments versus the charge detected at mid-
rapidity for specified beam energies ...................

The entrance channel mass dependance of the average multiplicity
of intermediate mass fragments versus the total transverse kinetic
energy for specified beam energies ....................

The system mass dependance of the emission of intermediate mass
fragments as a. function of the total transverse kinetic energy of the
particles ...................................

The entrance channel mass dependance of the average multiplicity of
intermediate mass fragments versus the number of hydrogen isotopes
for specified beam energies ........................

The entrance channel mass dependance of the average multiplicity of
intermediate mass fragments versus the charge of the hydrogen and
helium isotopes for specified beam energies ...............

The beam energy dependance of the average multiplicity of interme-
diate mass fragments versus the number of charged particles for each
of the four systems ............................

The beam energy dependance of the average multiplicity of interme-
diate mass fragments versus the charge detected at mid—rapidity for
each of the four systems .........................

The beam energy dependance of the average multiplicity of interme-
diate mass fragments versus the total transverse kinetic energy for
each of the four systems .........................

The beam energy dependance of the average multiplicity of interme-
diate mass fragments versus the number of hydrogen isotopes for each
of the four systems ............................

The beam energy dependance of the average multiplicity of interme-
diate mass fragments versus the charge of the hydrogen and helium
isotopes for each of the four systems ..................

vi

12

13

14

16

18

19

21

22

23

24

3.12

3.13

3.14

4.1

4.2

4.3

5.1

The beam energy dependance of the means and the widths of the
intermediate mass fragment multiplicity distributions for the events

selected by 10% cuts on each of the five centrality variables separately 27

The average values, widths, and reduced widths of the multiplicities of
intermediate mass fragments in the events selected by ~10% cuts on
the largest values of six centrality variables versus the beam energies

for 2°Ne+27Al and 40Ar+45Sc entrance channels ............

The average values, widths, and reduced widths of the multiplicities of
intermediate mass fragments in the events selected by ~10% cuts on
the largest values of six centrality variables versus the beam energies

for 84Kr+93Nb and 129Xe+139La entrance channels ...........

The mean number of intermediate mass fragments versus the beam
energy for the events selected by the two most efficient centrality

variables according to Reference [1] ...................

The mean number of intermediate mass fragments versus the beam
energy for the events selected using the two centrality variables that
were seen to be free from auto-correlations with the number of inter-
mediate mass fragments and of relatively high efficiency according to

Reference [1] ................................

The mean number of intermediate mass fragments as a function of

projectile energy for all of the systems .................

A qualitative comparison of the average number of intermediate mass
fragments versus the entrance channel mass in symmetric (this work)

versus asymmetric (Reference [3]) entrance channels ..........

vii

32

37

39

45

Chapter 1

Introduction

To better understand the processes involved in nuclear collisions, experiments are
performed which collide a projectile nucleus and a stationary target nucleus and uti-
lize electronic devices to measure the particles and fragments that are emitted. An
analysis is then conducted using the detected particle information to understand the
mechanisms governing nucleus-nucleus collisions. There many general theories that
may describe the nature of the processes controlling the emission of detected parti-
cles. These models differ, however, in terms of the predicted number of intermediate

mass fragments, i.e. particles with 3 3 Z ,E, 20 [6] [10,11].

In one such description, the first stage of the reaction is a compression of the
projectile-target system followed by an expansion to low densities. Density fluc-
tuations are then responsible for the formation of detectable intermediate mass
fragments [13—15]. Another possibility is that the collision compresses and expands
leaving an excited nuclear residue that decays via the emission of particles [16,17].
This decay of such a residue may occur in one of two general ways: “sequentially”
through the emission of predominately light charged particles [21] such as hydro-
gens or heliums over long time scales, or “simultaneously” through the emission of

relatively large numbers of intermediate mass fragments [22] (IMF’S for which 3 _<_

1

Z ,S, 20) over much shorter time scales.

Sequential decays have been observed in excited nuclear systems with relatively
low total excitation energies. These relatively slower decays (~1000 fm/c) [18]
are generally understood to behave statistically with well defined transition rates
based on the conservation of energy and angular momentum. Such excited systems
periodically emit particles that reduce the excitation energy of the system by discrete
amounts. These amounts are determined by the internal energy needed for the decay
and the kinetic energy of the exiting particles. The mass and kinetic energy of the
particles emitted are determined by the available excitation energy of the system
as well as the height of the Coulomb and angular momentum barriers over which
the particles must pass in order to escape. In this type of decay, large fragments
may be emitted only in the first stages of the decay; however, due to generally lower

barriers overall, light particle emission is always favored.

Sequential decays are reminiscent of the cooling of liquids by evaporation. This
type of decay should not be confused with the complete vaporization of the sys-
tems that occurs at beam energies on the order of hundreds of MeV / nucleon. This
“gaseous” phase is also characterized by the experimental detection of large numbers

of predominately light charged particles.

At higher excitation energies than those exhibiting sequential decays, multifrag-
mentation becomes an important process characteristic of the “liquid-gas” phase
transition region. These decays are more rapid (~100 fm/c) [19,20] than sequen-
tial decays and result in the emission of relatively many more IMF ’3. There have
been many theories attempting to describe the processes governing multifragment

emission, and these models have met varying degrees of success.

The number of intermediate mass fragments (Nimf) emitted in a nucleus-nucleus

collision is thus expected to be sensitive to the characteristics of the mechanisms
governing the evolution of the excited nuclei formed in heavy-ion collisions. Since
the transition from the liquid phase to the gaseous phase is expected to correspond
to maximal numbers of lMF’s [3] [8], IMF multiplicity distributions can be used to

locate such transition regions if they occur.

It is the purpose of this work to accurately measure the average number of inter-
mediate mass fragments (< Ninlf >) detected in central nucleus-nucleus collisions.
A wide range of beam energies and entrance channel masses are available for this
study. From this, it is hoped that particular regions dominated by sequential or
multifragmentation decays will be identified, as well as possible transition regions

between the two.

The most central collisions of the projectile and target nuclei result in the most
violent collisions and lead to the largest possible excitation energies and systems for
a given entrance channel and beam energy. The excitation energies in these collisions
should also be more equilibrated, as compared to peripheral collisions, due to the
larger number of nucleon-nucleon collisions. The most central collisions also result
in systems with low angular momentum and hence approach conditions for which

the center of momentum frame is the rest frame of the excited nuclear system.

Information concerning the impact parameter is then needed to select these most
central collisions. Unfortunately, the impact parameter cannot be directly measured
in a nuclear collision - it must be inferred from the experimental observables. Six
such observables will be assumed to be correlated with the impact parameter, and

cuts of each of these will be used to select the most central collisions.

Precaution will be taken to assure that the observables used to determine the

impact parameter are not otherwise related to the multiplicity of IMF’s. Any “auto-

correlation” between the observables used to select central collisions and the mul—
tiplicity of the IMF’s is a source of bias. Great care will be taken in this study
to locate the observables correlated with the impact parameter that are not also

auto-correlated with the mean number of intermediate mass fragments.

This thesis is organized as follows: Chapter 2 will contain experimental details in
which the specifics of the detector, targets, and projectiles will be discussed. Chapter
3 will determine the most appropriate centrality variables for this analysis. Chapter
4 will display the results of the thesis - the mean number of IMF’s as a function of
projectile energy and entrance mass for the most central collisions. Chapter 5 will
compare these results to previous work, and Chapter 6 will provide a summary, as

well as draw some conclusions based on this analysis.

Chapter 2

Experimental Details

The experimental data from nucleus-nucleus collisions was collected with the Michi-
gan State University 47r array [2] at the National Superconducting Cyclotron Lab-
oratory (NSCL) using beams extracted from the K1200 cyclotron. Four systems
are analyzed over a wide range of beam energies. The (nearly) symmetric systems
studied are: 20Ne+27Al at 55, 75, 95, 105, 115, 125, 135 MeV/nucleoni; 40Ar+45Sc
at 15, 25, 35, 45, 65, 75, 85, 95, 105, 115 MeV/nucleon; 84Kr+93Nb at 35, 45, 55,
65, 75 MeV/nucleon; and 129Xe+139La at 25, 35, 45, 50, 55, 60 MeV/nucleon.

The targets consisted of 1 mg/cm'2 of natural aluminum, 1.6 mg/cm2 of natural
scandium, 1 rug/cm2 of natural niobium, and 1 mg/cm2 of natural lanthanum. The

projectile beam intensities were approximately 100 electrical pA [7].

The outer layer of the main 47r ball consists of 170 fast/slow plastic scintilla-
tor telescopes known as “phoswich” detectors and a forward array of 45 phoswich
telescopes. The detectors in the main ball are arranged in 20 hexagonal and 10
pentagonal subarrays. In front of the main phoswich detectors are an additional 30
Bragg curve counters. The five forward most Bragg curve counters are subdivided
into six subelements each for a total of 55 separate BCC’s. These Bragg curve de-
tectors use an ionizing chamber of either P5 (95% argon, 5% methane) or CgFG gas

5

at pressures of 500 and 100 torr, respectively.

The scintillators use the well known phoswich gating technique to determine the
kinetic energy and charge information of particles that pass though the fast plastic
and stop in the slow plastic. The BCC’s provide similar E and AE information
for those particles that pass through the BCC’s and stop in the fast plastic. The
overall kinetic energy thresholds are ~20 MeV for protons, ~3 MeV/nucleon for
Z=3 fragments, and ~5 MeV/ nucleon for Z=12 fragments. The 471' array is capable

of efficiently detecting particles of charges ranging from Z=1 to Z ,S, 20.

An event was defined as collisions resulting in at least two hits registered by
any two phoswich detectors. Hits with Z values that fell ”between bands” in the
calibration of the electronic signals or that stopped in the fast plastic but contained

no BCC information were rejected.

The overall geometrical acceptance of the detector is approximately 85% of 47r.
The acceptance in. the main part of the ball consisting of the scintillators and Bragg
curve counters is greater than ~90%. This region covers laboratories angles of
~180 to ~1600. The forward array covers polar angles from N60 to ~180 and
has a geometrical acceptance of ~50%. Due to the relatively poor acceptance for
forward angles, the most central collisions will be selected for this analysis. For
these, the experimental results are less significantly compromised by the geometrical

inefficiencies of the apparatus.

Chapter 3

Selection of Central Collisions

3.1 Centrality Variables

In order to determine the impact parameter and hence select central collisions, the
assumption was made that particular experimental observables are correlated with
the impact parameter [I] [4]. The following six experimental observables can be

used to select the most central collisions in this data:

(i) the number of charged particles (Nchgd) detected in each event.

(ii) the total charge of the detected particles in a software gate about

“mid-rapidity” (ZMR). This is thus defined to be the sum charge of all

particles in the events for which:

-75yT S ypart < 753””

where the rapidity, gm”, of the particle is given by:

y _ 1 In Eparz + PPM, cos (9
part — ‘— '—
2 Epart — part (:08 6

 

y.

In this equation, Ew” represents the total energy, PPM, the total momen-

tum, and 0 the polar angle of the particles in the center of momentum

7

frame. The rapidity of the center of momentum (y) with respect to the

laboratory frame is given by:

1 1+6 Pp

=—1-——— b:——-—
y 2111—12 ’ Ep+MT’

in which MT represents the mass of the target (in MeV).

(iii) the transverse kinetic energy (KET) of the particles in the events.
This variable is defined as:

KE ar .
KET 2 Z —K——Ep—P—tsm20 * 100,

where 0 is the polar angle of the particle with respect to the beam axis,
KEPW is the kinetic energy of each particle, and KB}: is the projectile
kinetic energy in the laboratory. The factor of 100 is used for conve-

nience.

(iv) the total number of hydrogen isotopes detected (NH), which is the
sum of all of the protons (A21), deuterons (A22), and tritons (A23)

detected.

(v) the total charge of all of the fragments registered in the detector

(248;) in one event.

(vi) the total charge measured for hydrogen and helium isotopes (Zch),
which is the sum of the charge of all light charged particles detected in

an event.

A previous study conducted by Llope et al. [1] determined which of these cen-
trality variables was the most efficient in selecting central collisions in these data.
This study has linked the emission patterns or “shapes” of the emitted particles with
the impact parameter through an observable called the sphericity. The analysis was
based on the assumption that central collisions would result in the most isotropic

or spherical emission patterns of the events in the center of momentum frame.

The conclusion was drawn in Reference [1] that the variables that most efficiently
allow the selection of the most central collisions vary with both the entrance channel
mass of the system and the incident energy of the projectile. The best two variables
for the selection of the smallest impact parameters from Reference [1] are shown in
Table 3.1 for the systems and energies used herein. It should be noted that auto-
correlations between these centrality variables and the event shapes were shown to
be negligible if “moderate” centrality cuts were used (see Section 3.2). This does
not mean that these variables are not auto-correlated with the observable used in
this study, i.e. the number of intermediate mass fragments.

Table 3.1: The most efficient centrality variables for these data taken from the event
shape analysis of Reference [1].

 

 

 

 

 

 

 

 

 

 

 

system Projectile Energy (MeV/nucleon)
15125135145 55=>
2ONe+27Al KET
(Asysz47) NH
40Ar+4SSC NH NH KET
(A3,,8285) Zch KET NH
84Kr+93Nb ZMR KET
(Asys2177) NH ZMR
129Xe+139La NH ZMR ZMR
(Asys=258) ZLCP ZLCP KET

 

 

 

 

 

 

10

As mentioned above, the number of IMF’s detected in a heavy-ion collision is
expected to be dependant on the mechanisms governing the nucleus-nucleus colli-
sions. In order to study the most extreme conditions in nuclear matter, the most

central collisions are of interest.

The mean number of IMF’s is plotted versus values of each of the centrality
variables in Figures 3.1- 3.11. These mean values are plotted for four overlapping
projectile kinetic energy regions for each of the centrality variables. Figures 3.1- 3.6
Show the variations in the mean number of IMF ’s as a function of entrance channel
mass. In these Figures 20Ne+27Al is represented by asterisks, 40Ar-i-“sSc by circles,
84Kr+93Nb by triangles, and 129Xe+139La by stars. The error bars are statistical

only, and these results have not been corrected for the experimental inefficiencies.

Figure 3.1 represents the experimental dependance of < Nimf > on the charged
particle multiplicity, Nchgd, for four beam energies: 35 MeV/nucleon (upper left),
45 MeV/nucleon (upper right), 55 MeV/nucleon (lower left), and 75 MeV/nucleon
(lower right). In these frames, there is a significant difference of IMF multiplicities
for the various entrance channels of the same beam energy. The heaviest systems
display the largest values of < [Vim] >, and the lighter systems display the smallest.
The values of < Nfinf > in the lighter systems, Ne+Al and Ar+Sc, display maximum
average values near ~2, while the heavier systems, Kr+N b and Xe+La, show much
larger numbers of IMF ’s, on average. The largest mean number of IMF ’3 occurs for

Xe+La collisions at the largest available beam energy (~55 MeV/nucleon).

These frames show that the average numbers of IMF ’s detected depends on the
projectile - target combination and the beam energy. The heaviest systems emit the

largest numbers of lMF’s while the lightest systems emit the fewest.

11

In most cases in Figure 3.1, no maxima are statistically significant for a particu-
lar value of a centrality variable. Such maxima would indicate that an intermediate
impact parameter would lead to maximal emissions of IMF’s rather than a cen-
tral collision. It was decided that the possible maxima in the Xe+La system at
55 MeV/nucleon and the Kr+Nb system at 75 MeV/nucleon were not statistically

significant in these data.

Figure 3.2 represents the experimental values of < Nfinf > versus the total
detected charge at mid-rapidity, ZMR. This plot is quantitatively different from the
< Ni-mf > versus Nchgd plot of Figure 3.1 in that there is no significant variation
of the mean number of IMF ’s with the entrance channel mass. This is notable
considering that the entrance channel masses for the four systems range from 47 to
268 nucleons. The large number of IMF’S from ZMR cuts and the invariance with
respect to the entrance mass could be the result of auto-correlations between ZMR

and < NM, >.

Figure 3.3 presents the experimental values of < NM; > versus the total trans-
verse kinetic energy, K ET, of the events. Overall, the KET dependance illustrates
a larger variation of the mean number of IMF’s detected versus the system mass
at specific beam energies than that observed versus the Nchgd or ZMR variables in

Figures 3.1 and 3.2.

The slopes of the mean number of IMF’s as a function of KET in Figure 3.3
are observed to increase in apparently discrete increments for increasing entrance
channel mass. Further investigation into this effect is illustrated in Figure 3.4.
The top two plots in this Figure present the system mass dependance of the slopes,
and the bottom four plots present the system mass dependance on < Nimf > for

specified values of KET.

12

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/\ 6 -
'é : ll 35 MeV/nucleon; 45 MeV/nucleon
Z. : ”It .—
V 5 r—- oAr'I'SC -- * oAr'I'SC
: AKr-i-Nb : t [ AKr+Nb
: if. * Xe+La : .: a Xe+La
_ .* __ :44
_ *[+ _ AA
: t I *
I— * h-
3 _ *f '—
: f : if
L I. 2:
2 — g i— kg
_ 1.
Z a : 8
l ":0? ['38
~55” _ca
fl I I I I I I I I I I I I I I Ii L J I I L I I I LL I I I I I J
/\ 6 2 —
E : _ 55 MeV/nucleon: 75 MeV/nucleon
2" ~ **
V 5 7’ if ] xNe+Al - {Ne+Ai
: * AKF'I’Nb : A oAf'I'SC
_ :43 ,, Xe+La - iii A Kr+Nb
4 3— 29 — A #1
Z *A I A
_ if _ A
3 — *A -— A
I A : A
._ A _ A
h 8 ’— A
2 .. a L— Ag
: a : 5559
.. a ..
l ;- f i :‘ sf
I I I I I I I I I I I I I I I I _ I I I I I I I I I I I I I I I I
O 20 4O 60 O 20 4O 60
Nchgd Nchgd

Figure 3.1: The entrance channel mass dependance of the average multiplicity of
intermediate mass fragments versus the number of charged particles for specified
beam energies is plotted above. The symbols represent the systems available for
the beam energies: 35 MeV / nucleon (upper left), 45 MeV/ nucleon (upper right), 55
MeV/nucleon (lower left), and 75 MeV/nucleon (lower right). Corrections due to
auto-correlations and experimental acceptance have not yet been made.

< Namf>

12

10

13

 

  

I

35 MeV/nucleon

 

 

 

 

45 MeV/nucleon

 

 

 

 

  

 

 

 

 

 

  

 

 

 

 

 

— oAr-i-Sc - OAr-i-Sc
_ AKr+Nb __ [ AKr+Nb
_ * Xe+La * Xe+La
i l
_ [ flit l
. , I
MllilllllliiilllllllllJ_LIL111IJIIJILIIIIl
f 55 MeV/nucleorr- 75 MeV/nucleon
1— *Ne+Al — *Ne-i-Al
._ AKr+Nb _. oAr+Sc
_ l *XC'I'LC AKF'I'Nb
_ .Il _
it}.
it __
I I ‘i+
-
LLIIIIIIIIIJI_I_IIIIII IIIIIIIIIJLJIIIIIIII
25 50 75 100 25 5O 75 100
ZMR ZMR

Figure 3.2: The entrance channel mass dependance of the average multiplicity of
intermediate mass fragments versus the charge detected at mid-rapidity for specified
beam energies is plotted above. The symbols represent the systems: 20Ne+27Al
(asterisks), 40Ar+“53c (circles), 84Kr+93Nb (triangles), and 129Xe+139La (stars), for
the beam energies indicated above. The data above has not been corrected for
auto-correlations between the observables or for the experimental inefficiencies of
the detector.

l4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

A? ,0 _ 35 MeV/nucleon__ Z[5 MeV/nucleon
Z- A +S A +3
— r c *- r c
V :Krmb :Kr-i-Nb
8 — i * xe+LO — I" * XB'I‘LC
i
_ Hi] *- {ff
6 — ”tiff I] — i
t
_ I: A _ * M11315]
t
4 __ * AA 4+ + * AA I 4
* AAA +j’ — * AA
AA i ¢¢ * AA
r- ** AA 6) — *AA @
t A 000 * A 000 I
2 _ *A 00000 + ¢ +4 _*A
_tg; A A _mf A
fi‘IIIIIIIIIIIJJIJIIJ JIIIIIIIIIIIIIIIIII
/\E 10 _ 55 MeV/nucleon_ 75 MeV/nucleon
Z. N Al N Al
*' e+ " e+
V :Kr+Nb :Ar-i-Sc
8 2 +1 *Xe+Lo _' AKr+Nb
*#
.* l [l
6 — . e
* A4] AAAA+
_ * AA _ A
* AA i AA
4 _ * AA __ AA
* A A ¢+
__ * AA _ AA ¢
*A ¢
2 _ * A [if __ AA 0 000000 +4,
*A **+* A 00 * +
i— A W* _ 00 H +*
«A W . ’l‘ A0°
MIIIIIIIJIIIIIILIJ IJIIIIIIIIIIIII

 

0 1o 20 30 4o 10 20 so 40

Figure 3.3: The entrance channel mass dependance of the average multiplicity of
intermediate mass fragments versus the total transverse kinetic energy for specified
beam energies is plotted above. The symbols represent the systems available for
the beam energies: 35 MeV / nucleon (upper left), 45 MeV/ nucleon (upper right), 55
MeV/nucleon (lower left), and 75 MeV/nucleon (lower right). Corrections due to
auto-correlations and experimental acceptance have not yet been made.

15

The top left plot is the slopes of < Nim; > versus KET lines from Figure 3.3
as a function of incident projectile energy for the four systems. The Ne+Al system
is represented by asterisks, the Ar+Sc system by circles, the Kr+Nb system by
triangles, and the Xe+La system by stars. As illustrated in this plot, the heavier
two systems, Kr+Nb and Xe+La, exhibit the largest variation in the slopes of the

mean values.

The top right plot is the slope of < Nimf > versus KET plotted as a function
of the charge (or mass) of the four systems. The three beam energies depicted
in this plot are: 35 MeV/nucleon (squares), 45 MeV/nucleon (diamonds), and 55
MeV/nucleon (open crosses). This plot indicates that there is a general linear re-
lationship between the d< Nmf > /d1{ ET and the charge of the system for a
35 MeV/nucleon beam energy. This linear relationship breaks down, however, for
higher projectile kinetic energies, perhaps due to the uncorrected experimental in-

efficiencies.

The bottom four plots in Figure 3.4 represent the relationship between the
mean number of IMF ’s and the total entrance channel for several specified values
of the total transverse kinetic energy. Four beam energies are plotted separately
as indicated. Three different KET values were chosen: 8 (solid circles), 12 (solid
squares), and 16 (solid triangles). As illustrated by the four bottom plots, for one
value of the total transverse kinetic energy per event there is a linear relationship

between average number of IMF’s and the entrance channel charge.

In similarity to Figures 3.1, 3.2, and 3.3, the experimental values of < Nimj >
versus the number of hydrogen fragments, N H, and the total charge of the hydrogen
and helium fragments, ZLop, are plotted in Figures 3.5 and 3.6, respectively. These

Figures display large differences in the mean number of IMF’s detected at different

16

 

x Ne + Al E335 MeV/nucleon
0 Ar + Sc :0 45 MeV/nucleon
A Kr + Nb £955 MeV/nucleon
* Xe + Lo

 
  

d<N.,,,,>/dKET
O O O O O
-* N (:4 -> U"
XIIIIT
IIIIIRII’IIIII11

III

 

 

 

IIIIlIIIIIIIIII[_IIIIIIJIIILIIIIIIIIIIII

 

 

 

  
 
 
  

 
 
  

0
4O 80 120 25 50 75 100
Projectile Energy (MeV/nucleon) A8,,
Ag 35 MeV/nucleon 45 MeV/nucleon
E _
2'" 6 _0 KEv=8 _. KE1=8
V e KE.=12 e K£,=12
2 _ __
IIIIIILIIILIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIJII
55 MeV/nucleon 75 MeV/nucleon
6 k0 KE.=8 #. KE,=8
. KE,=12 ‘ KE1=12
4 L‘ “5“” _A KE1=16
2 .2. _.
Jill lillllllllllilllli ILL lllililllllIlIlllI

 

 

 

 

25 5O 75 100 25 50 75 100
A

sys

Figure 3.4: The system mass dependance of the emission of intermediate mass
fragments as a function of the total transverse kinetic energy of the particles is
plotted above. The top two plots are the slopes of the mean number of intermediate
mass fragments versus the total transverse kinetic energy of the particles per event.
The top left plot is the slopes as a function of beam energy, and the top right plot is
the slopes as a function of the charge of the system. The bottom four plots are the
mean number of intermediate mass fragments for one total particle kinetic energy
value as a function of the charge of the system.

17

beam energies. These plots are also unique in the fact that plateau regions are
observed which are not seen for the Nchgd, ZMR, or KET dependance of < Nimf >
shown previously. The variables NH and Zch are closely related by definition,
and they each indicate that there is either a NH or ZLCP threshold where the
number of IMF’s emitted remains constant (a plateau region) or a threshold where
the additional production of light particles results in a decrease in < Nimf > (a
maximum). Also, these plots display the lowest average number of IMF’s generated

as compared to the curves shown in Figures 3.1- 3.3 of the other centrality variables.

It should be noted that < Niynf > from events with small NH or ZLCP values
in Figures 3.5 and 3.6 are considered to correspond to events with large impact
parameters. These values are then the result of experimental inefficiencies that are

the most important in peripheral collisions.

The experimental < Nan; > values are also plottedversus the values of the var-
ious centrality variables in Figures 3.7- 3.11 for each system separately at specified
beam energies. The symbols each represent one beam energy: 15 MeV / nucleon, solid
triangles; 25 MeV/nucleon, diamonds; 35 MeV/nucleon, circles; 45 MeV/nucleon,
open crosses; 55 MeV / nucleon, squares; 75 MeV/ nucleon, open triangles;

115 MeV/nucleon, stars; and 135 MeV/nucleon, asterisks. The error bars depict the

statistical errors of the mean values.

The plots of < Nimf > versus Nchgd in Figure 3.7 illustrate a significant beam
energy dependance in the lighter two systems, Ne+Al and Ar+Sc, but a lesser beam
energy dependance for the two heavier systems, Kr+N b and Xe+La. This effect is
in part due to a visual illusion caused by the smaller range of beam energies covered
by the larger systems. The Ne+Al system results in the smallest numbers of IMF ’s

of the four systems. The number of IMF’s for this system was seen to decrease for

18

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/\ I c
E35 :_ 35 MeV/nucleon? 45 MeV/nucleon
Z“ E :
V 3.2 :- oAr+Sc :‘ oAr+Sc
E A Kr+Nb : l A Kr+Nb
2.8 x +Lo :- Xe+La
[é— * e E *WTIIII] ] t
._ *- *-
2i4 : *fl'w I E— ‘*‘ l
C + ‘ l :
2 :- -* [i :— _ AL‘F‘A’Ai‘C‘iAA
: ' ' : *A _ +
1.6 :— -i~A-'A"A"A‘ ] :‘ .A' ¢ ¢+
_ 'A- -A- :
_ A ._ +
I .2 :— ‘A' ¢+ :— ‘A‘
c + :
a ‘A' - «r
0'8 7.0 3:5” 0000
o 4 *— Oo :—
: I I I I I I I I I I I I I I : I I I I I I
/\ 3 :
E35 E- 55 MeV/nucleon} 75 MeV/nucleon
z" E _ E
V 3 2 :- *.*+* LII“ ,‘ Ne+Ai _— x Ne+AI
E ,3" -+-'. I. A Kr+Nb E o Ar+Sc
2.8 L— ‘*‘ l i, Xe+La :' A Kr+Nb
: a.» l— ”A
L .._ .AA'A. .
2.4 : “fl" : ‘A’A- 4+
: . :A-A‘Aa . . ' C 'A' ‘ ' '
2 C.— -t~ AA AI _— ‘A'
E A + E ‘A’ o ' '
1.6 E.— '*"A' :— 'A'
1 .2 E- -A- E- '4'
r- '*' .—
0.8 :1; ' E‘ .A-OOOOOOOO+
O . 4 E‘aaeWx-xfi a] §:*Wm*
7 JI I I I E. I I I I I I I I I I I I I I If I I I I I I I I
O l O 20 O ‘l O 20

Figure 3.5: The entrance channel mass dependance of the average multiplicity of
intermediate mass fragments versus the number of hydrogen isotopes for specified
beam energies is plotted above. The symbols represent the systems: 2°Ne+27Al
(asterisks), 40Ar+45Sc (circles), 84Kr-i-93Nb (triangles), and 129Xe+139La (stars), for
the beam energies indicated above. The data above has not been corrected for
auto-correlations between the observables or for the experimental inefficiencies of

the detector.

19

 

_-

IIII
g;

 

 

 

I I I I I I l

I

A? _ 35 MeV/nucleon; 45 MeV/nucleon
5 : oAr'I'SC f: oAr'I'SC
- AKr+Nb "' AKV'I'Nb
:- *Xe+Lo _— *Xe-rLo
e :—
i will 3 :Me 1l
** ”n+1 __ * I l 1r
. . [NH : {4 “‘4‘“ I
. _ *A i
s A 4‘31 s w J.
l A
t— A” _- ‘A I
. .. . ,1
] ¢

 

 

 

 

 

I I I I I I l l l

 

 

IIIIII

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A? ; 55 MeV/nucleon__ 75 MeV/nucleon
\Z/ I x Ne+Al : x Ne+Al
— A Kr+Nb — o Ar+Sc
[_— l * XB'I'LO r A Kr+Nb
:_ fmfir‘ +1] ] [[1' L l
_ '. I C ]
- *t 4 ] - Mi# 5
{PM ”Mac A A
E” I. if] E A i]
l" 1AA *- A
“Ft ’25 If AA ]
.— *A *- A
: .2“ E
_ I I I I I I I I I I I I_ I I J+ I I I I I I I
O 20 40 O 20 40

Z LCP

ZLCP

Figure 3.6: The entrance channel mass dependance of the average multiplicity of
intermediate mass fragments versus the charge of the hydrogen and helium isotopes
is plotted above. The symbols represent the systems available for the beam ener-
gies: 35 MeV/nucleon (upper left), 45 MeV/nucleon (upper right), 55 MeV/nucleon
(lower left), and 75 MeV / nucleon (lower right). Corrections due to auto-correlations
and experimental acceptance have not yet been made.

20
increasing beam energies.

The Ar+Sc system is the only one of the four in Figure 3.7 in which the curves
representing < Nimf > for individual beam energies significantly cross at some point.
This is an indication of a possible change in the reaction mechanism for increasing
beam energies in nearly central collisions. Such transitions are characterized by

maximal average multiplicities of IMF’s.

Figure 3.8 depicts experimental < Ngmf > values versus ZMR for several beam
energies in each of the four systems. In similarity to Figure 3.2, there is very little
variation of the number of IMF’s detected at specific values of ZMR versus the beam

energy in all systems.

The experimental values of the < Nimf > versus KET are plotted in Figure
3.9. In contrast to Figures 3.7 and 3.8, no significant beam energy dependance
is seen for the Ne+Al system, but a significant dependance on the beam energy
is seen in the three heavier systems, Ar+Sc, Kr+Nb, and Xe+La. The average
number of IMF’s in both the Kr+Nb and Xe+La systems increases monotonically
with increasing beam energies. The opposite effect with respect to the beam energy

was observed for the Ne+Al system in Figure 3.7.

The most significant variations between the four systems for specified beam
energies occur in the plots of < Nan; > versus NH and Zch shown in Figures
3.10 and 3.11. These plots exhibit minima, maxima, and plateau regions. The
curves representing each beam energy for the Ar+Sc system also cross, similar to
that observed in earlier plots, indicating a possible transition. Overall, < Nimf >
versus NH and Zch result in the lowest mean numbers of IMF’s (see also Figures

3.5 and 3.6).

1.4

< Nimf>
it;

0.8

0.6

0.4

0.2

21

 

Ne+Al

 

MeV/ nucleon
use
A75
* 115

A ,“135

 

 

 

 

 

IIIIIIIflIIIIIlIIIIIIIIIIIIIIIRI

 

IIIIIII

 

 

20

CD

 

30

 

 

MeV/nudeon
o 35
A 55
A 75

[ Kr+Nb

 

 

42%-

IITIIIIIIIIIIIIIIIIIIIII]

III

 

fIIIIIIIIIIII

1

 

 

0 20 40

60

Nchgd

43 (F (D

NJ LN

 

V

 

IIIIIIIIIIIIIIIIIIRT]

O A
09 A;
f
U

IIII

 

+0],

0
0AA

i

IIIIIIIIIIIIII_L

Ar+Sc

 

 

MeV/nucbon
'15
o 35
A 75
* 115

 

l 1+

 

 

CD

10

20

3O 40

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

 

 

 

0 20

IIIIIIIIIIIIIIJ

40

Xe+La

 

 

MeV/ nucleon
o 25
o 35
e 45
A 55

 

 

6O
Nchgd

Figure 3.7: The beam energy dependance of the average multiplicity of intermediate
mass fragments versus the number of charged particles for each of the four systems
is plotted above, uncorrected. These systems are: 2oNe+27Al (top left), 40Ar+“53c
(top right), 84Kr+93Nb (bottom left), and 11"9Xe+139La (bottom right) for select
beam energies in MeV / nucleon represented by the symbols indicated above.

22

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/\ 4 _ 9 __
E36 _E__ Ne+A| E Ar+Sc
Z"- i E MeV/nucleon 8 E— MeV/nucleon
p. 55 : 15
V 3 2 E :75 7 :- :35
2 8 f. 1,,115 E A 75
‘ 5 g x 135 6 :- *115
2.4 E— E
E 5 1.—
2 E— gg”? 4 5- h+
1.6 :— DgA E i
5_ BA 3 E—
1.2 E W E
0.8 E— rm 2 E—
0.4 ” 1 :-
11l1111l1111l1111 -'111l1111l111
O 10 20 30 40 O 20 4O 60
A? 10 r Kr+Nb 14 — Xe+L°
z" MeV/nucleon - MeV/nucleon
“ 35 25
V :55 12 335
8 2 A75. (>45
_ 10 A55
6 2 8
4 _ 6
' 4
2 _
_ 2
II I I I I I I I I I I I I I I I II - I I I I I I I I I I I I I I I
0 25 50 75 100 0 4O 80 120

Z... ZMR

Figure 3.8: The beam energy dependance of the average multiplicity of intermedi-
ate mass fragments versus the charge detected at mid-rapidity for each of the four
systems is plotted above. The select beam energies are represented by the symbols
as: 15 MeV/nucleon (closed triangles), 35 MeV/nucleon (circles), 45 MeV/ nucleon
(open crosses), 55 MeV/nucleon (squares), 75 MeV/nucleon (open triangles), 115
MeV/nucleon (stars), and 135 MeV/nucleon (asterisks), for the four systems as
indicated above. Correction have not yet been made for auto-correlations or exper-
imental acceptance.

23

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

4 _ 7 _
AESG E Ne+A| : Ar+Sc
Z” . E MeV/gtgcleon 6 E— MeV/qgcleon
V3.2 E— 07 z V
- 5 _ 35
28 E. 2115 5 ;— :75

' E x135 - i *115

2.4 E- 3 +
P 4 .—
2 E— : i “i
1.6 E’— J’ +H 3 E" 1 (14+r
1.2 E- * E 2 E. +
: A :

0.8 E- “) :
E 1 -

0.4 r 4 I + +
IJIIIIJIIIIIIIIJI 111120I1111l1111
O 10 20 30 4O 0 4O 60

8 _ O _

A? ; Kr+Nb 9 E Xe+La
Z“ 7 T MeV/nucleon E MeV/nucleon
v E l 8 :— 1 1

6 :— 4%11 l:75 E. Q i :45

A1513 7 —
5 :— Ann :1 E 53° 555
.1 o1 62- .200.
'- O 1-
4 :— An 000 (I) + 5 :l- :00 i++
3 :— 530 : D00 00°06
: I300 I) 3 E. [300000
2 '_ 600 ¢ 3 [Foo0
: 80 E_ *3,"
1 is) l 1% ii 2 590°
:60 1 933°
fiumuuluuluu luluullullmr
O 10 2O 3O 4O 0 10 20 3O 4O
KET KET

Figure 3.9: The beam energy dependance of the average multiplicity of intermediate
mass fragments versus the total transverse kinetic energy for each of the four systems
is plotted above, uncorrected. These systems are: 20Ne+27Al (top left), 40Ar+453c
(top right), 84Kr+93Nb (bottom left), and 129Xe+139La (bottom right), for select
beam energies in MeV / nucleon represented by the symbols indicated above.

24

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
  

 

 

 

 

 

A? E Ne+A| 2.4 L Ar+Sc
Z‘O.7 _— MeV/nucleon : MeV/nucleon
O 5 L A 75 2 _— o 55
‘ : * 115 _ 1 A 75

_Q- X 135 : A. * 115
0.5 "_— 1.6 *—
0.4 32;... :
r— - 1——
E'A Q-AEM_ 1.2 :
O . 3 _— Q'fi' gfi - 'LI ‘*
: +.A *W *3 ‘1: O . 8 E)- AA-‘A' g. I
0.2 ;* + *9!- ‘il‘ L0 A' 4".“ . I11»
I + ** *"l" 11* h 0 30
0.1 :_ *4. (f 0.4 512:.» 04,
:LLIIIIIIIIIIIIJILJL :III'IVVI'IIIJIIIIIIIIIIIIII
O 4 8 12 16 20 O 5 10 15 2O 25
4.5
A? E Kr+Nb 5 ;_ Xe+La
Z.- 4 :— MeV/nucleon : MeV/nucleon
I 35 _ 25
V 3.5 _— 355 — Co) 35
3 E A 75 4 _— 0 45
:— _ A 55
2. 5 E— A gap-13:74:44- 3 E. 00W
.:_ KBD‘D'Q- . _
2 C 86- 0¢+ .+ : “00000” +
E a °<> I 900
1 E 8 i 1 ” $8°°°°°°°¢l
- 0
0.5 $38 no
:IIIIIIIIIIIIIIiIIIIIILIL IIIIIIIIIIIIIIIII
O 5 10 15 2O 25 O 10 2O 30
NH NH

Figure 3.10: The beam energy dependance of the average multiplicity of interme-
diate mass fragments versus the number of hydrogen isotopes for each of the four
systems is plotted above. The select beam energies are represented by the symbols
as: 15 MeV/nucleon (closed triangles), 35 MeV/nucleon (circles), 45 MeV/nucleon
(open crosses), 55 MeV/nucleon (squares), 75 MeV/nucleon (open triangles), 115
MeV/nucleon (stars), and 135 MeV/nucleon (asterisks), for the four systems as
indicated above. Correction have not yet been made for auto-correlations or exper-
imental acceptance.

0.6

0.5

<(Nmfi>

0.4

0.3

0.2

0.1

25

 

 

 

 

 

 

 

25

 

 

 

 

 

 

 

 

 

I Ne+A|
- MeV/nucleon
T A55

l" A 75

Z .115
_'Dr x L35

: .1933.

“ 13- A" .

r A. ‘35

_ A a:

1— -A- *fl**g

; * '=

_ ‘* afifi*¥-4J§§

E .. 4* ***

_ 1b * 461$?

:-*@* {SF
:1111l1111i1111IL1' 1111
0 5 10 15 20

L Kr+Nb
l; MeV/nucleon
_ (>35

— D_55

:' A'75

I i

Q

_ mi

3 m

'- I I I I I I 1% I I I I I
0 20 40

Z Lop

2.8

2.4

 

 

 

 

Ar+Sc

 

 

MeV/ nucleon

'15

 

1t

 

 

 

Xe+La

 

 

MeV/ nucleon

0 25
035
945
B55

 

 

 

 

 

2m

Figure 3.11: The beam energy dependance of the average multiplicity of intermediate
mass fragments versus the charge of the hydrogen and helium isotopes for each of the
four systems is plotted above, uncorrected. These systems are: 20Ne+27Al (top left),
40Ar-i-“Sc (top right), 84Kr+93Nb (bottom left), and 129Xe+139La (bottom right),
for select beam energies in MeV/nucleon represented by the symbols as indicated

above.

 

 

 

26

From the mass variation plots, Figures 3.1- 3.6, and the beam energy variation
plots, Figures 3.7- 3.11, it can be concluded that cuts on the different centrality
variables result in quantitatively different values of < Ngm; >. It is generally as-
sumed that small values of these observables correspond to peripheral collisions,

while the largest values correspond to the most central collisions.

There are wide fluctuations, however, in the observed average numbers of IMF ’s
emitted and the general shape of the mean number of IMF curves when central
collisions are selected using each of the various centrality variables. These variations
in the IMF multiplicities suggest that the variables are not all as efliciently related
to the impact parameter, or that auto-correlations exist between some of these

centrality variables and < Nimf >.

The selection of the top 10% of the spectra of any of the six centrality variables
generally results in the most cental events (i.e. those impact parameters (b) satis-
fying < b > /b,,,ar,$0.22) [1] [12). More stringent cuts, such as the acceptance of
a smaller percentage of events, are not expected to result in smaller selected im-
pact parameters. The top 10% of each variable would then correspond to collisions
resulting in the highest numbers of charged particles, the highest numbers of light
charged particles, or the largest transverse kinetic energy of detected particles per
event. These events might also exhibit the overall largest Z values registered in the
detector or in a mid-rapidity gate. All of these conditions are consistent with those
expected from the most violent of nuclear collisions. The central events are then
only those that fall within the top ~10% of the Nchgd, ZMR, KET, 21,012, and NH

values.

The mean number of IMF ’s of the events allowed by selecting the top 10% of

the spectra of each of the centrality variables is plotted in Figure 3.12 as a function

27

 

 

 

 

 

 

/\E ’- NCth ZEZ 1. Nchgd
._ _ xNC+AI __
Z4 oAr+Sc 4 _ fifl
V E #4 AKr+Nb —
.— *Xe-I-Lo 1 M
62 *- -: -
o - M
.5 1 W L -
+1 _.
$0”.mmcoflmHHlLudA,
O _— ZMR q)2 :- ZMR
U 1—
4 _ _ W
_ 1 _—
2 M :m
.. M 1:
O I I I I I I I I I I I I I I I I O I I I I I I I I I I I I I I I I
F KET 2 KET

I,
11 V

0....1M o ........11.....1

.. NH 2 NH

III

 

 

IIIFII

$1;
11

 

 

O 1 I 1 M O b“1 L 1 I 1 1 1 1 I 1 I 1 1 I 1 1
r
— ZLCP 2 - ZLCP

1

 

 

 

 

O 1I1de OnlIIIIIIIIILLIIII

4o 80 120 40 80 120
Projectile Energy (MeV/nucleon)

 

Figure 3.12: The beam energy dependance of the means and the widths of the
intermediate mass fragment multiplicity distributions for the events selected by 10%
cuts on each of the five centrality variables separately is plotted above. Central]
denotes that this one dimensional cut was made. The left column contains the mean
values uncorrected for experimental acceptance, and the right column contains the
widths of the distributions for the four systems: 20Ne+27Al (asterisks), 4°Ar+458c
(circles), 84Kr+93Nb (triangles), and 129Xe+139La (stars).

28

of projectile kinetic energy for each of the four systems. The symbols represent the
various entrance channels; Ne+Al by the asterisks, Ar+Sc by the circles, Kr+Nb by
the triangles, and Xe+La by the stars. The left column of plots in Figure 3.12 is the
experimental values of < Nimf > of events obtained through a cut on the top 10%
of each centrality variable. This is denoted Central] to indicate a one dimensional
cut is made on each centrality variable separately. The right column depicts the

widths of the IMF distributions calculated via:

 

ANimf = \/< N37,! > — < Nimf >2.

Figure 3.12 implies dramatic differences in the values of < Ngmf > of fragments
selected by the various cuts 011 the largest values of each centrality variable (as
noted above). The Xe+La system consistently results in the highest mean values,
and these numbers range from a maximum mean value of ~3 for cuts on NH to a

maximum mean value of ~5 for cuts on ZMR.

The general trends in Figure 3.12 summarize the behavior of the right most val-
ues in each distribution in Figures 3.1- 3.11. These trends indicate that with increas-
ing beam energy the experimental values of < Nimf > decrease for Ne+Al collisions,
reach a slight maximum in Ar+Sc collisions near beam energies of 45 MeV/nucleon,
and increase monotonically for the Kr+Nb and Xe+La systems. These distinctions
can result from differences in the sensitivity of the various centrality variables to
the impact parameter and from possible auto correlations between the centrality
variables and the mean number of IMF’s. To distinguish between these two pos-
sibilities, Section 3.2 will determine to what extent the mean values illustrated in

Figure 3.12 are manifestations of such auto-correlations.

29

3.2 Auto-Correlations

The selection of the most central collisions on the basis of a cut on the largest values
of a given centrality variable may, simply by definition, directly affect the values of
the number of IMF’s that are possible. This effect is a source of bias and needs to
be removed from the analysis. If two variables are correlated, a plot of one variable
versus the other would result in a distribution with a small width. If two variables
were completely uncorrelated, then such a plot would result in a distribution with

the largest possible width.

In this analysis, the multiplicity of the number of IMF’s as a function of six
separate centrality variables is used to determine a common unknown quantity —
the impact parameter. Complications could result in utilizing these observables due
to the fact that the variables could be auto-correlated with the number of IMF ’s
detected. Cuts on such a variable could suppress or enhance the number of IMF’s

counted.

Two types of auto-correlations are possible: “positive” and “negative”. Spe-
cific centrality cuts might artificially enhance the observed number of IMF’s sim-
ply by definition. This type of artificial elevation will be denoted as a “positive
auto-correlation”. On the other hand, biases between the variables that result in
artificially low values will be denoted as “negative auto-correlations.” Both posi-
tive and negative auto—correlations between the centrality variable and the number
of intermediate mass fragments are characterized by a relative suppression of the

widths of the IMF distributions.

To avoid using the mean number of IMF ’3 directly, the classification parameters

for auto-correlation will be based on the variance of the IMF spectra (ammf) defined

30

UNgmf =< Nzinf > — < Ngmf >2

and the variance normalized by the mean (aNgmf/ < NM; >). Events from cuts on
centrality variables resulting in a relatively small variance and a small normalized
variance will be classified as “positive auto-correlators” since their effect is to artifi-
cially enhance the number of IMF ’s that are observed. Conversely, events following
cuts on centrality variables displaying a small variance of the number of IMF’s and
a large normalized variance will be classified as “negative auto-correlators” since

their effect is to automatically select collisions with small numbers of IMF’s.

To determine to what extent each centrality variable is correlated with Nimf, the
means, widths, and reduced widths of the events selected through one dimensional
~10% cuts of the largest values of each centrality variable are plotted in Figures
3.13 and 3.14. The two lighter systems, Ne+Al and Ar+Sc, are shown in Figure
3.13, and the two heavier systems, Kr+Nb and Xe+La, are shown in Figure 3.14.
In these Figures, the mean values are shown in the upper frames, the variances are
shown in the middle frames, and the normalized variances are shown in the lower
frames. All frames are plotted versus the available projectile kinetic energy for each

system.

Auto-correlations are recognized by relatively small widths of the mean distribu-
tions. Furthermore, these auto-correlations between the variables can be classified
as “positive” or “negative” auto-correlations with the number of IMF ’3 based on
the normalized variances. Figures 3.13 and 3.14 thus indicate that three distinct
groups form in all four systems appearing approximately always in the same order.
Centrality cuts on the variables ZMR and Zdet consistently display the largest mean

values and the smallest reduced variances, NH and Zch consistently display the

31

 

 

 

 

 

 

       
 

 

    

 

   

Central1
2 LNe’I'AI CNN
’ [321111
E AKE!
15 M"
/\ ' - 41-12“
E E. *2“,
Z 1;
V :
0'5 EM
:IIILIIIIIIIIIIIIIIIII:IIIIllIIIIIIIIIIIIIII
1.2— —
.E E E
Z
b0.8~— —
PIIIIIIIIJIIIIIIIIIIIIIL-IlIIIlIIIIIIIIIIIIIII
/\ 1:— ,7,W
3:: _
Z075 ‘
V'
\.
£0.5_
b _
0.255— _
:IIIIIIIIIIIIIIIIIIIIII:IIIIIIIIIIIIIIIIIIIII

 

 

 

 

60 80 100 120 25 50 75 100
Projectile Energy (MeV/nucleon)

Figure 3.13: The average values, widths, and reduced widths of the multiplicities of
intermediate mass fragments in the events selected by ~10% cuts on the largest val-
ues of six centrality variables versus the beam energies for 20N e+27A1 and 40Ar+45Sc
entrance channels are plotted above. Central] denotes that this one dimensional cut
was made. The six centrality variables are labeled by symbols as indicated above.
This data is not yet corrected for experimental inefficiencies.

32

 

 

 

 

 

 

   

 

 

 

 

Centrgl1
5 Kr+Nb
r
/\ 4—
E
Z -
V
2_ :f5
C/
IIIILIIIIIIIIIIIIIIIIJII—I
.5 3 1 '—
b — .—
2 .— —
1:ILIIIIIIIIIIIIIIIIIIIIII:LII‘IIIIIIIIIIIIILIIII
1.5: _-
A. : :
.5 ._ _—
Z1'2 5W 5
V {WC
\ E _G/e\e—e—9—o
$0.8 ~W‘0 —
t) 3W:
0.41%:3 _—
”:IIIIIIIIIIIIIIIIIIIIIIIIFIIILIIIIIIIIIIIIIIIIII

 

30 40 50 60 70 30 40 50 60
Projectile Energy (MeV/nucleon)

Figure 3.14: The average values, widths, and reduced widths of the multiplici-
ties of intermediate mass fragments in the events selected by ~10% cuts on the
largest values of six centrality variables versus the beam energies for 84Kr+93Nb
and 129Xe+139La entrance channels are plotted above. Centrall denotes that this
one dimensional cut was made. The six centrality variables are labeled by symbols
as indicated above. This data is not yet corrected for experimental inefficiencies.

33
smallest mean values and the largest reduced variances, and Nchgd and K ET con-
sistently fall in the middle for both values. The middle frames, representing the
variances, are not as structured due to the fact the three groups cross and even
separate at some points. On average, the variables Nchgd and KET tend to have the
larger widths or variances while ZMR, Zdez, NH, and Zch tend to have the smaller

variances.

The fact that these groups are clustered independent of energy indicates that
they are the consequence of an internal effect rather than physical one. The con-
clusion is drawn that the formation of these groups reflects varying degrees of auto-
correlations. The relative extent to which each variable is auto-correlated with the

number of IMF’s can then be determined from these plots.

Applying the guidelines defined above to Figures 3.13 and 3.14, ZMR and
Zde; have small widths as well as small reduced widths, hence they can be classi-
fied as positive auto—correlators. Conversely, NH and Zch have small widths and
large reduced widths and therefore can be classified as negative auto-correlators.
The remaining two variables, Nchgd and [(137, have the largest possible widths
and intermediate reduced widths, hence these observables are the least affected by
auto-correlations. Therefore, only cuts on Nchgd and KET will select nearly central
collisions that are unbiased by auto—correlations with the observable, the number of

IMF’s.

New significance can now be attributed to the mean number of IMF plots in
Figures 3.1- 3.11. Special attention should be placed on the plots of < Nimf >
versus NH and Zch; Figures 3.5, 3.6, 3.10, and 3.11. Both NH and Zch
were determined in this section to be negative auto-correlators. As a result, plateau

regions are formed at low values of < Nimf > in these plots. This “negative” effect

34

can also be seen not only through the overall low mean values but also through
the artificial maximums in < Nimf > curves. These multiplicity maximums are the
result of the negative auto-correlations decreasing the true mean number of IMF’s

present for large values of NH and Zch.

Chapter 4

Most Central Collisions

The preceeding section utilized experimental observables to select small impact pa-
rameter collisions. It was determined that the largest values of these observables
generally corresponded to collisions with the largest numbers of IMF ’s. In order to
select the smallest impact parameters, only events that fell in the top 10% of each
of the six centrality variables were chosen. From the widths of these distributions,
the extent to which each observable was auto-correlated with the mean number of

IMF’s was also determined.

To improve the efficiency of selecting the most central collisions, a stricter cut
is now performed involving the selection of the top 10% of two different centrality
variables simultaneously in an event. The use of a “two dimensional” cut at least
recovers some of the loses in efficiency for impact parameters selection that are more

evident for one dimensional cuts.

One possible combination of two centrality variables for this “two dimensional”
cut are those shown in Table 3.1. These centrality variables are the ones that were
determined to be the most efficient in selecting central collisions solely based on the

sphericity study [I] which is free from auto-correlations of any type.

35

36

The results of “two dimensional” cuts on the centrality variables shown in Table
3.1 are shown in Figure 4.1. The solid line is < Nimf > of events satisfying the two
dimensional cuts. The axis is labeled Centralg to indicate that a “two dimensional”
cut was made. The symbols represent the two conditions that were used to make
the cut for each system and beam energy. These points are the same values shown
in Figures 3.12- 3.14. In Figure 4.1, NH cuts are represented by the diamonds,
ZMR (squares), KET (triangles), and ZLCP (stars). The mean values of the number
of IMF’s are plotted versus the beam energy for each system. The statical errors
in these mean values are negligible and are not drawn. The four systems, Ne+Al,

Ar+Sc, Kr+Nb, and Xe+La, are plotted separately.

Figure 4.1 indicates that the “two dimensional” < Nimf > curve falls between
the chosen variables for the two lighter systems, but mostly above the one dimen-
sional variables for the heavier systems. Assuming that the largest numbers of IMF’s
are seen in the most central collisions (see Figures 3.1- 3.11), if these two dimen-
sional cuts effectively selected smaller impact parameters, then one would expect
a larger number of IMF’s to result from the two dimensional cut (solid lines) than
from either of the individual one dimensional cuts (symbols). It is evident from this

plot that this is not the case.

Using the results of the auto-correlation analysis conducted in Section 3.2, dis-
crepancies between the the assumptions in the paragraph above and the curves
shown in Figure 4.1 can be understood. Figure 4.1 reveals that these discrepan—
cies are purely the result of positive and negative auto-correlations which become

important if the number of IMF’s is the experimental observable, as in this case.

In both the Ne+Al and Ar+Sc systems, NH and Zch are used as one of the

two variables for the “two dimensional” centrality cut throughout the energy ranges.

37

 

 

 

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 2 E
A. . Ne+Al 0 NH §Ar+Sc
v 0.8 E— A“ 1.6 E—
_N E *Zlcp E
8 0,7 -:_—A 1.4 E— A A A A A
E 0.6 E- A 1-2 E— A
(D : I
O 0.5 E— 1 '5‘
0.4 5'0 0.8 E—
0.3 :— 0.6 E-
0.2 :— 0.4 5
0.1 — 0.2 :
:LIIIIIIIIIIIIIIIIIIIII :IIIIIIIIIIIIIIIIJIIII
60 80 100 120 140 25 50 75 100
A 5 K 7 X +L
A : - e O
.5 4.5 :— ”Nb :
Z : 6 :-
V 4 E— B B :
_N _ A A L D.
:3 3 :_ A A I A A
C : [3 4 a.
(l) : 5
L) 2.5 :— I
2 E 3 :— o
1.5 f—* E
I 2 : 3
1 E— :
E 1 -
05 E- C
:IIIIiJIIIIII L_IIIIIIIIIIIIIIIIIIIIII
40 60 80 20 30 40 50 60

Projectile Energy (MeV/nucleon)

Figure 4.1: The mean number of intermediate mass fragments versus the beam
energy for the events selected by the two most efficient centrality variables according
to Reference [1] is plotted above. Central; denotes that this “two dimensional”
cut was made. The symbols represent the results of the one dimensional cuts as
indicated above. The solid line represents the more stringent cut hence the most
central collisions for the four systems: 20Ne-l-27Al (upper left), 40Ar+45Sc (upper
right), 84Kr+93Nb (lower left), and 129Xe+139La (lower right). This data is not

corrected for the experimental acceptance of the detector.

38

These variables were seen to be negative auto-correlators and thus suppress the mean
value of the number of IMF’s. This is especially evident for beam energies of the
Ar+Sc system that are less than 45 MeV/nucleon for which the variable ZLCP was
used. Conversely, the utilization of a positive auto-correlator, ZMR, in the two
variable centrality cut for the two heavier systems produces the opposite effect,

artificially increasing the values of < Ngmf > in the two dimensional cut.

It was determined in Section 3.2 that the variables Nchgd and KET are the
two centrality variables that are least sensitive to auto-correlations with the average
number of IMF ’s for the selected central collisions. These variables, however, are not
the two most efficient variables in choosing central collisions [1]. Auto-correlations
produce anomalous results and should be avoided, even at the risk of loosing some
sensitivity to the impact parameter. The only acceptable balance is then to maxi-
mize the efficiency of the selection of central collisions (i.e. from Reference [1]) while

minimizing the contributions from auto-correlations during the specification of the

variables upon which the cuts are made.

Taking both factors into consideration, a “two dimensional” cut is performed on
the data set using the observables Nchgd and K ET for all systems and beam energies.
Only events falling in the top 10% of both variables are defined as the most central

collisions.

Figure 4.2 depicts the average values of the multiplicity of IMF’S in the events
selected by this “two dimensional” cut on the centrality variables Nchgd and KET.
This cut is also referred to as Centralz, but unlike the two dimensional cuts used to
generate 4.1, these are free from auto-correlations for all systems and beam energies.
The solid line represents the mean number of lMF’s in the events accepted by the

two dimensional Nehgd and KET cuts. The symbols represent the mean number of

Centrol2 <Nimf>

Centrol2 <Nimf>

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

4.5

3.5

2.5

1.5

0.5

 

 

 

 

 

 

 

 

 

 

 

t. O Nchgd 1
Z IQKET
:— 1
E
:A 1
:‘o A 1
E O
7 ZS A
I O O A
E O 6 0.
E A
T O O
E- 0
E— O.
_LlllllllllLllLJllllll

60 80 100 120 140
E A A
E o O
L A A
r; O O
:_ A
I O
E
a 1 l 1 11 1 l L 11 L

40 60 80

N'A'ouoo—‘ixab'os'oom

 

 

 

 

 

 

 

 

20

:- A

: A A A

r- o O

E- 0 o o 6 A

EA A O 8

E 0

E0

:LLLlllllllllllLlllll
25 50 75 100

I A

r A O

L 0 o 6

E A

Z O

: A

—-O

_llllllllllllllllllllll

3O 4O 50 60

Projectile Energy (MeV/nucleon)

Figure 4.2: The mean number of intermediate mass fragments versus the beam en-
ergy for the events selected using the two centrality variables that were seen to be
free from auto-correlations with the number of intermediate mass fragments and
of relatively high efficiency according to Reference [1] is plotted above. Centralg
denotes that this “two dimensional” cut was made. The symbols are the one dimen-
sional values with Nchgd cuts represented by circles and K ET cuts represented by
triangles. The solid line represents the more stringent cut hence the most central
collisions for the four systems: 2ONe+27Al (upper left), 40Ari-"53c (upper right),
84Kr+93Nb (lower left), and 129Xe+139La (lower right). This data is not corrected
for the experimental acceptance of the detector.

40

IMF’s from each of the one dimensional cuts on Nchgd and KET separately that
are used in determining the thresholds for the two dimensional slice. Nchgd cuts are
represented by circles, and K ET cuts are represented by triangles. As illustrated
in the Figure, the selected events result in higher < Nimf > than either of the one

dimensional cuts.

Figure 4.3 summarizes the above results by plotting these mean values obtained
by the two dimensional cuts on the same axis for each of the four systems. The
top plot is the experimental values of < Nimf > versus the projectile energy for the
systems Ne+Al, Ar+Sc, Kr+Nb, and Xe+La as labelled. The bottom plot is the
variances of the number of IMF distributions as defined previously. Figure 4.3 is
then a representation of the experimental number of IMF ’s detected resulting from
the best possible selection of central collisions that is free of auto-correlations but

not yet corrected for experimental acceptance.

The probabilities for IMF emission are thought to be indicative of the phases
of excited nuclear matter [3] [8], and changes in the average multiplicity of IMF ’3
versus the beam energy is an indication of a transition region. By studying the IMF
multiplicity in Figure 4.3, it can be inferred that the Ne+Al system is probably
above this transition region, i.e. is in a gaseous stage, for the available projectile
kinetic energies in this study (above 55 MeV/nucleon). The Kr+Nb and Xe+La
systems, on the other hand, appear to be on the lower excitation energy side of
such a transition for all the available beam energies. For these systems, increases in
beam energy result in increases in the average numbers of IMF ’s in the most central
collisions. Figure 4.3 apparently indicates a transition region in the Ar+Sc system
occurring around a beam energy of ~45 MeV/nucleon. Such a maximum would be

characteristic of a liquid-gas phase transition, if the assumption is valid that such

41

 

 

 

 

 

 

6 1—
A\ 5 E-
.‘E :
Z _
V 4 :
.9 :
O 3 T
L ..
E) 2 i—
1 L

r a H

O _I A I l l l l l l l l I l I l I I l l I l l I l J L l l 1 I I I

20 40 60 80 100 120

 

 

 

 

 

2.25 E—
2 3—
E175 E—
Z :
P 15 5— /
__N E
I 4——/\
'E 1 3,,/
(I) I
00.75 E_ \\
0.5 ;—
o.25 :—
O :1 I l 144 I l I l I Ill I I I I I I l I ii I I I l I I I I
20 4o 60 80 100 120

Projectile Energy (MeV/nucleon)

Figure 4.3: The mean number of intermediate mass fragments as a function of pro-
jectile energy for all of the systems is plotted above for the most central collisions.
These are the mean values taken from Figure 4.2 and are not corrected for the ac-
ceptance of the detector. The bottom graph is the variance of the IMF distributions
for the four systems: 2ONe—l—T’IAI, 40Ar+“58c, 84Kr-l-93Nb, and 129Xe—l—lg‘QLa.

42

a critical point maximizes that number of intermediate mass fragments. It should
be noted that none of these curves have yet been corrected for the experimental

acceptance dependance on the beam energy and the system mass.

Chapter 5

Discussion

In the preceding chapter, a cut was performed on the top 10% of two centrality
variables in an effort to select the most central collisions. The variables Nchgd and
K ET were used for this selection process due to the fact that these variables are
the least auto—correlated with the number of IMF’s as well as reasonably efficient in
selecting central collisions according to Reference [1]. From these “two dimensional”
cuts, a representation of the number of intermediate mass fragments occurring in
the most central collisions unbiased by auto-correlation was obtained. From the
IMF multiplicity distributions, approximate regions of phase transitions in the de-
cay mechanism of matter could be roughly identified as a function of beam energy
although experimental corrections have not yet been performed. These results will

now be compared with similar studies of IMF emission that have been previously

reported.

The work of Li et a1. [5] was conducted with the same 40Ar+453c system that was
used in the present study. The centrality condition used in Reference [5] was an ob-

servable based on the number of charged particles, the total transverse momentum,

43

44

and the charge of the detected particles at mid-rapidity given by:

ZMRX Pi )(Nchgd
Ztot Pproj Nmaz:

1/3

C=( )3

 

where Z10; is the total charge, NW” is the maximum number of particles, and Pp,”- is
the total momentum of the projectile. Comparing the uncorrected data in Reference
[5] to that shown in Figure 4.3, the mean number of IMF’s are higher than that
found in the present analysis (Figures 4.2 and 4.3) for the same system and beam
energies. This is a result of the use of the variable ZMR for the selection of central
collisions in this previous work [5] which was found to be a positive auto-correlator

(Section 3.2).

The research of Bowman et a1. [3] was conducted with the Michigan State
University Miniball which is a detector with similar acceptance to the one used in
the present analysis. The work of Reference [3] involved a l2”Xe projectile beam
at a kinetic energy of 50 MeV/nucleon impinging on the targets: 12C, 27Al, 51V,
”"Cu, 89Y, and 197Au. The mean numbers of IMF’s as a function of the number
of charged particles in this study can be compared to the present analysis (Figures
3.1 and 3.7). It is relevant to compare the mean number of IMF’s reached in the
largest values of New for each experiment which is shown in Figure 5.1. This
plot is < NM; > versus the entrance mass of the target for both studies. The
results of Reference [3] are represented by the open circles, and the results of this
analysis are represented by the asterisks. This plot illustrates that for masses below
~180 mass units, symmetric systems result in larger number of IMF’s than the
asymmetric systems. The reverse effect is true for systems larger than ~180 mass
units; however, this difference could be purely the result of the slight differences in

the acceptance of the two experiments.

It should be noted at this point that after auto-correlations are taken into consid-

45

 

AE _ 50 MeV/nucleon
5 7 :0 Asymmetric Entronce Channels (Ref. [3]) a
6 E3“ Symmetric Entronce Chonnels (this onolysis)
L _
"E3 53 *—
q) _
U .-

5 _—

4 _-

3 —- 0

y.
2 _

_ 0

0

 

 

LllllllllLiIllllLllllllllllliAllllJllL

4o 80 120 160 200 240 280 520
/\sys

 

Figure 5.1: A qualitative comparison of the average number of intermediate mass
fragments versus the entrance channel mass in symmetric (this work) versus asym-
metric (Reference [3]) entrance channels is plotted above for a beam energy of 50
MeV/ nucleon. The results from this analysis are represented by the asterisks and
the results of Reference [3] are represented by the open circles.

46

eration, none of the plots of the mean values in Figures 3.1- 3.11 contain statistically
significant maxima. The absence of maxima in these plots indicates that for the in-
termediate beam energies used in this study, the most central collisions contain the
highest multiplicities of IMF’s. At much higher beam energies, such as those studied
by Ogilvie et al. [9], this was not the case. At these higher beam energies, the most

numbers of IMF’s were observed for intermediate impact parameters.

Further comparison with other work is limited due to the fact that results of
the present analysis are not corrected for the acceptance of the detector. This
sort of correction can be accomplished by generating events with model codes that
reproduce the basic features of the data. Once a reasonable match is made, the
efficiencies of the intermediate mass fragment measurements can be determined.

Such studies are underway.

Chapter 6

Summary and Conclusions

The purpose of this analysis was to obtain a representation of the average number of
intermediate mass fragments registered in a 47r geometry as a function of the kinetic
energy of the projectile. Many different entrance channels: 20Ne+27Al, 40Ar+45Sc,
84Kr-l-93Nb, and 129Xe+139La, were utilized spanning a wide range of beam energies.
Central collisions were of the most interest, and the centrality variables that were

least correlated with the number of IMF’S were identified.

To determine the extent of these auto-correlations, the mean number of inter-
mediate mass fragments as well of the widths and the normalized widths of these
distributions were plotted versus the beam energy. From these representations, it
was determined that the number of charged particles in one event (Nchgd) and the
transverse kinetic energy of emitted particles per event (KET) are the two centrality
variables that are least correlated with the number of IMF’s. Of the remaining four
variables, the charge of detected particles at mid-rapidity (ZMR) and the charge of
all of the particles registered by the detector (cht), per event, were classified as pos-
itive auto-correlators. Conversely, the number of detected hydrogens (NH) and the
charge of the light charged particles (ZLCP), per event, were classified as negative

auto-correlators.

47

48

These results were then used to obtain a representation of the average number of
IMF’s observed for each central collision as a function of target mass and projectile
energy that were unbiased by auto-correlations. Although the present results have
not yet been acceptance corrected, there is a qualitative indicator that the 2ONe+27Al
system results in a gaseous phase of nuclear matter for beam energies greater than
55 MeV/nucleon. The 84Kr—l—93Nb and 129Xe+139La systems appear to result in
nuclear systems in a liquid phase for energies less than 75 MeV/nucleon, and the
40Ar+4sSc system appears to undergo a phase transition from liquid to gas around
40 - 70 MeV/nucleon. In the interest of quantifying these statements, future plans
include the correction of the present measured average IMF multiplicities versus the

beam energy for the inefficiencies of the experimental apparatus.

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ARIES

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