Annalen der Physik. 7. Folge, Band 45, Heft 4, 1988, S. 258-264 VEB J. A. Barth, Leipzig Entropy of a Two-phase Mixture By H. SCHULZ, H. REINHARDT The Niels Bohr Institute, Copenhagen (Denmark) and Central Institute for Nuclear Research Rossendorf, Dresden (GDR) B. KAMPFER Central Institute for Nuclear Research Rossendorf, Dresden (GDR) Abstract. The isentropic expansion of a blob of nuclear matter towards the two-phase instability region is studied by considering two extreme conditions: the instantaneous development of the phase transition and the spinodal decomposition. We show that the experimentally observed entropy values of light clusters give evidence for the onset of a liquid-gas phase transition in heavy ion and proton induced reactions, but to be more conclusive the disassembly of relatively cold nuclear systems has experimentally to be investigated in more detail. Entropie einer nuklearen Zweiphasenmischung Inhaltsubersicht. Die isentrope Expansion eines nuklearen Feuerballs und das folgende Eindringen in die Region der Fliissigkeit-Gas-Phaseninstabilitatwird in zwei NLherungen untersucht : es wird entweder eine schnelle Phasemeparation oder die spinodale Entmischung angenommen. Es kann gezeigt werden, daB die experimentell bestimmten Entropiewerte der leichten Cluster Hinweise auf eine Zweiphasenentmischung in Schwerionen- und Protonen-induzierten Reaktionen geben. Die detailliertere experimentelle Untersuchung der Disintegration relativ kalter nuklearer Systeme ist notig, um zu weitergehenden Aussagen zu gelangen. One of the intriguing aspects of heavy ion and proton induced reactions would be the proof (or disproof) of the existence of a liquid-gas like phase transition in hot nuclear matter. Such a phase transition is predicted by different theoretical approaches for infinite nuclear matter (see [l--61 and references quoted therein). At present it is not fully understood what are the experimental evidence for such a phase transition. Following Fisher's droplet model [7] critical exponents z of a power law fit A-" to medium fragment mass spectra has been extracted [S], whereas the light cluster abundance8 have been fitted by considering the onset of a phase transition [3, 91. Additional informat,ion may be gained from entropy measurements because the entropy is the interesting quantity which provides a window into the earlier stage of the disassembly process. The total entropy of the system will grow rapidly when the collision process begins and remains nearly constant when the expansion of t h e entire system sets in. The subsequent multifragmentation process is very complicated and its time evolution proceeds through thermodynamical and hydrodynamical instabilities [5, 13-16, 23, 241. Since the total entropy is approximately conserved one can expect H. SCIIULZ et al., Entropy of n Two-phase Mixture 259 that the specific entropy values associated with the yield of light or medium clusters can give valuable information and signal the onset of the phase traiisition process. Such specific entropy values have recently been inferred from data on light fragment cross sections. I n ref. [lo] the entropy was extracted by using a quantum statistical motlel which assumes that a t nuclear disassembly chemical arid thermal equilibrium is established. The average value of the specific entropy for fragments with 1 2 A 2 4 was found to be s = 3.60 f 0.3 2 and turned out to be nearly independent of the beam energy ant1 projectile nucleus. A previously suggested model [ 1I ] has been generalized [ 121 to consider deuteron like pairs contained in larger composites and to infer entropy values from data on light fragment cross sections. It was found that the entropy decreased slightly with decreasing beam energy but never dropped below its value a t the critical point of about SIN = 3. I n what follouw we show how these results can be interpreted in the physical picture of the onset of a phase transition familar from the van der M'aals gas. I T Fig. 1. The nuclear phase diagram with the liquid and gas border lines pL and pG and the spinodnls s, and sl. The isentropes labelled by the specific entropy value are calculated either by considering an instantaneous phase transition (dashed lines) or by ignoring the phase separation. The numbers on the spinodal s2 and on the phase border lines give the specific entropy values. The break-up rcgion and the region where predominantly compound nuclei are formed are hatched. Fig. 1 shows a typical nuclear matter phase diagram [13] with the border lines pL and eGfor the liquid and gaseous phase and the spinodals s1 and s2 defined by the locus where the compressibility vanishes. Within the region bounded by the spinodals the , ~ respectively, nuclear matter is unstable, whereas in the region confined by s ~ and the nuclear matter is in a metastable state. As usual we assume that the intermediate hot nuclear system formed by a heavy ion or proton induced reaction expands along an isentrope (see Fig. I ) and finally reaches a stage where the system becomes metastable or unstable. The conditions for the initial heating-up and compression t o arrive a t these situations have been studied recently [13-161. Ann. Physik Leipzig 46 (1988) 4 2(iO Following the isentropic expansion towards the unstable region one can distinguish the following two extreme scenarios for developing the phase transition : (i) The system expands slowly enough so that, when reaching the metastable region, bubbles of the new phase are formed due to thermodynamic and quantum fluctuations. At the spinodal s1 the entire system becomes unstable and decays. (ii) The initially homogeneous system is driven rather rapidly along the isentrope into a situation where it becomes thermodynamically unstable and the spinodal decomposition occwrs. Given the fact that the spinodal decomposition represents a break-up of the initially homogeneous unstable system, it is natural to assume that this process happens in the vicinity of the break-up density Q w ~ , / 3(hatched area in Pig. 1). Let us first consider the case (i),i. e. a phase separation process which is fast compared to the expansion. The total entropy of the two-phase mixture is then given by = A (eL - e) (2) @G ( c L - - P f 2 ) is the mixing coefficient determined by the leverage rule. This coefficient. is nothing else than the ratio of the gaseous particles (N,) t o the total particle number ( N ) . At given temperature T the specific entropy values S, and SL are fixed and the mixing coefficient has to be adjusted to conserve the total entropy. The resulting path of the system in the T,e plane is shown in Fig. 1 by dashed lines. Fig. 2 illustrates how the total entropy values are composed by those of the two separated phases. The corresponding mixing coefficients are displayed in Fig. 3. Sote that only within the break-up region the fraction of the gaseous particles becomes noticeable. According to this picture the specific entropy value of the gaseous (liquid) phases are determined through their corresponding values along the phase border line. Kote that S, increases for decreasing total entropy values or temperature. -_-_ ___ . _-_ I 1 i 05 x I I 3 hm-31 0 0 05 01 3 [fm-?; Fig. 2 Fig, 3 Pig. 2. The sharing of the total entropy S into the entropy values S , and 8, d b function of the density g. The dotted line is calculated by considering the bubble formation explicitly. The numbers on this curve denote the partirle number of the critical bubbles. The break-up region is hatched. Fig. 3. The mixing coefficient x as function of the density at given total entropy values. H. SCHULZ e t al., Entropy of a Two-phase Mixture 261 Till i ~ o wwe hare considered the phase separation as an instantaneous process by ignoring the underlying microscopic nature of the two-phase formation via the growth of critical bubbles in the metastable region. The bubble formation probability has been estimated in ref. [16] by using a droplet model and in ref. [16] within a field theoretical approach. Following these results we illustrate for the SIN = 2 isentrope iri Fig. 2 the corresponding path of the system (dotted line). The figures on this path denote the number of nucleons in the critical bubbles. It can be seen that only close to the spinodal sl, wert' the size of the bubbles is much smaller than that of the system, the phase transition can develop. I n other words, only if the system has almost passed the metastable region the application of the picture of a n instantaneous phase transition (using the Maxwell construction) is justified because then the critical bubbles are so small that the phase transition is not anymore hindred. Compared to the picture of an instantaneous phase transition described above we have now the situation that the liquid becomes first superheated and afterwards the phase separation occurs. Furthermore, i t has been shown [13, 1.5, 161 that the phase transition may predominantly develop when the total entropy S is confined t o the interval Smi,, < S < S,,. For entropies S > S,,, the system remains always in a gaseous phase and, hence, is not influenced by the onset , , , m 2.81) (see also Pig. 1). of the phase transition a t all. The calculation show that S The lower limit Smin N 1.2 results from two effects. Due to the occurence of a negttive pressure in the system the collective expansion is slowed down and eventually comes t o halt before reaching the spinotlal s1 except for extreme initial compression energies (Ieposited in the system, which experimentally can hardly be achieved. Furthermore, the two-phase separation is strongly hindred because the critical bubbles exceed the size of the system. This qualitative picture is in good agreement with the findings of ref. [17], where the fragmentation process has been investigated by means of a statistical model including finite size effects. There it has been found that for total entropy values S < 1.3 only compound nuclei are formed which can lower their excitation energy by particle emission, garnnia rays and fission. SIN Fig. 4. The specific values of the gaseous phase calculated in different approximations as functioii of the total entropy. The curve labelled by 8LP.de'/N represents the entropy values when assuming spinodal decomposition, whereas SF9t/N is calculated by considering a n instantaneous phase transition. The curve Ssp gives the entropy values of the gaseous phase along the spinodal s2 (cf. Pig. I). The dashed line would give the results of the gaseous phase when there is no phase separation. Above the critical entropy, 8, coincides with the total entropy 8. I) Kapusta [ I d ] has obtained S,,,,/N m 3.3 with a slightly different equation of state. 268 Ann. Physik Leipzig 45 (1988) 4 Now let us turn t o the case (ii). We consider the other extreme situation, where the system is isentropically driven into the break-up stage (full lines in Fig. 1).Once the system is in such an unstable state it decays immediately. This is because fluctuations grow here exponentially, since there is no longer an energy barrier for forming critical clusters (bubbles). This process called spinodal decomposition is familiar from nucleation theory in statistical and fluid mechanics [18, 191. During the nucleation process in the vicinity of the growing clusters the available matter is exhausted for forming them and, therefore, no other clusters can be built up. This effect acts among the clusters like an excluded volume interaction introduced explicitly in statistical nuclear fragmentation models [20, 211. Compared t o the processes discussed before (case (i))the multifragmentatiori proceeding through an instantaneous spinodal decomposition in the break-up region would mean that the expanding fluid blob is more cooled down. The situation is illustrated in Fig. 4, where the specific entropy of the gaseous phase, S,, has been extracted by using the results of Fig. 1 for the different processes characterized by the items (i) and (ii) and drawn as a function of the total initial entropy S. It can clearly be seen that S, increases as the total entropy decreases. I n other words, one gets the result that the specific entropy associated with the gaseous phase increases as the beam energy (or deposited energy) decreases. Such an effect is a t present not seen from the existing experimental data and seems to be an artifact of retaining the thermodynamic equilibrium conditions for the whole separation process. I n ref. [22] the relatively strong increase of the values for 8, is obscured by considering an unphysically small break-up density, @bu -N eo/lO. (Note that in ref. [lo] a break-up density pbu = e0/3 has been used to infer the entropy values from the experimental data). The average values for the specific entropy for fragments with 1 5 A 5 4 were determined in ref. [lo] to be SIN = 3.60 f 0.12 and turned out to be independent of the incident energy and the projectile nucleus. I n ref. [22] these entropy values has been tried to fit by assuming pbu = po/10 and by invoking the two-phase picture, i.e. the specific entropy of the light clusters has been associated with that of the gaseous phase of the two-phase region (cf. Figs. 1, 2 and 4).It seems to us that this interpretation can not be applied, because one has to analyze simultaneously light and heavier composites which are emitted from the s a m e source. Usually, heavier clusters detected experimentally stem from the decay of spectator matter, which is often characterized by a much lower temperature than the participant matter source, which predominantly emitts the light particles with A 5 4. The experimentally found constancy of the entropy associated with the light particle production seems to indicate that in the participant matter the disorder has already achieved such a degree that it does not depend very much on how it was initially formed. This tendency is in agreement with recent results for entropy values extracted from 4n measurements of the light cluster production Ar a t energies 400 MeV/ yields [25]. Also cascade calculations [26] for the reaction Ar u 5 Elab5 2 100 MeV/u show that SIN increases only slightly in this large energy interval. It seems t o us that the rapidity dependent entropy values as extracted in ref. [ 101 are not conclusive for gaining safe information on the onset of the phase transition of the liquid - gas type. This is because the uncertainties coming from the experimental conditions are a t present too large t o be sure that light and heavier clusters are emitted from the same source which can be characterized by a common temperature and total entropy value. Experimental studies of light and medium cluster production with a 4n detector allowing for triggering central collision events would give us much more valuable information on the two-phase separation onset. + H. SCHULZ et al., Entropy of a Two-phase Mixture 263 Analyzing the experimental data within the static two-phase picture oue has to be aware that in a kinetic treatment the equilibrium situation will never be reached and therefore the final stage of the evolution will be characterized by a lower entropy value of the gaseous phase as, for instance, predicted by the leverage rule. The final values for S, and their precise values can only be calculated by solving kinetic equations which govern the evolution of a system which consists of droplets immersed in a gaseous nuclear medium and moves towards the final state. From Fig. 1 and especially from Fig. 4 it can be seen that along the spinodal s2 the specific entropy values remain fairly constant, S,JN = 3.2. This almost constant value represents a better estimate for the entropy of the gaseous phase and considers that some non-equilibrium aspects of the separation process are accounted for. Therefore, we would expect that due t o the existence of the two-phase separation the specific entropy of the gaseous phase should remain almost constant with S, ‘v 3.2 if the experimental conditions are such that the disassembling intermediate nuclear system can be characterized a t break-up by a temperature T 5 Tcrit.The average value found in ref. [lo] SIN = 3.60 f- 0.12 is very close to our predictions but to be more conclusive one has to consider the disassembly of even colder systems with T 5 8 MeV. The method of ref. [22] would predict then entropy values of SIN N 6 unless extremely low break-up densities are taken into account. To summarize, the interesting conclusion which can be drawn from our present analysis is that the behaviour of the measured specific entropy values of the light clusters gives some evidence for the development of a first order phase transition. But to be more conclusive one has to treat relatively cold systems. The phase transition manifests then itself by the fact that with decreasing beam energy (temperature) the specific entropy values inferred from light cluster yields does not drop below some critical value of SIN m 3 in contrast to the total entropy. The development of a theory which includes nonequilibrium aspects but having a t the same time predictive power seems to be a major challenge to the theorists. A first attempt in this direction has been undertaken in ref. ~41. Two of the authors (H. S. and H. R.) are grateful to the Niels Bohr Institute for the kind hospitality extended to them. References [l] LAMB,D. Q.; LATTIMER, J. M. ; PETHICK, C. J.; RAVENHALL, D. G.: Phys. Rev. Lett. 41 (1978) 1623; Nucl. Phya. A 360 (1981) 459. IS] DANIELEWICZ, P.: Nucl. Phys. A 314 (1979) 465. [3] SCHULZ, H.; MUNCHOW, L.; ROPICE, G.; SCHMIDT, M.: Phys. Lett. 119B (1982) 12. [4] CURTIN, M. W.; TOKI,H.; SCOTT, D. K.: Phys. Lett. 123 B (1983) 289. [5] BERTSCH, G.; SIEMENS, P. J.: Phys. 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[16] REINRARDT, 1171 BONDORF, J.; DONANGELO, R.; MISNUSTIN, I. N.; SCHULZ, H.: Nucl. Phys. A, 444 (1985) 460. [IS] BINDER,K.; STAUFFER, D.: Adv. Phys. 25 (1976) 343. [19] LANCER, J. S.: Lecture Notes in Physics 132 (1980) 12. CZO] RANDRUP, J.; KOONIN, S. E.: Nucl. Phys. A366 (1981) 223. [21] BONDORF, J.; DONANGELO, R.; MISHUSTIN, 1.N.; PETHICK, C. J.; SCHULZ, H.; SNEPPEN, K.: Nucl. Pliys. A 443 (1985) 321. [22] CSERNAI, L. P.: Phys. Rev. Lett. 54 (1985) 639. [ 2 3 ] LOPEZ,J. A.; SIEMENS, P. J.: Nucl. Phys. A 431 (1984) 728. [24] STRACK, B.; KNOLL, J.: Z. Phys. A315 (1984) 249. [25] GUTBROD, H. H.; LOHNER, H.; POSKANZER, A. M.; RENNER, T.; RIEDESEL, H.; RITTER,H. G.; WARWICK, A.; WIEMAN, H.: Phys. Rev. C 32 (1986) 116. [26] GUDIMA, K. K.; TONEEV, V. B.; ROPKE,G.; SCHULZ, H.: Phys. Rev. C 32 (1985), 1605. Bei der Redaktion eingegangen am 31. MXLrz 1986. Anschr. d. Verf.: Dr. H. SCHULZ Dr. H. REINHARDT Dr. B. KAMPFER Central Institute for Nuclear Research Rossendorf Postfach 19 Dresden DDR-8051

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