Theoretical approaches to statistical multifragmentation

217~

Nuclear Physics A488 (1988) 217c-232~ North-Holland, Amsterdam

THEORETICAL APPROACHES TO STATISTICAL MULTIFRAGMENTATION

D. H. E.

GROSS

Bereich Physik, Hahn-Meitner-InstitutBerlin. Glienicker Str. 100, 1000 Berlin

39,

West Germany

ABSTRACT: First a classical and explicit model for multifragmentationof *'Ne projectiles in collisions with heavy nuclei is introduced. All possible many particle correlations can be calculated. Then the theory of statistical multifragmentation at higher energies is ysl resented and compared to Weisskopf's theory of evaporation. The decay of Xe at excitation energies up to "2 GeV is discussed in terms of nuclear thermodynamics. Especially new phase-transitions are found which do not exist in macro systems.

1. INTRODUCTION Nuclear fragmentation is one of the most interesting new reaction in medium energy heavy ion collisions. For a long time fission was the only known fragmentation. However, higher bombarding energies offer the chance to split a nucleus into several big pieces. Even though this process is not yet found unambiguously, more sophisticated experiments are planned to search for this new decay mode "multifragmentation". From theoretical point of view multifragmentationis predicted by several models. However, the models are still controversal, and subject of hot disputes. Besides claiming the existence of multifragmentationsome of these models offer a new view of nuclear reactions. They present them as part of the general statistical mechanics of small systems with strong short range nuclear and long range Coulomb forces. Their interplay gives rise to new phase transitions in finite hot nuclei. Even though some of the open questions have not been settled yet, the theory presented here is a possible and interesting view of an important part of medium energy nuclear reactions.

2. LOW ENERGY DIRECT FRAGMENTATION, EXAMPLE 2oNe+1g7Au, 20.A MeV Only a few theoretical models exist which give a microscopic dynamical picture of nuclear fragmentation. The most prominent is BUU or

VUU which is

presented at this conference. Due to the many degrees of freedom which are treated explicitly these calculations usually suffer from the low statistics.

0375%9474/88/$03.50 0 Elsevier Science Publishers B.V (North-Holland Physics Publishing Division)

218c

L3.H. E. Gross / Sfafistical mMlfifragm~~faf~on

They are by far inferior to microcanonicalMonte Carlo calculations with their millions of events. Therefore, multiple differential cross-sections cannot be calculated in reasonable computer time. One has to pay a price here. We (i.e. Klaus Mahring and myself) decided to reduce the number of degrees of freedom by considering only the break up of e-nuclei into a-nuclei('). So we consider the collision of *'Ne with "'Au at 20 MeV*A. The *'Ne is assumed to be composed out of 5 u-particles interacting via a Lennard-Jonespotential which is determined in a microscopic ATDHF calculation(*) and which is found to describe low energy a-a scattering well. Technically the 5 a-particles of *'Ne are distributed uniformly over the microcanonical phase space at the experimental binding energy and then this 5 a-cluster is scattered by an inert a-Au potential including surface frietion. Only breaking of *'Ne into a-nuclei can be described, but we hope to obtain sufficient insight into the (true) many-body dynamics. (Experts will realize there is an essential difference to the initialisationin other many body theories like quantum molecular dynamicsc3). The kinetic energy of a single particle is not limited by the Fermi-energy in a local Thomas-Fermi approximation. All momenta are possible that are allowed by the conservation of the total energy of all particles. Thus our initial distribution is a proper eigenstate of the many-body Hamiltonian.) Experience with this model shows an interesting fact: The stability of 12Cf 160, or *'Ne clusters depends strongly on the angular distribution for small partial waves in a-a scattering. It is quite important that quantummechanically these are nearly isotropic in contrast to classical scattering. Therefore, we added a stochastic elastic scattering for central collisions at impact parameters less than about 3.5 fm. This helps considerably to equilibrate relative momenta for central collisions of two a particles. Classically, withaut the stochastic force there will be a strong forward focussing of elastic a-a

scattering when the ATDHF potential is used.

The model is astonishingly successful in describing multiple differential quantities like d*a/dEd@ (Wilczynsky plots) or angular correlations of different ejectiles at low total energy(1). More complicated reactions like multifragmentationat higher energy cannot be described by models simulating the explicit dynamics of the nucleonic constituents. This is true if one is interested at least in more detailed and complex quantities like two-fragment correlations etc. In such a case it is better to leave the direct reactions and to look for a subgroup of possible reaction mechanisms and to investigate purely statistical reactions.

D. H. E. Gross / Statistical

3.

multifragmentation

219~

STATISTICAL DECAY 3.1 What do we understand about statistical decay, and what makes it especially interesting? Here one assumes there is an equilibrated source. This means that "All

accessible quantum states of the system are equally likely". This is the fundamental assumption of statistical mechanics, see e.g. the famous books of Tolman(4) or by Guggenheim and Fowler(5). The number of accessible states is the microcanonical partition sum Z(A,Z,E,P,L,V). In semiclassical approximation this is the volume of the accessible phase space in units

of

h.

The

parameters A,Z,E,P,L are the globally conserved quantities of mass, charge, energy, momentum and angular-momentum (with respect to the At this point the sume

the

C.M.).

physical situation must be further specified. We

as-

following scenario: In a reaction induced by highly energetic pro-

tons the target nucleus becomes strongly excited in a short time and then expands and breaks into several pieces. During this phase a strong mixing takes place due to the strong interactions of the fragments. As long as they are sufficiently close to one another there is considerable exchange of nucleons and strong excitation. We suppose that during this early part of the expansion the available phase space becomes filled uniformly. However, when the average distance between the fragments grows to something like 2 fm, the exchange of nucleons ceases. This configuration is called the freeze-out. Here the system occupies the freeze-out volume V. During the expansion this is the last moment where the phase space is filled and the entropy is maximal. From here on the fragments separate on Coulomb trajectories and are otherwise free. Of course, they may be excited and evaporate particles later. For simplicity we assume further that all decay modes have the same freeze-out, a spherical configuration with a radius of about 2 fm Ali3. This concept of a freeze-out volume which is the same for all decay modes and all particles is certainly a strong oversimplification and must be improved in later work. The freeze-out configurations are the accessible states in the sense of the general physical picture above. States having no overlap with the freeze out volume are not accessible. The number of different linear independent configurations at freeze-out with the same value of A,Z,E,P, and L is the partition sum Z(A,Z,E,P,L,V).From Z(A,Z,E,P,L,V) all thermodynamic properties of our system follow. For example, by using relations like 'l/Tq dlnZ/dE thermodynamic quantities like the temperature can be defined for our microcanonical ensemble. Also thermodynamic relations like the equation of state

D.H.E.

22oc

may be deduced of

nuclei

which

even

result

2. This

from

though

study

the

mainly

of

models

the

3.2

decay

is

reason

the

E it

of

of

available

Two Models

for

statistical

the

be difficult

find

to

nuclei

with

reaction

processes

source.

equilibrated

of

decay

provides

important

the

chance

long-range

mechanisms

quite

to

forces.

often

test

space.

Statistical

The Classical

studying

less

systems

direct

phase

for

decay

small

investigating

multifragmentation

might

a more or

statistical

thermodynamics

Moreover,

Statistical

at higher

from the

The investigation

Gross /

Decay?

Weisskopf

Model

and the

Statistical

Model

of

Equipartition The quantum rate dNif/dt

If then

the of

= g is

where Mif

the

final

is

of

a nucleus

T matrix

the

and pf

are

just

folding

of

of(E)

loA_, (E-s)Pl

(c)de

into

d2Nif/defdt

q

the

$

final

state

one neutron

(Both

rate

the

A-l

particle.

The specific

from state

i to

f is

dNif/dt (3.1)

free(continuum) =

decay

(Mif120f(Ef)

states

just

to

times

(bound)

are

state

energetically

density. an A-l density

residue with

the

nucleus, one of

the

distinguishable). (3.2)

energy

interval

E: (3.3)

(Mif\2p,(~)pA_,b)

with p,(s)de

where Sz is d2Nif dsfdt Now the

dNfl

-=-

dt with

the

d3pd3r I (2lLtQ 3

q

the

(large)

= g

matrix

211

6 inverse

(3.4)

s

normalization

(Mif12

element

P4smpda

=

-$

volume

PpA_,(Ef-s)

can be determined

(3.5)

. from detailed

balance:

(3.6)

I

Mfi12 flux

jf

pi

q

= vf

Sfoflnv

l/Q

D.H.E. Gross / Statistical a*Nif

inv - m

-Z&U

aaat

The

ff

n2.h3

221c

multifragmentation

(3.7) %-l/p,

’

virtue of using detailed balance is in the cancellation of the (undeter-

mined) normalization volume 61. Without any further constraint we call this the generalized Weisskopf model. ulnV 1s taken from experimental data. The statistical model of equipartition, as defined above, amounts to the assumption of \Mif\ = const and consequently dNif/dt = pf, i.e. the population of the final phase space is uniformly, and inv a l/Vf "f d2Nif -

atasf

(3.8) =

const m p p f f A-l/pA

’

In the statistical model of equipartition one may characterize the final state-density pf by the accessible phase space (accessible under the constraints of global conservation laws of E,P,L,A,Z). Here various choices are possible. In the model which is

used

in the following the simplest assumption

was taken namely that only states are possible (accessible) where the fragments fill a predefined freeze-out configuration. Very often ofin' is taken to be xR* (classical Weisskopf model), even though in many cases the compound-formationcross-sections drop down with higher E and are not constant. In this case we have a*Nif -asmp

ataEf

f f A-l/pA ’

(3.9)

This, however, does not satisfy equipartition and is consequently different from the statistical model. Detailed balance is useful at low excitations but meets considerable problems when excitation energy is raised. If the final state of the daughter is an excited state then ofinv is not measurable and model assumptions are necessary. More difficult is the case that the daughter state f may decay further. Then the intermediate state density PA-1 is defined if the partial width I?of the secondary decay of (A-1) to (A-2) is smaller than the level spacing D of intermediate states. In the other extreme r >>D we have the situation of simultaneous production of both fragments.

222c

D. H. E. Gross / Statistical

Consequently, of

validity

the

of

both

following

The chance

porated

*pA_l not

large,

intermediate

the

a few nucleons

will

be rescattered

and reabsorbed

Once a particle

the

final

classical

states

be produced

is

is

Weisskopf

IO-**

created

model.

the

equipartition

the

same energy

decay of

I’ of

with

Classical

simultaneous

emission

regime

a level

of

where

equally

likely.

In between

the

the

the

mixing

order

density

PA-1 is

Weisskopf of

several

emitted is

theory

One should

particles.

particles

strong apply

not

enough the

to

to

all

to

find

in

earliest

treated

is

is

high,

as free

become

important.

a heavy

to

total

treatment At lower the

A. At excitations

residue

In addition

and only

the

Moreover,

is the

equipartition

significant.

states

is

quite

similar

the

mass of like

(as

we will

the

In ref. ansatz

system is

excitation

are Here of

frag-

and excitaR+K

justified.

Coulomb and nuclear of

5 MeV*A, say,

the

fragments

the

probability

7 a mean field of

see

results.

production

this

with

(statis-

there

mass-spectrum

below

no-interaction

there

for multifragmentation by Randrup and Koonin (6).

particles If

give

We have

down.

space

paper

interactions.

all

validity

models

phase

a grand-canonical

interfragment

over

both

the

A < 33 were considered.

U-shaped

tried.

the

are

interactions looks

where

use equipartition

One of

fragments

a region

of

established

populate

APPLICATION OF THE STATISTICAL MODELFOR MULTIFRAGMENTATION

chapter)

two regions

become

one approaches

4.

ignore

will

Moreover,

(i.e.

next

energy

states

breaks

the

tion

away un-

const*f

rates

in

ments with

eva-

a few MeV. Then an

model.

the

runs

to

tical)

old.

at a low

it

The final

Xe at E=400 MeV) the

nucleus

more rescattering

Attempts

region

by other

proportional

sets,

occurs.

decay

of

131

(e.g.

= 10Uz3 to

situation

quite

different

populated.

next

be also

of

by the

compound

the

to

negligeable.

excitations

dNif/dt

before

will

is

as formulated

be equally

only

a nucleon

The population

At high

the

energy

of

fragments

disturbed.

may characterize

approaches.

At low excitation rate.

arguments

multifragmentation

Randrup

model

and Koonin

was a term

C(Ai,Zi): C(Ai,Zi) was added nucleus

to

with

<$> 1

=

Zi(Z-Zi)

the

binding

charge

=

Zi

2 r

Bi which

irrespective

l/3

cA

0

energy

(4.1) is

the

of

the

’

pmk>Zk

average

fragmentation

Coulomb of

the

energy

of

rest.

Zk

1 k



i

+ A l/3) k

(4.2)

a

D. H. E. Gross / Statistical

and the

mean multiplicity



is

MAiT 3/Z ) ( -

=

223~

multifragmentation

S,(T)*9*exp{[Bi-Ci+~~i+~LNNi]/T}

(4.3)

2nfi2 where

Bi

spatial

is

ci(T)

the

=

binding

CAi

q

4.1,

4.2

M the

nucleon

internal

mass,

partition

T the

temperature,

;

establish

solve

was introduced.

shown in

fig.

fixed

by the

constraints

CZi = Ztot

isotope, to

Q the

sum,

(4.4)

I_~~,&.Jare

and 4.3

method

equations Au is

Atot

fragment

effective

the

yipi(e)e-E’Tde o potentials

each

energy,

and ci(T)

chemical

Eqs. for

the

volume

now a set

for this

(4.5) of

lg7Au

there

large

system

A typical

very

are

many coupled

about

of

equations

3000 eqs.)

coupled

highly

mass distribution

for

(One

In ref.

2 an

transcendential

the

disassembly

of

4.1.

Au KT=5.15MeV

100

!

20

0

40

60

Mass distribution lg7A Inspite fragment

of

its

feels

within

complication only

the

the this

field

of

80

100 A

120

140

FIGURE 4.1 for the

fragmentation

mean field

approach

is the

still other

a mean field fragments

(ref.

160

180

2 IO

of 7)

approach distributed

as every according

D. H. E. Gross / Statisrical

224~

muliifragmenlation

their average (statistical)distribution . Fluctuations of mk away from are ignored. These can be considered only if a full Monte Carlo calculation is done allowing for individual fluctuations of the Coulomb field. Obviously, one also should do a fully microcanonicalcalculation with sharp total A,Z,E,P,L. Such were presented by the Copenhagen group(8) and the Berlin

calculations

group(9) along very similar lines. Some of the differences are discussed in (lo). Since then a series of many papers have been published which come to the following main conclusion: a) Freeze-out is reached when the average distance between neighboring fragments is of the order of 2 - 3 fm. b) For a system of about 100-130 nucleons there is an anomaly in T(E*) at about T=S MeV. Here T is independent of E* or even dT/dE* < 0. The function T(E*) bends backward and after a while starts to rise again with E*. (Such a behavior is only possible in a microcanonical ensemble. In a canonical one the heat capacity dE*/dT is equal to the fluctuation of E, ( 2)/T2 and consequently always positive). This, perhaps, is the first example of the difference between microEX

E”lA

[Me'fl 1200--9 -0 ,OOO_

E"=T2A/B

.... calculated results

-5 600-

I

I

I

I

I

T[MeVI FIGURE 4.2 T(E*) for 131Xe, microcanonicalMetropolis simulation Experimental data ~~~~ vs TApp) TV : S32+Ag, V : 160+Ag (ref. 19)

D. H. E. Gross / Statistical muliifragmentation

and canonical at

given

temperature

tuations. chanics

treatment

closed

Recently nuclear

cases

effective

distribution of

Coulomb

repulsion

4.1

also

problematic

of

the

in

to

However,

c9)

years

volume

enters

excplicitly

as the

classical

it

container is

fluc-

decay. of

of

be first

this

me-

microcano-

a new and in

sampling

claims

a spherical

by ref.

nuclear

offers

Metropolis

the

statistical

same picture

paper

paper

of

show up in

to

one does

E and ignoring

details

as they

The first

and rearrangement.

Conventially

mean value

contributed

fragments

and solved

decay.

delicate

The other

hard-sphere

Density

the

such

(II)_

freeze-out.

the to

some

fragment solve

the

interacting

exactly

that

via

problem

which

before.

Dependence

The freeze-out similar

and Randrup

though

statistics

only

nuclear

may miss

systems

fragmentations

at

was addressed

which

finite

Koonin

nical

compound

T fixing

A procedure of

of

22%

parameter

1200.

-6

in the

inverse

*

205

=

270

*

320

multiplicities. 2

cross-section

I~R In the

It

is

-EE'z$T2

IOOO-

. 0

T(E*) for different transition I and II between two fission

0

2

L

6

a

Twd

FIGURE 4.3 freeze-out volumes. Observe the fact that phase only exist near R,=2 fm. = distance fragments with A q A/2 as a function of R,. R {fm} 107 2.05 2.7 3.2

{fm) -1.76 1.8 8.4 13.5

a

classical

D.H.E.

226~

Gross / Statistical

m~iti~ragme~tu?~~n

Weisskopf model. It gives the spatial extension of the system. As this is not well known it becomes a fit parameter which might be veried within reasonable values. In fig. 4.3 we show T(E*) for 13'Xe at various densities. At these densities the mean distance between neighboring fragments varies nonlinearlywith the density. It is easy to estimate this e.g. for symmetric fission (see the figure caption to fig. 4.3). Therefore, only a range of -2 < Ro < -2.2 is acceptable. Only in that range do the

fragments not

The curves

T(E*)

overlap

and the distance remains in reasonable limits.

for various densities behave similarly in this region. ~!?~/A

f

‘31x6

FIGURE 4.4

s (E*)

, total

entropy 5 = j 3 dE as function of the freeze-out volume.

5. THERMODYNAMICS OF HOT NUCLEI 5.1 Phase Transitions? In this chapter we discuss one of the most interesting aspects, the question whether a small system with long-range forces like hot nuclei show well marked phase-transitionsand whether these are of a new type. A microcanonical Metropolis-samplingwas used

to

simulate

these

decay

modes. Details of

the calculation are described elsewhere(y). Here we discuss the results for the decay of 131Xe.This nucleus was the key-system for experimental(")

D.H. E. Gross / Staristical ~u~~if~ag~enca~i#~ and theoretical(9) finite

nuclei.

tracted(“)

exploration

of a possible

From the experimental

which was lateron systemsf’3).

From the observed

the stability

of the temperature

distributions

fragmentation

transitions

and the isobaric

are excellently

in our microcanonical

=

= G

fragmentation

(5.1)

> f

ZE,’

entropy of the system, Nc, Nv are the numbers of the pro-

duced charged fragments and prompt neutrons gy of prompt neutrons The result

and fragments together

was plotted

4.2,

in figs.

tion

energy E* (points).

ref.

[ 91 turns out to be indistinguishable of fig.

4.2.

The statistical

The temperature

uncertainty

is smaller

at I:

5.1 as a function

of first

order.

consists

At transition

of the excita-

or singular

65OCE*C750

at these two

would be called

phase

heat is -150 MeV at transition

ensemble,

in

size.

I T(E*) bends even slightly

only in a microcanonical

ener-

of disas-

out of 2 - 6*106 events.

than the point

dE*/dT is large

The latent

kinetic

from the thermodynamic one on the

In the language of thermodynamics these

-‘TOO MeV at II. the entropy

is the total

in the configuration

3OO
MeV, T N 6 MeV. The heat capacity transitions

and E,’

of the prompt neutrons as defined

The sample at each point

We see two anomalies transitions.

model the thermodynamic

3(N,+Nv)-5

<

where S is the total

sible

reproduced by our micro-canoni-

T frorn(14)

as

1

scale

at T = 5 MeV.

in detail.

temperature

sembly.

occurs

Y(A) (12) it

distributions

Here we want to examine the nature of possible

model(“).

We calculated

r

for many other hot

power law of the mass-yield

that in 13’ Xe a phase transition

The isobaric

in

data a temperature of T = 5 MeV was ex-

.

was cbncluded

phase transition

found to be characteristic

;,;l;ar

cal

liquid-gas

227C

indicates

I and

backward. This,

a more sudden increase

posof

(opening

of new decay channels). A similar anomaly in T(E*) has (15) been found in our canonical calculations and also in microcanonical calculations by the Copenhagen-group (8) for A=lOO, Z=50, isospin-frozen, at T=5 MeV. In ref.t8),

however, the system was allowed

more fragments with rising of TfE*) with rising

E* and falling

gin and may have nothing Are these transitions clear

density.

A falling

p can very well be of a very trivial

ori-

to do with a phase transition. linked

to the expected

matter(16) ‘7 There are considerable

show a sharp liquid

to expand and to produce

E*, whereas ours is at constant

doubts,

to gas sphase transition.

liquid-gas

transition

as nuclei

Moreover,

in nu-

are too small to

the long-range

D.H.E.

228c

Coulomb-force all

leads

thermodynamic

quite

distinct

Fischer’s

Monte

phase

this

is

T(E*) from

indicating B = -

without

transition,

1103

Coulomb

like

This

a Van der for

found were

dependence

size.

inapplicable

anomalies

Waals

nuclear

the

entropy

gas.

Concepts

fragmentation

in (18)

,

Inf

fig.

and

systems as

systems.

in nuclear

given

of

makes nuclear

where

may not

a canonical

was performed.

verified

in

temperature

Coulomb-energies

system

ones

are

the

simulation

thermodynamical all

that

liquid-gas

In fact

model

multifragmentation

quadratic

on the

from macroscopic

arguments

Carlo

an essentially

potentials

droplet(17)

Early be the

to

Gross / Statistical

our

calculated

switched-off transitions

MeV to

exactly

(dashed

1495

on the

MeV if

average

the

curve).

disappeared.

B = -

follows

calculation.

The binding

the

is

points

we also

show the

same way as before Clearly

Coulomb

5.1

all

energy taken

but with

anomalies of

’ 31Xe

in shifts

off. However,

obtained

with

T(E*)

Coulomb

action.

E*

E*/A IMeVI

5 600

I

2

I

I

4 T IMeVl

I

I

6

I

1.

B

FIGURE 5.1 re T as function of E* Xe. Full line: standard dots: V the

result of the simulation with full but for F events same as the dots,

for

the

value

mic$ocanoniE*=T A/E;

Coulomb interactions; only (two fragments

Al,A2>10, the rest are small; + the same as the dots, but for only those events where there is one big fragment A>5 and all others have At4. Dashed line T(E*) for a microcanonical sampling with all Each point is an average over about Coulomb-energies switched off.

2 - 6.106

events.

inter-

D. H. E. Gross / Statistical

More detailed be gained

if

insight

one divides

pseudo-evaporation only

with

least

three

tive

yield

E*.

We see

the

transition

Al,

F,

A2 > 10,

in percent

the all

the

the

on at the into

many

the

are

5.2

fig.

I and at transition

for II

where

there

Al,A2,A3

in

of

E,F

the

event

two

mode where

The rela-

excitation of

energy

F modes at

cracking

to

can

E, the

each

are

> IO.

favor

from

in

cracking

as a function

a changeover

groups:

produced

C, the

E events

I and II

following

mode,

small;

many have masses

yield

transitions

fragments

pseudo-fission others

shown in of

going

channels

among

among the

is

drop

decay

where

fragments

a rapid

what is

the

mode,

one has a mass A>lO;

fragments at

into

229~

multifrngmentation

de-

cays.

131Xe

1200

1000

800

600

400

200

0

E*t[MeVl FIGURE 5.2 Heavy-mass correlations. E, events with only one fragment AlalO, the rest small; F, events with two fragments AI,A+lO, the rest small ; C (cracking), at least 3 fragments have masses A1,A2,A3>10 E,F events respectively the rest is small. En, and F,, denotes where all Coulomb energies have been switched off. Without Coulomb the surface tension is dominant. Xe decays by nucleon evaporation only. There is no fission, and there is always a big evaporation remnant.

The

of

energy,

131Xe.

average duced. -2

The

where

IO charged The entropy

units.

transition

new decay

mode

fragments has

Consequently

is

I appears, that

opens

and about

an extra

rise

here the

over

a huge additional

far is

same

above

a hot number

the

average

phase

space

the

fission of

trend

fission

barrier

where

neutrons

from

([expAS-l]expS)

on the are

about

pro75 by

becomes

230~

D.H.E.

available

to

the

tified.

Clearly

between

the

system these

surface

Coulomb-energies nucleons

to

this

the

switched

off.

(c.f.

like

In that Enc,

in neutral

transition’

ordinary

energies.

curves

specific

multifragmentation

name ‘phase

ruled

Coulomb

physics

are

that

modes are and the

are

Our transitions like

such

by evaporation

respondence

Gross / Statistical

seems

fission

by the

They disappear, case

fig.

macroscopic

phase-transitions

5.2).

small

interplay

just

There

systems

of

jus-

when all

a 131Xe nucleus

Fnc in

quite

like

looses

is

no cor-

real

highly

gases.

charged

systems

nuclei.

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

l‘**-:..*

. .. . . . . l::. ‘:

131Xe 360MeV

I

I

I

I

1

2

3

4

Ln(S2>

FIGURE 5.3 Microcanonical Monte Carlo simulation of ln(a,,x) 131Xe at E* q 360 MeV (c.f.13). At phase transition Transition I is remains in the coexistence phase. a vaporization of the system. It

is

tation sion in

to

fig.

5.1).

contrary

clei

are

the

Here,

its

make the

temperature

of

the

whole

observation

modes

(e.g.

the

in the

transition

however,

evaporated,

specific

that

pseudo-evaporation

T=5 MeV. This

lects

observe

or temperatures

mode F makes the

the

at

interesting

energies

in T(E*)

already

yield

only

is

where

events,

ensemble

pseudo

very

of

evaporation

ln for I the system not connected to

occur

at different

at about

T=4 MeV (triangles

a few percent

of

a-particles later

for

exci-

The pseudo

decay-modes

important

vs.

decay-modes:

only

in T(E*)

transition

is

transitions

different

w

decays.

and lighter

at

T=5.5

which

shows

experiments,

events)

all

fis-

MeV. It

On nu-

is

a transition where

by a trigger-bias.

one seThere

D. H. E. Gross / Siatistical

one possibly

will

perimental and

fit

see

values

studied

Especially amax is the

to

the

the

the

liquid

to

are the

all

from system

true

for

gies

E* -

to

remains

8 of

disappears

gas.

II

the

of

and a critical

the

seen(19)

can

be and has

are

nuclei

ex-

the

within

phase.

ln(amax)

the

binding

phase events.

energy is

of

the

liquid-gas this

from the to

vs.

channels

has two

Therefore

system

liquid

do with

ln

(fig.

Here

and is

one in all

correlation

systems

lattice. (21).

ln

channel

largest

vapor of

cubic

with

decay

this

coexistence

correlation

in

I has nothing

80% of

the

already

the

the

transition

about

T(E*)

ln(amax)

each

to

to

transition

in

for

order

in

The correlation

entirely

transition

of

except

am,,

the

First

on a finite

amax corresponds

nicely

places.

phase

calculations

0ur phase

liquid

gas

fragment fragments

one at small

phasec21).

the

correlations

In percolation

indicates

vapor

the

largest

Ca* of

and the

in

different

4.2).

calculations(20~21)

The one at high

correlation

sition

from the

same amax.

coexistence

the

(fig.

mass of of

a transition

T(E*)

interesting

branches.

of

at slightly

our

by percolation

mean value

with

transition

and signals

excellently

The transition been

the

231~

multifragmentation

the

shows

5.3).

that

This

is

Only at excitation 131Xe the

to

tran-

still

ener-

coexistence

branch

established.

SUMMARYOF CHAPTER V Two phase-transitions MeV. This heavy-mass

nuclear

Coulomb-forces

and finite are

macro-systems the

systems

results

novel

with

To W.A. Friedman sometimes

controversal

5.21,

size

effects

do not

I am very

(esp.

systems

within

the

surface

the

ranges

These

absent overall the

all

peculiar

long-range

energy).

totally

Changing

reasonable

are

between

and 6 The

with

transitions

competition

and are

T-5

transitions.

as a calculation

phase

by the

’ 31Xe at

of

on similar

as well

interactions.

volume)

disassembly

these

controlled

short-range

here

the

(7,8,18)

show that

in many-body

(freeze-out

presented

fig.

off,

They are

for

reports

(c.f.

switched

systems.

transitions

found

previous

correlations

Coulomb-energies to

were

strengthens

phase

in ordinary density

quality

of

of the

change. grateful

for

numerous

very

illuminating

though

discussions.

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