Dolly Seminar

ABSTRACT There are so many models to study the dynamics of heavy ion collisions. The TDHF model is used to study the collisions of heavy ions at very low energy. Here only mean field is taken into account, and here nucleon-2 collisions donot take place ,so Basically there are three different microscopically realizations: the Intra Nuclear Cascade (INC) Quantum Molecular Dynamics (QMD) and the Boltzman–Uehling–Uhlenbeck model (BUU) In the INC model the nucleus–nucleus collision is simulated as the sum of all individual nucleon–nucleon collisions without taking into account self consistent mean– field 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 two–body potentials.and the phenomenon of Multifragmentation is generally studied using QMD model.

Chapter 1 1. INTRODUCTION: The nuclear physics is one of the most extensively studied fields. It has been differentiated in 4 major branches: one of the branch discuss about deconfinement and quark-gluon plasma. Second branch deals with study of γ-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 as well as theoretically to understand the nuclear physics at intermediate energies which ranges between 10A MeV and 2A GeV.

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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 [1] Heavy ion physics has attracted much attention during the last three decades . The behavior of heavy nucleous under extreme conditions of temperature, 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 The energy of accelerated heavy ions is usually classified into following three groups : • Low energy (E

20 MeV/nucleons )

• Intermediate Energy ( 10MeV≤ E ≤2 GeV/nucleons ) • 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 applicable for collisions at low energies. INC model applicable at high energies, BUU and QMD models work at intermediate energies. 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- equilibrium physics. 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 2

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 semi classical 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) [2,] 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 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)[2-4] equation directly, instead one solves the classical Hamiltonian equations of motion for propagation of particles in mean field . Due to one body nature BUU cannot describe correctly, for example the Multifragmentation 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 Multifragmentation can be studied. [2-5] Multifragmentation The breaking of colliding nuclei in to large number of entities consisting of nucleons as well as light and heavy fragments is one of the interesting fields in present intermediate nuclear research. Several theoretical models have also been developed to study the

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multifragmentation. It is found that QMD model is able to explain the phenomenon of Multifragmentation quite well.

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 functions 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 Іψ(t)>

=

|ψ(0)>

(1) Where H is many body Hamiltonian given by H

=

T + V =

+

(2) The state Іψ> 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 fragments , their shapes and particle numbers. These quantities are determined from one body density matrix

4

with elements

=

<ψ|

|ψ>

(3) The expectation value of one body observable is given as =

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 we can show that the one body density matrix follows [16] Iћ

= [ h(ρ) , ρ ] + C(1 ,2 ) ]

(4)

Where h(ρ) is the HF single particle Hamiltonian with matrix element = < i| h(ρ) | j > =

< ψ | H |ψ>

(5)

And C is the correlated part of the two body density matrix. The eq. (4) is exact but has two unknown quantities: ρ 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 full solution of the time dependent Schrodinger equation. To reduce this equation to one body equation, we have to introduce some approximation. The TDHF assumes =

-

(6) The second step towards the TDHF equation is to neglect the second term of right hand side in eq(4). This can be done in two alternative ways: 5

The correlation C vanishes if we impose |ψ> to be an independent particles state at any time.  The truncation of the BBGKY hierarchy can also be done by neglecting the

residual interaction =Ĥ–

(i)

(7)

This is a mean-field approximation because the Hamiltonian is approximated by a onebody operator Ĥ ≃

. In this case, a system described by a Slater determinant

at an initial time will be an independent particles state at any time. Iћ

We finally get the TDHF equation

=

[h(ρ) , ρ ]

(8) Where ρ is now the one-body density matrix of an independent particles state. The operator associated to ρ acts in the Hilbert space of single-particle states. It is written ρ=

><

|

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 6

+

.. Fig .1 Fig .1 Density profile (x, y, 0) with TDHF for the

+

at different R. The iso-densities are plotted every 0.025 line indicates the separation plane.

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reacton of

= 100 MeV

. In each case, the vertical

Advantages 

The time dependent Hartree Fock theory is used to describe the low energy heavy –Ion collisions. The TDHF has been applied to study different physical processes with



bombarding energies up to 10A MeV. 

The TDHF Is also able to explain the fusion, 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 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. 8

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 Monte -Carlo sampling. The time evolution is followed by dividing the whole reaction time into the small intervals pass the point of closest approach

. Two nucleons are supposed to collide if they is

with

(

nucleon – nucleon cross section in their center of mass system and

as the total is the center of

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

Elastic: (9) Inelastic:

(10)

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 non interacting Fermionic system is given by S

=

∫ dγ [ 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

(11) being total number of events and N are the number of particles in a given cell The dγ in the phase space volume element given by

9

∫dγ

=

4

∫

.

(13) To calculate the f, the whole phase space is divided into cells i. The distribution function f is then estimated by relatation = (14) The INC gave excellent opportunity to extract information about several experimental observables [9]. As the INC does not contain the mean free field of nucleons, it is 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-UehlingUhlenbeck (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 10

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 of motion.

The BUU model solves the BUU equation[ 10-12]which reads as: + v.

f -

U.

f

=

∫

dΩ

×{ [ f )]

( 1 - )(1

-

(1 - f ) ( 1 -

(15) ×

( 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 ,

(16) By a Monte – Carlo integration =

(t),

(t)) ,

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

11

f(r,p,t)

=

(18) 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 Hamiltonian’s equation of motion. = (19) = (20) 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 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) model The quantum molecular dynamics (QMD) model is based on an event by event method. Here each nucleon interacts via two or three body interactions that preserve the nucleon -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. 12

QMD model simulations deals in three 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 discuss each of these parts.

 Initialization Here each nucleon is represented by Gaussian wave packet or by a coherent state of form (r

,

(t), (t))

=

exp

[

(t).

r

-

]

(21) 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. Φ

=

(r,

,

, t)

(22) 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.[13-14] 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 ( 13

+

) nucleons is given by

(r, p, (t) Where

(t) and

(t))

=

(23)

(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 ,

=

(24)

To initialize a nucleus, we have to assign the coordinates and momenta of all nucleons. In three dimensional space 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 uniformly distributed in polar coordinate by: r =

R

,

Cosθ = 1 - 2 Φ= Where

,

,

(25)

2П

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: = (26) Where U ( ) is local potential. The center of each Gaussian wave packet in momentum space is uniformly distributed in polar coordinates by: =

(

Cos θ = 1 - 2

, ,

(27)

Φ = 2П

14

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

.

In other words, we demand ≥

(28)

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

S

=

φ, φ*] dτ

(29) With the Lagrange functional � = <φ|ίћ

- H|φ>

(30)

The time evaluation is obtained by the requirement that the action is stationary under the allowed variation of wave function δS

= δ

*]dt =0

The Hamiltonian H contains a kinetic term and mutual interactions

(31) . The time

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

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under allowed variation of wave function. This yields an Euler – Lagrange equation for each parameter. We obtain for each parameter λ, an Euler- Lagrange equation:

-

=0

(32)

If the true solution of the Schrödinger 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 wave function in the restricted parameter space which comes closest to the solution of Schrödinger 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,

-

(33)

= potential energy), the lagrangian and the variation can

easily be calculated and we obtain: �· = =

+

> >=

=With

=

+

,

(34)

,

(35)

>=-

t and = ∫

,

<

*|V (

(36) )|

> .

These equations represent time evolution and can be solved numerically. Thus the variational 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.

16

̇ = -

̇ =

;

.

(37)

The numerical solution can be achieved in the spirit of the classical molecular dynamics. [14-15]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

(38) are the two and

three body interactions. The total momentum in QMD is conserved because the Hamiltonian is well defined for the whole system. Apart from local Skyrme interaction and an effective coulomb interaction

, a finite range Yukawa term

are also included to account for various

effects. The final potential reads as: =

+

+

,

(39)

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 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 ,

(40)

are the free parameters.

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In order to investigate the influence of different compressibilities one can generate the above potential energy. = ( ) +

(41)

Depending on the values these parameters, one can have the soft(S) and hard (H) equations of state .

Fig –2.4.1 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] 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 change drastically if one 18

considers 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

19

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 QMDMD simulation no longer leads back to two well separated nuclei. All particles seem to stem from a single compact source. In momentum-space, however, the nonisotropic emission 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 20

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) 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 were dubbed as quantum molecular dynamics (QMD) Model.

In past decade several refinements and improvements were made over the original QMD. These new versions were named as IQMD (isospin QMD) GQMD (G-matrixQMD) etc.

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2.5 Isospin Quantum Molecular Dynamics Model (IQMD) Quantum molecular dynamics (QMD)[14] 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 elastic and inelastic cross sections for proton-proton and proton-neutron collisions used in IQMD are shown in Fig. 2.5. The cross section for neutron-neutron collisions is assumed to be equal to the proton-proton cross sections

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Fig 2.5 Elastic and inelastic cross sections for proton-proton (pp) and proton neutron (pn) used in IQMD

Summery 23

We have studied so many statistical as well as dynamical models to study heavy ion collisions, such as TDHF model 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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