A study of the disintegration of highly excited nuclei with the Vlasov-Uehling-Uhlenbeck equation

Nuclear Physics A468 (1987) 321-347 Nosh-Holland, Amsterdam

A STUDY OF THE DISINTEGRATION OF HIGHLY EXCITED NUCLEI WITH THE VLASOV-UEHLING-UHLENBECK EQUATION L. VINET’, C. GREGOIRE and P. SCHUCK2 GANIL,

BP. 5027, 14021 Caen-Cedex,

France,

B. REMAUD3, and F. SEBILLE Znstitut de Physique,

UnioersitP de Nantes,

44072 Nantes-Cedex,

France

Received 30 September 1986 (Revised 1 December 1986) The disintegration of hot and/or compressed nuclei is studied using (i) the Vlasovequation (VE) with imposed spherical symmetry, (ii) the VE in three dimensions (3D) and (iii) the VE in three dimensions supplemented by the Uehling-Uhlenbeck collision term (VUU). We find that case (ii) is slightly more unstable with respect to disintegration compared to case (i) whereas (iii) tends to make nuclei more stable. In all cases the thermal energies (15-20 MeV per nucleon) needed to totally disintegrate a nucleus seem to be higher than those found in static and hydrodynamic calculation. On the contrary, compressional energy very much helps disintegration. Some comments on the introduction of fluctuations and corresponding fragmentation are added.

Abstract:

1. Introduction The study of highly excited nuclei and their modes of d&excitation is presently

of great interest. Several approaches to the problem are being proposed. First come the completely static models lT3) where a nucleus is considered in equilibrium with a su~ounding gas treated in a “hot” Hartree-Fock-Skyrme approach or using semiclassical density functional theory “). With sophisticated subtraction techniques of the gas phase ‘) a limiting temperature can be defined beyond which the nucleus totally dissolves into the gas phase. With the SkM” force the temperature is found to be of the order of 5-6 MeV (when the Coulomb force is included ‘) which seemingly is in agreement with experiment “). One can ask the question of how well this static approach simulates the true dynamical situation of the deexcitation or disintegration of an &&fed nucleus. As a first attempt in this direction, hydrodynamical calculations have been performed and the deduced limiting temperature seems to be in agreement with the static calculation ‘)_ Hydrodynamics however supposes local equilibrium and it is not clear whether this is established in the course of the dynamical evolution. In a recent investigation, we therefore took the ’ This is part of a doctoral thesisby L. Vinet. * ISN, 53, avenue des Martyrs, 38026 Grenoble, France. 3 IRESTE, 3, rue Mar&ha1 Joffre, 44071 Names-Cedex, France. 0375-9474~87f $03.50 @ Elsevier Science Publishers B.V. (North-HoBand Physics Publishing Division)

322

L. Vinet et al. / Disintegration

more or less opposite point of view and treated the evolution with the (spherical) Vlasov equation (VE) “) which favors non-equilibrium (diabatic) evolution. With the modified BKN force ‘) (yielding a realistic compressibility of 228 MeV), we obtained a very high flash (limiting) temperature of -18 MeV. We attributed this to a very efficient and fast cooling mechanism due to a very energetic non-equilibrium spill off of nucleons. In this work we continue our investigation in several directions. First of all we release the constraint of sphericity. Though we still prepare the system initially in a spherical state (with some numerical fluctuation however) the subsequent evolution is in three dimensions (3D). Therefore if the system energetically prefers to break spherical symmetry (e.g. fragments) it will do so (this is due to the numerical fluctuations, since in principle symmetry is a conserved quantity, see discussion in sect. 4). We will see that where we had before (with imposed sphericity) bubble formation, now the bubble breaks into pieces probably due either to dynamical instabilities with respect to long wavelength excitation or to pure mechanical rupture. It is to be stressed that this break up occurs independent of the smallness of the numerical noise and we checked that the evolution pattern stays basically unchanged in doubling our basis size (number of pseudo-particles). We therefore demonstrate that, even within pure mean field theory, a fragmentation process can occur if it is energetically favored. The release of the symmetry constraint has as a consequence that nuclei disintegrate at a somewhat lower limiting temperature but this does not seem to be a dramatic effect. A second important and new ingredient is the addition of a Uehling-Uhlenbeck collision term to the 3D calculations and the study of its influence on the dynamical evolution of a hot and/or compressed nucleus. We find that two-body collisions, i.e. viscosity, slows down distintegration of the nucleus thus again increasing the limiting temperature beyond which no real remainder of the initial nucleus subsists. We used the modified BKN force mentioned before and the force employed by Aichelin and coworkers “) and found in all cases limiting temperatures in the range of 15-18 MeV. With respect to the fact that we also included Coulomb forces this seems to be a considerably higher temperature as has been found in the static or hydrodynamic calculation, though an unexpected strong dependence of the effect on the different forces used in the various calculations cannot be completely excluded, since, for instance, the surface energy can also influence the dynamical evolution appreciably. This work is organized as follows; in sect. 2 we give some details of our formalism, in sect. 3 we discuss dynamics of the Vlasov equation with imposed spherical symmetry whereas in sect. 4 this constraint is released and the system can evolve freely in three dimensions. In sect. 5 we finally introduce the collision term and study the expansion regime of a nucleus with VUU dynamics. In sect. 6 some remarks on the treatment and influence of fluctuations on multifragmentation are made and in sect. 7 we present our conclusions.

L. Vinet et al. j ~~s~ntegr~tion

323

2. General formalism

Our aim is to treat dynamically the evolution of highly excited nuclei. To this purpose the semiclassical limit of TDHF theory supplemented by a two-body collision term should be a reasonable first approach. Though TDHF theory (plus extension) has inherent difliculties since it does not represent a reaction theory for a definite channel, we think that for the study of certain global properties this shortcoming may become less important. Our equation therefore reads:

af

P -+-.---.-=&f-J,

at

m

fv av w

aR aR

ap

where f=f(R,p, t) is the phase-space distribution, V= V[p(R, t)] is the selfconsistent mean field potential of the Skyrme type and I[‘] represents the collision integral to be specified in sect. 5. We would like to call eq. (1) the Vlasov-UehlingUhlenbeck (VUU) equation whereas without the collision term it is the Vlasov equation (VE)*. Unfortunately the Vlasov equation is notoriously difficult to solve and surprisingly a somewhat coarse grained solution of the (quantal) TDHF equation, when it is put on a grid in r-space, seems to be more stable than the corresponding Vlasov equation lo). This feature could be due to the fact that the finite uncertainty of position and momentum of the particles in the quantum case has a smoothing and thus stabilizing effect on the solution. Of course we would not like to retain all the detailed shell effects of the quanta1 case which seems to be unnecessary at high energies and what would be nume~cally an unsolvable task. Some coarse graining of the quanta1 case in order to wash out shell effects but retaining the uncertainties of positions and momenta due to the finiteness of the system seems thus unavoidable. Since it is unknown how to do that in a definite way in non-equilib~um dynamics we here propose a different way which is inspired by the well established Strutinsky smearing procedure in the static case and which in some minimal way will reintroduce some quanta1 features into eq. (1). Indeed the problem can best be studied in the static case; a Strutinsky smoothed (or equivalently a semiclassically calculated**Wigner transform of the density matrix of a nucleus in its ground state looks as shown in fig. 1 [ref. ‘“)I. We see that the distribution is more infinite matter like in the interior than in the surface region, For a harmonic oscillator potential there is no B-dependence; we show in fig. 2 the corresponding distribution together with the B-averaged dist~bution of fig. 1. We see that the distribution of the harmonic oscillator represents very well the average diffusivity of the phase-space distribution of a real nucleus. * The nomenclatureof this equation in the literature is not uniform and we ourselves called (1) the Landau-Vlasov equation in previous publications. It however seems that this name is sometimes also employed for the streaming term (lhs) of eq. (1) alone 9, and to avoid confusion we will hence forth use the name VUU. ** This equivalence is well studied and documented in the literature ll).

324

Fig. 1. The Wigner function of ref. ‘3) is represented in polar coordinates (E, e) where E = 2(p’f ti2)/k@,, and tg6 = p^/l? with 2 mj? = pz and 2i’= mwt R2.We see that for B = fir, the distribution is similar to nuclear matter, whereas for 0 = 0 a quite strong diffusivity is exhibited (we are grateful to A. Ayachi for the preparation of the figure).

The static solution of (1) is the Thomas-Fermi fo= Nsr-K)

dist~bution: *

(2)

In fact the unit step function B is formally not the only possible solution of (2) but any function F(H,) of the classical energy H, is. However, as soon as one uses for instance some smeared out function for F like the one in fig. 2, one will populate the continuum for a finite (self-consistent) potential, i.e. there will be a finite density everywhere. Thus, for a finite isolated nucleus in its ground state, we have to reject any other solution of (1) than the unit step function. Any finite diffusivity of the distribution function is then due to quanta1 k-correction not contained in eq. (1). However, a quite fruitful procedure of the poor man to reestablish some quanta1 effects consists in a folding procedure 14):

“&CR, PI = with H,(R, p) =p2/2m + V(R) where V(R) is the mean field corresponding to the Skyrme force. The smearing functions are conveniently chosen to be (normalized)

325

L. Vinet et al. / Disintegration 1.5

I

I

I

1.0

fld

0.5

0.0

Fig. 2. Comparison of the Wigner function for the harmonic oscillaror (broken line) with ho, = 41 A-“3 (MeV) and the &averaged distribution of fig. 1 (dashed line). The e-dependent factor of the jacobian from (R, p) to (E, 0) has been included in the averaging (we are grateful to A. Ayachi for the preparation of the figure).

gaussians: d(R,p)=Nexp

-

The width parameters At and Ap are then adjusted such as to reproduce the average phase-space diffusivity shown in fig. 2 and which can be deduced from the harmonic oscillator (we will at the moment not care about more subtle effects like the skewness of the distribution in the surface layer I”)]. The second open parameter can be adjusted such that e.g. the surface energy or any other judiciously chosen quantity corresponding to the employed force has the right value. This extremely simple generalization of Thomas-Fermi theory yields, as a matter of fact, very satisfying results much improved over the pure Thomas-Fermi theory. We obtain for instance very nice smooth local densities which agree well in the surface region with more elaborate semiclassical theories, an example of which has been shown in one of our preceding publications 15). Another deficiency of pure Thomas-Fermi theory is that it over binds.“). That this is also largely cured by our improved TF theory can be seen from table 1 where we show the binding energies obtained from a self consistent solution for 40Ca with the Skyrme SkM*force using different approximations.

326

L. Vinet et al. / Disintegration TABLE 1 Binding energy for “%a calculated with the SkM* force in HartreeFock (HF), extended Thomas-Fermi (ETF), Thomas-Fermi (TF) and coherent state (CS) method [eqs. (3), (4)]

B(MeV)

HF

ETF

TF

CS

EXP

341.1

343.6

365.0

345.9

342.1

EXP is the experimental value.

Though the generalization of TF theory given by (3) cannot compete with more refined semiclassical theories ‘I) but for simplicity, its true virtue is revealed in dynamical situations. To this purpose let us first discretize expression (3): N A3RA3 f,(RP p) = nlz, ___Pe(&~-(H,(R”,pn)))d(R,-R,~~-~). Q&)3

(5)

In (5) the discretization has been performed in such a way that it corresponds to the widths of the gaussians and thus the explicit factors ARAp drop out through the normalization of the gaussians. The brackets on the classical energy means that we are not taking simply the classical energy of the gaussian but add to it its zero-point energy: d3R’ d3p’ (%(RP P)> =

@&)3

H,(R’, p’)d(R’-

R, p’-p)

.

This then corresponds to a projection of f(R, p, t) on a basis of coherent states 14). The discretization (5) is fine enough to ensure a completely smooth behavior of the static solution (about the dynamical case see below). The important advantage of (5) is now that we can let the centers of the gaussians evolve according to the time evolution of their mean values:

without destroying the stationarity of our problem. We therefore have in (5) with (6) and (7) a quite valuable self-consistent Thomas-Fermi theory which is based on a swarm of moving pseudoparticles (the frozen gaussians whose width is constant with time 16) which evolve according to Ehrenfest’s equations (7) without loosing the stationarity of our problem (it should again be noticed that because of the unit step function as a support for their centers no pseudoparticles can leave into the continuum). It is evident that this is an ideal preparation to initialize a dynamical process. The only modification being that the paths of the pseudoparticles are given

327

L. Vinet et al. / Disintegration

by an explicitly our method

time-dependent

of introducing

Thomas-Fermi

distribution

static case, first results

potential.

quantum

it is a priori not clear whether

Though

effects through

has as nice properties

of our procedure

for instance

corroborated

calculations

for slab collisions

a smearing

in dynamical

procedure situations

show that this is indeed

by a favorable

comparison

of the as in the

the case: this is

“) with corresponding

and r2C on “C at 84 MeV/u.

TDHF

Also we checked

that

basic quantities like mass, momentum and energy are conserved during the time. However, dynamical situations are so diversified that it is not clear whether the comparison is so favorable in all cases. In could for example be, just to mention one possibility, that in the dynamical break up of the nucleus into several smaller pieces (see sect. 5) the small pieces have a diffusivity larger than the one our gaussians were initially adjusted to. Though more detailed studies of our theory are thus certainly in order, it is our firm belief that the main ingredients are basically very sound and natural and it should be a promising tool for handling dynamical situations.

3. The spherical Vlasov equation: monopole oscillations compressed nuclei In the case of spherical symmetry the Vlasov equation partial waves of angular momentum L and we obtain:

8-L. pr dfL --3av at+------

(

ar

and expansion of hot and

can be projected

L2 afL -=o,

mr

)

dp,

where fL stands forf(r, p,, pi) at fixed value of L. The momentum to the radius vector is given by pI = L/r and p, is the component The folding

procedure

onto

(8) pl perpendicular of p parallel to r.

of eq. (3) allows us to write:

x @C&F-4r0,Pro,

LO)),

(9)

where IO is the lowest order spherical Bessel function. Expression (9) is then discretized as before and the time evolution of each coherent state is followed, allowing the computation of the physical observables of interest. We will now report on our results obtained by solving eq. (8) with (9). The nucleus is excited either by heating or by compression according to the method developed in ref. “). In this reference three regimes for disintegration of nuclei were obtained. For moderate initial temperatures (~8 MeV) an evaporation process is obtained

328

L. Vinet et al. / Disintegration

with formation of a residue. For larger excitation energies (initial temperature between 8 and 15 MeV), a bubble nucleus is formed, which is a particular solution due to the imposed spherical symmetry. Above 18 MeV initial temperature, complete vaporization is found. We will focus here on specific characteristics of the solutions. Essentially, two model cases have been considered. Indeed, the procedures used to initiate excitation energy in the nucleus shared in thermal and compressional components correspond in a first case (case I) to a sudden (diabatic) heating of the system already described in ref. “) and in a second case (case II) to a simultaneous compression and heating of the system. We also will briefly consider a third case where we enclose the nucleus initially in a large harmonic container, which at time f =0 is removed. Of course, all these initial conditions may not correspond very pl-ecisely to what happens in a heavy ion reaction but still we hope that these studies will shed light on physics of the deexcitation of highly excited nuclei. The preparation of the system is performed in a two-step process; (i) The self-consistent determination of the one-body distribution for the nucleus in its ground state ( T = 0 MeV) is performed. As mentioned above, the distribution is decomposed on a basis of coherent states by a folding method. The occupation numbers are those given by the Thomas-Fermi approximation for the single-particle energies of the coherent states. (ii) Sudden (diabatic) heating is performed by sudden modification of occupation numbers such that the local density remains unchanged from its ground-state value. Accordingly, the local momentum distribution is modified at each point r as follows: f,(r,p)+{I+exp(s(r’P:-“(I))}P’ I

(10)

with

~(r,p)-)$+ V(r). The chemical potential p(r) is determined by the prescribed value of p(r) and q is the initial temperature. For case II in addition to case I, some compressional energy is introduced in the initial condition by modification of the energy levels. The system is built selfconsistently in a one-step process at finite temperature by adding a Ar* constraint to the hamiltonian. The system being prepared by a sudden heating (case I) or by simultaneous compression and heating (case II), one assumes that it follows subsequently the dynamics of the Vlasov equation. In this section, the modified BKN force ‘) (with an incompressibility of 228 MeV) is adopted as effective interaction. Since the preparation procedures put some pseudoparticles (coherent states) into the continuum, one expects some evaporation process to occur. On the other hand,

L. Vinet et al. / Disintegration

329

the large amount of excitation energy will also induce collective modes. As a matter of fact, diabatic heating leaves the (local) density unchanged and in an infinite matter scenario the density does not correspond to its equilibrium value any longer which has a lower value. The finite nucleus therefore will start to expand and, for not too high initial temperatures, end up in a monopole vibrational mode. This we checked in calculating the mean square radius (r’) of 4oCa as a function of time for various temperatures. For Ti = 4 and 8 MeV we extracted monopole resonances at - 19 and - 15 MeV respectively what shows a temperature dependence which has similarly been found by other authors 18). Fig. 3 shows a comparison between a previous calculation of ref. “> for a sudden heating with q = 18 MeV and the case II described above with hi = 2 MeV. fm-* and Ti = 12 MeV. For the sudden heating case, the initial temperature exceeds the flash temperature, i.e. no residual nucleus is remaining after 2 - 10mz2s. It turns out that the same behavior is found in the second case (fig. 3b) where the initial temperature is much smaller than 18 MeV. This feature illustrates the efficiency of the compressional energy for disintegrating the nucleus. We will obtain a similar conclusion with the most general description in sect. 5 where no spherical symmetry is introduced and in which two-body residual interactions are taken into account. In fact the respective r6le of compressional and thermal energies has already been discussed in the framework of the hydrodynamical picture of ref. ‘1. The general conclusion is identical to our statement concerning the importance of the compressional energy in disintegration processes. On the contrary, the result obtained in solving the spherical Vlasov equation differs from the hydrodynamical picture by the anisotropy of the local momentum distribution which can reach extremely important values for the particles emitted into the continuum. Such a feature is absent in a hydrodynamical description based on a locally equilibrated velocity distribution. Examples are given in figs. 4 and 5. Fig. 4 shows two curves corresponding to the time dependence of the anisotropy parameter 2Q’flL for two radial distances I from the center of mass. The quantities K$ an n, are the parallel and perpendicular components of the pressure tensor “). At t = 3 = 1O-22s after the preparation stage (obtained with Ti = 0 and hi = 0.4 MeV fm-*) the respective spatial densities are 0.25~~ and 0.1~~ for r = 5 fm and r = 5.4 fm. For an isotropic momentum distribution (as for instance, in hydrodynamics) 2L711/17,is equal to unity. One can see that its values deviate substantially from the isotropic configuration with pronounced oscillations. The magnitude of the anisotropy increases strongly for increasing radial distances. This is shown in fig. 5 where for the same case as in fig. 4 we show at time t = 3 - lo-** s the momentum anisotropy as a function of radius with its dramatic increase in the external region of the nucleus as this was already qualitatively discussed in ref. ‘). We also investigated a third way of heating which corresponds to something like an adiabatic heating. This latter is supposed to be so slow that the density of the bulk always follows its value which minimizes the energy, i.e. it lowers its value

330

L. Vinet et al. / Disintegration

.20

. 15

. 10

.0.5

0.

2.

ft.

6.

8.

10.

12. I-

ffml

. 15

. 10

.05

0.

Fig. 3. Time evolution of density profiles in Masov dynamics for a ?Za eicited nucleus at initial time. (a) shows total disintegration of the system by sudden heating with an initial temperature Ti = 18 MeV above the flash temperature of ref. 6). (b) is displayed for comparison with fig. 3a. It has been obtained by simultaneous initial compression and heating. Initial temperature is Ti = 12 MeV and initial compression is determined by Ai = 2 MeV . fmF2 (see definition of case II in text).

L. Vinet et al. / Disintegration

331

r=!54fm

r=!SOfm

.8

I

I

I

I

I

I

I

I

1.

2.

3.

L.

5.

6.

7.

I

Fig. 4. Time dependence of the anisotropy parameter 211ii/Lr1in Vlasov dynamics. The full (resp. dashed) line corresponds to radial distance of 5 fm (resp. 5.4 fm) from the center of mass of 40Ca nucleus initially compressed with a compression parameter Ai = 0.4 MeV . fm-*.

1.3

-

t2

-

1.1 -

.9

-

1.

2.

3.

L.

5.

6.

rffml

Fig. 5. Radial distance dependence of the anisotropy parameter in Vlasov dynamics. The system is identical to that of fig. 4. The results have been plotted 3 . 1O-22s after the initial excitation.

332

L. Vinet et al. / Disintegration

with respect to the cold nucleus. For a finite nucleus this way of heating cannot be initialized statically because of the population of the continuum. However, confining the system into a large harmonic box Ar2 we may use as an initial weight for the pseudoparticles:

(11)

f,(r,p)={l+exp[(Hc(‘9~))-P]}-‘, I

where CLis now the global chemical potential determined from the particle number condition. The system being confined by the box we can perform a self-consistent calculation at finite temperature, a configuration which is used to initialize the dynamical evolution when at time t = 0 the confining field is suddenly removed (this is similar to the prescription used in ref. “)). In fig. 6, we show such an evolution for Ti = 15 MeV and A = 0.3 MeV - fm-‘. We see that at t = 0 the density at the center is quite low compared with its ground-state value (fig. 2 of ref. “)); also the dynamical evolution is quite different from the case of a sudden heating at the same temperature T (see fig. 2b of ref. “)). The decompression of the center is much slower and no bubble is formed. This comes from the fact that the bulk density in the scenario of fig. 6 is closer to its “equilibrium” value and no violent decompression is initiated. Rather the nucleus is losing particles by an evaporation like process and one could speculate that the flash temperature is even higher in the case of adiabatic heating than it is for diabatic (sudden) heating. A further interesting quantity which we studied is the evolution of the energy per particle of the remaining nucleus in the different scenarii with time. In fig. 7 we p Irllfm-3

)j

L.

6.

8.

10.

12.

rlfml Fig. 6. Same as fig. 3, but for an adiabatic heating. The system has been prepared in this scenario by confinement into a harmonic box hi? with Ai = 0.3 MeV . fmW2 and by simultaneous heating to an initial temperature Ti = 15 MeV.

333

L. Vinet et al. / Disintegration

0.

4.

8.

12.

16. time (10e2$

Fig. 7. Time variation of the energy per nucleon (upper part) and of the mass (lower part) for the residual nucleus in Vlasov dynamics for a 40Ca excited nucleus. The residual nucleus is defined by the part of the one-body distribution located within a sphere around the total center of mass with a radius of 8 fm. The full (resp. dashed) lines correspond to a sudden (resp. adiabatic) heating; the initial temperature is T, = 15 MeV and the stiffness of the confining harmonic box is Ai = 0.3 MeV . fm’ for the adiabatic heating.

show this evolution

for two cases: the first one is the case of a sudden

heating

to

Ti = 15 MeV. The corresponding bubble formation has already been displayed in ref. “). The second case is the one of adiabatic heating (Ti = 15 MeV) discussed in fig. 6. In both cases we see initially a very rapid drop of the excitation energy with which goes along

a sharp

drop in the mass. To define the residual

nucleus

and to

calculate the energy we choose a box of 8 fm radius. This is a bit small for case I of bubble formation and thus some oscillation can be seen which is due to the pulsating bubble. It is interesting to notice that the energy and particle loss has, on the time scale presented here, essentially stopped after 2-4. 10ez2 s. This is a somewhat intriguing fact because it is difficult to see why there should be such a sudden change in the regime of deexcitation at a point where the nucleus still has an excitation energy per nucleon of about 8 MeV per nucleon (which corresponds to a temperature of -11 MeV). It could be that this is a special property of the Vlasov equation; however, fully quanta1 calculations show similar features i9). Indeed the rapid drop of mass and energy seen in fig. 7 corresponds very likely to an early escape out of the considered volume of those pseudoparticles which are initially in the continuum but the particles which are in the well, i.e. essentially

334

L.. Vinet et al. / Disintegration

between the Fermi level and the top, have a much harder time to leave the nucleus. We will show in sect. 5 how the introduction of two-body collisions remedies this situation and a much more continuous drop of the excitation energy will be seen there. We conclude this section in summarizing and commenting on our principle findings: (i) for sudden heating, three expansion regimes (evaporation, bubble formation and flash) are found. (ii) Compressional excitations are more efficient for disintegration than thermal excitations (see also ref. ‘)). (iii) Adiabatic heating shows no bubble formation and a flash temperature which is similar as in case (i). These flash temperatures are considerably higher than those found in static calculations le3). 0 ne explanation could be that cooling due to particle emission is very rapid due to: (iv) Strong local momentum anisotropies, for instance among the emitted particles. It may be that relaxation of the spherical symmetry constraint lowers the limiting temperature at which total disintegration of the nucleus occurs. It is, for instance, clear that a bubble formation is an artifact of spherical symmetry. It is very likely that dynamical instabilities such as low frequency density or surface waves will break the bubble apart. We will study such questions in the next paragraph. 4. The Vlasov equation (VE) in three dimensions (3D) In the spherical case we had to deal effectively with three phase-space dimensions, a case where we still could pave the phase space uniformly with our coherent states. The six-dimensional phase space of the general case is numerically much harder to treat and at present a uniform pavement of phase space is technically not possible. We therefore selected the N coherent states (between five and ten thousand in the cases chosen here) at random in a phase-space volume defined by the static Thomasfermi limit. This method amounts to computing the folding product of eq. (3) by a Monte Carlo method. The number N is chosen by requiring the numerical stability of the solution against its variation. We have checked that taking N equal to 100 times the number of involved nucleons allows a good description of the dynamical processes we are considering here since doubling this number does not significantly change the dynamics. Nevertheless, its finiteness implies some numerical fluctuations which could affect the evaluation of some observables such as for example the high energy spectra. Also the multifragmentation we will discuss below is somewhat dependent on it, though the sampling is sufficiently dense to get completely smooth local quantities such as density and kinetic energy density; these are the only quantities which are in fact needed in our 3D calculation, since we will be using the very simple Skyrme-like force of Aichelin et al. *) (the limiting temperature

L. Vine? et al. / Disintegration

335

obtained in static calculations with this force is around 11 MeV*). This fact does unfortunately not allow at present to decide clearly whether the disrupture of the nucleus we will observe below is because of mechanical instability due to fluctuationinduced break up or whether the system runs into a dynamical instability. It is clear that the constraint of spherical symmetry which we imposed in the last paragraph represents a severe restriction to the dynamics. Though it is true that symmetry is a conserved quantity of TDHF il) and consequently also of VE (the same holds true for the extensions including two-body collisions) and thus starting out from spherically symmetric initial condition the solution should in principle stay spherical forever. However, there can appear dynamical instabilities such as they are well known from hydrodynamics and RPA theory 20,‘1) (which is the small amplitude limit of TDHF or VE). These instabilities occur whenever one of the collective frequencies approaches zero indicating a shape or phase transition. The slightest numerical noise suffices in such cases to let the instabilities grow exponentially. Such situation will very likely happen for the break up of the bubble which we described in the last section; but also without bubble formation simply by expansion of the nucleus, i.e. by lowering its central density as we saw in fig. 6 (or fig. 2c of ref. “)) the system will probably quickly run into shape instability and subsequent rupture of, for instance, quadrupole type. Indeed the isoscalar particle-hole interaction has a strong density dependence (this is known from Landau theory “)) being almost zero at normal density but becoming strongly attractive as one lowers the density. In this way the low-lying (hydrodynamic-like) 2+ modes will quickly generate an instability as a function of the degree of dilatation of the nucleus**. For illustration we show in fig. 8 for the Gogny force *‘) (but any realistic force should give similar results) the density dependence of the Landau parameter Fz” which is related to the nuclear incompressibility. In view of the very strong density dependence of this force parameter, it is easily conceivable that - as the system expands (see fig. 6) - as soon as the bulk density reaches values around half its saturation density or below, the nucleus ruptures into pieces because of the dynamical instability triggered by the increasingly low energy of the low-lying hydrodynamical quadrupole modes “). A closer study of this phenomenon will be presented in a future publication. One should for instance investigate the time scales on which such phenomena can occur before drawing definite conclusions. An example of such a scenario can possibly be seen on the 1.h.s. of fig. 9 where the 3D evolution of an 40Argon nucleus initially heated to Ti = 15 MeV with the sudden recipe is presented. We see that the nucleus after t = 200 fm/c has ruptured essentially into three pieces. Of course at this stage we cannot exclude the possibility that this is due to pure mechanical break up. We do l We thank E. Suraud for providing us with this information. *‘As an interesting side remark we would like to mention that, if we could constrain the density of a nucleus to lower values statically, there should appear a phase transition to quadrupole deformation even with no she21efects present, i.e. for a hot nucleus or a very large nucleus.

Fig. 8. Variation of the monopole Landau parameter FT as a function of local Fermi energy e&R)/,& for the Gogny force (ref. 2’)). A is the Fermi energy ~~(0) at R - 0 fm (we are very grateful to R.W. Hasse for furnishing the numerical values).

not claim here that the three pieces of nuclear matter shown on fig. 9 have anything to do with what one measures experimentally, first of all because the fragments may rupture further, evaporate particles and so on and secondly because our mean field theory does not project on a definite exit channel as we said earlier. Nevertheless we think that the onset of freeze out due to instabilities at t - 50 fm/c and p - 0.5~~ is quite realistic (this is to be weighted with the quite crude force we are using here} and that this kind of fragmentation process can easily enter into ~mpetitio~ with multifragmentation due to the fluctuations possibly present after violent ion-ion collisions (a proposal which has been made recently by Knoll et al. 22) and which is compatible with the percolation theory of multifragmentation put forward by Campi and coworkers 23); see however our remarks on ffu~tuations we will make in the following section). The result we can see on fig. 9 obtained with the 3D-VE at Ti = 15 MeV represents roughly also the case of limiting temperature beyond which no remainder of the initial nucleus persists; there exists however some transitional region where it is difficult to decide whether there is still a remainder of the initial nucleus or whether the system has ruptured into several pieces. Certainly a lower limit is given by Ti = 13 MeV and one can thus say that the release of the constraint of sphericity has lowered by 2 to 4 MeV the limiting temperature a nucleus can initially be heated to before it subsequently completely disintegrates; the

L. Vine? et al. / Disintegrafion

331

lot

ibl

I’

I*

I

’

1

D .‘,‘,,

“,

‘,. ..‘T’.

;:

.:‘.

:

’ ,.

‘f:.. .;

‘..

,,.’ .:..,

: . . :. -:,

:,,:.

j ‘,’

,;.

(I.

::.::-.‘;

.::

_‘;

. ...:

. .

‘.:;

I’” 11 -70,

I

-5.

1

I

R

5

IJ x3

zifm) Fig. 9. Two-dimensional density dist~butions ,f( z, x) as a function of time during the expansion of a .^ suddenly heated *“Ar nucleus (initial temperature T,= 15 meV). Fig. 10a shows results for a three dimensional Vlasov dynamics. Fig. lob is obtained by solving VUU equation.

introduction of the 3D evolution has however not dramatically altered the situation with respect to the spherical case. In order to push the comparison between spherical and 3D evolution further we also present the case of a sudden heating to Ti = 8 MeV in using the force of Aichelin et al. *). The corresponding time evolution of the local densities for 4oAr is presented in fig. 10. We see that the nucleus essentially undergoes a large amplitude monopole vibration of the same amplitude as well in the spherical as in the 3D evolution but the time evolution seems to be somewhat faster in the spherical than in the 3D case. This could be due to the fact that in the 3D calculation because of our phase-space sampling method some fluctuations of other than the monopole mode are present and exchange their energy in the course of time; this

I... Vinet et a!. / Disintegration

338

p

lrl lfmT3/ .I8

CA LCXJLA JION

2.

4.

6.

8.

ro. r

12.

ffmf

p Ir) 1 fm-‘1

. t8

CALCULAJION .06 .cJ2

Fig. 10. Comparison between spherical calculation (upper part) and 3D calculations (lower part) in Vlasov dynamics. Density profiles of an initially excited 40Ar nucleus are displayed for various times t (in 1O-22s). The initial conditions are those of sudden heating with an initial temperature Tj = 8 MeV. Calcuiations have been performed with the effective interaction of ref. ‘)_

can act like a (one-body) viscosity on the monopole mode a feature which is evidently absent in the spherical case. 5. DynamicaI evolution with the Vlasov equation plus collision term (VUU) Excitation af nuclei during the preparation of the system makes very likely a rearrangement of occupation numbers during dynamical evolution. As a matter of fact, excitation corresponds to a substantial opening of phase space where the residual two-body interaction is expected to play a r$Ie. For instance, Pauli blocking is less and less effective as temperature increases. In this context it is therefore very important to extend the 3D Vlasov equation by the introduction of a collision term as it was done for the microscopic description of intermediate energy heavy ion

L. Vinei et al. / Disintegration

339

In particular we want to reexamine the question of dynamical instabilcollision 14*15). ity occurring above an initial temperature threshold of Ti - 13 MeV in the 3D Vlasov dynamics (see preceding paragraph). For the collision integral of eq. (1) we use the form given by Uehling and Uhlenbeck24) which can be shown to be the classical limit of its corresponding quantum analogue 9*25).

I(r,p, t)=

d3p, d3p, d3p, JJJ (27rh)9

dg *(z

>

S(p,fp2-p3-p4)6(&1fE2-&3-E4)

cff [J’,&,f,f,--j’,.&fif2J. NN

(12)

The notations are E (resp. p) for single-panicle energy (resp. momentum); f = 1 -f, the effective nucleon-nucleon cross section (d~/d~~~~ is given by the scattering inside the medium and could be estimated by a G-matrix calculation 26). The calculation was performed by the particle in cells method applied to coherent states involved in the description of the one-body distribution function. More computational details can be found in refs. 14,15). The collision term makes the momentum distributions relax towards equilibrium along the dynamical paths. It reduces the mobility of pseudoparticles inside the nuclear medium due to a decreased mean free path of the nucleons “). Consequently nuclear expansion will take place with substantial two-body viscosity. The effect of this viscosity on the dynamics is shown on the right column of fig. 9 where it is immediately seen that, in comparison with the calculation without two-body collisions (left column), viscosity has a stabilizing effect: there clearly stays a central remnant of the initial nucleus in the VUU case whereas in the VE case the nucleus breaks up. Thus the limiting temperature 7; beyond which a nucleus flies apart has gone up again with respect to the 3D-VE case and closes up with the earlier findings of T,- 17 MeV of the purely spherical evolution described in sect. 3. Let us again study the most interesting evolution of the energy per particle corresponding to the VUU dynamics. In fig. 1la we present the case corresponding to fig. 9 whereas in fig. llb the time dependence for a lower initial temperature ( Ti = 12 MeV) is shown. We immediately see a striking difference with the VE case of fig. 7: two-body collisions help very much to evacuate thermal energy into the continuum and where, in the VE case, we had stabil~ation of the excitation energy after t = 2-4 - 1O-22s, the VUU case keeps cooling down quite rapidly; this in spite of the fact that the mass evolution is quite similar in both cases. So we can conclude that it is not only the viscosity which makes the evolution of the nucleus more cohesive but also the rapid cooling mechanism due to two-body collisions. Having now the full VUU dynamics at hand we can make a more complete study on the dynamical evolution of highly excited nuclei. As we discussed already in sect. 3, we have to distinguish between several ways of exciting a nucleus. Again let us first discuss thermal and compressional excitation. For the 3D-VUU case the thermal energy is introduced as in sect. 3 by a sudden heating. For the 40Ar nucleus

340

L. Vinet et al. / Disintegration E/u (Mel//u)

(a)

t

100

t Ifm/cl

300

200 TIME

CoAr 30

-

EXPA~SIGN T= 12 MeV

\ \ \ \

.

t

N

VW

/////////////~////

// GROUND

I TOO

“OAr I

I I 200

300

STATE I

>

f lfm/cl

TIME

Fig. 11. Same as fig. 7 upper part for the 3D expansion of an “Ar nucleus with Ti = 15 MeV (fig. 1la) and Ti = 12 MeV (fig. 1lb) for the force of ref. ‘). The time evolution of the energy per nucleon of the residual nucleus (defined by integration inside a sphere around the total center of mass with a radius of 8 fm, as in fig. 7) is given for Viasov dynamics (full curves) on the one hand and for VUU dynamics (dashed curves) on the other hand.

341

L. Vinet et al. / Disintegration

the excitation energy E* = E - E (ground state) per nucleon is displayed in fig. 12. as a function of T2. The corresponding level density parameter varies in this temperature range between &,A and &A. For the generation of compressional energy it seemed for us however convenient to do it differently from sect. 3 by switching on a velocity field in radial direction at initial time. The compressional field is defined by k = -hr where r is the radius vector and h the strength. At initial time this field is applied to each coherent state entering into the decomposition of the ground-state one-body distribution function. Fig. 12 shows the variation of excitation energy per nucleon versus the strength coefficient h making possible comparisons between the different modes of excitation at fixed excitation energy. For each of the above modes of excitation, one obtains two possible regimes for the expansion as already described in the previous section devoted to 3D calculations without the collision term. On the one hand, there is an evaporation process leaving a residual nucleus after removal of excitation energy by the more energetic particles; on the other hand, “multifragmentation” can occur due to dynamical instabilities (or mechanical rupture) above a certain threshold in excitation energy. The number of fragments produced within our theory is a measure of the degree of instability but, as already mentioned, the residual fragments cannot be identified with real nuclei due to the one-body character of our approach. For illustration, we show in fig. 13 the fragmentation pattern obtained by a sudden heating to 17 MeV as it is obtained from the VUU equation.

t i

h

e Lu

0.

I

1. .5

.05

.I

L

.15

I

I

*I

.0.5

t

.2

I

I

1

.25

.Of .003

i

)

h tfm-*j

.3 )

N/N,

Fig. 12. Excitation energy per nucleon for an 4oAr nucleus as a function of excitation parameters, namely: T* for a sudden heating, h for compressional velocity field, Nf iV, for fluctuations, where N is the number of coherent states per nucleon in our caiculations. IV, = 100 corresponds to a number beyond which retention of a numerically stable solution is guaranteed.

L. Vinet et aL / Disintegration

342

5. 0 -5.

-10.

- 5.

0.

5.

10.

-10. -5.

0.

5.

10. 2

Fig. 13. Multifragmentation Ti = 17 MeV. The pictures

(fml

pattern in dynamical expansion of an 40Ar nucleus, suddenly heated at are similar to those of fig. 9 and correspond here to VUU dynamics. Time I is expressed in fin/c units.

In fig. 14 the number of fragments (equal to one for the evaporative regime and larger otherwise) is plotted versus the initial excitation energy per nucleon for a system with a total mass 40. It turns out that also with the VUU approach the efficiency of the two modes of excitation to break the nucleus apart is very different. For compression, the onset of rupture is found at -6.5 MeV per nucleon, whereas

Fig. 14. Number of fragments (determined by visualization of plots identical to figs. 9 and dynamical expansion of @Ar nucleus. The three curves correspond to various initial preparations system described in the text, the initial excitation energies being given in fig. 12.

13) in of the

.L Vinei ei al. / Disintegration

343

LO

TIME

-

30

-

20

-

70

i fm/cl

30.

30

T=13 Mel/, na. = IOOi

20.

20

---.___ 10.

-10

T/ME

ffm/cj

344

L. Vinet et al. / Disintegration

sudden heating is less efficient since -17 MeV per nucleon are needed to make the system disintegrate. In comparison with the 3D case without collision term, this value of 17 MeV per nucleon for initial excitation energy generated by sudden heating is larger by -3-4 MeV of initial temperature indicating again a quite strong quenching of the multifragmentation process due to two-body viscosity and rapid cooling of the system in the VUU case. As far as the emitted particles are concerned, the one-body observables are similar for the two expansion regimes. Their mean kinetic energy decreases as function of time; the most energetic ones escape during the first stage of emission (figs. 15a, b for T = 13 and 17 MeV, i.e. below and above the instability point). Their mean neutron and proton numbers increase with time. If the time is large enough, their sum reaches the total mass for the multifragmentation regime but never exhibits any sudden variation. 6. Some remarks on fluctuations It is clear that the way we initialize our hot and/or compressed nuclei is not unambiguous. In principle the initialization should be the result of an adequate theory for the collision of two heavy ions. As a matter of fact we think that the VUU approach is, for the single particle distribution function, a quite reasonable approach to initialize the system and we have done such calculations the results of which are partially given in ref. i4). The conclusion we can draw from these studies is that up to energies of GANIL, i.e. -60 MeV/u the VUU approach does not permit to deposit more than T = 10 MeV thermal excitation energy and only very little compression of the order p/p,,= 1.2 can be reached. We therefore have to go to higher energies to reach the compression and limiting temperatures we have been considering in this paper. However, the VUU approach, as presented in this paper, has to be modified to include some relativistic effects before it can be used at these higher energies. The point we would like to make here is that the VUU equation should be realistic enough to generate in principle, via two-particle collisions, the fluctuations in the phase-space distribution which in the expansion phase then probably trigger very much the multifragmentation process. It is a different story to determine from which energies and under which conditions strong fluctuations in the one-particle distributionf(r, p, t) such as holes (Swiss cheese) and ripples are produced. It has been our experience that the initial fluctuations which are present in our 3D calculations due to our sampling procedure have rather the tendency to become washed out due to the two-body collisions. In any event, the question of to put them into the initial condithe fluctuations has incited some authors 22,28*30) tions more or less by hand and then let the system evolve according to some one-body dynamics. Indeed strong multifragmentation patterns are found which however will certainly depend again on the way the fluctuations have been put initially. We want to show here that it can lead to quite ambiguous results if the fluctuations are

345

L. Vinet et al. / Disintegration

introduced

into for instance

more initial

fluctuations

states which

creates

the VUU dynamics

in ad hoc manner.

into our theory consists

in rarifying

more and more holes in the phase-space

One way to put

the number

of coherent

distribution.

Indeed

one

tempting procedure could consist in putting just one coherent state per nucleon 27*28).Since the basis of coherent states solves the VUU equation via the semi classical equations of motion (7), this then very much resembles classical molecular dynamics 29) supplemented by the Pauli principle. We would like to stress the point however that there is no special reason to take the number of coherent states equal to the number of nucleons. On the contrary, strong initial fluctuations or not, in any case we should make sure that the subsequent dynamical evolution follows the hypothetically exact solution of VUU equation as closely as possible. It is our belief that this can only be achieved if we use at least of the order of 50 coherent states per nucleon. In order to demonstrate the ambiguity of the results obtained with different degrees of rarefication of the numbers of coherent states, we also display in fig. 12 the excitation energy per nucleon as a function of the decreasing number of coherent states and in fig. 14 the corresponding increase of the number of fragments. We see that the curves pass through the point N = 40 (our number of nucleons) very steeply and no definite conclusions can therefore be drawn from the point where the number of coherent states equals the number of nucleons. On the contrary, one should be careful that one does not calculate in this way approximately the correlations of different phase space points of the one-body distribution which have nothing to do with genuine two or more nucleon correlations.

7. Conclusions We have studied in this work the dynamical nuclei using increasingly general approaches

expansion of hot and/or compressed of the Vlasov and Vlasov-Uehling-

Uhlenbeck kind. In a first step we constrain the system to sphericity and considerably widen the investigations which we published earlier 19). We find that compressional energy is much more efficient to destabilize a nucleus than thermal energy, a result which is in qualitative agreement with the hydrodynamical approach of Barranco et al. ‘). Depending

on the initial temperature

we obtain

strong monopole

vibrations,

bubble formation or total vaporization. With the modified BKN force (compressibility of 228 MeV) a flash temperature i.e. total vaporization (corresponding to diabatic heating) of about 17 MeV is obtained for 40Ca, what seems to be higher than typical values obtained from static calculations ‘). The study of the energy per nucleon in a box around the nucleus shows that within 2-4 * 1O-22 s there is a drastic drop in excitation energy and mass number whereas, for later times, these values essentially stabilize. This comes from the fact that only the particles initially in the continuum can escape the nuclear volume quickly whereas the bound nucleons have a very hard time to escape. This finding is in qualitative agreement with fully quanta1 calculations “).

L. Vinet et al. / Disintegration

346

Releasing the constraint of spherical symmetry, i.e. going over to the 3D case, lowers by 2-4 MeV the limiting temperature at which a stable remainder of the nucleus subsists because of early break up of the bubble into fragments. We speculate that this break up is due to long wavelength dynamical instabilities but a purely dynamical rupture due to the initial fluctuations induced by our Monte Carlo sampling method in the three-dimensional case cannot be excluded. We have discussed the physics of instabilities of the low-lying hydrodynamical modes in some detail. The introduction of two-body collisions a la Uehling-Uhlenbeck still modifies the evolution process in a significant way; firstly we find that two-body collisions, i.e. viscosity, make the nucleus more reluctant to break up. The limiting temperature is again increased and reaches roughly the same value of -17 MeV as in the case where spherical symmetry was imposed. This value should be compared to the limiting temperature of - 11 MeV which is obtained from an equilib~um calculation (see footnote in sect. 4). Besides the viscosity which renders the evolution of the nucleus more cohesive (a feature which is intuitively unde~tandable: it is more difficult to splash into pieces a drop of honey than a drop of water) there is a second effect which goes in this direction and which makes differences with the case without two-body collisions: after the first spill off of those nucleons which have been initially in the continuum, the two-body collisions help very much to further transfer nucleons into the continuum and therefore cool down the nucleus much more efficiently than with pure mean field dynamics {which therefore has to be considered quite insufficient in this respect). The importance of the effect is shown in fig. 11. Having our most general evolution equation, i.e. 3D-VUU at hand, we make at the end a more quantitative study of the limiting temperatures as a function of compressional and thermal energies. It is found that, at low excitation energies, compressional energy is about three times as effective to break a nucleus as sudden heating, whereas at high excitation energies there seems to be a cross over (see fig. 14). We would like to thank E. Suraud and D. Vautherin for useful discussions. References t) 2) 3) 4)

S. L&it and P. Bonche, Nucl. Phys. A437 (1985) 426 E. Suraud and D. Vautherin, Phys. Lett. 138B (1984) 325 E. Suraud, Nucl. Phys. A462 (1987) 109 G. Auger, E. Plagnol, D. Jouan, C. Guet, D. Heuer, M. Maurel, H. Nifenecker, C. Ristori, F. Schussler, H. Doubre and C. Grtgoire, Phys. Lett. 169B (1986) 161 5) M. Barranco, H. g, J. Nemeth, C. Ng& and E. Tomasi, La Rabida, Summer School on Theory of nuclear structure and reactions (July, 1985); M. Pi, M. Barranco, J. Nemeth, C. Ng& and E. Tomasi, Phys. Lett. 166B (1986) 1 6) L. Vinet, F. Sebilfe, C. Gregoire and B. Mmaud, P. Schuck, Phys. Lett. 172B (1986) 17 7) H.S. Kiihler and B.S. Nilsson, Nucl. Phys. A417 (1984) 541

L. Vinet et al. / Disintegration 8) J. Aichelin and G. Bertsch, Phys. Rev. C31 (1985) 1730 9) L.P. Kadanoff and G. Baym, Quantum statistical mechanics, 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)

21) 22) 23) 24) 25) 26) 27) 28) 29) 30)

Frontiers

347

in Physics

(Benjamin,

1962,

New York) H.S. Kiihler, private communication P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980) P. Schuck, Contribution to Density functional methods in physics, NATO ASI Series B, vol. 123, p. 417 M. Durand, U.S. Ramamurthy and P. Schuck, Phys. Lett. 113B (1982) 116 C. GrCgoire, B. Remaud, F. Sebille, L. Vinet and Y. Raffray, Nucl. Phys. A465 (1987) 317 and Phys. Lett. 18OB (1986) 198 B. RCmaud, F. SCbille, C. Gregoire, L. Vinet and Y. Raffray, Nucl. Phys. A447 (1985) 555~ E.J. Heller, J. Chem. Phys. 64 (1976) 63 L. Vinet, PhD thesis, Orsay (1986) D. Vautherin and N. Vinh Mau, Nucl. Phys. A422 (1984) 140 P. Bonche, D. Vautherin and M. Veneroni, J. de Phys. C4 (1986) 339 J. Cugnon, Phys. Lett. 13SB (1984) 374; H. Schulz, B. Kiimpfer, H.W. Barz, G. Riipke and J. Bondorf, Phys. Lett. 147B (1984) 17; D. Vautherin, Proc. of 1st Int. Spring Seminar on Nuclear physics, Microscopic approaches to nuclear structure calculations, Sorrento, May 1986 D. Gogny and R. Padjen, Nucl. Phys. A293 (1977) 365 B. Strack and J. Knoll. Z. Phys. A315 (1984) 249 X. Campi, J. Desbois and E. Lipparini, Proc. XXI Int. Winter Meeting on Nuclear physics, Born-no, 1985) E.A. Uehling and G.E. Uhlenbeck, Phys. Rev. 43 (1933) 552 P. Danielewicz, Ann. of Phys. 152 (1984) 239 A. Lejeune, P. Grange, M. Martzolff and J. Cugnon, Nucl. Phys. A453 (1986) 189 A.H. Blin, R.W. Hasse, B. Hiller and P. Schuck, Phys. Lett. 161B (1985) 211 J.A. Rosenhauer, J. Aichelin, H. Stocker and W. Greiner, J. de Phys. C4 (1986) 395 J. Aichelin and H. Stocker, Phys. Lett. 176B (1986) 14 G. Gale and S. Das Gupta, Phys. Lett. 162B (1985) 35