Multifragmentation of nuclei : Dream or reality

Nuclear

Physics A488 (1988)233c-250~ North-Holland, Amsterdam

233~

MULTIFRAGMENTATION OF NUCLEI: DREAM OR REALITY

Christian Ng6

Laboratoire National Saturne 91191 Gif sur Yvette Cedex France

Abstract We give a brief overview of the experimental and theoretical situation about multifragmentationof nuclei which takes place when a sufficiently large amount of thermal excitation and compression is deposited in a nuclear system.

1.

INTRODUCTION

Heavy ion collisions allow to form and study excited nuclei. Around the eighties, one could easily produce nuclei with a thermal excitation energy per nucleon of

the order of

l-2 MeV. These species de-excite by

particle

evaporation and/or fission leading to evaporation residues and/or fission fragments. The new accelerators providing heavy ions at bombarding energies larger than about 20 MeV have given the possibility to create nuclei at excitation energies which can be a non negligeable part of their binding energy. At the very beginning of these studies the basic question was to know when nuclei boil off [I]. This question turned out not to be the right one because excited nuclei prefer to evacuate their excitation in a more economic way which is not the one where the nucleus breaks up into its constituants, the nucleons. Indeed, it is more efficient, from the energy point of view, that a nucleus breaks up into clusters because one gains the binding energy of the clusters. This transition is expected to occur at a lower excitation energy than the one corresponding to a

separation of

the nucleus into

nucleons. At this stage, an open question is to know whether this transition from an excited nucleus towards a configuration involving several clusters is basically an evaporation process or a new de-excitation mode. In the latter case one would like to know why and how nuclei break up. The process in which a nucleus breaks up into several pieces is usually called multifragmentation and we shall use this denomination throughout this contribution. Several years ago the inclusive mass distribution of fragments produced in proton induced reactions on heavy targets have been studied very carefully in

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

C. Ng6 / Multifragmentation of nuclei

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a systematic

way

up to very high bombarding energies (350 GeV) [21. It was

found an unusual production of medium mass fragments. Their mass distribution follows a power law A-T, where A is the mass of the fragment and T = 2.34. This result has been interpreted as a possible indication that these fragments could be produced in a gas-liquid phase transition near the critical point [2,31. If this is the case, one expects that the mass distribution follows a power law with T = 7/3. The multifragmentationof the target nucleus under the impact of the proton occurs as soon as the bombarding energy of the proton is larger than about a few GeV [21. Similar mass distributions of medium mass fragments have also been obtained in heavy ion induced collision, provided the bombarding energy is large enough. The mass distribution of the fragments follows also a power law but the exponent t takes values in a broad range of about 2-4 [41. The interpretation of multifragmentation in terms of a gas-liquid phase transition near the critical point is very appealing. However, we know from macroscopic systems that it is rather difficult to observe a system near a critical point and a small change of the experimental conditions moves usually the system away from the critical point. Therefore, it is hard to believe that one always observes a gas-liquid phase transition near the critical point over a very large bombarding energy domain and for a broad range of projectiles. As soon as one is not at the critical point, it is predicted that the mass distribution follows an exponential rather than a power law distribution. It is also difficult to explain the production of several medium mass fragments as a conventional evaporation process. Indeed, the probability of a simultaneous emission of heavy clusters is very small except at temperatures which are certainly too high compared to the amount of excitation energy which has really been really deposited in the nucleus [Sl. In this contribution we review the problem of multifragmentationof nuclei with a special emphasis on this process when it is induced in heavy ion collisions. gather than beeing exhaustive, we rather insist on the physics of the problem, the ideas which are involved, and the experimental problems which one still has to solve to have a better understanding of this process. In section 2 we briefly recall the pieces of experimental information that one has concerning multifragmentation of nuclei. In section 3

we show that

multifragmentation arises if low density nuclear systems are formed. In section 3 we try to classify and review some of the theoretical approaches to multifragmentation. Finally, we shall conclude by a short summary of the contribution and discuss which experimental results would be needed to have a better understanding of multifragmentation.

C. NgC / Multifragmentation

2.

of nuclei

235~

EXPERIMENTAL INDICATIONS ON MULTIFRAGMENTATION So

far there are only indirect indications about multifragmentation of

nuclei. They belong to different categories which we shall briefly examine.

2.1

Emulsion experiments

The most direct way to see that nuclei break up into several pieces is provided by the emulsion experiments where now a lot of data are available for many projectiles (protons and heavy ions) over a broad range of energy [6,71. Above a certain bombarding energy threshold, which depends upon the system, one observes several tracks associated to medium mass fragments. The energy, atomic number and the emission angles of the particles can be estimated with a reasonnable accuracy provided their atomic number is greater than 2. However, it is not possible to detect neutrons by this method. In fig.1 we display, as an illustration, the result of such an emulsion experiment from Jakobsson et al

[61

for a carbon beam of 852 MeV. For this particular event, no big residue

is present indicating a complete break up of the system. The emulsion data show that the system breaks up but it does not say if the fragments are emitted sequentially or simultaneously.

Projectile

Fig. Schematic

2.2

picture

of

an emulsion

data

1

obtained

by Jakobsson

et

al

161.

Electronic experiments

It is very difficult to make an exclusive measurement of all the fragments emitted in a multifragmentation process because many fragments and light particles are produced with a rather small kinetic energy in the emitting system. A full detection of a multifragmentation event requires a tracking detector with a large solid angle, a good atomic number and mass resolution and a small energy threshold. Some attempts have been made to show that more than two medium mass fragments are

emitted in some cases but a

full

measurement of all the characteristics of the products of the reaction is for the moment not yet available [81. So far the best results obtained about

C. Ng; / Multifragmentation of nuclei

236~

multifragmentation have been measured at high energies with the plastic ball t91 and interesting results have been obtained. 2.3

Disappearance

of incomplete

fusion

At bombarding energies just above the Coulomb barrier it has been found, in central collisions, that two nuclei can merge in a single system provided they are not too heavy IlOl. This process is called complete fusion because all the nucleons of the projectile and of the target remain in the composite system. As the bombarding energy increases, fusion becomes incomplete because only part of the projectile fuses with part of the target. Before the fusion process takes place, prompt particles are emitted which carry out linear, angular momentum as

energy. The fusion cross section, "r,

well as

is

experimentally defined as :

o-=0-

F

where (r

ER

and (r

flS

are

+#

ER

flS

the evaporation residues and

sections, respectively. Consequently, one incomplete fused

system

but

only

its

does not normal

the fission cross

detect directly the

de-excitation products,

evaporation residues and/or fission fragments. It has been found that the incomplete fusion cross section, defined by eq.(ll, decreases at intermediate bombarding energies and may vanish completely till. The bombarding energy at which incomplete fusion disappears depends critically upon the mass asymmetry of the system. For instance, for the Ar+Au or U systems 9

vanishes around

40-50 MeV/u [121 while it is different from zero for the C+AU and U systems at 86 MeV/u [131. The fact that the fusion cross section disappears does not mean that one has not formed a fused system but may indicate that this system de-excites differently than by light particle emission and/or fission. The open question is to know where goes the cross section which is expected to lead to incomplete fusion. The overall experimental data suggests that the fused

system,

which

is

highly

excited,

very

likely

undergoes

multifragmentation. If it is so, it means that the threshold at which incomplete fusion disappears should be the threshold where multifragmentation appears. Consequently, it is interesting to estimate the excitation energy above which this occurs. The experimental results have been analysed with reasonnable assumptions and

it has been found that the critical energy

deposition above which incomplete fusion disappears is about 3-6 MeV/u, depending on the system under consideration. This is schematically shown in fig.2 where, from the compilation of the data relative on linear momentum measurements, one has plotted the region where the transition occurs as a

C. Ng6 / Mulrifragmentation ofnuclei

237~

function of the mass of the compound system (shaded area). For heavy fused nuclei, one observes that the energy necessary to break up the system is smaller than for lighter fused nuclei.

0.6

I

0

I

I

I

I

50

100

150

200

1 250 AM

Fig.2 The critical excitation energy per nucleon E* above which multifragmentation takes place, divided by the binding energy per nucleon B, is plotted as a function of the mass of the nucleus. The two curves corresponds to different initial compressional energies per nucleon t141. The shaded area corresponds to the region where incomplete fusion disappears as

it can be deduced

experimentally Ill]. From ref.Ll51.

2.4

Inclusive mass distributions of medium mass fragments

The interpretation that multifragmentation could be a critical phenomenon has been triggered by the fact that the inclusive mass distribution of the final products obtained in proton induced reactions follows a power law dependence. A similar behaviour has been also observed with heavy ions. A compilation of several results obtained in heavy ions and proton induced reactions leads to power laws characterized by an exponent t which vary in the range 2-4 [41. It is interesting to note that the power law dependence of the inclusive mass or atomic number distribution seems not to be fulfilled in heavy ion collisions at very high bombarding energies. This is illustrated in fig.3 for the I60 + lg7Au system at 200 GeV/u measured by Berthier et al at CEFW 1161. The same measurement performed at 60 GeV/u with the same experimental set up is also shown. The full line corresponds to the power law 2-2."4which fits

C. NgG / Multifragmentation of nuclei

238~

well the proton data results 1171. The dashed curve corresponds to a fit to the heavy ion data. If the proton and heavy ion distributions have the same power law dependence at 60 GeV/u, one sees that this is no longer the case at 200 GeV/u. This results is interesting because it differs from what has been observed so fax. In the proton induced reactions it should be noted that many impact parameters may contribute to the multifragmentationprocess while this is no longer the case in the heavy ion collision situation (central collisions lead to a complete destruction of the target while peripheral collisions do not deposit enough excitation energy).

z Fig.3 Atomic

2.5

number

distribution

obtained

by Berthier

et

al

t161

at CERN.

Conclusion

All the experimental indications quoted above show that multifragmentation has not been studied very well in details. One gets a general trend that nuclei break up into several piece above a critical bombarding energy which depends on the mass asymmetry of the initial system, but more exclusive data are needed in order to draw deeper conclusions. A sensible question is to know whether the mechanism which is responsible for nuclei disassembly is a normal evaporation process [181, as we know from lower excitation energies, or whether it proceeds differently. Two remarks may suggest that the second solution is probably the right one :

l-

In order to get a mass distribution of the medium mass fragments which

follows the experimental one, one needs to have very large temperatures, i.e. very large excitation energies [Sl. In heavy ion induced reactions at very

C. iVg6 / Mulfijragmenfation of nuclei

239~

large bombarding energies it is difficult to evaluate the amount of the initial available energy which is transformed into excitation. This is easier to deduce at lower bombarding energies and one always infers energies of the order of a few MeV per nucleon. Galin et al have shown, for instance, on the system that the low energy neutron multiplicity saturates above a certain bombarding energy [lSl. This experimental fact means that the excitation energy that a nucleus can sustain cannot be larger than about 4-6 MeV per nucleon. It probably indicates that above this threshold nuclei de-excite differently, by multifragmentation.With such low excitation energy values one is unable to describe a large production of medium mass fragments within a conventional evaporation picture. However, one should stress that, below this threshold, clusters are produced by conventional evaporation. This has been carefully checked, for instance, by Charity et al [ZOI. 2- If conventional evaporation would be responsible of the production of medium mass fragments, one expects that they are produced sequentially. At high excitation energy, the emission time is reduced very much and particles can be emitted with a time difference of the order or smaller than 10-22s. In this case it is clear that the evaporation process is influenced by the emitted particles which have been emitted previously but are still in the interaction region. The conventional evaporation description does not take into account these effets which are certainly important. Therefore, they should be studied in order to discuss quantitatively the emission of clusters by very excited nuclei.

3.

Why

do

nuclei break up?

Experimentally, one observes that nuclei break up if one can deposit a sufficient amount of excitation energy in the system. However, we believe that this is not really the fundamental reason why nuclei break up into several pieces and we shall try to convince here the reader of this point of view. In the nuclear medium there are two kinds of forces, the long range repulsive Coulomb forces and the short range attractive nuclear forces (these latter are

repulsive at

small distances but

we

shall

not

need

this

characteristics in the discussion). At normal densities, one can consider, to a good approximation, that the nucleus is formed of a homogeneous fluid of nuclear matter governed by a mean field created by the whole set of nucleons. In this context, one can, to a good approximation, describe this system in terms of a local average density. In this situation the nucleon density varies smoothly from one point to the other. Now, let us suppose that, by some means, one decreases very much the average density of the system. Then, it is clear that, as we move in the medium, we shall have regions where there are

C. Ng6 / Mul~ifragmentation

240~

of nuclei

nucleons, i.e. the nuclear density is large, and regions where there are no nucleons, i.e. the nuclear density is zero. If such a configuration is formed, one easily imagine that it will not be very accurate to describe the properties of this system in terms of an average density, i.e. in terms of a mean field. In the first situation one says that the fluctuation of the mean field are small while

in the second case they are

large because the

correlations between the nucleons will become of outmost importance. In order to illustrate a little bit more the importance of these correlations let us consider a volume V in which we put A nucleons, neutrons and protons (fig.41 1211.

Fig.4 Schematic

presentation

clusterisation 1OW

of

density.From

of

the

nuclei ref.

at

1211.

If V is of the order of the volume of the nucleus in its ground state (fig.$a), each nucleon interacts by nuclear forces with all

its close

neighbours. In this situation the physics of the system is dominated by the mean

field

created by

the

whole set

of

nucleons and

the

associated

correlations are small. A nuclear signal can easily propagate through the whole system. If the volume V becomes bigger as in fig.4b. each nucleon no longer interacts with all its neighbours by means of nuclear forces because the nuclear forces are of short range. Clusters can be formed. The physics of each cluster is of course dominated by its own mean field but the mean field of a particular cluster cannot describe properly the physical properties of

C. Ng6 / Multifragmentation

of nuclei

241~

the whole system. If such a configuration is created, it will be very unstable because of the Coulomb forces which are repulsive. Consequently, the system is expected to break up into pieces. The above considerations are based on

very general arguments and should

apply to nuclear systems at low densities. The problem is to form such species. This is possible in heavy ion collisions, as well as in proton induced reactions, above a certain bombarding energy threshold. In heavy ion collisions one expects the following to happens. In a first phase the projectile and the target merge into a non equilibrium system. For not too large bombarding energies this system contains a lot of excitation energy which is not shared among all degrees of freedom. Several things will happen in order to sustain this energy. Prompt particles are emitted to remove part

of

the

available energy

while

two

body

collisions

lead

to

a

thermalisation of the system. Furthermore, part of the initial energy is stored in the compression mode of the system. After a time, which one can estimate to be of the order lO_"s from the point where the projectile and the target have merged, one is left with a system which has almost reached thermal equilibrium and

is compressed. This system expands, very

likely almost

isentropically [221, and cools down. The larger the compression and the thermal excitation, the larger the expansion. Consequently, it is easy to imagine that, above a certain bombarding energy threshold, one reaches a point where the mean field fluctuations become large and where the system breaks up into clusters. Protons

induced reactions may

proceed a

bit

differently since

the

compression of the system, which is essential to lead to an expansion, will be much smaller than in heavy ion collisions. Therefore, protons of relatively high bombarding energy (a few GeV) are required to trigger multifragmentation. At these energies the incident proton cascade through the target nucleus. This leads to a shower of secondary particles which may either leave the target nucleus, create mesons, or thermalize in the medium. In the end, one is left with a thermally excited system which has a lower density due to the emission of particles. Consequently, this system may break up. Because of this lack of compression, the bombarding energy required in proton induced reaction in order to have multifragmentation is larger than the total bombarding energy needed in heavy ion collisions. In conclusion, it is very likely that it is because one forms a nuclear medium at low density that one gets multifragmentation.Since these conditions cannot be made directly, one has to use heavy ion collisions or proton induced reactions to create a compressed and thermal excited system which subsequently expands and eventually reaches the point of disassembly.

242~

C. NgB / Multifragmentation of nuclei

4. THEORETICAL APPROACHES A

lot

of

work

has

been

and

is

devoted to

the

understanding of

multifragmentationof nuclei. This problem is difficult because one has to go beyond

the

conventional mean

field

approaches and

take

care

of

the

correlations between the nucleons. In this section, we shall only briefly describe the basic ideas of these approaches and present some results which may be relevant experimentally. We had to make a choice among the vast litterature and apologize for leaving off several important aspects and works on the subject.

4.1

Mean field dynamical approaches

It is now possible to rather well simulate what happens in a central collision of two heavy ions within the framework of mean field theories. For instance, the models based on the Boltzmann equation (called also Landau Vlasov equation with collision terms or Uehling Uhlenbeck equation in the heavy ion litteraturel has provided a useful framework to get a realistic picture of what might go on during the first and second phase of the collision [23,241. However, they require a huge amount of computational time which prevent them to be applied systematically to heavy ion collisions. One of the weak point of these approaches is that they are unable to provide a proper description of

the

last stage

of

the

collision, in

the

case

where

multifragmentationtakes places, because they neglect the fluctuations of the mean field. Even if the system does not fragment, these approaches cannot describe the formation of clusters which may be emitted by conventional evaporation or preequilibrium emission. Mean field approaches, applied to the description of infinite nuclear matter, show that the equation of state of nuclear matter is very similar to the one of a Van der Waals gas. Consequently, a phase transition is expected to occur at

low densities [25]. However, a

nucleus is finite and not

necessarily formed in an equilibrium state which make any extrapolation of nuclear matter calculations to finite nuclei very difficult.

4.2

Statistical models

Most of the models devoted to multifragmentation are static models where one starts with a

spherical nucleus in thermal equilibrium. Within the

framework of statistical mechanics one can study whether this system may break up into several pieces. In this approach, one does not consider the whole dynamical evolution of the system from its initial compressed and heated up state until the moment where it breaks up into fragments. These models concentrate on the disassembly stage only, as it takes place when the system

243~

C. Ng6 / Multifragmentarion of nuclei

reaches the break up volume and undergoes multifragmentation. In order to fragment the nucleus has to have a volume which is substantially larger than its initial one, i.e. it means that the system has expanded. If one uses equilibrium statistical mechanics to describe the nucleus at the breaking point one has to take care of the fact that it is a finite isolated system which contains a small number of particles and that long range forces are present in the medium. A proper description of this situation requires the use of the microcanonical ensemble [26,271. The practical use of this ensemble is however rather tedious and the first attempts were made in the framework of the grand canonical ensemble (mean number of particles and mean energy fixed) [281 or using the canonical ensemble (mean energy fixed) [291. E' E*IA IMeW

Fig.5 Microcanonical calculation of the

temperature

function of energy nucleus.

E*

T

as

a

the excitation for

From

the

131Xe

Gross

and

Massman L271.

In order to illustrate the statistical approaches we have chosen to show two examples from Gross et al

[271. In fig.5 the excitation energy per

nucleon, E*/A, is displayed as a function of the temperature T for the 131Xe nucleus. The points are the results of the microcanonical calculation while the full curve represents just the Bethe formula. For E*/A around 3 MeV and T around 5 MeV, one observes a change in the evolution which could be attributed to a phase transition in a finite system. Another transition, but not so

*

apparent, can be seen at larger E /A (a 5 MeV) and T (- 6 meV) values. A similar result has also been obtained by Bondorf et al using the canonical ensemble [293.

It

is interesting to look at the evolution of the number of fragments

produced in the de-excitation of a nucleus as a function of its excitation * energy E , This is illustrated in fig.6 for the 131Xe nucleus by the microcanonical calculation of Gross and Massmann [271. The evaporation like component (E) where there is one fragment of mass A~10 decreases as E* increases. First because binary fission (F) increases and second because three or more fragments with a mass A210 are produced. This latter component (Cl, refered to as cracking, corresponds in fact to a multifragmentation of the system.

600

800

1000

1200

E*lMeVf

Fig.6 Relative probability of evaporation (E), binary fission (F) and cracking CC) as a function of the excitation energy of the 13'Xe nucleus fmicrocanonicaf calculation). From Gross and Massmann I271.

As far as statistical approaches are concerned, instead of assuming that all the available microsates of the system are equiprobable (microcanonical ensemble), one may assume that it is the different break up partitions of the initial system which are equiprobable. This was assumed by Aichelin and Hiifner [301 who tried to describe the multifragmentationof nuclei in a similar way as the shattering of glass. In this approach, the assumption of partitions equiprobability implies that the available microstates are not equiprobables. Nevertheless, the

computed mass distributions of

the multifragmentation

products also agrees with the experimental measurements.

C. Ng8 / Multifragmentation

Percolations

approaches

multifragmentation

1311

of nuclei.

multifragmentation

has

that

Percolation

of nuclei

concepts

site-bond approaches

might

than

be

the

or

in phase

is to make a connection

physical

analysis

quantities.

Some

between

attempts

has

description

of the moments in emulsion better

to nuclei

space

245~

qualitative

gas-liquid

have been applied

percolation)

a

events observed

percolation

multifragmentation

provide

A careful

the different shown

also

of nuclei

The

describe

transition

approach.

main

(site,

problem

the percolation done

[321

to

on a lattice

[311.

been

of the mass of

[7l by Campi

suited

phase

of

bond or of

parameters

in

this

these

and the

direction

in

ref.[14,331.

4.3 Stability

of hot and compressed

The dynamical

evolution

issue in the disassembly what

happens

to

investigated

a

of nuclei.

hot

and

by Nemeth

et al

statistical

equilibrium

and

constraints

are

which

be representative

might

nucleus Fermi

of

expands model

This

a

performed

for heavy

are

very

lattice. used

where

into several

pieces

fragments

means

by

this

It is found

phase

of

the

This

it means

model.

is in These

features

and compressed

be

isentropic,

to

cascade energies

[221. At each

model

on

The

most

approach one can breaks

distribution important

field

a cubic

However,

the nucleus

mass

is

calculations

of the mean

suited.

Then,

Thomas

which

that the mean field

the primary

percolation

many

The hot

percolation

is perfectly

up

of the

results

of

: energy

is illustrated

associated

regards

is no longer the case.

that compressional

energy.

diagram

small, process

been

nucleus

by a time dependent

to

site-bond

to study has

evolution..

study

of the fluctuations

and one can obtain

this model are the following

than thermal

in

its

situation.

assumed

important

question

to

at higher bombarding

a 3-dimensional

the expansion

meet situations

is

the magnitude

As long as they remain

to describe

allow

has been described

ion collisions

using

during

but

of the real

This

that the initial

spherical

drastic

approximation

of the expansion,

evaluated

remains

a very

it is interesting

nucleus.

[14,331 assuming

rather

is probably

In this respect

expansion

good

nucleus

compressed

and this process

probably

stage

course

[341.

nuclei

of an excited

to

the

_

Pb

is more efficient for example

nucleus

as

a

to break up nuclei

in fig.7 which function

of

shows

the

the

thermal

L

excitation per nucleon the system This

energy

about

case

compressed

nucleon

c:. If the system

quantity

extreme

per

the

5.5 MeV/u

drops where

down

the

of the compressional

is not compressed

of thermal

to about

there

system,

E T and

is

no

lower

thermal the

in order

of compressional excitation.

amount

energy

(c:=Ol one needs to provide

excitation

1 MeV/u

excitation

that

it breaks

excitation

Therefore,

of excitation

energy

to up.

in the

the

more

needed

to

246~

C. NgG / Multifragmentationof nuclei

break it up. This effect can be easily understood if one remembers that thermal excitation corresponds to

desorganized energy while compression

corresponds to a coherent energy in the mode of instability. Another way to look at the results is shown in fig.2 which shows the amount of total excitation energy per nucleon c*/B (compressionalplus thermal) divided by the binding energy per nucleon B required to break up a nucleus as a function of the mass number for nuclei along the beta stability line. If there is no compression one needs to provide about 70% of the binding energy in order to break up the nucleus. This amount decreases to about 40% in case where 0.5 MeV/u are initially put in compressional energy. The shaded zone indicates the region

where

one

has

experimentally estimated that

incomplete fusion

disappears. If this model is correct, it means that heavier systems (which are formed experimentally with heavier projectile) are more compressed than light ones (which were formed with lighter projectiles). This looks reasonnable.

*u”

~u~tifmynentation

0

1

3

2 &;

4

5

(MeV)

Fig. 7 Multifragmentation and normal de-excitation region9 of the “‘Pb function of the thermal excitation per nucleon cT and of the excitation

energy

per nucleon

* cc.

nucleus as a compressional

From ref. f211.

More fundamental theories can be used to study nuclei disassembly as the one proposed in ref.[351 which follows the evolution of a linear combination of Slater determinants. However, due to the complexity of the approach, calculations have only been performed in a two dimensional world which is very likely to be different from the real one.

C. Ng3 / Multifragmentation

Towards better

4.4

A step the

further

break

describes

of up.

the

target. while

model

the projectile

nucleons

are

leads

initial

order

proposed

of

where one tries dynamical

the by

fused

Miller

Harp,

cooling

excitation

lO_” s the remaining

energy

Berne

down of

only,

In

well

This

flash

of

the system and only

into

system has reached

I381.

heat.

thermal

After

this of the place prompt

part

a time of

equilibrium

its

simple

the system takes

emission).

goes

the

in the potential of

and

systems

means of

and

a thermalization

(preequilibrium

to a substantial

by

to describe

expansion

to assymmetric

system

is assumed to be trapped

emitted

available

its

which applies

approach, of

in ref.[36,371 system,

Due to two body collisions

particles the

This

fused

equilibration

preequilibrium approach,

the

241~

descriptions

has been achieved

equilibration

possible

dynamical

of nuclei

of the

and will

z 5 20

50

100

E(MeV/u) 1.c

Fig.8 multifrag.

Incomplete e

e-

and

multifragmentationexcitation

-K 0

fusion

functions

for

different systems calcu-

200

lated by Cerruti et al

400 C ( MeVlu)

1371. The of

multiplicity

multifragmentation

fragments of mass larger than 4

is also shown.

From Cerruti et al [37J.

E 1MeV/u) start as

to expand. described

in

Its

stability

section

4.3.

towards This

multifragmentation

picture,

initially

can then be studied

developped

for

head on

C. Ng6 / M&fragmentation

248~

of nuclei

collision can be extended to non zero impact parameters as proposed by Cerruti et al [371. An important effect of the small number of particles involved in nuclear

collisions is

that

the

transition from

incomplete fusion

to

multifragmentation will be very smooth. A given impact parameter at a given energy leads to a certain probability of fusion and multifragmentation which can be evaluated with simple hypotheses. Therefore, it is possible to obtain the excitation functions for incomplete fusion and multifragmentation of various systems. The main result of ref.[36,371 is that the transition region from incomplete fusion to multifragmentation critically depends upon the initial mass assymmetry of the system. This is illustrated for example on the C, Ne and Ar + Au systems in fig.8. In addition to the excitation function, the multiplicity of fragments of mass larger than 4 is also indicated. One sees, for the Ar+Au system, that the transition occurs around 40-50 MeV/u while it occurs around 200-300 MeV/u for the C+Au system. These results are in agreement with the experimental data for the Ar+Au system 1121 but no experimental data is not yet available for the C+Au system between 200 and 300 MeV/u.

4.5

Full

dynamical

theories

One of the goals of theoretical approaches to the multifragmentation of nuclei induced by heavy ion collisions is to build a model which is able to treat on the same footing all the different phases of the reaction. In order to do that it should be a dynamical approach which includes in some way the many body correlations which are resposible for clusterization.First steps in this direction have already been made within the framework of molecular dynamics in which one takes into account of some quantum features like the Pauli principle and the Pauli blocking for instance [391. The first results of these approaches are very appealing but the computer codes need a very large amount of time which does not allow to use them in systematic studies.

5.

CONCLUSION

In

this

contribution we

have

reviewed the

pieces

of

experimental

information about multifragmentationof nuclei. We have seen that most of the experimental indications are very indirect and that there is a real need for more exclusive experiments. We have briefly described the main theoretical approaches to the problem. The statistical approaches allows to rather well describe the phenomena but they say nothing about the formation of the excited system they consider. One step further is made by considering the time evolution of very excited nuclei and their possible disassemby. Finally, some approaches try to include the whole dynamics but they are really at the very

C. NgG / Multifragmentation

of nuclei

249~

first stage. Before ending this contribution it may be worthwhile to try to point out some of the experimental data which are needed in the near future to test the models developped so far. One needs urgently to fully measure all the characteristics of the fragments produced in the multifragmentation process. Since their kinetic energy (which is about the Coulomb repulsion between the fragments) is small in the emitting system, the best way to do it is probably to use a heavy projectile on a light target (inverse kinematics) and look at the multifragmentation products which are focussed at small angles and have a high velocity in the laboratory system. It is also important to measure quantitatively the multifragmentationas well as the incomplete fusion cross sections for many systems at different bombarding energies and compare these quantities with the total reaction cross section. A knowledge of the impact parameters leading to multifragmentation would also be very useful. These quantities, which are the simplest one can imagine to measure, would put constraints on the different models existing on the market. These measurements will, however, need a large experimental effort but they are worthwhile to be done. Finally, one may conclude by saying that multifragmentation is probably not a dream but an exciting reality which is difficult to study.

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