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THEQQ

a col

This is to certify that the
thesis entitled

Interpretation of Nuclear Multifragmentation

Data in the Framework of Percolation Models

presented by

Marko Kleine Berkenbusch

has been accepted towards fulfillment
of the requirements for

Master' 3 degree in Physics

 

 

 

Wo fgang Bauer

Date May 7, 2001

 

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

LEERARY

Michigan State
University

 

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINE return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJCIRC/DateDuepes-ms

 

INTERPRETATION OF NUCLEAR MULTIFRAGMENTATION DATA IN THE
FRAMEWORK OF PERCOLATION MODELS

By

Marko Kleine Berkenbusch

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

2001

ABSTRACT

INTERPRETATION OF NUCLEAR MULTIFRAGMENTAT ION DATA IN THE
FRAMEWORK OF PERCOLATION MODELS

By

Marko Kleine Berkenbusch

Phase transitions occur in a great variety of physical systems on all length and
energy scales. One of the smallest systems which is believed to show critical behavior
in the sense of phase transitions is the atomic nucleus. Due to the size of the nucleus,
the experimental investigation of these phenomena is extremely difficult. One class

of experiments employed in such an analysis is multifragmentation reactions.

This work deals with the application of a statistical percolation model of mul-
tifragmentation to the analysis of an experimental data set obtained by the ISiS

collaboration.

We discuss the importance of considering physical limitations of the detection
procedure in modeling the experimental results. Strong indications of the existence
of a liquid—gas type phase transition in the data set are found. A method'for the
determination of two critical exponents, 0 and T, of the transition is introduced in
the framework of the percolation model. This method is applied to the experimental
data to obtain numerical values of the exponents. It is demonstrated that sequential

decay processes have to be taken into account in this analysis.

ACKNOWLEDGMENTS

First of all, I would like to express my thanks to Prof. Wolfgang Bauer for guiding
my research as my advisor and for giving me the opportunity to work in the nuclear
theory group at the National Superconducting Cyclotron Laboratory at Michigan
State University, East Lansing. I enjoyed the friendly and open atmosphere of this

facility very much throughout this year.

I would also like to thank the German National Merit Foundation, the Studiens—

tiftung des Deutschen Volkes, for the financial support I received during this year.

Finally, I want to thank my family for the support they gave me during my studies

throughout the years.

iii

Contents

LIST OF TABLES vi
LIST OF FIGURES vii
1 Introduction 1
2 Overview of Percolation Theory 3
2.1 Basic Definitions ............................. 3
2.1.1 Bond Percolation ......................... 4

2.1.2 Site Percolation .......................... 5

2.1.3 Site-Bond Percolation ...................... 6

2.1.4 Polychromatic Percolation .................... 7

2.1.5 Continuous Percolation ...................... 8

2.1.6 Percolation with long-range Interaction ............. 8

2.1.7 Percolation on the Bethe Lattice ................ 8

2.1.8 Percolation Threshold and the Percolating Cluster ....... 9

2.2 Phase Transitions ............................. 9
2.2.1 Definitions ............................. 10

2.2.2 Classification of Phase Transitions ................ 11

2.2.3 Order Parameter ......................... 12

2.2.4 Correlation Length ........................ 12

2.2.5 Critical Exponents and Universality ............... 13

2.3 Theoretical Results of Percolation Theory ............... 15
2.3.1 Phase Transition in Percolation Theory ............. 15

2.3.2 Cluster numbers ......................... 15

2.3.3 Critical Exponents in Percolation Theory ............ 17

2.3.4 Finite Size Scaling ........................ 19

2.4 The Nuclear Lattice Model ........................ 22

iv

2.4.1 The Model ............................. 23

2.4.2 Choice of pB ........................... 25
3 Application of the NLM to ISiS data 29
3.1 The ISiS Experiment ........................... 29
3.2 Implementation Details and Modifications of the NLM ........ 30
3.2.1 General Setup ........................... 30
3.2.2 The Filter Code .......................... 31
3.2.3 Charges 17-20 ........................... 34
3.2.4 The Tcl/‘I'k Interface ....................... 34
3.3 Results ................................... 35
3.3.1 Choice of Parameters ....................... 35

3.3.2 Excitation Energy Spectrum and Distribution of pre-equilibrium
emitted Particles ......................... 37
3.3.3 Charge Yield ........................... 38
3.3.4 Multiplicity Distribution ..................... 40
3.3.5 stidue Distribution ....................... 41
3.3.6 Zresidue Distribution gated on Multiplicity ........... 43
3.3.7 Second Moment Distribution ................... 43
3.3.8 Fisher Droplet Model (F DM) Scaling .............. 45
4 Summary and Conclusion 58
A Source Code 62
A.1 isisevent.h and isisevent.cpp .................... 62
A2 percolate.cpp .............................. 67
A3 filter.cpp ................................ 77
A4 varfdmparams . cpp ............................ 80
BIBLIOGRAPHY 86

List of Tables

2.1 Percolation thresholds for various lattices. ‘Site’ refers to site per-
colation, ‘Bond’ refers to bond percolation. Only nearest neighbor
interactions are considered. ....................... 10

2.2 Critical exponents in a liquid-gas system (dis the topological dimension
of the system under consideration) ................... 14

2.3 Percolation critical exponents for d = 2, 3, 4, 5 and in the Bethe lattice.
Rational numbers give (presumably) exact results whereas those with
a decimal fraction are numerical estimates ................ 19

vi

List of Figures

2.1

2.2

2.3

2.4
2.5

2.6

3.1

3.2

3.3

3.4

3.5

3.6

Example of a typical site percolation configuration with a density of

19:20.2 ................................... 6
Example of a typical site percolation configuration with a density of
1220.7 ................................... 7

Variation of the probability II (solid lines) that a cluster is spanning the
whole system for medium and large system sizes. The dashed line gives
%, proportional to the probability that at concentration p a spanning

cluster starts to appear. The width of the transition region or peak
should vary according to L1/”. ...................... 21

The Nuclear Phase Diagram (schematic) ................ 23

The fireball geometry: the lattice sites in the cylindrical channel with
radius r at impact parameter b are left unoccupied. .......... 25

Relation between pH and the temperature of the fragmenting nucleus
as given by Equation 2.38 (B26.6MeV) ................. 28

Spectrum of excitation energy per nucleon and frequency of pre-equilbrium
emitted charges in experimental data ................... 38

Inclusive charge yield spectrum. The experimental data (corrected by
the “17-20” events, see Section 3.2.3) is given by the circles, the filtered
and unfiltered model data by the solid and dotted lines, respectively.
The charge is given, as in all following diagrams, in multiples of the
charge of a proton. ............................ 39

Multiplicity distribution. The experimental data are given by the cir-
cles, the filtered and unfiltered model data by the solid and dotted line,
respectively. ................................ 40

Distribution of Zresidue (experimental data and filtered model data,
circles and solid lines) and of the largest fragment (unfiltered model

data, dotted lines). ............................ 41
Zm distribution for different multiplicities. In the case of the unfiltered
data the distribution of the largest fragment is plotted instead. . . . . 42

Second moment (M2) versus multiplicity. In the upper branch, the
second moments have been calculated including the largest fragment,
in the lower branch the largest fragment has been excluded. ..... 44

vii

3.7

3.8

3.9

3.10

3.11

3.12

Plot of the scaled fragment distribution as a function of the scaled
control parameter (multiplied by 196) for fragments of charge Z. (a)
shows the filtered data, (b) the unfiltered model data. In this plot,
a critical percolation probability pC = 0.7, a = 0.45 and 722.18 have
been chosen .................................

Density plot of X2 as a function of a and T for the unfiltered per—
colation events. The 4 innermost contours represent [X2 / (degrees of
freedom)]-values of 3.2, 3.5, 3.8 and 4.1, respectively. A critical break-
ing probability of 0.65 has been used in this plot. ...........

Density plot of X2 as a function of a and 7' for the experimental data
(uncorrected). The innermost contour corresponds to a [X2 / (degrees
of freedom)]-value of z 12 .........................

Plot of the scaled fragment distribution as a function of the scaled
control parameter (multiplied by Tc) for the experimental data (uncor-

rected). Values of a = 0.99, T = 2.13 and T0 = 9.5 MeV have been
used .....................................

Density plot of x2 as a function of 0 and T for the experimental data
(with filter— and sequential decay correction. The 4 innermost contours
represent [X2 / (degrees of freedom]-values of 3.3, 3.7, 4.1 and 4.6, re-
spectively. A critical temperature of 8.3 MeV has been used in this
plot. ....................................

Plot of the scaled fragment distribution as a function of the scaled
control parameter (multiplied by TC) for the experimental data (with
filter- and sequential decay corrections) Values of a = 0.5, T = 2.35
and Tc 2 8.3 MeV have been used. ...................

viii

48

51

52

53

55

Chapter 1

Introduction

Critical phenomena and phase transitions are common features in a wide variety of
physical systems. They carry extremely important functions on all time, length and
energy scales in our world. Everyone is familiar with two prominent cases of a liquid-
gas and a solid-liquid phase transition we encounter in everyday life: the boiling of
water and the melting of ice. There are also less obvious examples like the boiling of
an egg or other bio-chemical polymerization processes that are crucial for biological

life and that do not just take place in highly specialized laboratory environments.

What these phenomena have in common is that the system under consideration
undergoes a profound qualitative change when a freely adjustable parameter crosses
a certain threshold; in the example of boiling water or melting ice, the adjustable

parameter is the temperature.

Although the aforementioned phenomena seem to be significantly different from
each other, they share deep connections on a more abstract level. A good theoretical
understanding of these underlying similarities is not only of academic interest, but

may also result in practical applications of great importance.

One system in a length and energy scale that is not as easily perceived in everyday

experience is the nucleus of an atom. It is assumed that nuclear matter undergoes at

least two distinct phase transitions. Even though practical applications of a theory
of phase transitions in nuclear matter seem to be rare, such a theory might help to

solve some of the persistent problems in understanding the structure of matter.

In this thesis, we will deal with a phase transition of nuclear matter. In particular,
the framework of percolation theory will be used to analyze data that have been
collected in high energy nuclear reactions. The easily adjustable environment of a

model will help to interpret the experimental findings.

In Chapter 2, a short overview of percolation theory and critical phenomena in
general will be given, and a model of nuclear fragmentation reactions based on perco-
lation theory will be described. Chapter 3 will address the application of this theory
to the interpretation of a set of data obtained in proton/pion - Au collision experi-
ments at the AGS accelerator facility of the Brookhaven National Laboratory. Some
emphasis will be put on the question of whether traces of a phase transition can be
detected in the collected data. The appendix gives some parts of the source code used

to implement the percolation model and to analyze the data.

A typographical convention will be that important terminology is typeset in italic

style when it is introduced for the first time.

Chapter 2

Overview of Percolation Theory

Percolation models are some of the simplest physical models for systems with many
degrees of freedom which exhibit a whole range of critical phenomena that are widely
studied in statistical physics. In particular, percolation models provide an easy means
to study the typical behavior associated with phase transitions. While the first model
of this type was introduced by Flory [22] and Stockmayer [43] in 1941 and 1943,
respectively, the term Percolation Theory was only coined in 1957 by Broadbent and

Hammersley [8].

In the following sections, we will give a short introduction to the different types

of percolation models and present the most important results in this field.

2. 1 Basic Definitions

The main feature of most types of percolation models is a graph in d—dimensional
space. In many cases — including the nuclear lattice model, which will be discussed in
this thesis in further detail — the graph can be associated with Z“ (where Z denotes
the set of integers). We will call the points of this graph vertices or sites. In the
case of Z“, these vertices would be given by coordinate vectors :1: = (.121, .122, . . . ,er)

with integer components mi. The connection between two vertices :2: and y will be

denoted by (3,3)) and called edge or band. In standard percolation theory, only edges
between nearest neighbors (with respect to the l-norm) are considered. The number
2 of interacting neighbors of a site is called the coordination number of the lattice.

For Zd, for example, we have the coordination number 2 = 2 x d.

2.1 . 1 Bond Percolation

For typographical convenience, we will introduce the term percolation tuple for (at the
moment) a pair of two numbers (p, q) with the properties 19, q E [0, 1] and p + q = 1.

These numbers will be interpreted as probabilities.

In bond percolation, we start by introducing a percolation tuple (p, q). We then
randomly assign the property of being closed (with probability p) or open (with the
corresponding probability q = 1 - p) to all edges of the graph under consideration.
In the first case we will call the two end-vertices of an edge connected; in the second
case we will call them not connected. It is important to mention here that the states

of all the edges of the graph are statistically independent.

Based on these conventions, we define a bond-cluster as a subset of vertices of the
graph in which every pair {33, y} of two vertices of the subset can be connected by a
path of closed edges. We will write a: H y for this situation. In the case r 94) y, i.e.,
when there is no connecting path between the two vertices, they do not belong to the
same bond-cluster. The size .9 of a bond-cluster is defined as, as for all the following

definitions of clusters, the number of sites that belong to that specific cluster.
As an example of this rather abstract model one could think of the following:

Imagine an oil field. Typically the oil (or gas) is enclosed in porous rocks. In
our simple model we want to assume that the pores in the rock which hold the oil

can be represented by the vertices of our graph. The property of an edge between

two vertices to be closed will then be interpreted as an open connection between the
corresponding pores that allows the oil to flow between them. When one now starts
to drill into this rock and pump oil from one of the pores (or a group of pores), the
question arises: to how many other pores is this pore connected, i.e., how big is the
cluster the pore belongs to? As we will see later, percolation theory provides us with
information about the distributions and general properties of these clusters and thus

could help to evaluate the chances of “hitting” a big oil field.

2. 1 .2 Site Percolation

Weconsider a situation similar to the one above, but this time we assume that all
edges are closed a priori. Again, we choose a percolation tuple (p, q). Now we will
not apply the probability p to the edges of the graph to determine if they are open or
closed, but rather to the sites in order to assign the state of being occupied or empty
to them. A site-cluster will now be defined as a subset of connected occupied sites of
the graph. This means that every member vertex of a given cluster that consists of
more than just one single vertex has at least one occupied nearest neighbor belonging
to the same cluster. A single-vertex site-cluster must therefore be surrounded by
empty sites. This corresponds to an isolated black black square in Figure 2.1. For a

system with a higher density of occupied sites see Figure 2.2.

To illustrate this idea, we could think of two types of balls, say metal and plastic
balls, being tightly packed in one layer on a flat surface (thus forming a 2-dimensional
hexagonal lattice). We distribute these two types randomly according to the proba-
bilities p and q. In this situation, the question of electric conductivity of the lattice

is apparently directly connected to the structure of the “metal-ball site-clusters”.

Many results of percolation theory hold for both bond and site percolation. It

has been shown by Stauffer in 1979 [42] that the two models belong to the same

universality class, which means that they share the same set of critical exponents (see
Sections 2.2.5 and 2.3.3). Although, historically, bond percolation was developed first,
every bond percolation model is equivalent to a site percolation problem (possibly on

a different lattice and with consideration of correlations)

I'-

1: "-._'
I

 

 

i

 

 
  

 

:l'..*'

___I

Figure 2.1: Example of a typical site percolation configuration with a density of p=0.2

 

 

2.1.3 Site-Bond Percolation

The two aforementioned concepts of percolation can be combined into the more gen-
eral concept of site-bond percolation. In a model of this type, we choose two percola-
tion tuples (p, q) and (p’, q’). The probability p is used to determine the distribution
of closed bonds; the probability p’ is used to determine the distribution of occupied
sites. A site-bond-cluster, in this model, is defined as a subset of connected (in the
way explained in the context of bond percolation) and occupied sites of the lattice.
In particular, with the choice 17’ E 1 we find ourselves in the situation of bond perco-
lation whereas the choice p E 1 yields the case of pure site percolation. For further

references see [39, 45, 9, 2].

 

Figure 2.2: Example of a typical site percolation configuration with a density of p=0.7

2.1.4 Polychromatic Percolation

Polychromatic percolation can be considered as a generalization of the three other
models. In polychromatic percolation we relax the condition that every site has to be
in either one of two states (occupied or empty), but rather allow any (fixed) number n
of states to choose from for the individual sites. The different states could be thought
of as different colors, as the name of the model indicates. A percolation tuple does no
longer consist of two numbers, but of n numbers p1,p2, . . . ,p,1 E [0, 1] with p1 + . . . +
p" : 1 (also called a simpler). Now, clusters can be defined for each of the different
colors (states), and their mutual dependencies can be studied. One example for a
model of this type is the ‘ThreeComponent Reactive Percolation Model" introduced
by Halley and Holcomb [26]. They consider the situation of two types of atoms
that can react and form dimer molecules. Simulations of this situation in a 3—state
percolation model can reproduce the empirical results of resistivity measurements

fairly accurately.

2.1.5 Continuous Percolation

Since a lot of processes in nature do not necessarily take place on a regular lattice,
percolation models have been introduced that are not set on a lattice with a given
structure, but that rather utilitize a description with continuous random variables. In
3 dimensions, numerical evidence indicates that the critical exponents are the same
as in lattice percolation. Certain arguments from renormalization theory are also in
favor of this view. It is therefore in most cases sufficient to stay in the computationally

easier framework of lattice percolation.

2.1.6 Percolation with long-range Interaction

In our discussions so far, we only considered nearest neighbor interactions of lattice
sites. One could also think of a generalization in which long-range interactions be-
tween lattice sites that are not nearest neighbors are considered. This would result
in a higher coordination number and a redefinition of the term cluster. Taking into
account this redefinition, numerical simulations again show no difference (concerning

critical exponents) to standard percolation models.

2.1.7 Percolation on the Bethe Lattice

Percolation on the Bethe lattice, sometimes also referred to as percolation on the
Cayley tree, takes place on a lattice with a special structure, the Bethe lattice. To
construct this lattice, one starts with one point, the origin, and adds 2 neighbor
vertices to it, out of which 2 neighbors emanate again, one of which is the connection
to the origin, but 2 — 1 are new vertices. Therefore, no closed loops are possible on
this type of lattice. It can be interpreted as equivalent to percolation on a hypercubic

lattice in d dimensions in the limit d —> 00. The Bethe lattice has attracted special

attention because it is one of the few non-trivial cases in which the percolation problem

can be solved analytically.

2.1.8 Percolation Threshold and the Percolating Cluster

We will explain these two central terms of percolation theory in the context of site
percolation. A translation to the other types of percolation models will be obvious.
One qualitative feature of the dependence of the structure of site-clusters on the prob-
ability p (concentration of occupied sites throughout the lattice) is the following: The
closer the concentration p is to one, the more likely it is to find clusters consisting
of a large number of sites in the lattice. In fact, it can be shown that in the case
of a lattice of infinite extent, there exists exactly one number pc _<_ 1, the percola-
tion threshold, for which a cluster exists that extends the whole system. Below this
threshold, no such percolating cluster exists, above the threshold, at least one such
cluster occurs. Table 2.1 shows some examples of percolation thresholds for different
lattices. Thresholds for bond percolation are also shown in this table. In the case of
a finite, two-dimensional system, for example, the presence of a percolating cluster
is equivalent to the notion that a cluster that spans from one edge of the system to
the other exists. However, it must be noted that the existence of a definite threshold

holds only in systems of infinite extent (see also: Section 2.3.4).

2.2 Phase Transitions

One of the main objectives of studying percolation theory is, as mentioned above,
the ability of percolation systems to model phase transitions. Before we proceed with
our description of results of percolation theory, we give a short overview of phase

transitions. For further reference, see also [37].

 

[ Lattice Site Bond ]

 

Hexagonal 0.6962 0.65271
Square 0.592746 0.5
Triangular 0.5 0.34729
Diamond 0.43 0.388
Simple cubic 0.3116 0.2488
BCC 0.246 0.1803
FCC 0.198 0.1 19

d = 4 hypercubic 0.197 0.1601
(1 z: 5 hypercubic 0.141 0.1182
(1 = 6 hypercubic 0.107 0.0942
d = 7 hypercubic 0.089 0.0787

 

 

 

Table 2.1: Percolation thresholds for various lattices. ‘Site’ refers to site percolation,
‘Bond’ refers to bond percolation. Only nearest neighbor interactions are considered.

2.2.1 Definitions

One can categorize physical systems with many degrees of freedom, for which a tem-
perature can be defined, in two different classes: If the constituents of the system do
not interact with each other, it is always possible to apply the formalism of statistical
mechanics in a way that the partition function of the system separates into a product
of partition functions of the single constituents (eventually after a transformation to
normal coordinates). It remains analytical at all values of the temperature T. If,
on the other hand, interactions exist, the situation may occur that the thermody-
namic functions possess singularities for certain values Tc of the temperature T. This
phenomenon manifests in the undergoing of the system of a qualitative change in

macroscopic behavior.

Before this change occurs, the system is in one possible state of thermal equilib-
rium, which is denoted as a phase of the system. Throughout the system, certain
macroscopic observables assume constant values. After the change, the system will

be in a different phase, with different values of some of the macroscopic observables.

10

Prominent examples of such phase transitions are the melting of solids, boiling of

liquids or changes from ferromagnetic to paramagnetic states in magnetic systems.

2.2.2 Classification of Phase Transitions

In experimental observations, qualitatively different types of phase transitions are

encountered. Ehrenfest introduced the following classification in 1933:

A phase transition is said to be of nth order, if the first n — 1 partial derivatives of
the thermodynamic potentials with respect to their natural variables (e.g. G (T, p, N)
for fluid systems) are continuous at the phase transition point, whereas at least one

of the nth partial derivatives exhibits a discontinuity (jump of finite size).

A prominent feature of first—order phase transitions is the appearance of phenom-

ena like latent heat (as seen when melting ice or boiling water).

The Ehrenfest scheme is no longer used today because of several reasons:

0 Since the physical behavior of a system is mainly governed by the thermody-
namic functions and their first derivatives, the differences between phase tran—
sitions of high orders start to vanish, which makes the differentiation between

them uninteresting

0 Although this property cannot be verified experimentally, newer experimental
results seem to suggest that the observed discontinuities in the thermodynamic
functions or their derivatives are rather singularities than finite jumps. This is
also supported by the analytical Onsager solution for a phase transition in a

two-dimensional spin-system.

Therefore, today one usually differentiates between two different classes of phase

transitions: discontinuous phase transitions, which correspond to first-order phase

11

transitions in the Ehrenfest sense, and continuous phase transitions for all other

C3888.

2.2.3 Order Parameter

Another typical feature of continuous phase transitions is the appearance of a quantity
called the order parameter. An order parameter is an observable quantity of the

system that can only be defined for one of the two phases.

As an example consider a fluid system (e.g. water). In the temperature range
below the critical temperature Tc and above the melting temperature, two different
phases can appear in the system: liquid and gas. Above the critical temperature,
the liquid phase can no longer be assumed by the system. Therefore, the observable
Ap = puquid — pg“, can only be defined when the system is in the subcritical phase,

but not when it is in the supercritical phase.

2.2.4 Correlation Length

The correlation function of a quantity X of the system (for example the magnetiza-

tion) is defined as:
g(r,r') = (“fl-770'» - (110)) <fl?(l")> (2-1)
where 23(r) is the density of the quantity X:

X = [d3ra3(r) (2.2)

g(r,r’) is a measure for the correlation between the system properties of X at the
points r and r’. In a spatially homogeneous system we have g(r,r’) = g([r — r’ |),

i.e., the correlation is invariant under translations. In the vicinity of critical (phase

12

transition) points, g(r, r’) is approximated by

 

_JP_-'_'l
g(r 1" = C exp( HT) ) (2 3)
’ Ir rIId-2+n '

Equation 2.3 implicitly introduces a new quantity, the correlation length 5 (T) As
can be seen from the expression for g, it defines a length scale for regions affecting

each other and those that do not.

The definition of the correlation length leads us to another feature differentiating

continuous from discontinuous phase transitions:

In the case of a discontinuous transition the correlation length stays finite, while in
the case of a continuous phase transition it diverges when the temperature approaches

the critical temperature T c:

gUMEEm an

This means that close to the critical temperature we have fluctuations that extend
much further than the usual range of interaction in the system. The term critical

fluctuations is used for this situation.

2.2.5 Critical Exponents and Universality

In the critical region of continuous phase transitions, the behavior of many system
quantities can be characterized by means of critical exponents. It can be observed

very often that a physical property F depends on the reduced temperature

_T—n
_ n

 

c
in the following way:

F(€) =ac“°(1+bc$+...); :1: > 0. (2.6)

13

For 6 —> 0, i.e., T —> Tc, all terms in the parenthesis except for 1 vanish and F (6)

follows a power law. We write:

F(e) cc 6‘” (2.7)

The number (,0 governing F (e) in the vicinity of T c is called the critical exponent of F.
Critical exponents are chosen to be positive by definition. The critical exponent of a
quantity may depend on the side from which the critical temperature is approached.

The more general definition therefore is:

(\0 lne
,_. 1D|F(€)|
6 _ll/m) ln(—c) (29)

There is a widely accepted naming convention for critical exponents in statistical

physics. We will give some of the more important exponents in the following table:

 

 

 

[ Property Exponent Definition ]
Heat capacity Oz CV oc 6—0
Order parameter 5 Ap oc (—e)’3
Compressibility 7 mp oc 6‘7
Critical Isotherm (5 p — p0 oc (p - ,oc)‘s
Correlation Length V 6 or 6‘”
Correlation Function 77 g(r, r’) or [r — r’ rain—’7

 

 

 

Table 2.2: Critical exponents in a liquid-gas system (d is the topological dimension
of the system under consideration)

It is interesting to mention here that phase transitions contain a universality that
is manifested in the critical exponents: the critical exponents are almost universal,
which means they are the same for almost all thermodynamic systems. They depend

only on

o The dimensionality of the system

14

o The range of the microscopic interaction

0 The spin-dimensionality of the system

This universality hypothesis was first expressed by Grifliths in 1970 [24].

2.3 Theoretical Results of Percolation Theory

The results presented in the following sections do not depend, unless explicitly stated
otherwise, on the type of percolation system, in other words, it does not matter if
site, bond, or site-bond percolation is regarded. We will therefore just write “cluster”
and leave out the further distinctions introduced above. For purposes of simplicity,
we will restrict the discussion to a system represented by a Z“ lattice and use the

notations of site percolation.

2.3.1 Phase Transition in Percolation Theory

As already explained in Section 2.1.8, the properties of the clusters of a percolation
system undergo a significant qualitative change when the probability p crosses the
threshold pc. This phenomenon can be associated with a continuous phase transition.
The fact that this phase transition is continuous will be indirectly motivated by the

discussion of the existence of various critical exponents in the following sections.

2.3.2 Cluster numbers

We consider a percolation system of size Ld and a given percolation tuple (p, q) (or,
to be more precise, an ensemble of such systems). One system quantity of interest
is the number of clusters of a certain size 3, denoted by NS (p) To be independent

of the actual size of the system, we will rather consider the number of clusters of a

15

given size per lattice site,

NS (2?)
12d

and call this quantity cluster number. The notation already indicates that the cluster

 

713(1)) = (2-10)

numbers depend on the concentration p of occupied sites (or closed bonds in the case
of bond percolation). Since we are interested in the critical aspects of percolation,
the behavior of these cluster numbers for concentrations p close to the percolation
threshold pc is of special interest. From analytical solutions and numerical results, it

can be inferred that for p close to pC the cluster numbers behave as follows:

n30?) = S'Tf[(p - p080} (p -+ pa, 8 —+ 00) (2.11)

We see the power law dependence of n3 at the critical point that has already been
mentioned in the context of critical exponents. The scaling function or cutofir function
f accounts for the fact that a power law dependence is only correct in the case of
p 2 pc. This must be the case because for p < pC no system spanning cluster exists
and therefore n, (p) has to decay faster than the power law for high 3. The cutoff
function f (2) has the general form that it approaches a constant value for [z] < 1
and decays quickly for Iz] >> 1.

Implicitly introduced by equation 2.11 are two critical exponents of percolation

—1/a

theory: 0 and 1. With the definition Sg = (p — pa) , we can rewrite equation 2.11

mm = 8"f [(7)] (2.12)

This leads to the interpretation of 55 as a crossover size for the clustersizes from power

as:

law abundance for 3 << Sg to exponentially rare clusters of size 8 > 8,15.

In the case of the Bethe lattice, we can give explicit terms for the scaling behavior

of the cluster numbers:
713(1)) 0< 3‘5/2 expl-((p - pawl/02] (2-13)

16

We can immediately see the values of the critical exponents, o = 1 / 2, T = 5/2 and
the form of the scaling function: f (z) = exp(22). f obviously shows the asymptotic

behavior mentioned before.

2.3.3 Critical Exponents in Percolation Theory

We have already introduced two critical exponents in percolation theory in Section
2.3.2: o and T. In particular, T is believed to provide information about the nature
of the phase transition examined (see [20]). This is motivated by the fact that in
the phase transition observed in a Van—der-Waals gas the size of condensing droplets
scales analogous to the cluster numbers as described by Equation 2.11 (with a critical

exponent of T = 7/3).

It has been shown by Broadbent and Hammersley [8, 27, 28], as cited in [25],
that for systems with a dimensionality higher than 2 the percolation threshold fulfills
0 < pC < 1, i.e., we have a critical phenomenon where the critical point can be
approached from both sides. Despite this fact, a and T are the same for both sides
of the critical probability: in our notation from Section 2.2.5, we can write a = o’
and T = T’. The correlation function g(r) in percolation theory is defined as the
probability that a site at a distance r from an occupied site is also occupied and
belongs to the same cluster. The average number of sites to which an occupied site
at the origin is connected is therefore 2 g(r), the sum running over all lattice sites.

We then define the correlation length or connectivity length 6 somewhat differently

than the way it is introduced in statistical mechanics:

£2 = 221222") (2.14)

2. 9(7‘)

6 (p) shows the typical critical behavior close to the critical probability pc:

€0< Ip—pcl’” (2.15)

17

with the critical exponent 1/ (see also Table 2.2).

Let us also consider the fraction P of sites of the lattice belonging to the percolat-
ing cluster. This value can of course only be different from 0 for p 2 pc. It therefore
is an order parameter of the percolation phase transition. It can be shown that P

shows critical behavior:
P 0< 9(1) - pc)(p — 1905 (2-16)

with the critical exponent

527-2 (2”)

0'

 

Next we are going to consider the convergence of the mean cluster size S at the
percolation threshold:

5 0< [P -10c|_7 (2-18)

This introduces the critical exponent:

 

7 = 0 (2.19)

Another important quantity in the analysis of percolation systems is the kth moment

of the cluster size distribution:
00—1
M. = Z skn. (2.20)

The notation “oo ——- 1” is used to indicate that in the summation the infinite cluster
(if existent) is excluded (otherwise the sum does not necessarily converge). For Mk,

we find:

M. oc Ip — pcl‘T‘HW (2.21)

See Table 2.3 for some values of the critical exponents in systems of different dimen-
sionality (note that no lattice topology is given, since — as discussed _ according to
the principle of universality the critical exponents only depend on the dimension of

the system in this case).

18

 

[Exponent d=2 d=3 (1:4 (125 Bethe]
[3 5/36 0.41 0.64 0.84 1
43/18 1.80 1.44 1.18 1
4/3 0.88 0.68 0.57 1/2
36/91 0.45 0.48 0.49 1/2
187/91 2.18 2.31 2.41 5/2

 

 

 

 

 

‘iQQQ

 

Table 2.3: Percolation critical exponents for d = 2, 3, 4, 5 and in the Bethe lattice. Ra-
tional numbers give (presumably) exact results whereas those with a decimal fraction
are numerical estimates.

2.3.4 Finite Size Scaling

Let us consider a finite, til-dimensional lattice of size Ld. As pointed out in the
preceding section, several quantities of the cluster structure (like the mean cluster
size S or the higher moments of the cluster size distribution) diverge at the critical
probability pc. This can not happen in a lattice of finite extent. There are also some
other changes that take place when we deal with finite lattices instead of infinite ones.
The main quantity governing the qualitative and quantitative behavior of system
properties on lattices of finite size is the correlation length 6, or, to be more precise, f
in comparison to the linear system extent L. As has been discussed in Section 2.3.2,
the power law dependence for the cluster size distribution is only valid as long as the
clusters stay smaller than a cutoff size 85 that is directly connected to the correlation
length by the dimensionality D of the clusters: 85 = {D (D does not necessarily
correspond to the topological dimension of the system, since the clusters can be
fractal, so that D S d). The key point of this observation is that the functional form
of system quantities is basically the same as in the infinite case when the system size L
is larger than «5, L > 5, since the cluster structure is basically determined by clusters
with 3 << Sg. If the system size L is smaller than 5, L < «S, correlations exist that are

larger than the actual system. Therefore, we have to expect qualitatively different

19

behavior in this case. One can infer the following generalized scaling laws for systems
of finite size. It is assumed, that f scales with the percolation probability according
to f or [p — pcl‘”, and that a given observable X scales according to X o< [p — pclx in
the infinite system with the critical exponent x. We then get the following general

form of the scaling law:

. /V
X(L,§) : gx/ux1(L/§) o({ 2:” 2:: (2.22)

OF
X(L1p) : (p _ pol—X332 ((17— pc)L1/V) (2.23)

where 231 and x2, respectively, are “transition functions” for the quantity X that
describe the change in behavior for the transition L > 5 ——> L < f, with the

asymptotic form given by the last expressions in equation 2.22.

One “non-critical” quantity in percolation theory that is also affected by a change
from infinite to finite lattices is the percolation threshold pc. If we denote the proba-
bility for the existence of a percolating cluster as II, it is clear from the discussion in

Section 2.1.8 that 1100(p) must be a step function for the infinite lattice:

0, < c
11.0) =[ 1 g); (2.24)

For the finite lattice, of linear extent L for example, there can exist a system—spanning
cluster for the density p slightly lower than pc; also, no system-spanning cluster can
occur for p slightly greater than pc. Instead of having a sharp step at p6,. I'IL(p) will
rather be given by a smooth function approximating a step function — the larger the

lattice, the better the approximation (see Figure 2.3).

It can be inferred from the analytically solvable, one-dimensional case that IIL(p)

should behave as follows:
TILL”) = 9 ((1? — pc)L‘/") for In — pcl << 1, L > 1 (2.25)

20

 

 

 

 

 

 

10
11

CL“

dp
0 pc p

Figure 2.3: Variation of the probability II (solid lines) that a cluster is spanning

the whole system for medium and large system sizes. The dashed line gives 5%,

proportional to the probability that at concentration p a spanning cluster starts to
appear. The width of the transition region or peak should vary according to LU”.

g(x) is a scaling function increasing from 0 to 1 as its argument x increases from —00
(far below threshold) to +00 (far above threshold). The derivative of the expression
in equation 2.25 then gives the probability density for the existence of a percolating

cluster in the system:

(111 _ 1/u W
a; —— L g, (0 — p.)L ) (2.26)

The average concentration pm, at which, for the first time, a percolating cluster ap-

pears can then be defined as:

Pay E [p (dd?) dp (2.27)

From equations 2.26 and 2.27 we find:
pav — pc (X L_1/V (2.28)

With the probability density 2.26, we can also calculate the width A of the p—interval

in which the transition takes place on average:

42 2 [(19 — p.)2 (Ell-13%

21

= (p2) - (iv)2
The width scales according to:

A or L’l/V (2.29)

Sometimes, the fractional shift C(L) and the fractional rounding 6 (L) are introduced

as follows:

€(L) s W (2.30)

Pa

ML)
19.

 

Qq
A

bi
v

I“

(2.31)

They apparently also scale preportional to L‘l/V.

2.4 The Nuclear Lattice Model

The Nuclear Lattice Model (N LM) introduced by Bauer [6] (see also [7]) makes use
of several of the aspects of percolation theory discussed so far. It was introduced as
a model of the production of complex fragments in nuclear collisions at intermediate
and high energies. Experiments carried out in this area have shown many seemingly
different phenomena over the years. One class of results that has received special at-
tention is multifragmentation reactions (MFR). The experimental data seem to show
traces of critical behavior. In recent years, experiments concerned with the produc-
tion of “Quark-Gluon-Plasmas” (QGP) and the corresponding phase transition (or
quark deconfinement) have caught some public interest. However, there are indi-
cations that infinite nuclear matter also supposedly undergoes a “liquid-gas” phase
transition, that is somewhat better understood than the QGP transition (see Figure
2.4). Since the assumed critical point for the liquid-gas transition lies in a region
of the phase diagram that is physically well attainable in a variety of contemporary

research facilities, it can examined quantitatively to some extent.

22

 

1000
Big Bang Querk—Gluon
Plasma

(HOV)

Temperature

sear

 

 

1
Density

Figure 2.4: The Nuclear Phase Diagram (schematic)

It is one of the objectives of the NLM to provide a tool to study the region around
this critical point in the nuclear phase diagram on a theoretical basis. It has to
be mentioned, however, that it is by no means clear that MFR can be described
correctly by a phase transition model at all. In particular, all theoretical models of
phase transitions assume a system of infinite extent. Therefore, special attention has
to be paid to the fact that we deal with systems of relatively small size (of the order

of 102 constituents) in MFR experiments.
2.4.1 The Model

The NLM describes, as mentioned earlier, reactions of the type
Y + AT —> Ap + X (2.32)

where Y is the projectile, which will, in our case, usually be a proton, AT is the
target nucleus, AF is the remainder of the nucleus after the fast, first-stage “pre-
equilibrium” particles X have been removed. ‘AF thermalizes, meaning that the energy

deposited in the collision is distributed homogeneously over the system, and breaks

23

up into fragments. The AT nucleons of the target nucleus form, in this model, an
approximately spherical distribution on a simple cubic lattice. Arbitrarily deformed
nuclei can also be considered, but assuming that the target nucleus is reasonably
compact, the results are essentially unchanged. The simple cubic lattice is chosen
since it is particularly amenable to computation. The results of percolation theory
show that the lattice topology should have no influence on the cluster properties. This
is certainly only true for infinite lattices, but it is assumed that the changes caused
by considering a finite system are small enough not to have a profound impact on the
results derived from this model. For a given impact parameter b, those nucleons are
removed from the lattice that lie within a cylindrical channel of radius r (radius of
projectile) at impact parameter b (see Fig. 2.5 for the fireball geometry). For proton-
induced reactions typically 6-8 nucleons are in the fireball, so the effect on the results
are slight, i.e., one obtains very similar results Without removing the nucleons from
the fireball channel. Since the impact parameter introduces a physical length-scale in
the model, a lattice spacing d has to be chosen. d can be computed approximately

from the nuclear saturation density p0 (p0 z 0.15 nucleons/fm; see [6, 4])

d = —1— z 1.81 fm. (2.33)

1 3
.00/

Using the breaking probability pH (which would correspond to q in the terminol-
ogy of our earlier discussion of bond percolation) as an input parameter, a Monte-
Carlo algorithm decides for each bond individually whether it is broken or not. This
procedure is followed by a counting algorithm that looks for clusters and evaluates
their size. Finally, we integrate over all impact parameter b. This happens with the

geometrical weights that different impact parameter intervals have:
dN(b) oc bdb (2.34)
(area of a ring of thickness db at radius b: 27rbdb). By this procedure, an inclusive

24

"participant"

   

"spectator"

Figure 2.5: The fireball geometry: the lattice sites in the cylindrical channel with
radius r at impact parameter b are left unoccupied.

mass yield distribution is obtained that can be compared to experimental data. At
this stage, the only adjustable parameters that enter the model are the breaking
probability p3 and the mass AT of the target nucleus (which, of course, is given by

the experimental setup).
2.4.2 Choice of p13

Next, we will discuss three different ways to choose the breaking probability 113. This

is crucial in order to reproduce the inclusive spectra obtained by typical experiments.
Constant p3

In this approach, a constant p3 is chosen for all events and all impact parameters b.
Reconsidering the earlier discussion of the thermalized source with energy deposited
all over the nucleus, it seems to be reasonable to assume that the individual bonds
are more likely to break for higher excitation energies of the source. On the other
hand, the fact that the excitation energy certainly depends on the impact parameter

of a collision shows us that the excitation energy varies from event to event (which is

25

also supported by experimental results). Therefore, the constant breaking probability

approach seems to be too simple to reproduce inclusive experimental data.

The Woods-Saxon Approximation

In order to take into account the dependence of p3 on the excitation energy (and
therefore on the impact parameter), p3 is assumed to be larger for central collisions
than for grazing collisions. An ansatz analogous to the Woods-Saxon approximation

of the nuclear density profile is chosen:

 

_ P o
pB(b) — 1+ exp[(ll: — R)/a] (235)

where R is the radius of the target nucleus and a parameterizes its “diffuseness”.

a should therefore be an experiment-independent fit parameter whereas p30 is the

adjustable parameter for the actual experimental situation.

The Glauber Approximation

This approach can be considered an extension of the Woods-Saxon approach. It is still
assumed that p3 depends on the impact parameter or, equivalently, on the deposited
energy. Now the deposition of energy in the nucleus is modeled by the ansatz that the
breaking probability should be proportional to the integral over the nuclear density
along the path traveled by the projectile. It is assumed to be spatially constant over

the whole lattice, i.e., the situation of total thermalization is assumed.

pBoZledeR

 

123(1)) = (2.36)

fpiRdeR

For the nuclear density p(R), one of the standard parameterization (like Woods-

Saxon) can be used. Again, p30 (in addition to other parameters from the used

26

model for the density) is left as the adjustable parameter for the experiment. pB(b)
is a monotonically decreasing function with pB(0) 2 p30. However, this model can
only be correct for small projectiles, since for larger projectiles (like in heavy ion-ion

collisions) the overlap integral of the two nuclear densities should be used instead.

p3 from experimental excitation energies (Hybrid Model)

If the results produced by the NLM are to be compared to experimental data on
an event-by-event basis, and if the excitation energy for the single events in the
experimental data is known, another approach to obtain p3 is possible (see [33]). We
assume that the energy distributed into each bond of the lattice, 6),, can be described
by a Boltzmann distribution with mean energy (6b). Each site of the lattice has an
average of or bonds (on an infinite, 3—dimensional lattice, or would be 6/2, but we
have to take into account that the lattice is finite which causes surface effects). The
average deposited excitation energy per site then is (E,) = a (69,), and the binding
energy of the initial nuclear system is B 2 QB), (where E, is the binding energy per
bond). When the system undergoes the multifragmentation reaction, any bond which
has an energy greater than Eb will break. Therefore, the bond-breaking probability
is:

Eme‘Es/TdES

fave/ME.
0

 

PB (2.37)

Here, T is the temperature of the nuclear system which has to be introduced in
an appropriate way. It is through this temperature that the actual experimental
excitation energy E" enters the breaking probability: T = T(E*). The average
binding energy per nucleon B is an adjustable parameter of the model (within the

restrictions given by the possibilities to measure this quantity). With the help of the

27

generalized, incomplete Gamma function I‘(x, 20, 21), equation 2.37 can be rewritten

 

 

 

 

 

as:
(E*)-1—2r 550—3— N? (238)
p3 " 2’ ’T(E*) '
1 . fl
0.8 -
0.6 .
E
8°
0.4 -
0.2 -
0 1 1 1 L l l 1 1 l
0 2 4 6 8 10 12 14 16 18 20

T in MeV

Figure 2.6: Relation between p3 and the temperature of the fragmenting nucleus as
given by Equation 2.38 (B=6.6MeV)

In this approach, p3 no longer explicitly depends on the impact parameter b.
Since it is not possible to measure the impact parameter of the reactions likely to be
described by this model, one can no longer apply the removal of the fireball channel
without further assumptions about the correlation between the impact parameter and

the excitation energy. This will be discussed in further detail later (see 3.3.1).

28

Chapter 3

Application of the NLM to ISiS
data

3.1 The ISiS Experiment

The experimental data analyzed in this thesis were obtained with the Indiana Silicon
Sphere (ISiS) 47r charged-particle detector array [16] [32] in two experiments (E900
and E900a) at the Brookhaven National Laboratory AGS accelerator. In E900 un-
tagged secondary positive beams at 5.0, 8.2 and 9.2 GeV/c incident on a 197An target
were employed. In E900a a tagged negative beam of 8.0 GeV/c Tr‘ and antiprotons
incident on the same target was used. The Au target consisted of foils of 10‘5 purity,

which were prepared by vacuum evaporation onto a glass slide (KCl substrate).

The detector array consisted of 162 triple-detector telescopes arranged in a spher-
ical geometry. The telescopes span the polar-angle range from 14° - 86.5° in five
segments in the forward hemisphere and 935° - 166° in four backward-hemisphere
segments. The detector telescopes consisted of a gas-ionization chamber operated
at 16-18 Torr of C3F8 gas, a 500 um passivated silicon detector, and a 28-mm CsI

scintillator with photodiode readout.

In this thesis, we focused on the analysis of approximately 1.5 x 106 p —+ Au events

29

at 10.2 GeV/c in the E900 experiment.

3.2 Implementation Details and Modifications of
the N LM

3.2.1 General Setup

Since only the charges were detected in the inclusive mass yield spectra of the ISiS
experiment (see Section3.1), we had to incorporate the fact that no information about

the fragment masses was available in our model. We followed two approaches:

Lattice with 79 nucleons (protons): We just modeled the protons of the Au nu—
cleus by distributing 79 sites approximately spherically on the lattice (as de-
scribed in Section 2.4.1). Therefore, we could compare the sizes of the clusters
obtained in the model calculation directly to the experimental data. However,
as described in Section 2.3.4, finite-size effects have to be considered, and these

effects certainly depend on the actual size of the system.

Lattice with 197 nucleons (full Au nucleus): This approach is certainly more
likely to incorporate the finite-size effects correctly. On the other hand, since the
model assumes isospin-symmetry, we had to derive the charges of the produced
fragments (clusters) in order to be able to make a comparison with experimen-
tal data. We did this by downscaling the fragment size with the mass-charge
197/ 79 ratio of the Au nucleus. Since we are scaling from one discrete set of
numbers ({1, . . . , 197}) to another ({1, . . . , 79]), Moiré effects occur in a simple
multiplication of the fragment size by the scaling factor. Therefore, we imple-
mented a Monte-Carlo algorithm to avoid these effects. This approach does not
take into account binding energy effects, since the ratio of neutrons to protons

in stable configurations changes from small fragments (1 : 1) to larger fragments

30

(more neutrons than protons). Especially for small fragments, it is certainly not

equal to the Au ratio 197/ 79.

A comparison of the two aforementioned approaches shows that the respective
results vary only insignificantly. We used the approach that works with a smaller
lattice because it is less time-consuming in numerical simulations. The part of the
simulation that takes most time is the cluster-counting algorithm. Its time complexity
is on the order of N 3, where N is the number of the lattice sites (however, algorithms

with a time complexity on the order of N can be implemented).

3.2.2 The Filter Code

One peculiarity of the experimental data is the existence of “residual fragments”
(in the data file, at most one per event), which contain the sum of all undetected
fragments in a given event. The charge of these fragments can be determined by
comparing the number of detected particles with the size of the thermalized source.
Thus, the residual fragments do not describe fragments that physically appeared in

an event. The reasons for the appearance of undetected residuals are as follows:

0 The 47r geometry of the detector cannot be perfect, since parts of the surface of
the detector-sphere cannot be “actively detecting” because of the area occupied

by the beam pipe and “seams” between individual detectors;

0 Some of the 162 detectors were found to be defective or operating improperly

after the experiment was completely assembled;

e The detectors were only constructed for a certain mass and energy range, and

therefore can not detect fragments that lie outside these boundaries.

31

To incorporate these effects in our model calculations, we “piped” all percolation
fragments through a filter code that decides whether the fragment is detected or not.
This code has been provided by the ISiS group. With the help of this filter, we can

introduce artificial residual fragments in the model data in a controlled way.

The input that is required by the filter code is the charge, mass, energy and
direction (in the laboratory frame of reference) of the fragment for which the successful
detection has to be determined. Since the NLM in the way we implemented it (see
above) only provides us with information about the charge of the fragments, the

missing information has to be determined in the following ways (see. A.2):

Mass of Fragment

Here we use the ratio of neutrons to protons in the Au nucleus to determine a prelim-
inary mass A’ from the charge Z by A’ = or x Z. a is the ratio of protons to nucleons
in the nucleus. In general, this will be a non-integer number. To choose an integer
mass A of the fragment, we assign [A’] or [A’] to A with the probabilities [A’] —— A’

or A’ — [A’], respectively.
Direction of Particle Emission

Since the energy of the protons is high enough to merely remove the fireball channel
from the target nucleus (see Fig. 2.5) without transferring considerable linear mo—
mentum [32], the angular distribution of fragment emission in the multifragmentation
reaction is assumed to be isotropic. Therefore, a random direction with respect to an

isotropic distribution is chosen for each fragment.

32

Fragment Energy

The energy of the fragments is assumed to be distributed according to a Maxwell-
Boltzmann distribution [30] in the rest frame of the target nucleus [44]. It can be

written as:

 

(120' (AF) _ 0'0 * Elf
dE*dQ* _ 2(7rT)3/2‘/’jTexp (_ T ) (3'1)

where E“ is the kinetic energy that is given by, after a correction for the Coulomb

barrier and the recoil,
AT

E*=__'__.
AT—Ap

E’ — K. - B (3.2)

E’ is the kinetic energy of the fragment in the moving frame of reference, K. is an
adjustable parameter < 1, and B is the Coulomb barrier. The barrier B, for two
fragments touching each other with charges Z p and Z R = ZT — Z,» and masses AF

and AR '2 AT — AF, is:
62ZFZR
To ' (Aff/3 '1' 14:2/3)

By transforming equation 3.1 to the laboratory frame one obtains the energy of the

B:

 

(3.3)

fragments in this frame:

(120' (AF) __ \/—_l_3: (120' (AF) (3 4)
dEdQ ‘ E’ dE’dQ’ ' '

The connection between E’ and the laboratory energy E is given by

, 1
E = E '1‘ 52415132 — fly/2AFE ° GOSH-3101,) (3.5)

where B is the velocity v / c of the emitting system.

Since in our case high accuracy is not necessary and B is close to 0, we assume
[3 = 0. This is supported by the findings described in [32]. Therefore, equation 3.5 is
simplified to E’ = E, and we can substitute E’ by E in equation 3.2. The value of

E’“ to be used in equation 3.1 is therefore E shifted by —n - B. From earlier analyses

33

conducted by Hirsch et al. [29] and Bauer et a1. [3], we use their value of T=12
MeV. For It we chose a value of 0.6 to reflect the lowered freeze-out density pic of the
nuclear matter before breaking up into fragments. Bauer in [3], for example, finds it
to be

We z 0.36p0 (3.6)

for the reaction 300 GeV p+Xe ——> 12C + X.

3.2.3 Charges 17-20

Another peculiarity of the experimental data is the fact that while the detectors were
able to detect fragments with charges up to 20, it was only possible to resolve them
elementally in the charge range 1-16. The fragments with charges between 16 and
21 were therefore assigned a charge number on a Monte-Carlo basis, extending the
assumed power law of the mass yield curve. Since the provided filter code does not
implement this feature - all fragments with charges of 17 and higher are declared
undetected — we removed those fragments both in the experimental data and in the

model calculations.

3.2.4 The Tcl/Tk Interface

The experimental data as well as the data derived from the model calculations, con-
sist of a list containing the single events of the multifragmentation reactions. In order
to simplify the fitting procedure, we implemented an interactive user interface that
is able to display the difl'erent characteristics of the derived events (mass yield, mul-
tiplicity distribution, etc.) and that allows for immediate adjustment of the model
parameters. As an easy and system-independent way to realize this, we implemented
an interface in the script language Tcl/Tk. This interface serves as a front end to the

percolation simulation and data acquisition routines that are written in C++. It dis-

34

plays the experimental data and the data from the model calculations simultaneously
to make a visual comparison of the two data sets possible. With this setup, changes

in the model can be evaluated quickly and easily.

3.3 Results

Unless stated otherwise, all diagrams in the following sections were created with
500,000 events produced with the C++ implementation of the NLM (see Section 3.2
and appendix A2) and the same number of events from the experimental data (see
Section 3.1). In particular, the bond-breaking probabilities for the percolation events

have also been chosen from these events (in the final setup, see Section 3.3.1).

3.3.1 Choice of Parameters

In the beginning, we worked with the Glauber approach for the pmak - impact pa-
rameter dependence. As a fitting procedure we tried to reproduce the inclusive charge
yield spectrum of the experimental data. In order to reach a reasonable correspon-

dence of the two data sets, we used the following setup:

Projectile Mass 3 4 For good correspondence in the high mass region, a projec-
tile mass of 4 (as opposed to one, which would be realistic for proton-induced
reactions) had to be chosen. This might be due to the fact that the projectile
energy of 10.2 GeV is comparably low, and therefore the cylindrical fireball—
channel degenerates to a “trumpet-shape”, which can be approximated by a

cylindrical region of larger radius.

Impact Parameter Range restricted to 0-4 fm The range of impact parame-
ters used to integrate the inclusive data had to be restricted to 0-4 fm (while the

largest possible impact parameter for a grazing reaction would be 7 fm). The

35

reason for this might be that the fall-off of the Glauber approximation is not
steep enough. The approximation does not take into account the production
of pions (rest mass of 140 MeV, which is certainly attainable in that projectile
energy range) which decay within the nucleus and deposit their kinetic energy

+ rest energy.

Undercritical p30 The adjustable model parameter p30 had to be chosen to be
p30 = 0.6. Since this is the maximal breaking probability (realized in central
collisions), no critical events appear in the model calculation (critical pmak for
three dimensional bond—percolation on a cubic lattice: 0.7512, with finite size

correction: z 0.7).

Even though with this model setup a fair agreement of experimental and model
data could be achieved (at least for the inclusive mass yield spectrum), some of the

adjustments seem to be rather arbitrary and finally led us to a different approach:
pbreak from Hybrid Model and experimental Source Size

In our final setup for the analysis, we used the determination of pm“), by means of
the hybrid model (see Section 2.4.2). In particular, pbreak is given by Equation 2.38
with T determined from the experimental excitation energy. The value of the binding
energy per nucleon, B, was adjusted to 6.6 MeV (to achieve best correspondence of the
inclusive mass yield spectra). This binding energy matches well previous research (see,
for example, [5]). It is assumed that the relation between the excitation energy E" of
the fragmenting nucleus and the temperature is given by E“ = aT'2 with a = A0 / 13
(corresponding to the high temperature limit of a model for a degenerate F ermi—gas;
A0 is the mass number of the nucleus, see [17, 15, 14]). This is done on an event-

by-event basis and the corresponding percolation events are generated. As already

36

mentioned in Section 2.4.2, in this approach, we have no straightforward method to
determine the impact parameter of the incident particle. Therefore, no controlled way
of removing the fireball channel exists. Instead, we use the information about the
size of the fragmenting thermalized source, which is obtained by adding the charges
of the individual fragments and the charge of the residue (as given in the event file),
and use this as the actual size of the percolation lattice (see Section 2.4.1). Thus, the
size of the lattice generally changes from event to event, but matches exactly what is

found in the experiment.

Working with a smaller spherical lattice representation of the thermalized source
rather than with the fireball geometry described earlier obviously decreases the effec-
tive surface area of the system. Since nucleons at the surface have fewer bonds to ad-
jacent neighbors, this method might introduce significant changes in the model data.
However, nuclear transport theory calculations in the BUU (Boltzmann-Uehling-
Uhlenbeck) model have shown that the channel tends to “heal” before the actual
breaking up occurs. Therefore, no meaningful systematic errors should be introduced

in the model by our method of using a compact thermalized source.

3.3.2 Excitation Energy Spectrum and Distribution of pre-
equilibrium emitted Particles

In Figure 3.1, the spectrum of the excitation energy e“ = E*/A0 per nucleon of the
thermalized source (circles) and the distribution of pre-equilibrium emitted charges
(lines) are plotted. In this diagram, all available 1818 events have been plotted.
As already explained (see Sections 2.4.2 and 3.3.1), the first set of data is used to
determine a bond-breaking probability for our model, while the second set gives the
size of the lattice used in the calculations. The distribution of the pre-equilibrium

emitted charges is fairly wide and could not be produced in this form by using the

37

Excitation energy per nucleon in MeV

2.5 5 7.5 10 12.5 15
80000 1 r

 

 

rpre—equilibritlm emitted charges

excitation energy spectrum 1—6—4

70000 -

60000 -

50000 ‘-

40000 -

Yield

30000 F

20000 t

10000 -

 

 

 

 

 

o 10 20 30 40 I 5060
Charge

Figure 3.1: Spectrum of excitation energy per nucleon and frequency of pre—equilbrium
emitted charges in experimental data.

simple fireball geometry model. Using this distribution we could calculate an average
thermalized source size of approximately 65 charges, i.e., on average about 14 “fast”
charges are emitted before the source equilibrates. However, it should be mentioned
that the distinction between pre—equilibrium and equilibrium emitted particles is by

no means exact and unambiguous [32].

3.3.3 Charge Yield

Figure 3.2 shows the resulting inclusive charge yield spectra. In this figure, the
experimental yields are plotted as circles, the filtered model data are given by the

solid line and the unfiltered by the dotted line.

The model calculation and the experimental data are in close agreement. In the

38

 

 

 

 

 

 

 

 

ie+O7 , 1 r I 1 l . 1 1 4
. expenmental data +—-6—l ,
. model calc., filtered
’9 model calc., unfiltered ------------------
1e+06 .- g: ':
100000 _- 1
10000 ,- 1
1! » «
.2
>— >
1000 r 1
. J
100 r .21
p m 1
10 r 1
[r d
. l
1 l 1 l l l l l l
O 10 20 30 4O 50 6O 70 80

Charge

Figure 3.2: Inclusive charge yield spectrum. The experimental data (corrected by the
“17-20” events, see Section 3.2.3) is given by the circles, the filtered and unfiltered
model data by the solid and dotted lines, respectively. The charge is given, as in all
following diagrams, in multiples of the charge of a proton.

experimental data, the fragments with charges between 17 and 20 have been omitted,
as described earlier in Section 3.2.3. Therefore, the yield in the range 17-79 is given by
residual fragments. One can clearly see the “gap” that is produced by this method at
Z = 16. This gap is not present in the unfiltered calculations. Besides this artificially

introduced difference, the filtered and unfiltered model data differ significantly for

small and large charges.

That the experimental data do not follow an exact power law for small charges is
probably due to the fact that the primary yield of fragments is altered by sequential
decay of unstable fragments before their detection. For instance, the small jump at
Z = 4 is most probably caused by the lack of a stable 2B nucleus (see also Section

3.3.8).

39

3.3.4 Multiplicity Distribution

 

 

70000 , I I . . I
experimental data r—e—r
model calc., filtered
model calc., unfiltered ------------------
60000 — ° .
Q o c
0
50000

40000

Yield

30000

20000

10000

 

 

 

 

Multiplicity

Figure 3.3: Multiplicity distribution. The experimental data are given by the circles,
the filtered and unfiltered model data by the solid and dotted line, respectively.

Figure 3.3 shows a plot of the multiplicity distribution for the experimental data,
filtered model data and unfiltered model data. Again, the experimental data is plot-
ted with circles, and the filtered and unfiltered model data with solid and dotted
lines, respectively. Although the correspondence between model calculation and the
experimental results is not as good as in the case of the charge yield spectrum, one
can clearly see that the filtered data are considerably closer to the experimental data
than the unfiltered data. Since the filter basically “eliminates” fragments by assign-
ing them to the residual fragment, the filtered multiplicity spectrum is to be shifted
towards smaller multiplicities in comparison to the unfiltered spectrum. This can be

clearly seen in the diagram.

40

3.3.5 stidue Distribution

 

 

    
   

100000 . 1 . . T I l 1 .
E 2 Z",s dist. exp. data l—e—l 3
Zres dist., model calc. .
largest fragment, model calc. .
l 1 ' i : _ . _e- e 9
10000 : .. . . . . . _ - 1.: _ i. ....... -5 . _:

Yield

1000 :-

 

 

 

. i :
0 1 0 20 30 40 50 60 70 80
Charge

 

Figure 3.4: Distribution of Zresmue (experimental data and filtered model data, circles
and solid lines) and of the largest fragment (unfiltered model data, dotted lines).

The distribution of Znndue is shown in Figure 3.4. Since for the unfiltered model
data no residual fragment is created, the distribution of the largest fragments is
plotted for this data set instead. For the experimental data and the filtered model
data, the yield of fragments with a charge greater than 16 corresponds exactly to the
charge yield spectrum, because no regular fragments with a charge > 16 can pass
the filter. A comparison with the largest fragment of the unfiltered data shows that
the yield for the high charge region is dominated by single fragments per event. The

effect of the filtering can mostly be seen in the range of 0—30 charges.

41

’ exp. data ~—o—~ . . . _
0.1 _— meodel calc. multrplrcrty—3
largest frag. model calc. - ..... -

 

 

0.01
0.001
0.0001

1e—05

0.1

 

 

.. - l
0.01

0.001 ,

0.0001

Yield

0.1 -

 

 

0.01

0.0001 -

1e-05

T

q»

 

multiplicity=15

0.001 -

0.0001 ,

 

 

1e—05

20

80 40 60 ' 70
Charge

 

 

80

Figure 3.5: Zre, distribution for different multiplicities. In the case of the unfiltered
data the distribution of the largest fragment is plotted instead.

42

3.3.6 Zresidue Distribution gated on Multiplicity

In Figure 3.5, the Zreswue (or, in the case of the unfiltered model data, the largest
fragment) distribution for different values of the event multiplicity is plotted. It is
obvious that the use of the filter has a much larger overall impact in these plots than
for the inclusive Zresme plot (Figure 3.4). This has to be seen in connection with
the shift in the fragment multiplicities caused by the employment of the filter code
(Section 3.3.4). This fact can be of some importance for a critical exponent analysis

using higher moments of the fragment distribution.

3.3.7 Second Moment Distribution

Figure 3.6 shows the second moments of the fragment distribution (see also Section
2.3.3) as a function of the event multiplicity. In the upper branch of the diagram, the
largest fragment is included in the calculation of the second moment for each event,
whereas the lower branch is obtained by excluding the largest fragment. In the case
of the experimental data and the filtered model data, the largest fragment is usually
(but not necessarily) given by the residual fragment. The correspondence between the
experimental data and the model data is not as good as for the charge yield spectrum;
however, it is apparent that the filtered model data match the experimental results

considerably better than the unfiltered data.

It has been proposed by Bauer [4] that the existence of a maximum in a M2/M1
versus M0 plot can be interpreted as a trace of a phase transition in the percolat-
ing system (i.e., that events with critical and overcritical percolation probability are
present). This is a manifestation of the fact that the value of M2/M1 diverges accord-
ing to |p — pc|‘1/". This holds only for infinite systems. Also, as described in Section

2.3.3, the infinite cluster has to be excluded in the calculation of the moments. In [4],

43

 

10000 1 , , . 1 ‘
: eXperimental data 1—9—1

model calc., filtered
model calc.. unfiltered ------------

 

    
    

largest fragment included

1000 _~ 7

.......

100 _ ............... ‘ _

M2

largest fragment excluded

 

 

 

1 I l l l l 1 1
0 5 10 15 20 25 30 35 40
Multiplicity

Figure 3.6: Second moment (M2) versus multiplicity. In the upper branch, the second

moments have been calculated including the largest fragment, in the lower branch the
largest fragment has been excluded.

44

it is shown that for finite systems, excluding the infinite cluster translates into ex-
cluding the largest cluster (or, in our case, the largest fragment). Since, on average,
the multiplicity M0 of a percolation event increases with increasing bond-breaking
probability monotonously, a maximum can be expected in a Mg/Ml versus A110 dia-
gram if events with undercritical, critical and overcritical percolation probability are
present in the data. For practical purposes, a cutoff at about 1/3 of the system size

should be used when evaluating the moments of the distribution.

It cannot be inferred unambiguously that a phase transition is present either from

the experimental data or from the filtered model data in Figure 3.6 (lower branch).

3.3.8 Fisher Droplet Model (FDM) Scaling

In 1967 Fisher introduced a model of droplet formation and condensation [21], which,
since then, has been applied to nuclear multifragmentation in different ways. It is
based on the assumption that the non-ideal gas can be approximated by an ideal gas of
clusters. Elliot et a]. [12] applied the basic scaling ideas from the Fisher droplet model
both to percolation theory (a connection that can be based on theoretical reasoning)

and to nuclear multifragmentation data from the EOS collaboration [10, 15, 17].

One of the basic features of the droplet model and its forerunners [23, 35] is that

the mean number of droplets of a given size A can be written as:

(3.7)

AAu]
T

(NA)o<[

where Au = u — a, and ,a and a, are the actual and liquid chemical potential, respec-
tively. In order to describe the distribution for intermediate values of A, Equation

3.7 was modified to include the influence of the surface of the droplets:

 

C 2/3
(NA) 0< [Ag/1 — (T); ]

45

where c(T) is the surface free-energy density. To account for the properties near crit-
icality, Fisher introduced an explicit term for c(T) and a topological factor resulting

in an expression for the normalized droplet distribution:

_ NA _ _T AA” (30an
(”Al — <A_o> — (10A exp [ T T (39)

 

This scaling behavior will be referred to as “FDM scaling”. In Equation 3.9, A0
is the size of the system and go is a normalization constant depending only on the
value of T, qo 2: 1 / C (T — 1) (with Riemann’s Zeta function C) [36]; T, the topological
critical exponent, depends on the dimensionality of the system with origins that lie
in considerations of a n-dimensional random walk on a surface closing on itself. In
percolation theory, T depends on the scaling behavior of the normalized cluster dis-
tribution close to the critical percolation probability (see Section 2.3.2 and Equation
2.11). For three dimensions 2 g T g 3. coeA" is the surface free energy of a droplet
of size A; Co is the surface energy coefficient; 0' is the critical exponent related to the
ratio of the dimensionality of the surface to that of the volume (for percolation, see
Section 2.3.2); and e = (T C —— T) / Tc is the scaled control parameter that measures the

distance from the critical point, TC.

It becomes clear that, when we substitute 6 2 (pc — pbreak)/pc and T = pbreak,
Equation 3.9 takes on the form of the cluster scaling relation 2.11 in Section 2.3.2

with the special cutoff function

 

 

(3.10)

f [(pbreak — pC)A0] : exp [AA/j, _ CDC/40]

pbreak pbreak

This once again shows that percolation phenomena, with a geometrical phase tran-
sition, share with thermal critical phenomena the same scaling behavior (as well as

the same renormalization group and features of universality [13]).

It has been found by Elliot et a1. [12] that the bulk factor exp[AAu/pbreak] for

percolation and exp[AAu/T] for the droplet model is very close to unity, i.e. that

46

Au z 0 in the vicinity of the critical point. With this information, we can rewrite
Equation 3.9 as:

(306/10

 

 

Tc—T)

= _ A0
1 ml C“ (Trc

(40/qu” = exp [—

 

(3.11)

Therefore, a logarithmic plot of (n,.()/q0A‘T versus the scaled control parameter,
A"e/Tc or Aae/pbreak, respectively, for a given A, should show a straight line crossing
(0, 1) for critical events. Furthermore, all these graphs for different A should collapse.
This, of course, only holds for events close to the critical temperature or probability,

i.e., for data points with the scaled control parameter close to 0.

Percolation Model

Figure 3.7 shows a plot of the scaled fragment distribution versus the scaled control
parameter for the percolation (model) events. Instead of the control parameter in
Equation 3.9, we use the scaled parameter multiplied by pc in order to work with a
dimensionless quantity: Z " (pC — pbreak)/pb,.eak. Part (a) shows the filtered model data
and part (b), the unfiltered model data. The diagrams include plots for Z values
ranging from 3 to 8. A critical breaking probability of pC = 0.7 has been chosen.
This value differs from the theoretical value of 0.7512 (see Table 2.1, page 10) since
we have to take finite-size scaling effects into account. For this purpose, we used the

parameterization for the fractional shift given by Bauer [6] for simple cubic lattices:

(19av — 120/19. = 6(a) "s 020/710'96 (3-12)

Here, n is the linear extent of the system, for which we chose \/3 79. This yields a pm,

value of approximately 0.7.

As critical exponents in this plot we used the values 0 = 0.45 and T = 2.18
(independent of the lattice topology, see Section 2.2.5), which are published, for

example, in [34].

47

 

 

 

 

0.1

 

<n(Z)>/qOZ “

 

0.01

 

 

 

 

 

0.001

 

 

 

 

 

 

 

 

 

 

 

1 .5 2 0 0.5 1
26(pc-pbreak l/pbreak

Figure 3.7: Plot of the scaled fragment distribution as a function of the scaled control
parameter (multiplied by pa) for fragments of charge Z. (a) shows the filtered data,

(b) the unfiltered model data. In this plot, a critical percolation probability pc 2 0.7,
o = 0.45 and T=2.18 have been chosen.

48

It can be seen that the graphs for different Z values both for the filtered data and
the unfiltered data collapse onto single curves in the plotted Z range. This is not
surprising, since FDM scaling behavior is predicted by standard percolation theory.
This feature is obviously not profoundly affected by applying the filter code. The fact
that the unfiltered data curve does not cross the “critical point” (0,1) might be due
to an imprecise estimate of pC and the necessity to adjust the values of o and T for the
filtered data (to account for the filter effects). Besides these effects, the “bend-over”
of the curves for small values of the scaled control parameter is a strong indication of
the fact that overcritical events are produced in the simulation —— independent of the
explicit choice of pc. This deviation from a pure exponential dependence is due to
limitations of the FDM to describe droplet formation correctly far beyond the critical

point.

To sum up, it becomes clear that only with the right choice of the critical exponents
o and T and the critical bond-breaking probability pc, the curves for different Z values
will collapse and follow an exponential curve in the vicinity of the critical point. This
circumstance can be used to introduce a new method of finding the critical exponents

o and T, and pc of the phase transition.

For a given choice of pc, the “quality” of the F DM scaling is calculated (in terms
of a scalar value decreasing with increasing correspondence with the assumed scaling
behavior) for points in the o — T plane. The result of this calculation can be visualized
in a contour plot indicating the agreement of the data with the assumed scaling

behavior. The best values for o and T are then given by the minima of the profile.

To evaluate the quality of the scaling for a given pair of values for o and T, we
analyzed the (supposedly) exponential part of the scaled distribution close to the

critical point for the individual charges Z. The logarithm of the distributions should

49

have a linear functional form crossing the point (0,ln( 1)):

1n ((n(Z))qOZ'T) = (12 ' [200% — pbreak)/pbreak)] + bZ (313)

with the slope oz and the offset bz. Ideally, bz should be 0 and az should equal oz:

for all pairs Z and Z’ for which data are available.

Instead of considering theg data points for different Z individually, we combined
them into one set of points {(x,,y,~ :l: 0,)[2' : 1 . . . N} and determine the degree of
correspondence with a linear dependence by applying a “least x2” fit. In this method,

the function

N -- —ax- 2
x2(b,a) E 2: (2+) (3.14)

1

with two free parameters a and b is minimized. For details, refer to [41]. Since we
already know that the data should run through the critical point (0,1) (or, in our

consideration, (0,ln(1)=0)), we used

 

X2(0,a) = Z (3“ — my) (3.15)

i=1 01'
as a measure for the quality of the fit.

In Figure 3.8, this analysis has been conducted for the unfiltered model data. A
critical breaking probability of p0 = 0.65 :l: 0.2 has been found to yield the lowest X2
values. This value is lower than the value of 0.7 estimated earlier. One reason for this
discrepancy could be that the size of the lattice varies from event to event (see Section
3.3.1). For T and a, we find the values T = 2.18:l:0.01 and a = 0.5i0.1. These values
are in good agreement with the theoretical values of 2.18 and 0.45, respectively. It is
quite apparent that the determination of T is more precise than the determination of
o. This is probably due to the fact that the Z values in this analysis only range from
3 to 8 so that the variation of Z" with o is smaller than the variation of Z" with T.

Also, the normalization constant qo depends on T, but not on o.

50

 

 

 

 

 

Figure 3.8: Density plot of X2 as a function of a and T for the unfiltered percolation
events. The 4 innermost contours represent [X2 / (degrees of freedom)]-values of 3.2,
3.5, 3.8 and 4.1, respectively. A critical breaking probability of 0.65 has been used in
this plot.

51

 

 

 

 

 

Figure 3.9: Density plot of X2 as a function of a and T for the experimental data
(uncorrected). The innermost contour corresponds to a [X2/ (degrees of freedom)]-
value of z 12.

Experimental data

The true potential of the method described above is to provide a way to determine
the critical exponents in the case of experimental data. Unlike other methods, as the
analysis of higher moments of the fragment distribution for example, this methods
allows to find Tc, 0 and T at the same time. As discussed in Section 2.3.3, the two

other critical exponents 13 and 'y can be expressed as functions of o and T.

Figure 3.9 shows the result of the determination of a and T using the “least X2”
method. It is apparent that the minimization procedure fails to produce meaningful
results. Figure 3.10 (using the “best fit” determined in the analysis visualized in Fig.
3.9) shows that the yields for Z=4 are lower than the yields for Z=3. Since these

differences should be compensated for by the factor Z" in the scaling of the cluster

52

 

 

 

 

 

 

10 1 1 1 1 12 3 .
1 . ' _: f = F—dr-‘r—i I
‘ Z=4 ”*4 ‘
2:5 1---Ir---°

... 2:6 ;. ...... a ...... .. 1
I; u" '2. O" ’ ‘

~1'a-141sw-w 31-1 ~..: .1... .. ~

1&3! ‘ 1‘51“. 1 ., .- .; nun-3:3... an“: I"! ”an” “u“ 11W .
1 _ g £1 a '1 .‘ “01"" u it” : . . ,. ,. ..-. . ...
: 1 ‘0" . . .l””“”u”u"“"“"”. “u” i
p )J’J 00””.guuaullfluun»u"” " ‘
I lulunuu 4
M»‘ 4..
it : We...

0 I 1...
2' J
A O 1 r .
A 4
N .
v d
C 1
V .

a
0.001 1 . 1 1 1
-1 -0.5 0 0.5 1 1.5 2

Z°(Tc-T)/T

Figure 3.10: Plot of the scaled fragment distribution as a function of the scaled
control parameter (multiplied by Tc) for the experimental data (uncorrected). Values
of o = 0.99, T = 2.13 and TC = 9.5 MeV have been used.

distribution, it becomes clear that the order of Z = 3 and Z = 4 cannot be changed

without increasing the discrepancy between the yields of Z = 5 and Z = 6.

The reason for this problem is that the fragments of the nuclei that are produced
in the experiment can undergo sequential decays before they are detected. The energy
of the incident particle in the experimental setup described is high enough to induce
isospin-symmetric fragmentation reactions; that is, at the time a nucleus breaks up,
the actual composition of a fragment (in terms of isospin) barely affects the probability
for its emission. Thus, both fragments with stable and unstable neutron / proton ratios
will be produced. Therefore, the primary yield of fragments, which would carry the
signatures of the phase transition, i.e., which would exhibit FDM scaling behavior, is
altered (through decays) before its detection and is not directly attainable from the

experimental data. For a more detailed discussion, see for example [19].

Furthermore, the “non-perfect” detection process introduces changes that are not
related to the phenomenon we want to study, and should be compensated for. The
latter problem is overcome by extracting correction factors for individual charges Z
accounting for detection effects from the model data before and after filtering (see
Figure 3.2). From a comparison of part (a) and part (b) of Figure 3.7, it can be
seen that the shift in the data for different Z is mostly coherent, i. e., all curves
are approximately shifted by the same amount. However, this shift impacts the
determination of the critical breaking probability or temperature, respectively, and

should therefore be considered in the analysis.

In order to obtain correction factors for the sequential decay effect, we used the
“EASY” (Exact Nuclear Statistical Yields) code written by Pratt [40]. The simulation
requires the size of the nucleus and the excitation temperature as inputs. We used
the average value of 65 for the number of protons in the decaying nucleus (see Section
3.3.2) and a number of neutrons corresponding to the neutron/proton ratio of an
Au nucleus (assuming isoospin symmetric emission of pre—equilibrium particles). We
found that a temperature of 7 MeV yields the best results. The EASY code produces
data for primary (after fragmentation) and final (after sequential decay) yields for
different (p,n) configurations. We summed over 11 in order to calculate yields and
ratios depending only on the charge p. Unfortunately, since the EASY simulation
was originally not intended for analyses of the low mass region and because it is still
considered as “work in progress”, we could only obtain useful correction factors for

charges up to 6.

Figure 3.11 shows the results of the X2 fitting procedure for the critical exponents
o and T (and the critical temperature T) using the sequential decay correction.
A critical temperature of 8.3 i 0.2 Mev has been used in this plot. This value is

comparable to theoretical estimates [18]. Again, it is difficult to determine the value

54

 

T 2-

 

 

 

 

Figure 3.11: Density plot of X2 as a function of a and T for the experimental data
(With filter- and sequential decay correction. The 4 innermost contours represent
[X2 / (degrees of freedom]-values of 3.3, 3.7, 4.1 and 4.6, respectively. A critical tem-
perature of 8.3 MeV has been used in this plot.

55

of o exactly. In this case, the range of Z values is even smaller than for the percolation

calculation. We derive the numerical values a = 0.5 i 0.1 and T = 2.35 i 0.05.

However, it should be mentioned here that the errors are estimated from the
fitting procedure. The actual values of Tc, 0 and T to a large degree also depend
on the correction factors for sequential decays. Therefore, a precise determination of
these factors is crucial in order to obtain quantitative results and certainly also adds

systematic uncertainties to the given values.

In this analysis, we are only interested in the critical temperature and the critical
exponents of the phase transition. In order to derive values for other parameters in
the Fisher Droplet Model, like the surface free energy coefficient c0 (see Equation
3.9), it is necessary to use the masses A of the fragments instead of the charges (see
[11]). One possibility to do this (since no information about masses is available in
the data set) is to scale all charges Z by the mass/charge ratio of the Au nucleus.
Under this redefinition of the fragment sizes, one also has to rescale the normalized
droplet distribution (nz) to (n A) by dividing through the mass/charge ratio. Oth-
erwise, coherent shifts in the FDM scaling plots yield different values of the critical

temperature.

In Figure 3.12, an F DM scaling plot is given with the optimized parameters Tc,
(7 and T found above. It can be seen that the data collapse very well onto a single
Curve. Again, this agreement with Fisher’s model must be interpreted as a strong

indication at a liquid-gas type phase transition in excited nuclei.

As for future analyses, it would be desirable to determine sequential decay cor-
rection factors for particles with charges greater than 6. A broader Z range should

help to alleviate the significant uncertainties in the a determination.

56

 

_—J

NNNN

03011180)
... f
1

  

 

0.1

<n(Z)>/qOZ"

 

 

 

 

 

 

 

 

 

0.5
2° (T c-TVT
Figure 3.12: Plot of the scaled fragment distribution as a function of the scaled control

parameter (multiplied by TC) for the experimental data (with filter- and sequential
decay corrections) Values of o = 0.5, T = 2.35 and Tc 2 8.3 MeV have been used.

57

Chapter 4

Summary and Conclusion

Percolation theory represents a simple statistical model for systems undergoing a
phase transition. Despite the simplicity of the model, it shows all the features of
critical behavior, universality and critical exponents. Unlike most other (thermal)
models in statistical physics, the phase transition dealt with in percolation theory is
of a purely geometrical nature. The role of the adjustable parameter “temperature”
is played by a probability (the bond-breaking probability in bond percolation or the
probability for a site to be occupied in site percolation, for example), and almost all
properties and functions that mark the critical behavior of the system are geometrical.
One of the most important quantities describing all crucial features of the system,
for instance, is the (normalized) distribution of cluster sizes. As in thermal phase
transitions, simple power laws govern the system behavior in the vicinity of the phase
transition point. The critical exponents introduced in this context have been shown
to be independent of topological details of the model and to depend only on the
dimensionality of the system. This can be interpreted as a strong indication that
results derived from (certainly simplifying) models based on percolation theory can

also be of practical relevance for “real world” applications.

Consequently, percolation theory has been successfully applied to all kinds of

58

different areas inside and outside of physics (for examples, see [1, 38, 31, 46]). Even
though hardly any analytical results can be derived, the percolation models’ simplicity
makes them ideally qualified to be analyzed in Monte Carlo simulations with rather

moderate demands in computational performance.

To summarize, Chapter 2 of this thesis presented a short introduction to the basic
ideas of percolation theory. Furthermore, the general concepts and features of phase
transitions were explained in this chapter, and the atomic nucleus as a system which
is believed to undergo a liquid-gas type phase transition was introduced. 'ITaces for
this transition are expected to be found in nuclear multifragmentation reactions. A
percolation model for nuclear multifragmentation reactions presented by Bauer [6]

was described.

Chapter 3 dealt with the application of this model to the interpretation of a data
set of proton and pion induced multifragmentation reactions that has been collected
by the ISiS collaboration at the AGS accelerator facility at the Brookhaven National
Laboratory. In trying to reproduce the experimental data by model calculations, it
was shown that some modifications of the model have to be made. In particular,
the comparably low energy (for this kind of reactions) of the fragmentation—inducing
projectiles made it necessary to carefully analyze the size of the thermalized source,
which could no longer be determined by simple geometric considerations (see Sec-
tions 2.4.1, 3.3.1 and Figure 2.5). Also, the fact that the ISiS data set provided
information about the excitation energy of the source allowed us to determine bond-
breaking probabilities that are directly correlated to the experiment (Section 2.4.2)
by employing a pbreak (T) relation of the hybrid model. In a more detailed analysis of
the data, we found that it is important to take into account effects that arise from
inevitable experimental constraints in the detection process. In the model, this was

done by introducing a filter that simulates the non-detection of fragments (Section

59

3.2.2). Especially quantities depending on the multiplicity of events and moments of
the cluster size distribution were shown to be sensitive to the changes induced by the

filter.

Finally, the data were analyzed with respect to their scaling behavior according
to the Fisher Droplet Model. Since this model has been introduced to describe phase
transition behavior in liquid—gas systems, the actual analysis allowed us to conclude
that traces of a phase transition are present in the experimental data. Also, the
FDM scaling behavior can be described in terms of two critical exponents o and T,
which characterize this type of transition. A method for the determination of these
two exponents, together with the critical temperature T6 of the transition, based on
FDM scaling was explained in this work. In conducting this analysis, we also tried to
compensate for the filter effects discovered in the model calculations and for sequential
decay processes inherent in this type of reactions. It has been shown that the latter
have a great impact on the quality of the analysis. We derived values of Tc 2 8.3 $0.2
MeV, o = 0.5 :l: 0.1 and T = 2.35 :l: 0.05.

We believe that this method can also be applied to other data sets. Its advantage
over other methods is that it allows for the determination of three important quantities
characterizing the phase transition at the same time. It would be interesting to
compare the results achieved with this method with numerical values derived from

other methods for other sets of data.

To conclude, it is obvious that with recent experimental data, new information
about the nuclear phase diagram can obtained. The transition is still far from being
understood. Therefore, studies of the nuclear phase diagram provide numerous op-
portunities for future research, both in trying to interpret experimental results and

in trying to develop new theoretical models or refine existing ones.

60

APPENDIX

61

Appendix A

Source Code

In the appendix, we will present the most important parts of the source code used for
the simulations discussed in this thesis. All programs were written in C++ and were

compiled with diflerent C++-compilers on different architectures.

A.1 isisevent.h and isisevent.cpp

The C++-class isisEvent is used to interface the data file provided by the ISiS corpo-
ration. Since the results of our percolation simulations are stored in the same format,
they are also accessed using this class. The class contains routines for input/output

of events and basic analyses.

Headerfile:

#ifndef ISISEVENT_H
#define ISISEVENT_H

#include <slist>
#include <iostream>
#include <fstream>
#include <string>

#define isisnum float

#define max_frag_num 79
#define buffersize_b (32768+4)*2

62

class isisEvent

{

public:
isisEvent (string Typ = "")
~isisEvent () ;
isisnum mult ;
isisnum ExA ;
isisnum Zsrc ;
isisnum Zres ;
isisnum moment(int m) ;
isisnum moment(int m, int cut) ;
isisnum moment_zres (int m) ;
isisnum moment_zres (int m, int cut) ;
isisnum moment_zres_ex16_20 (int m) ;
isisnum moment-zres_ex16_20 (int m, int cut) ;
isisnum sourcesize() ;
bool fission() ;
void rearrange() ;
isisnum realmult() ;
bool tagged() ;
isisnum Zthermfmax_frag_num] ;
string eventtype ;

private:

friend istream& operator>>(istream& is, isisEvent &iE) ;
friend ostreamh operator<<(ostream& os, isisEvent &iE) ;
friend ifstreamh operator>>(ifstream& ifs, isisEvent kiE) ;
friend ofstream& operator<<(ofstream& ofs, isisEvent aiE) ;

} ;

#endif

Implemenation:

#include "isisevent.h"
#include <cmath>

#define min(a,b) (((a) < (b)) ? (a) : (b))

isisEvent::isisEvent (string Typ)

{
eventtype = Typ ;
}
isisEvent::”isisEvent ()
{
}

isisnum isisEvent::moment (int m)
{

isisnum mom = O ;

if (m <= 0) return mult ;

63

else {
for (int i=0; i<mu1t; i++)

{
mom += pov(Ztherm[i], m) ;
}
return mom ;
}
}
isisnum isisEventzzmoment (int m, int cut)
{
isisnum mom = O ;
if (m <= 0) return mult ;
else {
for (int i=0; i<mu1t; i++) if (Ztherm[i] <= cut) mom += pow(Ztherm[i], m) ;
return mom ;
}
}

isisnum isisEvent::moment_zres (int m)

{
isisnum mom=0 ;
if (m <= 0) return mu1t+1 ; // +1 because of zres
else {
for (int i=0; i<mu1t; i++) mom += pow(Ztherm[i], m) ;
mom += pow(Zres, 1n) ;
return mom ;
}
}
isisnum isisEvent::moment_zres (int m, int cut)
{
isisnum mom=0 ;
if (m <= 0) return mu1t+1 ; // +1 because of zres
else {
for (int i=0; i<mu1t; i++) if (Ztherm[i] <= cut) mom += pow(Ztherm[i], m) ;
if (Zres <= cut) mom += pow(Zres, m) ;
return mom ;
}
}
isisnum isisEvent::moment_zres_ex16_20 (int m)
{
isisnum mom=0 ;
if (m <= 0) return mu1t+1 ; // +1 because of zres
else {

for (int i=0; i<mu1t; i++) if (Ztherm[i] <= 16) mom += pow(Ztherm[i], m) ;
mom += pow(ZreS. m) ;
return mom ;

64

}
}

isisnum isisEvent::moment_zres_ex16_20 (int m, int cut)

{

isisnum mom=0 ;

if (m <= 0) return mult+1 ; // +1 because of zres
else {
for (int i=0; i<mult; i++)
if (Ztherm[i] <= min(cut, 16)) mom += pow(Ztherm[i], m) ;
if (Zres <= cut) mom += pow(Zres, m) ;
return mom ;

}
}
isisnum isisEvent::sourcesize ()
{
return moment_zres(1) ;
}

bool isisEventzzfission ()
{
bool temp = false ;
for (int i=0; i<mu1t; i++) temp = temp | (ZthermEiJ == 25) ;

return temp ;

}
void isisEventzzrearrange ()
{

int i, j ;

isisnum temp ;

if (mult > 1)
for (i=0; i<mu1t-1; i++)
for (j=i; j<mu1t; j++)
if (ZthermEiI > Ztherm[j]) {
temp = Zthermfj] ;
Ztherij] Zthermfi] ;
'Ztherm[i] temp ;
}

Zres = 0 ;
for (i=0; i<mu1t; i++)
if (Ztherm[i] > 1000) Zres += Zthermfi]-1000 ;
}

isisnum isisEvent::realmult ()
{

int i ;

isisnum mul=0 ;

65

for (i=0; i<mu1t; i++)
if (Ztherm[i] < 1000) mul++ ;

return mul ;

}

bool isisEvent::tagged ()
{

int i ;

bool temp = false ;

for (i=0; i<mu1t; i++) temp = temp | (Ztherm[i] > 1000) ;

return temp ;

}

istreamh operator>>(istream& is, isisEvent &iE)

{

short dummy, i ;

is >> dummy ; iE.mult = (isisnum)dummy ;
is >> dummy ; iE.ExA = (isisnum)dummy ;
is >> dummy ; iE.Zsrc = (isisnum)dummy ;
is >> dummy ; iE.Zres = (isisnum)dummy ;

if (iE.mu1t < max_frag_num)
for (i=0; i<iE.mu1t; i++) {
is >> dummy ;
iE.ZthermEi] = (isisnum)dummy ;

}

return is ;

}

ifstream& operator>>(ifstream& ifs, isisEvent &iE)
{

short dummy[max_frag_num] ;

ifs.read(&dummy, 4*2) ;

iE.mult = (isisnum)dummy[0] ;
iE.ExA = (isisnum)dummy[1]/100.0 ;
iE.Zsrc = (isisnum)dummy[2] ;
iE.Zres = (isisnum)dummy[3] ;

if (iE.mult < max_frag_num)
{
ifs.read(&dummy, (int)iE.mu1t*2) ;
for (int i=0; i<iE.mult; i++) iE.Ztherm[i] = (isisnum)dummyEi] ;
}

return ifs ;

}

ofstream& operator<<(ofstream &ofs, isisEvent &iE)

{

66

short dummyEmax_frag_num] ;

dummyEO] = (short)iE.mult ;
dummyEl] = (short)(100*iE.ExA) ;
dummy[2] = (short)iE.Zsrc ;
dummy[3] = (short)iE.Zres ;

ofs.write(dummy, 4*2) ;
if (iE.mult < max_frag_num) {

for (int i=0; i<iE.mult; i++) dummyEi] = (short)iE.Ztherm[i] ;
}

ofs.write(dummy, (int)iE.mu1t*2) ;

return ofs ;

}

ostream& operator<<(ostream& os, isisEvent &iE)

{

03 << iE.eventtype << endl ;

08 << iE.mu1t << endl ;

03 << iE.ExA << endl ;

03 << iE.Zsrc << endl ;

03 << iE.Zres << endl ;

for (int i=0; i<iE.mult; i++) {
os << iE.ZthermEi] << endl ;

}

return os ;

A.2 percolate . cpp

The program percolate . cpp contains the actual simulation implementing the NLM.
It writes its results to a datafile in the same format in which the data of the ISiS
group was provided. The parameters are read from the standard input, which al-
lows us to write batch files via redirection of the standard input. percolate.cpp
makes use of the external routines Gamma(double a, double 20, double 21) and
filter(double theta, double phi, double 2, double &a, double e, double
&ring, double &det, double &cod) implemented in the files gamma.cpp and

filter. cpp respectively.

67

/***************************************************************************
percolate.cpp - description
begin : Tue, Oct 10, 2000
*******************#t#*********#*****#*#***#*****************************##/

#ifdef HAVE_CONFIG_H
#include <config.h>
#endif

#include <iostream.h>
#include <fstream.h>
#include <stdlib.h>
#include <cmath>

#include "isisevent.h"
#include "isisreadbuffer.h"
#include "filter.h"
#include "gamma.h"

#define PI 3.141592654

#define min(a,b) (((a) < (b)) ? (a) : (b))
#define max(a,b) (((a) > (b)) ? (a) 2 (b))
int startbuffer = 1 ;

int endbuffer = 726 ;

ifstream datafile ;
ofstream outfile ;
char outfilename[256] ;

isisEvent eventlist[eventbuffersize] ;
isisEvent percevents[eventbuffersize] ;

int evcount, aevent=0, abuffer=startbuffer ;
bool clstmb[12][12][12] ; // Clustermembers
bool con[10][10][10][3] ; // Bonds
bool nonuc1[10][10][10] ; // sites
long int newpts[1000][3] ;

long int nucnum[10][10][10] ;

long int mevent[1000] ;

int A1 ; // target charge

int Ar ; // target mass

int Alr ; // backup for target charge
int Mass ;

int nx, ny, nz ;

int multip ;

int noruns ;

bool fixed ;
bool tag_undetected ;

68

double pbreak ;

double r1, r2 ;
unsigned long int irandO ;

bool Filter ;

int mlow, mhigh ;
double isisT, isisE ;
double B = 6.7 ;

double ran_ (unsigned long int *i)

{
return (double)rand()/RAND_MAX ;
}

void
{
int h ;

input_params ()

>> A1 ;

= A1 ;

>> Ar ;

>> B ;

>> irandO ;

>> noruns ;

>> mlow ;

cin >> mhigh ;

cin >> h ;

fixed = (h != 0) ;

-cin >> h ;

Filter = (h != 0) ;
cin >> h ;
tag_undetected = (h != 0) ;
cin >> startbuffer ;
abuffer = startbuffer ;
cin >> endbuffer ;

cin >> outfilename ;

}

cin
A1r
cin
cin
cin
cin
cin

inline double sqr (double x)
{ return x*x ; }

void shape ()
{

const double xs = 0.05 ,
const double ys = 0.07 ,
const double 28 = 0.13 ,

double D[11][11][11] ;
// upper array bound 11 -> 1..10 can be

double x0, y0, 20 ;

// target / projectile radius

// cutoff masses

// Binding energy per nucleon

// Target Mass

// Binding energy per nucleon [MeV]

// ov &= (onoruns == noruns) ;
// break fixed number of bonds
// Model detector efficiency

// just mark undetected,
// without creating zres...

used (no messing with index shift)

69

}

int ix, iy, iz, ixdmin, iydmin, izdmin ;
int n1 ;
double dmin, dxyz ;

for (ix=0; ix<10; ix++)
for (iy=0; iy<10; iy++)

for (iz=0; iz<10; iz++) nonuc1[ix-1][iy-1][iz-1] = true ;

r1 = pow((0.75*(double)A1/PI), 1.0/3.0) ;
nx = (ny = (nz = 2*((int)r1+1))) ;

x0 = ((double)nx+1.0)/2.0 + xs ;

y0 = ((double)ny+1.0)/2.0 + ys ;

20 = ((double)nz+1.0)/2.0 + 23 ;

// Distribute nucleons approx. spherical in cubic lattice
for (ix=1; ix<=nx; ix++)
for (iy=1; iy<=ny; iy++)
for (iz=1; iz<=nz; iz++)
D[ix][iy][iz] = sqrt(sqr(ix~x0)+sqr(iy-y0)+sqr(iz-20)) ;

ixdmin = iydmin = izdmin = 0 ;
for (n1=1; n1<=A1; n1++) {
dmin = 999 ;
for (ix=1; ix<=nx; ix++)
for (iy=1; iy<=ny; iy++)
for (iz=1; iz<=nz; iz++) {
dxyz = D[ix][iy][iz] ;
if (dxyz < dmin) {
ixdmin = ix ;

iydmin = iy ;

izdmin = iz ;

dmin = dxyz ;
l

}
nonuc1[ixdmin-1][iydmin-1][izdmin-1] = false ;
D[ixdmin][iydmin][izdmin] = 9999 ;
}

Mass = A1 ;

void break_bonds ()

{

// extern double ran_ (unsigned long int *) ;

int nbreak, nbonds = 0 ;

int idir ;

int ix, iy, iz, ixnext, iynext, iznext, i, n ;
int mix[2000], miy[2000], miz[2000], mid[2000] ;

for (idir=0; idir<3; idir++) {

ixnext = 0 ;
iynext = 0 ;

70

iznext = 0 ;

if (idir == 0) ixnext = 1 ;
else if (idir == 1) iynext = 1 ;
else iznext = 1 ;

for (iz=0; iz<nz; iz++)
for (iy=0; iy<ny; iy++)
for (ix=0; ix<nx; ix++) {

if (nonuc1[ix][iy][iz]) {
con[ix][iy][iz][idir] = false ;
continue ;

}

if (nonuc1[ix+ixnext][iy+iynext][iz+iznext])
con[ix][iy][iz][idir] = false ;
else {
if (fixed) {
con[ix][iy][iz][idir] = true ;
nbonds++ ;

mix[nbonds-1] = ix ;
miy[nbonds-1] = iy ;
miz[nbonds-1] = iz ;
mid[nbonds-1] = idir ;
}
else {
if (ran_(&irand0) < pbreak) con[ix][iy][iz][idir] = false ;
else {
con[ix][iy][iz][idir] = true ;
nbonds++ ;
}
}

}
}

}

nbreak = (int)(pbreak*nbonds+0.5) ; // <- "round"
if (fixed) {
for (n=1; n<=nbreak; n++) {
do {
i = (int)(0.5+ran_(&irand0)*nbonds+0.5) ; // "round"
} while ((i <= 0) ll (i > nbond8)) ;
if (con[miin-1JJ[miyEi-1]][miz[i—1]][midEi-1JJ)
con[miin-11][miyEi-1]][miz[i-1]][mid[i-1]] = false ;
else {
i++ ;
if (i > nbonds) i = 1 ;
}
}
}

71

void newtst (int ipx, ipy, ipz, icx, icy, icz, idir, int& imax)
{
if (lclstmbEipx][ipy][ipz])
if (con[icx-1][icy-1][icz-1][idir-1]) {
newpts[imax-1][0] = ipx ;
newpts[imax-1][1] = ipy ;
newpts[imax-1][2] ipz ;
imax++ ;
clstmb[ipx][ipy][ipz] = true ;

}

bool not_detected (int 2)

{

#define epsilonO 8.85419e-12

#define e0 1.602e-19*1e15*1e-6

// 1e15 correction due to fact that r given in fm, 1e-6 for MeV !

double theta, phi ;

double r, E ;

double bmax ;

double a, ring, det, cod ;

double kappa = 0.6;

double T = 12.0 ; // Temperature in MeV
double Barrier ;

if (!Filter) return false ;

// determine ejection angle...
phi = ran_(&irand0)*360.0 ;

do {
theta = ran_(&irand0)*PI ;
} while (ran_(&irand0) > sin(theta)) ; // <- rejection method !?

theta *= 360.0/(2.0*PI) ;

// determine Energy-distribution plus Coulomb-barrier
bmax = sqrt(0.5*T)*exp(-0.5) ;

r = 1,2*(pow(z*Ar/A1r, 1.0/3.0)+pow(Mass*Ar/A1r, 1.0/3.0)) ;
Barrier = e0*z*(Mass-z)/(4*PI*epsilon0*r) ;

if ((Ar*z/A1r-(int)(Ar*z/A1r)) > ran_(&iran0)) a = (int)(Ar*z/A1r)+1.0 ;
else a = (int)(Ar*z/A1r);

if (a == 0) a == 1 ;
do {
do {

E = ran_(&irand0)*20*T ;
} while (ran_(&irand0) > sqrt(E)*exp(-E/T)/bmax) ;

72

E = (E + kappa*Barrier)*(Ar-a)/Ar ;
} while (E < 0) 3

filter(theta, phi, z, a, E, ring, det, cod) ;
if (a == 0) return true ;
else return false ;

}

bool process_isisevent ()

{

isisEvent ev ;

if (aevent < evcount) {
ev = eventlist[aevent++] ;

if (ev.fission()) return false ;

isisT sqrt(13.0*ev.ExA) ;
isisE = ev.ExA ;
A1 = (int)ev.sourcesize() ;
if (A1 > 79) return false ;
return true ;
}
else {
if (abuffer < endbuffer) {
evcount = isisreadbuffer(datafile, &(eventlist[0])) ;
abuffer++ ;
aevent = 0 ;
}
else {
datafile.seekg((startbuffer-1)*buffersize_b, ios::beg) ;
evcount = isisreadbuffer(datafile, &(eventlist[0])) ;
abuffer = startbuffer ;
aevent = 0 ;
}
return false ;
}
}

void percolate ()
{
int ix, iy, iz, iix, iiy, iiz ;
int imaxo, imino, imax, imin ;
int nrun, i ;
int nxpl, nyp1, nzpl ;
int ifrg ;
double zres ;
short ivord ;
int events, overallevents, buffers ;

events = 0 ;
iword = 1 ; // 1 for iword

73

overallevents = 0 ;
buffers = 0 ;
// ++++++++++ Main loop over events ***********
for (nrun=1; nrun<=noruns; nrun++)
{
// Experimental data
do {} while (lprocess_isisevent()) ;
pbreak = 1.0-2.0*Gamma(1.5, 0, B/isisT)/sqrt(PI) ;
L10:

// Determination of nucleon distribution on the lattice
shape() ;

nxpl = nx + 1 ;
nyp1 ny + 1 ;
nzpl n2 + 1 ;

for (ix=1; ix<=nx; ix++)
for (iy=1; iy<=ny; iy++) {
clstmb[ix][iy][0] = true ;
clstmb[ix][iy][nzp1] = true ;
}

for (ix=1; ix<=nx; ix++)
for (iz=1; iz<=nz; iz++) {
clstmb[ix][0][iz] = true ;
clstmb[ix][nypIJEiz] = true ;
}

for (iy=1; iy<=ny; iy++)
for (iz=1; iz<=nz; iz++) {
clstmb[0][iy][iz] = true ;
clstmb[nxp1][iy][iz] = true ;
l

// breaking of bonds
break_bonds() ;
zres = 0 ;
// Cluster recognition algorithm:
// 1) Initialization of array clstmb
for (iz=1; iz<=nz; iz++)
for (iy=1; iy<=ny; iy++)
for (ix=1; ix<=nx; ix++)

clstmb[ix][iy][iz] = nonuc1[ix-1][iy-1][iz-1] ;
multip = O ;
// 2) Cluster recognition and size determination:
for (iz=1; iz<=nz; iz++)

for (iy=1; iy<=ny; iy++)

74

for (ix=1; ix<=nx; ix++)
if (!clstmb[ix][iy][iz]) {
// new cluster, --- start ---
clstmb[ix][iy][iz] = true ;

newpts[0][0] = 1x ;
newptSEOJEIJ = iy ;
newpts[0][2] = i2 ;
imino = 1 ;
imaxo = 1 ;

L1400:

imin = imaxo + 1 ;

imax = imin ;

for (i=imino; i<=imaxo; i++) {
iix = newpts[i-1][0] ;
iiy newpts[i-1][1] ;
iiz newptsEi-1][2] ;
newtst(iix-l, iiy, iiz, iix-1, iiy, iiz, 1, imax) ;
newtst(iix+1, iiy, iiz, iix, iiy, iiz, 1, imax) ;
newtst(iix, iiy-1, iiz, iix, iiy-1, iiz, 2, imax) ;
newtst(iix, iiy+1, iiz, iix, iiy, iiz, 2, imax) ;
newtst(iix, iiy, iiz-1, iix, iiy, iiz-1, 3, imax) ;
newtst(iix, iiy, iiz+1, iix, iiy, iiz, 3, imax) ;

}.

if (imax != imin) {
imino = imin ;
imaxo = imax-1 ;
goto L1400 ;

}
if (imaxo == A1) goto L10 ;

if (imaxo > 79) goto L10 ;

// Determine if fragment was detected
if (not_detected(imaxo)) {
if (tag_undetected) {
multip++ ;
mevent[mu1tip-1] = imaxo + 1000 ;
}
else {
zres += imaxo ;
}
}
else {
if ((imaxo >= mlow) && (imaxo <= mhigh)) {
multip++ ;
mevent[mu1tip-1] = imaxo ;
}
}

// New cluster, -- end --

}

percevents[events].mult = (isisnum)multip ;
percevents[events].ExA = (isisnum)isisE ;

75

(isisnum)A1 ;
(isisnum)zres ;

percevents[events].Zsrc
percevents[events].Zres

for (ifrg=0; ifrg<multip; ifrg++)
percevents[events].Ztherm[ifrg] = (isisnum)mevent[ifrg] ;
if (tag_undetected) percevents[events].rearrange() ;

iword += multip + 4 ;
events++ ;
overallevents++ ;

if (iword > 32768-(79+4)) {
outfile.write(&iword, 2) ; // Fortran?
outfile.write(&iword, 2) ; // Fortran?
outfile.write(&iword, 2) ;
for (i=0; i<events; i++) {
outfile << percevents[i] ;
}
outfile.write(&(percevents[0]), (32768+2-iword)*2) ;
buffers++ ;
iword = 1 ;
events = 0 ;
}
}
if (iword > 0) { // write last buffer, if necessary
outfile.write(&iword, 2) ;
outfile.write(&iword, 2) ;
outfile.write(&iword, 2) ;
for (i=0; i<events; i++) outfile << percevents[i] ;
outfile.write(&(percevents[0]), (32768+2-iword)*2) ;
buffers++ ;
}
cout << "Written " << overallevents << " events in " ;
cout << buffers << " buffers." << endl ;

}

int main (int argc, char *argVEJ)

{
input_param8() 3
datafile.open("pAu102.bin") ;
datafile.seekg((startbuffer-1)*buffersize_b, ios::beg) ;
evcount = isisreadbuffer(datafile, &(eventlist[0])) ;

outfile.open(outfilename) ;
srand(irand0) 3

percolate() ;

datafile.close() ;
outfile.close() ;

76

A.3 filter.cpp

This source code contains the filter used to model the detector efficiency of the ISiS

detector. It is a C++-port of the Fortran version provided by the ISiS group.

#include "filter.h"
#include "baddet.h"

bool first = true ;

extern unsigned long irandO ;
extern double ran_ (unsigned long int *) ;

double seuilsim[16] = {3., 1., 0.85, 0.89, 0.9, 0.83, 0.85, 0.93, 0.94,
1.0, 0.96, 1.08, 1.04, 1.17, 1.22, 1.31 } ;
{12.5, 38.6, 65.9, 100.9, 136. } ;
double lim_tet[11] {14, 22, 33, 52, 69, 86.4, 93.6, 111, 128, 147, 166} ;
double thepp[9] {18.0, 27.5, 42.5, 60.5, 77.5, 102.5, 119.5, 137.5, 156.5} ;
double limtet[18] = {14.0, 21.5, 22.5, 32., 34., 51., 53., 68., 70., 86.4,
93.6, 110., 112., 127., 129., 146., 148., 166.} ;

double Esimax[5]

double e_th[9][20] ;

void filter (double theta, double phi, double 2, double &a, double e,
double &ring, double kdet, double &cod)

{
double ir, iz, et, ea ;
double phimin, phimax, ip ;
int j, ic ;
bool out ;

if (first) {
ifstream eth("eth.dat") ;
while (leth.eof()) {
eth >> it ;
eth >> iz ;
eth >> at ;
eth >> ea ;
e_th[(int)ir-1][(int)iz-1] = ea ;
}
eth.close() ;
first = false ;
}
det = 0 ;

if (theta < lim_tet[0]) det = -100 ;

if (theta > lim_tet[10]) det = -100 ;

if ((theta >= 1im_tet[5]) && (theta < lim_tet[6])) det = -100 ;
if ((int)det == -100) {

ring = 0 ;

77

det = 0 ;
a = 0 ;
return ;

out = true ;
for (j=1; (j<=10)&&(out); j++) {
if (j <= 5) {
if ((theta >= lim_tet[j-1]) && (theta < lim_tet[j])) {
ring = j ;
out = false ;
}
}
else if (j >= 7) {
if ((theta >= lim_tet[j-1]) && (theta < 1im_tet[j])) {

ring = j-1 ;
out = false ;
}
}

}
ir = ring ;
iz = (int)z ;
if ((theta >= limtet[1]) && (theta < limtet[2])) det = -100 ;
if ((theta >= limtet[3]) && (theta < limtet[4])) det = -100 ;
if ((theta >= limtet[5]) && (theta < limtet[6])) det = -100 ;

if ((theta >= limtet[7]) && (theta < 1imtet[8])) det = -100 ;
if ((theta >= limtet[9]) && (theta < limtet[10])) det = -100 ;

if ((theta >= limtet[11]) && (theta < limtet[12])) det = -100 ;
if ((theta >= limtet[13]) && (theta < limtet[14])) det = -100 ;
if ((theta >= 1imtet[15]) && (theta < limtet[16])) det = -100 ;

for (j=1; j<=18; j++) {
phimin = 20.0*j - 10.0 - 1.0 ;

phimax = 20.0*j - 10.0 + 1.0 ;

if ((phi >= phimin) && (phi < phimax)) det = -100 ;
}
if ((int)det == -100) {

ring = 0 ;

det = 0 ;

a = 0 ;

return ;
}

if ((ring >= 1) && (ring <= 9)) theta = thepp[(int)ring-1] ;
else {

ring = 0 ;

det = 0 ;

a = 0 ;

return ;

78

out = true ;
for (j=0; (j<=17)&&(out); j++) {
phimin = 20.0*j - 10.0 ;
phimax = 20.0*(j+1) - 10.0 ;
if (phi > 350) phi -= 360 ;
if ( (phi >= phimin) && (phi < phimax)) {
det j+1+(ring-1)*18 ;
out false ;
}
}

ip = (int)det - (18*(ring-1)) ;
phi = ((double)ip-1)*20.0 ;

ic = baddet_e900[(int)det-1] ;

if (ic == -1) a = 0 ;
if ((ic == -2) && (2 >= 3)) a = 0 ;

if ((int)z == 1) {
if ((ic==-2) && (e <= Esimax[(int)z-1])) a = 0 ;
if ((ic==-3) && (e > Esimax[(int)z-1])) a = 0 ;
}

if ((int)z == 2) {
if ((ic==-2) && (e <= Esimax[(int)z—1])) a = 0 ;
if ((ic==-3) && (e > Esimax[(int)z-1])) a = 0 ;
}

if ((int)z == 3)
if ((ic==-3) && (e > Esimax[(int)z-1])) a= 0 ;

if ((int)z==0) a = 0 ;
if (z > 16) a = 0 ;

if ((int)z==1) {
if (((int)a==2) && (e <= 10.6)) a
if (((int)a==3) && (e <= 12.5)) a
if ((int)a==4) a = 0 ;

}

II II
H
- .

if ((int)z==2) if (((int)a==3) && (e<=28)) a=4 ;
if ((int)a != 0)
if (2 <= 16) {
if ((e/a) <= e_th[(int)ir-1][(int)iz-1]) a
}
if ((int)a != 0)
if (2 >= 4) if ((e/a) > 12) a = 0 ;

if (((int)z==1) || ((int)z==2) || ((int)z==3)) {

79

ll
0

if (e <= Esimax[(int)z-1]) cod=10 ;
if (e >= Esimax[(int)z-IJ) cod=20 ;
}

if (((int)z==4) ll ((int)z==5)) {
if (e <= Esimax[(int)z-1J) cod=10 ;
if (e >= Esimax[(int)z-1]) a=0 ;

}

return ;

A.4 varfdmparams . cpp

This section contains the C++ source code of the routine that varies the two critical
exponents a and 7' for a given pc to evaluates the quality of the “FDM-scaling”

behavior.

#include <iostream.h>
#include <fstream.h>
#include <cmath>

#include "isisevent.h"
#include "isisreadbuffer.h"
#include "gamma.h"

#define min(a,b) (((a) < (b)) ? (a) 2 (b))
#define max(a,b) (((a) > (b)) ? (a) : (b))
#define PI 3.141592654

#define Euler 0.577

ifstream datafile ;
ifstream ratiofile ;

isisEvent events[eventbuffersize], ev ;

int i, j, k, x, evcount, evts ;

double scaled_low = -0.3 ;
0.7

double scaled_high =
double scaled_range = scaled_high-scaled-low ;

#define Alow 3

#define Ahigh 10

#define Arange Ahigh-Alow+1
#define bins 50

double sigma = 0.65 ;
double tau = 2.1 ;

80

double xl[bins*Arange], yl[bins*Arange], yerr[bins*Arange] ;
int points ;

#define runs 500000
double B = 6.6 ;
double Tc =

double pc
double q0

5
/7.730171158 ;

9.
0.
1.

00501

double A1 = 65 ;

double scaled_dist[Arange][bins+2] ;
double scaled_err[Arange][bins+2] ;
double scalingEArange][bins+2] ;
double chi2 ;

double scaled_start = -0.1 ;
double scaled_end = 0.4 ;

double sigma_low = 0.2 ;
double sigma_high = 0.8 ;
double tau_low = 2.01 ;
double tau_high = 2.61 ;
int sigma_steps = 61 ;
int tau_steps = 61 ;
int 3, t ;

double ratios[11] ;

int startbuffer = 1 ;
int endbuffer = 409 ;

double minchi2=1E15, minsigma, mintau ;

void init_stuff ()
{

int i, j ;
datafile.seekg((startbuffer-1)*buffersize_b, ios::beg) ;
evts = 0 ;

for (i=0; i<Arange; i++)
for (j=0; j<bins; j++) {
scaled_dist[i][j] = 0 ;
scalingEiJEJJ = 0 3
}
}

void process_event (isisEvent ev)

{
double scp, T, pbreak ;

81

T = sqrt(ev.ExA*13.0) ;
// pbreak = 1.0 - 2.0*Gamma(1.5, 0, B/T)/sqrt(PI) ;

for (i=Alow; i<=Ahigh; i++) {
scp = pow((double)i, sigma)*(Tc-T)/T ;
if ((scp > scaled_low) && (scp < scaled_high)) {
scp = (int)((double)(scp-scaled_low)*(double)bins/(double)sca1ed_range) ;
scaling[i-Alow][(int)scp]++ ;
}
}

for (i=0; i<ev.mult; i++) {
if ((ev.Ztherm[i] >= Alow) && (ev.Ztherm[i] <= Ahigh)) {
scp = pow((double)ev.Ztherm[i], sigma)*(Tc-T)/T ;
if ((scp > scaled_low) && (scp < scaled_high)) {
scp = (int)((doub1e)(scp-scaled_low)*(double)bins/(double)scaled_range) ;
scaled_dist[(int)ev.Ztherm[i]-Alow][(int)scp]++ ;
}
}
}
}

inline double sqr (double x)
{ return x*x ; }

void chi_square ()

{
int A, i, lowbin, highbin ;
double S, Sxx, Sxy, Sx, Sy ;
double b, temp ;

lowbin = (int)((scaled_start-scaled_low)*bins/scaled_range) ;
highbin = (int)((scaled_end-scaled_low)*bins/scaled_range) ;

points = 0 ;
for (A=Alow; A<=6; A++) {

for (i=lowbin; i<=highbin; i++) {
if (scaled_dist[A-Alow][i] > 0) {
temp = scaled_dist[A-Alow][i] ;
xl[points] = (double)scaled_low+(i*scaled_range)/(double)bins ;
yl[points] =
log10(temp*ratios[A]/(A1*max(scaling[A-Alow][i],1)*q0*pow(A,-tau))) ;
yerr[points] = yl[points]
- log10((temp-sqrt(temp))*ratios[A]/
(A1*max(scaling[A-Alow][i], 1)*q0*pow(A, -tau))) ;
points++ ;

82

S = 0 ;

Sx = 0 ;
Sy = 0 ;
Sxx = 0 ;
Sxy = 0 ;

for (i=0; i<points; i++) {
S += 1/sqr(yerr[i]) ;
Sx += x1[i]/sqr(yerr[i]) ;
Sy += yl[i]/sqr(yerr[i]) ;
Sxx += sqr(x1[i])/sqr(yerr[i]) ;
Sxy += xl[i]*yl[i]/sqr(yerr[il) ;
}

b = (S*Sxy - Sx*Sy)/(S*Sxx-sqr(SX)) ;

chi2 = 0 ;
for (i=0; i<points; i++) chi2 += sqr((yl[i]-b*xl[i])/yerr[i]) ;
}

void determine_q0 ()

{
q0 = 1.0/(1.0/((tau-1)-1)+Euler) ;
}

int main (int argc, char *argEJ)
{

int i ;
cin >> Tc ;

ratiofile.open("ratiosT7_f.dat") ;
i = 0 ;
while (i < 10) {

ratiofile >> i ;

ratiofile >> ratiosEi] ;

} ;

ratiofile.close() ;

datafile.open("pAu102.bin") ;

for (s=0; s<sigma_steps; s++) {
for (t=0; t<tau_steps; t++) {

sigma = sigma_low + (sigma_high-sigma_low)*s/(sigma_steps-1) ;
tau = tau-low + (tau_high-tau_low)*t/(tau_steps-1) ;

determine-q0() ;
init_stuff() ;

for (i=0; i<(endbuffer-startbuffer)+1; i++) {
evcount = isisreadbuffer(datafile, &(events[0])) ;

83

for (j=0; j<evcount; j++) {
ev = events[j] ;

evts++ ;
if (evts == runs) break ;
process_event(ev) ;

}

if (evts == runs) break ;

}

chi_square() ;
cout << sigma << "\t" << tau << "\t" << chi2 << endl ;
if (chi2 < minchi2) {
minchi2 = chi2 ;
minsigma = sigma ;
mintau = tau ;
}
}
cout << endl ;
}
cout <<"#\t"<<Tc<<"\t"<<minsigma<<"\t"<<mintau<<"\t"<<minchi2 ;
cout << endl ;
datafile.close() ;

84

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