NUCLEAR FRAGMENTATION AND LIQUID-GAS PHASE TRANSITION

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NUCLEAR FRAGMENTATION AND LIQUID-GAS PHASE TRANSITION

H. Schulz

To cite this version:

H. Schulz. NUCLEAR FRAGMENTATION AND LIQUID-GAS PHASE TRANSITION. Journal de

Physique Colloques, 1987, 48 (C2), pp.C2-125-C2-134. �10.1051/jphyscol:1987219�. �jpa-00226485�

JOURNAL D E PHYSIQUE

Colloque C2, suppl6ment au n o 6, Tome 48, juin 1987

NUCLEAR FRAGMENTATION AND LIQUID-GAS PHASE TRANSITION

H. SCHULZ

Central Institute for Nuclear Research, Rossendorf.

DDR-8051 Dresden, D.R.G.

Abstract: On the basis of a statistical multifragmentation model for finite nuclei disassembly the possible signatures of the liquid-gas phase transition predicted for infinite nuclear matter are discussed and compared 'with data.

1. Introduction

In this contribution I wilJ discuss some aspects of the fragmenta- tion of hot nuclei in the light of a statistical multifragmentation model (SMFM) developed in the last years in Copenhagen (see refs.l-9)) and put some emphasis on the possible signatures of the liquid-gas phase transition predicted for infinite nuclear matter. The SMFM is in- its spirit similar to that of Koonin and ~andru~lO-l'

Fai and

~ a n d r u p l ~ - ~ 3 ) and Gross and co~orkers~~-'~). Due to the limitation of space for this contribution I will not point out in which details these models differ and not give a general review but concentrate only on some qualitative predictions of SMFM and compare them with experi- mental data.

The SMFM can be viewed as an approach for the transition state of a hot nucleus in which the possible final states are determined by appropriate statistical weights. In doing so we consider at the disas- sembly stage an ensemble of hot droplets with mass number A>4 and gaseous-like particles (A&), which are assumed to be in thermal equilibrium and confined in a fixed volume. The further decay of the hot fragments is considered in a second step by calculating the possi- ble evaporation processes including gaseous particles and light frag- ments.

The main ingredients of SMFM are as follows. At disassembly the multiplicity of the fragments is given by M ~ N * , ~ , where (A,Z) denote

A 3

mass and charge of the fragments and the set ofC~~,~)characterizes the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987219

partition of the system. The conservation laws for total mass, charge

and energy are 2 N A , ZA

A,, FA, ^ZZ

^{Zo, ~ N ,Z}

^{E,. ~} The probabi- ^{~ ~ E ~} lity of a configuration is given by A

In calculating W the momentum and center of mass conservation have been included.The quantity S stands for a normalization constant, SA,z is the entropy of a fragment, ~ A Z is the corresponding degeneracy factor and NA,z! gives the number of permutations of N A , ~ identical fragments. The energies EA,z and entropies follow from the liquid drop model generalized to finite temperatures. For the energies one has

E ~

, Etrans ~

Ebulk

Esurf

Esym

E ~ o u l . (3) Here Etrans=3T/2 is the translational energy; Ebulk=(Wo + T2/&,)A for A>4 and Ebulk=-BxTz for A64 with

€,=l6 MeV, WO=-16 MeV. and BE78 stands for the experimental binding energy). The surface energy reads

d!! ²

Esurf=(P(T)- dT)A 13 ^with ~ ( T ) = I ~ M ~ v ( ( T ~ - T ~ ) / ( T $ + T ~ ) ) ~ / ~ , where the critical temperature is Tc=16 MeV. The intrinsic excitation energy is therefore partly contained in surface and bulk energy. The symmetry energy takes the usual form E ~ ~ ~ = ~ ~ M ~ v ( A - ~ z ) ~ / A and the Coulomb energy is ECoul= $ (Z~~~/R~,)(I-R~~/RPZ). where RIZ is the radius of the Wigner-Seitz cell in which the fragment is embedded. In the case of few fragments (M<5) one could also calculate the Coulomb energy exactly. For the explicit form of the entropy and further details in calculating the energy of the fragments the reader is referred to the original papers1-9).

A peculiarity of the model is that the free volume Vf available for the fragment's free motion is choosen multiplicity dependent

Here 2d=2.8fm is the so-called crack width, which has been estimated from the intersurface distance between spherical nuclei at the Coulomb barrier and V, is the volume of the initial hot nucleus.

For a given observable Q its average value is given by

< Q > =

~ Q ( [ N ~ , ~ S W($N~,~'J

all partitions

whereby due to the very large number of partitions a Monte Carlo sampling has been used in calculating <Q>.

2. Predictions of the SMFM and Comparison with Data

For briefness I will confine myself only to a short discussion of some qualitative results obtained from SMFM and compare some of them with data. I will begin with the representation of mass and isotopic distributions and comment on the so-called power-law fits to the inclusive distributions. Thereafter I will say some words about the connection of the percolation approach with SMFM. The comparison with experimental data is finished by explaining SMFM prediction for the nuclear temperature and by discussing the energy spectrum of frag- ment

(a) Mass and Isotopic Distributions

. . .

It is well known that the statistical multifragmentation models are able to describe the trends of the existing inclusive experimental mass distributions changing from U-shaped forms at low excitation energies to exponentially decaying ones at sufficiently high excitation energies. The U-shaped distributions are observed when the fragmentation process sets in. In this

p + A g , E p = l G e V

case all intermediate mass fragments

have approximately the same propability for their formation. A typical example for a U-shaped fragment distribution obtained from 1 GeV proton induced reaction on Ag is shown in fig 1.

Pig.? Experimental mass distributions (open circles) for the reaction p+Ag at 1 GeV, (solid lines: after evaporation, dashed lines: primordial distributions).

The fit to the experimeiit and primordial ribution is shown (adap-

0

*

ted from ref.

The lower histogram is calculated without taking into account the decay of hot fragments, whereas in the upper one evaporation is included. The data are well described by assuming an excitation energy of ~*/A=4.2 MeV corresponding to a temperature of T=5.1 MeV.

In fig.2 isotopic distributions from high energy proton induced

reactions on Kr are compared with the predictions of SMFM. The

description of the data is satisfactory. It is not illustrated but

worthwhile to note that only after considering the evaporation of the primordial hot fragments the final distributions for the isotopes come in agreement with the data (for details see ref

The temperature governing the isotopic distributions is again about T=5 MeV indicating

I 1 - 1 that in the vincinity of these _ _ - -

loot

temperature values a relative- ly broad mass and isotopic distribution exists.

Fig.2 Calculated isotopic distributions for different charge numbers Z are compa

PO M

with the data %P

rmrr number A

(adapted from ref.

(b) Power Law Fit

---

As seen from fig.1 the lighter fragment yield (A<20-30) can reaso- nably well be reproduced by the so-called power-law fit (cf. ref

19- 21)), whereby the exponent has been brought in close connection with the critical one following from Fisher's droplet theory2*). In fitting the data one has to keep in mind that they represent the final ones of cold fragments, i.e. after secondary and evaporation processes have taken place. The results in fig.1 indicate that the exponent is about one unit below the critical one (ZcritZ2.3) when considering the primordial mass distribution. This suggests that a n interpretation of the inclusive data in terms of Fisher's droplet model is questionable.

(c) Connection of SMFM with Percolation Approach

The percolation models 23-26), which are based on simple criterions for the linkage of nucleons to form complex fragments, are also able to describe the trends of the existing inclusive experimental mass distributions. To understand this fact better we compare in fig.3 the predictions of the percolation approache4) with those of SMFM for the multiplicity of fragments with A>4. One sees obvious Similarities between these two approaches. The reason is that also in the SMFM, which contains a series of physical assumptions, the percola- tion phenomenon persists. It is governed by the crack width d in our model and by the free volume in others. The parameter choice 2d=2.8fm implies that compound nuclei of excitation energy of

MeV can exist. Lowering the crack width

the frazmentation onset is shifted towards larger excitation energies. In the SMFM the crack width has been related to the range of the nuclear forces and its precise value should therefore be investigated thoroushly by comparing with data.

But one should also keev in mind that the inclusive mass distributions do not con-

rain e n o u ~ h physical information to disentangle the purely statistical elements of the data from the important physical content.

'-1

F i g . 3 u p p e r p a r t : P r e d i c t i o n s of SFM,for t h e a v e r a g e m u l t i p l i c i t y MDROp of f r a g m e n t s w i t h A*. l o w e r p a r t : MD

c a l c u l a t e d

h i n t h e p e r c o l a t i o n model ( a s a p t e d f r o m r e f .lf ^1.

( d ) T e m p e r a t u r e v e r s u s E x c i t a t i o n Energy and Fragment Energy S p e c t r a

. . .

I n f i g . 4

shown how t h e tempe- r a t u r e v a r i e s w i t h e x c i t a t i o n e n e r - gy p e r p a r t i c l e . Two d i s t i n c t r e - gimes c a n be o b s e r v e d ( s e e a l s o r e f

A t t h e c r a c k tempe- r a t u r e ~ * " c 5 - 6 MeV t h e r e e x i s t s a r a t h e r s h a r p t r a n s i t i o n f r o m t h e compound n u c l e u s regime t o t h e m u l t i d r o p e n s e m b l e .

F i g . 4 T e m p e r a t u r e of t h e compound n u c l e u s and m u l t i d r o p ensemble a s

f u n c t i o n of t h e e 3 g i t a t i o n e n e r g y ( a d a p t e d from r e f .

A

p l a t e a u l i k e - b e h a v i o u r

r e a c h e d f o r e x c i t a t i o n above t h a t n e c e s -

s a r y f o r c r a c k i n g t h e n u c l e u s . T h i s p a r t i c u l i a r b e h a v i o u r i s due t o

the fact that the gain in surface energy when forming droplets is compensated by the deposition of excitation energy into translational motion. For still higher excitation energies (E*/A>,B MeV) the free gas limit is approached. Fig.5 illustrates that this model prediction seems to be in line with experimental findings 2 9 ) for the apparent temperature deduced from the fragment's energy spectra by using Maxwell- Boltzmann distributions.

Fig.5 lower part: Entropy and temperature of the compound nucleus and multidrop ensemble as a

W

function of the excitation energy for a nucleus

o 2 L 6 8 U Q

with AO=lOO; gpper part: apparent t

€*/A ( M )

function of E /A (adapted from ref.

Recently it has been shown3') that such an apparent temperature of T C 12-15 MeV is compatible with a nuclear temperature of T z 5 - 6 MeV, because the consideration of Coulomb expansion and evaporation proces- ses has as net effect that the resulting energy spectra become much flatter as one expects from a nuclear temperature as low as 5 MeV.

This is illustrated in fig.6. One sees that the data are well described up to fragment energies of about 40 MeV. However, our calcu- lations do not reproduce the tail of the experimental spectra. I think that these highly energetic fragments do not originate from multifrag- mentation processes as described within SMFM but have to be attributed to non-equilibrium processes.

Calcy$,ated energy spectra of

T35-6 ^{and 0 compared to the}

expeq4rnentally observed

f

ref. (adapted from ref. 303?es

3. Comparison to Infinite Nuclear Matter

Nuclear matter shows in bulk equilibrium approximation for temperatures below the critical one a typical van der Waals behaviour familiar from macroscopic systems. A phase diagram of nuclear matter calculated by means of an effective Skyrme interaction is displayed in fig.7. The phase border lines F L ^and 9G confine the region where the liquid and gas phase can coexist. From the theoretical analysis of infinite nuclear matter it follows that the conditions leading to

During the disassembly process the sizes of the fragments will be randomly determined quantities, because they are produced by uncontrollable (chaotically) grown instabilities (cf. ref

31-36)).

This process is to some extent simulated by our Monte Carlo method allowing for a random splitting of the considered system.

Into the fig.7 (see also ref . 5 ) ) is inserted the density- temperature plot as calculated by means of SMFM. It is seen that for S/AsI (T 6 5-6 MeV) the system will remain as a heated-up compound nucleus. This result agrees with the predictions for infinite nuclear matter37y38). Above the crack temperature the fragmentation sets in by giving large fluctuations in the break-up density. The resulting break-up density is close t 0 3 ~ / 3 and lies in the region where infinite nuclear matter becomes dynamically unstable.

Concerning the critical temperature Tc=16MeV for infinite nuclear matter the SMFM predicts that already for significantly lower temperatures only gas particles (Ac4) exist. This result is a consequence of the inputs to the model and is illustrated in fig.8, fragmentation are such that after the initial collision process has taken place the hot nuclear system expands isentropically (dashed lines labelled by the specific entropy values in fig.7) until it reaches the stage where the system becomes 20-

[M1 i

15 - ^/'

- dynamically unstable.

10 -

5

3 g . 7 Phase diagram for infinite nuclear

matter and the temperature-density path as calculated with SMF and labelled by

I

the excitation energy per nucleon and 005 010 0.15 the entropy (in brackets)

E ~ - ~ I (adapted from ref.

l

5.30,' /&25

-

I

I )I

/

^,,f

r' I'

,

where the total multiplicity M and M, for gaseous-like particles are shown as a function of the excitation energy. One sees that for Ao=lOO and E*/A~I 5 ~ corresponding to T=11 MeV the multiplicity of the e ~ gaseous particles begins to increase

A. = 100

/enormously. The resulting transition

30.

>

c Y

It would be an interesting experimental task to find out whether such a marked increase of the light particle yield does appear or not.

The associated temperature could be interpreted as the "critical temperature" of a disassembling finite nuclear system.

The situation is, however, much more involved, because evapora- tion effects might obscure the possible signatures. This is illus- trated in fig.9, where the multiplicity of hot and cold (after evapo- ration) droplets (A>4) are shown as a function of the excitation energy for an initial nucleus with Ao=lOO. It is seen that as a consequence of the secondary decay processes the multiplicity of the massive fragments does not exceed a value 3-4. This implies that high multiplicity events of massive fragments should be rather rare and evaporation effects will dominate the fragmentation pattern at sufficiently high excitation energies. This prediction one could verify with 4x measure-

Fig.9 Multiplicity of massive fragments (A>4)

S

a

before and after consideration pQ));econdary

ETA

decay effects (adapted from ref.

A. = 60

. /temperature varies monotonically with

0 A. = 20 .-.M<

. j ^Tc. The transition temperature to the gaseous phase in SMFM would be equal to

P

3

m - I'

W

3

I

4. Concluding Remarks

Tc if nucleons are accepted as the only constituents of the gaseous phase.

'

o , " " " "

Fig.8 Multiplicities as a function of

the excitation energy. The solid line

I

shows the multiplicity (M

for gaseous

Despite of some obvious similarities in the prediction of the calculation for the infinite nuclear matter and the finite nucleus o

partkyles for Ao=lOO fadapted from

ref.

EXCITATION ENERGY €*/A* ( MeV)

disassembly clear signatures of the liquid-gas phase transition have not yet emerged. One of the reasons might be that the signatures of the liquid-gas transition are considerably obscured, because experimentally measured distributions of fragments and their energy spectra contain rather significant imprints of the collision dynamics, which have been treated poorly in SMFM. There are, however, first attempts of a microscopic formulation of nuclear fragrnentati0n3~-~3) (for a recent review see ref .44)). The corresponding calculations are becoming sufficiently realistic so that a comparison with data becomes feasible. It may be one of the challenges of such approaches to pre- dict clear signatures of the liquid-gas transition, which can be proven experimentally.

It is a pleasure to thank

colleagues Hanns-Werner Barz, Jakob Bondorf, Raul Donangelo, Igor Mishustin, Chris Pethick and Kim Sneppen for collobaration and valuable discussions.

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