PHYSICAL REVIEW C MARCH 1997VOLUME 55, NUMBER 3

Resilience of nuclear matter in light ion induced reactions

M. Colonna, J. Cugnon,* and E. C. PollaccoCEA DAPNIA, CE Saclay, F-91191 Gif-sur-Yvette Cedex, France

~Received 23 July 1996!

Cavitation and heating of the target nucleus in the first instances of3He-induced collisions in the GeV/nucleon range are investigated in an intranuclear cascade model for the formation of this structure and astochastic one-body dynamics calculation to study its evolution. The hard collisions having essentially ceasedwhen the structure is fully developed, the latter model is particularly suited to study the possible breakup of thesystem. It is shown, however, that the target recovers a spherical shape rather rapidly, and has thus a goodchance to decay by standard evaporation, justifying the use of a cascade1 evaporation model to analyze thedata. It is also shown that the system has to be much more modified to break up into pieces instead ofrecovering a compact shape: in these reactions, it is thus expected that nuclear matter is resilient to shapedeformation and thermal excitation. Arguments are given to explain that expansion of the system, not impor-tant in these reactions, is required to overcome this resilience.@S0556-2813~97!03102-6#

PACS number~s!: 25.55.2e, 21.65.1f, 24.10.Lx, 25.70.Pq

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I. INTRODUCTION

Heavy-ion collisions in the incident energy range coving 100 MeV/nucleon to 2 GeV/nucleon are often discusin terms of a two stage scenario; the first stage correspoto the formation of thermalized sources and the secondto their decay@1–3#. In general, to simplify the phenomenological investigations, these sources are assumed to acquspherical structure. The latter hypothesis is supported bycroscopic transport calculations@2#. In accordance, statisticapartition @4–7# or evaporative models@8# are applied to de-scribe the evolution from spherical and thermalized emitteIt is, however, understandable that once the dynamicaltures of the first stage become significant, the thermalergy, angular momentum, shape distortion, and perhaps oeffects have to be consistently taken into account in theond stage.

Several theoretical works have recently pointed out tsources with exotic shapes~such as doughnuts, for instanc!could be formed in low energy (.20 MeV/nucleon! heavy-ion collisions@9–11#. However, the conditions under whicthese shapes could occur in real world are not well knoand seem to be so special that, very likely, this should crespond to infrequent events. Formation of transient uneshapes can very well occur in light ions induced reactionsRef. @12#, it was suggested that 1 GeV protons drill a conichole in the target during the first moments of the collisioActually, it would be more appropriate to speak of a waleft behind by the incoming proton, since the nuclear denis not vanishingly small in this wake, but is depleted to t2/3 of its normal value. Antiprotons could give a simileffect , even at very lower energy@13#. Recently, Wanget al.@14# have suggested, in the light of Boltzmann-UehlinUhlenback~BUU! calculations, that this wake effect is amplified by light ions. These predictions depend markeupon the BUU model. Two models are employed, wh

*Permanent address: Physics Department, University of LieB-40009 Sart-Tilman Liege 1, Belgium.

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basically differ by the surface tension. In the first modwith the smaller surface tension, the hole may cover a lafraction of the nuclear volume and last for a relatively lotime. In the second model, the hole has a smaller extenand closes, say, 30 fm/c after the beginning of the collision

The authors of Ref.@14# concentrate on the3 He 1 Agsystem, for which experimental measurements have bperformed in the 0.6 – 1.8 GeV/nucleon domain@15–18#.Analyses of the measurements have been performed inframe of the following two-stage model: an intranuclear ccade~INC! model for the formation of the excited source bthe hard scattering processes and an evaporation modeits deexcitation~alternatively, an expanding emitting sourcmodel @19# has also been used!. The INC output ~mass,charge, and excitation energy of the residue! is used as inputfor the evaporation model. It has to be remarked thatjunction in time t jun when to stop the INC and pass ondecay is of consequence. Similar difficulties are encountewith other ~BUU, QMD, . . .! descriptions and relate to thfact that INC does not have all the ingredients to describedecay. The problem of the determination oft jun has beencarefully investigated in Ref.@20#. In a study of the temporaevolution of a number of parameters~excitation energy,number of ejectiles, and so on! a change in their rates iobserved at about the same time. Beyond this time the taphase space density becomes uniform and the system enucleons at relatively low energy and with an isotropic dtribution. This observation strongly suggests that thet junshould be identified with this time. This two-stage coupliprovides a good description of the bulk of the data@17,18#.

The above considerations raise two interesting questio~i! Does the hole close fast enough for the two-stage moto be applicable~the evaporation model implicitly assuminthat the target remnant is spherical at the end of thestage!? ~ii ! What are the frequency and characteristicsevents in which the target remnant retains an exotic shwhen it decays? We address these questions in this paOur strategy is the following. We first use the INC modelRefs. @21,22# to investigate the density distribution in thvery first moments of the collision. This model using a stae,

1404 © 1997 The American Physical Society

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55 1405RESILIENCE OF NUCLEAR MATTER IN LIGHT ION . . .

and constant mean-field potential is not appropriate enoto study the evolution of the shape of the system whenhard baryon-baryon collisions are over and when the hstarts to fill in. To study this stage reliably we employstochastic one-body approach~SOBA! @23#, which has beenproven to be an adequate tool for describing the possfragmentation of the system coming from the amplificatiof unstable fluctuations by the mean-field dynamics. Accoingly, the paper is divided as follows. In Sec. II, we presethe INC calculations for the 3He ~1.6 GeV/nucleon!1 107Ag collisions. We so determine the shape, the mass,excitation energy and the~hydrodynamic! velocity field ofthe target remnant at the time when the hole has the larextension. In Sec. III, these characteristics are used as iin the SOBA calculation to follow the evolution of the remnant. We extend this analysis to different choices of the inparameters to study under which circumstances exotic shwould fragment before reforming a spherical remnant andlook for a possible characterization of such fragmentationSec. IV, we shortly discuss the validity of the two-stage pture for light ion induced collisions. Finally, Sec. V containour conclusion.

II. CAVITATION IN THE FIRST STAGEOF THE COLLISIONS

In order to investigate the deformation of the target dsity distribution during the first moments of the collisiowe performed an INC calculation of the collisions of 1GeV/nucleon3He projectiles on107Ag targets. The detail ofthe INC model can be found in Refs.@21,22#. The averagedensity profile in the reaction plane forb50.1 fm collisionsis given in Fig. 1 and was obtained by considering 4000 ruThe wake left behind by the projectile is clearly visible ais mainly developed fort.15220 fm/c. At this time, thedensity depression reachesr0/2, r0 being normal nuclear

FIG. 1. Nucleon density distribution in the reaction planeb50.1 fm collisions of 1.6 GeV/nucleon3He projectiles on a Agtarget, at various times, as given by the INC calculation. Ft&15 fm/c, the nuclear density ahead of the projectile is onslightly modified ~the slightly lighter appearance of the targett56 fm/c, compared tot50 fm/c is an artifact of the plottingroutine!. The wake left behind by the projectile is clearly visiblThe average density att530 fm/c is slightly less than normanuclear matter density. For sake of clarity, the particles outsidetarget volume are not displayed.

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matter density. Incidentally, we want to point out that wused the experimental density distribution for3He ~see Refs.@24,25# for details!. The more important wake predicted ithe first calculation of Ref.@14#, that is called cavitation, maybe due to the choice of a too compact configuration for tprojectile. In any case, the formation of a wake in both BUand INC calculations is due to the initial nucleon-nuclecollisions. The mean field plays a minor role in the formatiprocess. In our calculation, the wake disappears progsively after 15220 fm/c because nucleons are running inthe partially empty space. At 30 fm/c, the density is almostuniform ~this corresponds to the timet jun at which the rate ofvariation of the number of ejectiles and of the excitatienergy is changing slope, as we indicated in the Introdtion!. This development is favored by the presence, inINC model, of a constant potential well in the target volumNucleons traveling backwards can be reflected by the potial wall. It is expected that, in reality the potential well, omore exactly the mean field, is deformed according todensity distribution although it is not clear that it followinstantaneously the density distribution, when the latterperturbed by very hard collisions, since the mean-fieldnamics is basically associated with soft processes. In thesection, the evolution of the wake is studied by incorporatthe dynamics of the mean field.

The time evolution of the target excitation energyE* andof the target remnant massArem, defined as the excitationenergy and mass number inside the original target volu@12#, is displayed in Fig. 2. As noted in the Introductiothere is clearly a change in the corresponding curves arot;30 fm/c, which can be identified ast jun . We will comeback to this figure in Sec. IV.

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FIG. 2. Time evolution of the excitation energyE* ~lowercurves, scale on the left! and of the target mass numberArem ~uppercurves with triangles, scale on the right! in b50.1 fm collisions of1.6 GeV/nucleon3He ions on a Ag target. The open symbols corespond to the INC calculation and the black symbols to the SOcalculation using input data extracted from the INC resultst;20 fm/c. The curve with the open squares is obtained fromcurve with the open circles by removing the mass energy ofD particles. See text for more detail.

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1406 55M. COLONNA, J. CUGNON, AND E. C. POLLACCO

To investigate the average~or hydrodynamical! currentinside the target, we divide the space in cells and compthe following quantity

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r~rW !.

The linear size of the cells is about 0.6 fm. We discussmain properties of the velocity field. At the beginning of thcollision, the velocity field is concentrated on the projectiAs time evolves, the longitudinal current spreads considably, reflecting the sideward diffusion of the particles ivolved in the current~see Fig. 1!. It is also largely slowingdown. This does not mean, however, that there are noparticles. But the latter are contributing only slightly to thmacroscopic current. As expected, the velocity field acqua perpendicular component. This current is much smathan that along thez direction, even when the wake is fulldeveloped. Just to give an idea, let us mention that, attime, thez component of the velocity field is maximumlittle bit ahead of the apex of the cone, with a valueb;0.2. The perpendicular component has a maximum ablute value of;0.07.

The formation of a wake is observed for impact paraeters up to 3 fm, with an amplitude that decreases withcreasing impact parameter. For all higher impact paramethe density is only slightly modified, and the nucleus esstially retains a spherical shape. Therefore, we can safelyan upper limit on the ‘‘cross section for the wake formationof the order of 270 mb for this system. The total cross stion being of the order of 2 barns, the fraction of this kindevents is of the order of 10 to 15 % at the most. This fractis expected to be the same for medium heavy and hetargets.

III. RESILIENCE TO SHAPE DEFORMATIONAND THERMAL EXCITATION

A. Cascade shapes

As noted in the previous section, INC calculations areadapted to describe the nuclear shape once the hardsions have ceased and dynamical mean-field effects becimportant. To study this question, we consider a SOBA cculation with input corresponding to the time when the wais the mostly developed, i.e., around 15–20 fm/c in the caseof Fig. 1. The detail of the calculation can be found in R@23#. It is sufficient to say that this transport model usestest particle method~here, we use typically 50 test particleper nucleon! and that a noise is introduced at the beginnof the calculation. The noise is suitably tuned in orderaccount for the thermal fluctuations associated with the rdom nature of nucleon-nucleon collisions, as described

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the Boltzmann-Langevin theory@26#. This method has beenshown to be accurate for describing the dynamics of the fltuations of the phase space distribution function, especiwhen the system enters the so-called spinodal region@27#.More precisely, in this Ref.@27#, it is explicitly shown thatthis method gives similar results for the spinodal decomsition, as the two-dimensional complete BoltzmanLangevin calculation on a lattice.

Furthermore, it has been shown that the treatment ofmean field requires some special care. For simple Skyrlike effective potentials, the mean field is a function of tlocal density and the latter is calculated by providing the tparticles with a Gaussian of width parameters. As shown inRef. @28#, in semiclassical calculations, a value ofs50.9 fmis necessary to provide a good approximation to the quadescription of spinodal instabilities. Since we are primarinterested in the bulk instabilities of the remnant, we chothis value. We stress the fact that a smaller value of tparameter ass50.5 fm is recommended to get a good dscription of the nuclear surface in the ground state withclassical test particle method, but it is not suitable for traing the possible bulk instabilities.

In order to simplify the transition between the two calclations, we idealized the state of the system as predictedthe INC. To start, we construct a Ag nucleus, from whichcone has been removed, with the apex at the center ofnucleus and with opening angle similar to the one ofwake observed in Fig. 1. The system is initialized at teperatureT58 MeV. This corresponds to an excitation eergy slightly less than the thermal excitation energy pdicted by the INC calculation. The spatial distribution of thtest particles is disturbed randomly~noise! in concordancewith the known density fluctuations of a Fermi gas at thtemperature. The evolution of the density, in a representarun, is given by Fig. 3. It can be seen that, in less thanfm/c the system loses a number of test particles and rapforms a compact spherical shape. Our results shownuclear matter is resilient to the deformation and heatproduced in the INC stage, inasmuch as, by comparing

FIG. 3. Nucleon density distribution in the reaction planevarious times, as given by the SOBA calculation for a configuratcorresponding to a Ag target atT58 MeV after removal of a coneThe Gaussian parameters is taken as 0.9 fm. The system contai91 nucleons. The density scale is arbitrary, but the darkest zocorrespond to the largest values of the density.

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55 1407RESILIENCE OF NUCLEAR MATTER IN LIGHT ION . . .

figure and Fig. 1, one can realize that the idealization ofstarting configuration used in the SOBA calculation is ctainly less favorable for shape resilience, since the deninside the cone is not vanishing in the INC calculation.

In addition, we also performed a SOBA calculationimposing the velocity field, as given by the INC calculatioat t.15 fm/c. For this case, we usedT56 MeV, which inprinciple is more favorable to the development of spinoinstabilities@29#. The resilience of the nuclear system holfor this case also, as shown by Fig. 4, where a typical evis displayed. Compared to the previous case, the shapmore perturbed at the beginning, because of the velofield, but the spherical symmetry is also basically recovewithin t;60 fm/c. It is, however, interesting to note that thsystem size grows much more than in the previous case~Fig.3!: atT58 MeV, the system cools by an important emissiof energetic particles leaving behind a ‘‘core’’ moderateexcited and rather stable; atT56 MeV, this emission is con-siderably diminished since the energetic particles leavesystem less easily, thus developing, apparently, moreciently a macroscopic pressure. The ensuing expansion,gered by the latter creates larger fluctuations in the voluof the system, leading to more instability. This motivates oremark above about the choice of the temperature. Howein the present case, this effect is not strong enough to athe breaking of the system.

For the sake of comparison, we have considered SOcalculations with the parameters50.5 fm. The resilience isthen less apparent. ForT56 MeV and the velocity field thesystem recovers sphericity, somewhat slower, in 85% of40 events that we calculated. It breaks down, mainly in tpieces, in the remaining events. ForT58 MeV and the ve-locity field the system recovers sphericity in 2/3 of thecalculated events. We think this higher proportion comfrom the fact that atT58 MeV, with a small surface tensionthe system loses test particles at a higher rate, approacvaporization.

B. Exotic shapes

In view of the results obtained so far, we addressedfollowing questions: To what degree should the systemdistorted in order to break down into pieces instead of go

FIG. 4. Same as Fig. 3, withT56 MeV and the velocity fieldextracted from the INC calculation. See text for detail.

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back to the spherical shape? Is the shape~that we suspect tobe! realized in 3He induced reactions far from the onsetthis multifragmentation process? These questions arehard to answer as there are many ways of distorting thetem. We restrict our study only to one of the possible etrapolations of the distortion realized in the INC, but wthink it is already meaningful. We considered a much moexcavated configuration, with only one quarter of the diaeter along the incident direction remaining in the target~seeFig. 5!. The so-prepared remnant contains 70 nucleonstemperature of 6 MeV. Furthermore, the velocity field dcussed in the previous subsection is superimposed. Asshape is characterized by a large surface-to-volume ratio,expected that the bulk instabilities give way, at least ptially, to the surface instabilities. Therefore, taking thGaussian parameter ass50.5 fm is expected to reveal thlatter more easily. But, for the sake of comparison, we aperformed SOBA calculations for this starting configuratiwith s50.9 fm. The results for a typical event are illustratin Fig. 5. They are rather surprising as the system remaconnected~a single part! for a rather long time, although it isnot really going rapidly to a spherical shape. A closer insption of the results seems to indicate that the relaxation ofsystem is very long because there is a near cancellationtween the Coulomb forces, which tend to deplete the sysalong the collision axis, leading to an almost annular shaand the surface tension, which tends to make the shape mcompact. The velocity field works against the surface tsion, as the Coulomb forces. The results are radically difent in a SOBA calculation, with this times50.5 fm. In thiscase~see Fig. 6! the system breaks down into pieces rathrapidly. In this typical example, the system breaks first inpiece traveling forward and a not complete annular rwhich breaks into two pieces that are seen in Fig. 6 by thcross section in the reaction plane. Once again the velofield is instrumental in this breaking. It is interesting to looat the mass distribution of the fragments, in the case oflast calculation. It is given in Fig. 7~for 50 events! andshows that the mean mass of the fragments is of the orde12. This is significantly smaller than the average mass offragments coming from a spinodal decomposition~about 20–

FIG. 5. Same as Fig. 4, but for an initial configuration corrsponding to an excavated Ag nucleus. The maximum thicknesthe longitudinal (z) direction is equal to half of the original radiusThe system contains 70 nucleons. See text for detail.

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1408 55M. COLONNA, J. CUGNON, AND E. C. POLLACCO

30!, as predicted in Refs.@23,29#. This observation, whichalone can by no means be taken as a clear signature, isertheless consistent with an interpretation of the fragmetion observed in this calculation as due to a surface instaity @10,30#. The role of the Coulomb force~between the endsof the system! and of the velocity field need to be investgated carefully before drawing a definite conclusion.

IV. VALIDITY OF THE TWO-STEP PICTURE

The considerations of Sec. III A indicate that the systbasically recovers a spherical shape after a relatively stime (&60 fm/c), before continuing to deexcite presumabby standard evaporative processes~or by the emission ofintermediate mass fragments when the remaining excitaenergy is large enough!. It is interesting at this point to maka comparison between the~average! evolution of the systemin INC and SOBA. This is provided by Fig. 2. The evolutioof the thermal energy~the mass energy of theD particles, notintroduced in the SOBA calculation, has been subtractefacilitate the comparison! and of the remnant mass are give

FIG. 6. Same as Fig. 5 fors50.5 fm.

FIG. 7. Average numberm of fragments of massAf ~upper part!and probability distribution for havingm fragments~lower part!, inthe SOBA calculation of Fig. 6.

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by open symbols for INC and black symbols for SOBThere is a remarkable similarity between the two calcutions when the time scale for SOBA is compressed byfactor three relative to the INC time scale. A possible expnation lies in the fact that, although the matter and momtum distributions in the INC are fairly homogeneoust.15220 fm/c, the target has still small packets of faparticles, which, by escaping on a short time scale, rapcool the system. In SOBA, smoothing gives rise to a slowenergy depletion. More importantly, there is a good corspondence~translated by the arrows in Fig. 2! between thevalues of the excitation energy and the mass of the remnat the time the system is spherical again in the INC1SOBAcalculation (t.60 fm/c) on the one hand, and the valuesthe same quantities in the INC calculation alone at the ti(t.30 fm/c) when the system is spherical and more or lehomogeneous, on the other hand. These considerationstify to some degree the use of a cascade1 evaporation~orthe like! model, with the choice of the stopping time recommended in Ref.@20#, to analyze the data. Note that the agrement between the properties of the spherical configuratiin the two calculations shown in Fig. 2 is obtained wiextreme conditions, namely a maximum development ofwake and an idealization of the hole in SOBA calculation

V. DISCUSSION

In this paper, we investigate the appropriateness ofcascade1 deexcitation model for analyzing the experimendata relative to light ion induced reactions in thGeV/nucleon range. A necessary condition is that the sysis thermalized and basically spherical when the hard cosion phase~cascade! comes to an end. We have seen that tis the case for the3He 1 Ag system at 1.6 GeV/nucleonincident energy. Consequently, we consider that this shoalso be the case for lower energy and smaller projectilesto at least the same energy. Note that the wake predicted2 GeV protons corresponds to a density depleted of o2/3r0 and reaches its full development at 8 fm/c @12#. Forthese systems, the possibility of having a fragmentationfore the evaporation or an evaporation from an exotic shremnant is very unlikely. Taking into account the resultsSec. III A for the 3He 1 Ag system, we can put a verconservative upper bound of;90 mb on the cross sectiofor the occurrence of these special events, taking intocount the least favorable~for resilience! of our results andthe fact that the wake is of decreasing importance asimpact parameter is increasing. Our results with the prochoices50.9 fm show that the cross section for these scial events is, very likely, vanishingly small. In any case, oinvestigations indicate that nuclear matter is particularlysilient to deformation and to thermal excitation in this kinof reactions. Note that in central collisions, the excitatienergy of the remnant after the cascade phase could bhigh as 600 MeV, i.e.,;7 MeV/nucleon, as indicated in Fig2 and confirmed by the experimental data@17#. The situationis rather contrasted with the one prevailing in central heaion collisions @31#, where multifragmentation occurs frequently for a similar estimated excitation energy per nucleAlthough it is not yet clear whether the emission of seveintermediate mass fragments is a rapid process or simp

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55 1409RESILIENCE OF NUCLEAR MATTER IN LIGHT ION . . .

slow evaporative process, there are good indications@32# thatthe system is then less resilient to a rapid breakup in inmediate mass pieces. A possible explanation may rest onfact that the system is not significantly compressed in liion reactions. On the other hand, in heavy ion collisionssystem is presumably more compressed, leading to a cotive expansion. This in turn will lead to volume instabilitiewhich facilitate the the breakup of the system@23#. This

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issue deserves extended investigations.Coming back to the3He1Ag system, the possibility of

having deformed shapes leading to unusual decay pattercertainly increasing with energy. One may speculate onformation of very excavated shapes such as the one wesidered in Figs. 5 and 6. They may provide an appeaopportunity to investigate possible surface instabilities@30#of nuclear matter. This point also deserves further analys

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