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NUMERICAL SIMULATIONS OF RELATIVISTIC HEAVY-ION REACTIONS

By

Frank Cecil Dafﬁn

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1998

ABSTRACT

NUMERICAL SIMULATIONS OF RELATIVISTIC HEAVY-ION REACTIONS

By

Frank Cecil Dafﬁn

Bulk quantities of nuclear matter exist only in the compact bodies of the universe.
There the crushing gravitational forces overcome the Coulomb repulsion in massive
stellar collapses. Nuclear matter is subjected to high pressures and temperatures as
shock waves propagate and burn their way through stellar cores. The bulk properties
of nuclear matter are important parameters in the evolution of these collapses, some

of which lead to nucleosynthesis.

The nucleus is rich in physical phenomena. Above the Coulomb barrier, complex
interactions lead to the distortion of, and as collision energies increase, the destruction
of the nuclear volume. Of critical importance to the understanding of these events is

an understanding of the aggregate microscopic processes which govern them.

In an effort to understand relativistic heavy-ion reactions, the Boltzmann-Uehling-
Uhlenbeck[Ueh33] (BUU) transport equation is used as the framework for a numerical
model. In the years since its introduction, the numerical model has been instrumental
in providing a coherent, microscopic, physical description of these complex, highly

non-linear events.

This treatise describes the background leading to the creation of our numerical

model of the BUU transport equation, details of its numerical implementation, its

application to the study of relativistic heavy-ion collisions, and some of the experi-

mental observables used to compare calculated results to empirical results.

The formalism evolves the one-body Wigner phase-space distribution of nucleons
in time under the inﬂuence of a single-particle nuclear mean ﬁeld interaction and a
collision source term. This is essentially the familiar Boltzmann transport equation

whose source term has been modiﬁed to address the Pauli exclusion principle.

T wo elements of the model allow extrapolation from the study of nuclear col-
lisions to bulk quantities of nuclear matter: the modiﬁcation of nucleon scattering
cross sections in nuclear matter, and the compressibility of nuclear matter. Both are
primarily subject to the short-range portion of the inter-nucleon potential, and do

not show strong ﬁnite-size effects.

To that end, several useful observables are introduced and their behavior, as BUU
model parameters are changed, explored. The average, directed, in-plane, transverse
momentum distribution in rapidity is the oldest of the observables presented in this
work. Its slope at mid-rapidity is called the flow of the event, and well characterizes

the interplay of repulsive and attractive elements of the dynamics of the events.

The BUU model has been quite successful in its role of illuminating the physics of
intermediate energy heavy-ion collisions. Though current numerical implementations

suffer from some shortcomings they have nonetheless served the community well.

This work is dedicated to the holy pursuit of the truth, and the love of the chase.

iv

ACKNOWLEDGMENTS

I wish to thank my parents for their stubborn, unﬂagging love and support of this
gross underachiever. Without their Jobian patience I would undoubtedly be pumping

gas and watching pro wrestling. Thank you.

Of the academics I have known, none have been more inﬂuential than Gerald
Chatwood, my high school science and mathematics teacher. His class was the one

bright spot in an otherwise dreary high school adolescent nightmare.

Professor Wolfgang Bauer deserves my thanks for his wise guidance and sharp
insight during my study for the Ph.D. While it goes without saying, it should be
nonetheless, that without Professor Bauer’s support this work and the incalculable

intellectual growth from which it springs would not have been possible.

Professor PawelDanielewicz is one of the brightest people that its been my fortune
to know. Much of this work would not have been possible without his stimulating

discussions.

Professor Vladimir Zelevinsky deserves acknowledgement for his astonishingly
clear understanding of the fundamentals of modern physics. His presence in the

faculty is a rare gift.

The ear of David Brown has been bent more than once during the ﬁnal stages of

this work. To him I owe a debt of gratitude for his objective attention.

V

Finally, I thank Professor Dan Stump for his efforts in the classroom. Learning

quantum mechanics has never been so easy.

vi

Contents

LIST OF TABLES ix
LIST OF FIGURES x
1 BUU and the Nuclear Equation of State 1
1.1 Introduction ................................ 1
1.2 The BUU Model of Nuclear Matter ................... 3
1.2.1 Nuclear Mean Field Parameterizations ............. 6
1.2.2 Nucleon Scattering Cross-Section and the
Collision Integral ......................... 8
1.3 Numerical Implementation
of the BUU Equation ........................... 8
1.3.1 The Parallel Ensemble ...................... 9
1.3.2 Initialization ............................ 11
1.3.3 The Collision Integral ...... , ................ 11
1.4 Summary ................................. 16
2 Collective Phenomena 18
2.1 A Picture of the Nucleus-Nucleus Collision
from BUU ................................. 18
2.2 Directed Transverse Flow ......................... 24
2.3 Impact Parameter Dependence of Flow
for Momentum-Dependent and
Momentum-Independent Nuclear Mean Fields ............. 33
2.3.1 The Development of Density in Time .............. 35
2.3.2 Sphericity Analysis ........................ 52
2.3.3 Total In—Plane, Transverse Momentum Versus
Rapidity .............................. 55
2.3.4 Concluding Remarks ....................... 75
2.4 Balance Energy .............................. 76

vii

2.4.1 Effects of Iso—Spin and In-Medium Corrections
on the Balance Energy ...................... 79

2.5 Summary ................................. 91

BUU, Coalescence and the Radially Expanding Thermal Model 93

3.1 Introduction ................................ 93
3.2 Models and Calculations ......................... 95
3.3 Results ................................... 98
3.4 Temperature and Microscopic Features of BUU ............ 103
3.5 Concluding Remarks ........................... 108
Summary and Outlook 109
4.1 Summary ................................. 109
4.2 Outlook .................................. 112

viii

List of Tables

1.1

2.1

3.1

Parameter sets used for the density-dependent mean ﬁeld U .....

The constants in used in the nuclear mean ﬁeld. Note that “sof ” refers
to a compressibility K of 215 MeV and “stiff” refers to a compressibility
of 380 MeV for the momentum-dependent mean ﬁelds, whereas “soft”
refers to a compressibility K of 200 MeV and “stiff” refers to a com-
pressibility of 380 MeV for the momentum-independent mean ﬁelds. .

Effects of the microscopic features of BUU on apparent temperature
and radial ﬂow velocity. .........................

ix

34

List of Figures

2.1

2.2

2.3

2.4

2.5

2.6

Momentum-space projection upon the Pz-Pz plane for 197Au + 197Au
reaction at 50 MeV/ A. Impact parameter is zero, soft (compressibility
= 200 MeV) mean ﬁeld and vacuum nucleon cross sections were used.
Each panel is separated by 10 fm/c. Initial state is the upper left panel,

and time passes from top to bottom. ﬁnal-state is the lower right panel. 20

Conﬁguration-space projection upon the reaction plane for 197Au +
197An reaction at 50 MeV / A. Impact parameter is zero, soft (compress-
ibility = 200 MeV) mean ﬁeld and vacuum nucleon cross sections were
used. Each panel is separated by 10 fm/c. Initial state is the upper
left panel, and time passes from top to bottom. ﬁnal-state is the lower
right panel. ................................

Momentum-space projection upon the Pz-Pz plane for 197Au + 197An
reaction at 50 MeV / A. Impact parameter is 3.5 fm, soft (compressibility
= 200 MeV) mean ﬁeld and vacuum nucleon cross sections were used.
Each panel is separated by 10 fm/c. Initial state is the upper left panel,

21

and time passes from top to bottom. ﬁnal-state is the lower right panel. 22

Conﬁguration-space projection upon the reaction plane for 197Au +
197Au reaction at 50 MeV/A. Impact parameter is 3.5fm soft (com-
pressibility = 200 MeV) mean ﬁeld and vacuum nucleon cross sections
were used. Each panel is separated by 10 fm/c. Initial state is the
upper left panel, and time passes from top to bottom. ﬁnal-state is the
lower right panel. .............................

Momentum-space projection upon the Px-Pz plane for 197Au + 197Au
reaction at 500 MeV/A. Impact parameter is 3.5fm. Each panel is
separated by 10 fm/c; soft (compressibility = 200 MeV) mean ﬁeld and
vacuum nucleon cross sections were used. Initial state is the upper left
panel, and time passes from top to bottom. ﬁnal-state is the lower
right panel. ................................

Conﬁguration-space projection upon the reaction plane for 197Au +
197Au reaction at 500 MeV/ A. Impact parameter is 3.5 fm, soft (com-
pressibility = 200 MeV) mean ﬁeld and vacuum nucleon cross sections
were used. Each panel is separated by 10 fm/c. Initial state is the
upper left panel, and time passes from top to bottom. ﬁnal-state is the
lower right panel. .............................

23

28

2.7

2.8

2.9

2.10

2.11

2.12

2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
2.24

Momentum-space projection upon the Pz-Pz plane for 197Au + 197Au
reaction at 500 MeV/A. Impact parameter is 3.5fm, stiff (compress-
ibility = 380 MeV) mean ﬁeld and vacuum nucleon cross sections were
used. Each panel is separated by 10 fm/c. Initial state is the upper
left panel, and time passes from top to bottom. ﬁnal-state is the lower
right panel.

Conﬁguration-space projection upon the reaction plane for 197Au +
197Au reaction at 500 MeV/A. Impact parameter is 3.5 fm, stiff (com-
pressibility = 380 MeV) mean ﬁeld and vacuum nucleon cross sections
were used. Each panel is separated by 10fm/c. Initial state is the
upper left panel, and time passes from top to bottom. ﬁnal-state is the
lower right panel. .............................

Evolution of the in—plane, transverse momentum for forward-going nu-
cleons. Impact parameter is 3.5 fm. Two different mean ﬁelds were
used, as indicated.

Average in-plane, transverse momentum versus reduced rapidity. The
system is 197Au + 197Au at 500 MeV/A, impact parameter is 3.5fm.
Two mean ﬁelds were used, as indicated. ................

Flow results from M. J. Huang, et al.[Hua96]. In the Figure exper-
imental data are the cross-hatched regions, the rest are calculated
ﬂow values with detector acceptances. “S” and “H” refer to the soft,

momentum-independent and hard, momentum-independent mean ﬁelds,

respectively. “SM” and “HM” refer to soft, momentum-dependent and
hard, momentum-dependent mean ﬁelds, respectively. The BUU re-
sults denoted by “(0.8mm)” were calculated with a 20% reduction in
the free nucleon-nucleon cross section.

Average in-plane, transverse flow as a function of impact parameter.
Upper panel shows ﬂow calculations for 197Au on 197Au at 400 MeV/A
for the indicated impact parameters. The lower panel shows the same
observable, but for 197Au on 197Au at 200 MeV/ A ............

Calculated density contours.
Calculated density contours.
Calculated density contours.
Calculated density contours.
Calculated density contours.
Calculated density contours.
Calculated density contours.
Calculated density contours.
Calculated density contours.
Calculated density contours.
Calculated density contours.
Calculated density contours.

Numbers are normalized densities:
Numbers are normalized densities:
Numbers are normalized densities:
Numbers are normalized densities:
Numbers are normalized densities:
Numbers are normalized densities:
Numbers are normalized densities:
Numbers are normalized densities:
Numbers are normalized densities:
Numbers are normalized densities:
Numbers are normalized densities:
Numbers are normalized densities:

xi

P/PO-
P/Po-
P/Po-
p/po-
P/Po-
P/Po-
p/po-
P/Po-
P/Po-
P/Po-
P/Po-
P/PO-

30

32

39
40
41
42
43
44
45
46
47
48
49
50
51

2.25

2.26

2.27
2.28
2.29
2.30
2.31

2.32

2.33

2.34

2.35

2.36

Flow angle versus flow ratio f3/f1,2 for protons. Squares are for soft
(K=200 MeV) equation of state without momentum dependence. Cir-
cles are for soft momentum-dependent equation of state. Numerals
beside the points indicate the impact parameter in fm. ........

Flow angle versus ﬂow ratio f3/f1,2 for protons. Squares are for soft
(K=200 MeV) equation of state without momentum dependence. Cir-
cles are for soft momentum-dependent equation of state. Numerals
beside the points indicate the impact parameter in fm. ........

Total transverse momentum as a function of rapidity-ratio .......
Total transverse momentum as a function of rapidity-ratio .......
Total transverse momentum as a function of rapidity-ratio .......
Total transverse momentum as a function of rapidity-ratio .......

Total in-plane, transverse momentum p; in arbitrary units as a func-
tion of rapidity from hypothetical ellipsoids. The upper panel shows
results from a Gaussian—ellipsoidal density, whereas the lower panel are
results from the uniform ellipsoidal density. Each panel shows 0(y) of
ellipsoids which have kinetic energy ratios of 5, but rotated various
angles qbﬂow relative to the beam axis ...................

Total in-plane, transverse momentum pm in arbitrary units as a func-
tion of rapidity from hypothetical ellipsoids. The upper panel shows
results from a Gaussian-ellipsoidal density, whereas the lower panel are
results from the uniform ellipsoidal density. Each panel shows 0(y) of
ellipsoids rotated various angles 43,10", relative to the beam axis, but
which possess identical kinetic energy ratios. ..............

Total in—plane, transverse momentum p; in arbitrary units as a func-
tion of rapidity from hypothetical ellipsoids. The upper panel shows
results from a Gaussian-ellipsoidal density, whereas the lower panel are
results from the uniform ellipsoidal density. Each panel shows 0(y) of
ellipsoids rotated various angles dmow relative to the beam axis, but
which possess identical kinetic energy ratios. ..............

Total in-plane, transverse momentum pm in arbitrary units as a func-
tion of rapidity from hypothetical ellipsoids. The upper panel shows
results from a Gaussian-ellipsoidal density, whereas the lower panel are
results from the uniform ellipsoidal density. Each panel shows 0(3)) of
ellipsoids rotated various angles ¢flow relative to the beam axis, but
which possess identical kinetic energy ratios. ..............

Total in-plane, transverse momentum pa, in arbitrary units as a func-
tion of rapidity from hypothetical ellipsoids. The upper panel shows
results from a Gaussian-ellipsoidal density, whereas the lower panel are
results from the uniform ellipsoidal density. Each panel shows 0(y) of
ellipsoids rotated various angles ()3 ﬂow relative to the beam axis, but
which possess identical kinetic energy ratios. ..............

Balance energies for 58Fe + 58Fe from BUU model with and without
the Coulomb ﬁeld. Stiff mean ﬁeld and vacuum nucleon cross sections
were used. .................................

xii

54

56
58
59
61
62

70

71

72

73

74

77

2.37

2.38

2.39

2.40

2.41

2.42

2.43

3.1

3.2

3.3

3.4

Balance energies for 58Zn + 58Zn from BUU model with and without
the Coulomb ﬁeld. Soft mean ﬁeld was used and in—medium nucleon
corrections as indicated. .........................

Mean ﬁelds due to Bao-An Li, et al.[Li95, Li96a], left panel, and
Sobotka[Sob94]. 6 is deﬁned in Equation 2.30, and is often referred
to as the neutron excess. ,6 is the normalized symmetric density p/po.

Balance energy versus reduced impact parameter for 58Fe + 58Fe, solid
points, and 58Ni + ”M, open points. ..................

Balance energy versus reduced impact parameter for 58Fe + 58F e, solid
points, and 58Ni + 58Ni, open points. ..................

Balance energy versus reduced impact parameter for 58Fe + 58Fe, solid
points, and 58Ni + 58Ni, open points. ..................

Balance energy versus reduced impact parameter for 58Fe + 58Fe, solid
points, and 58Ni + 58Ni, open points. ..................

Calculated balance energies using the BUU model. Solid points are
58Fe + 58Fe and open points are 58Ni + 58Ni. Diamonds represent

calculations using Equation 2.26 as the mean ﬁeld with a = —0.2,
and squares represent calculations using the stiff mean ﬁeld without
asymmetry corrections and with or = -—0.2 ................

Spectra of BUU + coalescence for impact parameter averaged Au + An
collisions with b_<_3fm. The 197Au + 197An phase-space is calculated
using a momentum-dependent stiff equation of state from Table 2.1 and
the free nucleon cross sections[PDG88]. Global temperature and radial
ﬂow velocity are obtained by ﬁtting the radially expanding thermal
model[Sie79] to deuterons, tritons, 3He, and alphas simultaneously.
Dotted lines are the global ﬁts for a radial ﬂow velocity of zero.

Excitation function of radial ﬂow velocity 6 and apparent temperature
from BUU + coalescence for impact parameter averaged 197Au + 197Au
collisions with b_<_3 fm. The Au + Au phase-space is calculated using
a momentum-dependent stiff equation of state from Table 2.1 and the
free nucleon cross sections[PDG88]. ...................

The single-particle, reaction-plane momentum distributions for central
Au on Au collisions using the stiff, momentum-dependent mean ﬁeld.
a is the in-medium cross section reduction factor and the angular cuts
are illustrated as white lines on the graphs. The right-most panel is
the kinetic energy distribution of the systems after 30 fm/c for various
cross section reduction factors. Solid lines are calculations using a = 0
dotted lines a = —0.5 and dashed lines a = —0.9. ...........

The single-particle, reaction-plane momentum distributions for central
Au on Au collisions using the soft, momentum-independent mean ﬁeld.
oz is the in-medium cross section reduction factor and the angular cuts
are illustrated as white lines on the graphs. The right-most panel is
the kinetic energy distribution of the systems after 30 fm/c for various
cross section reduction factors. Solid lines are calculations using a = 0,
dotted lines a = —0.5, and dashed lines a = —0.9 ............

xiii

78

83

85

86

88

89

90

100

101

107

Chapter 1

BUU and the Nuclear Equation of
State

1 . 1 Introduction

It is primarily the accessibility of ordinary matter, such as water, that has fostered
such a well-developed, mature science of its bulk properties. After all we are macro-
scopic, Newtonian beings immersed in a world of inches and miles per hour. Our very
biology is a reﬂection of the relationship we have with the slow, macroscopic world.
It is little wonder that we have such difﬁculty studying elements of nature which are
very far removed from our own scales of time and space. The very fast—the relativistic
world of Einstein—and the very small—the world of Bohr—are very strange to us. We
must fashion elaborate machines to communicate with these worlds, and apply expert

skill in gleaning the effects of nature from the effects of the machinery.

Extracting the bulk properties of nuclear matter is thus a wonderfully challenging
endeavor. Protons and neutrons make up the vast majority of the mass of our solar
system, including the thesis you are now reading. Yet because these particles are
so small, and ordinarily so tightly bound to one another, studying them and their
interaction is difﬁcult. They are quantum-mechanical systems—systems so small, the
mere act of measurement signiﬁcantly alters their state.

1

The only empirical observations that can be made of large quantities of nuclear
matter are astronomical, since the Coulomb repulsion among protons renders macro-
scopic quantities of nuclear matter unstable. Only when huge amounts are brought
together does the gravitational attraction overcome the Coulomb repulsion. However,
such large masses cannot be formed in our solar system without catastrophic results
(not to mention the technical challenges in doing so), thus we are relegated to being
the local voyeur—peeking in from a distance on events over which we have no control.
We can neither choose nor can we know their initial states. In addition, the great dis-
tance between us and these systems imposes other uncertainties, themselves objects

of current debate.

Another option is to study the small, stable packages of nuclear matter we have
here on earth: the nuclei of atoms. Far more accessible, they nonetheless pose their
own problems in the effort to know the nuclear equation of state. There are strict
limitations placed by the nature of these systems on the information we can extract
from them, and on the mathematical feasibility of extrapolating this information to
explain inﬁnite nuclear matter. The systems are very small: with fewer than 1000
nucleons in any single interaction and collision volumes of the order of 10"40m3. In-
teraction times are typically on the order of 10‘21 seconds. Apart from the Heisenberg
uncertainty Principle Ax.- Apj Z 6,)- ’1, AE At ,2 h, where Arc, Ap, A E, and At

are the uncertainties in length, momentum, energy and time of an event, respectively,

and where 653- is the Kronecker delta, there are technical limitations.

To be correct, thermodynamic quantities such as temperature are well deﬁned only
for inﬁnite matter. That is, the differential volume 61) is inﬁnitesimal compared to the
volume. In addition, the matter contained within 611 is to be continuous. Macroscopic
systems of ordinary matter do not meet these criteria, but it can be shown that they

deviate in—substantially from the ideal. In real systems of ordinary matter, one may

expect to study a mole molecules. So the ideal of continuous matter becomes a fair
approximation. Such is not the case for even the compound nuclei from heavy ion
collisions. One must then be careful when applying the language of thermodynamics

to the science of heavy ion physics, or any other quantum process.

At intermediate energies, roughly between 50 MeV per nucleon and 1 GeV per
nucleon, the constituents of the heavy-ion collisions (protons, neutrons and mesons)
are relativistic, quantum-mechanical and strongly coupled. The Coulomb and the
strong nuclear forces are important in the evolution of these systems. It is no surprise
that a model derived from ﬁrst principles is currently, and for the foreseeable future,
computationally infeasible. To create a useful model, one is forced to make judicious

approximations.

The effort to understand the nuclear equation of state at intermediate energies
has for several years been focused upon the interplay between empirical data and
observables generated from numerical models. The information gathered from the
comparison tells us something about these models and nuclear matter in the context

of those models.

1.2 The BUU Model of Nuclear Matter

Perhaps the ﬁrst suggestions of an independent, free particle description of what was
then called “high energy” (around 100 MeV per nucleon) nuclear collisions were set
forth in the late nineteen forties [Ser47]. There the transparency of ions to incident
protons and neutrons was argued from the the degeneracy of fermionic matter in the

nucleus.

This simple model, eventually known as the Intranuclear Cascade (INC) model,

formed some of the foundation upon which the application of more sophisticated

transport models to heavy—ion collisions would be built. It would be almost thirty
years until practical numerical techniques could be implemented on computers acces-
sible to the nuclear physics community [Bon76, Smi77, Cug81]. While the INC was a
good ﬁrst step towards a practical model of heavy-ion collisions, it featured only hard
collisions among the nucleons and neglected the attractive portion of the internulceon

potential.

Another model which saw early application to heavy-ion collisions was the time-
dependent-Hartree—Fock (TDHF). Some calculations were quantitatively promising,
but serious theoretical problems existed [Dan97, Neg82]. Chief among these was the
lack of two—body correlations and collisionless dynamics, which led to slow thermal-
ization. It saw better success in describing the low—energy excitation modes of heavy
ions. It is easy to see why this is so: a principle approximation of Hartree—Fock theory
is that the Hamiltonian can be written as a single particle term and a weak two-body
term. The assumption that two-body dynamics are unimportant is better suited for

low-energy processes. At high energies the approximation is poor.

A treatment of the long-range interaction among nucleons can be found in Vlasov
transport theory. In fact one can obtain the Vlasov transport equation from TDHF
equations by applying the random phase approximation and recasting the result using
the Wigner transform [Koon79, Ber88]. The result, as one might guess, is a collision-

less approximation.

These models are incomplete compared to the Boltzmann-Uehling—Uhlenbeck trans-

port (BUU) equation, Equation [Ueh33]:

g—f + D(f) = / d¢1 / gwwgwmf'f; (1 + 0f)(1+ 0h) -

ff1(1+ 0f’)(1 + 61%)}. (1.1)

where primes denote quantities that are to be taken after collision, f is the Wigner

phase-space distribution, g is the relative velocity of the colliding particles (binary
collisions only), 19 is the change in the direction of g, 0 takes the value 0 for classical
statistics, —1 for Fermi-Dirac statistics, +1 for Bose-Einstien statistics, and d5? is the

usual solid angle. The expression
D(f) = r,(Bf/8x,) + X,(8f/8r,) (1.2)

describes the transport of mass and energy into and out of a unit volume in phase-
space, where z is a summation index, and X, represents the three components of the
force per unit mass. Here one ﬁnds role of the long-range interaction and collisions.

In more recent papers Equation 1.1 has a more familiar form [Ber84, Ber88]:

:7: + r;- V. f — v.U . v, f = 7—2717)? / d3p2d3p21dQ-ggu12
Xilff2(1— f1’)(1- fz') - f1'f2'(1— f1)(1- f2)l
x (2”)363(17+172 — 51' “ 177)} (1.3)

where U is the nuclear mean ﬁeld, da/dﬂ is nucleon scattering cross section (both
are discussed below), primes denote quantities to be taken after the collision between
particles 1 and 2. Note that the two colliding particles change their momenta, but
their positions in conﬁguration-space remain unchanged (they will be propagated to
new positions using Hamilton’s equations in the mean ﬁeld). An important feature
of Equation 1.3 are the terms (1 — f) in the collision integral. They facilitate an
approximate treatment of the Pauli Exclusion Principle, since f is a measure of the

occupancy of phase-space.

Collisions which have ﬁnal states in highly occupied regions of phase-space are
stochastically forbidden at a rate proportional to f’. This allows partial transparency

of nucleons incident upon bulk nuclear matter found in Serber’s work[Ser47].

It can be shown that the BUU formalism is a truncation in the BBGKY (Bogoliubov-

Born-Green—Kirkwood-Yvon)[Won77, Bau86a] hierarchy. The full N -body theory is

quantum-mechanically exact. However, the full N-body is beyond us today. The
truncation allows a numerical solution, but cannot address multi-particle dynamics.
Nonetheless, the BUU formalism is very well suited to calculation of the single-body

phase-space distribution.

1.2.1 Nuclear Mean Field Parameterizations

Equation 1.3 features U, a mean ﬁeld which is the average of the potentials of the
surrounding matter on a single nucleon. Frequently a parameterization of the Skyrme

many-body potential [Skyr59] is used:

U(p) = A (i) + B (if (1.4)

where p0 is the normal, zero-temperature, zero-pressure nuclear matter density, A is
attractive, B is repulsive, and a > 1. The parameters A, B and a are partially de-
termined by the nuclear saturation density of inﬁnite nuclear matter p0 = 0.16 f m’3,
and saturation binding energy E / A = —16 MeV. In the studies presented in this work,

the compressibility [Ber88]:

2
_ P F
K _ 9(——3m + A + 03) (1.5)

is treated as a “free” parameter to unambiguously ﬁx all three parameters:

 

 

K+44.73
A = —2. — .
981 4690[K_166.32]
K+255.78
B = .
23 45 [K - 166.32]
a _ K+44.73
211.05

Note that pp is the Fermi momentum of nuclear matter.

The mean ﬁeld allows the nucleons to be initialized with Fermi momentum— other-

wise the system would quickly disassociate—as well as ﬁlling an important role in the

 

 

 

 

Model K (MeV) A(MeV) B(MeV) a
Soft 200 -—356 303 7/6
Medium 235 —218 164 4/3

Hard 380 — 124 70.5 2

 

 

 

 

Table 1.1: Parameter sets used for the density-dependent mean ﬁeld U

evolution of phase-space. Typical numbers for the compressibility K are 200MeV,
often referred to as a “soft” equation of state, and 380MeV, a “stiff” equation of
state, Ref. [Ber84]. Note that the “medium” mean ﬁeld was introduced by Bauer,et

al. [Bau86].

Development of momentum dependent mean-ﬁeld, which arises from the inclusion
of non-local interactions, is illustrated in some detail in Ref. [Jeu76]. Early implemen-
tations of this dependence appear in Ref. [Aich87, Gale87]. There are two popular
forms of this dependence, the one used in the studies presented in this work is from

Ref. [Wel88]:

 

U004?) = A (i) + B (if + %/ d3p'1i(€:2)2 (1-6)

In this work two parameter sets are used for the momentum-dependent mean ﬁeld:

A = —110.44 MeV, B = 140.9 MeV, C = —64.95 MeV, a = 1.24, A = 1.58 pp|T=0m=

Po ’

which gives a compressibility of 215 MeV, and A = —5.89 MeV, B = 36.21 MeV,

C = —64.91MeV, 0‘ = 2.45, A = 1.58 pFiT=0,p= which gives a compressibility of

Po’

380 MeV.

1.2.2 Nucleon Scattering Cross-Section and the
Collision Integral

Of vital importance is the role of hard, short-range interactions among the nucleons.
The mean ﬁeld is primarily responsible for the longer range interactions, both attrac-
tive and repulsive in nature, but the repulsive, short-range interaction is carried by
scattering. The collision cross sections used here in the BUU equation are param-
eterizations of empirical data by the Particle Data Group [PDG88]. The following

scattering processes are built into the computer code used in this work:

z>2222
++++++
Zl>l>l>22
llllll
szzzz
++++++
ZDDZDZ

i-meson and a nucleon, and thus it has four charge

A is a resonant state of a 7r
states: A++ , A+, A0, and A‘. It has a mass of 1232 MeV, does not contribute to the
Pauli exclusion of nucleons (it is a spin-% fermion), and has a mean lifetime of about
1.8 fm/c. This resonance plays an important role of in entropy production during
heavy-ion collisions. N“ is an excited state of a nucleon. However, its production
rate is only about four percent of all resonances produced in heavy-ion collisions at

800 MeV/ nucleon [Dan95]—this energy approaches the upper limit of applicability of

the BUU code featured in this work. Thus, this work will ignore this rare process.

1.3 Numerical Implementation
of the BUU Equation

There are numerous ways to execute the BUU equation in computer code. One may

attempt a numerical solution of Equation 1.3. However, the code used in this work

attempts a simulation of heavy-ion collisions rather than an explicit solution of the

phase-space distribution f (F, 15').

1.3.1 The Parallel Ensemble

The calculations of heavy-ion collisions which used the INC model amounted to Monte
Carlo solutions to the collision integral. The stochastic sampling of phase-space meant
that meaningful results would be obtained when an adequate portion of phase-space
was calculated. Here there the situation is modiﬁed by the inclusion of the nuclear
mean ﬁeld. Derivatives in coordinate-space (for density-dependent mean ﬁelds) and
derivatives in momentum-space (for density— and momentum-dependent mean ﬁelds)
take on spurious numerical ﬂuctuations due to two approximations, both of which

lead to a calculable model.

In BUU formalism the collision integral is treated as a continuous source function.
As the number of test-particles goes to inﬁnity, the BUU equation becomes exact.
However, the model is numerically executed on ﬁnite-state machines (computers), and
the microscopic collision process is never mathematically inﬁnitesimal. The collision
integral is then a ﬁnite sum of discontinuous elements. Its contribution to phase-
space is never continuous, and derivatives on the phase-space distribution, or parts
of it like the matter or momentum density, are divergent in general. Numerically the
derivatives are ﬁnite differences, and while not immune to divergence such as in the
case of a divide by zero, they are nonetheless susceptible to ﬂuctuations resulting from
the aforementioned approximation. In addition, the calculation of the Pauli-blocking
factors (terms of 1 — f (1", 13)) suffers as well. This causes problems at the low energy

limit for this model in the form of enhanced transparency.

Another source of numerical ﬂuctuation concerns the derivatives the mean ﬁeld

potential. Formulated from the Skyrme parameterization, which begins with

10

Dirac-delta functions, it is density-dependent and, if chosen to be so, momentum-
dependent as well; whenever derivatives are taken of this ﬁeld, they are naturally
subject to the same divergences as are derivatives of the phase-space distribution.
The nucleons are propagated each time step in the computer code using Hamilton’s

equations of motion:

dp’ _
a? _ —V.U (1.7)

it = 2 + WM (1.8)

dt ﬁ.p+m2

 

where VP is the gradient with respect to momentum-space. One can easily see in

Equations 1.4 and 1.6 where the derivatives in phase-space appear.

Efforts to address these spurious ﬂuctuations in the collision integral and phase-
space distribution have been implemented in the model studied in this work. An
ensemble of nucleus-nucleus collisions is evolved in parallel in time. Nucleon colli-
sions occur only among nucleons of the same member of the ensemble. There is no
collisional communication among the members of the ensemble. This independence of
the members of the ensemble is used to reduce the combinatorial burden. However,
in order to generate the mean ﬁeld, the mass density and momentum density are
averaged over the entire ensemble. This goes a long way to quell the stochastic noise

from the collision integral.

Another method employed to control spurious ﬂuctuations is to distribute the
matter of the nucleon over conﬁguration-space. The “position” of the nucleon marks
the centroid of the distribution, with a constant Gaussian fall-off in all directions.
Nuclear matter is smeared in the density distribution so that derivatives in this dis-

tribution are better behaved.

11

1.3.2 Initialization

A semi-classical picture of the nucleus is used with the BUU equation. A nucleus is
initialized by randomly distributing its nucleons throughout a spherical volume which

has a radius given by

where A is the number of nucleons in the nucleus. The form of this formula is a
result of the proﬁle of the inter-nucleon potential: a hard-core repulsion at short
distances (r S, 1 fm), a strong attraction at intermediate distances (1 S, r S, 2 fm),
and a weak attraction at long distances (1“ > 2 fm). The direction of a nucleon’s
Fermi momentum is randomly assigned, but its magnitude is calculated from the
local density given by the Woods-Saxon prescription. Thus those nucleons near the
surface of the nucleus, where the density decreases for real nuclei, have a smaller
Fermi momentum than those nearer the center of the nucleus. This procedure is
repeated for each of the members of the ensemble of pairs of nuclei to be used in the

calculation.

1.3.3 The Collision Integral

The basis of the treatment of the collision integral, the right hand side of Equation 1.3,
lies in previous attempts [Bon76, Smi77, Cug81] at a Monte Carlo solution. In this
work the baryons are assigned an index which in various arrays within the program
allow extraction of their position, momentum and identity. This information is used
in the routines which eﬂ'ectively constitute the collision integral. A procedural outline

will be used to describe the numerical implementation for two nucleons.

From the main program a collision routine is called which ﬁrst randomly chooses

a particle (baryon or 7r-meson). If it is a resonance, then whether, and if so how, the

12

resonance will decay is calculated; each time step the probability for decay is calcu-
lated from the width of the resonance; a random number is generated and compared
to the decay probability, and if it exceeded by the decay probability the resonance is
said to decay. Then the program calls routines which calculate the momenta of the
products of the decay. Since the product of resonance decays consists of a nucleon
and a pion, the exit channel of the nucleon must be checked against the occupancy of
phase-space of the rest of the spin-% matter of the system. This is the Pauli-blocking
mentioned earlier. First the occupancy, the Wigner term f in Equation 1.3, is mea-
sured by a number between zero for no occupancy, and one for full occupancy. A
random number between zero and one is generated and compared to the occupancy.
If that number is less than the occupancy, the decay is disallowed, and a new baryon

is chosen at random. If not, then phase-space for the nuclear matter is updated.

Assuming the particle is not a resonance, a new particle is chosen to complete the
pair to be considered for collision. If the pair consists of two n-mesons or if the pair
consists of a 7r-meson and a resonance, the collision is not allowed, and the particles
go on their merry way. In addition, if the pair are separated by a distance exceeding
a range of a maximum cross section (that of a 7r-meson and a nucleon) or if the
center-of-momentum energy of the collision is less than a predetermined cut off value,
the collision is disallowed and a new pair is considered. Pion—nucleon collisions result
in A and N* resonances and the absorption of that 7r-meson. The cross sections of

these processes were calculated according to Danielewicz [Dan91].

Nucleon-nucleon collisions dominate the aggregate of collisions and are thus very
important in the dynamics of the entire system. Once certain kinematic requirements
are met, such as whether the nucleons are moving toward each other, the cross section
is calculated according to the Particle Data Group ﬁt [PDG88]. Here the in-medium

effects of fermionic matter on the cross section can be introduced for study. The

13

eﬁect is illustrated in Briickner G-matrix theory [Brii55], where certain “sof ” or low-
momentum processes are disallowed by the Pauli exclusion principle. Briickner theory
forms the foundation for the on-shell propagation, ignoring the mean ﬁeld, of the
nucleons between collisions. In intermediate energy heavy-ion collisions, most of the
nucleon collisions scatter nucleons to the outside of the initial Fermi spheres. Thus,
the analogy is extended to “hard” processes in this model. Usually the form of the

effect is parameterized by

on" = afree(1+ a—e)
.00

(1.9)
[Kla93], where am, is the in—medium nucleon cross section, of,“ is the nucleon cross
section in the vacuum, p is the density of matter local to the collision, p0 is the density
of inﬁnite nuclear matter in the ground state, and a is a parameter between 0 and ——1.

Reducing the cross section makes nucleus-nucleus collisions less repulsive on average

as there are fewer nucleon-nucleon collisions.

More sophisticated treatments of in-medium corrections exist. Alm,et al.[Alm95]
used a thermodynamic T—matrix at ﬁnite temperatures to calculate the in-medium
nucleon cross section. Their cross section depended upon the local values of tem-
perature, density, collision energy and the momentum of the center-of-momentum
frame of the colliding pair relative to the surrounding matter. They found an en-
hancement of the cross section for a local temperature of 10 MeV and local density of
0.5m. At higher temperatures, however, a reduction was calculated. They found that
the enhancement in the cross section was due to Pauli blocking in their calculations,
whereas the reduction in the cross section was due to self-energy effects. Nonetheless,
in BUU calculations using the cross sections of Alm,et al. nucleon collision rates were
found to be remarkably similar to nucleon collision rates in BUU calculations using

the parameterization of Klakow,et al.[Kla93].

14

More recent work by Schnell,et al.[Sch98], using a non-relativistic in-medium scat-
tering matrix with a realistic inter-nucleon potential, calculated the nucleon cross sec-
tion as a function of relative momentum of the colliding pair, their total momentum,
local density and local temperature. They then calculate an average cross section
which depends upon the energy of the scattering, the local density and local tem-
perature. They found this average cross section to be only weakly dependent upon

temperature. Their parameterization:

 

<a> __ 0, ﬁ_ 22
(Oneal—1+ (E,T)p0 ﬂ(E,T)(pO) (1.10)

includes a second-order correction in density, as well as a dependency in E, the
scattering energy. We do not expect this new parameterization to signiﬁcantly alter

the single-particle observables calculated with BUU.

If the transverse separation (impact parameter) of the colliding nucleons in their
center-of-momentum frame is within M, then the possible exit channels are
considered. Elastic and inelastic processes are stochastically selected according to
the ratio of the scattering cross section as determined by the Particle Data Group
ﬁt [PDG88] to a maximum, low-energy (J3 = 1.89 GeV/62) cut-off cross section. If
the two are the same, the ratio is 1 and the collision is certain to be elastic. If this ratio
is less than one, then there is some chance the collision is elastic. If the collision is to
be inelastic, exit channels for the A or N*—and n-meson if direct pion production is to
be included in the model—are generated. Finally, all spin-é- exit states are checked for
Pauli-blocking. If an exit is blocked, then the collision is disallowed, the pre-collision
phase-space of the nucleons is restored and a new pair is randomly chosen. It is no
surprise that most of the given computational resources are devoted to sampling the

collision integral.

The geometrical arguments involved in the numerical implementation of the

15

collision integral are tantamount to assuming the nucleons to be point-like. At dis-
tances less than a fermi, the inter-nucleon potential is very repulsive. Thus one
may suggest the inclusion of this hard-sphere behavior in the scattering algorithm.
Such an effort has been made by Kortemeyer,et al.[Kor95]. This work featured the
Direct Simulation Monte Carlo approach, see Lang,et al.[Lan93] and Danielewicz,et
al.[Dan91]. The hard-sphere advection gives a van der Waals behavior in the collision
integral. Corrections were made to scattering probabilities. The modiﬁcations to
the collision integral pushed calculated observables further from their experimental
values, although it should be noted that the inclusion of in-medium corrections to the

nucleon cross section would improve the results.

Phase-Space Occupancy f (F, 15')

Measuring the occupancy of phase-space is necessary to approximately treat the Pauli
exclusion principle. In this implementation of the BUU model, coordinate—space
and phase-space densities are separately maintained. The 6-dimensional phase-space
information is ingeniously stored in a character array. Storing this information in a
character array minimizes the amount of memory the information consumes, although
it also necessarily places a limit on the number of nucleons which may occupy a single
phase-space element. Characters are represented by 8-digit binary numbers giving a
total of 28 = 256 permutations. Thus, 256 is the maximum number of nucleons which
may occupy a given cell in phase-space. Normally, a priori knowledge of the size
of the collision volume and its distribution in momentum—space allows adjustment
of the lattice parameters as long as the volume element (cell) dx dy dz dpx dpy dpz
remains h3. Here the coordinate-space lattice parameter is 2.73 fm and that for the

momentum-space lattice is 0.18 GeV/c.

The size of a cell comes from considerations of low-energy theories of the nucleus

16

and the Wigner transform, a classical analog of which is used here. Wigner functions
composed from quantum wave packets are not positive-deﬁnite. In an eﬂ'ort to remove
these non-classical negative values, a Gaussian smoothing is often applied with a
length parameter of h3, and so this is taken to be a proper lattice parameter for
phase—space.

Within the program whenever it is necessary to check the occupancy of a spin-

1 exit channel, the index of the nucleon is passed to a routine which returns the

NI

occupancy of phase-space in that nucleon’s neighborhood. First the nucleon’s position
and momenta are extracted from the appropriate arrays and are decretized on lattices
for momentum- and coordinate-space. These become the indices of the phase-space
lattice site. A linear interpolation is then executed on the 26 nearest neighbors. The
result is normalized to the volume element, spin and iso-spin degeneracies, and the
number of parallel members nucleus-nucleus events in the ensemble. In the end a
number between 0 and 1 is returned, the measure of the occupancy of that region of

phase-space.

1 .4 Summary

The BUU formalism forms the foundation for one of the most successful numerical
approaches to relativistic heavy-ion theory. The truncation of the BBGKY[Won77,
Bau86a] hierarchy, from which BUU model can be derived, relieves one of the huge

combinatorial burden that the full N -body calculation carries.

However, what one gains in calculability, one loses in physical completeness. In
part this means the systems modeled do not condense into the highly correlated
shell-like structures found in low energy nuclear excitations. Nor are quark and

gluon degrees of freedom explored at high energies. These limitations place upper

17

and lower limits in energy upon the viability of the model. Finally, the system’s
relativistic nature is only partially addressed; relativistic kinematic are used, but

ﬁelds are propagated with inﬁnite velocity.

The strong anti-correlation due to the Pauli exclusion principle is approximated
in parts of the model. However, nearly all dynamical and kinematical elements must

include consideration of Pauli—blocking.

The inter-nucleon potential is split among two mechanisms: hard scattering for
modeling the strong, repulsive, high-momentum processes, and a nuclear mean ﬁeld

for soft, low-momentum processes.

In spite of these approximations, the BUU model has been successful in providing

science insight into the rich dynamics of relativistic heavy-ion collisions.

Chapter 2 describes two directed, collective observables which can be used to dis-
criminate among the various sets of model parameters of BUU. Presented is the dis-
tribution of the average, in-plane, transverse momentum in rapidity, the derivative of
which at mid-rapidity is called the ﬂow of the collision. The other is the distribution
of the total in-plane, transverse momentum in rapidity. Both of these observables
are sensitive to the dynamics of the collision and, in turn, sensitive to the model

parameters of BUU.

Chapter 2

Collective Phenomena

Studying the behavior of groups of nucleons liberated in heavy-ion collisions gives
one a peek into the collective behavior of large amounts of nuclear matter. Today
we use several experimental observables of collective phenomena to characterize var-
ious equations of state. They are, to one degree or another, sensitive to important
elements of the model: compressibility, in-medium effects, momentum dependence.
And the search for cleaner probes continues. In this chapter some of the most often-
used observables in heavy-ion physics are presented in addition to the inﬂuence the

equations of state have on their values.

2.1 A Picture of the Nucleus-Nucleus Collision
from BUU

At intermediate collision energies, the nucleus behaves somewhat like a droplet of
liquid; as it collides with another nucleus, its shape distorts from an initial sphere to
an ellipsoid, compressional energy builds as some of the beam energy is converted,
primarily through hard nucleon collisions, to locally random or thermal energy. Some
of the matter is stopped, or nearly so, in the center-of-momentum frame of the nuclei

and forms a rotated elliptical hard core.

18

19

The evolution of the phase-space of a 197Au + 197Au collision with a beam energy of
50 MeV/ A is shown in Figures 2.1, 2.2, 2.3, and 2.4. The ﬁrst is a series of projections
of the momentum space of 10% of the nucleons of the entire ensemble every 10 fm/c.
As one looks at each panel, one sees the Px-Pz plane, where the z-direction denotes
the beam direction. The ﬁrst panel in the upper left corner shows the initial state of
the nuclei: two Fermi spheres the centers of which are separated by beam momentum.
Here the two spheres are not completely separated since the radius of each sphere is
only about 267 MeV/c whereas the beam momentum per nucleon is only 310 MeV/c.
As time advances from the upper left corner down the page, the initial Fermi spheres
dissolve as the beam momentum is thermalized. Nearly complete thermalization

results in the ﬁnal panel.

Figure 2.2 shows the conﬁguration-space projection of the same nucleons. Here the
view is onto the reaction plane, which is deﬁned by the centers of the nuclei and the
impact vector. Again the z-direction is collinear with the beam. Beginning with the
initial state in the upper left corner and progressing in time down the page, one sees
the well-deﬁned spherical nuclei approach each other. By about 90 fm/c the system
is near maximum compression. Here a large fraction of the beam energy has been
converted, through hard scattering, repulsive contributions from the nuclear mean
ﬁeld, and the Coulomb ﬁeld, into potential energy. The very next panel shows the
evidence of the release of this pent-up potential. Most of the nucleons have left the
calculation volume. The system becomes quite diffuse. This violence is typical of

central (zero impact parameter) collisions.

The amount of the initial longitudinal momentum transformed into momentum
perpendicular to the beam is often used as a means of characterizing the impact
parameter of the collision. This is accomplished with the help of theoretical models,

like BUU, wherein one has a priori knowledge of the impact parameter. A mapping

20

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_l(m.0 111111111111111111L‘L111111111111111111 _1mX).O
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PZ (MeV/C)

Figure 2.1: Momentum-space projection upon the Pz-Pz plane for 197Au + 197Au
reaction at 50 MeV/A. Impact parameter is zero, soft (compressibility = 200 MeV)
mean ﬁeld and vacuum nucleon cross sections were used. Each panel is separated by
10 fm/c. Initial state is the upper left panel, and time passes from top to bottom.
ﬁnal-state is the lower right panel.

Rx (fm)

21

 

 

 

 

 

 

 

 

 

 

 

15.0 ‘ l l l 15.0
5.0 5.0
-5.0 —5.0
15.0 15.0
5.0 5.0
—5.0 —5.0
—15.0 ‘ 1 ‘ ‘ ' L ‘ ~15.0

—40.0 -— 0.0-40.0 —20.0 .
0 0 —30.0 —10.0
R2 (fm)

Figure 2.2: Conﬁguration-space projection upon the reaction plane for 197Au + 197Au
reaction at 50 MeV/A. Impact parameter is zero, soft (compressibility = 200 MeV)
mean ﬁeld and vacuum nucleon cross sections were used. Each panel is separated by
10 fm/c. Initial state is the upper left panel, and time passes from top to bottom.
ﬁnal-state is the lower right panel.

22

 

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1W.0 IIII IITI IITI III IIII IIII IIII IIII 1W.O
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PZ (MeV/C)

Figure 2.3: Momentum-space projection upon the Px-Pz plane for 197Au + 197Au
reaction at 50 MeV/ A. Impact parameter is 3.5 fm, soft (compressibility = 200 MeV)
mean ﬁeld and vacuum nucleon cross sections were used. Each panel is separated by
10 fm/c. Initial state is the upper left panel, and time passes from top to bottom.
ﬁnal-state is the lower right panel.

23

 

 

 

Rx (fm)

 

 

 

 

l

 

 

 

—40.0 —20.0—40.0

—30.0 —30.0

RZ (fm)

-20.0

—15.0

—10.0

Figure 2.4: Conﬁguration-space projection upon the reaction plane for 197Au + 197Au
reaction at 50 MeV/A. Impact parameter is 3.5fm soft (compressibility = 200 MeV)
mean ﬁeld and vacuum nucleon cross sections were used. Each panel is separated by
10 fm/c. Initial state is the upper left panel, and time passes from top to bottom.

ﬁnal-state is the lower right panel.

24

can then be constructed from the theoretical calculations of the transverse momentum

in the ﬁnal state. Other observables may be used as well.

Figure 2.3 is the momentum-space projection for a 197Au + 197An collision with
a 3.5 fm impact parameter. One can immediately see that the momentum space
distribution appears qualitatively similar to that of the central collision, Figure 2.1.
The next section will address some of the differences between central collisions and
the far more common off-center collisions. The initial state consists of two Fermi
spheres not quite separated by beam momentum, and the ﬁnal-state appears well

thermalized at 100 fm/c.

The evolution of the conﬁguration-space distribution is strikingly different. Fig-
ure 2.4 shows a more complex picture. The ﬁrst panel shows two distinct nuclei offset
by the impact parameter. In the next few panels the two nuclei touch and their
constituent nucleons collide. These nucleon collisions begin to transform the initial
low temperature system into a hot compound nucleus. This nucleus is unstable. An
elliptical distribution is quickly formed and the initial angular momentum (there was
none in the previous example) results in a rotation of this ellipsoid. The rotation is
complete in ﬁnal panel after which residual beam momentum tears the system apart.

In experiments the remnants end up in the detectors.

2.2 Directed Transverse Flow

Collisions with non-zero impact parameter develop differently from central collisions.
Here the symmetry of the central events is broken. Of course in experiments one
cannot choose the impact parameter, and selecting on impact parameter is not a

trivial exercise. All events will have non-zero impact parameters.

Figures 2.5 and 2.6 are good examples of the evolution of a heavy-ion collision and

25

more typical of what is observed in experiments. The momentum-space distribution
of the nucleons is more vividly realized here because the beam energy of 500 MeV/ A
well separates the Fermi spheres. And, unlike the lower energy collision in Figure 2.3,
one can clearly see the characteristic ellipsoidal distribution of momentum in the
(ﬁnal state. With a beam energy of only 50 MeV/ A the ellipsoid is too small to reach

beyond the thermal sphere in Figure 2.3.

One can get a feel for the development of the deﬂection of matter in this off center
collision. Figure 2.6 shows the conﬁguration-space development of the collision. N o-
tice the sigmoidal “S”- shape of the distribution in the third panel (t=20 fm/c). The
sigmoid is an intermediate form which consists of spectator matter (nucleons which
have not signiﬁcantly scattered) in the periphery, and participant matter (nucleons
which have experienced several hard scattering events) occupying the volume common
to both nuclei. The participant matter is signiﬁcantly slowed [Bau88] and compressed.
The initial angular momentum carries the spectator matter into the forward and back-
ward hemispheres in the center-of-momentum frame (t=30 and 40 fm/c) and rotates
the compressed core. The ellipsoidal distribution follows from the expansion of the
compressed participant matter and the deﬂection of the spectator matter off of the
core, both along the major axis of the ellipsoid. Notice that in later times the system
is very diffuse, with only a few of the original nucleons remaining. This is a testament
to the release of the pent-up compressional energy in the system. The matter expands

quickly, and much of it has left the calculation volume.

In 1985 Danielewicz and Odyniec[Dan85] proposed a new observable: average
transverse in-plane momentum as a function of rapidity. To calculate this observable

from experimental data, one needs to establish the reaction plane. They proposed

26

the vector:

M
Q = Zwllpi—I (2’1)
u=l

where u is the particle index, p”i is the transverse momentum of the particle, 01,, is
a weight for the particle. Usually 0),, is -1 for y,, < yc.o.M., that is backward going
in the center-of-momentum frame, and is +1 for yu > yaaM, Thus for particles
deﬂected to positive rapidities, positive-x in Figure 2.6, their transverse momentum
makes a positive contribution to Q. Those deﬂected to the negative rapidities will
get a negative weight. Q is used to determine the reaction plane in experimentally
measured transverse momentum distributions. With an estimation of the reaction
plane, one can calculate the average per nucleon, in-plane, transverse momentum as
a function of rapidity. Figure is such a distribution for the 197Au + 197Au collisions in
Figures 2.5 and 2.6. The slope of the curve at mid-rapidity is the ﬂow[Dan85, Ogi89]

for the system.

Figure 2.7 shows a projection of the momentum-space distribution of a 197Au +
197An collision at 500 MeV/ A and an impact parameter of 3.5 fm. Unlike Figure 2.5
this calculation used the “stiff” mean ﬁeld. The two momentum distributions are

similar. Both exhibit the ellipsoidal shape of a system under directed ﬂow.

The observable “ﬂow” is deﬁned as:

(Km)
dy ’

31:0
where ( pm) is
N u
Emil)“:
N (y) ’

N (y) is the number of fragments (protons in this work) with rapidity y, and V(y) is

the particle index.

27

 

1000.0W..1W.., 1000.0

 

 

 
 

 

 

 
 

 
 

 

 

 

   

500.01 11 1500.0
001* .11 10.0
—500.0 f l 0 1f— 1 —500.0
1000010000
500.051 1_— 1500.0
0.051 1'1 10.0
—500.0 f1 1’— 1 1500.0
1000.0 E:H1:11:11:111111113311111111:{1:Minn: 1000.0
6 500.0;— 15000
E 0.0:— 1:1 10.0
2 —500.0 :1 1i 1 —500.0
V : 1; :
‘35 1000010000
$00.03— 1_— 1500.0
0.0 '1 1'— 1 0.0
—500.0 ‘1 if 1:1 1 —500.0
1000-0EHHl1H111==11w=I1H=l1mlm1l=1Hi1000.0
500.051 11— 1500.0
0.0 :1 1_— 10.0
—500.0 1 g 1_— ‘ 1 —500.0
—1ooo.0""'1—1000.0

 

 

 

 

—1000 0 1000 0 1000
—-500 500 -500 500

PZ (MeV/C)

Figure 2.5: Momentum-space projection upon the Px-P, plane for 197Au + 197Au
reaction at 500 MeV/A. Impact parameter is 3.5fm. Each panel is separated by
10 fm/c; soft (compressibility = 200 MeV) mean ﬁeld and vacuum nucleon cross sec-
tions were used. Initial state is the upper left panel, and time passes from top to
bottom. ﬁnal-state is the lower right panel.

28

 

 

 

Rx (fm)

 

 

 

 

 

 

." 1‘. '

—40.0 —20.0-40.0
0.0

 

‘ .1'. “ —15.0
—20.0
-30.0 —10.0

RZ (fm)

Figure 2.6: Conﬁguration-space projection upon the reaction plane for 197Au + 19"’Au
reaction at 500 MeV/ A. Impact parameter is 3.5 fm, soft (compressibility = 200 MeV)
mean ﬁeld and vacuum nucleon cross sections were used. Each panel is separated by
10 fm/c. Initial state is the upper left panel, and time passes from top to bottom.
ﬁnal-state is the lower right panel.

1000.0
500.0
0.0
—500.0
1000.0
500.0
0.0
—500.0
1000.0
500.0
0.0

-500.0

PX (MeV/C)

1000.0

500.0

0.0

—500.0

1000.0

500.0

0.0

—500.0

-1000.0

29

 

.......,.........., 10000

llll [luv—[ﬁVWIlwrv

1

500.0

0.0

vvrlrvtr‘l'r‘rr

—500.0

iiiiliiiilLJLIerr

v—I—rrT v

..

1000.0

uvrlrrvwlvvyxlyvv1
1"]
IlllllAlllllllIAlll

 

 

 

 

 

   

     

'1 1:1 1: 0.0

c .. -

' 4C I

:y- 1;— -_ -500.0
3 1F =: 1000.0
E :5 3

L" 4;- j 0.0

E - 22- 2

f 1E— __ —500.0
:HH'THHI'HHlHHf: “:1000.0
t it :

:1 €E - 500.0
: ’EE ' 3

:— 2t 9 : 0'0

r 1‘ , j

P > d} 1 -500.0
: 2t :

:: 1000.0
:— 1:— —: soo.o
:— “I— ‘2 0-0

:— 1:— —: —soo.o

 

 

 

l...l....l....l..l. ....l....l....l.... _1m00

 

—1000 0 1000 0 1000

—500 500 -500 500
P2 (MeV/C)

Figure 2.7: Momentum-space projection upon the P,,-Pz plane for 19"Au + 19"Au
reaction at 500 MeV/ A. Impact parameter is 3.5 fm, stiﬂ‘ (compressibility = 380 MeV)
mean ﬁeld and vacuum nucleon cross sections were used. Each panel is separated by
10 fm/c. Initial state is the upper left panel, and time passes from top to bottom.
ﬁnal-state is the lower right panel.

30

 

 

 

 

 

 

  

 

 

 

 

150......w. 150
5.0 - — 5.0
—5.0 ~ 1 —5.0
15.0 % : 15.0
r -
5.0 ~ — 5.0
4
_5_() — J —5.0
15.0 = : . : : 15.0
A 5.0 — — 5.0
E 4
i. —5.0 — — —5.0
‘34 l
15.0 % 1 15.0
5.0 ~ — 5.0
—5.0 — — —5.0
. i+i 1 15.0
— 5.0
— —5.0
—15.0 m i“! I 1 . ' - . ' —15.0
40.0 —20.0—40.0 —20.0
—30.0 -30.0 —1o.o

R2 (fm)

Figure 2.8: Conﬁguration-space projection upon the reaction plane for 197Au + 197Au
reaction at 500 MeV / A. Impact parameter is 3.5 fm, stiff (compressibility = 380 MeV)
mean ﬁeld and vacuum nucleon cross sections were used. Each panel is separated by
10 fm/c. Initial state is the upper left panel, and time passes from top to bottom.
ﬁnal-state is the lower right panel.

31

Average In Plane Transverse Momentum Vs. Time

 

 

 

 

 

Au + Au, 500 MeV/A
100.0 f . . . . . . T
Q
> 80.0 ~ .
0
g .a-vr‘GOOGCD
§ '4 5 l
"‘3 60.0 — 0 ~
2
’8 0
S
g 40.0 — 4
E
§ H K: 380 MeV
e , 9—9 K=200 MeV
g 20.0 — ‘ -
5
E-'
[0'
0.0 _ l i l i l . l i
0.0 20.0 40.0 60.0 80.0 100.0

Time (fmlC)

Figure 2.9: Evolution of the in—plane, transverse momentum for forward-going nucle-
ons. Impact parameter is 3.5 fm. Two different mean ﬁelds were used, as indicated.
Figure 2.8 shows a projection of the conﬁguration-space distribution of the same
collision as in Figure 2.7. Contrast this with Figure 2.6. The two collisions look the
same until about 50 fm/c where the calculations using the stiff mean ﬁeld and the
soft mean ﬁeld diverge. The stiff mean ﬁeld is more efﬁcient in converting the beam
energy into transverse degrees of freedom. The diffuse appearance of the latter stages
of the calculation in Figure 2.7 bears witness to the nucleons leaving the calculation

volume.

The time development of the system using different nuclear mean ﬁelds is illus-

trated in Figure 2.9. Notice that the more repulsive ﬁeld converts more of the beam

32

Average In-Plane Transverse Momentum Vs. Rapidity Ratio
Au + An, 500 MeV/A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

400.0 ' . ' T r
E 200.0 — - =’= ' 1
E
a d
5
E
2 0.0
2
d.)
3.
[-
0 —200.0 * I
5
E J -— K=200 MCV
------------ K=380 MeV
—400.0 ‘ ‘ ‘ I I 1 l
-10 —1.0 0.0 1.0 2'0

Rapidity Ratio (ypmmlym)

Figure 2.10: Average in-plane, transverse momentum versus reduced rapidity. The
system is 197Au + 197Au at 500 MeV/ A, impact parameter is 3.5fm. Two mean ﬁelds
were used, as indicated.

33

momentum to transverse momentum than the soft mean ﬁeld. The slopes of the
plots of Figure 2.10 are the ﬂows of the two calculations. The stiff mean ﬁeld yielded
a ﬂow of 225i6 MeV/c whereas the soft mean ﬁeld yielded a ﬂow of 172i6 MeV/c.

A X2-minimization of a straight line was applied to the region near mid rapidity.

2.3 Impact Parameter Dependence of Flow
for Momentum-Dependent and
Momentum-Independent Nuclear Mean Fields

The inter-nucleon interaction is in general momentum dependent[Ga1e87, Gale90].
This stems primarily from the Briikner treatment of the behavior of nucleons in nu-
clear matter and non-local effects from nucleon-nucleus potential scattering[Gale90].
Soft or low~momentum transfer interactions are blocked by Fermi terms in the scat-
tering equation. On average, nucleons streaming into hot, well-thermalized regions
of the collision volume are blocked from scattering into these regions. This makes
the bulk dynamics more repulsive on average. In the BUU model used here, the
momentum-dependent of mean ﬁeld of Welke,et al.[We188] is used:

: g 3;” Q 3, NW")
U015) A(po)+B(po) +po/dp1+(5—j,ﬁ)2 (2.2)

The mean ﬁeld can be modeled with various combinations of force parameters lead-
ing to different compressibilities in addition to toggling the momentum dependence.

Values for the parameters in Equation 2.2 are shown in Table 2.1.

For most collisions the addition of the momentum dependence in the mean ﬁeld

yields more repulsive behavoir overall. Figure 2.12 shows ﬂow values calculated for

r!

.
._-_‘ A 4
A-

34

 

EOS A (GeV) B (GeV) 0 C (GeV) A (GeV/c) p0

Soft -0.109 0.082 7/6 0 — 0.168
Stiff -0.062 0.03525 2 0 - 0.168
Softﬁ' —0.11044 0.1409 1.24 —0.06495 0.416 0.168
Stifffi -0.0589 0.03621 2.45 —0.06591 0.416 0.168

 

 

 

 

 

 

 

 

Table 2.1: The constants in used in the nuclear mean ﬁeld. Note that “sof ” refers
to a compressibility K of 215 MeV and “stiff” refers to a compressibility of 380 MeV
for the momentum-dependent mean ﬁelds, whereas “sof ” refers to a compressibility
K of 200 MeV and “stiff” refers to a compressibility of 380 MeV for the momentum-
independent mean ﬁelds.

197Au on 197Au at 400 MeV/A (upper panel) and 200 MeV/A (lower panel). The

calculations used an in-medium nucleon cross section reduction factor of 20%, and

the compressibility K was ﬁxed at 200 MeV for all calculations.

Notice that for all but the smallest impact parameters, the ﬂow values are larger
for calculations using the momentum-dependent mean ﬁeld than those for calculations
using the momentum-independent mean ﬁeld. However, for the 197Au on 197An colli-
sion at 200 MeV/ A, something unexpected occurs at the lowest impact paramters: the
momentum-independent mean ﬁeld produces a larger flow than does the momentum-

dependent mean ﬁeld.

This effect was ﬁrst brought to our attention during calculations of 84Kr on 197Au
at 200 MeV/A, which appear in work by M. J. Huang,et al.[Hua96]. Figure 2.11
shows ﬂow results from BUU calculations compared with experimental data. There
the impact-parameter—averaged, ﬁltered ﬂow values for momentum-independent mean
ﬁelds were larger than those for momentum-dependent mean ﬁelds. The trend was

reversed at an impact parameter of about 4fm.

Why is this so? What mechanism is responsible for this counter-intuitive result?

Is there an experimental observable which may be more sensitive to this effect and

 

 

 

 

 

 

150
3
3.
.100
£
3.
250
I .
0’41-.-L....i.1..|..‘
0 2 4 6
Nb)

Figure 2.11: Flow results from M. J. Huang, et al.[Hua96]. In the Figure experi-
mental data are the cross-hatched regions, the rest are calculated flow values with
detector acceptances. “S” and “H” refer to the soft, momentum-independent and
hard, momentum-independent mean ﬁelds, respectively. “SM” and “HM” refer to
soft, momentum-dependent and hard, momentum-dependent mean ﬁelds, respec-
tively. The BUU results denoted by “(0.80%)” were calculated with a 20% reduction
in the free nucleon-nucleon cross section.

thus may be used to discriminate among the model parameters of BUU? The following

will attempt to answer these questions.

2.3.1 The Development of Density in Time

In order to give the reader a sense of the differences in the evolution of reactions using
the momentum-dependent mean ﬁeld and reactions using the momentum-independent
mean ﬁeld, Figures 2.13—2.24 were developed. These Figures show the lab-frame evo-
lution of the nucleon density of 197Au on 197Au collisions in time. Each Figure shows
the normalized contours of density projected onto the reaction plane. The normal-
ization constant is p0, normal nuclear density. The upper panels show the contours
from calculations using the soft, momentum-independent mean ﬁeld; the lower panels

show the contours from calculations using the soft, momentum-dependent mean ﬁeld.

36

All of these calculations utilized the same model parameters used to generate the
ﬂow values in Figure 2.12. The times at which the contours are extracted from the
BUU calculations were selected to demonstrate the salient features of the differences
between the dynamical evolution of reaction using the momentum-dependent mean

ﬁeld and reactions using the momentum-independent mean ﬁeld.

Figures 2.13—2.16 show central 197Au on 197Au collisions. For time t =1.0 fm/c,
one can discern the Lorentz contraction of the projectile nucleus. At this time step
the upper and lower panels are almost identical. At t =5.0 fm/c, Figure 2.14, the
nuclei begin to touch. Here one can just see the beginings of differences in the
evolution of the systems. This is quite clear by t =10 fm/c, Figure 2.15. The neck
region separating the centroids of the two nuclei are markedly different in proﬁle; the
momentum-independent calculation showing substantially steeper density gradients
than the momentum-dependent calculation. As time passes the gradients in the
momentum-dependent calculations mitigate. By t =60 fm/c, Figure 2.16, the systems
approach thier respective ﬁnal states looking remarkably similar given their previous
divergent behavoir. Apparently the momentum-independent mean-ﬁeld has had time

to relax to the smooth, diffuse proﬁle seen in Figure 2.16.

Figures 2.17—2.20 show density proﬁles for an impact paramter 0 =2 fm and beam
energy 200 MeV/ A. This is the same impact parameter wherein Figure 2.12 the
momentum-dependent ﬂow enhancement is strongest. Begining at t =1.0 fm/c the
nuclei are very similar. By 10 fm/c density gradients for the two systems begin
to differ. Around t =20 fm/c, Figure 2.19, the momentum-dependent calculation
shows maximum compression. However, it will be about another 10 fm/c before the
momentum-dependent calculation reaches maximum compression. Figure 2.20 shows
the momentum-independent system near maximum compression. Note the steep gra-

dients and the enhancement of the central density in the momentum-independent

' ' ..-_.._._=_"_“.I

37

system over those of the momentum-dependent system.

As time goes on the dense central cores of both systems rotate under the inﬂuence
of the cold spectator matter farthest from the beam axis. An important clue reveals
itself during this phase of the evolution: the momentum-independent system is re-
leasing its pent-up compressional energy more slowly than the momentum—dependent
system. It is the timing here that is important. If a system releases the energy
too early, that is, before the rotation past 90° to the beam axis, the motion of the
spectator matter up and down the beam axis will be impeded. In momentum-space,
where directed ﬂow is measured, this will shift a portion of the matter closer to mid-
rapidity resulting in a more spherical distribution. This appears to be the case for the
momentum-dependent calculations, and could offer an explanation for the suppression

of the ﬂow.

Figures 2.21—2.24 show density proﬁles for 197Au on 197Au collisions at 400 MeV / A.
At time t =1.0fm/c one can see the pronounced Lorentz contraction of the projec-
tile nucleus. From Figures 2.22 and 2.23 one can see that the two systems reach
their respective maximum compression at roughly the same time, although it is clear
that higher densities and gradients persist in the momentum-independent calculation.
Substantial release of the compressional energy seems to occur when both systems

have rotated past 90° to the beam axis. By 55 fm/c the two systems are similarly

diffuse.

Inspection of Figures 2.13—2.24 and Equation 2.2, one gets the impression that
something else in addition to the timing of the release of compressional energy may
be at work. The momentum dependence in Equation 2.2 is likely to be smaller for
the well-thermalized matter of the core. However, the gradients in momentum-space,

which contribute to the velocity ﬁeld, see Equations 1.7 and 1.8 in Chapter 1, tend

. Amy- m -'I

38

to deﬂect the in-coming matter perpendicular to the beam axis. This would tend to
reduce the oblateness of the momentum ellipsoid and lead to a general supression of

the ﬂow signal. Evidence of similar effects have been seen[Gut89, Gut90].

“r___'rza‘ N.J.I

39

Flow Vs. Impact Parameter
‘”Au + mAu, K = 200 MeV

 

I T f T

280.0 r

G——€) Momentum Dependent
H Momentum Independent

180.0

80.0

     
   
 

 

O

f. 400 MeV/A

3 . ' ' I ' I _
E 2800 r G—Q Momentum Dependent

g H Momentum Independent

0

>

a: 180.0 " F
g .
f-‘

80.0

 

200 MeV/A

 

l n l 1 l

0.0 2.0 4.0 6.0
Impact Parameter (fm)

Figure 2.12: Average in-plane, transverse flow as a function of impact parameter.
Upper panel shows ﬂow calculations for 197Au on 197Au at 400 MeV/ A for the indi-
cated impact parameters. The lower panel shows the same observable, but for 197Au
on 197Au at 200 MeV/A.

40

Au on An @ 200 MeV/A, b=0.0 fm, t=1.0 fm/c

 

 

 

 

 

 

-20 —10 0 10 20
==,=-=ﬁ,....,....,....,..
10(- -
’ Soft EOS
0? ‘.
_10 -— _
/|\ —20 - j
I .
| >
E -30:' -
g ;
:03 _4O’:¢}::4:} 4141.;+:= {.=
g j —20 —10 0 10 20 j 10
ES » j
E . _
g r 80“. P EOS .. 0
m r
e — —10
. 0.5
_— - —20
— d —30
» 0.3
L
L . d
i l . 4 i . l . L4 1 i . i r 14 i i . 14 A _40

 

-20 -10 O 10 20

X Direction (fm)

Figure 2.13: Calculated density contours. Numbers are normalized densities: p/po.

41

Au on Au @ 200 MeV/A, b=0.0 fm, t=5.0 fm/c

 

 

 

—20 —10 O 10 20
. . l . . . . l . . . . I . . . . I . . . . TT .
10 -'
i Soft EOS
0 _ ..
L
_10 — _.
I I
/l\ _20 >— a
I L
| L
E —30 — —
:3 .
c >
.9. f
6 -40 : : 1 : : ¢ : 1 1‘
o F -20 1 10
.E:: *
:3 l .
a - <
0 . . O
m .
F
- — —10
- - —20
- — —30
r A l r . 1 r l A i 1 1J i . . . l i L i . l L . —4O

 

 

 

-20 -10 0 10 20

X Direction (fm)

Figure 2.14: Calculated density contours. Numbers are normalized densities: p/po.

Beam Direction (fm) —-—>

Figure 2.15:

42

Au on Au @ 200 MeV/A, b=0.0 fm, t=10.0 fm/c

 

 

 

 

 

 

-20 —10 O 10 20
10L..,.=..,....,-..=,....,.._

, .

I Soft EOS 1

, J

0.. .—
407 i

l .

1 .
‘20? f
-30_ _

[ J
.._40 3+1+=+%%::::%¢¢¢+%:¢H%t:

f —20 —1O 0 10 20 j 10

L Soft—P EOS _‘ 0

F .

L

L 0.3 P 0.5 _ _10

1 9) *
_ ( 0.9 _ _20
1.5
L ' '3
1.2 ‘ ‘30
- . i I L. . . I . . . . l . . . . l . . . i l ,4 1 —40
--20 -10 O 10 20

X Direction (fm)

Calculated density contours. Numbers are normalized densities: p/po.

Beam Direction (fm) ———>

Figure 2.16:

43

Au on Au @ 200 MeV/A, b=0.0 fm, t=60.0 fm/c

 

 

 

 

 

 

 

—20 —1O 0 10 20
. . I I . . . I . . . . I . . . . I. . . .jﬁ .
10*- n
2 Soft E08
0_ _
-10- 4
_20I— _
_30— _.
L
_40 tiI It: : I : It: I : t r t I t t t t I t:
:- —20 —10 0 10 20 7 10
: Soft—P EOS 1
- - 0
: i
f - —10
_— — —20
i
— - —30
I. .
.rl..rnm....1....1....1.L -4O

—20 —10 O 10 20

X Direction (fm)

Calculated density contours. Numbers are normalized densities: p/po.

any! $17.17*“

.1-

44

Au on An @ 200 MeV/A, b=2.0 fm, t=l.0 fm/c

 

 

 

 

 

 

 

—20 —10 0 10 20
-f,...-,=.-.,....,..ﬁ-,-
10- -
. Soft EOS
0... ..
_10— _
A —20- e
l . .
I I
| L
’5‘ -30-— —
a : :
.2 . I
45 _4O :tItt ‘ . 4 : I f:
23 ~ -20 20 j 10
'5 :
8 ‘ S ft P EOS J
8 7 ° T 0
m b
_- — —10
r “720
I a
I
- — —30
1 . l . . . . l L . .LLIJ . L l I . L4 1 . . 1 —40
—20 ——10 0 10 20

X Direction (fm)

Figure 2.17: Calculated density contours. Numbers are normalized densities: p/po.

45

Au on Au @ 200 MeV/A, b=2.0 fm, t=10.0 fm/c

 

 

 

 

 

 

-20 —10 0 10 20
.ﬁ--.,....,....,....,ﬁ.
10— -
r
i SoftEOS
0.. _
—10_-
I —20~
|
I
E —30~
a i
.2 :
g —40_
(D
.5 ~
9 I
g .
a: F
11114.111..rl..llltma.l.11_4o
—20 -10 0 10 20

X Direction (fm)

Figure 2.18: Calculated density contours. Numbers are normalized densities: p/po.

46

Au on Au @ 200 MeV/A, b=2.0 fm, t=20.0 fm/c

 

   

 

 

 

 

—20 —10 0 10 20
..,.-..,....,....,-.-.,-.
1°.” 7
C SoftEOS
0.” i
_10I— _
t
I .
/I\ —20_— r
I i
I 1
’5‘ —30— 4
o I I
.2 :
45 _40 4:I#¢—¢¢Itt¢¢I¢¢ttI¢tt¢Ii:
g f—zo —10 0 10 20 j 10
5' : .
S ; Soft—PEOS _‘ 0
0 . .
CD
1
0.3 0'5 '
- --10
‘
q
- £9; ~20
V 09
1
- 0.1 *-—30
J
1.41....1.-..1...41....1..1_4O
—20 —10 0 10 20

X Direction (fm)

Figure 2.19: Calculated density contours. Numbers are normalized densities: p/po.

47

Au on Au @ 200 MeV/A, b=2.0 fm, t=30.0 fm/c

 

 

 

 

 

 

 

-20 -10 0 10 20
..I..=.I....I....I....I..
10" "
L Soft E08 1
0' i
. 1
u 4
-10_” j
. 1
I —20~ I
I I .
I .
’{3‘ ‘30? 4
g .
"-0 F 1
.8 _40 AI : t .L: I t t 4 t I t : : t I : : : : I t t
a F -20 -10 0 10 20 f 10
5 : .
g L Soft—P EOS _‘ 0
m I I
: 1
- n -10
t 1
.- n -20
- r —30
P L. l . . 1 . l r . . . l . I . . l . . . 11. . 1 _40
—20 —10 0 10 20

X Direction (fm)

Figure 2.20: Calculated density contours. Numbers are normalized densities: p/po.

48

Au on Au @ 400 MeV/A, b=2.0 fm, t=1.0 fm/c

 

 

 

 

 

 

-—20 -10 0 10 20
..I....I..-.I....I....I..
10- -
C Soft EOS
I
O:- _‘
_10-— _.
I
/|\ —20-— -
I .
| 4
’g‘ —30-— —
o i I
.2 ' 1
8 -40 :I: t
o) " —20 ‘“ 10
.b 1 *
Q :
g r
a) ,— _ 0
m .
f - -10
' ‘1—20
L 1.5 {160 _1 —30
0.1 (”\d 0.5 I
i .11 . J Ll . . . . l . i . i l i . i . l . i _40
—20 -10 0 10 20

X Direction (fm)

Figure 2.21: Calculated density contours. Numbers are normalized densities: p/po.

49

Au on Au @ 400 MeV/A, b=2.0 fm, t=30.0 fm/c

 

 

 

 

 

 

-20 -1O 0 10 20
III....I...-I....I....I..
10" “I
L SoftEOS ‘
0‘ i
’ 0.9 ‘
p 1 4
- '1‘ «
-10_— @12 1
/\ —20; ;
I . 0.1 0.5 03 .
|
’g‘ -30— -
c:
.2 - 1
*5 _40 :kI+4.::I¢t¢¢I¢:¢:I+t:+ItI
2 - —20 --10 0 10 20 ‘: 10
a , :
E - ,
o _ Soft—PEOS _ 0
a) . .
m 0.3
' “—10
— ‘I—ZO
“ “—30
“"“"““"“““‘"‘*1—40
-20 —1O 0 10 20

X Direction (fm)

Figure 2.22: Calculated density contours. Numbers are normalized densities: p/po.

__..‘-

50

Au on Au @ 400 MeV/A, b=2.0 fm, t=35.0 fm/c

 

   

 

 

 

 

—20 —10 0 10 20
-.,..-.,....,....,.-..,.-
10— -
I SoftEOS
O._ .—
C
-10I— _
I .
/I\ -20.* r
I
l
’g‘ -30— -
a I
.2 1
+5 _40¢+IH1+IﬁﬁIttttIttttItt
3 f—20 —10 0 10 20 j 10
a _ i
_ 0.3 .
g _ 0-5 Soft—P EOS _ 0
m . .
L
0.1 I
- ——20
1
— --30
..1....1....1....1....1.n1_4O
—20 —10 0 10 20

X Direction (fm)

Figure 2.23: Calculated density contours. Numbers are normalized densities: p/po.

Beam Direction (fm) ———>

Figure 2.24:

51

Au on Au @ 400 MeV/A, b=2.0 fm, t=55.0 fm/c

10

-10

—20

—30

-4O

Calculated density contours. Numbers are normalized densities: p/po.

 

 

 

 

 

-20 -10 O 10 20
-.I..ftj....I....I.ﬁ.I..
L .
1 Soft EOS :
L. 4
I 0.1 1
L .1
. 0.3 .

0.3 .
L l L l L L L l L l L L L l L
4 I . pf - I . I . . . I .
b -20 -10 O 10 20 ‘
Soft-P EOS I
— o
: 0.1 :
r 0.1 -
i 0.3
I .
L
L L l L L L L l L L L4 1 L L L L l L L L L 1 L4
-20 —10 O 10 20

X Direction (fm)

10

—10

—20

—30

1 -40

"ﬂ

52

2.3.2 Sphericity Analysis

A useful means of characterizing the shape of the ﬁnal-state momentum-space dis-
tribution is Sphericity analysis. Here we follow closely the works of Gyulassy, et

al.,[Gyu82], Gutbrod, et al.,[GthO] and Danielewicz [Dan95].

A weighted ﬂow tensor is calculated from the ﬁnal—state momenta:

F..- = Z Loo-(aw) (2.3)

u=l
where the sum is taken over all N fragments each of which possess an index 11. The
weight w” is given as 271;, where mm is the mass of the fragment. In the analysis
the tensor is diagonalized and the Eigen vectors are extracted. The largest of the
three will correspond to the major axis of the momentum ellipsoid. Its length f3 is
the square of the major radius, and its direction deﬁnes the ﬂow angle for the event.
The remaining two vectors deﬁne the size and orientation of the minor axes. Their
directions can be used to deﬁne the reaction plane in the analysis of experimental
data. However, since we possess a priori knowledge of the reaction plane in the BUU

model, they simply serve to describe the oblateness of the ellipsoid. Total kinetic

energy is the trace of the diagonalized tensor: f1 + f2 + f3.

Figures 2.25 and 2.26 display the results of the Sphericity analysis for 197Au on
197Au collisions at 200 MeV/ A and 400 MeV/ A, respectively. The abscissa is the ﬂow
angle and the ordinate is the ratio, f3/f1,2, called the kinetic flow ratios of the events.

The number beside each point is the impact parameter in fm/c.

In Figure 2.25 the trajectories of the two systems, one momentum-dependent and
the other momentum-independent, are well separated for the larger impact parame-

ters. It is little surprise that this separation is accompanied by a difference in the ﬂow

,5

“*0 14

53

values (see Figure 2.12). Note that for the lower impact parameters the momentum-
dependent calculations yield a ﬁnal-state momentum distribution of protons that is

more spherical than that of the momentum-independent calculations.

The picture is not so simple; one cannot explain the differences in flow values
using only Sphericity arguments. It is true that the more spherical the momentum
distribution, the lower the ﬂow value, but the orientation of the momentum ellipsoid
plays a part as well. And while the momentum-dependent calculations Show a more
spherical momentum distribution, they also have higher ﬂow angles. Thus there are

two competing event characteristics: the flow angle, and the Sphericity.

Figure 2.26 shows the same calculations, but at 400 MeV/ A. Here the trajectories
are more alike. The points from the momentum-dependent calculations and the
momentum-independent calculations are closer to one another for the lower impact
parameters than those in the previous Figure. Returning to Figure 2.12 one can see
that the two systems exhibit similar ﬂow values until an impact parameter of 4fm.

Thus the ﬂow enhancement appears energy dependent.

'.a

54

Flow Angle Vs. Flow Ratio
Au + Au, E=200 MeV/A

90.0 f 0 . l

 

 

80.0 1
70.0 _
60.0 1 j

. H Soft Momentum Independent EOS I
50 0 _ I—I Soft Momentum Dependent EOS _

Flow Angle (degrees)

 

 

 

 

O'Q l 310 1 Q

Flow Ratio

Figure 2.25: Flow angle versus ﬂow ratio f3/f1,2 for protons. Squares are for soft
(K=200 MeV) equation of state without momentum dependence. Circles are for soft
momentum-dependent equation of state. Numerals beside the points indicate the
impact parameter in fm.

"‘1

.r --

 

55

2.3.3 Total In-Plane, Transverse Momentum Versus
Rapidity

Flow is deﬁned as

 

where (pz) is
N01) 11
Eve) 1)..
N (y) ’

N (y) is the number of fragments (protons in this work) with rapidity y, and u(y) is

the particle index. Rapidity is deﬁned as

_1 E"+p‘,’

where

 

E” = (Mp: + 12522; + p522: + mi-

The z-direction is usually taken to correspond to the direction of the beam, the x-
direction corresponds to the impact parameter vector, and the Speed of light is taken
as unity. This often-used parameter offers the convenience of simplifying velocity
addition for relativistic systems. Once fragments are assigned their rapidities relative
to an inertial frame, their velocities relative to another inertial frame can be calcu-
lated by treating their rapidities and the rapidity of the inertial frames as Galilean
velocities—a much more intuitive operation. Their correct velocities relative to the

new inertial frame are then obtained by inverting their new rapidities.

During discussions with PawelDanielewicz, he suggested that the distribution of

the total in—plane, transverse momentum in rapidity may be more sensitive to the

56

Flow Angle Vs. Flow Ratio

Au + Au, E=400 MeV/A
0 1

 

90.0
80.0 A
70.0 a

60.0 _
, O—C Soft Momentum Independent EOS

50.0 r H Soft Momentum Dependent EOS

Flow Angle (degrees)

c—d

I
.I

7 .
0Q? 1 3:0 ‘ Q

Flow Ratio

 

 

 

 

Figure 2.26: Flow angle versus flow ratio f3/f1,2 for protons. Squares are for soft
(K=200 MeV) equation of state without momentum dependence. Circles are for soft
momentum-dependent equation of state. Numerals beside the points indicate the
impact parameter in fm.

57

ﬂow enhancement seen in momentum-independent calculations. This observable can

be written as

N (y)

00) = 2p; (2.5)

110/)

Figures 2 .27 and 2.28 Show the total transverse in—plane momentum for 200 MeV / A
197Au on 197Au collisions. Two curves are shown in each panel of each Figure: one
for the soft momentum-independent mean ﬁeld, and one for the soft momentum-
dependent mean ﬁeld. Figure 2.27, upper panel, shows 0(y) oscillating around zero
for central collisions. At an impact parameter of 2 fm, Figure 2.27, lower panel, the ob-
servable differentiates between the momentum-dependent mean ﬁeld calculations and
the momentum-independent mean ﬁeld calculations. In the lower panel of Figure 2.27
near mid-rapidity, the SIOpe of the momentum-independent curve is larger than the
slope of the momentum-dependent curve. The slopes of the two curves are almost
identical in appearance for an impact parameter of 4fm, Figure 2.28, upper panel.
However, the lower panel shows the mid-rapidity slope of the momentum-dependent
curve exceeding that of the momentum-independent curve. All of this correlates well

with the trends in the calculated flow values of Figure 2.12, lower panel.

hr- --*:'_--.--"~;

58

Total Transverse Momentum Vs. Rapidity Ratio
Au + Au, 5:200 MeV/A, b=0 fm

 

 

400.0 . . . A . I A
200.0 — ~
0.0 --_—~ ALA—L LnnA'”:L-L ﬁlleﬁ J11“. will;T .LAL-L “A

Transverse Momentum (MeV/C)

 

 

 

 

 

 

 

 

 

—200.0 s 2
400.0 L l L l L l L
-2.0 - l .0 0.0 1.0 2.0
Rapidity Ratio (ymm/ym)
Au + Au, E=200 MeV/A, b=2 fm
400.0 . L . T . L
—— Soft Momentum Independent EOS
------------ Soft Momentum Dependent 1308

Q 200.0 ~ ~
>
0
E
E
a
g 0.0
o
E
D
E
D
3.
§ -200.0 I ~
[—

400.0 L l L l J L L

-2.0 —l .0 O. 0 1.0 2.0

Rapidity Ratio (ymm/ ybum)

Figure 2.27: Total transverse momentum as a function of rapidity-ratio.

59

Total Transverse Momentum Vs. Rapidity Ratio

Au + Au, E=200 MeV/A. b=4 fm

 

 

 

 

 

 

 

 

 

 

 

 

 

400.0 . r . L I
— Soft Momentum Independent 1308
------------ Soft Momentum Dependent EOS
Q 200.0 ~ ..... r
>
D
E
E
E!
g 0.0 .......
o
2
8
B
3.
5 -200.0 - -
I-
L
—400.0 * ‘ * 1 1 ‘
—2.0 —l.0 0.0 1.0 2.0
Rapidity Ratio (me/yhm)
Au + Au, E=200 MeV/A, b=6 fm
400.0 . ﬂ . L . L -
—-- Soft Momentum Independent 1308
. ----------- Soft Momentum Dependent EOS
Q 200.0 ~ ~
>
0
g .
E
a
I:
0.0
E
o
E
0
t2
0
5'.
a —200.0 . ~
I—
—400.0 ‘ J * L ‘ ‘ =
—2.0 —l .0 0.0 1.0 2.0

Rapidity Ratio (me/thn)

Figure 2.28: Total transverse momentum as a function of rapidity-ratio.

r “71--mJ-s

60

The calculated (9(y) for 197Au on 197Au collisions at 400 MeV/ A, Figures 2.29 and
2.30, begin much the same; at zero impact parameter, 0(3)) oscillates about zero.
However, for an impact parameter of 2 fm, the mid-rapidity slopes of the curves are
very similar. For 4fm and 6fm, Figure 2.30 the momentum-dependent calculations
result in larger mid-rapidity slopes for the total in-plane, transverse momentum than
those from the momentum-dependent calculations. Again, this correlates well with

the calculated ﬂow values from Figure 2.12, upper panel.

The 200 MeV/ A calculations Show that the high- and low-rapidity peaks of 0(y)
are shifted more toward mid-rapidity for the momentum-independent mean ﬁeld than
for the momentum-dependent mean ﬁeld. This effect persists throughout the range
of impact parameters studied. The effect is not as dramatic in the 400 MeV/ A calcu-
lations. This could mean there is more overall stopping of nuclear matter in the
momentum-independent calculations than the momentum-dependent calculations,
a counter-intuitive result. However, the effect appears energy-dependent, as does
the enhancement of ﬂow in the momentum-independent calculations over ﬂow in
momentum-dependent calculations. The two are probably related, but not in a sim—

ple way.

One may write the total in—plane, transverse momentum, 0(y), as:

Fancy) : N(y)(pz)(y),

where N (y) is the number distribution in rapidity. The slope of the distribution is
then:
19 (y) _ dN(y)

dy— dy

 

(me) + WM». (2.6)

Thus, if the distribution is symmetric about mid-rapidity, which within ﬂuctuations
it is, the Slope of 0(y) is enhanced over the ﬂow signal. However, the slope of this

observable is more sensitive to acceptance cuts than is transverse ﬂow.

61

Total Transverse Momentum Vs. Rapidity Ratio

Au + Au, E=400 MeV/A, b=0 fm
500.0 e L a L = L
Soft Momentum Independent 1308
’ ------------ Soft Momentum Dependent EOS

 

 

300.0 :- 4

on ..L.
t

M Ankh—A4“;

A-.. . A n 003 ﬂnnnf’HAUmA
WUULIIWUUWH‘U

 

Transverse Momentum (MeV/C)

 

 

 

 

 

 

 

 

 

 

-100.0 * 4
,
-300.0 r r
1
-5m.0 L I L I L 1 A
-2.0 — l .0 0.0 I .0 2.0
Rapidity Ratio (Ym/Ym)
Au + Au. E=400 MeV/A, b=2 fm
500.0 . x . I . 1
Soft Momentum Independent EOS
’ ------------ Soft Momentum Dependent EOS
300.0 r A
Q
> b
0
E
E 100.0 * r
E!
r:
E
o
2 —100.0 ~ .
0
(3
“>5.
P- 4000 A E": —
t
_5000 L I L i L 4 A
-2.0 -I .0 0.0 1.0 2.0

Rapidity Ratio (ymm/ym)

Figure 2.29: Total transverse momentum as a function of rapidity-ratio.

62

Total Transverse Momentum Vs. Rapidity Ratio
Au + Au, E=400 MeV/A, b=4 fm

 

 

 

 

 

 

 

 

 

 

 

 

 

500.0 . 1 I I =
Soft Momentum Independent EOS
‘ ------------ Soft Momentum Dependent EOS
300.0 - _
Q
> F
U
E
8 100.0
3
E A
Q "71'
E
o
2 -1000 ~ -
8
E
E
I" -300.0 s ‘
L :2
_500.0 L 1 A 1 L L 4
-2.0 —l.0 0.0 1.0 2.0
Rapidity Ratio (ymm/yhm)
Au+Au.E=400MeV/A,b=6fm
500.0 . . . , . ,
—— Soft Momentum Independent EOS ._
’ ------------ Soft Momentum Dependent EOS g": 1": 1
300.0 — 1 -
Q
>
U
E
E 100.0 " ‘
B
G
E
o
2 -100.0 - ~
8
E
g 1'? :“1”
_5m.0 L I A l L 1 A
—2.0 -I .0 0.0 1.0 2.0

Rapidity Ratio (ymmlym)

Figure 2.30: Total transverse momentum as a function of rapidity-ratio.

 

63

To investigate the behavior of 0(y), one may attempt to ﬁnd an analytical expres-
sion which relates 0(y) to the characteristics of the momentum distribution. Outlined

below is such an attempt.

Analytical Exploration of 0(y)

Consider an inversion of Equation 2.4:

21_ 2 2
pz(2277)_212_3-2§=1, (2.7)
m1] m m

where 17 = tanh(y). In momentum-space constant rapidity yields hyperboloids of two
Sheets. 0(y) is then the sum of all values pm on surfaces of constant rapidity:

00;) = ] 10.9020 d8, (28)

S 6 y=const.

where the weight is

and f (13', 7", t) is the Wigner phase-space density distribution evolved under the BUU

formalism.

Figure 2.3 is typical for heavy-ion BUU collisions: the ﬁnal state momentum dis-
tributions are ellipsoidal in shape, though tilted about the 1),, axis at an angle equal
to the ﬂow angle from the Sphericity analysis. An ellipsoid rotated (film, about the

py-axis has the form:

 

2 ' 2
102 608%?“ + 3111(2'2‘”) ) + (2.9)

° 2 2
p: (“(32:10) 1.0%,...) ) +

 

 

2 . o
p cos(¢;zow) Sln(¢;zow) COS(¢;sz) Sln(¢;zow)
y + 2pxpz ( B2 — A2 : 1,

.LA. L~9hc ‘0.» 1
.L L

64

where C is the major axis, and A,B are the minor axes in the rotated system. Then
the total in-plane, transverse momentum for a distribution uniform over a rotated

ellipse can be written as:

0(y) = // pxxdf'dﬁ' (2.10)
56mm
where
6 E {pmpwpz : (2-11)
2 C05(¢ftow)2 Sin(¢fzow)2
Pa: (‘73— + T +

' 2 2
(——sm<:gw> +__ws<g;w> ).

 

 

P: + 2p 1) (COS(¢ftmu)Sin(¢fzow) _ COS(¢fzow)SiIl(¢fzow)) <

2 2 2
p2 1- n p. p
f.) E {p$)py’pz: ( ) — _ y :1}, (212)

and where x is a constant.

The surface integral of Equation 2.10 is confounded by the requirement that it
must be carried out over the intersection of the rotated momentum ellipsoid and
the surfaces of the hyperboloids of constant rapidity in momentum-space. These

intersections, surfaces really, take the form:

 

2 (1 - 772) (608(¢ﬂow)2 Sin(¢flow)2) Sin(¢flow)2 COS(¢ftow)2
z + —— + —— + —— +
[ 772 A2 32 A2 32 (213)

 

which does not have a closed-form solution.

65

A uniform density within the ellipse is admittedly artiﬁcial. A more natural

approach would be to use a Gaussian-ellipsoidal distribution:

2 _. 2 2

pm py p2
_ __ , .4
g(p‘)—rcexp(20%)exp(203)exp(2 3) (21)

Surfaces of constant probability density are ellipsoidal.

 

 

With this density 0(y) may be written:

0(y) = / mo?) as. (2.15)

865’)

Here the integral may be transformed into:

0(y) = / p.9(p.,py,h(p.,py))seem/4, (2.16)

A61); —pyplane

where

 

 

(m2 + 193 + I)§,)172
h(Px,Py) : i\/ 1__ 772 a

and 7 is the angle between the outwardly directed unit normal to the surface of one
hyperboloid—constant rapidity-and the z-direction. A solution to this integral is not

attempted here.

Numerical Analysis of (9(y)

In an effort to understand better the behavior of the total in-plane transverse momen-
tum in rapidity in the absence of a closed-form relationship between the observable

and hypothetical momentum-space distributions, a numerical study is presented.

Computer programs were written to generate momentum—space ellipsoidal distri-
butions for calculation of 0(y). Several different distributions were generated, but
they all fall under two distinct classes: Constant-density ellipsoid, and Gaussian ellip-

soid. Each class was studied separately, and within each class both the total kinetic

66

energy and particle number were conserved. The goal was to explore the charac-
teristics of the total transverse in-plane momentum distribution in rapidity as the

momentum ellipsoids changed orientation and oblateness.

The total kinetic energy for each class could best be achieved numerically by ﬁrst
calculating the criteria for conservation using continuous distributions. This can be

done analytically:

1
ICE = E, ”f (p: + p: +223) g(p.,p.,p.) dp. dp. dpz, (2.17)

where g(p$, py, pz) is the number density of the protons in momentum-space, and m
is the proton mass. For a uniform distribution of protons throughout an ellipsoid the

integral becomes:

ICE 2 [ff (pi +1): +p3) xdpz dpy dpz, (2.18)
p2 p2 2
35+], +5359
where
x _ 3N
— 4ABC7r’

N is the number of protons in the ellipsoid, C is the major axis of the ellipsoid and

A, B are the minor axes of the ellipsoid. Note that ICE is rotationally invariant.

Using the transform:

~ pm:
191—]
~ P
py:‘l—;"
~ Pz

67

one can rewrite Equation 2.18 as a spherical integral:

1
[(5 = .2}; f// ABC(A215'$2 + 3213312 + 021322))!‘11’1 dﬁy (1132

17-1: 2+p-y2'i'ﬁz 2 g1

(2.19)

_ 1 47r 2 2 2
— 2m (15)ABC(A +B +C )x
N(A2+Bz+C2)
10m '

 

The calculation of the kinetic energy due to the elliptical Gaussian distribution is

simpler:

N

N

1 _ p2 _ p _ p2
K5 = %-///(p§+p§+z?§)(e 27% ””6 a’i")wdpmdpydzvz (2-20)

 

3
1 w7r§ (“——

1 4N 7r; 3 3 3
=27; .ﬁ—Waaﬂya —8_ ( ”may”:+\/":"v"z+ 0,0,0.)

1 2 2 2
= 2—mN(0'x + 0y + Uz)‘

 

68

Figures 2.31-2.35 show the total in—plane transverse momentum as a function of
rapidity. Within each panel, both kinetic energy and particle number are conserved
to within about 2%. Note the high- and low-rapidity tails of 0(y), upper panels,
compared to the more abrupt patterns from the constant density ellipsoids, lower
panels. This is to be expected since the Gaussian has essentially inﬁnite extent
in momentum-space whereas the constant density ellipsoid makes no contributions
outside of its ﬁnite volume. Notice also that in all cases the peaks in the distributions

are sharper for the Gaussian ellipsoids.

Figure 2.31 shows the variation of 0(y) with ﬂow angle for a constant kinetic

energy ratio:

A _ XL pip:
_ N N V,
f1,2 2.511733% = Zuzl pgpy

 

since by symmetry and the absence of fragments 23;, pip; = 25:, 1251);. In both
panels a pronounced shift toward mid-rapidity develops as the ellipsoids are rotated.
Simple geometrical arguments satisfy for an explanation of this behavior: Consider
the problem in two-dimensions where the constant rapidity are hyperbolas in the
px-pz plane. As the momentum ellipse is rotated from zero ﬂow angle, positive p,B
contributions to the 2 p; in the forward-going hemisphere overtake the negative pz
contributions. Note that this is not simply a projection of the ellipse onto the pz-axis

as the summation is taken over lines of constant rapidity (the hyperbolas).

Figures 2.32—2.35 show the changes in 0(y) as the shape of the momentum el-
lipsoids is changed for a given ﬂow angle. In the Gaussian ellipsoids in Figure 2.32
(upper panel) where (bﬂow =20°, the mid-rapidity slope of the curves remain sur-
prisingly constant as the kinetic energy ratio f3/f1,2 goes from 2 to 5. Note that

the constant-density ellipsoid (lower panel) shows a slight change is the slope near

69

mid-rapidity. This suggests that

d0(y)
7?— (2.21)

31:0

is relatively insensitive to the oblateness of the ellipsoid for small flow angles. The
same can be said, though the effect is less dramatic, for the Gaussian distribution for
a ﬂow angle of 40°, Figure 2.33. In both Figures the peaks in the distribution shift
only slightly. The value of Equation 2.21 seems more sensitive to the oblateness of
the constant-density ellipsoid. However, the constant-density ellipsoidal distribution
is un—physical. The Gaussian distribution is more realistic, and one should give more

weight to results derived from it.

For larger ﬂow angles, Figures 2.34 and 2.35 show Equation 2.21 nicely distin-
guishes among the various ellipsoids for both classes of distributions. The peaks,

however, appear over nearly the same rapidity.

i" "’""""’"“ '1

70

 

 

 

 

 

 

 

 

 

—0.40 -0.20 0.00 0.20 0.40

Figure 2.31: Total in-plane, transverse momentum pm in arbitrary units as a func-
tion of rapidity from hypothetical ellipsoids. The upper panel shows results from a
Gaussian-ellipsoidal density, whereas the lower panel are results from the uniform
ellipsoidal density. Each panel shows 0(y) of ellipsoids which have kinetic energy
ratios of 5, but rotated various angles 43,10,” relative to the beam axis.

71

 

 

 

 

 

 

 

 

l

-0.40 -0.20 0.00 0.20 0.40

 

Figure 2.32: Total in—plane, transverse momentum pm in arbitrary units as a func-
tion of rapidity from hypothetical ellipsoids. The upper panel shows results from a
Gaussian-ellipsoidal density, whereas the lower panel are results from the uniform
ellipsoidal density. Each panel shows 0(y) of ellipsoids rotated various angles (ﬁlm,
relative to the beam axis, but which possess identical kinetic energy ratios.

72

 

 

“— fz/fm = 2 r14 IL L
......... f3/fm = 3 Hr”; -

 

 

 

 

 

 

 

—0.40

—0.20 0.00 0.20

0.40

Figure 2.33: Total in-plane, transverse momentum pan in arbitrary units as a func-
tion of rapidity from hypothetical ellipsoids. The upper panel shows results from a
Gaussian-ellipsoidal density, whereas the lower panel are results from the uniform
ellipsoidal density. Each panel shows 0(y) of ellipsoids rotated various angles (pm,
relative to the beam axis, but which possess identical kinetic energy ratios.

73

 

_ f3/f1.2=2
""""" falflz = 3

~ --- f,/f,',=4
—— f/f =5

31.2

 

 

 

 

 

 

 

l

 

—0.40 —0.20

0.20 0.40

Figure 2.34: Total in-plane, transverse momentum p; in arbitrary units as a func-
tion of rapidity from hypothetical ellipsoids. The upper panel shows results from a
Gaussian-ellipsoidal density, whereas the lower panel are results from the uniform
ellipsoidal density. Each panel shows 0(y) of ellipsoids rotated various angles gbﬂow
relative to the beam axis, but which possess identical kinetic energy ratios.

74

 

Px Vs. Y
¢flow = 800
x r ' I r
— f3/fL2 = 2 F I 1'
--------- 13/13, = 3 J g: h
“‘ falfm=4 7|,” “lel
b — — f3/fl.2=5 IIJ ' '~L _

 

 

 

 

 

 

 

 

l . l 1 l

—0.40 -0.20 0.00 0.20 0.40

 

Figure 2.35: Total in-plane, transverse momentum pa, in arbitrary units as a func-
tion of rapidity from hypothetical ellipsoids. The upper panel shows results from a
Gaussian-ellipsoidal density, whereas the lower panel are results from the uniform
ellipsoidal density. Each panel shows 0(y) of ellipsoids rotated various angles ¢flow
relative to the beam axis, but which possess identical kinetic energy ratios.

75

Summary on 0(y)

The total in—plane, transverse momentum as a function of rapidity may be a useful
observable in the study of near-central heavy-ion collisions. In the numerical study
the slope of the observable near mid-rapidity shows some sensitivity to the shape
of the momentum ellipsoid at large ﬂow angles and its orientation. In addition,
Equation 2.21 correlates well with the traditional ﬂow values. The shifting of the
peaks appears to be more sensitive to changes in the ﬂow angle than changes in
the shape of the momentum ellipsoid. Nevertheless, as the flow angle increases for
constant ﬂow ratio, the peaks shift toward mid-rapidity. When the ﬂow angles are
small and held constant while the ﬂow ratio is increased, the peaks shift away from
mid-rapidity. While Equation 2.21 appears to be more sensitive to the particle yield

near mid-rapidity, it will also be more sensitive to detector cuts.

2.3.4 Concluding Remarks

One expects the stiffer mean-ﬁeld to produce more ﬂow at all impact parameters.
However, an unexpected enhancement of the ﬂow signal at low-rapidities from 1”An
on 197Au collisions at 200 MeV/ A was found in BUU calculations using momentum-
independent mean-ﬁelds. This effect appears energy dependent and practically dis-
appears by 400 MeV/ A. The density proﬁles from time t=1fm/c to t=60 fm/c were
calculated for both momentum-dependent and momentum-independent mean ﬁelds
at collision energies of 200 MeV / A and 400 MeV / A. These proﬁles provided important

clues to the dynamics underlying this unexpected enhancement in ﬂow.

The shape of the momentum-space distribution is responsible for the suppression
of flow in the momentum-dependent calculations—this in spite of their larger ﬂow an-

gles. Generally the larger the ﬂow angle, the larger the ﬂow signal. In addition, the

76

more oblate the momentum distribution for a given non-zero flow angle, the larger
the ﬂow signal. Thus there is a competition between the oblateness and the ﬂow an-
gle. In the momentum-dependent calculations it is the oblateness of the distribution
that carries the day for low-impact parameters. At higher impact parameters the
momentum-dependent calculations still show a more spherical momentum distribu-
tion at a given impact parameter. However the higher flow angles overcome this. The

end result is higher ﬂow at the larger impact parameters.

It should be noted that this study represents a considerable challenge for experi-
mentalists. Of primary concern are the detector acceptances and their effect on the
observables and the precision of their impact-parameter discrimination. Neither are

trivial.

2.4 Balance Energy

Flow in lower energy collisions is often negative. That is, the dynamics are attractive
overall. The collisions are not violent enough and / or numerous enough and densities
are not high enough to overcome the attractive term of the nuclear mean ﬁeld. As
energy is increased the flow signal gradually becomes positive. The point in the ﬂow
excitation function where it crosses zero is called the balance energy[Ogi90, Kr092,
We393, Kla93]. The higher the balance energy for a given system, the more attractive
the dynamics. Experimentally one cannot easily distinguish between attractive ﬂow
and repulsive flow. For experimental studies the ﬂow signal will approach zero at the
balance energy. Often the ﬂow data for beam energies less than the balance energy

is reﬂected across the abscissa to indicate that ﬂow there is negative.

Changes in the model parameters of the BUU model affect the balance energy.

The “stiffer” —higher compressibility—the mean ﬁeld the lower the balance energy, the

77

Balance Energy Vs. Reduced Impact Parameter

 

 
   

 

 

 

58Fe + “Fe
100.0 I I I I ' I ' I I r I I ' 7
t.
90.0 f -
H K = 380 MeV, No Coulomb
G—-€ K = 380 MeV, Coulomb
:E _
S 80.0 T
0
E J
g3 70.0 - —
c:
m r
8
E: 60.0 L ,
d
m L
0/6‘\e”'—/’e "j
50.0 - a
40.0 i l 1 l i l r l . l i l i l 4
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
b/bmam

Figure 2.36: Balance energies for 58Fe + 58Fe from BUU model with and without the
Coulomb ﬁeld. Stiff mean ﬁeld and vacuum nucleon cross sections were used.

smaller the nucleon cross section, the higher the balance energy, the more central the
collision, the lower the balance energy. One may thus use the balance energy to learn
about the role of the nuclear mean-ﬁeld and in-medium corrections to the nucleon
cross section[Ogi90, Kr092, We893, Kla93]. In addition, the mass of the system affects
the balance energy. Generally, the higher the mass of the system the lower the balance

energy[WesQB].

Figure 2.36 illustrates the suppression of the balance energy from the Coulomb
ﬁeld. The term “Reduced Impact Parameter” (b/bmax) is the impact parameter of

the collision divided by the maximum impact parameter (the sum of the radii of

78

Balance Energy Vs. Reduced Impact Parameter

 

 

 

$8Zn + 58Zn
100.0 I I I I ﬂ I I ﬂ I I ' I I I
90.0 ~ a
H K=200MeV, (1:0.0
G—O K = 200 MeV, 0t=—0.3
2‘ _ _
3 80.0
D
E
>
9.9 70.0 - ~
8
m
8
E 60.0 — _
(U
m ./.\.—//_.
50.0 - _

 

0.40 0.45 0.50 0.55 0.60
b/b

max

.0 r l L l J
0.20 0.25 0.30 0.35

Figure 2.37: Balance energies for 58Zn + 58Zn from BUU model with and without

the Coulomb ﬁeld. Soft mean ﬁeld was used and in—medium nucleon corrections as
indicated.

79

the nuclei in the collision). At every impact parameter, the balance energy is lower,

indicating a generally more repulsive collision, for 58Fe + 58Fe collisions.

The effect of in-medium corrections on the balance energy is shown in Figure 2.37.
Here the nucleon cross section is reduced according to Klakow,et al.[Kla93], Equa-

tion 1.9:

can = Ufree(1+ 03—),
P0

where am, is the free nucleon cross section. The effect is clear: reducing the cross
section raises the balance energy at all impact parameters. This is not surprising since
hard scattering is responsible for a great deal of the repulsive nature of collisions at

these energies.

2.4.1 Effects of Iso-Spin and In-Medium Corrections
on the Balance Energy

It is known that the inter-nucleon potential is iso-spin dependent. That is, it depends,
in part, upon whether the nucleon is a proton or a neutron. The T21 state is unbound

for the two-nucleon system, but the state T=0 (deuteron) is bound, if weakly.

Work by Miiller and Serot[Mii195] has shown that the liquid-gas phase transition
may in fact be of second order rather than ﬁrst order. Their work, based upon a ther-
modynamic approach, demonstrated that local chemical instabilities are responsible
for the transition when an asymmetric potential is used. This contrasts with the com-
mon view that mechanical instabilities are responsible for the onset of fragmentation
in excited, diffuse systems. The new experimental facilities coming on—line which will
probe heavy-ion dynamics near the drip lines affords an opportunity to explore these

results.

In stellar evolution the softening of the compressibility of nuclear matter as

80

neutron ratios deviate from 1 / 2 is critical for generating a supernova explosion for
massive stars (12 MG to 15 M®)[Bar85]. Core collapse cannot sustain a shock if the
nuclear EOS retains its equilibrium stiffness; the result is at least a delay of such

important processes as nucleosynthesis.

Neutron stars cool by emitting neutrinos. After an initial fast-cooling phase,
they enter a more sedate cooling epoch producing neutrinos via the modiﬁed URCA

process:

(mp) + p + e" —+ (mp) + n + 11.. (2.22)

(n,p)+n—>(n,p)+p+e"+178.

However, if the proton density exceeds a critical value between 11% and 15%, which
is dependent upon the symmetry energy, the direct URCA process can occur and

become the dominant cooling mechanism[Lat91]:

n ——> p + e" + 173, (2.23)

p+e‘—>n+1/e.

The role of iso—spin in collective observables stems from two sources: an asymmetry
in the nuclear mean ﬁeld, and an asymmetry in the nucleon cross section. The latter is
automatically included via the parameterization of the Particle Data Group[PDG88].

The role of the former is studied below.

The effects of an iso-spin asymmetry and in-medium corrections to the nucleon
collision cross section on the balance energy were recently explored. The BUU model
for intermediate energy heavy-ion collisions is used with mean ﬁelds featuring iso-
spin asymmetry to calculate the balance energies of 58Fe + 58Fe and “M + 58Ni for
a range of impact parameters. We ﬁnd better agreement with experiment with these

mean ﬁelds and a reduction in the in-medium nucleon cross section. Unfortunately,

81

we fail to ﬁnd signiﬁcant differences in the balance energies between the 58F e + 58F e
calculations and the 58Ni + 58Ni calculations. This is at variance with recent work

from R. Pak, Bao—An Li, et al.[Pak97].

In this study, we choose two mean ﬁelds: One recently used by Bao-An Li, et

al.[Li95, Li96a] (also see Ref. [Tsa89, Dan92, Li93]):

U:A(£) +B(—p—) +07. (”7p”), (2.24)
P0 P0 .00

where p0 is the normal nuclear density, pn is the neutron density, pp is the proton

 

density, and T, is the iso—spin factor which is 1 for neutrons and —1 for protons.
Coefﬁcients A, B and a are typically chosen to match the ground state pr0perties
of symmetric nuclear matter such as the saturation density and saturation binding
energy. The compressibility is a free parameter, and in this study we choose it to
be 200MeV. This gives A = -109MeV, B = 82MeV, and 0 = g. In keeping with
Bao—An Li’s work, C = 32MeV.

The other mean ﬁeld is derived from a Hamiltonian due to Sobotka[Sob94]:

40
71= 1C8 +- pa 32(1),. +bpnpp+pp )+ W —(po pp+pnpp )- (2-25)

Here the coefficients a = —3.66MeV, b = 15.0 and c = 23.4MeV and ICE is the kinetic

energy term. Neutrons are acted upon by:

Unzaﬂz8a—+4a b-p—”+ 8cp"p2——’i+4cp—”:, (2.26)
apn p0 p0 p0 p0

whereas protons are affected by:

U=ﬂ=8apl+4a +8cp"p"+4c-p—’:. 2.27
P

app po Po pa po

 

bp_n

In symmetric matter, this mean ﬁeld reduces to the often-called “Stiff” equation of

state (compressibility 380MeV). It should be noted that there is no transformation

82

of the Sobotka mean ﬁeld that will yield an equivalent “Soft” equation of state.

Thus direct comparisons between the two formulations are impossible. One may

simply alter A, B and a in Equation 2.24 to give the appropriate compressibility in

symmetric matter.

With
5 = 811’.
Po
and
~ _ P
p _ _
Po

Equations 2.24 and 2.26 can be written as:
U = Aﬁ+ Bﬁ" + C'rzd
and
U, = [5(4a + 2ab) + 6(4a — 2ab) — 062 + 3c,62 — 206,6,
for the neutron potential, and

Up = [5(4a + 2ab) + 6(2ab — 4a) — C52 + 3062 + 2am,

(2.28)

(2.29)

(2.30)

(2.31)

(2.32)

where a, b and c are as deﬁned above. Figure 2.38 shows Equation 2.30 as neutron

excess and normalized nuclear density are varied, left-hand plot, and Equation 2.31

as neutron excess and normalized nuclear density are varied, right-hand plot. One

can see in the Figure that Equation 2.31 is more attractive to neutrons in the midst

of proton-rich matter for all densities than Equation 2.30.

The nucleon-nucleon cross sections are parameterization from the Particle Data

Group[PDG88] with medium modiﬁcation implemented according to Equation 1.9:

0,”, = 0£f6(1+a 3),

.00

um

I

83

 

Figure 2.38: Mean ﬁelds due to Bao-An Li, et al.[Li95, Li96a], left panel, and
Sobotka[Sob94]. 6 is deﬁned in Equation 2.30, and is often referred to as the neu-
tron excess. 5 is the normalized symmetric density p/po.

where a is varied between -1 and 0, p0 is normal nuclear matter density and where
p is the nuclear matter density in the neighborhood of the collision[Kla93, Alm95].

This is a parameterization of the Pauli-blocking of intermediate states in Biickner

G-matrix theory[Brii55].

We begin by calculating the proton freeze-out (recall that “freeze-out” refers to
the point in the evolution beyond which few hard collisions remain) phase-space
distribution at reduced impact parameters 6 = b/bmax = 0.275, 0.375, 0.475 and 0.575,
beam energies 50,60, 70, 80,90 and 100 MeV/A, using Equations 2.24 and 2.26 and
various in-medium corrections 0. This is followed by calculating the flow for each

permutation, and ﬁnally the balance energy for each 5.

It has been shown that an in-medium reduction of about 20% tends to best re-
produce the ﬂow signals of symmetric collisions for a broad range of energies[Ogi90,

Kla93, We893]. Previous work[Ogi90, Kla93, Wes93] showed the BUU model

84

consistently under-predicted the balance energies of various systems. R. Pak,

et al.[Pak97] found BUU under-predicted the balance energies of 58Fe + 58Fe and 58Ni
+ 58Ni collisions. We introduce this correction to the nucleon cross section to our
calculations in anticipation of raising the balance energies at all impact parameters.
Figure 2.39 shows the effect of the in-medium correction at = —0.3 for BUU calcu-
lations using the mean ﬁeld from Equation 2.24. Notice that the effect diminishes
as the impact parameter increases. It can be argued that that this is due to the
diminishing role collisions tend to play in the overall dynamics of the nucleus-nucleus
collision. The calculation also suggests a contraction of the difference in balance en—
ergies between the relatively neutron-rich 58Fe and the relatively neutron-poor 58Ni,
although there is a widening at the highest impact parameters. At near central colli-
sions, the combination of in-medium reduction of the cross section and the inclusion
of the asymmetry energy in the mean ﬁeld better reproduces the experimental data
in Figure 2.40. However, the model still tends to under predict the experimental data

for more peripheral collisions.

The trend of the balance energies is more linear, suggesting that the loss of re-
pulsive dynamics from fewer collisions in the dense regions of the collision volume is
partially compensated by an additional repulsive contribution from the mean ﬁeld.
Since both species studied here are slightly neutron rich, then, as the density is driven
higher by ever smaller cross sections (remember for these calculations the cross sec-
tion is 30% reduced at nominal density and is smaller still at higher densities), the
asymmetry term, along with the usual Skyrme terms, in Equation 2.24 grow ever
more repulsive. This would tend to reduce the upward surge of balance energies at
higher impact parameters. Any further reduction of the cross sections would likely

result in a further divergence in this trend from experiment.

Figure 2.42 compares BUU calculations using the stiff mean ﬁeld without

I;

85

Balance Energy Vs. Reduced Impact Parameter

 

 

 

 

 

 

100.0 e a

’E

O

2

8

a 80.0 — q
>

o

E

as I
b

= -l
[LI

8

5 60.0 ~ d
'3

m ....................

400 I I L l ‘ 1 i 1 r
0.20 0.30 0.40 0.50 0.60 0.70
b/bm

Figure 2.39: Balance energy versus reduced impact parameter for 58Fe + 58Fe, solid
points, and 58Ni + 58Ni, open points. Diamonds are BUU calculations using Equa-
tion 2.24 with a 30% in medium reduction factor, a, for the nucleon cross section.
Circles are BUU calculations[Pak97] with the same mean ﬁeld but with the vacuum
nucleon cross section.

86

Balance Energy Vs. Reduced Impact Parameter

 

 

 

 

 

 

100.0 a , 3.-
l.
I;
”a
0 4
2
8
g 80.0 ~ ~
>
g
8
G
LL]
2 60.0 ~ +
75
m ,
L
40.0 i l I l r L L l I
0.20 0.30 0.40 0.50 0.60 0.70
b/bm

Figure 2.40: Balance energy versus reduced impact parameter for 58Fe + 58Fe, solid
points, and 58Ni + 58Ni, open points. Squares are empirical data[Pak97], diamonds

are from calculations using the BUU model with Equation 2.24 as a mean ﬁeld and
an in-medium correction factor a = —0.3.

87

asymmetry corrections and an in-medium reduction factor of 20% to experiment[Pak97].
Here too we ﬁnd a collapse of the 58Fe balance energies onto the 58Ni balance energies
for low impact parameters, and a progressive separation at higher impact parame-
ters. That this occurs in calculations using Equation 2.24 and those using the plain,
stiff mean ﬁeld could indicate a weakness of the overall effect the asymmetry terms
in Equation 2.24. However, in asymmetric matter the asymmetry term makes the
dynamics more repulsive overall. Thus, Bao-An Li’s mean ﬁeld, which is just the
soft mean ﬁeld plus a linear asymmetry term, behaves more like the stiff mean ﬁeld

without the asymmetry correction.

Figure 2.41 compares the balance energies calculated using the Sobotka mean ﬁeld
and an in—medium correction factor, a = —0.2, with experimentally measured[Pak97]
balance energies. At low impact parameters the BUU calculations perform well.
However, there is a curious ﬂip-ﬂop in the calculated balance energies as impact
parameters increase. At the lowest impact parameter, the BUU calculations are well
within the reported experimental uncertainties. The ordering switches at f) = 0.375;
the 58N i + 58Ni balance energies are higher than 58F e + 58F e balance energies, contrary
to experiment. The ordering switches again at f) = 0.475 and remains consistent with

experiment to the end of the range of the impact parameters studied.

Figure 2.43 illustrates the differences in balance energies as a function of impact
parameter between BUU calculations using the stiff mean ﬁeld without any asymme-
try correction and BUU calculations using Equation 2.26, Sobotka’s mean ﬁeld. Note
that both sets of calculations use a = —0.2. The two pairs of curves are similar for
the higher impact parameters studied. The two pairs of curves quite close to each
other at the lower impact parameters, except for the ﬂip-ﬂop in the ordering of the
balance energies of the species in the calculations using the Sobotka mean ﬁeld. The

similarity of the calculated balance energies may diminish when species near the drip

88

Balance Energy Vs. Reduced Impact Parameter

 

 

 

 

 

100.0 ~ — g"
”a
O
.2
8
g 80.0 e —
% .
E
E
0
ﬂ
LL} 1
8
g 60.0 ~ -
7a
m L

40.0 i l 1 l A l . l .

0.20 0.30 0.40 0.50 0.60 0.70
b/bm

Figure 2.41: Balance energy versus reduced impact parameter for 58Fe + 58Fe, solid
points, and 5“Ni + 58Ni, open points. Squares are empirical data[Pak97], and dia-
monds are BUU calculations using Equation 2.26 and an in-medium correction factor,
a = —0.2.

89

Balance Energy Vs. Reduced Impact Parameter

 

 

 

 

 

 

100.0 — __ ‘
’8
O
.2
8
g 80.0 L —
>
O
E
Q
Q)
t:
m
8
g 60.0 — —
'8
an I

40.0 r l 1 l 1 l 1 l i

0.20 0.30 0.40 0.50 0.60 0.70
b/bm

Figure 2.42: Balance energy versus reduced impact parameter for 58Fe + 58Fe, solid
points, and 58Ni + 58Ni, open points. Squares are empirical data[Pak97], and dia-
monds are BUU calculations using the stiff mean ﬁeld without

asymmetry corrections and an in-medium correction factor, a = —0.2.

lines are studied. This anticipates experiments using the new radioactive beams.

There is a curious feature of the Figure in the differences between the balance
energies calculated with the mean ﬁeld from Bao-An Li and those calculated with the
mean ﬁeld from Sobotka. Notice that the differences among the 58Ni + 58Ni calcula-
tions are greatest at low impact parameters and smallest at high impact parameters.

However, the reverse is true for the 58Fe + 58Fe calculations.

90

Balance Energy Vs. Reduced Impact Parameter

 

 

 

 

 

100.0 — ~
.
e
O
2
8
g 80.0 ~ J
>
0 +
E
>x
9.”
Q)
I:
[L]
8
g 60.0 — ~
3
m
40.0 L l 1 l r l 1 l .
0.20 0.30 0.40 0.50 0.60 0.70
b/bmax

Figure 2.43: Calculated balance energies using the BUU model. Solid points are
58Fe + 58Fe and open points are 58N i + 58Ni. Diamonds represent calculations using
Equation 2.26 as the mean ﬁeld with a = —0.2, and squares represent calculations
using the stiff mean ﬁeld without asymmetry corrections and with a = —0.2.

 

91

2.5 Summary

This Chapter began with a detailed discussion of the conﬁguration-space and momentum-
space evolution typical of relativistic, heavy-ion collisions, then introduced two di-
rected, collective observables. The more common of the two, introduced by Danielewicz
and Odyniec[Dan85] and called directed transverse ﬂow, or just “ﬂow”, is the slope
at mid-rapidity of the average in-plane, transverse momentum versus rapidity. This
quantity has demonstrated a sensitivity to the directed collective dynamics of inter-

mediate energy, heavy-ion collisions.

r. a. “9.13M ‘1

However, an unexpected result was found in the comparison of ﬂow values for
momentum-dependent and momentum-independent 197Au on 197Au collisions at
200 MeV/ A. Normally momentum-dependence adds to the repulsive elements in the
evolution of heavy-ion collisions. But the momentum-dependent calculations yielded

smaller ﬂow values than those of momentum-independent calculations.

A qualitative study of the evolution of the density of these collisions was un-
dertaken. It suggested that the momentum-space distributions for the momentum-
dependent calculations may be more spherical in shape than those distributions from

momentum-independent calculations.

Sphericity analysis was then used to get a quantitative picture of this hypothe-
sis. Indeed, the momentum-dependent calculations showed a more spherical distri-
bution than the momentum-independent calculations. In addition, the momentum-
dependent distributions showed greater flow angles than those of the momentum-
independent distributions. Thus, in spite of higher ﬂow angles, which usually gener-
ally give higher ﬂow values, the lesser oblateness of the momentum-dependent distri-

butions lead to lower ﬂow values.

This leads to a proposal of an apparently new observable: the total in-plane,

92

transverse momentum versus rapidity. This quantity was studied in detail, and it
was found to be sensitive to both the orientation and the shape of the momentum
distribution. However, the slope of this observable is also more sensitive to population

distributions. This could present difﬁculties if detector cuts become obtrusive.

Finally, this Chapter carried out a study of the behavior of the balance energy
for 58Fe on 58Fe and 58Ni on 58N i using an iso-spin dependent nuclear mean ﬁeld and
in-medium corrections to the nucleon-nucleon scattering cross section. Two iso-spin
dependent mean ﬁelds were studied: one from Sobotka[Sob94] and one from the work
of Bao-An Li[Li95, Li96a]. The inclusion of in-medium corrections improved the per-

formance of both mean ﬁelds, but Sobotka’s best matched experimental work[Pak97].

The next chapter introduces an hybridization of BUU and a simple coalescence
model. A study of collective, non-directed behavior is then presented and the conse-

quences discussed.

 

Chapter 3

BUU, Coalescence and the
Radially Expanding Thermal
Model

3.1 Introduction

The collective phenomena discussed thus far have been of a directed nature. In
other words it has a deﬁnite direction in momentum-space. However, isotropic radial
expansion is common in our universe. A system’s rate of expansion can tell us about
the forces driving its evolution. It is then easy to wonder what radial expansion can

tell us about the forces driving the evolution of heavy-ion collisions.

In experiments conducted at Berkeley, Mike Lisa and the EOS-TPC
collaboration[Lis95] found evidence for a radial component and a thermal component
in the velocity proﬁles of light fragments (deuterons, tritons, 3He’s and alphas) emit-
ted near mid-rapidity. The collisions studied were Au + Au at energies ranging from
0.025 - 2GeV/A. They choose the most central events, those which exhibited the

highest multiplicities, with an estimated range of impact parameters of Ofm to 3fm.

Recall that the BUU transport model, while having been quite successful in

predicting and explaining directed phenomena of single-particle observables[Ber87,

93

94

Gale90, Wes90], possesses no self-consistent means of treating two-body forces. The
Coulomb ﬁeld is treated classically, since this is easily done with today’s comput-
ers, and hard scattering among nucleons is well modeled. A single-particle mean
ﬁeld is used to model soft processes arising from the cumulative effects of multiple
two—body interactions. However, the BUU formalism has no self-consistent means of
treating the N—body phase-space. In an effort to correct this deﬁciency in BUU, we
chose a simple coalescence algorithm. When applied to the single-particle freeze-out
phase-space distribution from BUU, the algorithm will provide the momentum-space

distribution of light fragments.

Several questions suggest themselves regarding the hybridization of BUU and a
simple coalescence model: how well, if at all, can this combination of models perform?
Can they predict or reproduce experimental data? Will they give us insight into the
inner workings of heavy-ion collisions? The subject of this chapter is a study which

seeks to answer some of these questions.

The thermalization of nuclear matter in heavy-ion collisions is not complete. That
this is so is easily seen in momentum—space scatter plots of test particles, Figures 2.3
and 2.5 of Chapter 2. The common volume of the two colliding nuclei can be well
characterized as having a single temperature. However, the spectator matter is still
quite “cold”. This matter still carries much of its original beam momentum. The
few collisions that the nucleons within these spectator regions have suffered have
meant little opportunity for the conversion of longitudinal momentum to random or
thermalized momentum. Thus, one would not expect a strong radial velocity proﬁle
to manifest from regions of high or low rapidity. However, near mid-rapidity directed
signals can be expected to give way to a radial expansion, and it is near mid-rapidity
that the EOS-TPC collaboration focus its efforts. And this is where we focus our

efforts.

95

We use a coalescence model to produce the momentum-space distribution of free
protons, deuterons, tritons, 3He and alpha particles. Then, using Xz/V minimization,
we ﬁt the resulting spectra to thermal yet radially expanding distributions to assign

a common temperature and expansion velocity for the entire collision volume.

The next section details the models and calculations we use in the study; Sec-
tions3.3 and 3.4 are presentations of our results and comparison to experimental

data, and in Section 3.5 we draw our conclusions.

3.2 Models and Calculations

The nuclear mean ﬁeld we use in this study takes the form of Equation 1.6[Wel88]:

U(p,ﬂ=A(£)+B(£) +9 d319, f(rip)2
p0 p0 p0 1+(é'2Aﬂ)

 

It can be modeled with various combinations of force parameters leading to different
compressibilities, in addition to toggling the momentum dependence. Values for the

parameters in Equation 1.6 are shown in Table 2.1.

In an effort to exclude spectator matter, fragements with momenta outside
00m = 90° i 15° relative to the beam axis are excluded. The excluded matter will
undergo directed ﬂow and likely contaminate the thermal signature of the participant
matter. We mimic this acceptence in our efforts to test BUU and simple coalescence.
In addition, our calculations concentrate on impact parameters Ofm to 3fm. Exper-
imental efforts cannot easily characterize the impact parameter of an event. We use

a geometrical weighting of

bi

Nb,- = —ij (3.1)
bi

where Nb, is the weight of events at an impact parameter b,, and Nb). is the weight of

events at an impact parameter bj. Thus, there should be 3 times as many collisions

96

calculated at an impact parameter of 31m as there are at lfm. This does imply that
there will be no events at zero fm. Central collisions are often calculated for the

insight they provide by virtue of their symmetry.

Since the EOS-TPC collaboration focuses its attention on the spectra of light
fragments in addition to that of protons, a coalescence model is needed. The algorithm
converts the freeze—out phase-space distribution of protons and neutrons into a phase-

space distribution of light fragments.

The coalescence of light fragments followed a simple, and some may say naive,
prescription: should two or more nucleons of the right species fall within a critical
radius, A R, in conﬁguration-space and a critical radius, AP, in momentum-space,
the group will constitute a fragment. The critical radius in conﬁguration-space for
heavier fragments is simply increased according to R,- o< A3”, where z' is the fragment

species, while the critical radius in momentum-space remains 100 MeV/c.

This prescription for coalescence is not original[Kru85, Nag94], although it rep-
resents something of a departure from what is more common at these energies: a

momentum-space coalescence[But63, Mek77, Kap80, Das81, Sat81, Cse86]. The suc-

cess of momentum-space coalescence at intermediate energies is well documented[Sch63,

C0877, Lem79, Jac84, Tsa88, Wan95]. The additional constraint, A R, allows one to
explore coalescence calculations where the source size exceeds that of the fragments
emitted[Bar94, Sor95]. However, source sizes at energies ~1 GeV/ A are expected to
be of the same order as the fragments emitted from them, effectively negating the
need for a constraint in conﬁguration-space[Wan95]. Indeed, in our work to ﬁx the

coalescence parameters, we ﬁnd the spectra to have little sensitivity to changes in

AR.

The coalescence model we use here produces light fragments, all at the same

97

time coordinate, propagates them to inﬁnity free of mutual interaction or interaction
with the spectator system, and without the possibility of decay from excited states
to ground states. This is equivalent to making non relativistic and sudden approx-
imations, in addition to assuming that chemical equilibrium is reached for all the
fragments, regardless of species, at the same time. It is free, however, from assump-
tions about thermal equilibrium, local or otherwise, light fragment potentials and

source sizes.

The sudden approximation is probably a good one[Mek77] and has been used
extensively in the older models. It has the advantage of being relatively simple to
code and minimizes the combinatorial burden. There are more sophisticated coa-
lescence models[Kap80, Dov91, L1095] which are less cavalier in their presumptions
about the conditions under which fragmentation, coalescence and clustering occur.
And the imposition of a coalescence “after-burner” upon the phase-space distribution
evolved using transport codes barren of strong multiparticle interaction fails to an-
swer questions regarding the role of clustering before freeze-out, though alternatives

do exist[Dan91, Dan95].

The spectra of protons, deuterons, tritons, 3He and alphas are analyzed using a
radially expanding thermal model[Sie79]. In this model the fragments are assumed to
possess a thermal velocity distribution characterized by a temperature in Maxwell-
Boltzmann statistics, and an overall radial velocity. In the global rest frame the

resultant distribution is

Q ~ exp("7E/T) [(7 + %) Shim) — 'ECOSMO) a (3.2)

 

where 7 E 1/\/1 — ,82, a a "yﬂp/T, T is temperature and ﬂ E v/c is the radial ﬂow
velocity. The spectra are ﬁtted to Equation 3.2 by ﬁxing the overall normalization and

varying T and ﬂ to obtain a minimum x2/u. Global ﬁts constitute simultaneous ﬁts to

 

98

deuterons, 3He, tritons, and alphas, since we use the proton spectrum to ﬁx the critical
radii for coalescence. Critical radii are ﬁxed by minimizing the difference between the
ﬁnal-state proton spectra from the BUU calculation and that from the EOS-TPC
study[Li895] at 1 GeV/A. We ﬁnd A Rdeuterm = 1.5fm and A P = 100 MeV/c. These
values are of the same order of magnitude as length and momentum parameters used

in other coalescence models[Nag96, Jac84].

In addition to the impact-parameter averaged study, we perform calculations prob-
ing the relative importance of the various features the BUU transport model in terms
of their effect on radial ﬂow velocity and temperature. Calculations are again of
Au + Au, but restricted to b = 2fm, a beam energy of 1GeV/ A, using various nu-
clear mean ﬁelds, with and without Coulomb ﬁelds, and various reductions in the

nucleon-nucleon cross section.

The effect of A-resonances is also studied since the decay products receive an extra
kick, and this may be visible in the proton spectra. By tagging those protons the
last interaction of which before freeze-out was a recoil from A-decay, and by removing
them from the BUU output we are able to isolate their inﬂuence on the overall proton

spectra. Our results are presented below.

3.3 Results

Our results for the impact-parameter-averaged calculations for IGeV/ A appear in
Figure 3.1. The temperature and radial ﬂow velocities are consistent with those ob-
tained in experiment[LisQS] within uncertainties, and provides good evidence that
BUU+coalescence is capable of reproducing this combination of radial and thermal
motion in light fragments. Our simultaneous ﬁt to the fragment spectra with a

nonzero radial flow velocity gives a minimum X2 /1/ of 1.3. Forcing a zero radial flow

99

velocity yields a minimum X2/l/ of 1.8, in keeping with, though not as dramatic as,
the results of the EOS-TPC study[Li395]. Without absolute cross sections, normal-

izations are free parameters in our minimization of xz/u-ﬁt.

We extend our impact-parameter-averaged investigation to energies 250 MeV/ A,
600 MeV/ A, 1.5 GeV/ A and 2 GeV/A. The results are presented in Figure 3.2. Here
one can see that results from the calculations have signiﬁcant overlap with experiment.
Temperatures extracted from BUU+coalescence calculations agree well with those
extracted from experiment, whereas our results for radial ﬂow velocity agree less
well. They do suggest a saturation of radial ﬂow velocity as beam energy is increased
beyond lGeV/ A. Other models show a saturation at higher energies[Li96b]. This is

consistent with AGS data[Har95].

The microscopic features of the BUU transport code seem to have little inﬂuence
on the radial flow velocity. We calculate the temperatures and radial ﬂow veloci-
ties for reactions with an impact parameter of 2fm and an energy of 1GeV/ A. The
results of the study are presented in Table 3.1. Immediately, one can see the insensi-
tivity of radial ﬂow velocity to the equation of state and the in-medium modiﬁcation
of the nucleon-nucleon cross section. The numerical models used in the EOS-TPC
study[Lis95] also showed the radial ﬂow velocity to have little dependence on the
equation of state (EOS) used. Indeed, we cannot discern any signiﬁcant EOS depen-
dence. Radial ﬂow will not develop within 06m = 90° :t 15° of the beam axis without
nucleon-nucleon collisions. For calculations at b = 2 fm and Ebeamzl GeV/ A that are
allowed to reach maximum compression before 0,", is set to zero, almost no baryons
obtained rapidities low enough to meet the kinematic selection criteria. However, we
do see evidence of directed ﬂow. Thus we conclude that both nuclear mean ﬁelds
and nucleon-nucleon collisions are important in the deve10pment of radial flow, and

that it is likely that they provide roughly equal contributions to radial ﬂow. We ﬁnd

100

   

 

 

 

 

 

 

 

 

 

m on
T u I ‘3 g)
5?. 3 9. 44% a
H
' 32a gun
' °$H a”
- ; "bu u"z~u
D m H
- ‘ :U a a a
.In _ m ~ cg II\ '3 "\
H --' ‘ mus-3% nur—"x
-H _ «
O ’ ‘ v-t N on
0° " ‘ 'o 'o 'c
I’ " 1-0 H q-n
" ‘ Irr FWUIIIT'I'I'I'I'T'l—I—
a ' Odo
Q30 - / -"‘
p.“ a ' °' 1
.- ‘ A
- - >
i ._ w m
:> . 1° (.9
v
8 - r
_ - 3:3
v-i * H". O 0
:1 . .°' '3
<3 _ == . .
3- :8 / '3
pm ‘ .
<1 L " . .
O-O—O '
R u
I- .1
- -‘Q
_ .°
' ‘0.
c
on v. N n
I I I I I
c o o o
H H F. F! 'I'.

N
I I
O O
VII H

ma) swap/asp

Figure 3.1: Spectra of BUU + coalescence for impact parameter averaged Au +
Au collisions with b£3 fm. The 197Au + 197Au phase-space is calculated using a
momentum-dependent stiff equation of state from Table 2.1 and the free nucleon cross
sections[PDG88]. Global temperature and radial ﬂow velocity are obtained by ﬁtting
the radially expanding thermal model[Sie79] to deuterons, tritons, 3He, and alphas
simultaneously. Dotted lines are the global ﬁts for a radial ﬂow velocity of zero.

101

 

 

 

 

 

 

 

 

I T I I I I I I
0.4% I ......... l, 4
_ l l I .
_ . -
g o.2- l -
“=- - oEos TPC data -
” OBUU ‘

0,0 : I : I, : : t I
100— _,.--"§ -4
. “q ..... é .
g - _
g _. ..
Z 50 " {I 01103 TPC data “
h i 03W 1

" .L
O l l l I l l l l
O 1 2

Ebom (A GeV)

Figure 3.2: Excitation function of radial ﬂow velocity ﬂ and apparent temperature
from BUU + coalescence for impact parameter averaged 197Au + 197Au collisions with
b33fm. The Au + An phase-space is calculated using a momentum-dependent stiff
equation of state from Table 2.1 and the free nucleon cross sections[PDG88].

102

 

EOS [ Ti5 (MeV) I 6 :l: 0.05
Coulomb, a = 0

 

 

 

 

 

 

 

 

 

 

 

 

 

Stiff 70 0.35
Soft 80 0.35
Soft 15' 75 0.35
Stiff 15' 90 0.35
No Coulomb, a = 0
Stiff 70 0.35
Soft 70 0.30
Soft if 95 0.35
Stiff 15' 95 0.35
Coulomb, a = —0.20
Stiff 75 0.35
Soft 65 0.35
Soft if 80 0.35
Stiff 13‘ 75 0.40
Coulomb, a = —0.50
Stiff 70 0.35
Soft 65 0.35
Soft 13' 70 0.35
Stiff 13' 90 0.35

 

 

 

 

Table 3.1: Effects of the microscopic features of BUU on apparent temperature and
radial ﬂow velocity.
that the magnitude of the radial ﬂow, as opposed to the total radial kinetic energy,

is chieﬂy governed by the beam energy.

In contrast, we do see striking changes in the unnormalizd kinetic energy dis—
tributions of protons and light fragments as the mean ﬁelds and in—medium cross
sections are changed. This is especially pronounced in the high energy tails of the
light fragment spectra, with a’s showing the most sensitivity. In addition, there
is some sensitivity in the temperature of the light fragment spectra to momentum

dependence in the nuclear mean ﬁeld, as well as to the inﬂuence of the Coulomb ﬁeld.

103

3.4 Temperature and Microscopic Features of BUU

There are trends in the effects of the microscopic features of BUU on the extracted
temperatures. The strongest is the addition of momentum dependence in the nuclear
mean ﬁeld. We see an increase in temperature as the momentum dependence is
switched on in the calculations. We ﬁnd the greatest increase in calculations devoid
of Coulomb ﬁelds and using the free-space values of nucleon—nucleon cross sections,

and smaller increases in those calculations which include the Coulomb ﬁelds.

The momentum-dependent terms in the mean ﬁeld are repulsive at these energies.
Thus, the addition of a repulsive mechanism should lead to lower densities and fewer
collisions. One might expect this to decrease the extracted temperature. However,
it seems that the repulsive momentum dependence tends to increase the amount of
strongly thermalized matter splashing off of the hard, dense elliptical core that forms
as maximum compression is reached. An explanation is that this matter, initially
streaming in at beam velocity, is compressed and thermalized against this core. Since
both density gradients (which by themselves offer some contribution via diffusive
mechanisms) and momentum gradients are larger in the longitudinal directions than
in the transverse directions, this matter will be ejected into the mid-rapidity regions.
The ejection of this matter competes with the reduction of the collision rate to produce
this result. This mechanism is sensitive to both the beam energy, which will set
the relative importance of the mean ﬁeld and collisions, and the impact parameter.
Geometrical arguments imply that the angle of the major axis of this hard, dense

core relative to the beam axis is strongly dependent upon the impact parameter.

There are weak trends with temperature variations and the reduction of the in-
medium nucleon-nucleon cross section. We use the prescription[Kla93]

0,”, = 0,f,',,“(1+a ﬂ)

P0

104

where three values a = 0, —0.2 and —0.5 are taken. One might expect that as the
collision cross section decreases, the amount of beam energy converted from directed
and longitudinal to random and transverse kinetic energy decreases as well. This
should manifest a lower temperature. Indeed, we ﬁnd this to be true for those cal-
culations using a soft momentum-dependent and the stiff momentum-independent
mean ﬁelds. However, we ﬁnd no discernible change when we use soft momentum-
independent mean ﬁeld, and ﬁnd a slight increase in temperature while using the stiff

momentum-dependent mean ﬁeld.

The weakness of these trends is due to the dominant role the ﬁrst few nucleon-
nucleon collisions play in the ﬁnal single-particle kinetic energy distributions. Fig-
ure 3.3 shows single-particle momentum distributions for central Au on An collisions
at 1GeV/A and using 200 test particles per nucleon. The top row of graphs show the
distributions early in the calculations, after 10 fm/c. Here the two Fermi spheres of
the initial state are clearly seen; the clouds around the origin represent the nucleons
elastically scattered in these early stages. The lower row of graphs are the distri-
butions after 30 fm/c. The kinematic cuts are represented graphically as the white
lines intersecting the origins. Within these cuts, one can see the initial collisions’
strong inﬂuence on the intermediate- and high-energy portions of the kinetic energy
distributions after 30 fm/c. As a result, the temperature becomes sensitive to the
kinematics of the initial state, namely the beam energy. The weakness of the sen-
sitivity of the apparent temperature to the in-medium cross section is shown in the
right-hand panel. Here the slopes, and thus the apparent temperature, of the kinetic

energy distributions are similar.

Figure 3.4 shows the same information as Figure 3.3, but for calculations using
a soft, momentum-independent equation of state. As the ﬁgure shows, most of the

intermediate- and high-energy portions of the ﬁnal kinetic energy distributions are

(AGO/I) Kouonbard pazrpzuuoN

v—t
5. 8.
O C
[IV

. 0.1

 

    

 

M
0 01 02 03 04 05
Kinetic Energy (GeV)

1

0

a =—O.9

 

 

 

0
PZ (GeV/c)

 

 

 

 

 

 

1 1 1.1 -N

1 111111 |

O. ” ‘” ‘7

‘F _ __ ~o
ll

5 ~ —» ~T

f. L. 1 . 1 . ‘ . 1 . 1 . 1 . ‘ (\II

= I) =

3mm 1 (o/A99)xd mos 1

Figure 3.3: The single-particle, reaction-plane momentum distributions for central Au
on Au collisions using the stiff, momentum—dependent mean ﬁeld. a is the in-medium
cross section reduction factor and the angular cuts are illustrated as white lines on
the graphs. The right—most panel is the kinetic energy distribution of the systems
after 30 fm/c for various cross section reduction factors. Solid lines are calculations
using a = 0 dotted lines or = —0.5 and dashed lines a = —0.9.

106

dominated by collisions occurring after 10 fm/c. These collisions are subject to in-
medium effects, and as a result, temperature manifests a sensitivity to a. This sen-
sitivity on a, however, is surprisingly small. We believe the nucleons that scatter
elastically to 90° are constrained by momentum and energy conservation. Thus, to
ﬁrst order these kinematic constraints are only sensitive to the beam energy. It fol-
lows that even if we drastically decrease the scattering probabilities, the nucleons
that do scatter to 90° have similar slope parameters in their energy spectrum. The
net result is that the slope parameters (temperatures) only show limited sensitivity

to the magnitude of the in-medium cross section.

That this limited sensitivity does not materialize in the light-fragment spectra
is due primarily to the imposition of our coalescence model on the single-particle
phase-space distribution upon freeze-out. Since we ﬁnd the light fragment spectra to
be relatively insensitive to the critical coalescence radii in conﬁguration-space while
sensitive to the momentum-space radius, to ﬁrst order the coalescence we use in
this study is a momentum-space coalescence. Figure 3.4 shows that a momentum-
space coalescence radius of 100 MeV/c is too large to adequately resolve the nucleon
density gradient in momentum-space. This effectively integrates out the features of
the momentum distribution that would likely lead to different global temperatures in
the light-fragment spectra as the collision cross section is modiﬁed. However, we ﬁnd

this radius to most accurately reproduce the proton kinetic energy spectra from the

EOS-TPC experiment[Li395].

Finally, to study the effects of A-decays on the ﬁnal-state proton spectra, we
calculate the spectra with and without those protons coming from A-decays. There
is concern that the recoil protons receive from the decays would contaminate the
spectra. Protons created in A-decays as a ﬁnal interaction, are unlikely to contain

information about the radial ﬂow and the temperature of the system that created

107

(AQD/I) Aouanboid pazneuuo N

0.001

 

 

 

0 0 1 0 2 0 3 0 4 0 5
Kinetic Energy (GeV)

1

0

01 =—0.9

   

 

a = —0.5
0
PZ (GeV/c)

 

 

 

 

 

 

.1.111..1.1.1.(\ll
NHOI—‘(Tlv—(o'dN

emu 01=1 (o/A99)Xd O/IIIJ os=1

Figure 3.4: The single-particle, reaction-plane momentum distributions for central
Au on Au collisions using the soft, momentum-independent mean ﬁeld. a is the in-
medium cross section reduction factor and the angular cuts are illustrated as white
lines on the graphs. The right-most panel is the kinetic energy distribution of the
systems after 30 fm/c for various cross section reduction factors. Solid lines are cal-
culations using a = 0, dotted lines a = —0.5, and dashed lines a = —0.9.

108

the A—resonances. In an effort to isolate this effect, we tag those protons the last
interaction of which is a recoil from a A decay. We see little change in the spectra

when those recoiling protons are removed. This is somewhat contrary to what is

reported by the EOS-TPC collaboration[Li395].

3.5 Concluding Remarks

A clear nonthermal component has been observed in calculated light fragment spectra
in 197Au + 197Au collisions at beam energies of 0.25, 0.60, 1.0, 1.5 and 2.0 GeV/A
with the application of a simple coalescence model to the ﬁnal state phase-space from
the BUU transport model. This hybrid successfully reproduces the observed temper-
ature and radial ﬂow velocity, within estimated uncertainties, found in light fragment
spectra in an experiment by the EOS-TPC collaboration[Li395]. Furthermore, we ﬁnd
unfortunately that the radial ﬂow velocity shows little sensitivity to the microscopic

features of the BUU model.

The global temperature extracted from the ﬁnal-state light fragment spectra
showed weak dependence on in-medium modiﬁcations to the nucleon cross section.
Lower cross sections lead to lower temperatures in conjunction with a soft, momentum-
dependent mean ﬁeld and a stiff, momentum-independent mean ﬁeld. From our cal-
culations of the single-particle momentum distributions of central 197Au on 197Au
collisions at 1 GeV/A using a stiff, momentum-dependent mean ﬁeld, we ﬁnd the
ﬁnal kinetic energy distributions to be dominated by beam kinematics. For calcula-
tions using a soft, momentum-independent mean ﬁeld, we ﬁnd the momentum-space

coalescence radius to be too coarse to resolve the in-medium effects.

We ﬁnd protons from the decay of A-resonances to have little’effect on the ﬁnal-

state proton spectra at 1GeV/ A.

Chapter 4

Summary and Outlook

4.1 Summary

The BUU formalism has proven itself successful in providing insight into the detailed
and aggregate behavior of dense, excited nuclear matter. Through the comparison
of observables such as balance energy, directed ﬂow, and radial ﬂow to experimental
values, one can estimate such bulk quantities as the compressibility of nuclear matter,
and the in—medium eﬂects on nucleon-nucleon scattering. Both of these quantities rely
heavily on the short-range parts of the nuclear force, and are thus only weakly aﬂ'ected

by the ﬁnite size of the nuclei.

The truncation in the BBGKY hierarchy[Won77, Bau86a] leads to a calculable
model. Unfortunately, because of this truncation, BUU is devoid of self—consistent
N -body interactions critical to the formation of clusters of nucleons found in exper-
iments; it does not form fragments without hybridization. At low energies the BUU
model shows none of the long range correlations among nuclei that give the shell-
like effects in nuclei. At high energies it fails to yield quark-gluon degrees of free-
dom in addition to exhibiting such un-physical behavior as super-luminal momentum

transport [Kor95a].

The model’s forte is the evolution of the single-particle, intermediate energy

109

110

Wigner phase-space distribution. And it is here BUU shines.

Directed transverse ﬂow was presented, and its dependence on BUU model pa-
rameters discussed. For years this observable has served the relativistic heavy-ion
physics community with its sensitivity to the repulsive and attractive elements in the

dynamics of heavy-ion collisions.

Momentum dependence usually enhances the repulsive elements of the dynamics
of heavy-ion reactions. However, at 200 MeV/ A 197Au + 197Au ﬂow calculations give
higher values for momentum-independent nuclear mean ﬁelds. At 400 MeV/A the
effect disappears; thus, it is energy dependent. To understand this effect, Sphericity
tensor analysis was performed to characterize the shape quantitatively and orienta-
tion of the ﬁnal-state momentum distributions. The effect was found to be caused
by a more spherical, and thus less directed, ﬁnal-state momentum distribution for
momentum-dependent mean ﬁelds. However, these distributions did exhibit higher
ﬂow angles. Momentum-independent calculations have, by contrast, more oblate,
and thus more directed, momentum distributions. However, momentum-independent

distributions did exhibit lower ﬂow angles.

A new observable, total, in-plane, transverse momentum versus rapidity was intro-
duced, and it was studied using hypothetical momentum ellipsoids of various shapes
and orientations. The sensitivity of the slope of the new observable near mid-rapidity
was discussed as well as the location of the forward and backward peaks of the ob-

servable.

At high ﬂow angles the slope is sensitive to the shape of the momentum distri-
bution. Nevertheless, given a constant shape to the momentum ellipsoid the slope is
sensitive to the ﬂow angle, large and small. Thus, there is degeneracy in the SIOpe

at high ﬂow angles; the observable will not uniquely map changes in the oblateness

111

of the ellipsoid and changes in the ﬂow angle to its mid-rapidity slope. This is not
surprising given that the ﬂow angle and the oblateness of the momentum ellipsoid

are independent of one another and the slope of 0(y) is a scalar.

The peaks of the distribution shift toward mid-rapidity while the ﬂow angle is
increased and f3/f1,2 held constant. For small ﬂow angles held constant, the peaks
shift away from mid—rapidity as f3/f1,2 is increased. However, at high ﬂow angles

held constant, the peaks show no signiﬁcant shift as f3/f1,2 varied.

The effects of iso-spin asymmetry and in-medium cross section corrections on the
balance energy were examined. Balance energy is the energy at which ﬂow values
go through zero. Recall that low-energy collisions are generally attractive, giving
negative ﬂow values. But as the energy of the collisions is increased, the repulsive

elements overcome the attractive ones, and the ﬂow turns positive.

Two iso-spin dependent mean ﬁelds were used: one from the work of Bao-An Li,
et al.[Li95, Li96a] and one from the work of Sobotka[Sob94]. R. Pak, et al.[Pak97]
showed that BUU under-predicts measured balance energies. The performance of
BUU using each of the two iso-spin dependent mean ﬁelds was improved when the
in-medium correction, Equation 1.9, included. Unfortunately, the calculations show
little of the experimentally measured separation in the balance energies, at all impact
parameters, between 58Fe on 58Fe and 58Ni on 58Ni collisions. The Sobotka mean
ﬁeld did show some separation of the balance energies at higher impact parameters,
but almost none at low impact parameters. Bao-An Li’s mean ﬁeld showed little

separation between the balance energies for all impact parameters.

The hybridization of BUU and a simple coalescence model proved successful in
reproducing, within uncertainties, experimental data. The study investigated the

radial and thermal velocity components of near-central Au on Au collisions for a

112

range of energies. Radial ﬂow did not appear sensitive to BUU model parameters.
Temperature, or rather the inverse slope parameter of the kinetic energy distribution,
did show marginal sensitivity to the in—medium correction factor a, see Equation 1.9,

for central collisions, but little sensitivity to the compressibility of the nuclear matter.

4.2 Outlook

BUU theory has made great strides over the last decade in our understanding of
heavy-ion reactions. The insight the model has provided has spurred growth in the
interest of heavy-ion collisions in the ultra-relativistic regime. Currently many in the
community are working to understand the quark-gluon plasma (QGP) expected from

the Relativistic Heavy-Ion Collider at Brookhaven National Laboratory.

The approximations, numerical and otherwise, which enhance the model’s calcu-
lability yet fail the model at high and low energies can be addressed. Hard scattering
may be modiﬁed to include a kind of relativistic correction and suppress the super-
luminal transport. Coulomb ﬁelds may be treated as propagating with ﬁnite velocity,

though this will require greater RAM than BUU currently requires.

One other weakness lies in the numerical model’s inability to create clusters of
nucleons. Whereas it is possible to extend the theory to include N-body dynamics,

a practical numerical implementation is as yet beyond us.

Nonetheless, BUU has been and will continue to be the premiere model for inter-

mediate energy heavy-ion collisions for some time to come.

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