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Revista Mexicana de Física 38, SllplnTlcnlo 2 (1992) 95-101

Nuclear Spinodal Decomposition

JORGE ALBERTO LÓPEZ

Universily of Texas al El PasoEl Paso, Texas 79968, USA

ABSTRACT. The fragmentation of 1101amI dense Iluclci was sludied via computer silllulations.Using-a classical potentiai, the disassembly of lwo-dilflcllsional droplets was investigaled usingCahn'5 structurc fUllctioll. The c1assical drops were found lo break through spillodal decompositionas delermined by tite exponcntial growth of the structure fllnclion inside the spinadal line.

RESUMEN. Se estudió la fragmentación de núcleos densos y calientes por simulación numérica. Seinvestigó la desintegración de got.ita.<o;bi-dilOensionales IIsando la fUllción de estructura dc Cahll.Se encontró que las gotas clásica.~ se rompen por medio de una descomposición "spinodal" 1 estofue determinado por el crecimient.o exponencial de la función de estructura dentro de la línea"spinodal" .

rAes: 21.6.5.+f; 2.5.70.Np

[NTRODUCTION.

Nuclear properties, such as compressibility, energy and numbcr density, etc. a.re knownonly at one point of the phase diagram of nllclear matter: at zero ten',peratllre alld normalnllclear density. '1'0 obtain more illformation a stlldy of nuclear matter at other vallles ofdensities and temperatures is needed. This stlldy is being done witl, heavy-ion r('actions.

lIea.vy-ion eollisions at energies between ten s and hundreds of l\leV / A are expeeted toprobe regions of high temperatllre and density of the phase diagram of nuclear matter. [Itis known, for instanee, that nuclei r.olliding at these energies break into smaller fragmentsdllring the reaction. This collld provide inforlllation about ehanges of pha.se in nuclearmaller as wcll as its critica! tempcraturc, speeifie heat, etc.] Ilowever, the extraetion ofthese properties from slleh reactions is not an easy task dile to the faet that the laboratoryobservables refieet only the la.st stage of the reaetion and not the hot and dense one.

'1'0 conneet the final observables, IIsllally Iight particles and intPfmediate-mass frag-ments, to tIte wanted properties olle lIlust re50rt to model-dependent ca)clllations. Severaltechniques are bcing llsed for l!lis purposc, rangillg from reactioll silllulation lIsing Cjuan-

lum lnicroscopic kinetic eqlla.tions lo calculaliolls illvolving lllany-lJody tcchlliques [1-5].However, since the process being studicd involvcs fomplcx proccss('s (critical phenomena,dissipative e[fcets, etc.) not all of these theoretiral approarhes can be uscd with con-fidence. A tcchniquc that can describe changcs of phases whilc going, ul}(ler lhe r¡glllcircumstanrcs, lo any approprialC lilllit (non-equilibriulll kinematics ur hydrodynamicalfiow) is the so ralled molerular dynalllies (Iv! D) ralculation. By a ,Iirert solution of theconpled eflualions of mOlioll, this tcchnique can describe tlle evolnlion of the rcartion

96 J.A. LórEz

takiug into aeeouut all dissipative and relaxatioual effeets while ineorporating all thewauted partid •. eorr •.latious (that might carry iuformatiou about the earlier phases of theeollisiou ).

This t•.ehuique has beru used iu ll,e pasl lo sludy difrerenl asprels of uudear reaelious[2-5]. The pre,,'ul work follows the sleps of ReL [51and sluuies lhe disassemb!y of hol anudense droplrls of dassiea! partides resembling uudear maller. In the next seeliou, lhe MDleehuique is bri"l!y p[rsent"d and the polenlia! useu juslifi"d. Th"u lhe lheorrliea! loolslo aua!yze lhe MD ea!eulatiou are iulroduee,!. AmI fiually, some eharaelerislic results arepresenled aloug with some dosiug eouclusious and prospecls for fulure work.

TIIE MOLECULAR DYNAMICS TECIINIQUE.

The Mn approaeh lo the nudear dynamies eousidrrs lhe collidiug uuel"ons as dass;ealpartides ;ulerarliug lhrough a two-body pOleutia!. By solving the wuplrd equations ofmotiou of the many-body system numerieally. lhe ~[D teehuiqu" eau approximate lhetime evolulion of tite reartion and Sllldy its different stages frolll a lIIirroscopic point ofview.

The firsl objreliou against lhe use of molreulae dyuamies for lhe sludy of nuclearmaller, is lhat lhe above meulioued leehnique involves purely dassieal dyuamies while lhenuc1eus is a quantulII systrtll. I1owever, for hol antl df'llsf' lIuclf'iH mattcr tI,,::, large valuesof the momeuta iuvolv"d iu the eollisious are "xpret •.d to redue" ti", bloekiug ¡utrodueeuby the Pauli nriueipl" aud makr the meau free path shorter (¡.c. almosl dassieal) lhan allower ('xcitatioll f'1lf'rgif's. It is hoped that un<1f'f thesf' cirClIlIIstanc(>s thf' quantlllll effectsbecolllc leSos importanl amI a r1assical treatlll(,lIt is more rcasonahlc.

A key iugredient of the MD teehnique is thr iuteraetiou poten tia!' In the past, nuclearfragmeulatiou has beeu studied by aualogy with th" dis;L%embly of argo u droplets iuler-aetiug via a u-12 poteulial [u]. [These dropl"ts Were fouud to disassrmble iu a varielyof ways iucludiug adiabatir spinoda! deeompositiou.1 lIowrv"r, ti", behavior of argoumoleeu!rs differs eousiderably flOm lhal of uudeous aud a study iuvolviug a more realisliepoteutial is iu order. Iu thr presrul sludy a potrutia! that reproduces the expeeleu lher-modyuamiral propel'ties of hot aud dense nucl,'ar mattcr (as obtained by a Skyrme-lypeinteraetiou) is emp!oyed.

The poleulial used iu lhis work is givcu by

Ver) = 91 exp( -mI r)/r - 92 exp( -m2r)lr

wilh coclllcienls 91 = 1072u ~leV-fm, mI = 0.:1955 fm-I, 92 = 10215 MeV-fm aud1112 = 0.3582 fm-I. This poteulial, is more realistie thau u-12 poteulia! siuee, as showuin Ref. [7], reproduces, muJer different approxilIlatiolls, lIle expectcd thermodynamicalpropNlies of nnclear maller.

Fig. I shows lhe pressnre-densily curve fol' lhis pOlenlial al T=20 MeV as oblaineufrolll a solution of the hyper-nctted chain equatioll, the PerclIs-Yevick equation [1] alldwilh a molecular dyuamies ealrulation of a two-diml'nsional r;as (with periodie bound-aries lo simu!ale an iufinile syslrm). A!so shown for wmparison are lhe '1'=0 aud T=20

NUCLEAR SPINODAL DECOMPOSITlON 97

4

3~n

E-I> 2al::¡:~al•..:lU) 1U)al•..a.

o

-10.00

.....

•

". ".

0.05

MDHNCPYFP

". " . ....

T=20

..........

........'.......................

0.10 0.15Denslty [fm .3]

....• •..

T=O

0.20

FIGURE l. 1'=20 f\.leV Pressure-density isothcrm of the potentiai lIseu in this calenlatian. ThedifTerent.s curves w€'re ealc:ulated using tite hyper- netted chaill Eq. (IINC)! Percus-Yevick Eq.(PY), and molecular dyuamic5 (MD). AI50 5hown for comparíson are lhe T=O ami T=20 MeVísolherms oblained ín Ref. [7] usíng a Skyrme-lype calculatíon.

MeV pres5ure-densíty curves ealculated ín Ref. [G]usíng a Skyrme-type interaetíon. Theparameters of the potential were adjusted to provide a good fit to the T=20 MeV curveof Ref. [G] arguing that it is during the high-density and high-temperature stage of thereaetion that the expansion velocity is determined [8].

The phase diagram corresponding to this potentia! is shown in Fig. 2 (dotted lines). Itwas obtained using MD for the 400 partide system with periodie houndary conditions.The lahels CE and IS denote the coexistanee region and the isothermal spinodal line,respeetively.

SIMULATIONS.

To understand the hreakup proeess, we are interested in studying the hehavior of a hot

98 J.A. LÓPEZ

4011

11I30

~>(1) IV~-(1)•..::::l 20-al•..(1)Q.

E .~

...o' " ,• , ,• , ,,

10 • • ,• f ~ "-•• . , .,• o , .• o • ,• o , •. o , ,. •• • •

18 CEO

0.00 0.05 0.10 0.15 0.20 0.25

Denslty [fm .2]FIGURE 2. Expansion trajectories in the T-p plane. The dashed Iines show the coexistance r€':gion(CE) and the isathermal spinadal line (15).

and dense drop of ¡i'luid undergoing free expansiono For titis purpose, a two-dimensionalarea containing 400 partides is first e'luilibrated at some desired temperature and density.After reaclting e'lnilibrinm, a disk witlt appl"Oximately 270 partides is cut out and allowedto expand into free space. lJy solving tite e'luations of motion, tite tltermodynamicalpropertics of tlle centra.l particlcs are determincd. In particular for two dimcnsions, thctemperature and density are obtained by tite time averages over 50% of tite most centralpartides

T =< Ek;" > IN and

Fig. 2 sltows tite average trajectories (plotted on tite pitase diagram) followed by expan-

NUCLEAR SPINODAL DECOMPOSITION 99

sions starting from different densities and temperatures. Eacb curve represents tbe averageof 20 events starting from diITerent microscopic configurations hut e<¡ual macroscopictbermodynamical variahles. Points on the curves are 10 fm/c aparto As judged by thedistance between adjacent points, it can he seen tbat the overall expansion of tbe drop isslowed down considerably inside tbe spinoda! lineo

In addition to the T." trajectories, tbe bebavior of tbe density fiuctuations arisingduring tbe expansion of tbe drop was also studied. According to Cabn'8 tbeory of spinodaldecomposition, small density fiuctuations are governed by tbe diffusion e<¡uation which,upon linearization, bas a general solution of lbe form [9]

Óp = A(q, t) cos(q' r)

wilb ", <¡and t denoting lbe density, momentum and time. In lerms of Gihhs free energydensity f and lbe van der \Vaals constant n, lbe temporal parl of lbe amplitude is givenby

this results in lwo different kinematical regions. lf {)2f /{),,2 + lJ,P > O tbe fiucluationsare damped away, but if lJ2f /(),,2 + lJq2 < O tbe amplitude of Ibe inbomogeneities willamplify in time leading to tbe fragmentation of tbe syslem. Tbis second condition issatisfied inside Ibe spinodal region.

1'0 follow Ibe densily fiuclualions Ibe structure funclion is used. Tbis funclion is con-nected lo Ibe density f\ucl"alions hy

S(k) = V < "k,,-f> /"0wbere "k denotes tbe Fo"rier lransform of 6,,(r). Tbis funclion can be easily ohlainedfrom tbe radial correlation f"nction g(r) using

S(k) = 1+" J rPrexp(-ik '1')[r¡(7-) -1],

wbile the function g(r) is in turn ohtained directly from sampling tbe central region ofthe expauding drop. Tbe exponential growth of the density inhomogeneities inside thespinodal line willlead to a similar growth of the structure function.

Fig. 3 sbows tbe temporal hebavior of S(k) for diITerent val "es of k. As seen in thisfigure, tbe structure function grows exponentially after tbe expanding system bas reachedthe isotbermal spinodal line, sbowing, perbaps, tbe spinodal decomposition of tbe bo-mogeneous system. Also sbown, for comparison, is tbe Landau-Lifsbitz long-wavelengthhebavior of S(k) represented witb a dotted line (see Ref. [7J for details).

CONCLUSIONS ANn ÜUTLOOI<.

Aseen hy tbe above-descrihed exercise, spinodal decomJlosition is a real possihilityfor tlle breakllp of expanding Jluclear matter. lIowevef, scv('ral ullrcalistic assumptions

100 J.A. LÓPEZ

150

k2

k,

125

....................................................................

75 100Time [fm/cl

CE 151 I

50

IV

25

6

5

4-.:.: 3-en2

1

oo

FIGURE 3. Time evolution of the structure function S(k) for different values of k. The run heredepicled rorresponds to the expansion of that starts frolO point IV in Fig. 2. The labels T"CE, and IS denot.e the time at. whic:h the expanding system was at the critical temperature,entered the coexistan ce region and cross the isothermal spinodal line, respedively. The expectedlong-wavelength hehavior of S(k) is represented hy the dotted line.

hep this simple calcn]ation from being adeqnate. Apart flOm the dimensionality, realisticdensity f1uctuations, snch those induced by the collision, wOII]d have to be taken intoacconnt. [It is possib]e t¡mt lhe se]ection of lhe breakllp mode lakes place in the initia]stage of the reaction.] A collaboration with C. Dorso is IInderway to stlldy' the etrect ofinitia] seeded inhomogeneities on the breakup process.

This work was done in co11a1>oration with G. Lubeck. Financia] he]p provided by thePhysics Dept. of the Ulliversity of Texas at E] Paso for attendance to this meeting isgratefully acknowledged.

REFERENCES

1. G.F. Berlsch and S. Das Gupla, Phys. Rep., 160 (1988) 189.2. A.R. Bodmer aud C.N. Panas, Phys. Rev. C1~ (1977) 1342; L. Wilets, E.M. Henley, M. Kraft

and A.D. Mackellar, Nuel. Phys. A282 (1977) 341.3. D.H. Roal aud J.N. Glosli, Phys. Rev. C37 (1988) 91; Ann. Rev. Nuel. Parto Sci 37 (1987) 1.4. C. Dorso, S. Duarte ami J. Randrup, Phys. Lclt. D188 (1987) 287; J. Physiquc 48 (1987)

C2-143; C. Dorso aud J. Randrnp, Phys. Leti. D215 (1988) 611.5. A. Vieentini, G. Jacueci, V.R. Pandharipande, Phys. Rev. C31 (1985) 1783; R. Lenk and V.R.

Pandharipánde, Phys. Rev. C34 (1986) 177.6. B. Friedman and V.R. Pandharipallde, Nue/. Phys. A361 (1981) 502.7. J.A. Lopez alld G. Lubeck, Phys. Leti. D219 (1989) 215.8. 11. lIeiselberg, C.J. Pethiek aud D.G. Ravellhall, Phys. Rev. Lett. 61 (1988) 818.

NUCLEAR SPINODAL DECOMPOSITlON 101

9. F.F. Abraham, Phys. Rep. 53 (1979) 93; J.W. Caho, Acta Me/al/. 9 (1961) 795; J.W. Cahoaod J.E. lIilliard, J. Ghem. Phys. 28 (1958) 2,,8; M. I1illierl, Acta Me/al/. 9 (1961) 525.