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Chapter 1

1. INTRODUCTION:

The nuclear physics is one of the most extensively studied fields. It has beendifferentiated in 4 major branches:

One of the branch discuss about deconfinement and quark-gluon plasma.

Second branch deals with study of-ray spectroscopy.

The third branch led to the study of nuclear collectivity through giant resonances

and lastly, the branch that deals with the study of intermediate energy heavy ion

collision. In last two decades, a lot of efforts have been made experimentally aswell as theoretically to understand the nuclear physics at intermediate energies

which ranges between 10A MeV and 2A GeV.

1.1 HEAVY ION COLLISIONS

The term heavy ion is used for the nucleous more massive than helium. The branch of

physics which deals with the phenomenon that occur when two heavy nuclei are

brought into close contact such that nuclear forces that hold the neutrons and protons

together within the nucleous are felt by other nucleons is called heavy ion physics [Bha-

71, Pri-62,Holf-95 ,Bro-72]

Heavy ion physics has attracted much attention during the last three decades. The

behavior of heavy nucleous under extreme conditions of temprature.,density, angular

momentum etc. is a very important aspect of heavy ion physics .

A large no. of accelerators have been developed to study these heavy ion reactions

[Sche-05, Bart-05] The energy of accelerated heavy ions is usually classified intofollowing three groups :

Low energy (E 20 MeV/nucleons )

Intermediate Energy ( 10MeV E 2 GeV/nucleons )

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Relativistic Energy ( E 200 MeV/neucleons )

There are so many theoretical models to study heavy ion collisions at these different

energies such as TDHF model which is applecable for collisions at low energies. INC

model applicable at high energies, BUU and QMD models work at intermediateenergies.Here we will study these models in detail.

1.2 Review of theoretical models

Theoretically several models have been developed to explain some of observed heavy

ion phenomenon. The heavy ion collisions involve very complicated non- equilibriumphysics. Due to lack of free space at low incident energies about 98% of the attempted

collisions are blocked. The whole dynamics at low energies is governed by the mean

field or by the mutual two or three body collisions. At relativistic energy(2A GeV ) Pauli

principle play role quite small(roughly 4%collisions are blocked)and hence the

dynamics of reaction is governed by Cascade picture. On the other hand both cascade

and mean field picture emerges at intermediate energies.

The conventional theories like the time dependent Hartree- Fock(TDHF)or semiclassical version the so called Vlasov equation is suitable approach at low energies,

where nucleon- nucleon collisions are negligible. A suitable approach for intermediate

energy heavy ion physics should treat the nucleon- nucleon collisions and the mean

field on equal footing. Some attempts were made in the literature to extend the TDHF to

take care of residual nucleon- nucleon (NN) interactions which are responsible for two

body collisions.(this was dubbed as ETDHF). However, due to complications, this theory

could not be used for large scale investigations.

In first attempt, the semi classical version of ETDHF(i.e Vlasov equation)

[1,2,3,4,5]was coupled with nucleon-nucleon collisions and thus a new realization

named as Boltzmann-Uehling-Uhlenbeck equation (BUU) is used till date to study the

large deviating problems of low, intermediate and relativistic heavy ion collisions. Many

more names like Landau-Vlasov (LV) equation or Vlasov Uehling-Uhlenbeck(VUU) Or

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Boltzmann Nordheim equation also exist for same realization. The solution of BUU

equation provides the time evaluation of one body distribution function in six

dimensional phase space. In actual calculations , one does not solve the Boltzmann

Uehling-Uhlenbeck(BUU)[2,6,8,,9] equation directly, instead one solves the classical

hamiltonian equations of motion for propagation of particles in mean field . Due to one

body nature BUU can not describe correctly, for example the multi fregmentation

phenomenon which involve the correlations between nucleons. Recently, some

attempts were made to extend the equation by including stochastic two -body

correlations so that the N body phenomenon like multifregmentation can be studied.

[10,11,12,13]

Basically there are three different microscopical realizations: the Intra Nuclear Cascade

(INC) Quantum Molecular Dynamics (QMD) and the BoltzmanUehlingUhlenbeckmodel (BUU) In the INC model the nucleusnucleus collision is simulated as the sum of

all individual nucleonnucleon collisions without taking into account self consistent

meanfield potentials and Pauli blocking for the collisions.The QMD follows the same

scheme as the INC, but takes into account the Pauli blocking in the collisions and a

nucleus potential which is calculated as the sum of all twobody potentials.

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CHAPTER 2

2 Methodologies

2.1 TIME DEPENDENT HARTREE FOCK THEORY

The time dependent Hartree Fock theory is a quantum mechanical theory which is used

to describe the low energy heavy Ion collisions. A no of different physical situations

have been studied: fission, heavy ion fussion and heavy inelastic collisions

The time dependent Hartree Fock theory is based on the assumption of independent

particle behaviour for the near equilibrium situations if the excitation energy is less than

10A MeV .

The TDHF describes the many body wave function by a single slatter determinant.

Since the heavy ion collision is a time dependent process, it is more convenient to use

the time dependent schrodinger equation. The wave function for the system at any time

t is given

(t)> = |(0)> (1)

Where H is many body Hamiltonian given by

H = T + V = + (2)

The state > contains all the information of the system , which we need for a good

description of the dynamics .We often need only expection values of one body

observables , such as the position of the fregments , their shapes and particle numbers.

These quantities are determined from one body density matrix with elements

=

The expectation value of one body observable is given as =

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The first step towards the TDHF theory is to restrict the description to one body

observable, and to seek for an equation giving the evaluation to .

Starting from eq(1) and using the ( BBGKY) Hierarchy [14-15] we can show that the

one body density matrix follows [16]

I = [h[] , ] + C(1 ,2 ) ] (3)

Where h[] is the HF single particle Hamiltonian with matrix element

= < i| h[] | j >

= < | H |>

And C is the correlated part of the two body density matrix.

The eq.(3) is exact but has two unknown quantities : and C.

The equation of motion shows that one body density matrix is related to two body

density matrix. Similarly equation of motion for two body density matrix is related to

three body density matrix , and so on. This series of equations is called BBGKY

(Bogoliubov-bonn-Green-Kirkwood-Yvon) hierarchy which is equivalent to the fullsolution of the time dependent schrodinger equation. To reduce this equation to one

body equation, we have to introduce some approximation. The TDHF assumes

= -

The second step towards the TDHF equation is to neglect the second term of right hand

side in eq(3). This can be done in two alternative ways :

The correlation C vanishes if we impose |> to be an independent particles stateat any time.

The truncation of the BBGKY hierarchy can also be done by neglecting the

residual interaction

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= (i)

This is a mean-field approximation because the Hamiltonian is approximated by a one-

body operator . In this case, a system described by a Slater determinantat an initial time will be an independent particles state at any time.

We finally get the TDHF equation I = =[h[] , ]

Where is now the one-body density matrix of an independent particles state. The

operator associated to acts in the Hilbert space of single-particle states. It is written

= >< | where denotes an occupied single-particle

state.

The TDHF theory neglects the pairing correlations which are contained in C. In fact,

TDHF describes occupied single-particle wave functions in the mean field generated by

all the particles and obeys the Pauli principle during the dynamics.

RESULTS

In this work, we focus on head-on collisions of two heavy-ions and take the collision

direction as the x axis. Following Ref. [19], we define center-of-mass coordinateR, total

momentum P and mass number A of projectile-like (+) and target-like () fragments

by introducing the separation plane. The separation plane can be conveniently defined

as the plane at position where isocontours of projectile-like and target-like densities

cross each other. We indicate position of the separation plane, i.e., position of the

window at X = . Illustration of density profiles and separation plane locations are

displayed at different times of the symmetric reaction 40Ca+40Ca

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in Fig. 1.

Fig .1 Density profile ( x,y,0) with TDHF for the + reacton of = 100

MeV at different R. The iso-densities are plotted energy

Advantages

The time dependent Hartree Fock theory is used to describe the low

energy heavy Ion collisions.

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The TDHF has been applied to study different physical processes with

bombarding energies upto 10A MeV.

The TDHF Iis also able to explain the fussion , compound nucleous

formation, dissipation, shock wave propagation and fragmentation.

Disadvantages:

TDHF Theory does not include the effect of nucleon-nucleon collisions al low

energy.

Here fluctuations and correlations were not present because the nucleons

behave independently.

Some attempts are also made in in the literature to extend the TDHF equation to

include the residual nucleon- nucleon (NN) interactions which are responsible for

two body collisions. This is called as extended time dependent Hartree Fock

(ETDHF) equation. Unfortunately ETDHF is too complicated to be used for large

scale investigations in heavy ion collisions [17]

2.2 Intra nuclear cascade model (INC)

At intermediate energies, the mean field and the two body nucleon- nucleon collisions

play an equally important role in the evaluation of the system.

Assumption

The Cascade model simulates the heavy ion collisions as a superposition of

independent two body NN collisions. Naturally in the absence of mean field, the

nucleons move on straight line trajectories until they collide.

In INC model each nucleon is considered as a collection of point particles distributed

within a sphere without any fermi momentum.

When two nuclei approach each other, the position of each nucleon ( within a sphere) is

assigned by Monti -carlo sampling. The time evolution is followed by dividing the whole

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reaction time into the small intervals . Two nucleons are supposed to collide if they

pass the point of closest approach is with ( as the total

nucleon nucleon cross section in their center of mass system and is the center of

mass energy. The colliding particles can also scatter elastically or inelastically.

Elastic:

Inelastic:

The cross sections for channels (a) and (b) are taken from experiments . the cross

section for channel (e) is obtained by detailed balance method . The cross sections for

channels (b) and (c) are taken to be same as (a).

At the end of simulations, all s decay isotropically into nucleons and pions by

conserving the charge, isospin quantum number. In other words the number of s at the

end of reaction gives the number of pions.

One can also calculate the entropy generated in a nuclear system after the collision.

The entropy for noninteracting Fermionic system is given by

S = d [ f ln f + (1 - f )ln (1 - f ) ] ,

Here f is the occupation probability in the phase space which is given by f = With R

being totel number of events and N are the number of particles in a given cell

The d in the phase space volume element given by

d = 4 .

To calculate the f, the whole phase space is divided into cells i. the distribution function f

is then estimated by relatation

=

The INC gave excellent opportunity to extract lnformation about several experimental

observables [11]. As the INC does not contain the mean free field of nucleons , it is

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more suitable for high energy experiments . The other models which include the mean

field as well as the nucleon-nucleon collisions are based on Based on Boltzmann-

Uehling-Uhlenbeck (BUU) equation.

Merits and Demarits

The intranuclear cascade model is capable of describing the high energy heavy

ion collisions.

In this model the mean field is completely neglected and the nucleon-nucleon

(NN) Collisions are taken without Pauli blocking .

Note that this was the first microscopic dynamical model used to understand the

experimental data of heavy ion collisions.

2.3 Introduction to BUU model

The Boltzmann-Ueling-Uhlenbeck (BUU) model is a semi-classical transport model that

is used to study nuclear collisions. The colliding nuclei are represented as individual test

particles (nucleons) which follow classical trajectories in a nuclear mean field potential,

which contains information about the nuclear EoS. Classical billiard-ball collisions may

occur if two nucleons pass close enough to one another. The Pauli-exclusion principle is

employed and forbids any collision which would cause more than one nucleon to

occupy the same state.

The dynamical description of nucleus-nucleus reactions is based on the equation ofmotion.

+ v. f - U. f = d

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{ [ f ( 1 - )(1 - (1 - f ) ( 1 -

) ]

( p + ) }

The right hand side denotes the collision integral which also include the pauli blocking .

This equation is solved by test particle method .Here the phase- space of each nucleon

is represented by large number of pseudo particles( called test particles ).In its

numerical implementation , the above equation reduces to a set of 6 ( ) N

couple of first order differential equation in time . Here N is the number of test particles

per nucleon, and A are the target and projectile, respectively. The test particle method

replaces the expectation value of a single particle observable;

= f (r, p, t) O (r, p) r p ,

By a Monte Carlo integration

= (t) , (t) ) ,

With r(t) and p(t) denote the points in phase space which are distributed according to

f(r,p,t)

f(r,p,t) = lim (r-r(t))(p-(t))

It is obvious that a large number of test particles will be needed to avoid the numerical

noise. These test particles are treated as classical point particles. In recent calculations,

one has also succeeded to use the gaussians wave packets for test particles. These

particles are then propagated under the classical hamiltonians equation of motion.

= -

=

One should also keep in mind that the forces acting on test particles are calculated

from the entire distribution which includes the test particles of all events. In other words,

the n parallel events are inter- connected and an event by event correlations cannot be

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analyzed within these models. In the limit n the distribution of these test particles

represents a true one body distribution function. The BUU model is able to explain the

one body observables like collective flow stopping and particle spectra, nicely. Due to

the lack of fluctuations and correlations, the N body predictions are beyond the scope of

these models. The N body features can be described nicely with molecular dynamics

models. In the following we discuss the quantum molecular dynamics (QMD) model in

detail.

2.4 Quantum molecular dynamics (QMD) modelThe quantum molecular dynamics (QMD) model is based on an event by event m

ethod.Here each nucleon interacts via two or three body interactions that preserve thenucleon -2 correlations and fluctuations that are important for N-body phenomena like

multifragmentation. This is in contrast to the one- body dynamical models which are

suitable for one-body observable only.

QMD model needs these steps:

First, one has to generate the nuclei. This procedure is called initialization.

These nucleons then propagate under the influence of surrounding mean field.

This is termed as propagation.

Finally, nucleons are bound to collide if they come too close to each other . This

part is dubbed as collision.

In this we shall disscuss each of these parts.

Initialization

Here each nucleon is represented by Gaussian wave packet or by acoherent state of form

(r , (t), (t)) = exp [ (t). r - ]

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The parameter L, which is related to the extension of the wave packet in phase space,

is fixed. The totel N body function is assumed to be a direct product of the coherent

states.

= ( r , , , t)

Note that we do not use a Slater determinant (with ( + ) Summation terms) and thus,

neglect antisymmetrization. First successful attempts to simulate the heavy- ion reaction

with antisymmertrized states have been performed for smaller systems. A consistent

derivation of the QMD equation of motion for the wave function under the influence of

real and imaginary part of the constant cross section (G- Matrix), is however, missing.

Therefore we shall add the imaginary part as a cross section and treat the collision in

cascade approach.

The Wigner transform of the coherent states with ( + ) nucleons is given by

(r, p, (t) (t)) =

Where (t) and (t) define the center of gaussian wave packet in phase space,

whereas the squared width L is assumed to be independent of the time. The density of

particle is

(r) = (r ,p , (t) , (t) ) p ,

=

To initialize a nucleus, we have to assign the coordinates and momenta of all nucleons.

In three dimensional spaces inside a sphere of radius R= 1.14 , Where A is the

number of nucleons of nucleus under consideration], The center of gaussian wave

packet is uniformaly distributed in polar coordinate by:

r = R ,

Cos = 1 - 2

= 2

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Where , , are the random numbers. The coordinates of nucleons are rejected if

the distance between them is less than 1.5 fm. The local fermi momentum is determined

by relation:

=

Where U( ) is local potential. The center of each gaussian wave packet in momentum

space is uniformaly distributed in polar coordinates by:

= ( ,

Cos = 1 - 2 ,

= 2

We reject those distributions where two particles are closer than some distance .

In other words , we demand

Typically 1 out of 50,000 initializations is accepted under typically 1 out of 50,000

initializations is accepted under present criteria. The initial phase space distribution for

the colliding nuclei on QMD agrees fairly well with the experiments.

Propagation

The successfully initialized nuclei are then boosted towards each other with proper

center of mass velocity using relativistic kinematics. The center of each distribution

moves along the coulomb trajectories. This distribution is kept fixed until the distance

between surfaces of nuclei is 2 fm. The equation of motion of many body systems is,then, calculated by means of a generalized variational principle:

We start from the action

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S = , *] d

With the Lagrange functional

=

The time evaluation is obtained by the requirement that the action is stationary under

the allowed variation of wave function

S = *]dt =0

The Hamiltonian H contains a kinetic term and mutual interactions . The time

evaluation of the parameters is obtained by the requirement that the action is stationary

under allowed variation of wave function. This yields an Euler Lagrange equation for

each parameter. We obtain for each parameter , an Euler- Lagrange equation :

- = 0

If the true solution of the Schrdinger equation is contained in the restricted set of

wave function (r, (t), (t)), this variation of action gives the exact solution of the

schrodinger equation. If the parameter space is too restricted, we obtain the wavefunction in the restricted parameter space which comes closest to the solution of

Schrdinger equation. Note that the set of wave functions which can be covered with

special parameterizations is not necessarily a subspace of Hilbert space, thus the

superposition principal does not hold.

For the coherent states and a Hamiltonian of the form H = -

( = kinetic energy, = potential energy), the lagrangian and the variation caneasily be calculated and we obtain:

= - > - ,

= + > = ,

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= - > = - ,

With = + t and = < *|V ( )| > .

These equations represent time evolution and can be solved numerically. Thus thevariational principle reduces the time evolution of the N- body Schrodinger equation to

time evolution equations 6. ( + ). The equations of motion now show a similar

structure like classical Hamiltonian equations.

= - ; = .

The numerical solution can be achieved in the spirit of the classical molecular dynamics.

[27,28,29]The wave functions (other than the Gaussians ..) Yields more complex

equation of motion for other parameters and hence the analogy to classical molecular

dynamics is lost. The total energy of particle is the sum of kinetic and potential

energies:

= + = + + ,

Where refers to the kinetic energy of particle and and are the two andthree body interactions.

The total momentum in QMD is conserved because the Hamiltonian is well defined for

the whole system. Apart from local Skyrme interaction , a finite range Yukawa term

and an effective coulomb interaction are also included to account for various

effects. The final potential reads as:

= + + ,

The Yukawa term has been added to improve the surface properties of interaction. In

nuclear matters where the density is constant, the interaction density coincides with

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single particle density and as well as are directly proportional to . The three

body part of interaction is proportional to (

In nuclear matter, the local potential energy has the form

= ( ) +

Here , are the free parameters.

In order to investigate the influence of different compressibilities one can generate the

above potential energy.

= ( ) + Depending on the values these parameters, one can have the

soft(S) and hard (H) equations of state .

Fig 3 equation of state . the density dependence of the energy per particle in nuclear

matter at temperature T=0 is displayed for four different sets of parameters.[25]

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The resulting compressional energy is shown in figure for the soft (S) and hard (H)

equation of state and for the soft and hard EOS with MDI, (SM,HM). Note that all

equations of state give the same ground state binding (E/A=-16MeV at ) ,but

they differ drastically for higher densities . Here the hard EOS leads to much more

compression energy than the soft EOS at the same density the inclusion of momentum

dependent interaction leads for infinite nuclear matter at rest to almost no difference

between the cases S,SM and H,HM, respectively. This changes drastically if one

consider the heavy ion collisions the additional repulsion due to separation of projectile

and target in momentum space shifts the curve for the SM, (HM) interactions to higher

energies.

Fig 4

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Fig 5

Fig. 4, 5 Time evolution of the paricle distribution in configuration space for the

reaction Au (150 MeV/A) +Au for the impact parameter of b= 3fm. The projection of all

particles in the reaction plane (x-z) is displayed for four different times as indicated.

The time-reversal symmetry of the MD calculations shows that system follows an

isentropic path. It is tempting to use the concept of the back propagation in order to

time-revert the expansion of a simulated heavy ion collision, which is assumed to

conserve the entropy. A typical evolution of a single event is shown in Fig. 4.the QMD-

MD simulation no longer leads back to two wellseparated nuclei. All particles seem to

stem from a single compact source. In momentum-space, however, the nonisotropicemission pattern is present, even after the back propagation is completed. The future

evolution of a classical dynamical system in general is strongly dependent on the

distribution of matter in phase-space. In particular the initial correlation between

configuration and momentum space determines the dynamics. In order to study the

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physics of the expansion phase in more detail, we have calculated the time-evolution of

the one-body distribution function in QMD. Technically this is achieved by superposition

of many events with the same incident energy and impact parameter. The result of this

procedure is equivalent to a test particle distribution of a VUU/BUU calculation. It allows

for determination of the local velocity distribution with arbitrary precision, which is only

limited by the number of events. The first and the second moment of the local velocity

distribution are related to collective motion and the local temperature respectively.

Advantages

One would like to have the methods where correlations and fluctuations among

the nucleons can be preserved.

The classical molecular dynamics (MD)[61,62] approach ,in principle , is capable

of treating both the compression and fragment formation .

The molecular dynamics predicts the collective flow in a quantitative agreement

with the data.

It incorporates the complete N-Body dynamics which is necessary to describe

the formation of fragments . Naturally the simple classical molecular dynamics

needs more refinements which should also include the quantum features. The

quantum features play very important role at low energies..This approach was

latter extended to incorporate the quantum features was dubbed as quantum

molecular dynamics (QMD).[30,31,32,33,34,35,36,,37,38,39,40,41].Model.

In past decade several refinements and improvements were made over the original

QMD. These new versions were named as IQMD (isospin QMD)[39,42], GQMD(G-

matrix-QMD)[41,45,46,47] etc

2.5 Isospin Quantum Molecular Dynamics Model (IQMD)

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Quantum molecular dynamics (QMD) model contains two dynamical ingredients, the

density dependent mean field and the in-medium nucleon-nucleon cross-section. In

order to describe the isospin dependence appropriately, the QMD model should be

modified properly. Considering the isospin effects in mean field, two-body collision and

Pauli blocking, important modifications in QMD have been made to obtain an isospin

dependent quantum molecular dynamics (IQMD). The Isospin-QMD (IQMD) treats the

different charge states of nucleons, e.g deltas and pions explicitly, as inherited from the

VUU model. IQMD has been used for the analysis of collective flow effects of nucleons

and pions. As it has been developed from the VUU-model, its coding is therefore

independent of the original QMD. The isospin degrees of freedom enter into the cross

sections (here cross sections of VUU similar to the parameterizations of VerWest and

Arndt have been taken, see also) as well as in the Coulomb interactions. The elasticand inelastic cross sections for proton-proton and proton-neutron collisions used in

IQMD are shown in Fig. 2.1. The cross section for neutron-neutron collisions is

assumed to be equal to the proton-proton cross sections[44]

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Fig 6

Elastic and inelastic cross sections for proton-proton (pp) and proton neutron (pn)used in IQMD

Summery

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There are many statistical as well as dynamical models to study heavy ion collisions

such as TDHF theory which is applicable at low energy, in this model only mean field is

taken into account. But no nucleon nucleon collisions are involved here.INC model is

applicable at high energy. Here mean field is totally negligible, only cascade picture

comes to play role here. Whereas BUU and QMD models play role quite effectively at

intermediate energies where both mean field as well nucleon 2 collisions are treated

on equal footing. So there models are much more useful to study the dynamics of heavy

ion reactions at intermediate energy.

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