The distinction between multifragmentation and sequential binary decay

NUCLEAR PHYSICS A

NucIear Physics A551 ( 1993) 73-92 North-Holland

The distinction between multifragmentation and sequential binary decay W. Gawlikowicz Institute

of Physics, Jugellonian

and K. Grotowski

University,

Reymonta

Cracow.

4 and Institute

of Nuclear

Physics,

Poland

Received 2 December 1991 (Revised 7 July 1992)

Abstract: Following the idea of L6pez and Randrup, the influence of the Coulomb interaction on trajectories of fragments emitted from a hot nuclear system is investigated. The fragments are allowed to decay in flight with decay constants depending upon their mass and excitation energy. It is found that the Coulomb field of the two heaviest fragments focuses velocities of the other fragments into some toroidal region, when the hot system breaks up simultaneously, while in the case of the sequential binary decay these velocities exhibit a nearly isotropic dist~bution. The sensitivity of this effect and its limitations are discussed.

1. Introduction It is well known that the multiplicity of particles emitted in heavy-ion reactions increases with increasing collision energy ‘). At low excitation energy one observes partitions with one (or possibly two) heavy fragments accompanied of light particles. When higher energy is available an enhanced number

by a number of intermedi-

ate-mass fragments is observed. The lower-energy behavior is explained as fusion, or incomplete fusion, or deep inelastic collision followed by sequential evaporation or fission ‘). At sufficiently high energy prompt (simultaneous) multifragmentation processes

are expected.

The possibility

of the existence

of such processes

is very

exciting and a number of models have been proposed to explain them. Thus for example the classical concept of a liquid-gas transition was utilized in ref. ‘). By using the Skyrme parameterization of nuclear forces and the Hartree-Fock theory, the equation of state was calculated and the critical temperature estimated [see also refs. “,‘)I. Other models are based on the statistical 6,7), percolation ‘), dynamical mean-field “) or molecular dynamics lo) approximations. Merging of the percolation and the dynamical mean-field evolution ‘I), or coupling the dynamics of the collision stage with the subsequent statistical multifragmentation or statistical deexcitation ‘*) have also been proposed. Correspondence to: Dr. W. Gawlikowicz, Cracow.

Institute

of Physics,

Jagellonian

University,

Poland.

0375-9474/93/$06.00

@ 1993 - Elsevier

Science

Publishers

B.V. All rights reserved

Reymonta

4,

14

W. Gawlikowicz,

Unfortunately, verification experimental data is difficult ments cannot break

and Randrup by studying

of sequential

prompt

multifragmentation

binary

decays

13) to look for some kinematic the Coulomb

/ Multifragmentation

of the multifragmentation models by comparison with and frequently impossible. This is so because experi-

simply distinguish

up (a chain

K. Grotowski

trajectories

(SBD)).

difference

(PM) from a sequential It was proposed

between

of decay fragments

by Lopez

these two processes

in the early stage of the

disintegration process. They have found some differences in the proton energy spectra, velocity distributions and velocity correlations. It is expected that the relative-velocity distributions between the intermediate-mass fragments, measured at small relative angles, are determined by the interaction between the coincident fragments. Since that interaction is mainly the mutual Coulomb repulsion, the shape of the relative-velocity distribution should depend on the distance (in space as well as in time) between fragments. In line with conventional nuclear interferometry studies 14) that effect has been used to select slow binary sequential processes 15). On the other hand, the relative velocity between fragments at large relative angles is suggested to be sensitive to the time characteristics of the break-up configuration [Gross et al. 16) and ref. “)I. Much effort was devoted to searching for corresponding experimental differences. The results are encouraging but somewhat ambiguous I’). Recently, a recursion relation has been derived by Cole et al. I*) which relates the mean square momenta of the sequentially emitted light particles and the final residue. It can be used to distinguish between the PM and SBD alternatives. For the same purpose the azimuthal angle correlation between three intermediate-mass fragments has also been used 19). As another possibility one should mention a study of the event-shape distribution in the momentum space “) which is shown 13)to be sensitive to the dynamics In this work,

of the fragmentation process. following the idea of Lopez

and

Randrup

13), we would

like to

concentrate attention on trajectories of particles moving in the Coulomb field created by the two heaviest fragments. We seek here a focusing effect similarly as in ternary fission 2’512). Preliminary results of this work have been recently published 23). All the calculations presented were performed for the $%m hot nucleus based on an assumption of the existence of a compound system. No entrance

and are channel

effects were considered. In the following sections of the paper the SBD code and the PM calculations are described (sect. 2). Next, the simulation results are presented (sect. 3). Finally, (sect. 3 and sect. 4) the sensitivity and limitations of the method are discussed.

2. The method Our method is based on the work of Lopez and Randrup 13). However, as our approach contains some modifications, and also for the convenience of the reader, we give a brief outline of the calculational procedure. We begin with a hot system A,, 2, created in some unspecified way, with an initial excitation energy EX. It may disassemble promptly or sequentially. For each

W. Gawlikon+cz, K. Grtltowski / ~~~rr~f~grnenf~rio~

disintegration fragments

event

it is assumed

that

as well as the final total kinetic

the final

mass

(charge)

75

spectrum

of the

energy are the same for the PM and SBD

alternatives. Therefore, in both cases the SBD Monte Carlo code is used to calculate the mass (charge) partition of fragments. It also provides the value of the final total kinetic

energy

of fragments.

Now one can calculate

trajectories

of SBD and PM

particles accelerated in the Coulomb field of other participants, and at some effectively infinite distance from the initial hot system one can compare the velocity distributions of fragments for those two alternatives. The comparison at infinity is made for cold, deexeited fragments. The sequential binary decay proceeds according to some time scale. The time constants governing consecutive partitioning should depend on the excitation energy, asymmetry of splitting and other factors (see sect. 2.3). In the case of the PM process it is reasonable to assume that one observes a simultaneous break up into a set of pre-formed excited fragments which move in the mutual Coulomb field and deexcite in flight. Deexcitation takes place mainly by the sequential evaporation of particles and should proceed with the corresponding time constants. Deexcitation can be simulated for each fragment by switching on the SBD Monte Carlo code. In calculations of L6pez and Randrup 13) the acceleration process of excited fragments is separated from the final deexcitation. In our work the excited fragments are allowed to decay in flight and it is mainly in this respect that our calculatians deviate from ref. I’). Some justification is needed far the assumption of the same mass (charge) spectra in the PM and SBD processes. It is known that the mass distributions predicted by different multifragmentation models are similar [for comparison between the statistical and the percolation model see ref. *“)J. Similar for the sequential decay 75).

2.1. SEQUENTiAL To

describe

the sequential

BINARY

are also predicted

DECAY

the SBD process binary

mass spectra

we have used the program

decay and calculates

Coulomb

DBINFRA.

trajectories

It simulates

of decay fragments

along the disintegration path. The lifetime of each fragment is randomly taken according to the decay constant (see sect. 2.3). The binary decay is simulated by the Monte Carlo program BlNFRA 26) which is a subroutine of DBINFRA. It is based on the transition-state method and treats evaporation of particles and fission in the same way “). Starting from an excited parent nucleus with excitation energy Eg and angular momentum 3, the program selects consecutive pa~itioning along the evaporation path, until all the products become “cold.‘” The probability for a given binary scission into nuclei labeled i and j is written as: exp{2[a(E*-Es,,-E,-E,,,-2T)I”*} P(A; Ai, Aj)c~exp {2[ aE*]““}-”

,

where Es_, is the ground-state separation energy and E, is the Coulomb barrier, T is the temperature which was evaluated using a level density parameter a, and E,,,

16

W. Gawlikowicz,

is the rotational

energy

defined

K. Grotowski

by

E,,, = [~M;R: +~~jRf The scission about

configuration

their center-of-mass

/ Multifragmentation

is assumed

+ ~~RiRj]~w’ .

to be that of two touching

with angular

velocity

(2) spheres

w given by the parent

rotating J and the

total moment of inertia. The reduced mass of the system i -j is pcLii.The separation energy is calculated from the modified Weizsacker formula ‘): E,,,=E(A)-E(A,)-E(Aj)

3

(3)

where E(A)

= 15.56A-/3(T,

/3(7; T,,) is a correction of surface the surface tension vanishes:

Tc,)A2’3-0.7A2,J4A’13.

energy for the critical temperature

(4) (T,,) in which

P(T, %.)=P(o)(~)s’4* The total kinetic energy given by Moretto “> in (evaporation) fragments. average. The calculation

(5)

is randomized after each partition according to the formula order to simulate the c.m. energy distribution of fission In the randomization process energy is conserved on the is of the “spin-off” type, because all fragments are emitted

in the plane perpendicular to the initial angular momentum which, in turn, is taken to be uniformly oriented in the plane perpendicular to the beam direction. In each evaporation step the direction of fragments is randomly generated with the isotropic distribution (in the c.m. system). It should be emphasized that such a “spin-off” approach and other approximations used in BINFRA represent a major simplification compared to a Hauser-Feshbach calculation. They are invoked to reduce the necessary computing time, and more importantly, in order to treat the emission of intermediate-mass fragments in a consistent manner. The agreement between the predictions of BINFRA and the experimental data (the energy, angular and mass distributions of particles) is satisfactory 26). 2.2. PROMPT

MULTIFRAGMENTATION

The calculations

are performed

in consecutive

steps by the computer

program

MLJLFRA. In the first step the subroutine BINFRA performs the sequential decay of the original hot nucleus A,,, 2,. This decay is temporarily stopped when the average excitation of all fragments, (ET/A,), is equal to or less than some specified threshold {e.g. 20% of E$/A,). At this point the total kinetic energy of all fragments is KB’N, and the total potential Coulomb energy is VFIN. The total energy EBIN= K R’N+ VYN. In the second step the set of excited fragments obtained in such a way is used for simulation of the prompt multifragmentation explosion. For this purpose the initial configuration for this set of excited fragments is computed and then these fragments are accelerated by the Coulomb field and are allowed to decay

W. Guwlikowjc~,

in flight DBINFRA

with

proper

until

decay

K. Grotowski

constants.

all fragments

/ Muit~fragme~tation

This

deexcite

is carried

completely.

77

out

using

In this manner

the

program

we have the

same mass spectrum for the sequential and the simultaneous breakup. The “freeze-out volume” of the initial configuration is determined by randomly sampling produce

positions

of excited

the minimum

possible

ment shapes are approximated Myers formula 29):

fragments by spheres

R(A) = 1.128A”3(1 Fig. 1 presents

and

non-overlapping

the distributions

shifting

them

spherical

in such a way as to

configuration.

with half-density

-0.756A-“‘3)

radius

The fraggiven

by the

[fm] .

of the value of the “freeze-out

(61 volume”

resulting

from the above procedure*. As one can see the most probable value of the “freeze-out volume” is about three times the volume of the initial hot nucleus. It depends slightly on the partition of the initial system. In the following calculations all possible partitions were included. One can get the “freeze-out volume” which is only about twice the initial volume, when the packing procedure begins from the two heaviest fragments. Our calculations show that the final result is not sensitive to its exact value. It was assumed

that for each event the total energy should be the same for the EBiN is taken as the energy available in the initial

PM and for the SBD. Therefore l The “freeze-out volume” be initially enclosed.

is represented

COUNTS -ssynmetric

--- asynnetric

by the volume

of a sphere

in which all the fragments

can

s . : .

.

. ,

. .

*. / . .

Fig. 1. Distributions of the value of the “freeze-out volume” normalized to the initial volume, V,, of the hot ‘:$rn nucleus. The solid line represents the symmetric case (the two heaviest fragments have similar masses). For the broken line the two heaviest fragments have very different masses. (1000 probes. 1

W. ~~wiik~wicz. K. Crotowski f Mu~~if~agm~~ia~i~n

78

configuration.

Now we can compute

the total

Coulomb

potential

energy

of the

fragments, V,, and go to the next step, by assigning initial velocities to the fragments. To do this we distribute the kinetic energy K = KRiN - V, among all degrees of freedom

of the fragments.

As in ref. 13), we assume

a Maxwell

distribution

of initial

velocities. The following

conservation

laws are used:

(i) energy

conservation,

(ii) total

momentum conservation (E; = 0), (iii) total angular-momentum conservation (i = 0). We can easily satisfy the first two rules 13). If e.g. V, is too large, one can renormalize it by a uniform expansion of the system. To avoid a situation when the initial (and in consequence the final) neutron velocities are zero, which would lead to a significant difference in the final kinetic energy of charged particles in the SBD and PM channels, we assume that the total kinetic energy of the SBD and PM neutrons is the same. The total momentum of the system may be adjusted by a galilean transformation. The available kinetic energy K has to be changed afterwards, by renormalizing the fragment momenta by a common factor. The angular-momentum conservation presents a somewhat more complicated problem. First of all we assume zero total angular momentum of the system, as for a central collision. Such a collision is usually characterized by the largest fragment multiplicity and the highest temperature of the resulting system ‘“). Therefore, for each event, we use a procedure which minimizes the initial angular momentum (L < 2h) while conserving the energy, the momentum and the maxwellian distribution of velocities. It should be stressed, that the value of angular momentum influences the partition probability of the system (the initial mass spectrum). After defining the initial conditions the program numerically integrates equations of motion of fragments in the mutual Coulomb field allowing decays in flight. As a result of such a procedure one obtains an output file of events containing information on the velocities of all fragments of the prompt multifragmentation. For each event there exists also information about the velocities of fragments, produced with the same partition by the sequential binary decay. 2.3. DECAY

TIMES

IN

SEQUENTIAL

There is much discussion

DECAY

in the literature

OF

HIGHLY

concerning

EXCITED

FRAGMENTS

the time scales involved

in

the formation and decay of highly excited nuclei ‘I). It is expected that thermal equilibrium may more easily be established in central processes involving sufficiently heavy nuclei. The Uehling-Uhlenbeck type calculations show that the difference between the entropy of a colliding system and its equilibrium maximum value varies parameter of the approximately exponentially with time 32). The decay-time exponential function is defined as a relaxation time. Theoretical calculations as well as experimental results suggest that around the Fermi energy the energy relaxation time has a value of about 4 x 10mZ3s [refs. 33’4)]. On the other hand, there is evidence that below the initial temperatures of about 6 MeV the hot compact composite nuclei live suficiently long to behave like thermalized systems 35). According to Borderie 31)

W. Gawlikowicz,

the lifetime lo-”

of nuclei

s, to about

The dependence predictions

(for the emission

upon

/ Muliifragmentation

of the first neutron)

10mZ3s, for temperatures

increasing

the mass of the decaying

agree with the earlier

from the statistical the observed

K. Grotowski

theory

small-angle

estimation

changes

from about

from 1 to 10 MeV, respectively.

nucleus

is not a strong

of the hot-nucleus

of (A = 142, T = 4.4 MeV) which charged-particle

19

correlations

lifetime properly

in the reaction

one. These obtained reproduces 680 MeV

40Ar+Ag [ref. 3”)]. As was mentioned in the Introduction, hot nuclei deexcite emitting light particles, intermediate-mass fragments (IMF’s) and fission fragments. Fission and IMF emission can be treated as the two extremes of a single mode of decay, connected by the mass-asymmetry degree of freedom 27,28,37).Recent measurements of pre-scission neutron multiplicities suggest for symmetric fission after fusion, decay times in the range (3.5 f 1.5) x lo-” s, over a large set of projectile-target combinations 38). For increasing mass asymmetry of the fission fragments the decay time is reduced 38) showing a smooth transition to the value characteristic for neutron evaporation. The IMF emission times do not change too much in the collision energy range 18-84 MeV/u [refs. 38z39)]. The lifetimes for emission of neutrons 3’,36) and for emission of intermediate-mass as follows: fragments 38,39) can be parameterized 7=2e13/Te.4/S

[WC1 .

(7)

Here T is the nuclear temperature (in MeV) and A is the mass of the emitted fragments. In this work the decay times in the sequential evaporation are calculated from eq. (7). The temperature T in eq. (7) is taken to be 5 MeV for excitation energies E*/u > 3 MeV [see recent results obtained at Ganil 40), conclusions of ref. 3’) and statistical model calculations for multifragmentation “)I. For lower excitation energies it is evaluated using the level density parameter a.

Fig. 2. The decay times of nuclei for the emission

of the first neutron, text).

as a function

of temperature

(see

80

W. G~w~iko~iez,

I(. Grotowskj

! ~ult~frag~gntation

As an example fig. 2 presents decay times for the emission of the first neutron. 3. Simulation results Calculations were performed for the hot Sm nucleus (A0 = 150, 2, = 62). All results are plotted in the cm. system. Fig. 3 shows the mass spectra, provided by the SBD code. As one can see, heavy evaporation residue and very light particles dominate the mass spectrum at the lowest excitation energy (2 MeV/u). The emission of intermediate-mass fragments prevails at 5 MeV/u. At 10 MeV/u the heavier fragments disappear. The shape of the mass spectrum at higher excitation energy depends upon the value of the critical temperature, T,,. Calculated by different authors, T,, has values between 25 MeV [an infinite nucleus ‘)I and 8 MeV [a finite nucleus, Coulomb corrections included “)I. Here we use the value T,, = 13 MeV. As one can see, the mass spectra of fig. 3 are similar to those calculated by Lopez and Randrup 13). As a result of calculations we get a file, where for the PM and SBD the velocities of all fragments are recorded, event by event. Such information may be used in

HI4 103 102 101

0

50

100

CICanul

Fig. 3. Mass distribution of fragments for the initial excitation energy of the Sm nucleus 2, 5 and 10 MeV/u, respectively, provided by BINFRA (5000 events). The Monte Carlo statistical uncertainties are indicated (the same in the following figures).

81

W. Guwlikffwicz, K. G~~r~wski ,J ~uitifr~g~e~tu~i~n

many

different

ways. We now concentrate

in each event and on the Coulomb the distribution

of the charge,

to the total charge of fragments, initial

excitation

energy

trajectories

on the two heaviest

of other lighter particles.

2, + Z,, of the two heaviest

of the system,

just after prompt

attention

2,.

We assume

that the average

Fig. 4 shows as compared

initial

excitation

(E$/A,)=0.2E$/Ao.The is E$/A,= 2 MeV/u, 5 MeV/u and

multifragmentation,

of the Sm nucleus

fragments

fragments

is

10 MeV/u. As one can see the two heaviest fragments remove, on average, 90%, 50% and 15% of the total charge, respectively, and consequently will induce, for 2 MeV/u and for 5 MeV/u, quite a strong Coulomb field. Partition of the charge between the first and the second fragment displays a rather broad distribution (fig. 5). For 2 MeV/u the heaviest fragment is usually the evaporation residue and the distribution is peaked around the zero value of ZJZ, . The energy spectrum of the heaviest fragment is broad (see fig. 6, E$/A,= 5 MeV/u) and for the sequential decay shifted towards lower energies. It indicates a pre-scission emission of charge in some of the SBD events. The average angle between the two heaviest fragments (fig. 7, E$/A,= 5 MeV/u) both for the multifragment and binary breakup. This suggests

is smaller than 180” a mechanism which

perturbs the back-to-back push. It can be associated with the secondary evaporation of light particles after the sequential binary fission, and with the Coulomb acceleration from the rest of the charged fragments executed during simultaneous multifragmentation. As one can see from fig. 4 the two heaviest fragments pa~icipating in disintegration take a large part of the charge of the system. They are emitted almost back-to-back (fig. 7), and at the beginning should create an axial, very strong Coulomb field. It

COUNTS

0.0 Fig. 4. Distribution ZO. The fragments

0.2

0.4

0.6

0.8

(zltzP>/zc

of the charge, Z, +Z,, of the two heaviest fragments as compared to the totai charge, just after prompt multifragmentation have 20% of the initial excitation energy of the Sm nucleus, which is 2, 5 and 10 MeV/u, respectively. (5000 events.)

82

W. Gawlikowicz, K. Grotowski / Multifragmentation

0.0 COUNTS :

0.2

0.4

0.6

0.8

SdeU/u

’

I

I

Z2/Zl

04

0.0 COUNTS

0.2

0.4

0.6

0.8

lObleU/u’

I

I

Z2/Zl

(a

lo2

lo1

0.0 Fig. 5. Partition

of the charge

0.2

0.4

0.6

0.8

between the first and the second fragment. are as in fig. 4. (5000 events.)

22121 The initial excitation

energies

83

I

I

DBI NFRCIO MULFRCIIL

0

50

E EMeUl

I.00

Fig. 6. Energy spectrum of the heaviest fragment, predicted for the binary decay (DBINFRA code) and for the multifragmentation (MULFRA code). The initial excitation, E,T/A,, of the Sm nucleus is S MeV/u and the average initial excitation of fragments, (ET/A,), just after prompt multifragmentation is 0.2E$/A,,. (5000 events.)

is interesting to look for possible focusing effects induced by that field on other charged fragments. For the beginning we analyze velocity distributions of charged nucleus (Eg/Ao=5 MeV/u; (Eg/A,)= fragments emitted from the ‘:$rn O.ZE,“/A,). As the cm. frame has a rotational degeneracy we propose a coordinate system with the z), axis given by the vector 6, - i$. Here 6, , and & is the velocity

0 Fig. 7. Distribution decay (DBINFRA

30

60

90

120

150

BCDEGI

of values of the angle between the two heaviest fragments, predicted for tire binary code) and for the prompt multifragm~ntation (MULFRA code). E,*/A,=S MeV/u, (E~/A~)=O.ZE~/A~. (5000 events.)

84

K

~u~~~~k~wj~~

K. ~~oi~~~~

/ ~~~f~~r~g~g~?~fiQn

of the heaviest and of the second heaviest fragment, respectively. This means that for each event we are performing a rotation of the coordinate system to emphasize the axial symmetry of the Coulomb field, of the two heaviest fragments. Such a construction leaves a one-di.mensional rotational symmetry (the U, axis is the symmetry axis) which is clearly seen in fig. 8. The velocities of fragments 1 and 2 pop&ate two regions close to the v, axis (high density of dots). The velocities of other charged intermediate-mass particles are scattered around and show a striking difference between the prompt and the sequential decay. While the PM velocities are concentrated inside some torus-like region, the SBD velocities exhibit a quite isotropic distribution. In order to illustrate this phenomenon more clearly, we transform the dist~bution of fig. 8 (excluding the two heaviest fragments) into the plot dN(O)/dR, presented in fig. 9. Here 0 is an angle between the u, axis and the velocity vector, and N is the number of velocity dots in the velocity space. As one can see, the PM velocity distribution is represented in fig. 9 by a gaussian-tike curve, while the SBD one is much more isotropic. It suggests a focusing effect caused by the Coulomb field of the two heaviest fragments. The intensity of that Coulomb field drops down too quickly to enable the SBD particles, coming from the much slower evaporation process, to be focused. The strongest focusing is observed for fragments A > 4 (fig. 91, the proton velocity distribution presented in fig. 10 shows that they are much less focused. Probabty they leave the region of the strong field too early.

Fig. 8. Velocity vectors oi fragments (.4>4) are presented as dots in a three-dimensional picture, for the binary decay (DBINFRA) and for the prompt multifragmentation (MULFRA). The coordinate system is defined for each event by relative velocities of the two heaviest fragments (see text.) (Calculations were made for 5000 events, but 1000 only were used here for better legibifity of the picture.) E$,/A,= 5 MeV/u, ~E~/~~}=~*Z~~jA*. (IOQO events.)

85

0.20 0.15

;; Y 4 5 n f L Y

0.10

0.05

0.00

s 0g2= L @ 0.20 Y 0.15 0.10 0.05 0.00

0

30

60

90

Fig. 9. Angular distribution of the fragment velocity vectors (A > 1 and A 5 4) as predicted for the binary decay by DBINFRA code and for the prompt multifragmentation by MULFRA code. E$/AO = 5 MeV/u, (Eg/AF)=0.2E$/Ao. (SO00 events.)

; 0.20 Y srl 5 0.15

n : 0.10 L Y ;

0.05

: -

0.00 0

30

60

90

120

150

G,HtEGl

Fig. 10. Angular distribution of the velocity vectors of protons (A = 1f as predicted for the binary decay (DBINFRA code) and for the prompt multjfragmentation (MULFRA code). E$/A,=S MeV/u, (E$/AF)=0.2E$/A,,. (5000 events.)

W. Gnwlikowicz,

86

&0”

K. Grotowski

DBIIjFRfW f HUl;FRRO :

.:. 2tlelj/u

0.40

/ Multifragmentation

! :

! :

;.

.‘.

;.

j

i

i

i :

I

:.

$. 1)... +++

0.30

;

0.20

.,

.:.

.,.

.:.

4

1

: +i . .I . . . . . , . . . . . . . . . . ... . . . . . . . . . . ..:. .-t

..I

Y * 5

0.10

1 0.00 0.20

0.15

0.10 0.05

0.00

r-

1

I

30

$0

90

120

150

BJDEGI

Fig. 11. Angular distribution of the velocity vectors of fragments with A> 1, predicted for the binary decay (DBINFRA code) and for the prompt multifragmentation (MULFRA code). The coordinate system is defined event by event (see text). The initial excitation, Ez/A,, of the Sm nucleus is 2 and 5 MeV/u. (E~/AF)=0.2E,*/A,,.(5000 events.)

tlUl;FRRO j ,,,,. .,,,,, ,,....: lOrl&J/u

0

30

: ,.....:

:

: . . . . ..I

i

:

:

:

60

90

120

150

...

B”tDEG3

Fig. 12. Angular distribution of the velocity vectors of fragments with A> 1, predicted for the binary decay (DBINFRA code) and for the prompt multifragmentation (MULFRA code). The coordinate system is defined event by event (see text). E$/A, = 10MeV/u, (ET/A,)= 0.2E,*/A,. (2000events.)

W. Gawlikowicz,

K. Grotowski

/ Mul~fiagmentaiion

87

0.20

0.10 ! 4 5 0.05 ”

n L 0.00 f Y ; 0.20 : lJ

:

j

0.15

0.10

0.05

0.00 0

30

60

90

120

150

BJDEGI

Fig. 13. Angular distribution of the velocity vectors of fragments with A> 1, predicted for the binary decay (DBINFRA code) and for the prompt multifragmentation (MULFRA code). The initial excitation, E$/A,,, of the Sm nucleus is 5 MeV/u. The average initial excitation of fragments, (ET/A,), is 0.2 and 0.5 of the initial excitation. (5000 events.) _

0

30

Fig. 14. Angular distribution of the velocity decay (DBINFRA code) and for the prompt

60

90

120

150

B”CDEGl

vectors of fragments with A > 1, predicted for the binary multifragmentation (MULFRA code). E$/A,=5 MeV/u,

(E$/A,)=O.SE,*/A,,.

(5000 events.)

.

. . <* .\. . . . . _ .

:..

.

.

.

.

.I.

.

.

.

01):

. . . . . ,*. . . . . .

:

.-.

:..

.

.

1

.

I

,.

:m

.

I_..

.

.

.

.

W. Ga~~~ikuwicz,K. Gro~owskj f ~uizifrug~en~aiion

It should proportional one-half

be noticed

that

acceleration

to the charge-over-mass

for alpha

particles

of a fragment

The SBD curves of figs. 9 and 10 decline 180”, which may be explained 1 and 2, respectively.

in the Coulomb

ratio, and that &/A,

and most heavier as Coulomb

As the two heaviest

89

field is

is one for protons

and

0 approaching

0” or

fragments.

slightly shadows

fragments

for angles

of the two heaviest are emitted

fragments

not exactly back-to-

back, only partial shadowing is observed. The focusing effect depends also on the excitation of the initial system. As seen from fig. 11 the strongest focusing may be observed for the 2 MeV/u initial excitation, and it decreases slightly for 5 MeV/u. It is absent for 10 MeV/u (fig. 12). The corresponding location of the maximum in the PM distribution shifts from the region close to 180” (large 2, -2, asymmetry at 2 MeV/u) towards 90” (for the more symmetric Z, -2, case at 5 MeV/u). The Coulomb shadows of fragments 1 and 2 are much less pronounced at 2 MeV/u where the sequential decay proceeds with longer decay times. For all calculations presented above the average excitation of fragments, just after prompt multifragmentation, was assumed to be (Es/AF) = 0.2~~/A~. Fig. 13 shows that the focusing begins to deteriorate at (E~/A,)=OSE.$/A,. For (ET/A,) = 0.8E$/A,

(fig. 14) it disappears. As mentioned in sect. 2, Lopez and Randrup took infinite decay constants in the SBD calculations and therefore separated the Coulomb acceleration phase from the final deexcitation. In order to demonstrate how such an approximation could modify the reaction picture the calculations have been repeated with all time constants multiplied by a factor of 10, or alternatively by a factor of 0.5 and 0.1. As seen from fig. 15, only the SBD distributions show a distinct dependence on the time constants. For “long” time constants the Coulomb shadow effect is reduced and one obtains the same picture as in ref. “) where the Lopez and Randrup approximation was

used. The reduction of the time constants by a factor of one-half has little effect, but for time constants which are ten times shorter the difference between the PM and SBD curves is reduced, especially at large angles 0. 4. Summary and conclusions Simulations

performed

for the prompt

multifragmentation

of the hot *$Sm nucleus

suggest a focusing effect which may be observed in the veIocity space. The Coulomb field of the two heaviest fragments, with velocities 6, and &, focuses velocities of remaining ejectiles into a toroidal-like region centered around their relative velocity 6, - 2j2. The focusing is much weaker for protons. The effect depends on the initial excitation of the system, and slightly on the excitation of decay fragments. It offers a possibility to distinguish the prompt multifragmentation from the sequential, binary decay. The nucfei participating in the SBD and PM reactions which are excited, have to decay with appropriate time constants. The “realistic” time constants used in

W. Gaw~ikowiez,

90

this work depend exact values picture.

on the mass and excitation

influence

For “long”

approximation

I(. Gr~~~wsk~ / ~u~fi~ragmenra~~on

the shape

time constants

where the Coulomb

energy of the nuclei in question.

of the Coulomb

shadows

one gets conditions acceleration

observed

Their

in the SBD

of the Lopez and Randrup

phase was separated

from the final

deexcitation. In an eventual

experiment,

a coincidence

between

the two heaviest

fragments

should be used as a trigger. As a second trigger one should use e.g. the high multiplicity of particles in order to select central collisions. The system has then small or zero angular momentum and the highest temperature. As it has been tested the focusing effect is still present for angular momenta of the order of few tens of h. It means that the initial strong Coulomb field of the two heaviest fragments enforces a uniform rotation of the total system. One should be aware that such experiments will be rather difficult. We shall need velocity measurements, the 473. geometry, and a low threshold detection and identification of heavy fragments. One should also ask the question, if the picture of prompt multifragmentation presented in this paper has a proper theoretical basis, how well is the “freeze-out volume” defined, will a compound nucleus exist at such excitation energy, etc.? This kind of criticism which applies to our work applies also to the work of Lopez and Randrup r3). This subject needs further investigation. The authors are indebted to Drs. A.J. Cole and J. Randrup for critical remarks. We acknowledge interesting and fruitful discussions with Drs. J. Brzychczyk, T. Kozik, 2. Majka, R. Planeta and Z. Sosin. This work was supported by the Polish Committee of Scientific Research and by the Ministry

of Education.

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