TIME DEPENDENT HARTREE-FOCK CALCULATIONS OF HEAVY ION COLLISIONS

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TIME DEPENDENT HARTREE-FOCK

CALCULATIONS OF HEAVY ION COLLISIONS

P. Bonche

To cite this version:

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JOURNAL DE PHYSIQUE Colloque Cb, supplément au n° 11., Tome 37', Novembre 1976., page C5-213

TIME DEPENDENT HARTREE-FOCK CALCULATIONS OF HEAVY ION COLLISIONS

P. BONCHE

Service de Physique Theorique

CEN.SACLAY - BP n°2 - 91190 Gif-sur-Yvette, FRANCE

Résumé. L'objet de ce rapport est l'application de l'approximation Hartree Fock dépendant du temps aux collisions entre ions lourds. Après un bref rappel du formalisme, nous ana-lysons plusieurs difficultés d'ordre conceptuel qui sont liées à l'application à la dyna-mique nucléaire d'un modèle de particule indépendante et de l'approximation du champ moy-en. Nous présentons ensuite un résumé des résultats numériques suivi d'une discussion des limitations et des améliorations possibles des calculs actuels.

Abstract. The Time-Dependent Hartree-Fock approximation as applied to heavy ion colli-sions is reviewed in this report. We briefly recall the TDHF formalism and present seve-ral conceptual difficulties which arise in the application of a mean-field independent-particle model to nuclear dynamics. A survey of numerical results is presented and the limitations and possible improvements of current calculations are also discussed.

I. INTRODUCTION

Despite vigorous theoretical activity, there exists no unified microscopic description of heavy ion reaction phenomena. Recent calculations have shown that the Time-Dependent Hartree-Fock (TDHF) approach might provide substantial progress toward this goal. This report will briefly review and sum-marize the TDHF formalism, several of its conceptual difficulties, and the results of numerical calcula-tions.

There are several motivations for using a mean-field independent-particle model to describe heavy ion collisions. First, the static HF approximation [1] and the RPA [2] provide excellent descriptions of many near equilibrium nuclear properties when used with appropriate effective interactions. These two approximations are, of course, the static and small amplitude limits of the full TDHF approach. Second, many degrees of freedom are included in a TDHF calculation, as is crucial to any microscopic description of phenomena ranging from fusion to strongly damped collisions. The evolution of these degrees of freedom is determined not by any apriori

assumptions, but by only the initial conditions, as in the ordinaring time-dependent Schrodinger equa-tion. In addition, the one-body nature of TDHF in-corporates the intuitively appealing notion of the mean-field to provide a coupling between single particle motion and collective degrees of freedom. Finally, a TDHF calculation may be used as a time dependent basis for further microscopic calculations.

II. FORMAL CONSIDERATIONS

In this section we briefly review the TDHF approximation. More rigorous and detailed treatments may be found in the literature [3,4] .

The full many-body wave function for A nucleons, Y, satisfies the time-dependent Schrodinger equa-tion.

i-fi $(F,,...,rA,t) = H ¥ ( r1, . . .)rA, t ) , (1)

where the dot denotes time differentiation and fi is the Hamiltonian operator. This equation may be deri-ved by minimizing the action in a complete set of A-body trial wave functions,

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P. BONCHE

is a constant of motion as is the total angular momentum

When restricted to the subspace of Slater determinants, the variational principle leads to the time- dependent Hartree-Fock equations

where p is the,one-body density matrix associated with the determinental wave function @ :

- -

* -

-

X Q(r,r2,.

.

,FA,t) Q (rX,r2,.

.

,rA,t)

.

( 4 ) The Hartree-Fock one-body hamiltonian h may be expressed in term of the two-body antisymmetrized potential V

In terms of the single particle orbitals {mi(;,t)} from which Qis built, p is given by

The matrix equation (3) thus amounts to the set of A coupled partial differential equations in the single particle coordinate

Several quantities are conserved by the TDHF equations for any set of initial conditions. We list them without proof (see ref. [ 4 ] for details).

i) The scalar product of two single particle particle wave functions is constant in time

Thus the nietric of the set (Qi} is time independent and the total number of nucleons is conserved.

ii) The total Hartree-Fock energy, <Qlfil@>, is also conserved.

iii) The total linear momentum,

P,

defined by

The TDHF equations are also time reversal inva- riant and preserve Galilean invariance. If the single particle wave functions forming a stationary HF determinant are multiplied by the phase eikSr, the resultant determinant is a solution of the TDHF equations, representing the same stationary HF state translating uniformly with velocity

;

= tr Elm.

111. CONCEPTUAL CONSIDERATIONS

Before reviewing the numerical calculations, several questions should be asked concerning TDHF. Some of them are still unanswered.

The first question has to do with the superposition principle. If it were possible to solve the exact time-dependent Schrijdinger equation to study a collision, one might, for instance, define the initial condition as an antisymmetrized direct product of two internal wave functions with a plane wave :

and then integrate (I).Equivalently, one could start with shell model wave functions for nuc1,ei A and B, introducing explicitly an impact parameter b (see Fig. 1)

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TIME DEPENDENT HARTREE-FOCK CALCULATIONS

A suitable superposition of wave functions Yb will give at any time the wave function y defined by the

initial condition (11). The total cross section ex- = 0'

%

+

I

Cph '?h +

I

'zp2h @Zp2h. +

.'

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ph 2p2h

tracted from (11) at t + +m will be the sum over all

in the completemany-body basis Q. generated from impact parameters (ar angular momenta) of the cross

@ by multi-particle hole excitations on the time- section extracted from the corresponding linear com- O

dependent single particle basis @

bination of

Y

_b. A'

In the TDHF approach,because weare dealing with Slater determinants, i.e. shell model wave functions, we are forced to work with a wave function of structure Yb (12). In other words, for a given reaction, we have to solve the TDHF-equations for all impact parameters. There exists no Slater determinant having the structure given by (11). Moreover, because of the non-linear nature of the TDHF problem, we have no proof that the computation of the cross section

R

by

R

will give the correct answer for the total cross section.

Concerning the degrees of freedom and the effective force of TDHF, two questionsarise. First, though many degrees of freedom are indeed included, the TDHF approximation introduces special correl'a- tions between them (e.g. clustering effects). There- fore the "effective" number is smaller than might be apparent. Second, the effective force which is used imposes restrictions on the degrees of freedom per- missible. Effective forces have beenadequately de-

fined for static HF or RPA and have been used (in simplified form) for large amplitude TDHF calculations without further justification or adjustment. The question can then be phrased in the following way : Is a one-body average of a ~WG-body effective interaction adequate at nucleon excitation energies small compared to eF ? What is the range of validity of such an approximation ? Are the usual Skyrme- type interactions the best ones for dynamics ? These questions still require more work.

In the introduction we mentioned that TDHF calculations might provide a suitable time-dependent basis for further calculation. Some words of caution concerning this are necessary. By introducing a complete set of single particle wave function

,

one can evolve all of them in time through

The self c o n s i s t e n t : h a m i l t o n i a n h i s b u i l t o n t h e first i = I ,

...,

A orbitals which define the "ground state" determinant Q,. One might then attempt to expand the exact wave function

The time-dependent Schrijdinger equation furni- shes a set of coupled equations for the c's in terms of the residual interaction. However, this expan- sion is non-trivial, since the effective interac- tiondsed takes into account, at least p-artly, the two particle -two holes amplitudes.

Properly, one should first truncate the basis of single particle states, which obviously cannot be complete in any practical calculation. One should renormalize the bare interaction to account for the truncation and solve the time-dependent Schrijdinger equation for the c's. There is no proof that such a renormalization will be time-independent. Further- more the initial conditions have also to be determined consistently.

In addition to the above considerations,the question of deviations of the TDHF wave function from the exact solution has been raised [51. At any time, the TDHF equations approximate the exact time-dependent Schriidinger equation. While integrating in time, a systematic error might be made which could result in inaccurate or even unphysical final states. An estimation of the life time associated with such deviations has been made by Lichtner and Griffin [ 5 ]

but also suffer from the definition of the residual interaction vis a vis the choice of the effective force.

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C5-2 16 P. BONCHE processes. One must be extremely careful to formula- te the comparison between TDHF and experiment in terms of expectation values of few-body operators, such as the one-body density, the fragment transla- tional kinetic energy or angular momentum, and the nucleon number or charge operators.

To balance the pessimism expressed in this section, one should.recal1 the motivations given in the introduction and elaborated upon inthe literature [4,61

.

Indeed, as we shall see in the next section, preliminary numerical results are much rea- cher than would have been supposed a priori and even provide a qualitative description of the data.

IV. RESULTS.

We now turn to a presentation of results obtained in a one dimensional geometry [ 4

l,

which was the first application of TDHF to large amplitude nuclear dynamics. Collisionswere studied between semi-infinite slabs of nuclear matter. The slabs are described by Slater determinants whose single particle orbitals factorize into non trivial functions of the quantized direction, Z, and plane waves in the two remaining perpendicular directions. The numerics are therefore relatively simple. Slabs are uniquely specified by the number of nucleons per unit area, a quantity analogous to the total mass number in a 3-dimensional geometry.

The TDHF equations were solved using a model effective force which included a direct Yukawa term and a zero-range density dependent force.

In figure 2 we have the density profile as a function of time for the collision of two slabs

2

having 1.4 particles per fm with a C.M. bombarding energy o f 0,5 MeV per nucleon of the total system.

-2 1

The time is in units of 10 sec. which is a natu- ral scale for nuclear reactions. After an initial fusion and a maximum compression at a time

+ 0.4 X 10-~' sec., the system continues to oscilla-

te without reaching a scission configuration. Figu- re 3 shows a higher energy collision (E/A=3.5 MeV) where two fragments scission after the maximum compression. By an inspection of the density profiles as a function of time, one can see that the slabs move much more slowly as they come apart. This is

an example of deep inelastic scattering. In figure 4 ,

we have an intermediate situation : the fused system breaks up after two compressions. Figure 5 shows a summary of the different situations as a function. of incident energy for. the same system of two slabs.

Figure 2. Density profiles p(z,t) at various times

t for the collisions of two slabs with

1.4 nucleon per fermi square with a bombarding energy E/A = 0.5 MeV in the center of mass system.

Figure 3. Density profiles P(z,t) as in Fig. 2 for E/A = 3.5 MeV.

The macroscopic separation distance between the slabs is defined as

+m

d(t) = 2

1

p(z,t)

1 ~ 1

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TIME DEPENDENT HARTREE-FOCK CALCULATIONS CS-2 17 F i g u r e 4 . D e n s i t y p r o f i l e a s p ( z , t ) a s i n F i g . 2 . E/A = 1.5 MeV. F i g u r e 5. The fragment s e p a r a t i o n d ( t ) f o r t h e c o l - l i s i o n s shown i n F i g u r e s 2 (curve a ) , 3 ( c u r v e e ) and 4 ( c u r v e c ) and a t v a r i o u s o t h e r e n e r g i e s .

The f i n a l fragment k i n e t i c energy can b e d i r e c t l y e s t i m a t e d from t h e s l o p e of t h e c u r v e s , a f t e r s c i s - s i o n which g i v e s t h e v e l o c i t y of t h e c e n t e r of mass of each fragment. The dynamics i n t h i s low energy r e g i o n can a l s o b e summarized i n terms of t h e f r a c - t i o n of bombarding energy remaining i n r e l a t i v e motion ( c o e f f i c i e n t of r e s t i t u t i o n ) g i v e n by Ef i n a l C, = k i ( t = +m) E i n i t i a l =

(--ye

d ( t = 0 ) ( 1 6 ) k A v a l u e of z e r o f o r C i n d i c a t e s f u s i o n . I n F i g u r e 6 , t h e two peaks a t E/A % 1.3 MeV and E/A % 2 MeV i n d i -

c a t e r e s o n a n t p r o c e s s e s i n which t h e system undergoes f i s s i o n a f t e r one o s c i l l a t i o n of t h e compound system. Curves (b) and ( c ) of F i g u r e 5 and F i g u r e 4 i n d i c a t e

t h e s e same p r o c e s s e s . A t e n e r g i e s above 2 MeV, t h e f i s s i o n t a k e s p l a c e on t h e "f i r s t try" l e a d i n g t o high- l y e x c i t e d fragments. From F i g u r e 5 , we s e e t h a t a t most 12% of t h e k i n e t i c energy remains i n t h e out- going channel. Above 5 MeV, more complicated f r a g - m e n t a t i o n phenomena, occur.

F i g u r e 6 . The c o e f f i c i e n t of r e s t i t u t i o n c a s a

f u n c t i o n of EIA. C a l c u l a t e d p o i n t s a r e denoted by a c r o s s . A v a l u e of z e r o deno- t e s f u s i o n .

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C5-2 18 P. BONCHE

tions, energy is transferred between a to d, so that scission conditions may occur at a later time. Thus, the resonance depends ,crucially upon the interplay of the fragment elongation and fragment separation .modes. After scission, the fragments continue to

oscillate in the two-fragment "valley" of the po-. tential energy surface as they move apart.

5

= 1.0 MeV

2.6

I ' I I,

10 15 20 25

Frcgnmt Sepooti d ( f m )

Pigure 7. Trajectories in the separation-elongation plane (d-a) at various bombarding energies. Hash marks are at. intervals of 0.1xl0-2~sec.

Calculations with two or three dimensions have also been performed. The most simple generalization of I dimension is a head-on collision of spherical nuclei [71

,

in which case the three-dimensional geometry reduces to two dimensions due to the axial symnetry. Figure 8 shows snapshots of the density for the collision 160

+

160 at 2 MeV per nucleon in the center of mass (Elab = 128 MeV for the pro- jectile).

As the collision in Figure 8 proceeds, the system fuses at t % 0.2 X Io-~' sec. Separation then begins,. apparently initiated by a surface bulge which travels outward from the z = 0 plane

(t % 0.3- 0 . 5 ~ 10-~' sec.). Finally, a neck forms -2 1

and scission occurs (t % 0 . 6 - 0 . 7 ~ 10 sec.), the fragments moving apart with greatly reduced transla- tional kinetic energy.

Not supprisingly, the dynamics of Figure 8 are predominantly single particle. This result, also found in the one-dimensional calculations of ref.[41, manifests itself as follows. The initial HF description of a single 160 nucleus requires four orbitals :

a 1s orbital whose density is a gaussian about the center of the nucleus, a lpm = O orbita1,whose density is two lobes on the z axisoneither side of the 1s orbital, and the Ipm = orbitals whose densi- ties are a torus about the z axis encircling the 1s gaussian. Of course, before contact, the orbitals for each nucleus are confined by their respective self-consistent potentials. However, upon contact the barrier between the nuclei in the full HF, potential disappears. The wave functions from each nucleus then begin to move across the z = 0 plane toward the other side. As lpm,O orbital is associated with the highest momentum components in the z direction (because of its node in a plane perpendicular to the z axis), it is expected to lead the other orbitals across the compound system. This is indeed the case,

-2 1

as can be seen at time t=0.35x 10 sec, where the state from the left-hand nucleu,s is hitting the right-hand wall of the potential well while the 'pm= +l torus from the same side trails behind. At

-2 1

t = 0.45x 10 sec, the m = 0 ortibal has bounced and moves leftward through the torus, which continues rightward. This causes a "reflooding" of the

-2 1

neck at t=0.55x 10 sec. Finally, at scission the Ipm,kl wave functions are on the opposite sides from which they originated, while the lp orbitals

m = 0 have returned to their original sides. Throughout, the Is orbitals are obscured in the contour plots, but these are expected to move with their associated torus. Of course, because of the anti-symmetry of the determinental wave function, the single-particle wave functions themselves have no physical

meaning. However, it is clear that their dynamics govern the observable properties of .the system.

In an effort to compare TDHF calculations with experimental data, initial configurations with different impact parameters must be considered. In a rigorous treatment, such system would entail fully 3-dimensional numerical calculations. However, a simple approximation has been introduced [6,8] to simulate in .twod,imensionsthe effect of a non zero impact parameter, thus taking advantage of the numerical benefits of a calculation in a smaller number of dimensions. This modification accounts for the effect of the relative angular momentum of the two:ions by imagining the collision to occur in a frame rotating about an axis perpendicular to the z

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TIME DEPENDENT HARTREE-FOCK CALCULATIONS C5-2 19

change of 8 is determined by the instantaneous mo- The collision of two infinite flat discs in a ment of inertia, I, about the rotation axis. A com- two-dimensional plane (Hartree-puck approximation) plete description of the induced modifications of was also studied [10]. Here the angular momentum the TDHF equations may be found in Ref. [6] and [8]. is introduced consistently, without any ad hoc

assumption although one pays the price by calcula-

160 + 160 €/A = 2 MeV ting in a flat world. In addition, it is possible

Figu

1 = 0.00 X 10." sec 1 = 0 . 3 5

Ire 8. Density contour maps for a head-on 160+160 collision at E/A = 2 MeV. The collision is shown in the center-od-mass frame. Because of the rotational symmetry about the hori- zontal (z) axis and reflection symmetry through the vertical axis ( z = 0 plane), only the density for'z

2

0 is shown. The contour stripes mark density intervals of 0.04 nucleons per fm3.

The reactions 160 + 160 at a bombarding energy of 2, MeV was studied with the effective interaction of ref. [4],including a coulomb term. Figure 9 shows trajectories for different impact parameters with the separation distance defined by the analogue of(15). Themost salient features are theabscence of fusion events, (althoughtheydo occur at lower bombar- dingenergies),thedeep inelastic character of the low

partial waves, and the orbiting near R = 30, In a TDHF calculation, fusion manifests itself in the for- mation of an oscillating compound system which fails to scission 141. The neglect of other possible degrees of freedom in these calculations (e.g. axial and re£ lection asymmetry) limits possible "dissipa- tion" of energy out of collective modes and thus prevents fusion. More results concerning these cal- culationsare presented here by S.E. Koonin et al.

[91

.

to study the collisions of deformed discs. Again, as the initial conditions are varied a great variety of phenomena emerges including deep inelastic events, fusion, fusion followed by particle emission, frag- mentation, orbiting, and grazing collisions. These results are presented in detail in two communications to this conference [IO].

160 + '"0 E/i4 2 MeV

Figure 9. R-8 trajectories for 160+ 160 at E/A= 2MeV. For each partial wave, we list OC, the

scattering angle for a non-interacting point coulomb trajectory; 'af, the scattering anglefromthe TDHFcalculation ;and Ef the final center-ofmass relative fragment kinetic energy in MeV. Tick marks indicate time intervals of 10-22sec, while the dots delineatetimes during which the ions clutch. The dashed circle at R = 2.8 fm is the sharp surface liquid drop value for the compound system 32s.

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momenta kjust below grazing. With a more complete derived from Skyrme-type interactions which have force (that of ref. 4 + coulomb), similar calcula-

tions ware made by Koonin et a1 1121. Their study of 160 + 160 at E / A = 2 MeV is directly comparable to the rotating frame calculation of Figure 9. The 3 dimension calculations show significant modifications in partial waves. T h e R = 10partial wave remains a deep inelastic event although R = 20 fuses. This behaviour is in qualitative agreement with the calculation of Cusson et al. Trajectories in partial waves

R

_>

30 are adequately reproduced by the 2- dimensional calculation.

V. DISCUSSION AND CONCLUSION.

The calculation reviewed above clearly demons- trate the richness of the TDHF appraach as a unified description of nuclear dynamics. The following comments pertain to its future possibilities as a prac-

tical tool in heavy-ion physics.

1 . Most calculations have so far been I- and 2- dimensional, since the full 3-dimensions is costly. For example, an 160+ 160 collision with all possible symmetries frozen (spin, isospin, reflection) and a local HF potential costs $ 20 ( % l minute of CDC 7600 time) per impact parameter when calculated in the 2-dimensional rotating frame. The corresponding 3-dimensional calculationcosts at least $ 100

('1.5 minutes) per impact parameters. Of course, more realistic interactions or the breaking of these symmetries increases the time required. Therefore, assumptions such as the rotating frame which can reduce the problem to a lower number of dimensions is of great interest. No mention of the numerical technics used has been made here. They are adequately described in the literature [ 131.

2. A systematic study of effective forces in TDHF has yet to be done. Forces used at present are

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J.W. Negele, Phys. Rev. C

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a very high incompressibility modulus K > 300 MeV. Recent calculations [ 141 seem to indicate that forces with a lower incompressibility modulus (K< 250 MeV) give better results for the small amplitude dynamics (RPA). Such forces would quantitatively modify the TDHF results. Indeed preliminary results

1

151 indicate that the inclusion of an effective mass enhan- ces fusion in rotating frame calculations.

3. In addition to provide a vivid representation of heavy-ion collisions, TDHF is capable of providing useful information l61

.

Since TDHF solutions behave classically, trajectories .are well defined and lead directly to fusion cross-sections, inelasticity, and angular distributions. It is also possible to define ion-ion potentials and fragment mass and charge distribution within the TDHF framework. Many comparisons of quantities like these with experiment are clearly necessary before we may conclude that TDHF is a viable tool.

These last comments have illustrated the preliminary character of the results obtained so far. I hope that they will provoke new calculations which

include as many degrees of freedom as po'ssible and use the "best" effective force. Such calculations will certainly be "expensive" but will undoubtedly increase our understanding of the microscopic basis of Heavy-Ion reactions.

ACKNOWLEDGMENT.

I am grateful to Prof. S.E. Koonin for advice and assistance in preparing this manuscript. Thanks are also due to Dr. K. Smith for useful conversations and to the "C.1.T

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