Entropy and cluster production in nuclear collisions

ENTROPY AND CLUSTER PRODUCTION IN NUCLEAR COLLISIONS

László P. CSERNAI and Joseph I. KAPUSTA School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, U.S.A.

I

NORTH-HOLLAND-AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 131, No. 4 (1986) 223—318. North-Holland, Amsterdam

ENTROPY AND CLUSTER PRODUCTION IN NUCLEAR COLLISIONS László P. CSERNAI* and Joseph I. KAPUSTA School of Physics and Asfronomy, University of Minnesota, Minneapolis, MN 55455, U.S.A. Received March 1985

Contents: 1. Introduction 2. Dynamics of the formation of light nuclear clusters 2.1. Importance of light fragment production 2.2. Deuteron formation in proton—nucleus reactions 2.3. Coalescence model 2.4. Sudden approximation 2.5. Thermodynamic model 2.6. Sudden approximation for intranuclear cascades 2.7. Impact parameter averaging 2.8. Rate equations for cluster production 2.9. Determination of entropy from the deuteron to proton ratio 2.10. Summary and critique 3. Entropy production 3.1. Entropy as an observable 3.2. General characteristics of the nuclear equation of state 3.3. Nuclear matter below normal nuclear density, the liquid—gas phase transition 3.4. Nuclear matter at high density 3.5. General features of entropy production 3.6. Entropy production in shock waves 3.7. Entropy production in viscous fluid dynamics 3.8. Entropy production in microscopic models 3.9. Entropy production in nonequilibrium phase transitions 3.10. Entropy production during the expansion stage 3.11. Liquid—gas phase transition *

225 226 226 227 229 233 234 239 241 242 245 248 249 249 250 250 257 259 260 262 262 264 266 267

3.12. Summary and critique 4. Entropy from light fragment abundances 4.1. Effects influencing light fragment production 4.2. Unstable particle decays 4.3. Finite density effects 4.4. Finite source size effects 4.5. Mott transition 4.6. Extracting entropy from deuteron and deuteron-like correlations 4.7. Phase mixture 4.8. Dynamics of equilibrium phase transition 4.9. Nonequilibrium phase transition 4.10. Light fragment abundances in the phase mixture 4.11. Summary and critique 5. Nuclear fragment mass distributions 5.1. Introduction 5.2. Law of mass action 5.3. Droplet and bubble formation 5.4. Fluctuations 5.5. Coulomb effects on fragmentation 5.6. Rate equations for fragment formation in a gas 5.7. Evaporation from a hot nucleus 5.8. Cold shattering of nuclei 5.9. TDHF expansion and fragmentation 5.10. Summary and critique 6. Summary and conclusion References

268 270 270 270 273 275 277 280 283 284 289 290 292 292 292 293 295 303 304 307 308 309 310 312 313 314

On leave of absence from the Central Research Institute for Physics, Budapest, Hungary.

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LP. Csernaiand J.I. Kapusta, Entropy and cluster production in nuclear collisions

225

Abstract: When nuclei collide at beam energies from several tens of MeV to several GeV per nucleon considerable disorder is generated. Nuclear fragments ranging from nucleons all the way up in mass to the target and projectile nuclei themselves have been observed experimentally. Theoretical models for the dynamics of the formation and emission of these clusters of nucleons are reviewed. Most of the models, but not all, are statistical in origin, following from the assumption that the phase space available for cluster formation and emission is the dominant factor. The entropy generated during the collision may be studied in diverse dynamical models, such as intranuclear cascade and nuclear fluid dynamics. The entropy of the system may be estimated from the measured abundances of nuclear clusters, thus providing information on the properties of hot and dense nuclear matter. Critical analysis of both conventional and exotic interpretations of the data are given.

1. Introduction The motivation for colliding nuclei at high energy is the quest for knowledge of the properties of nuclear matter at high density or temperature. For a long time this quest was focused on the energy and angle distribution of nucleons emerging from the collision. Production of heavier fragments, such as d, t, 3He, etc., was thought to be a minor complication for the most part. Since many models, such as fireball, hydrodynamics and intranuclear cascade, predict directly the baryon number or electric charge distribution, all measured fragment distributions were simply summed together. Clearly information is lost in this procedure. Two especially provocative papers have changed this situation. In [SK79]it was argued that during the course of a nucleus—nucleus collision the entropy is generated at the point of maximum temperature and density and thereafter is conserved. Furthermore it was argued that the entropy at the end of the collision could be inferred from the relative abundance of deuterons. Since the beam energy is known, knowledge of the entropy S provides one of the two variables necessary to specify the equation of state. The other would be the density, or equivalently volume V, at which the system thermalized. For a fixed number N of particles, one needs to know S = S(E, V). In the other provocative paper [FA82] measurement of the yield of fragments up to mass number A = 30 was made with an in-beam gas jet target. It was observed that the yield decreased with A not exponentially but rather as a power of A, that is, Y(A) A~.This mild decrease is reminiscent of the behavior near a liquid—gas critical point in ordinary atomic systems where it is responsible for the phenomenon of critical opalescence. Since also nuclear matter is expected, on general theoretical grounds, to undergo a liquid—gas phase transition at moderate subnuclear densities this was suggested as the first evidence for the observation of a phase transition in nuclear reactions. These and other papers have generated a lively debate on the mechanisms responsible for, and the significance of, entropy and nuclear cluster production in high energy heavy ion collisions. In principle one may think that all relevant questions could be answered simply by solving Schroedinger’s equation for the colliding composite nuclei. In practice this is not feasible at the present time, due both to the well-known difficulty in reproducing the properties of cold infinite nuclear matter with a microscopic Hamiltonian and to the additional complexity of a time-dependent phenomenon. Even if we did have the power to carry out such a computation the results would likely be so difficult to assimilate that we may resort to simple models to help our understanding of the basic physics anyway. In this report we intend to review and critically analyze contributions made to this field. We hope that this report will be viewed as tainted not with prejudice but with judgement. It may seem at times that the data can be equally well understood in terms of different models with different assumptions. If the assumptions are unreasonable we will so comment. However, in the words of Ludwig Boltzmann: “It follows that it cannot be our task to find an absolutely correct theory, but rather a picture that is as —

226

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

simple as possible while representing the phenomenon as well as possible. It is even possible to imagine two different theories that are equally simple and equally good in explaining the phenomena. Both, although totally different, would be equally correct. The assertion that a theory is the only correct one can only be an expression of our subjective conviction that there can be no other equally simple and equally fitting picture.” The report is organized as follows. In section 2 we follow an essentially historical approach leading up to the relevance of entropy to heavy ion collisions and a suggested method of extracting it from the data. Section 3 contains a brief discussion of the nuclear equation of state and a more detailed discussion of dynamical mechanics of entropy generation. Section 4 deals with the problem of extracting entropy from the light fragment abundances in detail. In section 5 the nuclear mass distribution of fragments emitted in nuclear collisions is considered. We make some conclusions in section 6 concerning the status of the field. A final remark: With just a few exceptions we include only those papers which were published prior to 1 January 1985.

2. Dynamics of the formation of light nuclear clusters 2.1. Importance of light fragment production When large nuclei are smashed together at energies greater than about 20 MeV per nucleon in the center of momentum frame an enormous number of final states are possible. Any final state will be allowed so long as electric charge and baryon number are conserved. Energy, momentum and angular momentum conservation place no restriction on the relative abundances of n, p, d, t, 3He, 4He, etc. at such high beam energies: excess energy, momentum and angular momentum are simply carried by the 111111

00-

Ne

+

U

400 MeV/nucl

30

\~He ~Hp ~ 50 80 100 Ebb (MeV/nucl)

30

N~oc~l.

0.1

-

10

120

40

Fig. 2.1. Double-differential cross sections for fragments from the irradiation of uranium by 400 MeV/nucleon 20Ne ions. From [GS76].

L.P. Csernai and J.I. Kapusta, Entropy and cluster production in nuclear collisions

227

translational motion of the individual fragments in the final state. For example, for d + d collisions there are 5 possible final states, namely d + d, t + p, 3He + n, d + p + n, 2p + 2n (excluding pion production). For a person accustomed to low energy nuclear reactions the thought of having to enumerate all possible final states in a ~8U + 2~Ucollision simply staggers the mind! Such was the situation by 1976 when data for such reactions as 20Ne + 238U at 250 and 400 MeV per nucleon first became available [GS76]. See fig. 2.1. Therefore it was natural to step back and ask what had been learned about light fragment production in proton—nucleus reactions. 2.2. Deuteron formation in proton—nucleus reactions The most quoted theoretical model for deuteron production in proton—nucleus reactions is due to [BP631.The dynamics involves the pairing of a proton and a neutron from the cascade which develops within the struck nucleus. The p—n pair interacts with the static nuclear optical potential which, together with the usual p—n strong force, allows them to bind together to form a deuteron. The calculation involves second-order perturbation theory, see fig. 2.2. The result relates the density of deuterons in momentum space d3NdIdK3 to the density of protons in momentum space d3N~/dk3(assumed to be the same as the neutron density), y d3Nd/dK3 = I V~K(1

+

m2/k2) J(KR) (y d3N~/dk3)2.

Here m is the nucleon mass, K2/m

=

(2.1)

2.2 MeV is the binding energy of the deuteron, V

01 is the depth of the optical potential, and J(KR) is a dimensionless function which depends on the shape of the optical potential and the radius R of the nucleus. Also K = 2k is the momentum of the deuteron. Note that the nucleon distribution is evaluated at the same momentum per nucleon. The relative momentum of the proton and neutron must be small because the recoil momentum which the nucleus is able to absorb via 2/m2)112 is the usual relativistic gamma. theThe optical potential is small. Finally, y = (1 + k most important aspect of eq. (2.1) is that the deuteron distribution is proportional to the square of the nucleon distribution. This is just phase space. The coefficient of proportionality depends on the

~k2

(a)

(b)

(c)

Fig. 2.2. Diagrams (a), (b), (c) illustrate the simplest means of deuteron formation. k 1, k2 are the momenta of the proton and neutron in the initial state, q the recoil of the nucleus, and K the deuteron momentum in the final state. In case (a) the neutron and proton interact first with each other to form an intermediate deuteron state. This deuteron is then scattered by the nucleus into the final state. In case (b) the neutron is scattered into an intermediate state by an interaction with the nucleus. The scattered neutron and an unscattered proton then interact with each other to form a deuteron. In case (c) a scattered proton pairs with an unscattered neutron. From [BP63].

228

LP. Csernai and J.I. Kapusta, Entropy and cluster production in nuclear collisions

deuteron binding energy and on the nuclear optical potential. The coefficient of proportionality also contains a momentum-dependent factor, and this is important. Equation (2.1) is expressed in terms of the Lorentz-invariant distribution y d3N/dk3. In this case the momentum dependence is 1 + m2/k2, where k is measured in the rest frame of the target nucleus. This is easy to understand: The nuclear optical potential is soft, i.e. it is easier to absorb excess energy momentum from slower moving nucleons than faster moving ones. The prediction of this model may be compared with experiment. As an example we shall take the data from [SS65, CS67, Ci801. The reactions were 9 GeV p + Pb and 24GeV p + AgBr. In fig. 2.3 is plotted dNd/dk versus (dN~Idk)2for various momentum bins. The data is sparse because these were emulsion experiments, but the proportionality between dNd/dk and (dN~/dk)2is quite clear. The slopes are different for different momentum bins so that it is useful to also check the momentum dependence of the coefficient of proportionality. From eq. (2.1) we should find that dN

dN 2 F(k) (—~)

1 k2 3/2 F(k) = ~ (i + —i)

,

.

(2.2)

Putting in this momentum dependence of F(k) leads to fig. 2.4 which illustrates that all momentum bins now fall on the same straight line. This provides a rough experimental verification of the model. A large amount of work was done with counter experiments in the early 1960’s. See [PS661and references therein. As a whole the theoretical model works quite well. However, there are occasional discrepancies, which are usually attributed to other mechanisms, such as: evaporation, quasi-elastic knock-out of clusters, and indirect pick-up. These effects occur in nucleus—nucleus collisions too [Sc79, SR81, BC821. discrepancies, which are usually attributed to other mechanisms, such as: evaprotatio, C)

-

Q~q, (‘I

n~(k) Fig. 2.3. The number of deuterons per unit momentum plotted against the square of the number of protons per unit momentum, for various momentum bins. The experiment was high energy protons on emulsion. From [Ci80].

L.P. Csernai and fl Kapusta, Entropy and cluster production in nuclear collisions

F(k)n~(k)

229

(arbitrary units)

Fig. 2.4. Same data as in fig. 2.3 but now plotted to take into account the phase space factor F(k) as given in eq. (2.2) by the model of [BP63I.

2.3. Coalescence model In the context of proton—nucleus collisions it was suggested that, independent of the details of the deuteron formation mechanism, the momentum distribution of deuterons should be proportional to the product of the proton and neutron momentum distributions [SZ63]. This was argued on the basis of phase space alone. Namely, that the deuteron density in momentum space is proportional to the proton density times the probability of finding a neutron within a small sphere of radius Po around the proton momentum. Thus d3Nd

41T

3/

d3N~\/ d3N 0 (2.3)

Here Po is simply a parameter to be determined from experiment. In principle it could be momentumdependent, but if it is experimentally then the model has little predictive power. One may argue that Pa ought to be on the order of K, or the Fermi momentum of the target nucleus, or the pion mass m,~.In any case it is important to realize that there is no particular dynamics invoked in the power law “ansatz” of eq. (2.3). When nuclei collide at high energy they transform a large fraction of their initial translational energy into internal excitation energy. Clearly for violent heavy ion collisions it makes little sense to speak of a static nuclear optical potential. Therefore the dynamical model described in section 2.2 probably has little relevance for high energy heavy ion collisions. However, the phase space reasoning which leads to eq. (2.3) was appealing enough that it was decided to generalize the formula for a fragment of baryon number A and to apply the results to heavy ionLorentz-invariant collisions [GS76].momentum The motivation of this coalescence 3NN/dk3 be the distribution of nucleons model goes as follows. Let y d before coalescence into nuclear fragments. Assume that protons and neutrons have equal densities, but the formulae can be generalized to accomodate the unequal situations as well. Consider a sphere in momentum space centered at k and with a radius Po. The probability of finding one nucleon in this sphere is

230

L.P. Csernai and f.I. Kapusta, Entropy and clusterproduction in nuclear collisions

1

d3NN

4ir

(2.4)

P=~-~--p~y-~--,

where M is the mean nucleon multiplicity and we neglect corrections of order (p0/mN)2. The statistical probability for finding A nucleons in this sphere is

(~)

PM(A) =

pA

(1- p)M-A

(2.5)

If the mean nucleon multiplicity is high, M 1, if the mean number of nucleons in the sphere is small, M~P 4 1, and if the nuclear cluster is not too large, A 4 M, then ~‘

PM(A)

=

~

(MP)A.

(2.6)

In terms of eq. (2.4) this is d3NA 7

=

1 /41r \ ~ (~ Po)

/ d3NN\A ~, ‘~*‘

-~-)

(2.7)

where K = Ak is the total momentum of the cluster. We still need to take into account the spin of the fragment [Me77},and the isospin of the projectile, target and fragment. Letting N and Z denote the neutron and proton number of the fragment, and N~,Zp, NT, ZT respectively those of the projectile and target, we get d3NA 7dK~

2sA+1 1 2A

~

1 /Np+NT~”/41r \/

d~Np~

~~ 1Po)

~\Y—3-)

•

(2.8)

To obtainand thedivides densitybyin the momentum one normally takesabout the single-particle cross asection 3o/dk3 reaction space cross section o~.(More this in sectioninclusive 2.7.) Finally word dabout notation: The Pa defined by eq. (2.8) is usually denoted by j3~in the literature [Me78]. Single-particle inclusive cross sections were measured in a counter experiment for the collisions of Ne + U at beam energies of 250 and 400 MeV per nucleon, leading to the fragments p, d, t, 3He and 4He [GS76, GW821. Fits of the coalescence model to these data are shown in fig. 2.5. The agreement is quite satisfactory, perhaps surprisingly so. The numerical values of Po obtained [Me781from the fits are dependent on the fragment: 71 MeV/c for d, 94MeV/c for t and 3He, and 122 MeVIc for 4He, all at 400 MeV per nucleon. There is also a weak beam energy dependence. Note that the yield of 4He, for example, is proportional to pg, and so a 10% increase in Po increases the yield of 4He by a factor of 2.36! Unfortunately this means that important dynamical effects can be easily absorbed into the fitted values of Po. Notice that in order to extract accurate values of Po from data it is necessary to know the absolute normalizations of the single-particle inclusive cross sections. Several years after publication of the aforementioned data it was discovered that the absolute normalizations were in error [SG8O].The protons were in error the most by a factor of about 2.5. Therefore the values of Pa quoted above are not

LP. Csernai and H. Kapusta, Entropy and cluster production in nuclear collisions

20Ne I

I’

I~~’

I!

I

0.1

El

-

~

100

Ii..

-

4He

~300

60°

400 MeV/nucl. I

I!l!l!I,

60

I~ IC

I,... .0I>

II?~l,_

3He

t

~Z~30° I0-\600 90° I -

1~

U

+

IIII~

d

100-

231

1,1,

d

•

1.111

I

t

60°

I

~

I

4He

3He

-oIW 1°

10 -

-

\\900

\900

00

30° 60°

0.1 2b - 250 \1200 MeV/nucl. 0.01 601 2b 6d12b

60 f9O° ~

i~d

60°-

6b

Elab (MeV/nucl.) Fig. 2.5. Experimental points and calculated lines from the coalescence model for the double-differential cross sections of fragments from the irradiation of uranium by 20Ne ions at 250 and 400MeV/nucleon. From [GS76I.

Ar+KCI—d+X 800MeV/A ~

I

I I

I

(D

I

I I

I

I

I

I

I I

:1

I I I ~

~

0

I

I I I

I

-

100

I0~

00 0~

$

‘

600

-

V

_1~I-

I0~

-,

\800

—

-

—

V

~ io2 V

+

Expenment

°

HF LF1 Coalescence

+~

•

-

0

—

_____________________________________________________________________________

I I

°

S

—+

\lI00q~

4145°

4~

—

0~.

c~J 10 IIIIIIIIIIIIIIIIIIIII~~1IIIIIII w 0.5 I 1.5 2 2.5 ~—

3

3.5 0.5 P(GeV/C)

I I

I I

I

I

1.5

i

I 2

i

0.5

i.I i

I

i

1.5

Fig. 2.6. Comparison of the inclusive spectra of deuterons produced in the Ar + KCI collision at 800 MeV/A with the coalescence model. From [LN79].

LP. Csernai and f.I. Kapusta, Entropy and cluster production in nuclear collisions

liii

I

IIIIIII,I,II,

-~

10 2

Ar+.Au

37MeV/NUCLEON Ar+.Au

-

30 degrees

~

I

~~1~~~~

137 MeV/NUCLEON

I

-

90 degrees .. —S.....

-

-

-

10 I) 100

~

I

8Li(xl0-4)

~l0

~l06

‘He(xiO

~

6Li(x104)

7 ~

~ZI0

E c~

~ I0~

~

~l0~0

7Be (x b0~)

-

~i

Li(xIO

I

-Il -~

l0~2

-

7Be(x107)

-

‘°Be(x108)

-

lI~~)<~~ 9

T

t~ç

~I.

~B(x l0_10)

-

-

N(xIO~3)

I I

IlIllIIIllIIIII1II~l

50

-

H

j~~\cx10~12

0

-

-

T#~\~~N.~Be(XI08)

0

::

:

‘~N

bO8 T\

-

100

150

200

50

100

50

200

ENERGY (MeV/NUCLEON) Fig. 2.7. Double-differential cross section for fragments from 137 MeVlnucleon Ar+ Au at 30°and 90°. Not all the fragments are shown. The solid lines correspond to coalescence model fits. The values of p~are plotted in fig. 2.8. From [JF84).

L.P. Csernai and .1.1. Kapusta, Entropy and clusterproduction in nuclear collisions

233

too accurate. Nevertheless the power-law embodied in the coalescence model has been checked by several groups for a variety of projectiles and targets, fragment numbers A and beam energies. As examples see fig. 2.6 [LN79]and fig. 2.7 [JF84]. It has been seen to work for fragments from d to 160 and for 137 to 2100 MeV per nucleon lab energies. This is in spite of the fact that oftentimes the assumptions which go into motivating the coalescence model are invalid. For example, in regions of phase space where MP is on the order of unity (see eq. (2.5)), the final result of eq. (2.8) still holds empirically [GG77]. Thus, although the model predicts the proportionality between the observed fragment momentum distribution and the primordial (preclustering) proton momentum distribution, the data actually show a proportionality between observed fragment momentum distribution and observed proton momentum distribution. In addition to the above difficulty the coalescence model does not give any clues as to the dynamics of the nucleon clustering. That is, it does not predict a numerical value for Po or how Pa depends on fragment mass, projectile—target combinations or beam energy. Nor does it allow one to extract any useful information from Po, such as clustering dynamics or nuclear matter properties. Clearly there was a need for a dynamical basis for the coalescence model or, more precisely, for the power-law embodied in eq. (2.8). 2.4. Sudden approximation It is an obvious and accepted fact that during a central collision of large nuclei the nuclear matter is compressed and energetically excited. Nuclear correlations may develop at nuclear densities greater than normal nuclear matter density (n0 = 0.145 to 0.170 fm3) but it is not clear whether one ought to speak of nuclear clusters existing at high density. It was suggested that the complicated many-body problem of cluster formation may be treated approximately in the following simple way [BJ77]. Assume that a condensation of independent nucleons into nuclear clusters takes place on a time scale which is short compared to the inter-particle collision rate. Then the formation of a cluster from Z protons and N neutrons may be estimated from the sudden approximation of quantum mechanics. Letting K denote the total momentum of the fragment of baryon number A = N + Z one obtains WA(K, X) =

2sA + 1 2A

dk~dx3

J [iii W~(k Z

dk~dx3 3 W~(k1, (2i~)

I

A

]~[i

1,~

(2ir)~

‘1)

(2.9) The index i refers to a proton if 1 i Z and to a neutron if Z < iS A. The Wigner six-dimensional phase space density is denoted by W. The overlap of the cluster wave function, including both the internal wave function with center of mass located at position X and moving with momentum K, with the wave function for A independent nucleons is denoted by ( ). The formal similarity with the (empirical) coalescence model is already apparent in the phase space structure of eq. (2.9). A typical width for the nucleon momentum distribution as determined by experiment is 400 MeV/c. See fig. 2.1. The momentum widths of nuclear wave functions as estimated from the shell model are considerably less. We furthermore assume that the nucleon distribution is uniform over the spatial extent of the cluster. In that case

I(A

N+Z,K,XJk

2 fl(2~)3~(x 1,x1.. ~kA,xA)I

1—X)~(k1—K/A),

(2.10)

234

L.P. Csernai and J.I. Kapusta, Entropy and cluster production in nuclear collisions

and so details of the cluster wave function are not important. Inserting eq. (2.10) into eq. (2.9) one arrives at WA(K, X) = 2sA+ 1 ~W~(K/A,X)1z [W~(K/A,X)]N.

(2.11)

This is essentially in the coalescence model form. If we further assume that in fact the nucleon phase space distribution is independent of position within some (e.g. fireball) volume V then

—v—

3N W(k, x) = (21T)~d~

(2.12)

This leads to d3NA dK3

—

—

2sA + 1 1(2~)31.4~1 (d3N 3N~\N 0\z (d 2A L V ] ~ dk~) ~ dk~)

2 13 (.

)

is

where k = K/A the momentum per nucleon of the cluster. Comparing eqs. (2.8) and (2.13) we can make the correspondence /

1

\ 1/(A—1)

~N!Z!)

41T

(21T)3

Y~Po~7 7.

(2.14)

The left side is momentum dependent whereas the right side is not. However for typical heavy ion reactions at present energies y 1. Furthermore the derivation of eq. (2.13) was essentially nonrelativistic. Therefore this is a rather moot point in any case. Thus we have seen that the sudden approximation of quantum mechanics can give a dynamical explanation for the coalescence model. A measurement of the so-called “coalescence radius” is actually a measurement of the effective volume of the nuclear matter at the time of condensation of nucleons into nuclear clusters. An estimate of this volume as extracted from experimental data will be given in the next section. It is important to realize, though, that the nucleon Wigner phase space density appearing in eqs. (2.9) and (2.13) is actually the primordial (or pre-clustering) distribution. The sudden approximation model thus has the same difficulty in explaining the data as does the coalescence model when the density in phase space is high. 2.5. Thermodynamic model The thermodynamic model for light cluster production [Me77] assumes that, late in the collision following decompression of the nuclear matter, chemical equilibrium among protons, neutrons and light nuclei is attained. After some time particle collisions are so infrequent that the particles essentially follow straight line trajectories. Thus one characterizes the state of the system at the time of breakup by a temperature T, proton and neutron chemical potentials he., and i~n, and volume V. Given the baryon number B, electric charge Q, total energy E and baryon density n one can calculate the above quantities. For example [Ka77]

L.P. Csernai and J.l Kapusta, Entropy and cluster production in nuclear collisions

B=>~B1Nj=nV,

O=~Q1N1,

E=>~E1N1

235

(2.15)

where the index j refers to the particle species (e.g. p, n, d, t, etc.) and .T\~is the total number of particles of that species. This number may be obtained by integrating the momentum distribution 2— i}~. (2.16) ~ 2s~+ 1 V~exp~(~+ m~’ dK3 (2ir)3 I L T i

~1±

Here .s~is the spin of the particle, m 3 is its mass, ~ is the chemical potential, K is its momentum, and finally the + (—) refers to Fermi (Bose) quantum statistics. In chemical equilibrium the chemical potentials for nuclear clusters of bayron number two and greater are determined in terms of ~ and For example, the reactions p.s.

p+p+t~-p+a, imply that 2 1a~+U,

/Lp+SUa.

A consideration of all possible reactions leads to the conclusion that for a nucleus with proton number Z and neutron number N =

~

+ z~.

-

(2.17)

This is just a statement that all states consistent with electric charge and baryon number conservation are populated according to the Boltzmann factor in the partition function of statistical mechanics. Equation (2.17) holds even if the nuclear cluster under consideration is not in its ground state. Its excitation energy and spin degeneracy rather are taken into account in in3 and s1. The dynamical issue as to whether chemical equilibrium can actually be attained will be discussed in section 2.8. If the density of the system in six-dimensional phase space is high, that is if the system is rather degenerate, then the quantum statistics in eq. (2.16) will be important. Then there is no simple relationship between the nucleon momentum distributions and the light nuclear cluster momentum distributions. On the other hand if the phase space density is low then the quantum statistics is unimportant and one simply observes that 3NA = 2sA + 1 1(2~31~1 e_Eh/T (d3N~\~ (d3N d 0\N 2 18) 3 2A L V i \ dk~) ~dk~) dK for a cluster with A = N + Z and mass mA = Nm,, + Zm~,+ E 0. Apart from the factor exp(—Eo/T) the relationship between the volume V and the “coalescence radius” Po is the same as in the sudden approximation model, eq. (2.14). This extra factor tends to suppress clusters with a high excitation, although the spin degeneracy tends to offset this. These highly excited clusters may be significant near the critical point of nuclear matter (see sections 3, 4 and 5). Specific variations of the basic thermodynamic model tend to include most of the known excited states up to some mass cutoff. (See, for example, [BJ77, Me78, GK78, RK81, FR82 and SB83].)

236

L.P. Csernai and f.I. Kapusta, Entropy and cluster production in nuclear collisions

There are two significant differences between the coalescence model or sudden approximation model and the thermodynamic model. The first is that the thermodynamic model actually relates the observed cluster and observed nucleon momentum distributions, in agreement with the experimental findings and unlike the other two models. The second is that the thermodynamic model also predicts the form of the nucleon momentum distribution, namely an exponential of the kinetic energy in the rest frame of the emitting system. This may be considered a virtue or a vice depending on your point of view. Several groups have extracted the effective interaction volume at the time of condensation or breakup. The momentum distribution used is an average over all impact parameters, i.e. the singleparticle inclusive cross section dK3

~219

o~dK3’

where o~is the reaction cross section. On the basis of the relation (2.13), or (2.18) if the binding energy is neglected, the authors of [BJ771 fit the data of [GS761.It was found necessary to include nuclear resonances up to A = 5 to obtain consistency with the data for p, d, t, 3He and 4He fragments. A mean baryon density of n = 0.4 ±0.1n 3 was taken as normal nuclear matter 0 was density. (An absolute volume was not found, quoted.)where n0 = 0.17 fm An independent experiment was done [LN79] which also inferred effective source sizes. First the parameter Po was fit and then the corresponding volume V = 41TR3/3 was determined, again neglecting the binding energy of the nuclear fragments. Table 2.1 contains a summary of the results. The results are about as expected, R being on the order of 3 to 4 fm. That is physically satisfying. However, the influence of decay of nuclear resonances was not taken into account. A recent experiment was carried out at the lower beam energies of 92 and 137 MeV per nucleon Ar beams [JF84]. Fragments with 2s A 14 were detected, momentum spectra measured (see fig. 2.7) and effective sizes were inferred just as in the preceding experiment. The values of Po and R versus A are plotted in fig. 2.8. Again the numerical values are reasonable. One may question whether there is actually one value of R consistent with all fragments, but it must be kept in mind that (i) the effect of nuclear resonances was not included, (ii) these are impact parameter averaged results. Most models of relativistic heavy ion collisions do not yield a single source with volume V in which particle emission is isotropic. For example if some transparency [WG76, Da78, GM83] or collective flow [CG81]of the colliding nuclei is allowed for one may end up with two fireballs [My781.A hydrodynamic calculation [KS81, CS83] yields a multitude of fluid elements each moving with its own velocity. All of these situations complicate the nice power-law inherent in eqs. (2.8), (2.13) and (2.18). The momentum distribution for a fragment of baryon number A 1 is (2.20)

The sum is over all volume elements e, Ke is the value of K in the rest frame of e, and y~= (1 + K~/m.~)112. The empirical power-law no longer holds exactly because the product of sums is not the same as the sum of the products. Therefore comparisons with data must be made on a case by case basis. As an example consider the reaction Ar + KC1 d + X at a laboratory energy of 800 MeV per nucleon —~

L.P. Csernai and J.I. Kapusta, Entropyand cluster production in nuclear collisions

237

Table 2.1 p~and R deduced from the listed reactions. Typical errors are ±10%.From [LN79]

System C+C

Energy (MeV/A) 800

(mb) 939

po (MeV/c)

R

167 204

2.9 2.6

122 159 164

3.9 3.4 3.3

142 162 142 189 142 154

3.4 3.3 3.4 2.8 3.4

3He

113 150 144

4.2 3.6 3.7

d

113

4.2

145 137 113 95 138 144

3.7 3.9 4.2 5.0 3.9

143 130 185 173

3.3

123 112 153 144 157 145

3.9 4.3 3.5

Fragment d t, ~He

C+Pb

800

2964

d 3He

Ne + NaF

400

1301

d t, 3He

800

d t, 3He

2100

d t, ~He

Ne+Pb

400

3497

800

d

3He 2100 d 3He Ar+ KCI

800

2445

d

t, He

Ar+Pb

800

4545

d

He

(fm)

3.5

3.7 3.7 2.9 3.1

3.7 3.4 3.7

[LN79,NL81]. In fig. 2.9 is shown a comparison between experiment and two thermodynamic models: the two fireball model and the firestreak model [GK78].This figure is taken from the excellent review of the thermodynamic model [DM81]. The same experimental data is compared with the results of a relativistic 3-dimensional hydrodynamic model [KS81] in fig. 2.10. The agreement between theory and experiment may be judged to be adequate considering that there are no free parameters (the breakup or condensation density was fixed at 0.12 fm3 a priori). The empirical power-law fit generally does better, which is no surprise. It has one free parameter Po for each fragment. In addition, if the dynamical model does not get the nucleon momentum distribution right on, the light nuclear fragment momentum distributions will generally be worse yet since they are essentially the nucleon distribution raised to the Ath power. See fig. 2.11 and [LN79, DM81, NL81].

238

L.P. Csernai and fl Kapusta, Entropy and cluster production in nuclear collisions

-

a)

18040 C)

~lOC ~

I

I

b)

80-

+4*

-

40-

to:5i

5i

c) d)

l~

~

~+ itt,

I

A Fig. 2.8. (a) Coalescence radii, jib, for Ar + Ca, (b) coalescence radii for Ar + Au, (c) interaction volume radii, R, for Ar + Ca, (d) interaction volume radii for Ar + Au. Circles and crosses represent the 137 and 92 MeV/nucleon incident energies cases respectively. From [JF84].

Ar+KCI—°-d+X III

11I!

800 M~V/A

CII

>

________

I

II

~~~IIIIhII~iuI

60°

~

-

.0

~ ;

30°

~

—

102 -.

0~ .

b 10

Exp~ri~+4

- —

CII

———Two Fircballs

D

10

Fir~str~k

CII

-

-

IIIIIIIIIIIIIIIIIIIIII_IIIII

Lii

0.5

1

1.5

2

2.5

3

\t~ -

1 0.5

II

I I

1

I I

I’~t~A

I

1.5

2

I _______________f 145°

-

~I

0.5

I

‘~

I

1

I’~

1.5

P (G~V/c) Fig. 2.9. Deuteron cross section for Ar on KCI at 800 MeV/A compared with the two fireball model and the firestreak model. From [DM81].

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

239

a

b

Ar+KC1—-p+X i()

Ar+KC1-d+\

10

‘111111’’

30

°CM

S

1~ .

io3

ç

L~ld2

+

>

~ 10 2

:~O3O

~Ii)

~id .0

-.-.-.-5(11

60 xlO

1 ~1

60°~i

-••

~02.

.0

2

. 0

S

.~

•

°~‘

~

-010

10_i

b

.5_._..

b

~ 1O~

90°~1~4

.

‘a

.-..

102

30

i0

io~

— -

-..~

,I,.I,,,I,.I,,,,I

100

200

.1

+

300 400 500 T (MeV)

600

700

0

,,I,.I,,,l

100

200

300 400 500 T (MeV)

CM

600

700

CM

Fig. 2.10. The (a) proton and (b) deuteron invariant cross sections plotted in the center of mass frame for a laboratory beam energy of 800 MeV per nucleon. The data is from [LN79I.The curves correspond to the results of a relativistic, three-dimensional, hydrodynamical model [KS8I].The solid curve is for a thermally soft equation of state, the dashed curve for a thermally stiff one.

800MqV/A

Ar+KCI—÷t+X

III

‘~2.

10

E ~

ft~

a-

-

~WL

II

30°

-

Exp~rirrwrst

Fir~str~ak

1°~~1~&

———Two Firaballs 10.5 I_____________________________________________ I I I I 1 I I I I1.5 I I I I I 2I I I I I2.5 I I I I I 3I I

I~

I

-

102 _‘~6O0 10 \f~~fff \

1

-

0.10.5I

.\

I I I

11111111111111 1.5 2

80°

~ff

\\~t

II

\

-

-

0.5_______________________ I 115111 1 I I\ 1.5 III 2

P(G~V/c) Fig. 2.11. Triton cross section for Ar+ KCI at 800 MeV/A compared with the two fireball model and the firestreak model. From [DM811.

2.6. Sudden approximation for intranuclear cascades Another dynamical model of relativistic heavy ion collisions from which one would like to compute light cluster production is the (generic) intranuclear cascade model. A formalism has been developed for deuteron production [RS75, Re82, GF83] which is very similar to the sudden approximation model.

L.P. Csernai and fl Kapusta, Entropy and cluster production in nuclear collisions

240

Rather than assuming that all nucleons condense or cluster simultaneously one waits until just after both prospective cascading nucleons make their last (hard or cascade) collision. Thus deuterons may be formed continuously throughout the time development of the colliding nuclei. The result is d3N dK3

3 =

{8(K— k~—k,,) W,, 2(k~,x0 k~,Xfl)}ensembte

(2.21)

.

The 3/4 is the spin factor. W,~,is the (two-particle) proton—neutron Wigner phase space density evaluated immediately after the last collision suffered by either the proton or the neutron. All possible p—n pairs are summed over. The brackets {. . } denote the ensemble averaging of all cascade events. Note the similarity between eqs. (2.9)—(2.11) and eq. (2.21). Again approximations have been made to the effect that the nucleon phase space density does not vary appreciably over the spatial and momentum extent of the deuteron so that the sensitivity to the deuteron wave function drops out. It has been emphasized that the right hand side of eq. (2.21) refers to the production of deuteron-like pairings which in fact may be contained in a large nuclear cluster. As discussed elsewhere [BC81], and as will be discussed in section 4, simple spin—isospin considerations suggest that Ndljke

~

Calculations with the intranuclear cascade code of Cugnon [Cu80] were done for deuteron production as described above in [GF831.The results are shown in fig. 2.12 where comparisons are made with the data of [SG8O].As with the other models agreement between theory and experiment is quite adequate. During the course of the analysis of the cascade output it was found that there was no significant structure in the two-particle density on the scale of 2 ±1 fm. There were essentially only long range 400MeV/A lOO.N~I~C

0.1

Ne+U

400MeV/A

Ar÷Ca

sum charges

tjJ

I

(A)

I

I

(C)

I

I

I

CCdtC~

o.I~I>H~JZ~N(D) (B) I

0

I

100

I

I

200

I

0

ENERGY PER NUCLEON

I

I

100

I

200

MeV/A

Fig. 2.12. (a), (c) Comparison of sum charge inclusive data [SG8OIwith cascade predictions. In (b), (d) the primordial deuteron distribution as calculated (solid lines) using eq. (2.21) is compared to data (solid triangles). The free deuteron data are indicated by dots. From [GF831.

L.P. Csemai and fl. Kapusta, Entropy and cluster production in nuclear collisions

241

correlations on a scale of 4 fm due to finite size effects. Hence the independent particle approximation essentially inherent in all the previous models of light fragment production is alright, at least in the intranuclear cascade model. 2.7. Impact parameter averaging It is rather surprising that empirically the power-law holds between single-particle inclusive cross sections. Impact parameter averaging should destroy the simple scaling-law since the average of the product is not necessarily equal to the product of the averages. See eqs. (2.8), (2.11) and (2.18). Let us investigate the consequences of impact parameter averaging for several of the models cited previously [Ka80]. For simplicity we will restrict our attention to deuterons and to the nonrelativistic limit. Generalizations are obvious but notationally more cumbersome. Consider the coalescence model with Po independent of momentum and impact parameter. Define the proton—neutron momentum distribution as

dk~dk~hh1~ k2 b) = D~~(k1, k2 b),

(2.22)

where b is the nucleus—nucleus impact parameter. The single-particle momentum distributions are (k1 b)

=

D0(k1 b) = N,,(b) J dk~D,,~(k1,k2 b),

3N ~~-j~(k d

1

(2.23)

2b)= Dfl(k2b)=N(b)Jdk~DnP(kl~k2b)~

where N~and N~are the total number of protons and neutrons. The coalescence formula is 3Nd/dK3 = 7rpo~D,,~(k,q = 0; b), d where k = ~(k 1+ k2), q = k1 — k2. Integrating over the impact parameter gives 3o-d/dK3 = d

1TPO~J

d2b D~ 0(k,q

=

0; b),

and so on. Define a neutron—proton correlation function by 6u~~/dk~ dk~ C(k, q) = (d3u~/dk~ (d3u,,/dk~)—1. o0d Here ~

=

d3od

5

(2.24)

(2.25)

(2.26)

d2b is the reaction cross section. Combining the above equations we arrive at irp~d3u~d3o,,

(2.27)

242

L.P. Csernai and f.l. Kapusta, Entropy and cluster production in nuclear collisions

Previous analyses of data have been done neglecting impact parameter averaging, i.e. setting C(k, q = 0) = 0. The correlations may indeed be small but in general this is incorrect. It could lead to an anomalous momentum dependence of the effective Pa. As an aside, note that C(k, q) should be smeared over a finite range in q~, Jq~< 100 or 200 MeV/c corresponding to the momentum spread in the deuteron wave function. In the sudden approximation model and in the thermodynamic model the situation is somewhat different. In these models (neglecting exp(—Eo!T) in the latter) (2.28) Now the interaction volume may depend on the impact parameter. Let N(b) be the total number of participant nucleons in the volume V(b), and let n. denote the breakup or condensation density. Define a new correlation function by C (k,

UN

~

S

q; b) d2bN~(b)D,,~(k, b).fd2bD~(k 1b)

1,

(2.29)

where UN =

J d~bN(b) = uo~N).

(2.30)

Then 3crd

3

~ n~d3o-

3o-,,

—~=4(27r)—-~-~--—-,-~[C(k,q=0)+1]. d 1~d cTNUK

(2.31)

UK

= 0) is to be interpreted as C’(k, q) smeared over a small range in q~around q = 0. Dynamical correlations between two protons for large q have been observed experimentally [NA79, TL8OJ. These have been interpreted in terms of quasi-elastic nucleon—nucleon scattering [KR81] and in terms of collective fluid-like flow [CG8I, CG82]. For small q, correlations due to the Hanbury-Brown Twiss effect [HT56, Ko77] are widely used to determine the size and lifetime of the emitting source [ZS81, GG841. Collective fluid-like flow may also contribute to small q correlations [Pr841between protons and neutrons. All of the above indicate that C(k, q = 0) and/or C’(k, q = 0) may not be

Again C’(k, q

negligible in high energy heavy ion collisions. Note that in either case, eq. (2.27) or (2.31), the integration

over impact parameter could be

restricted to correspond to an experimental trigger. Thus it seems to be necessary to either (i) measure C(k, q = 0) and C’(k, q = 0) if all impact parameters are to be used experimentally or (ii) to use a trigger in the experiment to restrict events to a narrow window in impact parameter. Otherwise meaningful values of P(I and/or V may not be possible since the correlation functions could change the absolute normalization. 2.8. Rate equations for cluster production Let us now return to the question of chemical equilibrium for light nuclear clusters in central

L.P. Csernai and.!.!. Kapusta, Entropy and cluster production in nuclear collisions

243

collisions of massive nuclei. A scenario has been painted wherein the light nuclear elements are built up in a series of multi-body reactions during the expansion stage of the collision [Me781.For deuterons typical reactions are n+p+N-~d+N

d+n+p-~d+d

(2.32)

p+n+p+n-~d+d where N is a nucleon. For tritons n + n + p+ N-*t+N n+d+N-~t+N

(2.33)

d+d-+t+p and so on. If chemical equilibrium is achieved then the above reactions go in both directions and according to detailed balancing. The characteristic time scale for reaching chemicalequilibrium for a cluster of baryon number A is the inverse of n(o(N + A —~breakup)vrei) where n is the baryon density, a- is the breakup cross section for an incident nucleon, vrei is the relativevelocity and ( ) denotes averaging over the relative velocity distribution. A typical value for Vrel is 0.5c for densities on the order of nuclear matter density n0 and temperatures on the order of 50 MeV. At such high relativevelocities breakup cross sections —~

~—

are typically geometrical in magnitude. Taking a typical value of 100 mb one finds a characteristic time scale of 1.5 fm/c. This is sufficiently short compared to realistic estimates of expansion times [Me78,Cu80, CB8O, KS81] that attainment of at least approximate chemical equilibrium is not unlikely. Of course one must actually solve a coupled set of rate equations to determine the concentrations of various species of nuclear fragments. As the simplest example one may focus on deuteron formation [Ka80]. Initially the most important reaction will be the first in eq. (2.32). The rate equation for a nonexpanding system is dnd/dt

flN(tT(N +

d

fl +

p + N) VreJ)[flpnn(nd/npnn)eq

— ndl.

(2.34)

The number densities of various species are labeled by the subscript. The equilibrium ratio is 312. (nd/npnn)eq = ~(4nimT) For a constant breakup cross section

(2.35)

112, (2.36) o(16T/irm) with m the nucleon mass. All of these equations assume that nucleons and deuterons are in kinetic equilibrium at temperature T, and that nonrelativistic kinematics and Maxwell—Boltzmann statistics are applicable. (0~Vre

1)=

L.P. Csernai and J. I. Kapusta, Entropy and cluster production in nuclear collisions

244

The time dependence of the temperature and baryon density now need to be specified. A simple hydrodynamical model [BG781was used. The baryon density evolves according to n(t)

=

n(0)[1

+

(t/t0)21—312,

(2.37)

where the characteristic time t is expressed in terms of the total excitation energy E and baryon

0

number of the system, 3Am 1/2 3A t/3 ~0= (~j~) (4~n(O))

(2.38)

-

In a hydrodynamical model of course thermal energy is continuously converted into kinetic flow energy. In this particular model the thermal energy and kinetic flow energy are

=

1+ ~to)2’

EOOW =

(2.39)

~

The temperature evolves according to 2E 1 1 2 1 — fld(t)Ifl(t) T(t) = - — 3 A 1 + (t/t0) and the densities satisfy n~(t)= n~(t)=

nN(t),

n(t)

=

(2.40)

nN(t) + 2nd(t).

(2.41)

Thus the full rate equation to solve is d

(t) =

~

3

(m

4 3/2 T(t)) (n(t) — 2nd(t))2 — nd(t)] . [n(t)

—

2nd(t)]

16T U (—

~—~--)

1/2

—~

3

fld(t).

(2.42) Here n(t) and T(t) are to be taken from eqs. (2.37) and (2.40). The last term in eq. (2.42) is a dilution term reflecting the fact that the deuteron density will decrease due to expansion. Finally a value of a- = 100 mb was used in the computation. Some typical results are plotted in fig. 2.13. The deuteron abundance builds up to the equilibrium value very rapidly. Obviously it would build up even more quickly if the other reactions in eq. (2.32) were taken into account. There is one caveat to this conclusion however. Does it make sense to speak of deuterons existing at normal nuclear matter density and above? No allowance was made for the extended size of the deuteron. This point will be discussed in section 4. However it is clear that even if deuterons could not be formed until subnuclear densities were reached it is clear that they would still attain chemical equilibrium. Defining f to be the fraction of the baryons bound up in deuterons notice the curious feature in the figures that the asymptotic value of f is independent of the breakup density! This is a very nice feature which will be exploited in the next section. The origin of this feature is easy to see. From eq. (2.42) the

L.P. Csernai and f.l Kapusta, Entropy and clusterproduction in nuclear collisions

245

3)

03

n (baryons/fm 02 0 15

0.25

08 -/Hydro/equil 0.6

0 I

0075

37.5 MeV/nucleon excitation I60nucleons

-

~~-----~

f

-

~~Hydro/nortequiI 0.4

Statlc/equll

-

-

0.20

-

1.1

0

4

.3

x

.

ynyuro/equil

a I

12 1024s)

n (baryons/fm3) .2 IS

.25

~ ., -

I

8 t(fm/c =33

16

~II

20

~75

50 MeV/nucleon excitation 20nucleons

-

~ -

(fm/c 3 3x 1024s) Fig. 2.13. The fraction f of nucleons which are bound in a deuteron as a function of time and density, for two different initial conditions. The solid line is a solution of the rate equation in a hydrodynamically expanding fireball. From [Ka80].

instantaneous equilibrium value of f is feq

=

~g(4ir/m T(t))312 n(t) (1 _feq)2.

(2.43)

Moreover from eqs. (2.37) and (2.40) 3A

3/2

n(t) T312(t) = (1 — feq)] n(O). (2.44) Putting these two together we see that f~ depends only on the initial condition and not the breakup density. This is no accident. In (perfect fluid) hydrodynamic flow the entropy is conserved. For the ideal gas equation of state used here the entropy depends only on the single quantity nT_31’2. (See also section 3.) Since the entropy is independent of time so is the equilibrium concentration of deuterons. 2.9. Determination of entropy from the deuteron to proton ratio Given that light nuclear cluster production in high energy heavy ion collisions can be measured, and given that various dynamical models of light nuclear cluster production can reasonably well fit the data,

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

246

what can we learn about the properties of hot and dense nuclear matter? Solution of the rate equation for deuteron production in hydrodynamic expansion suggests the following semi-quantitative picture. For approximately central collisions of very massive nuclei something like a thermalized fireball should be formed at high density and temperature. Once formed there is nothing to hold this fireball together, so it will blow apart. As it blows apart frequent nucleon—nucleon collisions allow the system to maintain local thermal equilibrium. The expansion phase of the reaction is thus hydrodynamic independently of whether or not the initial stage of the reaction could be treated using hydrodynamics. In hydrodynamic flow both the particle density and the excitation energy per particle decrease in just the right portion to conserve entropy. For example, a weakly interacting gas expands according to T(t) n(t)213 as seen in a comoving frame. Internal thermal energy is continuously converted to collective expansive flow energy. Eventually the nuclear matter becomes so rarified, and particle collisions so infrequent, that hydrodynamic flow ceases. At that point the particles follow asymptotically straight-line trajectories out to infinity. However, Liouville’s Theorem guarantees that the particles’ density in phase space remains constant in the absence of collisions, so that no additional entropy is —

generated.

It is essentially during the transition from hydrodynamic flow to free-streaming flow that the dynamical models discussed in sections 2.4, 2.5 and 2.6 predict that nuclear clustering occurs. If the entropy is high enough or, equivalently, if the density in phase space is low enough, then the formation of large clusters is highly suppressed. In that case all three models yield a ratio of deuterons to protons of Rd~=

8 (We),

(2.45)

where l(

W,,) = J dk3 dx3 W~(k,x)/J dk3 dx3 Wn(k, x)

(2.46)

is the average phase space density of neutrons, and assuming that W~,= Wi,. The 8 in eq. (2.45) is just the Jacobian in momentum space. Just at the moment when collisions cease (this could happen at a global point in time or it could happen at different times for different comoving matter elements) the nuclear matter can therefore be approximated by a Maxwell—Boltzmann distribution locally. The phase space density of particles of type j is =

n

3’2 exp(—K2/2m 1(2irm1T)

1T),

(2.47)

where i-z~is the spatial density. The entropy of a Boltzmann gas is related to the average phase space density via =

—

~In 2— ln( W~,).

(2.48)

Putting together the above equations one calculates the total entropy per baryon carried by the protons, neutrons and deuterons to be S/A = 3.945

—

ln Rd~— 1.25RdP/(1

+

RdP).

(2.49)

L.P. Csernai and .1.1. Kapusta, Entropy and cluster production in nuclear collisions

247

(The last term is often dropped since the relationship was derived assuming high entropy, i.e. Rd~°~ 1.) Thus eq. (2.49) allows us to infer the entropy directly from the deuteron to proton ratio. According to the picture of the heavy ion collision described previously the nuclear matter expands adiabatically after it has thermalized at some high baryon density and temperature. Thus a measurement of the deuteron to proton ratio allows us to infer the entropy of the hot and dense nuclear matter even though the deuterons may not actually be formed until late in the expansion phase. Since we know the energy of the system at maximum temperature and density (we know the beam energy) we now have one relationship between excitation energy and entropy. This is one of the two relationships we need to specify the equation of state of nuclear matter! The missing piece of information is the baryon density at which the system thermalized. The first entropy analysis was carried out in [SK79]with the then available experimental data. Only single-particle inclusive data was available so the central collision component was maximized by integrating the area under d3a-/dk3 at 90°in the center of momentum frame for equal mass colliding nuclei. See fig. 2.14. The entropy so extracted is plotted in fig. 2.15. The width of the horizontal error bars corresponds to the energy carried by the pions. Notice that the entropy decreases for large systems or for more central collisions, which could be expected since the thermalization takes place at higher mean densities in these cases. Notice also that the entropy increases with beam energy, which is also expected. To judge the significance of the result predictions for entropy versus excitation energy for a very soft equation of state at various densities are shown in fig. 2.15. The energy density is given by e(n, s)= e(n

0, 0)+~K[no/n — 1+ ln(n/n0)]n + eFG(n, s)— eFG(n,0),

-

I -

4

I I

A1~+A.1.

—p-

I

(2.50)

I

d+x

;I:2:

b ~

14 Rdp=O. T~30MeV

-

0~ —

T~44MeV /3 0.37

-

/3~0.29 10~ 0

100

200

300

~

I 400

I 500

600

700

(MeV)

Fig. 2.14. Deuteron cross section plotted as a function of the kinetic energy at 90°in the center of mass frame. The curves are the result of a fit to the data of ILN79J by the radial explosion model of 1SR791 with the given temperature T and radial flow velocity /3. Comparison of the absolute normalization of deuterons and protons gives the ratio Rd~.From [SK79].

248

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions I

I I

AP+AT 8-

Ar+KCI Ar+KCI Ne+NaF

Q

-

_

I

—Nucleons in potential Including pions

-

0

I

d+x

{MTAG4

:~

I

I —‘

I

I

I

I

I

50

00

150

200

250

-

Energy per Baryon (MeV) Fig. 2.15. Entropy as a function of energy obtained by the use of eq. (2.49) applied to the values of Rd~obtained in fig. 2.14. The solid curves correspond to the equation of state as given by eq. (2.50) with a compressibility K = 200 MeV. From [SK79].

where eFG is the energy density of a free Fermi gas, evaluated in the limit of low degeneracy. We see that much more entropy is created than would be expected from even such a soft equation of state as this. Even the inclusion of real pions does not give enough entropy. This is especially surprising in light of the fact that such diverse dynamical models as hydrodynamics [AG78] and intranuclear cascade [Cu80] predict maximum compressions of three to four. Among the more interesting but speculative possibilities at this point are (i) there are strong attractive forces present in the hot and dense nuclear medium thus raising the entropy, (ii) many more mesonic and baryonic particulate degrees of freedom are excited at this energy than expected, or (iii) collective degrees of freedom, such as pion condensation, are the cause. 2.10. Summary and critique We have seen in this section that light nuclear cluster production is an important element in high energy heavy ion collisions. The dynamics of deuteron formation in proton—nucleus collisions is generally quite different, since in that case it is the scattering of the cascade nucleons from the nuclear optical potential which causes the binding of protons and neutrons into deuterons. In a heavy ion collision where the projectile and target are smashed to pieces there is no remaining optical potential. Experimentally this mechanism is ruled out because it predicts that the coefficient of proportionality between the deuteron momentum distribution and the square of the nucleon distribution has a one over

L.P. Csernai and f.l Kapusta, Entropy and clusterproduction in nuclear collisions

249

momentum squared dependence, which is observed in proton—nucleus collisions but not in heavy ion collisions. The so-called coalescence model is actually not a dynamical model at all; rather, it is an empirical law that (generally) the momentum distribution of a fragment of baryon number A is directly proportional to the momentum distribution of a nucleon raised to the Ath power. In fact a rigorous application of the coalescence idea shows a gross discrepancy with the data in those regions of momentum space where many light clusters are formed. This is because of the gross difference between the primordial (or pre-clustering or pre-condensation) nucleon momentum distribution and the observed distribution. In a crude sense the sudden approximation model and the thermodynamical model can be said to give an adequate physical interpretation of the experimental data. In a global view they even lead to essentially the same result, namely, that the density in phase space of a cluster of baryon number A is proportional to the density in phase space of a nucleon raised to the Ath power with a coefficient of proportionality which takes into account spin and isospin alignment. This is really no accident. Both models populate phase space in an essentially statistical manner, and once that happens it no longer matters how it was done. Conceptually the models are different. In the former it is the overlap between the nucleon and cluster wave functions which drives the clustering, whereas in the latter one envisages a complex of multi-body reactions occurring in the nuclear medium. On a finer level the precise dynamical mechanism may or may not be important for such points as the production of particleunstable clusters, clusters within a cluster, etc. These points will be addressed in sections 4 and 5. Also on a finer level one must compute nuclear clustering within the context of a specific dynamical model of the nucleus—nucleus reaction, such as fireball/firestreak, hydrodynamics or intranuclear cascade. Otherwise a detailed comparison between theory and experiment may be very misleading because of dynamical correlations among the nucleons or because of artificial correlations due to impact parameter averaging. The close relationship between entropy and nuclear cluster formation should be rather clear since entropy is just a measure of the degree of orderedness of the system. If the entropy is high the nucleons will be bouncing about rather independently of each other. If the entropy is low most of the nucleons will be bound together in cold nuclei. Up to this level of analysis there are two main criteria which must be satisfied in order to infer the entropy of hot and dense nuclear matter which is formed only for a brief time during a central collision between massive nuclei at high energy: (i) Once thermalized the matter must expand adiabatically to the point at which nucleon collisions cease. (ii) Phase space must be statistically populated at that point. The dynamical models of cluster production and their interpretation of experimental data suggest that the second criterion is at least approximately satisfied. Clearly much work remains to be done in order to conclude that we can really obtain valuable information on the equation of state of nuclear matter. This leads us to study, in the subsequent sections, what sort of behavior the nuclear equation of state may exhibit, how entropy is generated in nucleus—nucleus collisions, more detailed analyses of the relationship between entropy and light cluster formation and, finally, the significance of the fragment mass distribution including medium mass nuclei.

3. Entropy production 3.1. Entropy as an observable As we have seen in the previous section the particular importance of the entropy due to its monotonically increasing behavior was observed in relativistic nuclear physics quite early. Here we

250

L.P. Csernai and J.l Kapusta, Entropy and clusterproduction in nuclear collisions

discuss the different processes which lead to entropy production in a heavy ion collision. But before that we have to discuss the nuclear equation of state with special emphasis on the liquid—gas phase transition below normal nuclear density. In section 4 the connection between the entropy, the correlations in a dense gas and the light nuclear abundances are discussed. 3.2. General characteristics of the nuclear equation of state We have some information about the properties of nuclear matter from conventional nuclear physics. There is a stable equilibrium state at the normal nuclear density n0 = 0.145—0.17 fm3 [My76,Be711 with a compressibility of K = 180—240 MeV [BG76] and a binding energy of 16 MeV/nucleon. If we want to learn about the equation of state at other densities and higher temperatures we have to rely on theoretical estimates [e.g.: JM84, PR84, RV84, RM82, ST79, FP81, Wa75, Fr77] and on the first few experiments addressing this problem [SB82, BH851. The experiments, however, do not yield direct information about the equation of state, thus a considerable amount of theoretical guesswork is involved. Studies of the nuclear equation of state can be divided into two branches: those below normal nuclear density and those above. From the point of view of entropy production both are important. The high density high temperature part of the equation of state is decisive in how much entropy can be produced in the first, compression stage of the collision. The low density behavior of nuclear matter determines the observables and the reaction mechanism of the final expansion stage in a collision before the breakup. In this section we will concentrate mainly on the low density part of the nuclear equation of state, which is more directly related to the final fragmentation than the high energy part of the equation of state. 3.3. Nuclear matter below normal nuclear density, the liquid—gas phase transition After an energetic nucleus—nucleus collision, many light nuclear fragments, a few heavy fragments and a few mesons (mainly pions) are observed in the 100 MeV—4 GeV/nucleon beam energy region.

Thus the initial kinetic energy of the projectile leads to the destruction of the ground state nuclear matter and converts it into a dilute gas (n ‘~ no) of fragments, which then loses thermal contact during the breakup or freeze-out stage. These frozen-out fragments and momentum distributions can be measured by the detectors considering also the fact that some excited fragments can decay while reaching the detectors. In early theoretical studies it was assumed that the observed fragments formed an ideal gas mixture at the breakup [WG76, Me77, etc.]. This approximation seemed to be quite applicable at higher energy collisions E ~ 1 GeV/nucleon but at lower energies the numbers of composite fragments deviated from the theoretical predictions. It was realized that the ideal gas approach is too crude when many composites are present. In one approximation the volumina of the composite particles were taken into account [SC81].If there is only one type of fragment this approach would lead to a Van der Waals gas type equation of state. However, with many different types of fragments the situation is more involved. Some estimates of the entropy of such a mixture were given in [BB83]considering p and d fragments only and using classical statistics. The same is done with quantum statistics in [SC81] with a wider set of fragments considered: p—a.

L.P. Csemai and J.l Kapusta, Entropy and cluster production in nuclear collisions

251

Whether these assumptions lead to a first-order phase transition was not considered, and a consistent equation of state, with compression and finite volume effects was not constructed. It was realized that the finite volume effect, as well as the compressional part of the equation state, is a consequence of nuclear interactions. To treat both in a consistent manner the nuclear matter and the mixture of nucleons and deuterons were considered in quantum many-body theory [RM82]. An effective Skyrme nucleon—nucleon interaction was assumed and treated in Hartree—Fock approximation. It was shown that a Mott density nM exists at which the bound deuteron states which are at rest merge into the continuum (nM 0.001 fm3 at T = 0). This work was later extended to more light clusters, and it was shown that it leads to a first-order phase transition with a critical temperature of T~= 20.3 MeV at n~= 0.07 fm3 [RS83]. A further improvement by including Coulomb effects [RS84] leads to a fragment distribution differing from the law of mass action used in early models. Jaqamann et al. [JM841considered also the effects of the finite size of a nuclear system and found that if A decreases from A = 200 to 50 the critical temperature T~drops from 19.1 MeV to 16.5 MeV. The inclusion of Coulomb interaction decreases these values further to 15.7 and 15.5 MeV respectively. The density at the critical point was found to be n~= 0.054—0.065 fm3. A nuclear equation of state was also derived from scaling relations for solids [RV84].The liquid—gas phase transition was found here too, with T~= 20.5 MeV and n~= 0.33n 0. In the major properties of the nuclear equation of state for densities below n0 the theories converge: A liquid—gas phase transition is clearly predicted with T~= 15—20 MeV and n. = 0.3—0.5n0. More accurate information and further details can be obtained only from thorough experimental research and comparison of experimental and theoretical results. As an example which is used in the majority of the literature [SB83, SN8O, CB8O, CS83, Da79, GK84, Ka84, Ni79, MS83], we use an analytic parametrization for the nuclear equation of state, in order to discuss the basic features of the low density nuclear matter. To specify the equation of state we may define one thermodynamical potential as the energy density e = e(n, s) as a function of baryon density n, and entropy density s: e(n, s) = e0(n)+ e~G(n,s),

e~G(n,s)=

eFG(n, s)—

eFG(n, 0),

(3.1)

where e0(n) is the ground state energy density, and eFG(n, s) is the energy density of an ideal Fermi gas. We can parametrize the ground state energy density as [Ka84} e0(n)

=

n0

~

a

(—)

i=2

(3.2)

,

n0

where a1 = +21.1, —38.3, —26.7, +35.9 MeV for i = 2,. . . 5 respectively. This parametrization yields a binding energy eo(no) e0(n0)/n0 —8MeV (instead —16this to parametrization simulate finite size effects) 3. Note of that is used for and smalla nuclear nuclear compressibility K = 210 MeV at n0 = 0.15 fm densities n S 2n 0. At high densities the sound speed exceeds the speed of light. For the thermal part of the energy density we use the nonrelativistic ideal Fermi gas approximation because for the low density and temperature at the breakup relativistic corrections are negligible. Then the energy density eFG depends on the density n and specific entropy a- = s/n as [LL54]: 513/m) y(a-), (3.3) =

=

eFG(n, s) = (n

where y is a dimensionless quantity and it depends on the specific entropy a- (or ,a/T) which are

L.P.

252

Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

dimensionless also. y(a-) can be given in integral form [LL541,but in actual calculations usually practical analytic parametrizations are used. Here we use a parametrization given in [Ka841 for the inverse function as: cr(y) = 0.5213 + 1.5 ln(y + 0.7064) +

1+

1 809 1/2 y l.139y~2+ 1.417y

+

1.014y312

(3.4)

which reproduces the exact value with an accuracy of better than 0.01. The other thermodynamical quantities can then be calculated from standard thermodynamic relations. From de = T ds + 1a dn the temperature is T = T(n, s) =

119e\

9s

= ,,

213 n m

fl213 y’(c.r)

=

(3.5)

-

mo-(y)

The chemical potential is

(~)

~(n,s)=

=

The pressure is p = —e p

=

~

t9n

3~_2

+

eFG(n,

s)_ T(n,s)u.

(3.6)

Ts + ,an. This yields

5 ~ ia

p(n, s) =

a~(i+3)(ny/35 no 3

31=2

1/3±1

1 (~f~) n0

+ ~eFG(fl,

s) = p0(n) + ~e~G(n, s).

(3.7)

This equation of state represents a stable equilibrium configuration if the energy has a minimum. 2e/ai ak (where k, ionly = n, s) is positive definite, which This conditionleads is satisfied the matrix M1k = a requirement to twoif independent constraints on the derivatives of the thermodynamical parameters [LL54, section 211: c~=T(as/aT)~>0, KT

=

(3.8)

—(1/P)(8P/3p)T >0,

(3.9)

where v = 1/n is the specific volume, Kr is the isothermal compressibility, and c~is the isochoric specific heat. These two constraints lead to numerous other thermodynamical inequalities, which all are satisfied if eqs. (3.8 and 3.9) are. Our equation of state satisfies eq. (3.8). We have to discuss the consequences of the requirement eq. (3.9), however. This has an implication on the chemical potential that (a,L/an)~>0, and also on the isothermal sound speed

=(~ \apl.

1-

8n

This reads nonrelativistically:

=_~~~(~) =~--‘-->0.

=-i-(~) m

VT.

T

m

a~

T

P

KT

(3.10)

L.P. Csernai and .1.1

Kapusta, Entropy and cluster production in nuclearcollisions

253

If this requirement is satisfied then the adiabatic sound speed is already automatically positive because

(ap/a~)~ = (c~/c~)(3p/3~)~,

(3.11)

and consequently v~=(c~/c~)v~>0.

(3.12)

It follows from eqs. (3.8, 3.9), that c~> c~[LL54], so v~>v~-.

(3.13)

Thus the positivity of v~is a weaker requirement than eq. (3.10). In relativistic physics the definition of sound velocities are different. Nevertheless, as we will show, the stability requirements still remain valid, i.e. eq. (3.10) and eq. (3.13), will be modified. Relativistically the adiabatic sound speed is

ap

=

(i-)

ae

ap

=

(~-) / (—)

(3.14)

.

Using the specific energy e = r(,.’, a-), d~= T da- — p dv: u~.=(~)/[r+n(~)

an

where w = e + p = n(e similar way as

]=(~) /(e+pv)=~~(.~!~ ön w api

an ~,.

+

\

=~e~v2

w

(3.15)

pv) is the enthalpy density. The isothermal sound speed can be expressed in a

fap\ = I—I /3p\ //1I e+pv—vT(—1 /aU\ 1I. u2~=() \ae’~ ‘an/TI t.

(3.16)

Since =(ôP~~~

(~ \avJ.

1~ \aT/~

(3.17)

=--~-

T \a~i~ Tag’

where a0. is the adiabatic heat expansion coefficient, nm (3p~ n(e+pv+vc~/a0.) \aPJT

pv~

(3.18)

w+cja0.~

Since.p, w, c,,, a0. >0 are the equilibrium values the stability condition eq. (3.10) can be used with the relativistic isothermal sound speed u~->0.

(119)

L.P.

254

Csernai and fl. Kapusta, Entropy and clusterproduction in nuclear collisions

Combining equations (3.12, 3.15, 3.18) u~=~(1+_~_)u~,

(3.20)

which says that eq. (3.13) holds for the relativistic sound speeds too. Thus we can use the relativistic sound speed U~for the discussion of the stability of our equation of state given by eqs. (3.1—3.4): 2

—

uT—

(ap/an)~— dpo/dn + ~e~0/n —~TG(y) — 5 * 2 (ae/an)~ de0/dn + 3eFG/n

— ~

(3.21)

TG(y)

where G(y) = y’(o-)/y”(a-). The adiabatic sound speed is 2 dp0/dn+~e~0/n U,,— .

de0/dn +

(3.22)

SeFG/n

2 Calculating u~we observe that there is a region in our thermodynamical parameter space where u 7.< 0, fig. 3.1. That is, our equation of state does not represent a stable equilibrium. The region where u~<0 is contained within the unstable region. There are speculations [SB83, LS841 that the matter in a relativistic heavy ion collision might penetrate into the unstable region because of rapid expansion during the collision. This is quite exciting, but these approaches have still to be justified by studies of microscopic processes using reaction rates in this unstable region. How far the rules of equilibrium thermodynamics can be used is still an open question. It is well known that if we force the system into the unstable region, or even close to this region, it splits up into two phases. Theoretically this is also a consequence of the stability requirements. If we allow for two coexisting phases we have one more free parameter in our problem, the volume fraction

20 Critical Point

T[MeVI ~

0.01

3]

0.1

4 no

n in[fm Fig. 3.1. Phase diagram of the nuclear liquid—gas phase transition calculated the model presented in the text. Here the temperature is plotted versus the baryon density n. Above the critical temperature T, there is no distinction between liquid and gas. There is only nuclear vapor.

L.P. Csernai and .1.1. Kapusta,

Entropy and clusterproduction in nuclear collisions

255

of the phases 1 and 2: A1= V,/V,

i=1,2,

(3.23)

or equivalently the phase abundances a,

=

N1/N.

(3.24)

The sum of both is normalized to 1, a1 + a2 = N1! V~: n

A,

=

— a~

and

A2 =

n—n1

=

1, A1 + A2 = 1 and there is a relation among a1, A1, n and

(3.25)

.

2e/ai ak,

Now that the matrixleads of the derivatives of theequilibrium: energy density M~,, a i = n, the s, a requirement should be positive definite to second Gibb’s criteria of phase P1P2P

(3.26)

T 1= T2= T

(3.27)

/L1—/.L2—/2.

(3.28)

These requirements restrict the region of stability on the n, s plane to a line! This is the Maxwell construction line (fig. 3.1), and it lies in the stable region of the previous stability study. Outside the region confined by this line the matter is stable in one single phase. Within this line but outside the u~-<0 region the matter is stable too, but only if formation of the other phase is restricted. The region between the Maxwell construction line and the boundary of instability ~4< 0 is metastable: the matter can be stable in this region if the other phase is not present. This is the phenomena of superheating and supercooling which are quite common in relatively slow thermodynamic processes, so we can expect these phenomena to occur in relativistic heavy ion collisions for sure. If we solve the Gibb’s criteria for our equation of state (3.1—4) the extensive thermodynamic quantities are given along the Maxwell construction line as functions of one intensive parameter, say T In fig. 3.1 the density is plotted for the two phases in equilibrium n~(T)and n~(T).The critical point at (Ta, n~)separates the Maxwell construction line into a Liquid (L) and a Gas (G) part: n~(T) n~(T).The critical temperature, density and entropy for this equation of state are 3, a-,, = 2.55. T~= 14.9 MeV, n~= 0.063 fm If the temperature of our system is below T,, and its total density is n~(T)< n
n~(T)+AL n~(T).

(3.29)

A similar equation holds for all extensive densities, such as: s, e, w, etc. For specific extensives the same

256

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

type of equation is valid, but with the aG, aL phase abundances. For example =

a~v~(T)+a~v~(T),

(3.30)

and similarly for ~, a-, etc. Since we are interested in the final expansion stages of a heavy ion reaction when the temperature and density decrease it is imperative to study the low temperature or low density limit of the nuclear liquid—gas phase transition. The numerical study of our equation of state shows that s~j1>s~and s~(T) increases, while s~(T)decreases with decreasing T, see fig. 3.2. This result suggests that we might reach the phase mixture region with arbitrarily high energy collisions in the subsequent quasi-adiabatic expansion [CB8O]if the breakup density is sufficiently low. In the literature essentially the opposite was anticipated and the liquid—gas phase transition was expected to have an effect on very low energy collisions only. The low temperature limit of the phase transition can be discussed analytically. In this limit the thermal energy of the gas phase can be represented by the Boltzmann limit, the thermal energy of the liquid phase on the other hand by the degenerate Fermi gas limit. Similarly the compressional energy of the gas phase can be approximated by the first term of eq. (3.2) while for the liquid phase we can use the usual parametrization around the ground state: EL =

~

(-~~— i)~_B +

~- mn213T2+ m,

(3.31)

213 + ~ T + m, (3.32) 2(n/n0) where B = 8 MeV is the binding energy, and b = 1.809. From the Gibbs criteria then we obtain for the specific entropies: =

a

(3.33)

a-~(T) B/T+~,

4 Gas

n [fm3]

no

Fig. 3.2. Specific entropy cr versus the baryon density n. Phase equilibrium is possible even above the critical entropy u,.

L.P. =

Csernai

and J.L

~b2mn7~’(T)213T

Kapusta, Entropy and cluster production in nuclear collisions

~b2mn~2”3T,

257

(3.34)

and for the densities: n~(T)= 4(mT/2ii-)3”2 e~’T,

(3.35)

n~~(T) = — 3 b2rnT~n~/3.

(3.36)

Thus the entropy of the gas phase tends to infinity as BIT if T—* 0. Since at low T, n~~ n~and n~ n 0 the breakup temperature TBU is more characteristic then the nBU of the total system. Fortunately this is a measururable quantity also and lies in the 4—14 MeV region [PC85, PC84, MB84] for a wide range of beam energies. This suggests, by the help of eq. (3.33), that we reach the phase mixture in energetic reactions up to an initial entropy of a-~’= 3—4.5. If we are above the critical entropy a-,, 2.5 then this happens in the gas phase and droplet formation may start. Below the critical entropy the phase mixture boundary is reached in the liquid phase and initially small bubbles of gas phase are formed. 3.4. Nuclear matter at high density Much less is understood about nuclear matter at high temperature and density than at low temperature and density. From the point of view of entropy production it is of great importance because most of the entropy is produced in the first part of the collision where the matter is compressed. See section 3.5. The entropy inferred at the end of the reaction might carry information about the high density part of the equation of state. Up to about 2 GeV/nucleon beam energy there is as yet no unambiguous signature found for a phase transition or any exotic behavior IBo82, SC821 of the equation of state in heavy ion reactions. Thus if we continue smoothly the equation of state from normal nuclear densities upwards the basic parameter to determine is the compressibility K of the nuclear matter. This affects the entropy produced during the compression. If the matter is softer, K is small, and at the same energy density and baryon density the entropy is larger. A decrease of K by about 100 MeV leads to an entropy increase of ~a- 0.1—0.2 around 1 GeV per nucleon beam energy [SG81I.So from the point of view of entropy production in the 1—2 GeV/nucleon beam energy region the value of K is of importance. The first attempt to evaluate K from measurements was made by Stock et al. [SB82], and they have found a K of 240 MeV in the 0.5—1.8 GeV/nucleon beam energy region where the maximum compression in a nucleus—nucleus collision is expected to be n/no 2.5—3.5. The initial evaluation involved a considerable amount of theoretical predictions. The maximum compression and the pion yield without compressional energy contribution were evaluated in a cascade model. Further development is necessary to eliminate theoretical assumptions as far as possible from the evaluation of K. This may be achieved by measuring simultaneously more quantities like entropy, apparent temperature, pion yield, etc. There is progress in this field [see the review by Stock] and so we eagerly anticipate firm information on the equation of state in the n = (1—4)no baryon density region soon. From the point of view of entropy production the expected first-order phase transition into QCD plasma is even more important. Such a phase transition is a strong softening of the equation of state due to the excitation of many more degrees of freedom in the quarks and gluons, and so it gives rise to a large entropy increase. Thus it is widely expected [e.g. 5t84b, BK85, SK79] that the observation of a

258

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

sharp entropy increase could be an unmistakable sign of this phase transition. Since the QCD plasma is discussed elsewhere we do not go into the details of different calculations of the equation of state but illustrate this phase transition with a simple schematic example. Let us characterize the plasma by the bag model equation of state [CJ741for u, d quark and glue plasma:

37 2T4+~/L2T2+ 1 1

Pq=~1T

(3.37)

4~B,

1622,LL

where B is the bag constant. The nuclear matter can be characterized by the energy density: eh

=

Kin \2 ~——1) n + ~nT, 18 no /

nm + —

(3.38)

where we neglected the binding energy and for the thermal energy part we took the Boltzmann limit. This equation of state would lead to acausality at high densities (VSOUOd> c), but we will have a phase transition before we reach this limit. The corresponding pressure and chemical potential are: K in~2 Ph=1j) 9 no/ (n—no)+nT,

(3.39)

(i

(3.40)

=

m+

~

18

-

-f-) (i —3 -~-)+ /LB(fl, T),

no

where laB(n, T) is the chemical potential of the Boltzmann ideal gas:

B114

200

235 MeV, n 3 0~0.16 fm K~240MeV

n [fm3] Fig. 3.3. Phase diagram of the nuclear matter—quark matter phase transition calculated in the simplified model presented in the text.

L.P. Csernai and ii. Kapusta, Entropy and cluster production in nuclear collisions /-‘B(fl,

21T 3/2 T) = TIn [~n (~) ],

259

d = 4.

Solving the Gibb’s criteria (3.26—3.28) determines the thermodynamical variables of our two phases in equilibrium n~(T),n~(T),s~(T),s~(T),and the critical temperature~See fig. 3.3. This portion of the phase diagram is quite different from the nuclear liquid—gas phase transition. Even at n = 0, T~ 160 MeV there is a large discontinuity in the entropy and energy density. Extensive Lattice—QCD studies are performed to trace down the properties of this phase transition. Most of them are still restricted to the ~a= 0 case.

3.5. General features of entropy production Before the nuclei collide they are in their ground state, thus their initial entropy is zero. There is a large amount of kinetic energy available, part of which will be lost during the collision due to irreversible or dissipative processes. Independent of the model we use to describe the collision, such irreversible processes are inevitable. This was realized early in the one fireball [WG76], 2—3 fireball [Da78, CG81] and firestreak [My78,GK78] models where, without discussion of the dissipative effect, it was assumed that all available kinetic energy is thermalized at the breakup density. Given an equation of state the choice of breakup density determined the entropy increase in these models. Measurements showed that collective correlations develop and persist during the collision. Thus the assumption of global thermalization is unsatisfactory. More sophisticated models were developed to account for the details of the collision process. Different models assumed different grades of thermalization. One extreme is the perfect one-fluid dynamics, where immediate local thermalization is assumed. Since this model is reversible it is unable to describe a basically irreversible process. (It can be proven that the entropy increase is exactly zero in perfect relativistic fluid dynamics [LL53, section 126].) Therefore, no continuous solution of perfect fluid dynamics can be found which satisfies the boundary conditions in a supersonic nucleus—nucleus collision. To resolve this problem the perfect fluid dynamics should be supplemented with the relativistic Rankine—Hugoniot relations which describe changes across a discontinuity and account for the entropy production. This model is very simple and appealing because the equation of state can be used in the solution of the fluid dynamical equations and Rankine—Hugoniot equations give us the value of the entropy increase. Numerical perfect fluid dynamical models use the integral form of the equations, and a finite calculational grid. Thus the Rankine—Hugoniot relations are satisfied between each cell, which results in dissipation and entropy increase, or in other words in numerical viscosity. One step further is the viscous fluid dynamics where only approximate local thermalization is assumed. This model yields the shock fronts and the dissipation directly without any further assumption. However, knowledge of transport coefficients is also needed, not only the equation of state. Here dissipation or irreversibility is present because we allow heat and momentum transfer between neighboring fluid elements and allow the system to develop towards global thermalization. If we assume even less, i.e. no local thermalization at all, we would have to solve the relativistic transport equations. This is a very difficult task but some models, like the intranuclear cascade (INC) [Cu80, CJ83, Cu84, BC81, TG83, G179], the two and three fluid dynamical models [AG78, CLS2] and some transport models [Ma81,Ma84, Ma84b] yield reasonable approximate solutions. Since entropy can be defined for nonequilibrated systems the dissipation and entropy increase can be evaluated in these

260

L.P. Csernai and f.i

Kapusta, Entropy and cluster production in nuclear collisions

models. This is in spite of the fact that equilibrium and near equilibrium thermodynamics cannot be applied, nor can the system be represented by an equation of state. In these cases irreversibility or dissipation occurs because of processes driving the system to some kind of equilibrium not present originally in the system. There is one more source of entropy increase which may appear both in the compression and expansion stages of a collision if a phase transition is present. The phase equilibrium may not be instantaneous and a delay in the equilibration leads to dissipation also. 3.6. Entropy production in shock waves If we have central collisions of equal size nuclei the matter coming in with supersonic velocity will be stopped in the center of mass system. Neglecting any transverse flow this happens in a shock front which is an infinitely sharp discontinuity in perfect fluid dynamics. The matter comes to rest in this front. If the incoming matter has a uniform density (n0) and constant speed the shock front separating the stopped compressed matter from the incoming nuclear matter propagates outwards at constant speed, see fig. 3.4. This scenario is widely used to obtain predictions for density, temperature and entropy increase in a very simple way [CJ73, SM74, SS75, SG78, BS75, SG81, St84, St84b, SB83]. For a perfect fluid characterized by an equation of state its energy-momentum tensor is Ti” = (e + p)u~u”— pg’s”

(3.41)

Density profiles in the C.M. ~1 ~

n/no 4

<13

3 VshQck~

I

V—o..

I

I

~Vshock

~ 0

II

~1

~*—v

H ~—~-

t2 v~0

-a-—v

v0

Fig. 3.4. Time development of density profiles in a compression shock front along the beam axis in an idealized central synunetric nucleus—nucleus collision. The surfaces and the shock fronts are idealized as infinitely sharp. The shock fronts propagate outwards in the CM. frame.

L,P. Csernai and f.I. Kapusta, Entropy and cluster production in nuclear collisions

261

where u’~= (y, yv) is the four-velocity, y = (1 — v2)~’2 and g’~ is the metric tensor, g’~= diag(1 — 1 — 1 1). If there is a discontinuity (or sharp change) in the flow across a surface, with a normal vector A”, the energy and momentum should be conserved when the matter flows from side (1) through the shock front to side (2). In the frame of the front the normal component of the energy-momentum flow T”’’~AVshould be continuous: —

[T””~A~IT~A~ — T~A..= 0.

(3.42)

If there is a conserved charge, such as baryon charge, its current N”

=

nu”~should be continuous also,

[N”A~,] N~)J1~, — N~’I)A 5.= 0.

(3.43)

These are the relativistic Rankine—Hugoniot relations [Ta48]. Eliminating the four-velocity from these equations yields the Taub adiabat: [p] (X1 + X2) = [(e + p)X],

(3.44) 2 and the square brackets stand for the

where X of is the argument generalized specific volume[a]X =a( (e + p)1n difference across the front: 2) a(1). Given an equation of state of the high density matter reached in a shock wave eq. (3.44) yields the compression and entropy increase for a given beam energy. For a nuclear equation of state like (3.1—2), (3.31) or (3.38) with a compressibility K the entropy production tends to flatten out around 2 GeV/nucleon beam energy at a- 4 [SG81] (fig. 3.5). However, around 3—4 GeV/nucleon a new increase is expected [St84b]if the dense nuclear matter overgoes a first-order phase transition into QCD plasma described by the equation of state (3.37), plus perturbative terms. This phase transition could increase the entropy production by a factor of 2 in the 5—15 GeV/nucleon energy region rising to values of a-’=7—10! Consideration of phase mixture between the nuclear and quark matter does not smooth out this sharp transition. The transition happens in a narrow beam energy region t~E 1 GeV per nucleon above the threshold of the phase transition [BC85]. This sharp rise in the produced entropy should be an unmistakable signature that we passed the threshold energy of the phase transition [SK79,MM8O, St84b]. Other predicted signatures may not provide such an unmistakable signature just at the threshold where the QCD plasma is produced. —

E lob 1GeV/n) Fig. 3.5. •Entropy per baryon S(E) versus E for a simple mean field model of the nuclear equation of state. SFG is the exact Fermi gas result. The parameters along the curves are the nuclear compressibilities K = 270, 550 MeV. From [SG81I.

L.P.

262

Csemai and fl. Kapusta, Entropy and clusterproduction in nuclear collisions

3.7. Entropy production in viscous fluid dynamics Viscous fluid flow is not adiabatic. Entropy production is proportional to the transport coefficients shear viscosity ~, bulk viscosity ~ and heat conductivity K [LL53].As an example neglect K and consider a one-dimensional relativistic flow. The specific entropy increase per unit proper time is: ñ2 , 6- = (~ + f)—1 (—)

(3.45)

where a aa/ar [CB8O].This does not mean that if the viscosity is larger we produce more entropy in the initial stages of a nucleus—nucleus collision. Due to the special initial and boundary conditions a shock wave arises initially in the viscous flow too. If we start from a perfect fluid flow and “switch-on” the viscosity gradually what happens is that the initially sharp front thickens but the final compressed state remains the same. Thus both eq. (3.45) and the Rankine—Hugoniot relations (3.42—43) will be satisfied. In a stationary front energy, momentum, and baryon charge cannot be accumulated in the front or released from it, so eq. (3.42—43) hold across a stationary front of finite thickness too. Viscous numerical calculations have shown that the shock front is stationary and so the Rankine—Hugoniot relations are satisfied [BC81b, CB8O, etc.1. Deviations appear only when the size of the colliding system and the width of the shock front are comparable. Since eq. (3.45) implies that the width of the shock front is proportional to the viscosity the value of ~ and ~is of great importance. A recent theoretical analysis of Danielewicz [Da84]suggests that ~ is larger than previously thought, so the width of the shock front is also bigger and is on the order of 3 fm. This indicates that the applicability of shock equations is restricted to nuclei of large mass A > 80—100. Large scale numerical calculations in 2 and 3 dimensions [AG78, CS83, Da79, KS81, Ni79, NS81, SB82b, SC82] show that the maximum density and entropy increase is not reached simultaneously or uniformly by the whole colliding system, even if the collision is central. One can maximize the hydrodynamic behavior by choosing the heaviest possible target—projectile combinations to minimize the effect of finite shock front widths. One can note also that if a delayed phase transition is present in the compression shock front this can also contribute to the thickening of the front. The viscosity has an explicit contribution to the entropy increase in the expansion where boundary conditions do not fix the amount of produced entropy. This will be discussed in section 3.10. 3.8. Entropy production in microscopic models Results from microscopic models of heavy ion collisions are oftentimes characterized by a timedependent single-particle distribution, or a Wigner distribution, in the 6-dimensional phase space: W(x,p) where x = x” = (t, x, y, z) and p = p”’ = (p°, pX, p) pZ) We neglect now correlations which would require two and more particle distribution functions to be considered. Assuming classical particles the entropy four-flow in a system characterized by W is [Gl80]: s~=

—j~d3 ~ Po

W(x,p)[ln W(x,p)— 1],

(3.46)

LP.

Csernai

and J.L

Kapusta, Entropy and clusterproduction in nuclearcollisions

263

and the invariant scalar entropy density is s = s”u~,,.This can be measured in the local rest frame of the matter. In [Ra79] the relativistic Boltzmann equation for a spatially uniform Wigner distribution was solved and the speed of thermalization was studied. A rapid entropy increase was found which saturated in 3—5 fm/c, see fig. 3.6. The saturation value of the entropy was very close to the Rankine—Hugoniot prediction with zero compressibility, fig. 3.5. Quantum corrections did not effect the saturation value but merely slowed down the thermalization by 5-10%. Entropy production in the intranuclear cascade model was analyzed by means of eq. (3.46) in [BC81] for 40Ca + 40Ca central collisions. At 800 MeV per nucleon beam energy an asymptotic specific entropy of a- 4.4 was found. This value exceeds the fluid dynamical predictions by more than one unit. This large excess is probably due to the fact that the thermalization is reached at a lower average compression than in the fluid dynamical model. In the intranuclear cascade model thermalization is reached at about 10 fm/c, whereas the maximum compression was achieved earlier at about 8 fm/c. The results of this calculation are compared to two hydrodynamical calculations in fig. 3.7. Assuming thermal equilibration at the maximum density, 4n 0, would yield a- = 3.2. At the moment when the maximum density was reached the average density was, however, only 2n0 and thermalization at this density would have given a- = 3.8. So the reason for the high entropy is that complete thermalization is reached rather late at a low density, while in the one-fluid dynamical models thermalization at the maximum compression occurs. This difference should decrease with increasing size of the colliding system, because there the time to reach full compression is longer and may become equal to or even exceed the thermalization time. In fig. 3.7 the result of a fluid dynamical calculation is also plotted where the entropy increase exceeded the one obtained in the intranuclear cascade. The reason for this is that in the equation of

I

I

I

I

I

I

2.1 GeV 4 1.05GeV

, / / I

>-3-

LiJ

400MeV

/

cr

/ 2

— —

-

250 MeV

—

1-/’

-

I

0

I

I

2

TIME Fig. 3.6. The entropy

I

I

I

4

6

I

8

(fm/c)

S(t) given by the classical approximation in a transport

theoretical calculation. From [Ra79].

264

L.P.

Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

Ca

Central 6

800 MeV/,A

+ Ca

I

Collision I

I

I

I

I

I

hydro — soft 5cascade 4-

-

hydro- stiff

S 3-

-

2-

-

a 0

I

I

I

1

2

3

4

I

I

I

5

6

7

f ( x 1O~~ sec) Fig. 3.7. Entropy per baryon as a function of time after first contact. A thermally soft equation of state s(n, e) produces more entropy than a thermally stiff one in a relativistic hydrodynamic calculation [KS81].The nuclear interpenetration in a cascade calculation IBC81I causes a slower buildup of entropy. From [Ka84b].

state used in this calculation the excitation of many more (unspecified) degrees of freedom was allowed. This softened the equation of state considerably leading to a large increase of the entropy. This example shows the importance of the degrees of freedoms considered and indicates that cascade and hydrodynamical models should be compared only when the same degrees of freedom are allowed in both models. Multi-component fluid dynamical models [AG78, CL82] yield or may yield a thermalization time later than the maximum compression time, thus providing larger entropy increase than the one-fluid models. The entropy increase unfortunately was not evaluated in the above works. Dissipative processes in microscopic models are not separated into a compression part and an expansion part. Nevertheless the major part of the entropy is produced before the maximum compression is reached. Hence there is a similarity between fluid dynamical models and microscopic models in this respect too. 3.9. Entropy production in nonequilibrium phase transitions Dissipation and irreversibility may arise from phase transitions in a dynamical system. If the system has the possibility of forming two different phases there is always an optimum particle number (a1) or volume (A,) ratio (3.23—25) of the phases which corresponds to the energy minimum or entropy

L.P.

Csernai

and .1.1. Kapusta, Entropy and cluster production in nuclear collisions

265

maximum. In a rapidly changing system with finite reaction rates the system may never reach this equilibrium configuration and thus a dissipative nonequilibrium process ensues. Such dissipation due to phase transitions may occur during compression or during expansion. If the beam energy is high enough the system may enter into the QCD plasma phase during the compression, and then in the expansion back into the nuclear matter phase. During the late expansion states the system may encounter the liquid gas phase transition of nuclear matter. Let us discuss the dissipation in a dynamical first-order phase transition in general terms. Assume that the system is thermalized but the Gibb’s criteria are not necessarily fulfilled (3.26—28). The volume average of the energy momentum tensor in a small local volume element containing both phases can be written as T”’~= ~ ArT~

(3.47)

where A, is given by eq. (3.25) and T~ by eq. (3.41). Let us derive the entropy production in such a mixture in a manner similar to [LL53, section 126]. From T””,,.,u~= 0 one can derive the following transparent equation assuming that T1 = T2:

&=_~1;~2ái+~(pi_p2)Ai~

(3.48)

which simplifies further if the pressure difference between the phases is neglected [CL83]. Note that a~ and A~are related to each other via eq. (3.25). Equation (3.48) tells us that if in a dynamic phase transition the particles go over from one phase to another at nonequilibrated chemical potential, p.~ /.L2, or if the volume fraction of one phase changes at unequal pressures an entropy increase is obtained. According to nonequilibrium thermodynamics the generalized currents like a, or Ar are driven by generalized thermodynamical forces which act in the direction of restoring the equilibrium. Thus =

—(a, —

=

— (Ar — A ~)IT0.

and (3.49)

Consequently the dissipation is determined by the ratio of the time scale characterizing the dynamics of the process ‘rdyfl, and the chemical equilibration times T,,h and ri,. Let us return to the discussion of the compression stage of a reaction. The role of a delayed phase transition is similar to that of the bulk viscosity [DG85]. As long as the compression front is relatively narrow and stationary the phase transition delay simply thickens the front proportional to TchITdyfl, but the final entropy increase and compression are not affected. The thickening may become so significant that it is necessary to drop the narrow stationary shock front approximation. So far there is no estimate of the characteristic time Tch of the transition into quark—gluon plasma, thus it cannot be decided whether the shock approximation is applicable in the presence of this phase transition in nucleus— nucleus collisions or not.

266

L.P. Csernai and f.l. Kapusta,

Entropy and cluster production in nuclear collisions

If we take a characteristic time Tch on the order of the QCD time scale 1 fm/c one can make an estimate of the entropy increase [CL83, BK84, BK85]. In a fluid dynamical model, using the equation of state discussed in section 3.4, at 7 GeV/nucleon beam energy about 1.5 units of extra entropy can be obtained at most due to this process. The extra entropy is due to the fact that the equilibrium phase mixture is not reached in the short time available in the dynamic process. 3.10. Entropy production during the expansion stage While perfect adiabatic flow shows a scale invariance the viscosity introduces a length scale characterized by the distance viscous effects can reach in unit time [Ka81,BS84]. Thus the dissipation in a viscous expansion depends on the size of the expanding system and on the magnitude of transport coefficients. Entropy production in viscous expansion was first studied in [CB8O,Ka81]. Typically for A + A = 80 at 800 MeV/nucleon beam energy an entropy increase of z~a-= 0.3 to 0.6 is obtained, which is on the order of 10%. In these calculations the transport coefficients were taken from simple estimates based on Boltzmann statistics. Recent estimates of these coefficients [Da84] are essentially higher, but these will not result in an essential increase of the entropy production because the extra entropy produced in the expansion does not increase linearly with i~ but rather a saturation will occur [Ka81]. The average specific entropy increase becomes less if we increase the size of the system [Ka81, BS84]. In fig. 3.8 the extra entropy produced due to viscous effects is plotted for different size systems 0.40

I

I

I

0.35 —

0.30 -

C+C

u +u~~

-

0.25-

0.20 -

S

3.88 +

~

fl

2n~

0.15-

nf

0.4n

0.10 -

E

S

-

0

beam

—

800 MeV/A

0.05-

0

-

-

-

NS+NS

0

0.1

0.2

,~/R (at normal density)

0.3

0.4

Fig. 3.8. The effect of viscosity or finite mean free path on the value of the entropy generated during the expansion stage of central collisions 113.neutron From [Ka8ll. between carbon nuclei, calcium nuclei, silver nuclei, uranium nuclei and stars. The mean free path divided by the radius of the combined system is evaluated at normal nuclear density and is proportional to A

L.P. Csernai and fl.

Kapusta, Entropy and cluster production in nuclear collisions

267

[Ka81].An infinitely large system (like a neutron star NS) can be considered as perfect fluid since the viscous length scale is negligible compared with its size. There is a very weak dependence on the nuclear size so that this effect is not easily observable. To measure this deviation from the exact scaling [BS84] would enable us to gain experimental information about the magnitude of nuclear viscosity at high density and temperature. Viscous calculations in 2 and 3 dimensions have been performed in nonrelativistic approximation [BC81b, CS83, TW8O]. Geometry and the impact parameter influences the flow pattern. Strong shears occur dividing the participant and spectator domains, which influence the entropy production. Now eq. (3.45) cannot be applied. The dominant part of the dissipation still appears in the shock fronts. Since better estimates of the transport coefficients are now available [Da84,DG85b] than the ones used in the above calculations a repeated evaluation of viscous effects in 3D geometry would be highly appreciated. Relativistic 3-dimensional calculations confirm the general features of dissipation outlined above. They are discussed in detail in the reviews of Stocker and Greiner [SG86] and of Strottman and Clare [SC86]. Finally, it should be mentioned that the deviation from scaling due to viscosity (in other words that in small systems more entropy is produced) nicely coincides with the conclusion drawn from the results of INC calculations where such an effect should also occur. See section 3.8. 3.11. Liquid—gas phase transition As already was discussed in section 3.9 a first-order phase transition may contribute to the entropy increase of the system. The liquid—gas phase transition in nuclear matter below n0 provides such a possibility. Especially in lower beam energy heavy ion reactions (on the order of a few 100 MeV/nucleon) the breakup density and breakup entropy of the system may fall into the phase transition domain (fig. 3.2). In any case entering this domain leads to very interesting consequences for the fragmentation (see section 4), but if the phase transition has a finite relaxation time comparable to the time scale of the dynamics the phase transition leads to an increase of the total entropy of the system. This was observed in two different scenarios by [LS84,Cs85] recently. Lopez and Siemens [LS84]considered supercooling in the single liquid phase in adiabatic expansion. In fig. 3.2 this can be viewed as the system moving on a horizontal, constant entropy line towards smaller densities, and entering the supercooled metastable region in the liquid phase. The adiabatic expansion was assumed to be continued in a single phase even when the system entered the unstable region. That is, in fig. 3.2 we cross the thin full line corresponding to u~-=0. The system forms a phase mixture and equilibrates rapidly when u~becomes zero at the dashed line in this scenario. This rapid equilibration happens at constant density n and energy density e, thus giving rise to a considerable entropy jump of Ao- 0.2—0.6. To penetrate into the metastable or even into the unstable region in a rapid dynamical process Is not impossible because the perturbations restoring equilibrium develop as 2~’”~”~ et21~~I~ e_IL1SIA (3.50) .

e’°’~”’~ =

e

If u is imaginary this gives rise to a growing fluctuation rising with the time scale r = A/u. At the boundary where u~= 0, u2 7~~ —0.006 c so isothermal fluctuations of the size of A 5 fm may develop in 60 fm/c. This is somewhat longer than a typical reaction time. The entropy increase versus the initial entropy obtained in this scenario is plotted in fig. 3.9. The maximum extra entropy, i~o~0.6, is produced when the initial entropy of the expanding system is —

268

L.P. Csernai and J.L Kapusta, Entropy and cluster production in nuclear collisions 3.00 I

2.40 dO

~

~-

°

FINAL ENTROPY

0.60 CHANGE OF ENTROPY 0.00 0.00

I

0.60

1.20

1.80

2.40

3.00

INITIAL ENTROPY Fig. 3.9. Final entropy and extra entropy produced due to expansion of the liquid phase through the metastable region. The extra entropy is 2 <0, and suddenly the two equilibrated phases are formed. From [LS84I. produced when the system reaches the boundary of instability, u,, = 0. This is due to the fact that in the adiabatic expansion through the metastable region the pressure is negative, so we have to invest energy, p d V, to expand the system. This energy thermalizes and gives rise to a large entropy increase when the two phases equilibrate and the pressure becomes positive. In a relativistic heavy ion collision there is kinetic energy available in the expansion which can thermalize according to this scenario slowing down the expansion considerably. In the o~ 0 limit, however, there is no such extra kinetic energy available in a nucleus—nucleus collision because this corresponds to the EIab 0 limit too, so if we put this picture in the framework of a complete nucleus—nucleus collision we obtain u~—~ 0, Ao—~0, in the zero beam energy limit. In the other scenario, instead of supercooling a delay of the phase transition was considered [Cs851. It was assumed that the equilibration time for the pressure and temperature balance between the phases is short, but the chemical equilibration time Tch was assumed to be comparable with the time scale of the dynamics, Tch u? 1~. Only the first two of Gibb’s criteria (3.26—27), Pi = P2, T1 = T2 were satisfied. Thus the abundances of the liquid and gas phases should be evaluated by using the explicit equation from —~

—~

-~

(3.49) =

—

(aG

—

a~)/rCh.

(3.51)

Following the phase transition this way dynamically resulted in a continuous increase of entropy according to eq. (3.48). The entropy increase obtained this way was similar to that of the previous scenario Au 0.4—0.5. This dynamical assumption resulted in a supercooled liquid phase in coexistence with a gas phase. Due to the parametrization Tch u~-the liquid phase, however, never entered the unstable region. -~

3.12. Summary and critique The entropy production in a heavy ion reaction is due to many different mechanisms. These are not independent and in some cases the differences are only due to the different theoretical approaches, while the physical processes are the same (independent of whether we describe them in a cascade or in a fluid dynamical model). Most of the processes discussed are, however, near equilibrium phenomena, i.e.

269

L.P. Csernai and J.I. Kapusta, Entropy and cluster production in nuclear collisions

deviations from equilibrium enter linearly in the expressions describing dissipation. (Exceptions are only the processes discussed in sections 3.6 and 3.8.) The connection among different dissipative mechanisms is discussed in nonequilibrium thennodynamics [Pr67]. The basic formula of the thermodynamics of nonequilibrium processes describes the entropy production as u = ~ XIJE

(3.52)

where .1, represents the rates of various irreversible processes (chemical reactions, heat flow, diffusion etc.) and X, are the corresponding generalized forces (affinities, gradients of temperature and chemical potentials, etc.). For example this is the generalized form of eq. (3.48) too with rates: fi = = daildt, J2 = = dA1Idt. The corresponding generalized forces are X1 = —(~~ ~2)IT and X2 = (p1 —

respectively. In equilibrium all rates and generalized forces should vanish simultaneously ft The rates depend linearly on the generalized forces

—

=

0, X,

=

0.

(3.53) for near-equilibrium phenomena. Thus, they may be much more complicated than usually assumed in this field; see eq. (3.49) or (3.51). Inserting (3.53) into (3.52) we get ó~=~L~X,X1.

(3.54)

Thus from the requirement that &> 0 it follows that L,1 is positive definite, and from the microscopic reversibility follows the Onsager relation [Pr67]: L11=L1~.

(3.55)

These requirements coincide with a minimum of entropy production at constant generalized forces X1, or in other words, when boundary conditions prevent the system from reaching equilibrium the system settles down in the state of least dissipation [Pr67, Ja80]. The conclusion from this theorem is that the many processes listed in sections 3.5—11 are not additive and the interaction between different dissipative processes should lead to a minimization of total entropy production. So far only some basic dissipative processes are studied in relativistic nuclear collisions, and the connections among the dissipative processes were hardly even mentioned. Fortunately due to the theorem above we should not expect an essential increase in the predicted total entropy production by including newer and newer dissipative phenomena. Consequently the present estimates are sufficiently accurate from the point of view of total entropy production. If, however, we want to gain reliable information about the transport properties of the nuclear matter there is still very much to do both theoretically and experimentally. On the other hand there is a lower limit of dissipation. This is obtained in the shock wave limit when the thermalization takes place at the maximum compression and we assume a subsequent adiabatic expansion, adiabatic fragmentation and adiabatic breakup. This lower limit and the previously discussed upper limit may differ only by about 10—25% in nucleus—nucleus collisions according to our present knowledge.

270

L.P. Csernai and .1.1. Kapusta, Entropy and cluster production in nuclear collisions

That a rigorous lower limit of the entropy increase should exist, independent of the phenomenological theory describing the dissipation, was pointed out by Remler [Re841recently. He also pointed out that this depends on the particular subset of collisions under study (like peripheral or central). In this respect the above mentioned lower limit obtained in the shock approximation applies only for the participants in a central or nearly central collision. Peripheral collisions may yield a smaller total entropy increase while the entropy per participant nucleon may be larger. Having discussed the most essential processes leading to entropy production in a heavy ion reaction, we can conclude that the majority of the entropy is produced initially when the matter thermalizes and the initial kinetic energy gives rise to thermal excitations. This can happen when the maximum compression is reached, ~r somewhat later depending on the model. A late thermalization yields more entropy. After the thermalization a further entropy increase is possible due to different dissipative or nonequilibrium processes but these contribute only at most about 10—20% to the entropy increase. All models yield the conclusion that the produced entropy goes to zero if the beam energy tends to zero. This general tendency will lead to interesting considerations when the light fragment abundances will be discussed in the next section. They seem to be incompatible with entropy values tending to zero.

4. Entropy from light fragment abundances 4.1. Effects influencing light fragment production Basic theories of light fragment production were discussed in section 2. Due to developments in experiment and theory, and especially due to the difficulties in trying to understand the high entropy values obtained, recently some new effects were studied which modify the earlier predictions. It was observed that in the most powerful thermodynamic model (section 2.5) the final decay of excited states should be considered. Also finite volume and finite density effects influencing the equation of state too (section 3.3), influence the particle abundances at lower energies where composite fragments are copiously produced in a nucleus—nucleus collision. These latter considerations lead to the observation that the dynamical path through a first-order (nuclear liquid—gas) phase transition results in an essential change in fragment abundances. Some features could be understood based on this study of phase transition dynamics. Other effects like Coulomb interaction or nuclear fission influence somewhat the light fragment production but they are more important in determining the intermediate and heavy fragment abundances. These will be discussed in the next chapter. See also J. Hüfner’s report [Hü85]. 4.2. Unstable particle decays During the expansion stage of a nuclear collision the matter eventually becomes so dilute that collisions and interactions among the particles cease and thermal equilibrium cannot be maintained. The density and temperature at which this happens are called the breakup density nBU and the breakup temperature TBU. The thermal momentum distribution corresponding to TBU and ~BU freezes in and will be observed by the detectors. There is, however, an effect which should not be forgotten: In thermal equilibrium excited nuclear fragments A* are present and some of these may decay after the breakup, so that their decay products reach the detectors. The importance of this process was observed by [BJ77,GK78]. Some fraction of the

L.P. Csernai and J.I. Kapusta, Entropy and cluster production in nuclear collisions

271

excited fragments decay by emitting a proton or a neutron changing the observed deuteron to proton ratio. Thus the entropy of the thermal equilibrium state at the breakup cannot be related exactly to the observed deuteron to proton ratio by eq. (2.49). The most important excited states are listed in table 4.1. When calculating the fragment abundances these fragments probably ought to be considered in thermal equilibrium. The momentum distribution eq. (2.16) of these fragments yields the partial density of a certain cluster j =

(~~)

(2s~,+1)m~T ~ (±1)”e~~~~1/T K2

(4.1)

where m is the fragment mass, s1 is its spin and the ±stands for Fermi or Bose statistics. Later this approach was extended to a much larger set of isotopes 1FR82] where all excited states with a level width less than 1 MeV, I ~ 1 MeV, up to A = 16 were taken into account using classical Table 4.1 Grouping of nuclear resonances into effective resonances used in a statistical calculation [GK78] ~ (2S~+1)E~ ~1(2S1+1)

Nucleus

4He

4Li

5He

5Li

Excitation energy above g.s.

j~’

2.2

1~

1.7 0(—5.1)~ 4.1 20.1 21.1 22.1 25.5 26.4 27.4 29.5 30.5 31.0 33.0 0(_3.0)a 1.4 3.2 5.0 0(0.9)a 4.0 16.8 19.9 0(1.9)a 4.0 16.7 18.0 20.0

21 1 0~ 0 20~ 2101 1 2~ 2 1 0 1

~ ~ 1

Decay modes p+n 3H+n 3H + n 3H+n p p, n p,n •..

p, n p, n, y p, n p, n, y p,n,d p,n,d tHe+p 3He+p 3He+ p 3He+p n, a fl,a y,n,d,t,a n, d, t, a p, a p,a y, p, d, 3He, a y, p, d, 3He, a y, p, d, 3He, a

Effective decay mode

effective ex. energy

p+n

2.2

3~+ n

1.6

~(~He±n)

~,(2S1+1) effective (2S+ 1) Boson Boson 3 11

Boson

+ + p)

27.5 28

Boson 12

3He+p

1.9

4He+n

11.1

Fermion 14

411e+p

11.9

Fermion 16

The numbers in parentheses refer to the binding energy with respect to the constituent nucleons.

272

L.P. Csemai and fl. Kapusra, Entropy and cluster production in nuclear collisions

statistics. All these fragments could be considered as stable within a nucleus—nucleus collision since their half-life, t112 150 fm/c, is longer than the characteristic time of the collision. Where experimental data were not available an empirical level density formula was utilized. After the breakup excited states undergo a nucleon or alpha evaporation in a de-excitation stage. In the model this sequential evaporation was simulated according to the branching ratios, and led to the final observable spectra. A fit of the experimental data could be achieved with a breakup at 0.3n0 (x = 3) only when the energy in the collective flow was assumed to be zero. If a higher breakup density of 0.7n0 (x = 1) was assumed the model overestimated the measured deuteron to proton ratios, especially for symmetric colliding systems. See fig. 4.1. If the collective flow is taken into account the sensitivity to the breakup density will be less because the final expansion is close to adiabatic. Consideration of this effect made it possible to fit the experimentally observed deuteron to proton ratios [NL81] by assuming entropy production in a compression shock plus 20% increase due to the final viscous expansion [St84].Thus at this stage the agreement between experiment and theory seemed satisfactory and the puzzle of large entropies (arising from direct application of eq. (2.49) to the observed deuteron to proton ratios) seemed to be solved. See fig. 4.2. Recently new experimental data [JW83, JS84, JF84} at lower beam energies have become available. Although the final particle decay was taken care of an analysis of the entropy showed that the entropy inferred from fragment mass distributions depends very much on the mass range of the clusters considered [JS84] and drops from about 4.5 to about 2 when the mass range under consideration is increased. See fig. 4.3. At the same time by generalizing the interpretation of eq. (2.49), which was valid only at high entropies above 4—5, by relating it to the ratio of deuteron-like clusters to proton-like clusters in all observed light fragments (thus being less sensitive to whether the fragments decayed later or not) experimental data were re-analyzed [Ka84]. Similar to the other study large entropy values 4 were obtained at low energies. Thus, more detailed data and further theoretical study indicates that the final decay of excited particles after the breakup may not be sufficient to resolve the disagreement between the theoretical predictions of the produced entropy and the observed fragment abundances. Thus other effects should be considered too. The importance of final particle decays is, however, not diminished. It cannot be disregarded in any

c 2

0

50

00

+ Symmetric systems

50

200

Excitation energy per nucleon E (MeV) Fig. 4.1. Deuteron to proton yield ratio (dip) as a function of the available excitation energy per participant nucleon obtained for several values of the model parameters. The lower boundary of the different bands represents symmetric systems, while the upper boundary corresponds to I = (N — Z)/A = 0.2. x is the volume parameter. From [FR82].

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions I

3.0

I

I

1111111

273

I

(a)

111111

• C+Ag oC+Au Ne~Au

X

I

1111111

. Ne+LJ ,~p+Ag V p~U

HEAVY FRAGMENTS

6

1.2 ----

I

-

I

I I liii

I

I

II

I

1111111

I

I

I

I

I II’

~1~3

I

I

0

I

2

I

ELAB

(GeV/n)

Fig. 4.2. The bombarding energy dependence of the entropy is shown as calculated for the viscous (Sn) and inviscid (S.) fluids. Also shown are the “entropy” values, “S”, obtained from the measured FNL81I: 0, Ne + NaF, inclusive; •, Ne + NaF, 90°CM; ci, Ar + KCI, inclusive; •, Ar + KCI, 90°CM, and calculated deuteron to proton ratios RdP. The full (broken) curves represent the viscous (inviscid) calculation with nBu = 0.7n 0. Resonance formation has not been taken into account in the determination of S,. From [St84].

tO

I

1111111

I

tOO 1000 Elab(MeV/NUCleOfl)

11111’

5000

Fig. 4.3. Extracted entropy per baryon (S/A) using a quantum statistical model as described in the text. (a) Target rapidity fragments from the reactions 480 MeV p + Ag, 2.1 GeV/nucleon Ne + Au, 30 MeV/nucleon C + Au, 55—110 MeV/nucleon C + Ag, 250 and 2100 MeV/nucleon Ne + Au, 400 and 2100 MeV/nucleon Ne + U, and 4.9 GeV p+ Ag and U; (b) midrapidity fragments from the reactions 35MeV/nucleon C + Au, 42, 92 and 137 MeV/nucleon Ar + Au, 100 and 156 MeV/nucleon Ne + Au, 241 and 393 MeV/nucleon Ne + U, and 2.1GeV/nucleon Ne+Pb. The dashed and solid lines represent the weighted average for fragments with 1 A 3 and I A 4, respectively. The dot-dash line represents a power law fit for fragments with 1 A 14. The three grouped solid lines represent the entropy calculated using fireball geometry and a Fermi gas model at three different densities, n = iOn0, 2.0n0 and 3.0n0. From [JS84].

analysis and especially not if we want to extract the entropy of the system. The final particle decay plays a very important role in other studies too. For example, it contributes to the observability of collective flow phenomena [GG84}.As was pointed out in [FR83, CF84] the final particle decay helps to diminish random thermal and finite multiplicity fluctuations which otherwise could mask the collective flow. 4.3. Finite density effects At the same time when the importance of the final particle decays was recognized another effect acting in the same direction was observed. At breakup, even if the density is small, flBU 0.1—0.3n0, the number of deuterons should be suppressed. The deuteron is a weakly bound fragment having a large

274

L.P. Csemai and J.I. Kapusta, Entropy and cluster production in nuclear collisions

volume of about 27 fm3. Thus it cannot be considered as a point-like particle in an ideal gas. First the total volume available for the particles to move in was reduced by subtracting the volumina of each particle, and then chemical equilibrium was calculated at a corresponding higher effective density in a quantum ideal gas model [SC81]. This simplified model led to a reduction of the number of deuterons due to the higher effective density, but other light fragments were also effected similarly. So the relative abundances did not really reflect the differences in cluster volumina. The same excluded volume approximation was used in [FR821,where the volume available for the particles to move in was parametrized by x as 4ir V=~-~--A 0r~.

(4.2)

Each nucleon was assumed to occupy a volume of V0 = 47rr~/3= n~= r’o, and a fragment of mass A occupied a volume of A~0.The parameter x was related to the physical breakup density of baryons then by

x

=

(no/ne)(1 — n/no)~°’~,

(4.3)

where e = 2.7182. In the dilute limit this yields x = n0/n. At finite densities eq. (4.3) takes into account the effect that for the first particle the whole volume V is available, but for the next one it is reduced by po, and so on. Consideration of finite cluster volume is in other words the use of a Van der Waals gas equation of state. This was noted in [BB83]. It is straightforward to extend the derivation leading to the Van der Waals’ equation of state [LL54, Chapter VII] to a mixture of Van der Waals gases. The free energy density of this mixture is g = g~—~T

~ J n1n1

T—1),

(4.4)

d1’~(e~”

where g~is the free energy density of the ideal gas mixture, n, = N~/V is the partial density and U ~, is the interaction potential between clusters of type i and j. Splitting up the integral into a hard core piece of volume v 0 and to an attractive term a1/T (see [LL54,section 731) in the dilute gas limit, n~v~1 ~ 1 leads to g g~ T ~ n, ln(1 ~ n1v~/2)—~ n~n1a0.The equation of state of this Van der Waals gas mixture is then ~,

—

p

=

n~an1

—

—

g=

n1T n1v,,

_________— — ~

1

1 2

—

q

nn1a11.

(4.5)

Inclusion of the finite volumina modifies the entropy density of this mixture compared to the ideal gas mixture entropy s~: s=s0—~naln(1—~Enivi1)~

(4.6)

leading to an increase of the entropy at the same n and T. Correspondingly the fragment abundances are

L.P. Csernai and II. Kapusta, Entropy and cluster production in nuclear collisions

275

changing too. The deuteron to proton ratio is smaller than in an ideal gas. In the dilute gas limit, from

=

ag/an,

~u1,1 + T

~

n1v~

—

n,a~1,

the ratio between two fragments satisfying 1u, (n,/n71M~1)idealas n~ A/A1 fl1

ffl,\ =

I,, A/A1) n1

ideal

ç I exp1— ~ Thc ~Vik k

=

(A1/A1),a1 is deviating from that of an ideal gas

A. —

-fl

a~— -~— (t~j~ — a)k)]}.

(4.7)

Neglecting the interaction term and assuming that the k dependence of the volume difference AV0 = (vi,, A1 VJkIAJ)/Ak is negligible we end up with a modifying factor of —

exp(—n Av0)

(4.8)

where n = ~k nkAk is the net baryon density. This expression is somewhat different from that of the preprint of [BB83]. Equation (4.8) was used in [C583]in evaluating the fragment abundances, but the correction to the entropy eq. (4.6) was not taken into account. In the recent analysis [JS84], already mentioned, both the unstable particle decays and the finite volume effects were considered in the simpler way mentioned at the beginning of this section. The result that the light fragment abundances show a much higher entropy than the theoretical estimates or the entropy of all light and intermediate fragments indicates that even both effects mentioned so far are not sufficient to resolve the entropy puzzle completely. The correction to the entropy expression, eq. (4.6), would even enhance the difference further by increasing the empirical entropy values.

4.4. Finite source size effects The composite fragments having a finite volume comparable to the size of the source may be suppressed compared to fragments of negligible size. This is an effect that cannot be studied in the infinite volume, thermodynamic In heavy ionfragments collisions can where sizeorder of a fireball hot spot is 3 [GG84b] andlimit. the sizes of light be the on the of a fewortimes 10 fm3 about 200-1000 fm a change in fragment abundances of a few percent. this effect can cause This effect is studied in the density matrix formalism in [SY81]and the sudden approximation model (sections 2.4 and 2.6) was used in this framework. The effective coalescence volume was found to be dependent on the wave function of the fragments, e.g. !Pd(r), and on the proton and neutron density distributions, D~(r)and D~(r),each normalized to one. For example, the coalescence volume for deuterons is ~irp~=

(2ir)3

J

d3r d3r’ !1’~(r)P2D~(r r’) D~(r). —

Assuming Gaussian forms for I V1’dI, D~and D~,with mean square radii r~,r~= r~= r~,respectively, one finds

276

L.P. Csernai and f.l. Kapusta, Entropy and cluster production in nuclear collisions

3/ V(r~,Td), ~.rrp~o = (2rr) where V(r~,rd) = [~ ~(2r~+ r~)]312.

(4.9)

In the ideal gas limit Td ~ r 0 and V(r~,rd) tends to the source volume V Using eq. (2.8) this leads to the volume correction nd/nN

=

(nd/n~)~d00l V/V(r~,rd).

(4.10)

where (fld/fl~i)icteal is the ideal gas limit of the fragment ratio. Since V(r~,rd)> V this correction (4.10) leads to the suppression of larger volume composite fragments. For larger mass composite fragments j an expression similar to (4.10) applies but the nucleon density appears to the A1th power and the volume correction factor f = V/V(r0, 1,) to the (A1 1)th power [SY81].As far as the interactions among the different fragments are neglected one can use these expressions to estimate the source size. This source size may be different for different fragments because not all fragments may freeze out simultaneously. Applying this model to experimental data [SY81] gave a source radius somewhat (2—15%) smaller than without any finite source size correction. Thus, although the effect is certainly not negligible, the much larger deviations in the deuteron-to-proton ratio and in the inferred entropy cannot be explained by it. An interesting result of this analysis is that the breakup density of deuterons is smaller than of alphas or, in other words, the apparent source size decreases with increasing fragment mass. This may be due to the larger thermal velocities of the lighter fragments, so that they can maintain thermal contact in the expansion longer than the slower heavy fragments. It is of historical interest to recall that suppression of the deuteron abundance due to a finite source size was considered more than twenty years earlier by [Is59]and [Ha60]in the context of proton—proton collisions. The suppression factor was obtained by integrating the square of the deuteron wave function over a sphere of radius R —

J

3r ~d(r)I ~(R

-

d

For a proton—proton collision a value of R 1 fm was considered reasonable. A substantial suppression thus occurs, in rough accord with experiment. The source size in a heavy ion reaction depends on the impact parameter. Since small impact parameter collisions yield a higher charged particle multiplicity, the effect of the source size can be studied by analyzing multiplicity selected events. In [GL83] the multiplicity dependence of deuteron to total charge ratio as well as the deuteron-like to total charge (Z~)ratio was investigated. At high multiplicities M~(in 400 MeV per nucleon Ca + Ca reaction at M~= 30, in 800 MeV per nucleon Ne + Pb reaction at M~= 50) these latter ratios saturated. See fig. 4.4. This result implies that we should use multiplicity selected data and not inclusive data to obtain information on nuclear matter at the highest density and temperature. The multiplicity dependence of these ratios was used to estimate the source size via eq. (4.10) in [GL83]. The radius of the emitting source was assumed to be proportional to the charge to the one-third

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

277

Ca÷CaE/A=400MeV CENTRAL TRIGGER 0~R=d/p V 0.2

RdUke/Z

-

a)

2040

b)

0

2040

Multiplicity of charged particles Fig. 4.4. Ratio of deuteron to proton production (a) and ratio of deuteron-like to electric charge (b) as a function of the total observed charged particle multiplicity for the reaction 400 MeV/nucleon Ca + Ca. (c) and (d) correspond to event by event contour plots of the logarithm of the ratio versus charged particle multiplicity. Relative intensities are indicated by the contour lines. From [GL83I.

power as 113 ro(2Z~) A fit to the 400 MeV per nucleon Ca + Ca measurements yielded r 2.9. In this evaluation, instead 0 = rd!the d-like to p-like ratios (see of evaluating the multiplicity dependence of all fragments separately, section 4.6) were used with eq. (4.10). Since heavier fragments contain the finite source size correcting factor to a higher power this approximation is less accurate and leads to an overestimation of the breakup density: =

fld-Iike/fl~= (nd/n~)~d 001f + (fl~

~(flt/fl~)ideaI

e/fl~i)itieaif =

nNf

+

.

(nd!n~)idea1f+ ~(nt/n~)jdeal flNf +

Recently the multiplicity dependence of fragment abundances was measured for heavy colliding systems, Nb + Nb at 400 and 650 MeV per nucleon [DG85]. As a function of proton multiplicity the deuteron per proton and triton per proton ratios saturate while the He per proton ratios are getting close to saturation. See fig. 4.5. This is a very reassuring result, because the data indicate that we are near to some kind of thermodynamical, infinite volume limit. 4.5. Mott transition The Van der Waals gas approach is only an approximation to take into account the interactions between the nucleons and clusters. An essential feature of this mixture is that at high enough densities the composite particles cannot exist. The highly excited states merge into the nucleon continuum.

278

L.P. Csernai and f.I. Kapusta, Entropy and cluster production in nuclear collisions

I

10

~

AAIA N

5

At

x

0.1

A

A

••• •

•

A

•‘

Qooo~~t

00~

00

•

00 0 •U

0 10-’~

~

0 0

~He

~rii

1i

p\~3He r

0

[Ti

•

~

P

o

S

4He

IS

~p

-~

0

10-2 --

¶

93Nb + 650 Me V/nucleon

53Nb + 93Nb 400 MeV/nucleon

i0—~0

20

40

60

80

20

40

I 60

I 80

100

Fig. 4.5. Ratios of the produced composite particles x (x = d, 1, 3He, 4He) to protons as a function of the proton multiplicity (No) for the system Nb + Nb at 400 and 650 MeV per nucleon. From 1DG85].

The problem of two nucleons imbedded in the surrounding nuclear matter is considered in [RM821 with the help of the Bethe—Goldstone equation for the thermodynamic Green functions. The binding energy of a deuteron of momentum k is reduced due to the Pauli principle and the interaction with the surrounding matter of density ~N by: AEd(k, T)

=

2~1~

+ ~j Pauti(k

T) flN,

(4.11)

where in the first term zl ~ is the energy shift of a free nucleon state at the continuum edge (k = 0) and the second term is responsible for the energy shift of a bound two-nucleon cluster. The energy shift was

L.P. Csemai and fl. Kapusta, Entropy and cluster production in nuclear collisions

279

evaluated by using a zero-range Skyrme type nucleon—nucleon interaction in Hartree—Fock approximation. It was found that ~jPauli(~

T) = C (i

+ AT)

exp

[_~°~/(i + AT)]

(4.12)

where C is a constant characterizing the interaction (C = 2332 MeV fm3), AT is the thermal wave length, A~ (2ir/mT)112, and a characterizes the size of the deuteron, a = 7.7 fm. Thus the energy shift of the bound deuteron is smaller than the energy shift of the continuum and there is a density where the bound state merges into the continuum. This Mott density ~M0tt is T and k dependent: .

nN(k

T) = ~E~7~i”~(k,

T),

(4.13)

where Efj~eis the binding energy of the free deuteron (without the surrounding nuclear matter), E~ = —2.21 MeV. From eq. (4.12) it is evident that the Mott density exists even for a point-like deuteron, a 0. This is due to Pauli blocking. Consideration of the interaction changes the deuteron to proton ratio compared to the ideal gas expression (nd/n~)idealas: —~

nd!nN

(nd/n~)idealexp(~zil~a1ThnN/T).

(4.14)

These works were extended to other composite clusters and to different approximation schemes and nucleon—nucleon interactions [RM82b, RS83, RS84, SM82, SR83, SV83]. These considerations lead to an equation of state exhibiting a first-order phase transition (see section 3.3) and predicted fragment abundances for light nuclear clusters. In the extreme low density limit this approach tends to the law of mass action (4.14) but with increasing density first the deuteron, then the triton and alpha, abundances start to decrease at the Mott densities (with k = 0) n~~0ss,n~0St, n~°tt respectively. See fig. 4.6 [RS83]. It was found that the deuteron to proton ratio Rd 0 is always smaller than 0.5 due to the interactions. So the naive use of eq. (2.49) would result in an entropy bigger than 4.2, irrespective of breakup temperature or density. This fact indicates that in the presence of interactions eq. (2.49) cannot be applied at low temperatures. The experimental deuteron to proton, triton to proton and alpha to proton ratios were fitted satisfactorily [SM82]with the assumption that the breakup density is flBU = 0.2—0.3n0. Note that this coincides with recent experimental results of ref. [GG84b]. From this agreement is was concluded that there is a strong indication of a first order nuclear liquid—gas phase transition in nucleus—nucleus collisions. The entropy that may be inferred from this approach was discussed in [KM82]. It was emphasized that the formula eq. (2.49) is inapplicable when many other fragments heavier than deuterons are present. It was suggested that at the breakup deuteron-like correlations should be considered rather than point-like particles. A simple connection does not exist between the entropy and Rd0 but an entropy of o = 4.5 could be attributed to the experimental fragment abundances assuming a breakup density as large as nsu n0, which may be untenably large. The method introduced in this series of papers was very appealing but required quite extensive calculations. Thus the consideration of a large number of different mass clusters, let alone their excited states, was not possible. Consequently a large scale analysis of the connection of entropy and fragment abundances was not performed.

280

L.P. Csemai and fl. Kapusta, Entropy and cluster production in nuclear collisions

3Log

[gfm~

-

.30

Fig. 4.6. Phase diagram of hot nuclear matter. The border lines for the coexisting two-phase equilibrium are calculated in: (i) HF approximation without considering clusters explicitly; (ii) ladder HF approximation under consideration of the deuterons only; (iii) ladder HF approximation under consideration of all light clusters up to A = 4 (heavy line). The numbers on the heavy line give the Rd,, value along the phase border. From [RS83].

The time dependence of the deuteron to proton ratio RdP was also studied by means of a rate equation in a fluid dynamical model [B582]. Similar to [Ka80],section 2.8, a saturation of this ratio is obtained in the classical ideal gas limit. The consideration of quantum statistics and interactions decreased the equilibrium value of Rd~.The solution of the rate equation yielded an Rd 0 value close to the equilibrium one at the breakup. The importance of quantum effects and interactions were discussed also in [JD82] by means of the virial expansion [LL54]. The limiting cases of the classical thermal model as well as of the coalescence model were studied. It was pointed out that with increasing temperature the deuteron may still exist at higher densities than at T = 0. This statement essentially coincides with the behavior of the Mott transition density, that it increases with increasing temperature, found in [RM82]. It also led to a considerable suppression (-=35%) of the deuteron abundances compared to the ideal gas limit. This approach can be considered also as the quantum generalization of the excluded volume effect which led to a Van der Waals equation of state [BB83]. See section 3.3. 4.6. Extracting entropy from deuteron and deuteron-like correlations We have seen in previous sections that there are complications to inferring the entropy from the yield of deuterons when the entropy is not high, u> 5, since the number of heavier fragments is not negligible. A significant development to overcome many of these complications was made by [BC81]. The extended validity of the formula relating the deuteron yield to the entropy was first used in the context of an intranuclear cascade model. However it is of more general utility. The entropy of a system of noninteracting neutrons is 3rd3 Sn_2J rd ~ [W~ln W~—(1—W~)1n(1—Wa)], (4.15) (21T)

where W~= W~(r,p) is the single-particle Wigner function. The factor of 2 is a consequence of the spin

L.P. Csernai and .1.1. Kapusta, Entropy and cluster production in nuclear collisions

281

degrees of freedom of a neutron. If the density in phase space is low so that W~°~1 then eq. (4.15) becomes SnN_2Jdf[WnlnWn+~W~lN[1_(lnWnY~(Wn)1, (21T)

(4.16)

where N is the total number of neutrons. There is a similar expression for the protons. On the other hand the number of deuteron-like correlations is defined by [Be83] Jd3rld3Pi...~:rAdsPAWNZ(rl,pl;”

Ndlike”~~

i=ij=Z+1

(

rA,pA)

Wd(rI— r 1,p1—p1).

(4.17)

ir)

This is the sudden approximation as discussed in sections 2.4. and 2.6. Now it is written in terms of the full A-particle Wigner distribution, where the first Z coordinates refer to protons and the last N coordinates refer to neutrons, with A = Z+ N. That eq. (4.17) is a measure of the number of deuteron-like pairings even if they occur within a larger cluster, like an alpha particle, is clearly a conceptual extension of the original idea. Note that the remaining A 1 particles in eq. (4.17) may be integrated over immediately to give —

Ndlike =

3rd3p W~

—

3 fd 4.’ (2ir)

0(r,p) Wd(r, p),

(4.18)

where r and p are the relative coordinates between the proton and the neutron. Assuming as usual that W~0is essentially uniform in the region of phase space where Wd has its support (but see section 4.4) leads to ~W~0(0,0).

(4.19)

If one further makes the approximation that W~0is the product of single-particle distributions then W~0(0,0) =

~1

W0(r, ~p) W~(r,~p) = 16Z( We).

(4.20)

For a system near thermal equilibrium —(In W~)=—In(W0)+~(1—ln2).

(4.21)

Therefore, combining eqs. (4.16) and (4.19)—(4.21) leads to =

S/A

=

3.945

—

Ifl(Ndlike/Z)

—

~NdI~kC!Z.

(4.22)

This formula is similar in appearance to eq. (2.49). However, it involves the ratio of deuteron-like correlations to total charge instead of the ratio of deuterons to protons. On the basis of the shell model [BC81, Be83] it can be easily shown that

282

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions Nd1~k~= Nd +

~(N, + N~H~) + 3Na

+

whereas Z= Np+Nd+Nt+2(N3He+N~,)+”•. When the entropy is high then N~~° Nd ~‘ N~~‘ Na, etc., so that

Nd/N0. Thus the original formula is reproduced. But, when the entropy is not high, eq. (4.22) is expected to be more accurate. If Z N then eq. (4.22) has an additional term (Z!A) ln(N!Z). Even for such a neutron rich nucleus as uranium this is a very small effect. One would also like to have a formula for the number of real deuterons Nd. It is straightforward to calculate a lower bound, which in fact is also probably a very good estimate of Nd itself [DJ82]. The probability that a given neutron and a given proton are close enough together in phase space to form a deuteron is WI~(O,0)/NZ. The total number of pairs is NZ, and if we take into account that the pair must be in a relative 1 = 1 state leads back to (4.19). A real deuteron is a very weakly bound object. To emerge as a free deuteron not bound in a larger cluster the neutron—proton pair under consideration should not be close in phase space to any other nucleons. The probability of this is Ndtike!Z

.

2

W~0(0,0) NZ L ~

Wnp(0,0)1A_

NZi

In the limit A = N + Z ~ 1, and taking into account spin, one obtains Nd = ~W~ 0(0,0) exp

[—

~

W~~(0, 0)].

(4.23)

For convenience of notation define the correlation parameter x = ~W~~(0, 0)/Z.

(4.24)

NdItk~IZ= x,

(4.25)

Nd/Z = x exp(—~(A/N)x).

(4.26)

Then

Therefore two different two-particle correlation functions, which are measurable, are predicted on the basis of one independent parameter x. This is an important self-consistency check. In terms of x the entropy is o~3.945—1nx—~x.

(4.27)

Thus in eqs. (4.25)—(4.27) we have found a simple relationship among Nd!Z, Ndlike!Z and S/A. The fragment ratios are plotted as a function of S/A in fig. 4.7. Note that the number of deuteron-like correlations decreases monotonically with increasing entropy as expected. The number of free deu-

L.P. Csernai and 1.1. Kapusta, Entropy and cluster production in nuclear collisions 10

I

I

I

I

I

I

6

7

283

Nd~Iike/Z

Nd/Z

-

0.01~

~

3

4

5

8

S/A Fig. 4.7. Ratio of deuterons to electric charge and ratio of deuteron-like correlations to electric charge as functions of the entropy per baryon. From eqs. (4.23)—(4.25).

terons also decreases at large entropy, but it also decreases at low entropy because the probability to find a free deuteron unaccompanied by other particles is small. This is the same general behavior as found in the Van der Waals excluded volume model and in the Mott effect. The use of an independent particle model for the entropy in no way implies the absence of interactions. In fact it is consistent with most discussions of the nuclear liquid—gas phase transition [JM84, Ka84]. A limitation to this approach does appear for very low entropy. From eqs. (4.19), (4.20) and (4.25) one sees that the maximum possible value of x is 12. For nucleons in a harmonic oscillator shell model, the limit x = 12 corresponds to filling up all of the lowest available energy levels. This means zero excitation energy and zero entropy. Even though the semi-classical expression (4.25) for the entropy is not valid for low entropy, or high x, it is amusing that the entropy using this expression vanishes at x = 11.8. Fortunately, as we shall see, it seems that experimentally x <2 so that this limitation is of little consequence for relativistic heavy ion collisions. 4.7. Phase mixture It seems that the previously mentioned effects could not explain why the light fragments, 1 c A <4, exhibited a rather large entropy u 4 even at the low beam energies around and below 100 MeV/nucleon [KM82, JS84, Ka84]. So the entropy puzzle [SK79] seemed to persist although the value of u was smaller than that first mentioned of a- = 5—6. The expectation that the entropy should tend to zero with decreasing beam energy was not fulfilled. Several people suspected that the dynamical path through the liquid—gas phase transition may be responsible for the unexpected behavior of the light fragment distributions [SV83,Si83, SB83, Ka84]. It was shown that such a thermodynamical treatment of the phase transition is not too bad even for intermediate mass systems of A~= 100 [GK84], although the thermodynamical limit is strictly valid only if A~ This is discussed more fully in section 5.4. —~ ~.

284

L.P. Csemai and f.l. Kapu.sta, Entropy and cluster production in nuclear collisions

If a phase transition is sensible for small systems, the next question is whether or not there is a possibility to create a phase mixture in a heavy ion collision. This question is twofold, because it depends on the time scale or speed of the dynamics of the process, as well as on the breakup density of the system. The time scale of the development of larger size density fluctuations of 2—3 fm is approximately 20—30 fm/c as estimated in section 3.11. This is comparable with the typical expansion time of an average heavy ion collision. Thus if we have a large colliding system such as central Nb + Nb or U + U collisions, at the relatively low energies around 100 MeV per nucleon the expansion time is more than sufficient to form coexisting phases. In smaller systems the time might not be sufficient to develop a phase mixture, so supercooling may occur, or the system might even go through the unstable region being in one single phase all the time. In these smaller systems, however, the applicability of thermodynamical and fluid dynamical terms is doubtful anyway. The other question is breakup. As we can see in fig. 3.2 and section 3.3, a phase mixture can exist only below a certain density n~or n~’which decrease with increasing entropy. At low total entropy (i.e. at low beam energies) the phase mixture may exist at densities slightly below n0 while at higher entropies a-~0~4 the density should be below no/40. See fig. 3.2. This is already very small compared to the lowest estimates of the breakup density nBU = 0.1—O.3n0 [GG84bI. Consequently the system freezes out in high energy collisions where the average entropy o~ 4, before phase mixture might be formed. In the few 100 MeV per nucleon energy region and below the formation of a phase mixture may preceed the breakup. This is just the energy region of the entropy puzzle. According to the arguments above it is reasonable to assume that a phase mixture develops in the final expansion stage of a reaction when the beam energy is not too high. This phase mixture may be in equilibrium or it may not. Nevertheless the basic features of the phase mixture, such as a-L ~ o-~and n0 ~ nL n0, indicate that the entropy puzzle is related to the liquid—gas phase transition. In the next two sections we will discuss the features of the liquid—gas mixture that may be reached in equilibrium and in nonequilibrium phase transition dynamics. This will be done in the thermodynamical limit where surface effects will be neglected. The surface effects will be discussed later in section 5. 4.8. Dynamics of equilibrium phase transition In the late expansion stage of an adiabatic or near-adiabatic expansion the system may reach the boundary of the phase equilibrium n~or n~.Whether the system is in the liquid or gas phase depends on the entropy. If this is above the critical entropy, a-a. 2.5, the system is in the gas phase and the mixture develops by forming small liquid droplets. Below the critical entropy the system is initially in the liquid phase and the mixture formation starts with small bubbles. See fig. 3.2. Such a scenario was recently considered in [SK84, Cs85]. In an earlier work [Da79] the dynamics of the droplet formation was considered in an approximate way, while in [LS84, Cs85] two different nonequilibrium phase transition schemes were discussed. In order to study the dynamics of the phase transition the equation of state of the phase mixture should be coupled to the equations of fluid dynamics determining the time evolution of the expansion. The fluid dynamical equations provide us with the local density n(t) and entropy density s(t) if the equation of state p = p(n, s) or p = p(v, a-) is given. In the equilibrium phase mixture, where the Gibb’s criteria (3.26—28) are satisfied the equilibrium values of the extensives n, s, e or r’, a-, e depend on only one intensive which may be the temperature T but could also be the pressure p or the chemical potential Then the equation of state ~.

L.P. Csernai and J.I. Kapusta, Entropy and cluster production in nuclear collisions

285

p = p(v, a-) necessary to solve the fluid dynamical equations can be obtained as the solution of the following system of equations, ii = aG v~(p)+ aL v7.~(p), a-

=

(4.28) (4.29)

aG a-~(p)+ aL a-~(p),

a0+ a5~=1.

(4.30)

Note that v~(p),~7~(p),o’~(p),a-~(p)are known. Thus not only p(t) but also the unknowns aG(t) and aL(t) can be determined. a0 and a~.can be eliminated analytically from eqs. (4.28—30) and then the equation of state p(v, a-) is given as the solution of the following equation: [a-—o~’(p)][v~(p)—p7~(p)]= [~—~Np)][a-~(p)— o-~’(p)].

(4.31)

Having obtained p the phase abundances are given by a0—

p

—

eqj \ p1~p~

e

e

eq~

—

a=

e

e

vc~(p) ~~~(p) a-~(p) o~(p) —

—

These equations were solved in [SK84]and [Cs85].In [SK84]the time dependence of the density n was studied assuming abiabatic flow. At low initial entropy, a- = 1 and density n = n0, a very long expansion time of 80 fm/c was obtained assuming phase equilibrium. When the formation of phase mixture was forbidden the calculation showed an oscillating density solution, like the monopole giant resonance, due to the fact that the matter entered the unstable, negative pressure region of the phase diagram, fig. 4.8. When the formation of phase mixture was allowed a small amount of gas phase was formed until the breakup at ~ = n0/3 was reached. The abundance of this gas phase was found to be aG = 3% (20%) for an initial entropy of a- = 1(2). Formation of a phase mixture at higher entropies was not considered there. It was even argued that above T = 12 MeV phase separation is not probable because the pressure is positive. It was claimed that between 8—12 MeV there is a possibility for the formation of the mixture although it should be hindered by the potential barrier of the bubble formation. In [Cs85]the method of calculation was the same but the basic view was different. It was observed that since the expansion is near adiabatic even above the critical entropy o~.= 2.5 a phase separation is possible because the system cools down in the expansion. Thus not only bubbles but droplets of the fluid phase can also be formed. The temperature during the expansion is decreasing and only the breakup temperature can be measured. Consequently the discussion of the phase mixture formation in terms of temperature is not very adequate. The breakup temperature is rather low. It is below the apparent temperature, usually evaluated experimentally from the slopes of the energy spectra, because of the collective flow present in these spectra [SR79]. For example apparent temperatures are between 10—40 MeV for a beam range of 42-437 MeV/nucleon [JW83]. The apparent temperature or slope parameter was found be 8.5 6Li,to 7Li andMeV 7Be in a 35 MeV/nucleon N+ Ag excitation collision [MB84] but thetoexcited populations of order of 1 MeV. indicated a much lower local corresponding a localstate temperature on the Fluid dynamical calculations yield local breakup temperatures in the order of a few MeV. Even in a 400 MeV per nucleon Ne + U reaction the maximum breakup temperature obtained in a fluid dynamical model is only 10 MeV [CS83]. -

286

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

1.5

~

S/N-3

~ 05 O 1.5 ~Iso

1.0 Q5

\\\~

\

I

~

\ \~

~‘///~‘

S/N2

~://// Z/~ .1

0

2

I

I

I

S/N-i

t

[tmic]

Fig. 4.8. The time evolution of the density during the adiabatic expansion in two comoving volume elements ((1): peripheral cell at 0.8 times the radius when the expansion starts, (2): central cell) for different initial compressions K. The metastable region is indicated by hatched areas. For S/N = 1 and i = 1 there is a steady expansion (dotted lines) when the Maxwell construction is introduced or oscillations appear (dashed line) which are comparable with the monopole vibrations. From [SK841.

Thus we can utilize the low temperature limit of the equation of state in the phase transition region derived in section 3.3. If we assume that the expansion is slow enough that phase equilibrium is maintained we can obtain the change of the phase abundance versus the temperature or density by inserting eqs. (3.33—3.34) into (4.30): =

T a- ~b2mn~2~ B/T+ ~b2mn~2’3T~ —

~—

(4.33)

The density of the system at a given temperature can be calculated then by means of eqs. (3.35—3.36 and 4.28).

In fig. 4.9 the result of such a calculation is shown. At the breakup 3—7% of the particles are converted into the gas phase but this phase occupies 70—90% of the volume of the system in equilibrium. In this example the entropy of the expanding system is a- = 0.84 and the temperature initially at n = n 0 is T = 6 MeV. We see that, contrary to the expectations of [SK84],phase equilibrium can be established with evaporating only very few particles from the fluid phase. To break up the fluid into smaller pieces there is still sufficient flow energy available in the 40—100 MeV per nucleon beam energy region. So it seems that the possibility of phase mixture at the breakup of low energy reactions cannot be excluded.

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

~iiiII

287

5

~3.

Adiabatic Expansion

I_i

2

o~O~ a 0.84

0 I

5.~

~10

~4. -~3.

~—Breakup—--.I Fig. 4.9. Change of the temperature T, liquid and gas

3] n[fm 0c~0L~ and specific entropy,

d.1

~0

of the phase abundances, a

0 = N0/N, A0 = V0/ V in an adiabatic expansion assuming complete phase equilibrium calculated in the low temperature expansion of the equation of state.

However, the consideration of surface effects is important and may modify the estimates obtained in the quasi-equilibrium thermodynamical limit considerably. This indicates that at the breakup we see small fluid droplets embedded in a dilute gas which occupies most of the volume of the system. Bubble formation might be possible in the initial stages but even at these extremely low entropies at n 0.6n0 the larger part of the volume is filled with gas. The surface effects might really inhibit the formation of small bubbles but the final droplets may be formed through other mechanisms too. Note that at low entropies around the breakup c~ç>0, that is the liquid droplets are losing nucleons at these late expansions stages. Above the critical point the phase mixture formation starts in the gas phase and droplets are formed 50 ~o <0. If we study expression (4.31) we observe that even below the critical point a0 <0 is possible at the breakup because from a0 = (8a0/ÔT)i”

and

Ti” <0

(4.34)

we see that aG has a maximum in adiabatic expansion. After reaching this value <0. The final situation is always that we have growing liquid droplets, except that at very low entropy the system may break up before reaching the maximum of a~.The value of maximum a0 in the low temperature approximation is given in table 4.2. More detailed numerical calculations [Cs851 show the same tendency that late in the expansion &-~ becomes negative, before the breakup in phase equilibrium, if the energy is high enough. See fig. 4.10. When a-—~a-~=2.5,a~”—*1and the temperature at the maximum is the critical temperature. Phase equilibrium may be present in nonadiabatic expansion too. Let us assume maximum dis&-~

288

L.P. Csemai and J.I. Kapusta, Entropy and cluster production in nuclear collisions Table 4.2 In a low temperature adiabatic expansion of specific entropy o~the maximum fraction of particles that form bubbles of gas phase is a~X.This maximum is reached at the given temperature T Tata~ 0.39 0.27 0.17 0.09

2.0 1.6 1.2 0.8

6.0 MeV 4.6MeV 3.4MeV 2.3 MeV

sipation, that after reaching the boundary of the Maxwell construction n~or nt” no more internal energy is converted into collective flow. Then the specific internal energy e remains constant, while the specific entropy of the system a- increases. Now it is possible to make use of the relation =

a~3~‘(T)

+

a~e~(T),

(4.35)

since e = constant. We get a total entropy increase of L~a- 30%, as may be seen in fig. 4.11. This is primarily due to the faster production of the gas phase while the entropy of the gas phase is actually smaller than in the adiabatic expansion. In both cases a-0 4 is much higher than the total entropy or the liquid phase entropy. It was shown [SR83} that if phase equilibrium is assumed at the breakup the light fragment abundances up to alpha particles can be fitted quite satisfactorily by assuming that only the gas phase contributed to their production. It was assumed that the breakup is at 0.1n0 and the breakup temperature was extracted from the experiments [CT83]. The p, d, t and alpha abundances were then calculated from the model showing the Mott transition, see section 4.5.

5Ofm/c

0.3

-

3Ofm/c

2Ofm/c

lOfm/c

aG:2

T[MeV} Fig. 4.10. Change of phase abundances, a0, versus temperature in an equilibrium adiabatic expansion and in a nonequilibrium expansion ECSSSI. The parameters along the curves are the times from the moment when the system reached the boundary of the phase transition.

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

289

>a-G

I soergic Expansion

H

a-tot

O~t~~r0.84 0

n[fm 1

0

Fig. 4.11. Same are fig. 4.9 but for an isoergic expansion.

4.9. Nonequilibrium phase transition As it was mentioned already in several different contexts, the phase mixture, if it is formed in a heavy ion collision, is probably not in equilibrium. In other words the Gibbs criteria (3.26—28) are not satisfied. This does not mean that a phase mixture cannot be formed, only that it is somewhat more complicated to describe. Deviations from equilibrium lead to entropy production. This is true for a delayed phase transition as was discussed in section 3.11. Delay of the phase transition may cause a supercooling or supersaturation in a gas phase [GK84]or a superheating in a liquid phase [Si83, LS84]. Interestingly enough, delay of the phase transition will not change the feature that the entropy of the gas phase a-0 will always be larger than the entropy of the liquid phase a-L since a-0> a-a> a-L. In [LS84]it was assumed that after expanding for a long period in the superheated unstable liquid phase the system suddenly reached phase equilibrium when u~= 0. Thus, apart from the entropy increase due to nonequilibration during the expansion, the final state did not differ from that of the equilibrium expansion. The system may not reach equilibrium even in the final state. This possibility was also considered in [Cs85]. In the nonequilibnum expansion scenario studied one of the Gibbs criteria requiring the equality of the chemical potentials of the 2 phases, (3.28), was relaxed and replaced by an explicit rate equation for the creation of the newly formed phase (3.49). The rate of the phase growth was 2 characterized by the relaxation time Tch = qu 7.. Apart from the extra entropy production discussed already in section 3.9, this delayed phase transition resulted in some interesting effects. The cooling was

290

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

slowed down compared to the phase equilibrium by a factor of two (for q = 10~fm/c3), and the gas phase had a much smaller abundance (20—30%) at the breakup, fig. 4.10. The extra entropy produced appeared in the larger entropy of the gas phase a-G> a-~.Since a-L did not change much, now, since eq. (4.29) is valid in nonequilibrium mixtures too a-+ z~sa-= a~a-~j+ a~a-r a~a-~ + aLa-p.

(4.36)

Subtracting eq. (4.14) from this we get ne

—

eq

a-o—a-G

j

aGaG / __________

a~,

,

5

eq — eq ~oG a-L

-

a0

Thus although the total entropy increase ~a- due to the delayed phase transition is not much, the increase of the gas phase entropy can be observable because a~~ 1, and the third term is also positive. It is reasonable to assume that even if /Lo j.t5 the other two intensives, T and P, can be equilibrated faster between the phases [Cs85}.Low barrier penetration probabilities across the surface of the liquid droplets do not prevent the pressure and temperature balance. It is not easy to estimate the difference of the chemical potentials in a nonequilibrium expansion since T~his not known. We have seen however that a~<0 late in the expansion stage even if the total entropy of the expanding system is somewhat below the critical point. Since the entropy can only increase, a-> 0, using eq. (3.48) we get (4.38)

at the final breakup stages of a nonequilibrium expansion. This will have interesting consequences when the surface effects are being discussed in section 5. 4.10. Light fragment abundances in the phase mixture As we have seen both in equilibrium and in nonequilibrium expansion the final two phases are very different. The gas phase is very dilute and has a large entropy o~ 3.5—4, while the liquid phase has low entropy a-L 1—2 and density close to n0. What is the fragment distribution in such a phase mixture? The gas phase, having large entropy, consists of very light fragments with an exponentially decreasing mass spectrum. In this limit there is not much difference in the model estimates. From the experimentally observed light fragment (proton to alpha) abundances all previously discussed models extract an entropy value on the order of 4, up to a few 100 MeV beam energy [Ka84,JS841, see fig. 4.3. This coincides with the assumptions of a phase mixture at the breakup, fig. 4.12 (from [Cs851). The liquid phase is present in the form of heavier fragments at the breakup. The prediction of the mass distribution of these heavier fragments is a more involved problem. The thermodynamical limit, V—~c~,does not yield a definite prediction. Surface effects, nuclear size, reaction geometry, fission, final state decays and even the collective flow pattern may influence the intermediate and heavy fragment mass distribution. Some of these effects, which have relevance from the point of view of entropy production and phase transition, will be discussed in section 5.

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

I

‘0

~

291

0(ne) 20

GAS

~

~

5iSOergic

.

....0di~bofjc 7

40

60

80 (ne) 62.__.._—~ ‘~40

I

-

~isoerntc

“

LIQUID

~ A~ -3 4r

I

50

I

100

I

I

I

ISO

200

250

1-14 •

I

I

300

EL[AMeV] Fig. 4.12. The entropy of the system at the breakup n = nBu = 0.015 fm3 for different beam energies. Full lines represent the entropy of the gas/liquid phase in isoergic and adiabatic expansion. The hatched area is the range where the entropy of the whole system may vary. Its minimum is given by the adiabatic, its maximum by the isoergic expansion. Note that at the breakup nG 4 nBu! The experimental points are taken from [JSS4I. Diamonds (o) indicate the gas, total, and liquid entropies at breakup for the nonequilibrium expansion (ne). The parameters along the liquid gas entropy curves and at the diamonds are the phase abundances, aL/aG at the breakup (in percent). From lCs85].

The light fragment distributions are not independent of the liquid phase. The final decay or fission of the heavier fragments can change the light fragment distributions too. While for the light fragments the grand canonical treatment is acceptable, the behavior of intermediate mass fragments is already strongly influenced by the limited nucleon number. To account for this problem a microcanonical treatment was developed recently [FR83] where light and intermediate mass fragments are created randomly in a Monte-Carlo calculation according to microcanonical expectation values. The final decays of excited states are also considered. To some extent surface and other nuclear effects are also included because the empirical parameters of the known nuclear fragments are used. In [JS84] entropy values were inferred from intermediate mass fragments up to A = 14. The theoretical method used accounted for quantum statistics, unstable particle decays, and finite volume effects. The entropy values obtained this way were a- 2, that is much below the entropy of the light particles. Up to about 300 MeV per nucleon beam energy no significant energy dependence of the extracted entropy values was seen. The entropy of the heavy fragments was found to be less than 2, fig. 4.3. The possibility of phase mixture, with components of different specific entropy, can explain these observations. Especially the energy independence of the entropies of a mass subset resembles the features of a final state phase mixture very much. Other explanations are not excluded. Since the data are inclusive geometrical effects and the contribution of different local regions may lead to this observation. However it is worth mentioning that all the fragments have similar apparent temperatures. This observation supports the idea that coexisting phases are present near equilibrium.

292

L.P. Csemai and J.L Kapusta, Entropy and cluster production in nuclear collisions

4.11. Summary and critique Although these experiments, as well as the fragment mass distributions discussed in section 5, suggest the existence of a phase transition and the formation of a final phase mixture, further studies are very important. If the phase mixture is present the energy dependence of the total entropy may be regained when the phase abundances aG and aL are measured or at least estimated. Further development in the field of fragment mass distributions is necessary too. Experimentally the selection of high multiplicity central collisions and the separation of events contaminated by fission and other effects is necessary. Theoretically, to establish a more solid link between the thermodynamical parameters of the system and the fragment mass distributions especially for intermediate and heavy mass fragments would be appreciated. The numerous effects influencing the light fragment abundances discussed in this section are all representing real physical processes. The majority of these effects, however, are only different formulations of the fact that interactions and quantum effects should be included. It is important to note that these effects do not modify the predictions concerning a high entropy low density gas phase which is very close to the ideal gas limit. Therefore the fact that light particle abundances exhibit a high entropy even at very low beam energies irrespective of the method this entropy was extracted from the data, is consistent with the existence of such a gas phase. The inclusion of heavier fragments in the entropy analysis is already a more involved problem. It is influenced by effects not discussed so far. The next section addresses this problem.

5. Nuclear fragment mass distributions 5.1. Introduction The dynamical mechanisms for light fragment production and entropy generation and their interrelationship have been discussed in the previous sections. In this section we shall be concerned with the nuclear fragment mass distribution up to about mass number 50. The mass distribution in the range of 4< A <50 is being hotly debated at the present time. This recent surge of interest was initiated by measurements performed by the Purdue—Fermilab collaboration of high energy protons on a gas jet target of krypton and xenon and their interpretation of the data in terms of droplet formation at a thermodynamic critical point [FA82, MA82, HB84]. We begin the theoretical analysis in terms of an ideal gas of nuclei and excited nuclear clusters using the law of mass action. This is essentially a nuclear vapor. It is illustrated how a critical phenomenon is reached in this approach. This leads naturally to the concepts of droplet and bubble formation associated with the nuclear liquid—gas phase transition. Further effects to be considered are: fluctuations in a finite system, contamination of multi-fragmentation by fission, Coulomb energies, solution of rate equations to determine whether chemical equilibrium is a meaningful approach, evaporation of hot droplets and the relative importance of conservation of energy. An approach which is very promising and which seems to incorporate most if not all of the above effects is ensemble averaging the expansion of relatively low energy fireballs using TDHF computer simulations.

L.P. Csemai and fl Kapusta, Entropy and cluster production in nuclear collisions

293

5.2. Law of mass action The law of mass action as applied to ideal gases was already mentioned in section 2.5. Ignoring relativistic, quantum and isospin effects (these are easily put back in) the number density of ground state nuclei of mass number A is ngs(A) = g~(

mTA 3/2 ) eA~±~T, 2ir

(5.1)

where g~is the spin degeneracy, m is the nucleon mass and B >0 is the binding energy per nucleon. The nonrelativistic chemical potential per nucleon ~‘ is related to the relativistic chemical potential ~ by ~a’= — m. For cold nuclear matter at equilibrium density n0, ~‘ = —B. See eq. (3.6). At T = 0 eq. (5.1) is no longer accurate; quantum statistics must be used and interactions are generally important. For T = 0, ~‘ may be greater or less than —B. If ~.t’< —B then the number density is a decreasing function of A. If ~‘> —B the number density is an increasing function of A. The latter case is actually unstable. Once /2’ = —B the nuclei would like to coalesce to form uniform liquid nuclear matter. Another way to see this is to put the quantum statistics back. Then the point at which = —B is the point where Bose—Einstein condensation of the bosonic nuclei would occur. Not only nuclei in their ground states but also nuclei in various excited states will be present. To count them as well we multiply by the density of states c exp[S(E*)], where c is a constant and S(E*) is the entropy of the nucleus of mass number A at total excitation energy E*, and integrate ~

n(A) =

flgs(A)

J

dE* c exp[S(E*)

—

E*/ T}.

(5.2)

The integration is usually done by saddle point approximation. The mean excitation energy E* of the cluster is related to the temperature by Sl(E*)= (dS/dE*)E.~= 1/T.

(5.3)

This leads to 2 (_ -~-S”(E~))-1/2 exp{/2 ‘A — [E*

—

BA — TS(E*)]}/T.

(5.4)

n(A) = cgA (rnTA)s/’ Now a liquid-drop expansion may be employed to obtain an analytic expression for the quantity in brackets [..~]. Or one may choose to count only known excited states. Several groups have written computer codes to calculate the nuclear fragment mass distribution using the law of mass action for ideal gases. In [RK8I, FR82, RF82] all known particle-stable nuclear states with A 16 were included explicitly, and those with A > 16 were included according to the following prescription. The binding energy was taken to be: 23

BA=(a 1A—a2A

/)(1_K

N-Z2 Z Z2 ( A ) )+cs~_c4_~~.

(5.5)

294

L.P. Csemai and .1.1. Kapusta, Entropy and cluster production in nuclear collisions

The density of states was taken to be 61/4

Ag0

~2

t/2

4 exp [2 (~

where

]

goE*)

1

g

12 (goE*)S/

0 = 50 MeV~

(5.6)

A sample result is shown in fig. 5.1. The number densities were calculated for isospin symmetric matter at excitation energies of 5 and 40 MeV per nucleon. The volume in which the fragments move is parametrized as V= 4irr~A~, where r0 = 1.l5fm, A~is the total baryon number of the system and xis a parameter taken to be 0.3 in this instance. Notice that for a fixed volume the magnitude of the slope of the cluster size distribution is a rapidly increasing function of the excitation energy. As the excitation energy is increased the likelihood of finding a heavy fragment (e.g. A = 12) is very much decreased. This is as expected on the basis of simple energy or entropy considerations. Another computer code [SB83]based on the one developed in [GK78] retains the quantum statistics, I

101)

I

I

I

I

I

I

I

X=03

~E=4OMeV

1=0

-

10_i

L_..r—,

I

—

0)

E cc

-2

I I I I

U

10— E5MeV -

I I I

L.a ~~1

-

I

LL

-

I

—~

~

o

r——~~ •~~~1

I—

0) .0

E z

i~-~—

—

I

i0~ 1

I 2

I

I 3

4

I

I 6

7

8

9

Fragment mass

10

11

number

12

13

14

15

A

Fig. 5.1. Mass yield curves for several values of the excitation energy per nucleon. From a statistical model of [RK81I.

L.P. Csernai and f.l Kapusta, Entropy and cluster production in nuclear collisions

295

and the known particle and particle unstable nuclei for A 10 and the known particle stable nuclei for 10< A 20. Comparisons of this model with data have already been discussed in section 4.10. Obviously it becomes very tedious to include even higher mass fragments in the code. Furthermore it is not obvious that cluster—cluster interactions are negligible at baryon densities as large as one-half of normal nuclear density. 5.3. Droplet and bubble formation A difficulty with the previous approach is that it does not take into account interactions between clusters or between clusters and nucleons. Without interactions it is not possible to correctly describe the nuclear liquid—gas phase transition as discussed in section 3.3 [Da79]. Suppose that in a central collision between heavy nuclei an intermediate state of high temperature and density is reached and that subsequently it undergoes an adiabatic expansion as discussed in previous sections. Then, no matter what the entropy per baryon is, it will eventually intersect the Maxwell curve separating liquid and gas phases. See figs. 3.2 and 5.2. What happens next depends on whether the system hits the Maxwell curve from the liquid side (n> n~)or from the gas side (n < ne): if from the liquid side bubbles begin to form, and if from the gas side droplets begin to form [5i83].The initial formation of droplets or bubbles can be studied using statistical methods. For definiteness we shall consider the formation of droplets in a gas. Bubble formation is analogously studied by interchanging the liquid and gas labels. The probability of droplet formation is estimated by calculating the change in the Gibbs free energy of the system when a droplet appears in the gas. See, for example, [LL54] and [Re65]. The Gibbs free energy is the relevant one since the droplet and gas should be in kinetic equilibrium (equal pressures and temperatures) but not necessarily structural, or chemical, equilibrium. Thus suppose that a spherical droplet containing A nucleons spontaneously forms in a gas consisting originally of a total number A + A’ nucleons, G~0drop

=

/.L0(A +

A’), 2o- + TT

GW~~hdrop =

(5.7) (5.8)

ln A.

~i0A’+ ILLA + 4ITR 7C 60 50 -

-

ir,K,A

T(MeV)

thermal break—up?

40

abundances? phase

30

separation?

20 10

cluster

-

U~
production?

-

%

-

‘~‘k 0

1

n/no

2

3

Fig. 5.2. Curves of adiabatic expansion. The initial condition is determined by the Rankine—Hugoniot shock condition.

296

L.P. Csernai and .1.1 Kapusta, Entropy and cluster production in nuclear collisions

Here /2o and /2L are the nucleon chemical potentials in the gas and liquid phase respectively at pressure p and temperature T The third term in eq. (5.8) is the surface free energy for a droplet of radius R and with surface tension a- = a-(T). The last term in eq. (5.8) is added to take into account the fact that the droplet surface closes on itself which reduces the total entropy associated with surface fluctuations (magic carpet effect). The probability of formation of the droplet is proportional to exp(—i~G/T)where E~Gis the difference between eqs. (5.8) and (5.7). The yield is /20/2L 4irr2a- 213 — rin A]. Y(A) = Y (5.9) 0 exp [ T A— T A Here Y 3 where n~’= 4irr3/3 is the density 0 is an phase undetermined constantT.and droplet Tradius R = rA” of the liquid at temperature Thetheconstant is Fisher’s constant [Fi67, Fi71] which is 7/3 in mean field approximation. Apart from the internal nuclear pressure p,,~divided by the liquid density of nucleons inside the nucleus n 0, eq. (5.4) is essentially equivalent to eq. (5.9). The primary difference is that the law of mass action treats the nuciei as if their surfaces bordered the vacuum whereas the droplet formula treats them more correctly with surfaces bordered by nucleon gas at the same p and T. The first application of the droplet-bubble phenomenoiogy was to high energy, 80—350 GeV, proton—nucleus reactions by the Purdue—Fermilab group [FA82, MA82, HB84]. Mass and charge distributions for A up to 30 were measured with higher precision than ever before possible because of the use of an in-beam gas jet target. Arguments based on emulsion experiments [Ta77] and on temperature measurements (see below) suggest that these fragments come from a common thermalized source. The fragment mass distribution for a krypton target is shown in fig. 5.3. It was noticed that a IlI~8

I

I

I

I

I

I

P+Xe-~’A~+X I0~-

0

~lO6~~

-

I0~-

0~ 0

I0~ 0

I

I

I

I

I

5

10

IS

20

25

1°

30

35

Fig. 5.3. Mass yield for 80—350 GeV protons on a gas jet xenon target. From [FA82].

L.P. Csernai and f.l. Kapusta, Entropy and clusterproduction in nuclear collisions

297

power law A265 fit the data better than an exponential e~A. The novel interpretation was that the target nucleus was almost instantaneously heated by the passage of the ultrarelativistic proton, and that subsequently the heated nucleus expanded in size until it passed through the critical point, T = T~and n = n~,of the liquid—gas phase transition. At that point the distribution of droplets is Y(A)=

(5.10)

YOA~,

from eq. (5.9), because the volume and surface free energies vanish. There is no distinction between liquid and gas at the critical point, only long range fluctuations. It is generally accepted that the above experiment is probing some critical phenomenon, but there are at least two difficulties with the specific interpretation given. First, why should one be so lucky to hit the critical point of nuclear matter accidentally with proton energies ranging from 80 to 350 GeV and with targets so different in size as krypton and xenon? Second, according to Fisher’s version of the droplet model, 2< i- < 2.5, whereas the data imply that r = 2.65, which is outside that range. The above phenomenology was generalized to allow for the fact that one may not always be so lucky as to hit the critical point T~,but that nevertheless one may get phase separation at some T < T~[Si83]. If the phases are in chemical equilibrium, /2G = /2L, then the usual Maxwell conditions apply. If so then from eq. (5.9) Y(A) = Y 0A~ exp

{—

~

A2/3].

(5.11)

For a finite range in A, say from 4 to 30, it is virtually impossible to distinguish between eqs. (5.10) and (5.11). A nonzero value of the surface tension a-(T) can mimic an unphysically large value of r(T >2.5). The theoretically expected value of r lies in a narrow range, being 2.33 in mean field approximation and about 2.2 in Wilson’s renormalization group approach [Wi71].Therefore, if one is able to determine T from the data, one ought to be able to infer o-(T), which is an important property of nuclear matter and an essential ingredient in calculations of gravitational collapse and supernovae. A compilation of proton—nucleus and nucleus—nucleus data was made to test the hypothesis that the critical point could be located experimentally [PC84]. First the temperature of the source in a given experiment was estimated. When possible the temperature was inferred from the kinetic energy distributions. An example is shown in fig. 5.4 which is taken from [HB84]. The curves represent a Maxwell—Boltzmann kinetic energy distribution with a fragment-dependent temperature Tfragment =

(1 A/A5)T,

(5.12)

where A. is the nucleon number of the emitting source, which takes into account recoil of the rest of the system when A is not negligible compared with A~.Then the fragment mass distributions, generally in the range 10< A <30, were fit with a power law Y(A) = V0 A’~.

(5.13)

The effective exponent Teff is just 2t3 a means parametrizing the data. to eq. the yield below ofT~, while according to eqs.According (5.1), (5.4) and (5.11) (5.9) the yield should fall-off exponentially in A

”

298

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions 3200

3600

I,\~

120

0

20

40

60

80

I00

FRAGMENT KINETIC ENERGY (MeV) Fig. 5.4. Kinetic energy distribution for carbon isotopes corresponding to the expertment of fig. 5.3. The smooth curves are a fit to a Maxwell—Boltzmann distribution with a temperature T= 15MeV in eq. (5.12). From 1HB84].

should fall-off exponentially in A above T~.According to these arguments Teff should reach a minimum of r at T~and should be greater than r at T< T~and T> T~. The initial compilation was criticized in [Bo84] and so we show the latest compilation in fig. 5.5 [PC85]. There it is seen that initially Teff does decrease as T increases. There is then a gap in the data for 9 MeV T ~ 12 MeV. If indeed there is a minimum in that range then it appears that T~ 12 MeV and T 2. This would be quite an impressive result if it stands the test of time. Future experiments hopefully will fill in the gap. However, it must be said that the data are also consistent with a monotonically decreasing Teff with increasing T In particular there is no evidence for a rise in Teff if attention is focused only on a particular projectile—target combination. A first-order phase transition requires a finite amount of time for completion. The nuclear system

L.P. Csernai and fl Kapusta, Entropy and cluster production in nuclear collisions

4.7

Zf5 II

~

299

‘~

3,0a-

.7

____________________________________________________________________________

4

6

8 10 12 14 TEMPERATURE (MeV)

16

Fig. 5.5. The apparent exponent, Teff, of the power law fit to the fragment distributions as a function of the temperature T The colliding systems are: •, p+Ag; x, p+U; •, p+Xe, p+Kr; A, C+Ag; v, C+Au. From [PC85].

may expand so rapidly that it passes the coexistence curve [B583] and leads to a supercooled liquid or superheated gas, either of which may be metastable. Deviation from equilibrium was discussed in section 3.9. There we derived the result that if cr 0 <0, that is if droplets are formed, then ILo> ILL. Above the critical point this is always the case. Even below the critical point a0 becomes negative eventually, although initially it is positive. Therefore we expect /20> ILL is the rule at breakup, except at very low energies where the system may pass out of thermal contact and break apart before the sign of a~3is reversed. Thus in the most probable situation where /L0> ILL eq. (5.9) predicts that the yield curve [GK84] 213 — ‘r ln A] (5.14) Y(A) = V0 exp[aA — bA has a minimum located at A~, a~~bA;lI~3~r/A*=0.

(5.15)

When ILG = ILL, a = 0 and A~ ~. The droplet A~at the minimum has the interpretation of being a critical droplet: smaller droplets are too small to overcome their relatively large surfaces, whereas larger droplets can and so will grow in an attempt to engulf the whole system. A number of reactions were fit with the form (5.9) allowing ILG ILL [GK84].An example is shown in fig. 5.6 for Ne + Au at 250 and 2100 MeV per nucleon from the experiment of [WW83].The critical size droplets were estimated to be A~= 61 and 233 respectively. Actual minima were seen in the reactions of C + Au and C + Ag at 15 and 30 MeV per nucleon by [CF83]. The most dramatic behavior is —*

300

L.P. Csernai and J.I. Kapusta, Entropy and cluster production in nuclear collisions

ir

103

I

Ne+Au—~A-*-X

-

102

-

Elab =

2100 MeV/nucleon

I 30

40

102_

100

~L 10

~L

20

I 50

60

70

A Fig. 5.6. Droplet model fit to the data of [WW83].From FGK84].

exhibited by the C + Au collisions where the yield has a minimum at Z = 12 and then a rise by two orders of magnitude on either side, fig. 5.7. The droplet model is able to fit the data very nicely, in fact, with only one free parameter. The free parameter is the chemical potential difference between the phases or, equivalently, the degree of superheating of the system. It is very difficult to predict dynamically what this parameter should be, but at least it lies within the physically allowed region [GK84] (there is a limit to the amount a system may be superheated or supercooled). So far only the droplet model is able to simply describe such a dramatic minimum in the yield curve. It would, if subsequently supported, be dramatic evidence for the liquid—gas phase transition in nucleus—nucleus collisions. Many of the phase space calculations described in this report may be capable of reproducing yield curves with such a shape as long as they include excited nuclei with charge up to at least Z = 24. Thus, in the law of mass action eq. (5.4), the volume term in the Boltzmann factor is (jil— /2o)A. Here /2’ = IL’(flBU, T) is determined by the overall density of the system at breakup. The ILo = /L0(n0, T) is the chemical potential for nucleons inside a nucleus and at normal nuclear matter density n0. As long as flBU < n0 it usually follows that /2’> ILo and so the volume term in the exponential is positive. Therefore the droplet formula is essentially the same as the law of mass action with the important difference that in the former the droplets or excited clusters have an interface with nucleons at the same pressure and temperature and not the vacuum. This alters the surface free energy, and also means that the density of nucleons in a droplet nL is less than normal density n0 as assumed in the law of mass action.

L.P. Csernai and J.I. Kapusta, Entropy and cluster production in nuclear collisions

I

—

I

I

I

I

—

C+Au-~Z+X

-

301

-

AQ: 50°

120°

—

io~

E lab

-

=

30 MeV/nucleon

-

-

I-

In

01

102

—

:

—

0

-

0

-

0

-

0

-

0 l0 •0

E lab = 15 MeV/nucleon

-

1

0

I

4

I

8

I 16

12

I

20

-

I

24

z Fig. 5.7. Data from [CF83].Smooth curves: droplet model from [GK84].Histograms: evaporation model from [CF83,FL83, FL83b].

Actually it is premature to claim that fig. 5.7 is conclusive evidence for a phase transition. It is possible that the decrease in the yield of Z < 12 is due to bubbling, vaporization and droplet formation throughout the volume of the system, while the increase in the yield for Z> 12 is just the binary fission tail. Unfortunately the experiment was single-particle inclusive and so binary fission events could not be removed from the data. One could argue that the phase space calculation includes even fission if it happens before or during breakup and therefore it must be included with the rest of the events. This may be pushing the model a little too far! However, there are three pieces of indirect evidence that the rise beyond Z = 12 is not due to binary fission. (1) The fission peak should occur at approximately one-half the charge and mass of the fissioning system. Thus in fig. 5.7 the peak should be near Z = 40. Is it possible for the fission tail to extend all the way down to Z = 26, or even to Z = 12? In a recent experiment [WF85] a gold beam with an incident energy of 900 MeV per nucleon was put through an emulsion which contained targets of H, CNO and AgBr. Nearly 500 unbiased collisions were scanned. In fig. 5.8 is plotted the largest charge observed

302

L.P. Csernai and fl Kapusta, Entropy and cluster production in nuclear collisions

80

~\

-

-

a

Fiss

on

Particles

60

I.

•

~

.2

2. ~

.

2.

-

~ 4Q

0

.~

1

•

2 .4

00

.2 - .2 4: .22 2 .2-2 .2 ~ • .2

-

i—i

0

-0 2

°°~

•

.2 •

.2

.2

• 20

• 3•—2 •

2.2 ••4

.22

-

.4 ~

.2

2. .2

2

•3 2

3~ 3 ~

•2~ 2

: _~.

0

313 4. .5 •~ ~

~ 2;~ .;~ 2

10

.4

•~

_4 .3 . .2 2..2 .2:2 •.— .2 • 3• 2-.3-2

-

-

0

.3

I

3..4

t~• 53t 4 1.5 ~

4

.4

5••4

.7 I

I

I

20 30 40 50 >Z 3 Fig. 5.8. Results from the experiment of [WF85]for a 900 MeV per nucleon gold beam through emulsion. The largest observed charge is plotted against the sum of the remaining observed charges, excluding hydrogen and helium isotopes. The numbers indicate that more than one event of that type was observed.

against the sum of the remaining charges excluding charge one and two fragments. The point is not that only 7% of the collisions lead to fission since this percentage undoubtedly is beam energy dependent. Rather the point is that the fission is primarily symmetric with essentially no chance of fissioning into a heavy and a light with the lighter fragment having Z ~ 24. In the earlier experiment [WW83]of Ne + Au at 250 MeV per nucleon, fig. 5.6, coincidences could be measured. It was found that for fragments in the range of 80 A 89 binary fission contributed approximately one-half the yield in a particular range of fragment kinetic energy. For other fragments binary fission contributed less than 10% to the yield. (2) From fig. 5.7 it is apparent that the cross section at Z = 12 rises by more than an order of magnitude when the beam energy rises from 15 to 30 MeV per nucleon. A fission cross section should not be that sensitive to the beam energy. A multifragmentation cross section could well be that sensitive since it requires a lot of energy to break up nuclear matter into many small fragments. Note that the two curves may cross around Z = 30 where the fission tail might actually begin, which is consistent with (1). (3) Similar but less dramatic minima have also been seen by the same experiment [CF83] using Ag instead of Au targets, and have been interpreted in the same way [GK84]. Binary fission of an energetically excited silver nucleus is a rather rare occurrence. In a relatively low statistics experiment

L.P. Csernai and f.I. Kapusta, Entropy and cluster production in nuclear collisions

303

with a carbon beam of 55—100 MeV per nucleon on AgBr emulsion only events with at least 12 charged tracks were kept [JJ821.Thus binary fission is clearly excluded, yet the results are qualitatively similar to the counter experiment of [CF83].The minimum in this charge yield curve is quite evident as remarked on in [5B831.However, the yields for Z>8 are only indirectly determinable because the fragments move so slowly through the emulsion, in contrast to [WF851where gold is the beam and carbon is the target. 5.4. Fluctuations For an infinite system fluctuations are important only near a critical point. For a system with a finite number of particles fluctuations can be important even away from a critical point. Could it be that nuclear systems are so small that these fluctuations completely wash out the first order liquid—gas phase transition below T~?This question was first addressed in the context of heavy ion reactions in [GK84]. Consider a system held at fixed temperature and pressure. We are interested in density fluctuations of this system. Instead of a nuclear system it may be helpful to think of a finite number of particles placed in a cylinder which is maintained at a fixed temperature T One end of the cylinder is sealed. The other end holds a movable, frictionless piston which exerts a constant pressure p on the gas particles. Due to the fact that only a finite number of particles per unit time collide with the piston, the position of the piston will fluctuate with time about some mean position. Thus the density of gas will also fluctuate with time. The ratio of probabilities for a system to be at density n2 or n1 is P(n2)/P(n1) = exp[—(G(n2)— G(n1))/T],

(5.16)

where G(n) is the Gibbs free energy at p and T For an infinite system in equilibrium the density n is determined by the equation of state once p and T are specified. It is necessary therefore to know G(n) for densities not permitted by the equation of state. This is provided by the Landau theory [LL541in which n is treated as an independent variable not restricted by p and T Such an analysis was carried out in [GK84]. A most interesting result is plotted in fig. 5.9. It shows the relative probability R for the system of 100 nucleons to be at density n compared to the thermodynamically favored densities of flL (liquid density) and nG (gas density). For T not too close to T~there are two well-defined peaks corresponding to a separation of liquid and gas phases, thus exhibiting a reasonably sharp first-order phase transition. As T approaches T~from below the valley separating the two peaks gets filled in and the distinction between liquid and gas gets washed out. At T~the distribution is flat at the top. These large density fluctuations at the critical point give rise to the phenomenon of critical opalescence in atomic systems. To find the relative probability for a system composed of N nucleons, N not necessarily 100, one simply scales the results of fig. 5.9 to the power N/100, R”~°°, because in eq. (5.16) the Gibbs free energy was taken to be proportional to the total number of particles. For the density midway between flL and n0 the relative probability assumes the simple form 2NI1T~] (5.17) R(n = ~nL + lfl~) = exp[—~(T— T~) Thus a larger number of nucleons N sharpens the distinction between liquid and gas phases. The lesson learned here is that for a finite nuclear system statistical fluctuations are important not -

304

L.P. Csernai and .J.l Kapusta, Entropy and cluster production in nuclear collisions

1.2

I -

~

1.0

-

0.8

-

0.6

-

0.81

0.91

I

I

I

T/T~ 1.0

I

I I

N~ 100

_____

0.4

0.2

0

-

0

0.2

0.4

0.6

I

I 0.8

1.0

n/n0 Fig. 5.9. R is the relative probability for the system to be at density n compared to the thermodynamically favored values n0 and nL. The pressure is the equilibrium vapor pressure. The number of nucleons is 100. From [GK84].

only at the critical point, but in some neighborhood of the critical point. These fluctuations provide a mechanism for the system to enter the metastable and unstable regions of the phase diagram. For 100 nucleons and 0.9 T/TC 1 the fluctuations are large and it is improper to speak of a sharp first-order phase transition. As the temperature is lowered the fluctuations diminish in amplitude and the first-order phase transition gradually emerges. 5.5. Coulomb effects on fragmentation The Coulomb force is known to play a very important role in nuclear structure physics. Therefore one may expect it to also have some effect on the disassembly of nuclear matter when the temperature, or mean excitation energy per nucleon, is not large. There have been several estimates of the Coulomb influence on the mass yield during fragmentation which we will briefly review here. The thermodynamic model as described in section 5.2 was also used in [GS82] with inclusion of the Coulomb energy C(Z) as felt by a fragment of charge Z. This Coulomb energy was included in the Boltzmann factor for the fragment as exp[— C(Z)/T].

(5.18)

The Coulomb energy of the given fragment in the presence of the other fragments was written as 2Z(Z C(Z) = e

5— Z)/(R(Z)),

(5.19)

L.P. Csernai and fl. Kapusta, Entropy and cluster production in nuclear collisions

305

where Z. is the total charge of the exploding nuclear matter and (R(Z)) is an average distance to the other fragments. The effect of eqs. (5.18) and (5.19) on the charge yield is clear: The greatest suppression occurs for Z ~Z5.The least suppression occurs for Z 1 and Z Z~. The mean distance (R(Z)) is determined by an approximate, but self-consistent, mean field type of calculation. Several experimental curves were fit in [GS82] after adjusting the baryon radius parameter to r0 = 1.5 fm and the Coulomb radius parameter to re,, = 2.36 fm (!). Typical behavior of the calculated results as a function of temperature is shown in fig. 5.10. For low temperature there is simply not io~—

Au KT=3.OOMeV—--KT~5.15MeV— K1i7.OOMeV

0

—~

20

40

60

80 100 A 120 140 160 180 200

Fig .5.lOa. Calculated limiting mass—yield distributions for high energy proton—Au reactions at three different temperatures.

iO~ -

__________________________________________________

Au KT~5.15MeV 102

-

101

100.

0

20

40

60

80

100 A

120

140

160

180

200

Fig. 5.lOb. The baryon and Coulomb radius parameters, as well as the temperature, are adjusted to reproduce the data of 1KW76]. From 1GS82].

306

L.P. Csernai and II Kapusta, Entropy and cluster production in nuclear collisions

enough energy available to break up the system into many fragments. For high temperature many small fragments are preferred due to their high entropy. For a narrow range of intermediate temperature one obtains a U-shaped distribution due partly to the influence of the Coulomb energy as discussed earlier. In principle, it is problematic to include long range forces in the bubble/droplet analysis of infinite homogeneous systems as discussed in section 5.3. It is possible to do so for a finite system. Consider a gas of A + A’ nucleons occupying a volume with radius R’ determined by 3ITn0R=A+A.

(5.20)

Consider also a liquid droplet consisting of A nucleons with radius r, 3-7rnLr=A

(5.21)

surrounded by a shell filled with a gas of the remaining A’ nucleons, and with an outer radius R determined by 3 = A’ + (nG/nL)A. ~pn0R Then the difference of the Coulomb energies of these two systems is ~C= (ze)~~{flL—

nG)r

[3(2nL—3n

2+ 5n 0) r

2— 5 (nt— no)~]+n~(R5—R’5)}.

(5.22)

(5.23)

0R

Here z is the charge to baryon ratio of the system. This would then provide an estimate of the Coulomb suppression of droplet formation since it is the difference of the free energies which enters in the Boltzmann factor, eq. (5.9). As the critical point is approached, flL~*n~and n 0—~n,~,so that ~C—~0. This is explicit in eq. (5.23) but it is clear that this is generally true no matter how droplets are spatially distributed; no difference in charge density, no difference in Coulomb energy. Even though the Coulomb energy should not influence the fragment mass distribution near the critical point (if it is possible to achieve such in nuclear reactions!), this does not imply a total absence of Coulomb effects on the fragmentation process. For example, yield curves are often obtained 2a-/dE dfl. Experimentally thereby is integrating the single particle inclusive differential cross section d always a low energy cutoff. As the nuclear system disassembles the mutual Coulomb repulsion among the fragments will cause a depression of the kinetic energy spectrum at low energy. If the experimental cutoff is too high it may happen that the measured part of the spectrum is incorrectly extrapolated to zero momentum. Another way of gauging the influence of the Coulomb energy is to imagine that the observed fragments must first pass through a Coulomb barrier set up by the residual nucleus. If the temperature of the pre-fragments within the residual nucleus is small very few of them will penetrate the barrier. As the temperature goes up a larger percentage of them could escape. This leads to an additional energy dependence to the effective mass-yield power Teff (see section 5.3) and may [Bo84b] or may not [PC851 be sufficient to account for the energy dependence seen in the data. In our opinion, though, this barrier penetrability has no meaning for those multi-fragmentation events where the whole system blows apart simultaneously.

L.P. Csernai and 1.1. Kapusta, Entropy and clusterproduction in nuclear collisions

307

5.6. Rate equations forfragmentformation in a gas The discussion in this section so far has focused on phase space calculations. One likes to imagine performing calculations in the microcanonical ensemble where energy and baryon number are exactly fixed. Due to computational complexity most calculations are actually performed in the grand canonical ensemble where the temperature fixes the mean energy and the chemical potential fixes the mean baryon number. It is not immediately obvious that the time scales involved in nuclear collisions are long enough to populate all of phase space statistically. To go one step further than phase space estimates for fragment production one can imagine solving a large (!) number of coupled rate equations [Me771.We may refer back to section 2.8. Rather than just deuteron production, as emphasized there, one would like to investigate the rate equations for all possible fragments up to the maximum possible, 1 A s A~.A first shot has been taken by [Bo841.The rate equations to be solved numerically were assumed to be dnk

(t)

~ n,(t)n1(t) a~51÷1, k — ~ n1(t) nk(t) O~k.

(5.24)

The first term is a gain term which arises when two smaller nuclei i and j fuse to form nucleus k. The second term is a loss term which arises when the nucleus k fuses with another nucleus i to form a larger nucleus i + k. Thermal averaged cross sections were used with the form 312dv v3 a-~ =

4~(m~1/2~T)

2/2T),

(5.25)

1(v)exp(—m11v

where v is the relative velocity and m~ 1the reduced mass. The cross section is 1”3+j113)+ A]2

(5.26)

a-~(v)= ir[ro(i with r 0 = 1.2 fm and A the de Broglie wavelength. Not all reactions lead to fusion so the velocity integration was cut off at the Fermi velocity of nuclear matter at normal density. The initial conditions are chosen to be n1(0) = ~n0and n.(0) = 0 for 1 < i ~ A~.The equations were solved numerically, and the data of [HB84] (see fig. 5.3) were reproduced if T = 25 MeV and a breakup time of about 12 fm/c was chosen. It would be interesting to see how well the model could reproduce other data but no further calculations have been done. Clearly the rate equation approach can be considerably improved [Mc62,Mc63] and holds promise for the future. An incomplete list of possibilities may be mentioned: (1) In the form (5.24) the equilibrium distribution cannot be reached because the inverse reactions, the breakup reactions, have not been included. Detailed balance determines these for us in terms of the equilibrium densities. See in this regard section 2.8. (2) Many-body reactions may be important since at the assumed density of ~n0the fragments are nearly touching each other. (3) The rate equations ought to be solved in an expanding systism with T(t) and n(t) decreasing with

308

L.P. Csernai and .1.1 Kapusta, Entropy and cluster production in nuclear collisions

time, as discussed in connection with deuteron formation. This may eliminate the need for a breakup time for the fragment distribution. 5.7. Evaporation from a hot nucleus Another phase space model for fragment production assumes a compound nuclear system at temperature T and allows it to evaporate particles from the surface [FL83].The level density of the excited system is essentially that of a Fermi gas of nucleons in a potential well. The emitted nuclear fragments may be in the ground state, or they may be in an excited resonant state as long as the decay rate of the fragment is small compared with the emission rate from the system. Fragments with baryon number up to twenty were included in the calculation. Only sequential emission of fragments was allowed and not rapid multifragmentation as in some of the other statistical approaches. More explicitly, the compound nuclear system C may decay into a smaller compound nuclear system C’ and an emitted fragment A. Then d2NA/dE dt represents the rate of emission of fragments of baryon number A with energy E. After solving for the time dependence of the temperature, the proton and neutron number, and the recoil energy of the compound system, one simply integrates d2NA/dE dt to get the mean multiplicity and the kinetic energy distribution for fragments of baryon number A. Sample results of applying this formalism are shown in fig. 5.7. Initial temperatures assumed were 3.4 and 4.9 MeV for incident energies of 15 and 30 MeV per nucleon, respectively. These temperatures correspond to a fused, equilibrated compound nuclear system and are the same values as used in the droplet model fit to the data. Even the absolute normalization of the calculation was determined according to a fusion cross section of 500 mb. Agreement with the data is very good, including the strong energy dependence. Comparisons were also made [FL83b] with the high energy proton—nucleus experiments of [FA82, MA82]. Good agreement was obtained for the kinetic energy distribution of carbon fragments, but less success was achieved for the mass yield distribution. See in this regard also fIgs. 5.3 and 5.4. Similarities and differences between thermodynamic models and evaporation models are discussed by 1GD841. A possible limit to the applicability of this evaporation model to high temperatures is the slow rate of decrease of T(t). As an example see the cooling curve illustrated in fig. 5.11. Note that it takes

~

IOU

10’

TIME

102

Io~

(fm/c)

Fig. 5.11. The monotonically decreasing dependence of nuclear temperature on elapsed time for evaporation is shown for a nucleus with initial parameters Zc(Tmax) = 55, A,(T,,,,.,,) = 133, Tmax = 18MeV and EF = 38MeV.

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309

approximately 100 fm/c for T to drop from its initial value of 18 MeV to 9 MeV, and approximately 1000 fm/c for T to drop to 3 MeV. The problem is that the residual compound nuclear system is kept at normal nuclear density. In reality a system with such a high temperature will also have a high internal pressure which will cause it to blow apart. Different dynamical expansion models give slightly different expansion rates, but the typical model shown in fig. 2.13a results in a decrease in density from n0 to ~n0 in about 5 fm/c. This evaporative emission model therefore will tend to underestimate the amount of rapid multifragmentation. At the very least it should be modified to take into account also the time development of the baryon density n(t), and it should include emitted fragments up to perhaps A = 50 instead of only A = 20. We should also mention some recent TDHF calculations in which spherical symmetry is enforced [DD84]. The compound nucleus is started off with zero temperature but at some central density greater than n0. One finds that the initial compressional energy is quickly converted into collective flow energy. The nucleus initially undergoes expansion, but due to the constraint of spherical symmetry and lack of sufficient energy, it does not undergo any rapid multifragmentation. Rather it oscillates radially, like a breathing mode, with some small evaporation of particles during each cycle. Thus, although the model is unrealistic in the sense of the constraint imposed (see section 5.9) nevertheless it illustrates the effect of conversion of internal energy into collective flow energy, even for low energies. We remark that the decay of excited clusters described in section 5.2, or the decay of hot droplets described in section 5.3, could very nicely be handled by this evaporation model. 5.8. Cold shattering of nuclei One may ask the question: What is the result if in a nuclear reaction the nucleus shatters like a piece of glass or a drop of mercury? One approach to this has been taken by LAH84]. They consider tossing protons into bins, each bin corresponding to a nuclear fragment with charge determined by the number of protons tossed into it. The calculation is very simple. Define P(M, Z) as the probability that one collision leads to M fragments of charge Z One then maximizes the entropy subject to the constraint of charge, but not energy, conservation MZ. (5.27) P(M, Z) = (1— e~Z)e~ Replacing a sum over Z by an integral the constant D is determined to be D = 1.28/Z~2,where Z, is the total charge of the shattered nucleus. The mean multiplicity of fragments of charge Z is M(Z) = 1/(e~ 1). —

(5.28)

Points to notice are that (1) there are no free parameters in eq. (5.28) and (2) the yield goes like Z’ for small Z and like e’~ for large Z. The form (5.28) does fit a variety of proton—nucleus data but does not do so well when compared with such nucleus—nucleus data as in fig. 5.7. The argument is given in [AH84} that no input, such as energy conservation, other than the maximization of entropy subject to charge conservation is necessary to reproduce a lot of data. However other statistical calculations do show a strong dependence on the energy available for fragmentation, such as in figs. 5.1, 5.7, 5.10. Also both proton and heavy ion induced reactions do show a dependence on the beam energy, such as in figs. 5.5, 5.6 and 5.7. Therefore there ought to be some possibility to take account of energy availability in this minimal model.

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L.P. Csernai and fl Kapusta, Entropy and cluster production in nuclear collisions

Actually eq. (5.28) is none other than the Bose—Einstein distribution function. This interpretation is confirmed by detailed examination of its derivation. Note that it treats all nuclear fragments as if they were bosons (the sum over M goes from 0 to ccl) whereas a fragment is a boson only if it contains an even number of nucleons, otherwise it is a fermion. In the latter case the distribution would be ~(Z)

=

(eD’Z + 1)’

(5.29)

where D’ is some other constant. Clearly eq. (5.29) will not fit the data very well because it goes only like M = as Z becomes small, which is far too soft compared to Z~’. Finally in the Maxwell— Boltzmann limit M = exp(—D”Z), D” another constant determined by charge conservation. The fact that energy conservation was not invoked is equivalent to saying that all partitions have the same energy. If energy conservation is taken into account then the distribution function is M(E Z N) =

exp[13(E

—

/22Z- IL~N)1±1

(5.30)

where /3 = T’, and ~ and /.L~,are the proton and neutron chemical potentials, and the + (—) sign refers to fermions (bosons). 3ILp(charge Thus eq. (5.28) follows from if one sets E = 0 (all partitions have the conservation) andeq.~,, (5.30) = 0 (neglect of baryon number conservation). samepeculiar energy), result D = ~/ The that D is not an intensive quantity arises from neglect of energy conservation. Normally when summing over momentum modes one obtains an overall factor which is an extensive quantity, namely the volume V, M(Z, N)

=

vJ

-~-c

A~i(E(p),Z, N).

(5.31)

(21T)

Then charge conservation does lead to chemical potentials which are intensive, e.g. Z~=~ZM(Z,N)

-=

V.

(5.32)

Overall this model probably contains too little information to describe the rich structure of multifragmentation. 5.9. TDHF expansion and fragmentation Most of the previous discussions in this section have relied heavily on the phase space available for fragment formation and multifragmentation. Undoubtedly the available phase space does play an important role. However, the expansion and breakup of a finite size quantum system does require a quantum dynamical framework for a detailed description. An exact nonrelativistic treatment would require, first, knowledge of the nuclear Hamiltonian for A~nucleons. This task is presently beyond our knowledge and computational ability. A first strike in this direction has been taken in [SK84b,KS84]. Instead of the exact Schroedinger’s

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311

equation, the Time Dependent Hartree Fock (TDHF) equations were solved numerically. In this exploratory study all calculations were performed in a two-dimensional world; in fact, on a mesh measuring 60 fm by 60 fm, or 200 by 200 grid points. The system consisted of forty nucleons. An ensemble of pure states, or Slater determinants, was assumed initially. This statistical ensemble allows a description of excited nuclear systems with a high entropy of 2 to 4 units per nucleon. The initial one-body density matrix W(x, p) is parametrized in momentum space as a Gaussian with an effective temperature T The width in position space determines the initial central density. As opposed to previous mean field theories at finite temperature which evolve the whole statistical ensemble in a common mean field, in these calculations each state in the ensemble evolves in its own one-body field. Thus fluctuations in the one-body field are accounted for, and this in turn allows the system to fragment into smaller nuclear clusters. The time development of one particular computer run is illustrated in fig. 5.12. The central density was taken to be 3n0 (with n0 suitably defined in two dimensions) and the momentum spread was taken as T = 50 MeV. Fluctuations in density develop as the system expands below n0, which is a clustering of nucleons into fragments. In this case ten stable fragments emerge. Note that the dynamics is definitively not a process of evaporation at the surface. There is droplet and bubble formation at the surface and within the interior. Typical mass yields at different temperatures T are shown in fig. 5.13. For large T the yield is a steeply falling function of A. At T = 20MeV there is a hint of a U shaped distribution. At still smaller T the distribution is rather flat, with however a big peak near A = 40. This approach is an extremely interesting and valuable one. Hopefully future development will lead to a better interpretation of the parameter T Also to be expected are an extension to three dimensions and inclusion of the Coulomb force. On the other hand, the fluctuations in density which develop are causal consequences of the

48

72

5.12. A sequence of density plots over the two-dimensional configuration space for two values of T, T = 50 MeV lower sequence, T = 25 MeV upper sequence at four time steps from 0 to 72 fm/c. The initial configuration (t = 0) is identical for both cases. The initial momenta of the two sequences emerge from one another by a scale transformation as to allow a comparison among events of affine space momentum correlations. The number of stable fragments is 10 and 8, respectively in these cases. From [SK84b]. Fig.

312

L.P. Csernai and LI Kapusta, Entropy and cluster production in nuclear collisions

MASS SPECTRA 100

tj•J

1=25 MeV

50

I

20

r—r

0~~

‘—fli

—

1=20 MeV

15

0

5

1)

15

20

25

30

35

~0

MASS A Mass spectra at four values of T and initial compression of 3 resulting from the ensemble average of thirty runs for each T, and with 40 wave functions. Based on TDHF dynamics. From [SK84bI. Fig. 5.13.

momentum fluctuations present in the initial state. This is especially enhanced by the absence of particle—particle collisions in TDHF. Supplementing the model with a dynamical process which accounts for the initial density distribution and the initial momentum fluctuations is still necessary to provide a full microscopic picture of heavy ion reactions. 5.10. Summary and critique The multifragmentation, or breakup of nuclear matter into smaller pieces, in heavy ion reactions is a very complicated process in a microscopic sense. In principle, it can be handled by solving Schroedinger’s equation for the many-nucleon system. In practice this is not feasible with current

computing capabilities. Therefore we must rely on intuitive pictures and mathematical models to develop some insight into the dynamics. This would be desirable anyway even if Schroedinger’s equation could be solved numerically. As a first very rough approximation one may assume that the available phase space is the most important aspect of the problem. Thus, if the excitation energy is very low the system simply cannot fragment into very many pieces. Then it makes sense to imagine a quasi-static source evaporating smaller fragments, as in section 5.7. As the excitation energy is increased internal pressure within the excited nuclear matter will cause it to expand to subnuclear densities. Then density inhomogeneities will

L.P. Csernai and LI Kapusta, Entropy and cluster production in nuclear collisions

313

develop within the interior of the system, and so one might apply the law of mass action to the total fragmentation of the system, as in section 5.2. At typically assumed breakup densities of 1/4 to 3/4 of n0 the surfaces of the nuclear fragments overlap. Properties of nuclei in this nuclear medium are undoubtedly different than nuclei in free space, and this brings in the discussion of bubbles, droplets and the liquid—gas phase transition as discussed in section 5.3. Excited nuclear matter formed in heavy ion reactions is composed of a finite number of nucleons evolving over a finite time interval. Therefore one must address the question of whether the available phase space can be populated statistically, as discussed in section 5.6, and whether fluctuations are so big as to smooth out a possible first-order liquid—gas phase transition, as discussed in section 5.4. Since nuclei are electrically charged one must take into account the influence of the Coulomb energy on the modes of fragmentation, as in section 5.5. In the future we should expect to see more microscopic dynamical models applied to multifragmentation such as the TDHF model discussed in section 5.9, to provide additional insight into this fascinating field.

6. Summary and conclusion Those of us working in the field of relativistic heavy ion collisions know that the subject of entropy and light fragment production is a source of continuing vigor. At international conferences the discussions after a talk on this subject are among the most heated and excited. The history of these studies is worth a few words. Six years ago a major push was provided by [SK79]. As discussed in this report a novel idea was presented in a rather general framework. Its title “Evidence for a Soft Nuclear Matter Equation of State” was provocative enough to arouse a lot of interest in the field. Subsequently many publications analyzed the basic idea in more detail and from different perspectives [BC81,FR82, KM82, BB83, St84, etc.] observing that one or another effect should also be taken into account. Nevertheless the heavy ion physics community was stirred up and did an admirable job. Due to the common effort of experimentalists and theorists numerous questions were cleared up, and the results strongly suggest that new and perhaps even more important discoveries might come soon. But first let us summarize what we have learned in recent years. Theories of relativistic heavy ion collisions basically agree that a large entropy increase occurs during the short lifetime 10_23_10~sec of the reaction. The produced entropy is a consequence of dissipative processes which convert a large portion of initial ordered motion of the nuclei into random thermal motion and into meson degrees of freedom. The specific entropy increases with beam energy. Notwithstanding this basic agreement there are some differences in the predictions of different models. In fluid dynamical models most of the entropy is generated by shock heating and then increases by perhaps another 10% during the expansion as a result of viscosity and heat conduction. In microscopic models local thermalization is delayed and is reached later in the collision when the density is somewhat less. Thus more of the incident energy is randomized and the resultant entropy is slightly more than in fluid dynamical models. The differences among theoretical models are not large: taking into account all essential effects the differences are within 20 to 30 percent. The main conclusion of fragment production theories is that the available phase space is the dominant factor which determines the fragment mass distribution and kinetic energy spectra. Sometimes the theories are formulated in different ways but a thorough analysis reveals that there is a

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common element to all, namely, the statistical mechanics of strongly interacting nuclear clusters. Since the system under consideration is not macroscopic in the sense of being composed of 10~particles, surface and finite particle number effects are not negligible. Most essential features are discussed in the literature. What is missing is a unified theory of all these features in a calculationally feasible framework. The relationship between theory and experiment is very interesting. The first generation of experiments adopted the ideas of coalescence from proton—nucleus experiments. A search for the dynamics of coalescence led to applications of thermodynamics and the quantum mechanical sudden approximation to fragment production in heavy ion collisions. This was followed by the discovery of a close correspondence between entropy and fragment abundances. New experiments lead to new theories and vice versa; a nice example being [FA82]. There are two very important experimental results. (1) The entropies inferred from light fragment abundances are much too large, o- 4, at low beam energies, and do not tend to zero as the beam energy decreases, to be understood without invoking the existence of a nuclear liquid to gas phase transition. (2) The fragment mass yield does not fall exponentially as most theorists predicted. This result may also be understood in terms of the liquid—gas phase transition. We believe that it will require more experimental and theoretical effort to develop a clear consensus concerning the origin and interpretation of the aforementioned points. When this is done not only will the final conclusions be important, but how and why we pursued them as well. Since we began this report with a quotation from Ludwig Boltzmann it is fitting that we also close with one. “Until now the most lively struggle of opinions has been going on; each thinks his is the right one, and well may he do so if it is with the intention of testing his own against the others. Rapid progress has strengthened expectations to the utmost; where will it all end?... Oh immodest mortal! Your fate lies in the joy of observing the surging battle!”

Acknowledgments We gratefully acknowledge all of our colleagues with whom we have had discussions over the years. In particular we would like to point to the role of the Nuclear Science Division of Lawrence Berkeley Laboratory for its part in nurturing the development of this field and which has been the mecca of relativistic heavy ion physics since its inception more than ten years ago. This report was supported by the U.S. Department of Energy under contract DOE/DE-ACO2-79ER-10364.

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