Statistical simulation of the break-up of highly excited nuclei

Nuclear Physics A475 (1987) 663-686 North-Holland, Amsterdam

STATISTICAL SIMULATION OF THE BREAK-UP OF HIGHLY EXCITED NUCLEI

AS.

ILJINOV’,

I.N.

MISHLJSTIN’,

The Niels Bohr Institute,

J.P.

Unioersity

BONDORF,

R. DONANGELO’

of Copenhagen,

DK-2100

Copenhagen

and

K. SNEPPEN

Q, Denmark

Received 29 March 1987 (Revised 24 June 1987)

Abstract: We employ a model of multifragmentation of highly excited nuclei based on the statistical approach and a liquid-drop description of hot fragments. We consider the principal equations of this model and the methods of solving them in the canonical and macrocanonica1 approximations. We then calculate the multif~gment break-up of nuclei taking into account the de-excitation of hot fragments both by Fermi break-up and by evaporation of clusters. Finally, we analyze the main features of the multifragment break-up and compare them with those of the evaporation decay of highly excited nuclei.

1. Introduction In recent years the means to study nuclear matter under extreme conditions have become much more numerous. There appeared a new generation of accelerators which produce high-intensity beams of protons, pions, antiprotons, and heavy ions of intermediate and high energy. The deep inelastic interaction of fast hadrons and heavy ions with nuclei may result in the formation of highly excited nuciear matter with excitation energy -10 MeV/nucleon. Today the properties of such hot nuclear systems and the mechanisms of their decay have become an object of intense study, in particular in connection

with the problem

of determining

the maximum

energy

E,, a nucleus

can

absorb as a whole. In the quite well studied range of excitation energies E,,- 1 MeV/nucleon the heating of a nucleus has practically no effect on its global properties, while the main mechanism of its decay is successive emission of particles (evaporation) by the ’ Permanent address: Moscow, USSR. ’ Permanent address: ’ Permanent address: Janeiro. Bras&

Institute

for Nuclear

Research,

I.V. Kurchatov Institute of Atomic Institute de Fisica, Universidade

0.3759474/87/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

Academy Energy, Federal

of Sciences

of the USSR,

Moscow, USSR. do Rio de Janeiro,

21944

117312

Rio de

664

A.S. Botvina

et al. / Statistid

sirnulatinn

compound nucleus. On the other hand, it is clear that the existence of a relatively long lived compound nucleus is impossible when the excitation energy is close to its total binding energy (E,,, - 8 MeV/nucleon). In this case the evaporation mechanism of de-excitation should give way to an explosion-like process which leads to the total disintegration of the nucleus. The high excitation of a nucleus leads to a drastic change of its properties. The first systematic studies of the temperature dependence of global properties of nuclei ‘J ) and the Hartree-Fock have been done using the Thomas-Fermi ‘) methods. The most important result was the prediction of a liquid-gas phase transition in nuclear matter. This first order phase transition is caused by the attractive nucleon-nucleon interaction, which leads to clusterization of nucleons in nuclear matter when the density p becomes smaller than the normal nuclear matter density p0 while the temperature T falls below the critical value T,. In the recent years many aspects of the liquid-gas transition in nuclei, including finite particle number effects, surface and Coulomb contributions, and the various dynamical effects, have been studied in the framework of various models ‘--I4), However, different models give quite different predictions for the main characteristics of this new phenomenon. For instance, the values of the critical temperature T, below which the liquid and the gaseous phases can co-exist range from 10 to 20 MeV. The considerable uncertainty in the theoretical predictions concerning the features of the phase transition in highly excited nuclear systems calls for a detailed experimental study of this phenomenon. The specific effect that can serve as a signal of such a transition is the multiple production of nuclear fragments (multifragmentation) with the break-up of a hot nuclear system. Both the total disintegration of nuclei and the multiple production of fragments have been observed many years ‘5*‘h) in nuclear reactions induced by intermediate and high energy hadrons. in ago heavy ion reactions such phenomena have been discovered only recently I’). Extensive experimental information on fragment production in hadron-nucleus and nucleus-nucleus interaction (mostly the inclusive cross sections) is available now I”). In the last few years a wide variety of models has been proposed to describe nuclear multifragmentation. One group of such models is based on very general probability concepts. For instance, for determining the most probable state of a information (maximum entropy) fragmenting system in refs. ‘9.20) the minimum theory ‘3) was used. principle was employed, while in refs. 2’.22) the percolation Another approach considers the direct cleavage 24) or shattering I”) of the target nucleus by an incident projectile (cold fragmentation). In this last model the combinatorial relations for probabilities of different partitions are used. All these models employ very simple assumptions concerning the formation mechanism and properties of a fragmenting system which are very far from a realistic model of a nucleus. Another group of models (the macroscopic models) is based on the analogy between fragmentation and the macroscopic condensation in a liquid-gas system. For instance, the model of refs. 2h*27)employs the formulae for yields of molecular

A.S. Bot~ina

clusters

in the limit of an infinite

et al. / Siati.dcnl

system

obtained

665

simulation

in ref. “) in the framework

the statistical macrocanonical approach. In the very interesting mentation of a hot drop containing a finite number of particles

of

study 79) the fragis described using

the classical molecular dynamics approach. Finally, the most extensive and, in our opinion, the most suitable for solving the problem of nuclear fragmentation is the group of microscopic statistical models. Among them one can distinguish a group of equilibrium statistical models based on the assumption that before disintegrating, a hot nuclear system gets into thermodynamic equilibrium with respect to all or some of its degrees of freedom 3”-3hf. These models differ in the statistical ensembles considered (macrocanonical, canonical, or microcanonical) as well as in the methods of calculation and in the description of individual fragments. Another group of more precise microscopic models 37) use the methods of nonequilibrium statistics and attempt to make an explicit account of the dynamics of separation of the liquid and gaseous phases in nuclear matter. Such a variety of models reflects our lack of understanding of the nature of so complicated a phenomenon as the multifragment decay of nuclei, Multifragmentation can be pictured as a process occurring in several stages 34), the most important among which are (i) the formation of a hot intermediate nuclear system; (ii) the formation of fragments and disassembling of the system into il~dividual fragments; and (ii) the de-excitation of primary fragments. Even if one leaves aside the stage at which the hot nuclear system is prepared in a given nuclear reaction one still has to go a long way in developing any of the microscopic statistical models of multifragmentation in order to be able to compare it directly with experiment. First of all, since multifragmentation is essentially an exclusive process, it is necessary that one formulates the statistical simulation of the break-up process in such a way that the characteristics of all fragments can be calculated on an event-by-event basis. Furthermore, one should take into account the de-excitation of primary fragments. The main aim of the present work is to develop the statistical model of multifragmentation formulated in refs. 3’.35) along these lines. In the next section we present the principal equations of this model and the methods of solving them within the canonical and macrocanonical approximations. Special attention is paid to the application of the Monte Carlo method to the calculation of multifragment decay taking into account fragment de-excitation. in sect. 3 we present the results of our calculations and analyze the characteristics of multifragmentation processes, comparing them with those of the evaporation cascade in order to find those characteristics which are most sensitive to the specific type of decay mechanism. In sect. 4 we give a brief conclusion and outline the prospects of further development of our study.

2. The statistical

model of multifragment

The formulation, substantiation and statistical model of multifragmentation,

disassembly

of hot nuclei

detailed description of this microscopic within which the fragments are regarded

AS.

666

as liquid

drops,

of the principal

2.1. THE

MAIN

Botuina et al. / Sfatisticai

in refs. 34*35),so here we give only a brief summary

are presented tenets

and relations

RELATIONS

simulation

OF THE

of this model.

MODEL

We assume that as a result of some dynamical process a portion of excited nuclear matter was formed, and that this system may be characterized by its excitation A o, and total charge Z,. If the excitation energy is energy E,, number of n&eons su~ciently high, the nuclear forces can no longer keep the nucleons together, and the system will begin to expand. Eventually the nuclear density falls below a certain critical value and the system will start breaking down into pieces. We assume that there is an intensive mass, charge and energy exchange between these prefragments, so that by the time they become isolated fragments, there is a partial or complete thermodynamic equilibrium. With further expansion of the system each fragment evolves individually, either emitting nucleons or experiencing secondary break-up. A fragmentation event is characterized by a partition vector .f= {N,,} where NAz denotes the number of fragments of the kind (A, Z), i.e. with A nucleons and charge Z. In order to find the statistical weight associated to a given partition j; we assume equipa~ition of energy over all degrees of freedom. We consider the fragments as Boltzmann particles moving in a volume V,.r. This free volume can be parametrized as Vr,= KV~, where K is a model parameter and V,, is the volume of the system corresponding to normal nuclear matter density po. The individual properties of fragments are described in the liquid-drop approximation 34), so that the free energy of a fragment may be presented as FAz = Fj;, + F;, + F;iV,”+ F;= t Faz ,

(1)

where the terms in the r.h.s. are respectively the volume, surface, symmetry, Coulomb interaction and translational motion contributions. For these contributions we have 3‘%=): F;,

= (- W, - T2/ q,)A ,

(2a) (2b)

F:;

= y(A -2Z)2/A,

FzZ=- :

W)

gqI-(l+K)-‘:3],

(2d)

AZ

+ Tin (N,,!)/N,,. Here W,, = 16 MeV is the binding energy of nuclear matter; eO is the inverse nuclear level density (for the Fermi gas model Q= 16 MeV); PC= 18 MeV is the surface

667

A.S. Botvina ef al. / Staricticai .~j~ui~ii~n

tension

of a cold nucleus.

the symmetry

energy

Tc = 16 MeV is the critical

coefficient

temperature;

y = 2.5 MeV is

formula;

RAZ = roA”’ is

in the Bethe-Weizsacker

the radius of a fragment corresponding to normal nuclear matter density po, i.e. ri,= 1.17 fm; A,= (2d2/mNT)” IS . the thermal wave length; mN is the nucleon mass; and g,, is the spin degeneracy the energy conservation equation

factor.

The temperature

T is determined

3 Z.‘e2 f C 6, + &round = - __ 5 R n,z

where Egroundis the ground separated volume

the Coulomb

state

energy

(3) of the (A,, Z,,) nucleus

energy of a homogeneous

v=

by

charge distribution

and

we have

in the break-up

(4)

v,l(l+K).

Evidently, the liquid-drop approach is not very good at describing light fragments. In principle, within this model it is possible to single out the nucleons and the light clusters with A = 2, 3 or 4, as it was done in ref. 34). In this case all these fragments are regarded as elementary particles with no internal excited states. However, this procedure is not totally correct since, owing to the inAuence of hot surrounding matter, the properties of light clusters are quite different from what they are in vacuum “1. With this in mind, in our present study we will confine ourselves to describing ail fragments other than nucleons in the liquid-drop approximation. Since those light nuclei either have no excited states at all or the energy of their first excited state lies very high (e.g. 4He) we roughly take into account the considerable decrease of the share of energy spent on internal excitation of lightest fragments “) making the parameter F!) dependent of A: ~,,(A)=~t~tl+3/(A-1)1,

A>l.

(5)

We should remark that we have compared the Coulomb energies obtained in the Wigner-Seitz approximation utilized here and in ref. 34) with exact calculations in which the spherical fragments were randomly packed into a spherical volume V,,. Our results have shown that this approximation provides good accuracy. 2.2. THE CANONICAL

APPROXIMATION

In the canonical approximation all the partitions are characterized a heat bath. In this approximation

we consider an ensemble of partitions in which by the same fixed temperature T maintained by the probability W, of a given partition f equals

W, =iexp(----F;./T), where 5 is a normalization

constant

(the partition

<=IT)exp(-F;/T).

(6) sum): (7)

AS. Botvina ef al. / Statistical

668

.simulatiort

An important property of all partitions in the canonical approximation is the strict conservation of the charge and mass of a finite system. For instance, for the partition {NAz} we have

Knowing the probabilities of all partitions for a given ensemble it is easy to calculate all average characteristics. For instance, if some physical. quantity Q (E, S, Naz, etc.) has the value Qr- for the partition.f, its value averaged over the ensemble {f} will be CQ,)=iz;; For example,

for the average multiplicity

and the corresponding

dispersion OAZ

Q,w,.

(9)

of fragments

of a given kind (A, Z) we have

equals =JW~,,HJ%Z)~.

If the quantity Q,- is additive with respect to fragments (i.e. if Qf =Ca,= QAzNAz), its average value is found by simply taking the sum over all the fragments

(Q/j= The multiplicity

of fragments

C

QAZ(NAZ)

with a given number

.

(11)

of nucleons

while the total multiplicity M is defined as M = I,, the average multiplicity of fragments of mass number

A equals

NA,z. It is easy to show that A is (12)

their dispersion

is v~=J(N;)-(N,,)‘,

their charge

is

(13)

and their charge

dispersion

is a; = J(z;)

Our calculations

- (Z/J’.

show that at low temperatures

(15)

( T-C 5 MeV) the main contribution

to the partition sum (7) comes from partitions with a small number of fragments. In this region of temperatures one can get good accuracy by directly looking through all the possible partitions with M between 1 and 5 for A,,- 100. the Iiowever, at higher temperatures (T > S MeV) one can no longer disregard contribution of partitions with a large number of fragments. Because of the very large number of partitions in the ensemble, in practice we select a certain number of partitions. Then for each partition j; one calculates its probability WV according to formulae (6) and (7), and after that, using the formula

/T Wri

( Qr>= $ W,.iQt.<

(16)

one finds the ensemble-average value of the physical quantity Q. The accuracy of such calculations depends on how representative the sample of partitions considered is. The algorithm of selecting the partitions was described in ref. j4). Some modifications to the Monte Carlo sampling procedure, and especially the inclusion of isotopic degrees of freedom are discussed by Sneppen 3J).

2.3. THE

MACROCANONICAL

APPROXIMATION

Let us now consider the grand canonical ensemble of partitions, where it is assumed that there is an infinite number of fragments and that the latter may have any given value of A and Z. Naturally, one should be careful when applying the macrocanonical approximation to a finite nuclear system. On the other hand, within this approach one is able to obtain analytical formulas for the ensemble-average characteristics of the system, which makes the macrocanonica1 approximation more transparent and more convenient than the canonical approximation. Instead of requiring exact conservation of the total mass and charge numbers (constraints (8)), within the macrocanonical approximation one requires only the conservation of their average values:

C A(N,A.z)= A,, , .A,.7

In order to account for the condition (17) one introduces two chemical potentials p and v, which are responsible for conservation of the mass number and of the charge, respectively. After that the multiplicities N,, are considered to be independent.

A.S. Borvina

670

In this approximation,

instead

et al. / Stati.~ricuf simufation

of expression

(6), for the probability

of a partition

we have

so that the grand

partition

sum is

[(

5=~lexp - 5-w

Z: ANAz-u

T

C Znr,,

AZ

AZ

>i

1 (19)

where w,z,Z

=

V, A3”

gAzF

T

--+$-~A-

ew

uZ)

1

(20)

and F$‘> is the free energy of a fragment (A, 2) minus the translational term, i.e. the sum of the contributions (2a)-(2d). Now one can easily obtain all the characteristics of the system. In particular the average multiplicity of fragments is given by (NM) = The inclusive

mass distribution

(iVA)=3A”‘exp

of fragments

(21)

WAZ.

with A > 1 is given by the expression

{-_:[(-w,-~)A+v(~-2~)‘a

T

II (22)

+/3(T)A2~3+~;(l+(l+~)-“3)(Z~)‘A-”3-~A-~(Z~) 0 while the charge distribution Gauss function 35)

of fragments

-

NA(Z) - exp L with an average

(Z -(Z,>)’ 2((T32

A>

1

1 is described

by the

(23)

charge

GA) = and a charge

with a given

,

(4~ + v)A 8y+2(1-(l+~)-““)A*/”

(24)

dispersion

a~=J{z~>-(zA)2-

J$f.

The chemical potentials p and Y are found by substituting of equations (17) and solving it by iteration. The temperature

(2% (21) into the system T is found from the

671

excitation

energy

equation -;F+ 5, + Egrounc, -

2.4. MASS

AND

CHARGE

x (NAz) A.2

DISTRIBUTIONS

OF

F,,-T-/

PRIMARY

3F

.

(26)

FRAGMENTS

The calculations of refs. 34335)s h ow that at temperatures higher than a certain 5-7 MeV the dominant process is that of multifragmentation while below Tiim Timthe evaporation evolution of the compound nucleus dominates. This is illustrated in fig. 1, where we show the temperature dependence of the relative probability of the break-up into a given number of fragments, calculated in the canonical approximation. We see that at low temperatures (T = 2 MeV) the probability of a break-up with large multiplicity M rapidly decreases with increase of M, and the most probable channel, with M = 1, corresponds to the compound nucleus*. At higher temperatures the probability of a large-M break-up becomes greater, and beginning

w

I

A, = 100 2 * = 44

0.7:

0.25

IO

20

M

Fig. 1. Relative probability of a system with A,, = 100 and to = 44 breaking down into M fragments for different values of the temperature T. Curves show the results of calculations in the canonical approximation with parameters K = 1 and a0 = 16. Numbers near the curves indicate the temperature in MeV. l Certainly, this excited compound nucleus is unstable with respect to evaporation of particles. Compared with the characteristic time of explosive processes, the evaporational evolution of the compound nucleus, leading to the increase of M, takes an infinitely long time and therefore is not considered in this section.

612

A.S. Botvina et al. / Statistical .simulation

at T > T,im one observes of multifragmentation. At moderate

the characteristic

temperatures,

in nuclear

many-body systems

break-up

of the system typical

with medium

mass A,,= 100 one

observes “quasi-evaporation”, i.e. the break-down of the system into a large residual nucleus and one or two small fragments. This gives the characteristic U-shaped inclusive mass distribution (NA) (see fig. 2). In the case of multifragmentation at high temperatures the mass spectrum monotonically decreases with increase of A (fig. 2). In heavy nuclear systems with A,= 200, in addition to “quasi-evaporation” one also observes “quasi-fission”, i.e. the break-down of the nucleus into two fragments with approximately equal masses and, possibly, one or two light clusters (fig. 3). With the rise of temperature the “quasi-fission” peak in the mass distribution disappears, and, both in heavy and in medium-weight nuclei, (NA) monotonically decreases with A. It should be noted that the mechanisms of “quasi-evaporation” and “quasi-fission” are different from the ordinary evaporation and fission processes, despite their superficial similarity. In our present study we have also considered the charge distributions of fragments. Both in the canonical and macrocanonical approximations the charge distribution for a mass-A fragment is close to a gaussian form (23) with mean charge (Z,)AZJA,. The corresponding charge dispersion C$ [eqs. (15) and (25)] grows both with the rise of temperature and with the increase of A (see fig. 4).

A0 = 100,

50

Zo= 44

-

T=5MeV

----

T=lOMeV

100

1

A

Fig. 2. Inclusive mass distributions of fragments (N,) produced with disintegration of a system with 2, = 44. The histogram and the smooth curves show respectively the calculations in the canonical and the macrocanonical approximations with parameters K = 1 and q, = 16 MeV. Solid lines, T = 5 MeV; dashed lines, T = 10 MeV. A, = 100 and

A.S. Bofuina er nl. / Stnf;.h~~/ .simulntion

Ao=zll, -T=

Fig. 3. Inclusive

100

Z,=

85 3 MeV

------T

50

673

150

= 5 MeV

200 A

mass distributions

A,,= 211and 2;, = 85. Calculations T = 3 MeV; dashed

of fragments (IV,,) produced with disinte~~tion of a system with were made with parameter values K = 3 and F,, = 16 MeV. Solid lines, lines, T- 5 MeV. Other notations are the same as in fg. 2.

As one can see the predicted inclusive mass distributions (NJ in the canonical and macrocanonical approximations are noticeably different at low temperature T s 5 MeV (figs. 2 and 3). Furthermore, the macrocanonical calculation systematically overestimates the charge dispersion P: within the whole range of temperatures studied (fig. 4). This discrepancy is due to the finite size of the nuclear system, owing to which the multiplicities of fragments are not large (N,,, - l), whereas in the macrocanonical approximation we take the sum over all the values of N,, from 0 to co. As a rule, the experimental mass and charge distributions are analyzed using the fitting expressions obtained in the macrocanonicai approximation (see e.g. ref. ‘“)). However, the incorrectness of these formulae can lead to conflicting results. For instance, there is a certain disagreement between the temperatures obtained by fitting the inclusive isotopic distribution. The width of the isotopic distribution corresponds to a lower temperature than that needed for describing the mass spectrum. This discrepancy is explained by the mentioned overestimation of the charge dispersion in the grand canonical approximation. In the temperature region T > T,i, the canonical and macrocanonical approximations give close results for the inclusive mass distribution. Owing to the smallness of the charge dispersion we conclude that the inclusion of the isotopic degrees of freedom has Iittle effect on the mass spectrum (NJ and other thermodynamical characteristics.

674

A.S. Botvina et al. / Statistical

simulation

/ / / / / /

:

: . . . :. : .. . . ..: :.. ... ..

P

h--l t tf

0.5

.:::

Ao=loo,

......

Z.

. . . . . . . . . . . -t-s

5

MeV

T=IO MeV -------

T=21 MeV

25

50

Fig. 4. Charge dispersion ~5 of fragments of a given kind A for a system wit1 4,) = 100 and Z, = 44. ne, T = 5 MeV; solid Calculations were made with parameter values K = 1 and Q= 16 MeV. Dotted lines, T = 10 MeV; dashed lines, T = 21 MeV. Other notations are the sz le as in fig. 2.

2.5. THE

ENERGY

SPECTRA

OF THE

PRIMARY

FRAGMENTS

Besides the average thermodynamical quantities and the inclusive mass yield one also has to know the energy spectra, the angular distributions and the various correlations between these quantities. For this reason one needs to know, besides the mass and charge numbers, the momentum and the excitation energy of each fragment produced in a given event. Let us then calculate the excitation energy and momentum of fragments for a given partition {NAz}. The internal excitation energy of a fragment (A, 2) is given by its bulk and surface contributions /3(T)-T---/-

W(T)

PO A2”

1 and its average

initial

translational

(27)

energy E&

=;T.

(28)

When the fragments fly away to infinity their total kinetic energy may be approximated by the sum of the energy of their translational motion and the contribution

A.S. Botoina ef al. / Statistical

to the energy

arising

from their Coulomb E:;

Let us distribute the total kinetic 0 occupied by the fragments: J-2=

VM d3p, .

1 .

repulsion

= E;,,+

energy

simulakwt

E&

evenly

675

to .

(29)

throughout

the whole phase space

d”p, (30)

f27rh)3b’

where V is the normalization volume and ~PZ,~.~and Pzt,z are respectively the mass and momentum of the (A, 2) fragment. The Monte Carlo generation of fragment momenta within the available phase space (30) is done using the method presented in detail in ref. “). When using this method one should take into account that the chargeless fragments are not affected by the Coulomb field (i.e. for them I&, = E,). Let us note that in this way we make only a rough account of the Coulomb repulsion of fragments, for the uniform distribution of the energy throughout the whole phase space does not take into account the details of redistribution of the Coulomb energy among fragments at the expansion stage.

2.6.

ME-EXCITATION

OF

PRIMARY

FRAGMENTS

After the break-up and the isolation of excited primary fragments, one should consider the stage of their de-excitation. In general, the excitation energy of primary fragments is smaller than that of the initial nuclear system, since multifragmentation is an endothermal process. At the same time, depending on the initial conditions (E,,, A,) and Z,J, the mass A and the excitation energy Ef of primary fragments may vary within broad limits. The specific values of A and E’ determine the mechanism of primary fragment de-excitation. Below we consider the two most important mechanisms. (a) T&e Fermi ~reffk-~p of fight primary fragments. For light primary fragments (with A =G16) even a relatively small excitation energy may be comparable with their totat binding energy. In this case we assume that the principal mechanism of de-excitation is the explosive decay of the excited nucleus into several separate clusters

and isolated

nucleons.

(This statistical

approach

was first used by Fermi 40)

for describing the multiple production of particles in NN and rrN collisions.) The probability of a nucleus with mass m disintegrating into n clusters masses mi (i = 1, . . . , n) equals 40*4’)

with

where g = nF=, (25, + I) is the spin factor, G = [I,“=, nj !, where “j is the number of identical particles of the jth kind, is the permutation factor, T(X) is the gamma function, and Eki, is the total kinetic energy of all the clusters at the moment of

A.S. Botvina

676

break-up. magnitude

et al. / Statistical

,simulation

The free volume V,, is a model parameter and is of the same order of as the volume of the primary fragment V. If the initial excitation energy

of a primary

fragment

of the (A, Z) kind is E”, then

>

-B,-C

I

tzi,

(32)

where E, is the excitation of the cluster i, and BC is the Coulomb given channel. It can be found in the Wigner-Seitz approximation:

barrier

for the

(33) Since in our present study the multifragment decay is described with the Monte Carlo method, it is natural to use the same method for describing the subsequent stage of Fermi break-up of primary fragments. To this end we calculate in turn the probabilities W (31) for all channels of the primary fragment breaking down into different combinations of cluster. After that, as in the case of multifragment break-up, we calculate the energies and the final momenta of particles produced, taking into account energy and momentum conservation laws, as in sect. 2.5. Here we consider the formation of the light fragments in their low-lying excited states, stable to nucleon emission. For this reason we consider them as individual particles with their own masses and charges (the characteristics of excited states of light nuclei are taken from the compilation 42). (b) Evaporation ofparticlesfrom heavy primary fragments. Heavy fragments have in general an excitation energy which is much smaller than their total binding energy. For this reason their principal decay mechanism is the successive emission of particles by the compound nucleus. In order to describe it we will use the statistical model of Wiesskopf 43). The specific nature of multifragmentation studies calls for a generalization of the standard version of this model 4’) to the case of emission of In this study we have taken into account the clusters heavier than a-particles. evaporation of fragments which are heavier than a-particles and are in excited states stable to nucleon emission. This corresponds to a limitation on the excitation energy ei of these states: &i< E,,, where F, = 7-8 MeV. The higher excited states are usually so short-lived that the assumption that there exists a sufficiently long lived compound nucleus becomes problematic. In this situation the secondary break-up process seems more adequate. The probability for the compound nucleus to emit fragment j can be written as E*PR,-F: g-‘,E.Lia;(E) p,(E*-B,-e(n-2h'

E)

EdE.

(34)

P&E*)

Here the sum is taken over all the excited states F, (i = 0, 1, . . . , n) of the fragment j, g{ = (2S, + 1) is the spin factor of this fragment in its ith excited state, pj and B, are the mass and separation energy of the emitted particle, 0, is the cross section of the inverse reaction, pi and pCN are the level densities for the final and initial

A.S. Borcina

et al. / Statistical

compound nuclei, and E is the kinetic system. Since primary fragments usually of shell effects on the level density by a simple Fermi-gas formula

rimulation

677

energy of products have high excitation

can be disregarded;

in the center-of-mass energy, the influence

this quantity

is described

p(E”)-exp2Az

(35)

with the level density parameter a = (0.1 to 0.14)AMeV’. In our present study the cross sections of the inverse reaction oj( E) were calculated within the optical model using the nucleus-nucleus interaction potential of ref. “‘), which gives a good description of the experimental data on fusion cross sections in heavy ion reactions. Within the whole energy range, including the subbarrier region, the results of these calculations can be approximated as

(l-Y/E),

EzV,+lMeV

(y+l)-‘expa(E-v-l), where Us is the geometric

E
(36)

cross section: (37)

$,,

MeV

Fig. 5. Cross section for heavy cluster emission at backward angles (8 = 120”-160”) as a function laboratory kinetic energy of ‘He. The data are from ref.4x), and the ctfrves show the results evaporation calculations mentioned in the text.

of the of the

AS. Botoina ef al. / Statistical simulation

618 V,

is the Coulomb

barrier: I$=

ZfiZje2 ro( A.;/” + A:‘3)

A,, 2, and A,, Zj are the mass numbers the fragments, cr = 0.869 + 9.91Z.fi’, and r, = 2.173

1 +6.103

(38)

’

and charges

. 10-m’ZjZ,j

1+9.443 . lo-‘zjz,j

of the residual

nucleus

and

fm .

The calculations for the evaporation cascade were done using the Monte Carlo method (see the detailed description in ref. “‘)). In actual calculations we took into account the emission of fragments no heavier than “0; the experimental energy levels of light nuclei were taken from the compilation 42). In recent years a number of modifications of the evaporation model for describing the emission of heavy clusters by excited nuclei 45-47) have been developed. Here we do not enter into the details of the differences between these modifications. We just note that we have checked that our modification of the evaporation model gives a good description of the recently obtained data on evaporation of heavy clusters 48), in what concerns the production cross sections and the energy spectra of different fragments (see fig. 5).

3. Discussion The statistical mode1 of multifragmentation presented in this paper allows us to make one more step towards a consistent and systematic analysis of experimental data.

Let us analyze

this mode1 using

the observable

characteristics

of multifrag-

mentation, such as the mass and charge distribution, the energy spectrum and the correlations in the yields of cold fragments. We first discuss the effect of the model parameter Ed on the mass and energy spectra. In general, the increase of go leads to lowering of primary fragment masses and consequently to an increase of their total multiplicity 35). This is illustrated by fig. 6, in which we compare the inclusive mass distributions in the case of “hot” multifragmentation ( E(,= 16 MeV which corresponds to statistical equilibrium over translational and internal degrees of freedom) and in the case of “cold” multifragmentation (Ed = co and p ( T) = p,, , which corresponds to a complete suppression of the excitation of internal degrees of freedom). As one can see in this figure, the difference in mass distributions between cold and hot fragmentation is more noticeable at relatively low excitation energies (when the mass distribution is U-shaped). In fig. 7 we observe that the energy spectra of medium-mass fragments are not very sensitive to the details of the fragmentation mechanism, but they differ drastically from the predictions of the pure evaporation model.

A.S. Bofuina ef al. / Statbtieal

679

.simuiation

E,

Eo=12.5 MeV

q

2.5 MeV

‘0’

100

c ? ;\.\

!t \\ iI i\ i \ i ’\

Id’

Iii2

\ \

~ 0

\ 25

5u

u

25

50

75

100

A

Fig. 6. Influence of primary fragment de-excitation on the inclusive mass distribution of fragments. The curves show calculations with the parameter K = 1 for a system with A,,= 100, Z,,= 44 and E,, = 12.5 MeV/nucl. (left) and EO = 2.5 MeV/nucl. (right). Dashed and solid lines represent calculations of the yield respectively before and after primary fragment de-excitation, with E,,= 16 MeV. Dot-and-dash line represent the calculation of the yield of nonexcited fragments (“cold” multifragmentation), with Fg = S-.

Let us now consider how the observable characteristics are influenced by the parameter K. It plays a double role. First of all, it defines the free volume V, available for translational motion of fragments. In addition, it determines their Coulomb interaction energy EC. Generally speaking, these two characteristics are of different physical nature, since V, is given by the con~guration integral over the coordinate space, while EC is determined by the shape and arrangement of fragments at the moment of the break-up. In further development of our model this fact can be taken into account by introducing two different parameters that would determine V, and E,,. A realistic value of K should be determined by the physicai conditions of the system near break-up. In ref. 34) it is argued that a realistic free volume could be determined by the total inner surface between the fragments times a break-up separation distance of the order of the interaction range between the fragments. This leads to a strong dependence of K on the partition, in particular its multiplicity. We do not choose K according to this analysis in the present paper in which we rather want to focus on the calculation of fragment distributions for various fixed ValUeS of K. As it was shown in ref. ‘“), an increase in K leads (via the increase of the free volume V,) to an increase in the fragment multipiicities and to a lowering of their

A.S. Rotuina et al. / Statistical simulation

680

W(E) MeV-’ 5

A0 = 211

Z. = 85 Eo = 6A.

MeV

2

1e2

5

2

Id3

5

150

E MeV

Fig. 7. Energy distribution of fragments with A = 27 (which before de-excitation had a mass A = 40) emitted with disintegration of a system with A,= 211, Z,,=85 and E,=6 MeV/nucl. The solid and dashed lines indicate calculations with F,) = 16 MeV and with K = 1 and 0.2, respectively. The dot-and-dash line is a calculation with K = 1 and F()=CG (“cold” multifragmentation). The dotted line shows the evaporation spectrum for fragments with A = 27. The area under the curves is normalized to unity.

masses. The “quasifission” peak in the mass distribution can be obtained only for rather large values of K (see fig. 3, K = 3). At smaller values (K = 1) the peak disappears and one observes the usual U-shaped spectrum of fragment masses since the Coulomb barrier for the channel M = 2 proves to be too high. Let us note that this situation is different from what one gets with the statistical description of the usual evaporation and fission, where the normalization volume is smaller and does not affect the results of calculations. At the same time, this calls for a more correct description of the freeze-out conditions in the multifragmentation model, especially for the channel M = 2. The parameter K also has a noticeable effect on the energy spectrum of fragments. With increase of K the kinetic energy of fragments becomes smaller (see fig. 7). This is because at the break-up moment the fragments are separated by larger distances, which diminishes the available Coulomb energy, and also because of the distribution of the thermal energy among more fragments. In their analysis of experimental data the authors of ref. 26) have noted a disagreement between the values of the temperature obtained (i) by fitting approximate

AS.

macrocanonical

expressions

BoltGnu ef al. / Siatisticul

.~irnu~~ii~ln

to the mass distributions,

681

and (ii) by fitting a Boltzmann

distribution W(E) - E exp (-E/ T) to fragment energy spectra. Whereas in the first case T = T,im, in the second case T z 15-20 MeV. This discrepancy can be remedied by a correct account of the Coulomb repulsion between fragments at the expansion stage. In that case the slope of the high-energy part of the calculated spectrum of fragments emitted from the system at T = 6.1 MeV corresponds to the Boltzmann distribution with T = 20 MeV (fig. 7). Let us now discuss how the de-excitation of hot primary fragments affects the observable characteristics of multifragnlentation. The calculations for the “cold” multifragmentation presented in figs. 6 and 7, in our opinion, allow one to make only a rough, qualitative estimate of the yield of cold fragments observed in experiment. So in these figures we also show the results of more complete calculations in which primary fragment de-excitation was included. The de-excitation of primary fragments increases the fraction of light clusters in the observable yield (NJ and results in a steeper slope of the mass spectrum at high temperatures (fig. 6). At the same time the charge distribution is shifted from the neutron-rich region to the beta-stability line (fig. S), which is a reflection of the natural fact that de-excitation leads to production of more stable nuclei. One also u p on the nucleon composition notices a strong dependence of the dispersion of cold fragments (see fig. 9). A similar dependence is also observed in the mass and charge distribution of cold fragments (fig. 10) if one does not average over some

W

E0=12.5 MeV

A = IO

A= 30

E0=12.5 MeV

I

I -

A=10 E0=2.5 MeV _

2

4

A=30

6

Fig. 8. Charge distribution of fragments with the mass A disintegration of a system with A,, = 100, Z,, = 44 and En dashed and solid histograms indicate calculations before parameters P()= 16 MeV and K = t. The area under

I2

E0=2.5

14

MeV

16

z

= 10 (left) and A = 30 (right) emitted from the = 2.5 ibottom~ and 12.5 {top) MeV/nucl. The and after primary fragment de-excitation with the histograms is normalized to unity.

A.S. Bofoina et al. / Sfafisiicuf .simulafiun

682

aZA

/ Eo= 12.5 MeV,

I -

/

I

/

I

/

/

/

/

/’

Eo= 2.5 MeV

/

!i I

IO

20

30

10

/

/

/ ,

20

4. , /

A

3o

Fig. 9. Influence of primary fragment de-excitation on the charge dispersion C$ of fragments. The curves show the results of calculations with parameters F”= 16 MeV and K = 1 for a system with A,= 100, Z,=44 and I$,= 12.5 (left) and 2.5 (right) Metl/nucl. The dashed lines are the results before primary fragment de-excitation. The dot-and-dash and solid lines show the results after de-excitation and respectively with and without the account of shell effects in de-excitation.


16’

ii\!

IO

I/

i,!

’

I

20

30

40

A

Fig. 10. Inclusive mass and charge (insert) distributions of fragments produced by a disintegrating system with A,= 100, 2, = 44 and EO= 5 MeV/nucI. All calculations were made with I( = 1 and q,= 16 MeV. The dot-and-dash and solid lines are calculations with and without taking into account of shell effects in de-excitation of prefragments.

AS.

Botvina et al. / Statistieai

simulation

683

interval AA and AZ, as is the case in fig. 6. This dependence of the fragment yield on the nucleon composition reflects the influence of pairing and shell effects at the stage of primary fragment de-excitation. If one neglects these effects, then the dependence of fragment yields on nucleon composition becomes much weaker (fig. 10). Since de-excitation leads to production of a broad variety of nuclei with different E”, the actual values of the yield are between the values corresponding to the two extreme cases of complete preservation and complete destruction of all shell effects. It is interesting that “cold” multifragmentation and the “hot” multifragmentation with subsequent de-excitation of primary fragments give very close mass and energy spectra (see figs. 6 and 7). One can distinguish the two cases by the charge characteristics of fragments. In the case of “cold” multifragmentation the charge distribution and the charge dispersion of fragments are the same as those of primary fragments. So the “cold” multifragmentation is characterized by production of neutron-rich fragments (A/Z = A,,/ZJ with a much greater value of CT; than in the case of “hot” multifragmentation (see fig. 9). Since in real nuclear reactions induced by hadrons or heavy ions the nuclei produced usually have a wide distribution in excitation energy E,, one will observe


P-

\

2,=44,

A~= loo,

-----

Eo=5AoMeV

evaporation

\

Fig. Il. 2,,=44

Average multipIi~ity of fragments with charge 2 emitted by a disintegratjng system with A, = 100, and E,=5 MeV/nucl. The solid and dashed lines, indicate calculations for multifragment break-up and for the evaporationat cascade, respectively.

684

A.S. Botuina

Aa= 100,

2

,simuiatiotr

et al. / Statistid

Zo=

44

~

fragmentation

----

evaporation

4

6

6

K

Fig. 12. Probability of emission of a given number K of large fragments with Z > 3 in a single break-up event for a system with A,, = 100 and Z,, = 44. The notations are the same as in fig. 1i.

the products of their decay corresponding to different mechanisms, the most important among which are multifragmentation and the evaporational cascade. In order to distinguish between these two mechanisms let us compare the results obtained with them for the decay of a system having given values of A,,, Z,,, EC). In general multifragment break-up gives an inclusive mass spectrum which decreases with A much slower than in the case of evaporation. The inclusive charge spectrum illustrated in fig. 11 shows a similar behaviour. Although modifications to the evaporation model (e.g. those of refs. 45,46)), could somewhat change this situation, the two mechanisms may be distinguished not by the mass yields only, but also especially by the energy spectra (see fig. 7) and also by the various correlations in the yield of heavy fragments. The simplest among the last one is the probability of multiple emission of several fragments in a single break-up event. The results of its calculation are presented in fig. 12. As one can see, the selection of break-up events with three or more heavy fragments can serve as a reliable criterion for distinguishing multifragmentation.

4. Conclusion In the present paper we have further developed the statistical model of multifragmentation presented in refs. 34,3s). This model has a number of advantages. First of all, it enables one to calculate the characteristics of all the fragments in a given break-up event, i.e. both inclusive and exclusive break-up characteristics. Furthermore, it makes a correct account not only of the conservation of mass, charge,

AS.

energy

and momentum

Bntvinn et al. / St~t~.stira~.~i~M~~tj~r~

of a finite nuclear

system,

but also of the fluctuations

685

of

other quantities such as the temperature, mass, charge and energy af fragments. Furthermore, inclusion of fragment de-excitation allows one to extend the analysis to that of the production of the cold fragments which are observed in experiment. As a result, our model can be used for analyzing the experimental data on production of fragments in deep inelastic interactions of hadrons and heavy ions with nuclei. For that purpose one needs to know the distribution of &, 2, and EO for the highly excited nuclei produced in these interactions. It seems quite natural to combine this mode1 OfmuItifragmentation with the intranuclear cascade model 4’)3 which describes the dissipation of the energy transferred to the target nucleus by an incident particle and gives a distribution in A,, Z,, and E, of the thermalized residual nuclei produced in this process. According to our preliminary calculations “1, this unified cascade-fragmentation evaporation model gives a rather good description to the experimental yields of fragments produced in high-energy protonnucleus reactions. in a following study we will use this model to make a systematic analysis of the experimental data on fragmentation reactions induced by hadrons and heavy ions at intermediate and high energies. Two of the authors (A.M., RD.) express their gratitude to the NBI for its hospitality during our stay at the Institite, when part of this work was performed_ R.D. also acknowledges support from the Danish Natural Sciences Research Council.

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