Theory of nuclear multifragmentation

NUCLEAR PHYSICS A

Nuclear Physics AS71 (1994) 379-392

North-Holland

Theory of nuclear multifragmentation * III. Pre-transition

nucleon radiation

Jorge A. Lopez Department of Physics, University of Texas at EI Paso, El Paso, TX 79968-0515, USA

Jorgen Randrup Nuclear Science Division, Lawrexe Berkeley Laboratory, Universityof California, Berkeley, CA 94720, USA

Received 24 August 1994 Abstract Within the framework of a previously developed transition-state theory for nuclear multifragmentation, we study the effect of pre-transition radiation of nucleons on the partial widths for breakup into specified multifragment channels. Typically, a considerable portion of the mass and energy of the source is radiated during the evolution of the system from a spherical compound nucleus towards the conditional significant reduction of the multifragmentation width.

saddle configuration,

thus causing a

1. Introduction Anew generation of complex fragment detectors is beginning to produce more exclusive data coming from nuclear collisions at intermediate energies. While this advance will help to better characterize the final stage of the reaction, a detailed understanding of the dynamics is still needed in order to allow a connection to be made between the data and the elusive properties of excited nuclei. This is particularly true in energetic heavy-ion reactions where the colliding nuclei have sufficient energy to cause matter to be emitted during the entire evolution, from the initial contact to the final detection. The study of light-particle emission in fragmentation reactions has been mostly concentrated either on the early stage of the reaction

(pre-equilibrium

emission)

or on the final

expansion stage when the fragments are receding. However, since the source of the fragments can radiate particles while the fragments are in the process of coming into being, the ensuing loss of matter and energy continually changes the source and may therefore affect the course of the reaction. With a nomenclature consistent with our earlier work[ 1,2], * This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Nuclear Physics Division of the US Department of Energy under contract no. DE-AC03-76SF00098, Associated Western Universities through an EMCOM Faculty Fellowship, and by the National Science Foundation under grant PHY-930-3847. 0375-9474/94/$07.00 @ 1994 - Elsevier Science B.V. All rights reserved XSD10375-9474(93)E0518-D

380

J.A. L&a,

J. Ran&xp

/ Nuclear multifagmentation

III

we denote this phenomenon upre-transition nucleon radiation”. In the present study, we explore how the pre-transition radiation may affect the multifragmentation process. The term “multifragmentation” has been employed to denote a variety of different physical processes leading to several or many complex fragments. In the present study we picture multifra~entation as a specific decay mode of a highly excited compound nucleus. Thus, in analogy with the Bohr-Wheeler description of nuclear fission[3], the shape of the compound nucleus performs a brownian-like motion as the system explores its available phase-space and the partial width for passing across the conditional barrier for a specific m~tifra~entation channel can then be calculated by statistical means. Such a transition-state treatment of nuclear multifragmentation was developed in ref. [ 11. After the system has passed over the conditional barrier, a complicated further dynamical evolution occurs, during which some of the prefragments may recombine or otherwise react, but eventually a number of distinct nuclear fragments emerge. This post-transition dynamics was subsequently studied in ref. [21 by treating the evolution of the interacting prefra~ents as a generalized damped nuclear reaction. In the present study we focus on the dynamics of the system prior to the crossing of the conditional multifragmentation barrier. This pre-transition stage of the multifragmentation process is important, because the system may radiate a considerable number of nucleons during the relatively slow shape evolution. As a consequence, the system typically suffers a significant loss of mass and energy before arriving at the conditional barrier, so the potential-energy surface is continually being modified and the breakup widths are co~es~ndin~y affected. In other words, at the time when the irreversible transition into an assembly of interacting prefragments occurs, the total system is smaller and colder than the original one and the statistical weights governing the branching ratios between the various multifragment channels must be modified accordingly. As the system evolves from a compact shape to one with well developed pre-fragments, the rate of particle emission and energy loss changes. Consequently, any study of mass and energy loss by radiation during the pre-transition stage has to be coupled to the dynamics of the fragmenting compound nucleus. In the present study, this is done within the framework of the tmnsition-state theory developed earlier [ 1,2], and we start with a brief review of this approach. Next, we describe a simple way of estimating the speed of the macroscopic evolution from the initial spherical shape to the conditional barrier where the irreversible transformation to an assembly of prefragments occurs. The equations governing the rate of proton, neutron and energy radiation are then presented along with two alternate prescriptions to estimate the rate of particle emission, namely the so called thermionic emission and the standard Weisskopf evaporation. The resulting coupled differential equations are solved nume~c~ly for a number of illustrative cases, and the results are presented and discussed in the conclusion. The problem of radiation losses prior to the actual breakup process is a general one, affecting any specific model for nuclear multifragmentation. The purpose of the present study is to elucidate the effects of the pre-transition nucleon emission and for that we employ for convenience the transition-state model developed earlier [ 11. It is to be ex-

J.A. Lbpez, J. Randrup / Nuclear multifragmentation III

petted that the general lessons learned will apply to other multifragmentation

381

models as

well and thus be of broader interest.

2. Transition-state theory We consider a highly excited “nucleus” characterized by its total nucleon number A and its total energy E. In actual applications the charge number Z is of course also included, but we omit this extension here for simplicity. We also ignore throughout the conservation of angular momentum, which has little bearing on the present study. We assume that any particular manifestation F of the system can be described as an assembly of N distinct (but interacting) prefragments, with masses, positions and momenta denoted by mn, r, andp,, respectively. As shown in ref. [ 11, this is a reasonable approximation near the top of the conditional barrier, where the transition current is calculated and beyond. For more compact configurations, which are of interest in the present study, such a description is more questionable. However, since only a few overall characteristics of the system are needed, it is possible to devise a reasonable treatment of the pre-transition configurations as well. For this purpose we employ r.m.s. size q as a measure of the spatial size of the system. For the spherical compound nucleus this quantity is given by qi = $R& where RO = roA’i3 is the nuclear radius. Near the barrier top, where the system can be approximated by N spherical prefragments, we have

m,(ri + $Rf,), where Rn = roA!,13 is the radius of the prefragment n and mo = c,, m,, is the sum of the prefragment masses. The corresponding conjugate variable is the radial momentum p = rn,-,i and is given by

Our first task is to devise a potential energy function I’ (q ) associated with the gradual transformation into a given multifragment channel A + A1 A2 ’ . . AN. Since the specific details are unimportant, we construct V (q ) by performing a simple interpolation between the known binding energy of the spherical compound nucleus and the conditional barrier top, V(q)

=

(3)

where qtOp is the r.m.s. size of the transition configuration and I&P = V(qtOp) is the corresponding deformation energy. This latter quantity is determined as in ref. [ 1 ] by writing the total binding energy on

J.A. Lbpez, J. Randrup /Nuclear multifragmentation III

382

the form (4) and assuming of the transition tributions,

that the interaction configuration

energy Vff!~N,for configurations

in the neighborhood

and beyond, can be written as a sum over pair-wise con-

For the interaction potential vp between the two prefragments i and j we employ the expression used in ref. [ 1] based on the parametrization made by Swiatecki [ 41; it gives a reasonable overall reproduction of the fission barrier heights and the associated nuclear deformations. The top of the conditional barrier for the particular multifragment channel considered is then determined by finding the maximum of the above expression (5) when scaling the prefragment positions r,, by a common factor. This procedure is expected to yield a reasonable estimate of the conditional transition surface (see ref. [ 1 ] for a further discussion) but fails seriously for more compact shapes where a pair-wise decomposition of the form (5) cannot be made. That is why we prefer to use the simple spline (3) which increases monotonically from its minimum at q = qo to its maximum at q = qtOP. In order to formulate the transition-state approximation to the disassembly problem, one considers the outwards probability current, i.e. the number of elementary multifragment states that pass by a given value q = q’ per unit time,

(6)

The flux factor p / mo in the p integration

can be thought of as arising from an integration

over values of q extending from q’ to q’ + p/ma, the distance covered by q per unit time. After division by h, the p integral then yields the number of elementary states that pass the specified value of q per unit time. Since (p/m0 )dp = dk, where k is the kinetic energy associated with the outwards flow, the integrations over k and E may be interchanged, so the former one can be performed analytically. The partial width for the system to pass over the conditional barrier (which forms a hypersurface in the multifragment configuration space) can then be written as rA,A2...AN (E)

= h uA’$jA;y)

,

X

( ( moq’i$l”‘N) ‘N-2 P1...N(El...N)T

L... N j,

(7)

383

J.A. Ldpez, J. Randrup /Nuclear multifragmentation III

where p(A, E) is the total level density

of the system

(which includes

all states lying

inside the transition hypersurface). We have here employed the stationary-phase approximation to evaluate the integration over the internal excitation energy E, and el...~ denotes the total excitation energy of the system when positioned at the barrier (and rt...~ is the corresponding temperature). The quantities in the bracket should be evaluated at the transition point (where the current v has its minimum) and then averaged over all directions in the hyperspace of fragment positions {r,} describing multifragment configurations that have been constrained to have a fixed center-of-mass position and an overall r.m.s. extension. (If angular-momentum conservation is included, this additional constraint must also be satisfied when performing the average.) The corresponding microcanonical sampling of fragment positions is easily done by means of the method described in ref. [ 5 1.

3. Pre-transition dynamics 3.1. INTRODUCTION

It is a quite complicated task to describe the detailed dynamics of the system inside the transition surface, because well-developed prefragments do not yet exist. Fortunately, for our present exploratory purposes it suffices to be fairly rough and we proceed as follows. The amount of kinetic energy associated with the outwards motion of the fragmenting system is equal to half the temperature, on the average,

The temperature is related to the internal excitation energy E in the usual manner and, using the simple Fermi-gas relationship, we have C/A = r*/eo, where we use e. = 8 MeV. Moreover,

the internal

excitation

energy can be obtained

E = V(q) +

by energy conservation,

;mo4*+ E ,

ignoring the energy associated with non-radial collective motion. We now make the rough assumption that the rate of outwards motion 4, at any point in the evolution, is given by the thermal speed in accordance with Eq. (8))

(10) where the relatively small energy associated with the radial motion has been ignored in the last relation. This should yield a lower bound on the time spent by the system before reaching the transition surface, since it is assumed that the radial motion is always directed outwards. We are now in a position to express the evolution of the mass and charge of the developing prefragmenting system as functions of the source size q by replacing dt by dq/q, using 4 from the above expression.

3.2. PARTICLE RADIATION

The evolution of a particle-emitting compound nucleus can be described by keeping track of the mass and energy being emitted. In particular we study the rate of loss of protons and neutrons as well as the rate of change of the temperature of the compound system. The rates at which the evolving source is losing neutrons and protons can be expressed in the following form, from the following expressions $N

= -EN&,

(11)

x

(12) Here the sums run over all ejectile species x that are being considered, with NX and Zx denoting the corresponding neutron and proton numbers. In the present study, only neutron and proton emission is being considered, as more massive ejectiles contribute relatively little. Furthermore, k, denotes the rate at which the specie x is being radiated from the source. The time evolution of the temperature of the emitting system can be obtained from the requirements of conservation of energy during particle emission 16 1. Thus consider a compound nucleus c (of mass number Ac and temperature rc) emitting a particle X, thereby transforming itself into the daughter nucleus d (of mass number Ad and temperature rd). By conservation of energy, the initial excitation energy of c, E’ (A=, 7c), should then be equal to the final energy, E*

(AC, 7~)

=

E’

b&f,

%f)

+ Bx

(13)

+ &

where E* (Ad, rd) denotes the total excitation energy of the daughter nucleus, B, is the separation energy for species x and KX is the kinetic energy in the final channel. The above expression ( 13) can be used to obtain temperature t, by noticing that

the time evolution

of the source

(14) where C, is the specific heat given by C, = &* jdr, with E* denoting the excitation energy per nucleon. Thus, adding up the ~nt~butions from the different ejectile species considered, we obtain the following expression for the evolution of the temperature, dr di=

--

1 AC”

(Bx -Axe*),&

+ &Kx

In the present work we use the specific heat of a Fermi-gas, denoting the Fermi energy.

1 .

C, =: r&/&r,

(15) with er

385

J.A. Ldpez, J. Randrup /Nuclear multifagmentation III

The rate at which kinetic energy is generated $Kx

J

=

mK;$K;

can be obtained

as [ 6 ]

= 22 (1 + $tx,

(161

v,

where VXdenotes the Coulomb barrier for the specie x. A complete description of the evolution of the system can then by obtained by coupling the rate equations ( 11) and ( 12) with expression ( 15) and solving them numerically. To relate the temporal evolution to the shape dynamics we turn these coupled equations into differential equations with respect to the generalized coordinate q using the formalism of sect. 2.2. In the resulting coupled equations the particle emission rates k, are needed and for this purpose we employ two simple alternative techniques models, thermionic emission and Weisskopf evaporation. For a comparison of these treatments techniques, see ref. [6]. 3.2.1. Thermionic emission We assume that, for each species, the emission rate can be obtained as k, = S,&, where S,, (q) is the effective total surface area of the prefragment n, calculated at the time when the system has expanded to the size q and A, is the rate of emission of species x from a standard nuclear surface, at the temperature rn of the prefragment. The effective surface area is estimated by simple interpolation between the compound configuration and the transition point, S,(q)

= 4nrz

qtopgn~(t)2/3

(

qtop

Here a,, (t ) = Ail31 c,, Ail3 is the compound nucleus and the second the fragment masses A,, all change at each time step. This procedure

-

+

40

4-n(t)2/3 @OP

-

. 40

(17)

>

prefragment’s share of the surface area of the spherical term represents its area at the transition point. Since with time, the quantities qo and qtOpare recalculated estimates S, (t ) as the surface area of the emerging

prefragment n corresponding to the radiationless multifragmentation of a source with the mass number A (t ) into fragments with masses A,, (t ), at the point when the size of the fragmenting system is equal to q (t ). The specific emission rate A, for nucleons is obtained in the standard manner using the Richardson method for thermionic emission (see ref. [ 7 1, for example). Considering the emission rate per surface area from a semi-infinite Fermi-Dirac gas of nucleons,

vn Px(En) - px)/rl

Gf!

“&=g J

v.>o

h3 1 + exp[-(6,

= Zsmra

Jde.1,

(1 + exp [--I)

V 2nw =

-77

Vx

2 exp

[ -~

+

Bx

7

Here g = 2 is the spin degeneracy for either neutrons

1 .

(18)

or protons, m is the corresponding

386

J.A. Ldpez, J. Randrup /Nuclear multifragmentation III

nucleon mass and pX is the chemical potential (equal to minus the separation energy Bx ). The temperature T should of course be taken to be the temperature of the particular prefragment responsible for the emission, T,,. The quantity PX(cn ) is the penetrability coefficient which depends on the energy of the nucleon in the normal direction, c,,. Neglecting quantum reflection and tunneling, we take this quantity to vanish below the barrier VXand to be unity above. The barrier height vanishes for neutrons, V, = 0, and for protonsweuse V, = exp(2(Z,, - 1)/R,,), where&(t) = r~A~(t)‘/~ istheradiusofthe prefragment. In deriving the last relation above, we have assumed that the exponential in the denominator is small compared with unity, as will be the case when the temperature is below V, + Bx MeV. The above expression ( 18) yields the rate of either neutron or proton emission, when the appropriate values of the separation energy Bx and the barrier Vx are inserted. Although thermionic emission lends itself to the introduction of particle reabsorption by neighboring fragments, this effect is not considered in the present study, since it was shown by exploratory studies to be a small effect. 3.2.2. Weisskopf evaporation As an alternative to the above described thermionic treatment, we have also carried out calculations based on the Weisskopf evaporation formula [ 8 1. The evaporation rate of particles x per unit energy is then given by

dkx-_ gxKxmx

cpd(E,‘)

-yaw

_,,x+d

a2?i3

dKx

(19)

where gX = 2SX + 1 denotes the spin degeneracy of the emitted particle x, while Pd (Ed ) is the density of states for the daughter nucleus at the resulting excitation energy Ei and pC(E,* ) is the similar quantity for the original nucleus c at the initial excitation energy. Finally, eX+&c is the cross section for the inverse reaction, the formation of the compound nucleus c by the absorption of x into d. Using a Fermi-gas approximation, the level densities can be related to the entropy function S, lnp(E*) = S(E*) and to the free energy f = (E’ - TS)/A. Using this and the geometrical cross section, the matter evaporated per unit time can be found by summing over the allowed energy interval [ 61, k, =

xi, J v,

xd& x

where y = m,r~/nh3 and r-0= 1.2 fm. The function

Kc(t) =

A~(A;-A~)

c

(20)

= ~2ew(-K/7)yE;;(rL

Fx(z)

is given by

1

(2Sx + 1)[(AE-AX)1’3

+

AY312 (21)

An expression

for the free energy can be obtained 2r2 f

5or2r2 C

=E*___ e0

from the literature [ 9, lo],

A;‘3

(7;

+ .r2)

T.5 72 ~. [ 7:: + 72

1

514

(22)

387

J.A. Lbpez, 1. Randrup / Nuclear multi~~agmentationIII

Again, e* = E*/A is the excitation energy per particle. We use eo = 8 MeV for the level-density parameter, u = 18 MeV for the surface tension and zc = 18 MeV for the critical temperature.

4. Results We are now in a position to illustrate the effect of the radiation loss on the various key quantities, namely the heights of the conditional barriers, the total mass and temperature of the system, and the partial widths for multifragment breakup. In line with refs. [ 1,2] we have considered the breakup of “‘Sn* into channels consisting of iV fragments of equal size. Figs. 1, 2 and 3 were obtained, for illustration purposes, assuming a symmetrical placement of the 3 fragments on a plane. In Figs. 4 and 5, for each value of the initial excitation energy E* and each mass partition AlAz ..I AN, a Monte Carlo simulation provided samples of 4000 set of fragment positions, in the manner described in ref. [ 11, which were then used to evaluate the constrained average in Eq. (7) for the breakup width r. A typical example is shown in Fig. 1 for a ternary breakup at an initial excitation of Ef = 3 MeV per nucleon. The curves show a steady decrease from the initial mass of A = 120 at the compound nucleus size qo toward the final mass value at the transition state, characterized by the size qtoP_The solid curve shows the development of the source mass as obtained with thermionic emission, while the dashed curve results when Weisskopf evaporation is used. The corresponding reduction of the source temperature z is

120 + 40 -I- 40 + 40 at 3 &V/A

4 8 2 2

115 -

s E z s ::

Thermionic

Emission

110 -

I 5

I

I

Source

I rms

I

I 6 size

I

I

*

I

I1

s

7 q (fm)

Fig. 1. Mass loss. The gradual reduction of the mass number A due to radiation of nucleons, as a function of source r.m.s. size q, as it evolves from a 120Sn* compound nucleus with an excitation of e* = 3 MeV per nucleon towards the conditional barrier for symmetric ternary breakup. Solid curve: thermionic emission; dashed curve: Weisskopf evaporation.

388 120 + 40 + 40 + 40 at 3 MeV/A

_ Thermionic

Emission

4.0 5 Source

6 rms size q (fm)

7

Fig. 2. Energy loss. For the same process as in Fig. 1 ( lzOSn* at E* = 3 MeV) is shown the gradual reduction of the temperature 7 as a function of the source r.m.s. size q. Solid curve: thermionic emission; dashed curve: Weisskopf evaporation.

shown in Fig. 2. In accordance with the results for the mass loss shown in Fig. 1, the use of thermionic emission produces a more rapid cooling of the system. The qualitative features of the two sets of results are the same, but there are signifkant quantitative differences, which serves to remind of the considerable uncertainty associated with predicting emission rates from highly excited nuclei. As mentioned above, the mass loss affects the position and height of the conditional barrier. To illustrate this effect, we show in Fig. 3 the effective potential as it would appear to the system at various stages along its evolution, as calculated using thermionic emission. Each curve shown is associated with the system as it appears when it has reached a given extension q and the corresponding point (q, V f q f ) is indicated on each curve by a solid dot. Individual curves are thus associated with a constant total mass, namely the value acquired at that particular source size. The first curve, associated with the solid dot closest to the lower-left comer, is then the potential energy of the initial system and thus it represents the barrier that the disassembling system would have had to overcome if no particle emission were to take place. Conversely, the last curve, associated with the dot closest to the upper-right comer, shows the potential energy barrier of the final system, when it has reached the size of the transition con~guration, qtop, and after it has radiated away a significant fraction of its mass. The sequence of dots thus outlines the effective barrier felt by the system on its way to the transition point. It is noteworthy to observe that the main effect of the particle loss on the potential barrier is to reduce the spatial extension of the transition configuration, while the overall height of the barrier remains fairly constant. The main effect on the multifragmentation width therefore arises from the loss of excitation energy and a suppression would be

J.A. Ldpez, J. Radar

/ Nuclear ~~tifrag~gntati~n III

389

120 + 40 + 40 f 40 at 3 MeV/A

Thermionic % 3

2O

P $

60

Emission

’ f:

G p 40 z “d J 20 !z 4

5 Source

6 rms size

7

8

q (fm)

Fig, 3. Change in barrier. FOTthe same process as in Fig. 1 (lzoSn* at E* = 3 MeV) is shown the potential energy function V(q) corresponding to various stages along the evolution from the initial sphere towards the top of the conditional barrier leading to symmetric ternary breakup. The nucleon radiation loss is calculated with thermionic emission; Weisskopf evaporation yields rather similar results. Each curve corresponds to the particular mass partition encountered when the system has reached the specified size q. The corresponding points (q, V (4)) are indicated by solid dots.

expected_ Since the radiation causes the transition configuration to be more compact, the resulting prefragments feel a larger mutual Coulomb repulsion, which might provide a diagnostic tool. The above illustrations pertain to one particular initial excitation energy and a single breakup channel placed in a spatially symmetric manner. In order to give a more global impassion of the effect of pre-transition emission, we show in Fig. 4 the average mass number of the system at the transition point, as a function of the initial excitation E* = E”/A and for a range of fragment multiplicities N. We see again that thermionic emission produces a larger mass loss (roughly about 5-30%) than the Weisskopf evaporation (a 510%). An inspection of the thermionic and Weisskopf expressions for the emission rates [ Eqs. ( 18) and (20) ] shows that one of the main differences between these methods lies in their dependence on the total surface area: whereas the thermionic emission treatment takes advantage of the enhanced oppo~unity for radiating as the surface area increases, the Weisskopf evaporation neglects this effect. This effect is most clearly borne out by the larger nucleon emission observed for higher fragment multiplicities. One might plausibly expect that this difference is the main cause for the differences in the two sets of results. As mentioned earlier, it should be expected that the change of nuclear potential and excitation energy will reduce the fragmentation probability. The combined effect of these factors on the breakup width &+..A~ (E) is shown in Fig. 5. To illustrate the conesponding suppression factor of a given channel as a consequence of the emission, the

J.A. Lbpez, J. Randrup /Nuclear multifragmentation III

390

L

2

D

60;

: Thermionic

Emission

Weisskopf Evaporation

:

Fig. 4. Final mass. For the same source as in Fig. 1 ( t20Sn* at c* = 3 MeV) is shown the average of the final mass number of the system, A(qt,), as it arrives at the top of the conditional barrier leading to breakup into N fragments of equal size; based on a sample of 4000 events. Left panel: Weisskopf evaporation; right panel: thermionic emission.

ratio of the calculated width and the width obtained without emission is presented, as a function of the initial excitation energy in the source. As expected from the previous results, a net reduction of the transition width is observed. The magnitude of the reduction ranges from factors of ten for low multiplicities to millions for larger multiplicities. It is reasonable to expect this reduction of the fragmentation width to depend significantly on the spatial configuration of the system. 5. Discussion In previous work, the transition state treatment of nuclear fragmentation was developed and applied to the calculation of multifragment breakup widths [ 11, and to describe the complicated dynamical evolution the fragments undergo after passing over the condi-

d

210-Q

Tbermionic Emission I....I....I.....I~~..I..~~I~~~.

Weisskopf Evaporation 6

2

Exc~ation

tnergy

2 EL/A ‘(f&V)

Fig. 5. Breakup width. As a function of the initial excitation energy in the source tzoSn* is shown the average relative effect of pretransition nucleon radiation on the breakup width I’ corresponding to breakup into N fragments of equal size. Left: Weisskopf evaporation; right: thermionic emission.

J.A. Lbpez, J. Randrup /Nuclear multifragmentation III

tional barrier [ 21. The present study addresses the dynamics

of the disassembling

391 system

prior to its arrival at the conditional saddle point, for the purpose of exploring the effect of pre-transition radiation on the multifragmentation process, via modifications in the potential-energy barrier, the masses and excitations of the prefragments, and the statistical weights of the different multifragmentation

channels.

Within the transition state formalism, we formulated a set of coupled equations for the mass and energy loss from an evolving source. Invoking two alternative radiation mechanisms, thermionic emission and Weisskopf evaporation, the loss of matter and energy was calculated for tin nucleus with an initial excitation energy of 3 MeV per nucleon. Both mechanisms lead to mass losses ranging from 10 to 30% by the time the system reaches the conditional barrier. The energy loss was expressed by the steady reduction of the temperature of the compound system as it moves through the pre-transition region and typical temperature drops of 20% were obtained. The main modification of the multifragmentation barrier was found to be a reduction of the r.m.s. size of the transition configuration. These effects of the radiation combine to produce a net reduction of the decay widths that varied in magnitude from a factor of 10 to lo6 for the combination of system, channels and energies considered. Since the fragmenting system thus experiences a significant cooling on its way towards the transition surface, the number and variety of available breakup channels is reduced. Although a relatively small effect at the higher excitations, this channel suppression can be significant at lower excitation energies (2-3 MeVIA) and it grows as the fragment multiplicity is increased, because of the associated increase in the height of the corresponding conditional barrier. Two factors were found to have a large effect on the reduction of the multifragmentation widths, namely the loss of excitation energy and the contraction of the transition configuration. A typical source formed in a central nuclear collision may possess a certain amount of ordered outwards motion, as a result of the initial c6mpression. It is relatively straightforward to include such a collective radial motion in the transition-state expressions for the partial widths and the effect will be analogous to the effect of imposing the strict conservation of angular momentum [ 21. The radial flow will shorten the time it takes the system to develop from a sphere to the transition configuration. Consequently, there will be less time for the pretransition radiation to occur and the effect of its effects will therefore be smaller. We therefore expect that the present type of treatment is likely to provide an upper bound on the effect of pretransition nucleon radiation. This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Nuclear Physics Division of the US Department of Energy under contract no. DE-AC03-76SF00098, Associated Western Universities through an EMCOM Faculty Fellowship, and by the National Science Foundation under grant PHY-930-3847. We also wish to acknowledge the warm hospitality extended to us by the theory groups at LBL (JAL) and GSI (JR) while this work was in progress, and to thank the authors of ref. [9] for lending us a code to test our implementation of the

392

J.A. Lbpez, J. Randrup /Nuclear muhifragmentation III

Weisskopf emission formula. References [ 1] [2] [3] [4] [S] [6] [7] [8] [9]

J.A. tipez and J. Randrup, Nucl. Phys. A503 (1989) 183 J.A. Lbpez and J. Randrup, Nucl. Phys. A512 (1990) 345 N. Bohr and J.A. Wheeler, Phys. Rev. 56 (1939) 426 W.J. Swiatecki, Aust. J. Phys. 36 (1983) 641 J. Rat&up, Comput. Phys. Commun. 59 (1990) 439; Nucl. Phys. A522 (1991) 651 W.A. Friedman and W.G. Lynch, Phys. Rev. C28 (1983) 28 G.P. Hamwell and W.E. Stephens, Atomic physics (McGraw-Hill, New York, 1955) V.F. Weisskopf, Phys. Rev. 52 (1937) 295 H.W. Barr, J.P. Bondorf, R. Donangelo, I.N. Mishustin and H. Schulz, Nucl. Phys. A448 (1986) 753 [IO] E. de Lima Medeiros and J. Randrup, Phys. Rev. C45 (1992) 372