distinction between multifragmentation mechanisms from imf-imf correlation functions

N U C L E A R P H Y S I C S A

ELSEVIER Nuclear Physics A 594 (1995) 156-174

Distinction between multifragmentation mechanisms from IMF-IMF correlation functions

S u b r a t a Pal 1 Saha Institute of Nuclear Physics, I/AE Bidhannagar, Calcutta 700 064, India

Received 6 May 1995; revised 28 July 1995

Abstract

In intermediate energy heavy ion collisions, hot nuclear systems formed may disassemble by any of the three distinct mechanisms, namely, sequential fission, prompt multifragmentation or sequential evaporation. For identical multifragment channels considered for these three mechanisms, the correlation functions for intermediate mass fragments (IMFs) which are sensitive to the spatial-temporal extent of the emission process are exploited to delineate the different possible break-up scenarios.

1. Introduction

In recent years, intense effort has been devoted to the study of the formation and decay of hot and compressed nuclear systems in intermediate energy heavy ion collisions.

Such excited nuclear systems decay predominantly by the emission of multiple intermediate mass fragments (IMFs: 3 <~ Z ~< 20), a process commonly referred to as nuclear multifragmentation. To unravel the underlying reaction mechanism pertaining to multifragment final states, a large number of experiments have been performed [ 1-8 ] and various theoretical models [9-16] have been constructed. These models may be broadly classified into two groups: dynamical and statistical. In the dynamical Boltzmann- Uehling-Uhlenbeck (BUU) model [9], each nucleon traverses in the ensemble averaged field of all the other nucleons and experiences two-body collisions. As a result, the correlations and fluctuations are considered only partially through binary collisions and therefore the model may not be reliable beyond one-body observables. The microscopic

1 E-mail: subrato @saha.emet.in

0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9474(95)00367-3

S. Pal~Nuclear Physics A 594 (1995) 156-174 157

quantum molecular dynamics (QMD) calculations [10] overcome this limitation, but they lack a proper description of fragment formation. On the other hand, the statistical models, in spite of disregarding the initial dynamics of the reaction, are found to be quite successful in explaining several observables related to multifragmentation phenomena. The inherent assumption in these models is that prior to multifragmentation the hot nuclear complex is in thermodynamic equilibrium. The statistical models in use can be classified into three main groups. In the first two groups, together referred to as sequential binary decay (SBD), the excited nucleus decays either by a sequence of binary fission events [2] (which hereafter will be referred to as sequential fission (SF)) , or evaporates one fragment after the other [ 15,16] (which will be referred to in the following as sequential evaporation (SE)) . The third group of statistical models assumes a one-step prompt multifragmentation (PM) within a certain expanded freeze-out volume. In spite of considerable efforts, the study of the extracted physical observables, however, fails to provide an unambiguous delineation of the reaction processes to date. This warrants more exclusive experimental information as well as theoretical studies to find clear signatures to discriminate between the different underlying reaction mechanisms in multifragmentation.

This has motivated us to examine the kinematical differences for identical multifragment channels produced by the three decay scenarios with the aim to delineate the underlying reaction dynamics. This approach is similar to that followed by L6pez and Randrup [ 17] where an attempt was made to find some differences in the proton energy spectra, the event shapes from the velocity distribution and the folding angle distribution between the heaviest two fragments as a signature to distinguish between PM and SF. In a later study [ 18], the focussing effect of charged particles in the Coulomb field of the two heaviest fragments in PM and SF was investigated for a possible signature.

The relative velocity distributions between the intermediate mass fragments are sensitive to the space-time characteristics of the fragmentation process. Contrary to the proton-proton correlation function [ 19-27] where quantum statistics and the mutual interaction between the coincident proton pair are important, for the IMF-IMF correlation function [28-38], the quantum effects have a minor role to play; their mutual interaction as well as the interaction with the neighbouring fragments govern the shape of the correlation function. Moreover, the IMFs carry on the average large momentum which induces dynamical correlations arising out of energy and momentum conservation. For the PM scenario, the notion of temporal evolution of the source does not arise, instead correlations in the source, i.e., pre-emission correlations between all the fragments play a crucial role in dictating the shape of the correlation function [36,38]. In contrast, for SF, in addition to the temporal effects, the presence of multiple sources of IMF emission may manifest in the final shapes in the correlation functions. For the decay of a nucleus by SE, since a heavy charged fragment is present at nearly all stages of the decay, its Coulomb effect on the evaporated charged fragments may reveal features in the correlation function, which may be distinct from those of the other two decay processes. This suggests that the IMF-IMF correlation may be a key to understanding the multifragmentation mechanism. It is therefore instructive to explore the IMF-IMF

158 S. Pal~Nuclear Physics A 594 (1995) 156-174

correlation functions from the decay of a hot nucleus by SE PM and SE for a possible distinction between these break-up mechanisms.

The remainder of the paper is organised as follows. The theoretical framework is described in Section 2. In Section 3, the results and discussions are presented. The summary and conclusions are given in Section 4.

2. The models

We consider the decay of a thermally equilibrated nuclear complex characterised by its baryon number A, charge Z and excitation energy E*. In this section, we present an outline of the sequential fission model and therefrom the construction of the events in prompt multifragmentation and sequential evaporation models. In the last part, the construction of the IMF-IMF correlation function employed in the paper is discussed.

2.1. Sequential f ission

In this decay mechanism, the initial excited nucleus breaks up into two fragments. The daughter nuclei may be sufficiently excited and therefore undergo further binary decay. The decay chain continues till all the sequentially generated fragments are stable against any further particle emission. It is instructive to note that though we refer to this process as sequential fission, all possible binary channels from nucleon evaporation to symmetric fission have been considered.

According to the transition-state model of Swiatecki [39], the decay probability of a nucleus with charge Z, baryon number A and excitation energy E* into two fragments (A1, Z,) and (A - A1, Z - ZI) with relative kinetic energy K at the saddle point is given by

o( exp [2~/a (E* - VB - K) - 2 v / ' ~ ] . P ( A , Z , E * ; A I , Z I ) (1)

The barrier height, VB, in the two-sphere approximation may be written as

Va (Ts) = Vc + VN + Esep (T0, TS), (2)

and its position and height are calculated numerically. The temperature of the fissioning nucleus is given by To = V/-~-/a, a being the level density parameter taken to be a = A / I O MeV -1. The temperature Ts of the system at the saddle, which is also the temperature of the fragmented daughter nuclei, is given by

Ts = x / ( E * - VB -- K ) / a . (3)

As is evident from Eq. (3), the evaluation of Ts requires the knowledge of the kinetic energy of the relative motion K of the two fragments which is assumed to follow a thermal distribution P ( K ) ,,~ v/-g e-K/rs. Because of the complicated interrelationships among VB, K and Ts, a fully consistent determination of Ts is quite involved. This


problem can be circumvented by replacing K in Eq. (3) by its average value 1.5Ts, and then Ts is evaluated by an iterative procedure with To as the starting value. This is expected to be a good approximation as the dispersion in kinetic energy is ,-~ Ts and E* - VB is generally much greater than Ts. The so-extracted value of Ts is used only to evaluate the barrier VB from Eq. (2) and thereby the decay probability from Eq. (1) and the thermal distribution. In Eq. (2), the Coulomb interaction Vc is taken to be that between two uniformly charged spheres. The nuclear interaction VN between two fragments of masses Al and A2 is classified in three groups depending on their masses [40,41]. (i) A1 ~< 4, A2 ~< 4; the nucleon-nucleon interaction is taken to be a gaussian with range parameter 1.5 fm in close parallel to the one-pion exchange potential, and the depth is determined by reproducing the deuteron binding energy. The interaction among the fragments is then calculated by folding the nucleon-nucleon interaction with a gaussian density distribution. (ii) Al ~< 4, A2 > 4; the interfragment interaction is taken to be the real part of the optical potential [42]. (iii) In the mass range given by Al > 4, A2 > 4, the proximity interaction of Blocki et al. [43] has been used.

The temperature dependent separation energy is taken as

Esep(T0, Ts) = B ( T o ) - B l (Ts) - B2(Ts), (4)

where B, B1 and B2 refer to the temperature dependent binding energies of the parent nucleus and the daughter nuclei. The binding energy given by the Weizs~icker formula, modified to include the temperature dependent surface energy, is taken as

zr2Z 2 B ( T ) = 15.677A - 28 (N - A Z)2 1 8 " 5 6 ° ' ( T ) A 2 / 3 - 0"717A~3 + 0.245 ~ .

(5)

The surface tension constant o-(T) is taken as [44]

~ ( r ) = ,~(0) 1 + 5 T c 1 - Tc} ' (6)

with

o-(0) 0.951711 1.7826 (A ZA-2Z)2 ] = - . ( 7 )

The critical temperature T¢ is taken to be 16 MeV. Once Ts is known, the relative kinetic energy K of the fissioning fragments lying

in the range 0 ~< K ~< E* - VB is generated in a Monte Carlo technique obeying the thermal distribution mentioned earlier. Energy conservation is ensured by plugging this kinetic energy into Eq. (3) and recalculating the temperature of the daughter nuclei. The fragment kinetic energies and hence their velocities are obtained from energy- momentum conservation. The direction of emission of one of the fragments is generated randomly, the other moves in the opposite direction. The position coordinates of the


fragment centres at the saddle are dictated by the direction of the relative fragment velocity. They are so taken that the centre of mass of the daughters is the same as that

of the parent. The trajectories of the fragments are calculated in the overall centre of mass frame, and

if the generated fragments have sufficient energy, they may decay in flight. The lifetimes of the excited nuclei against binary decay could be calculated from the transition-state

model itself. However, the absolute value for the decay width predicted in this model is generally found to be too small though the branching ratios in various channels are

reasonable. In Ref. [45] the decay lifetimes for fragments have been calculated in the transition-state model and are found to be occasionally larger by orders of magnitude

compared to those in the Weisskopf model. In the absence of any better prescription,

for the lifetimes of emission of a fragment of mass A~ from a source at temperature T, we have taken a simple parametrised form [ 18],

r = 2e l3 / re &/8 [ fm/c ] , (8)

obtained from a good fit of the available data for the lifetimes for emission of different particles. The lifetimes given by Eq. (8) are in general found to lie in between those calculated in the transition-state and Weisskopf models. Because of the uncertainty involved in the lifetimes, we have also performed calculations with wide variations of

r. The generated fragments are allowed to evolve in the overall centre of mass frame till the total Coulomb interaction energy is very small ( ~ 1 MeV) and the excitation energies of all the fragments are below particle emission threshold.

2.2. Prompt multifragmentation

In order to facilitate comparison between the sequential fission event and that from prompt multifragmentation which differ only kinematically and contain same multifrag-

ment yield, i.e., have the same total energy and multiplicity distributions, we follow the

approach of L6pez and Randrup [ 17]. In this method, the cold multifragments generated in a SF event are placed randomly and most compactly but in a non-overlapping manner within a spherical configuration. The total Coulomb interaction energy V in this configuration is then calculated. ( I f it exceeds the asymptotic kinetic energy K ~ of the

corresponding SF event, V is reduced by a uniform expansion of the system to V = K ~ ) . The excess kinetic energy K = K ~ - V is distributed in the translational degrees of free-

dom of the fragments by employing a maxwellian distribution. The momenta of the fragments are boosted in the overall centre of mass system and are finally normalised to the total available kinetic energy K. To avoid a situation where the neutrons would posses zero velocity, we assume that the total kinetic energies of the neutrons in SF and PM are the same [ 18]. The integration of the trajectories of the fragments under mutual Coulomb repulsion from the freeze-out state is continued until they have attained their asymptotic velocities.

It is instructive to note that the particles emitted from the freeze-out state in PM are assumed here to be particle-stable, whereas in Refs. [ 17,18] the fragments in the

S. Pal/Nuclear Physics A 594 (1995) 156-174 161

expansion stage contain some excitation energy and are allowed to deexcite by sequential binary decay. In general, a fragment can be excited to a level with a lifetime larger than the explosion time and decay subsequently. With this constraint on the excitation energies, it is seen [ 11 ] that inclusion of particle-unstable states does not critically affect the observables, which validates our assumption. Moreover, the IMF-IMF correlation functions in the decay by PM was observed to have a significant sensitivity only to the dynamical correlation in the source [36,38]; evaporation from the fragments primarily by light particles at very late stages of the evolution at relatively small temperatures is expected to play a minor role in determining the shape of the correlation function.

2.3. Sequential evaporation

In contrast to SF and PM, the decay mechanism by sequential evaporation considers the successive evaporation of particles starting from the initial excited nucleus. Unlike SF, the emitted fragments are all assumed to be cold and therefore do not undergo any further particle emission. To obtain a SE event for each event generated by SF (or PM) which consists of exactly the same fragments and has the same total energy but differs in the velocities of the individual fragment, the following scheme has been employed.

Starting with a nucleus of baryon number A, charge Z and excitation energy E*, the decay probabilities of all the fragments but the heaviest one, generated in the corresponding SF event, are calculated using Eq. ( 1 ). These fragments are then allowed to evaporate one by one, which sequence is determined through a Monte Carlo procedure employing the decay probabilities calculated at every step of the particle evaporation. The positions and velocities of the particle and the residue at each decay are obtained in a similar manner as in the SF model. However, in contrast to SF, the evaporated particles are assumed to be particle-stable; the total remaining excitation energy after overcoming the barrier at every decay stage is then contained in the residual nucleus. At any stage of the evaporation process, the residual nucleus and the evaporated particles evolve under mutual Coulomb repulsion. The time interval between two successive evaporations is governed by Eq. (8). The largest fragment obtained in SF is not taken into account in the decay probability calculation in the SE model so that it is constrained as the final evaporation residue. The asymptotic velocities of the fragments are normalised so as to match the total kinetic energy generated in the corresponding SF event.

For all the three decay mechanisms, the effect of angular momentum at any stage of the calculation has not yet been considered. Inclusion of angular momentum alters the decay widths for particle emission. However, since the same isotopic yield has been employed in all the three models, it is expected that the influence of the angular momentum on the correlation functions would not be very significant as has been shown in the later part of the paper. To avoid further complications, we have not considered the recombination of the particles in close proximity during their dynamical evolution [40,41 ]. The effect of neighbouring fragments on the fission barrier in SF and SE [41] has also been

neglected.


2.4. IMF-IMF correlation function

The IMF-IMF correlation is studied as a function of reduced velocity Vred of the IMF pair given by

Urel Ured = ~ , (9)

where Z, and Z2 are the charges of the two IMFs and Vre, is the asymptotic relative

velocity. The use of reduced velocity has the advantage over Vrel that the IMF-IMF correlation functions for different IMFs would display a similar dependence on Vred [28,29]. Because of the use of IMFs in these correlation functions, quantum effects are expected

to be negligible but the energy-momentum conservation and the interaction with other fragments in the disassembling system seem to play a very important role.

Using the classical approximation of the Koonin-Pratt formula for intermediate mass fragments [28,29], the IMF-IMF correlation function may be written as

E r'2(Ured) (10) 1 + R(Vred) = ~ Yback(Vred) '

where I~2(Vred) is the coincidence yield and Yback(Vred) is the background yield constructed by event mixing, i.e., by randomly selecting each of the fragment pair from

different events. It may be pointed out that in many studies of the two-particle correlation function, an a priori arbitrary normalisation of the correlation function is used so

that R(Vred) tends to zero at large Vred. This may be true for proton-proton correlation functions, but the IMF-IMF correlation functions are sensitive to energy-momentum conservation and therefore, in general, the correlation function R(Vred) may not con- verge asymptotically to zero [ 36]. So we have not introduced any arbitrary normalisation of the correlation functions.

Additional information on the break-up scenarios for multifragmentation may be

obtained by employing directional cuts on the two-fragment correlation functions. It has been observed [29,34] that the directional correlation functions are sensitive to the strength of the final-state Coulomb interaction of the coincident IMF pair with the residual system. Recognizing its potential utility, the directional correlation functions for the decay mechanisms are also studied in this paper. The directional correlation

functions are constructed by employing cuts on the angle ~ = c o s - ' ( IP. Vredl/PVred) between the reduced velocity Vred and the total momentum P = Pl +/72 of the two coincident IMFs with momenta p~ and P2. In this paper, the longitudinal and transverse correlation functions are calculated by taking 0 ° ~< $tong ~< 40 ° and 75 ° ~< Ctrans ~< 90 °, respectively.

3. Results and discussions

To delineate the different reaction models pertaining to nuclear multifragmentation through IMF-IMF correlation functions, we have performed calculations with 197Au as


1021Z i I I I J I

101 ~o

_ _

D io -2 ~ " • ~E o ". %

10 .3 ~ % % • %

I ° I 1 % I I I ° 10 20 30 40 50 60 70

Z

Fig. 1. The charge distributions from the decay of 197Au in sequential fission at excitation energies per nucleon of 3 MeV (open circles), 6 MeV (solid circles) and 9 MeV (squares).

a representative system. In Fig. 1, the inclusive charge distributions from multifragmentation by sequential fission are displayed at three excitation energies. At a low excitation energy of 3 MeV/nucleon, the nucleus decays predominantly by light particle emission and symmetric fission. The charged particle multiplicity distribution P ( N c ) , depicted in Fig. 2 for this energy, is nearly gaussian in shape and peaks at Nc ~ 10. The corresponding multiplicity distribution P (NIMF) for intermediate mass fragments, shown in Fig. 3, is Poisson like and is peaked at NIMF = 0, indicating that IMF emission is very improbable at this low energy. At a moderate excitation energy of 6 MeV/nucleon, the production of heavy fragments is suppressed at the expense of light and intermediate

100

ulO Z

O-lff~

I I

o 0

o 0

0

o

l I I I I I I

.'.. ooo~o 0 0

• • 0 0

• • 0 0 o 0

0

0 0

O Q • 0 0

O -

o Oo O"

I i [ I I I I i i 0 5 10 15 20 25 30 35 40 45 50

Nc

Fig. 2. The probability distributions of the charged particle multiplicity P ( N c ) from the decay of 197Au in sequential fission. The symbols have the same meaning as in Fig. 1.


10 0

_1

10

:7- 10-: o_

I

o o

10

?° 0

I i I I I i I

o • ~ 0 @ Q

O 0 o O

o o O •

o o

o o 0

I I I I I I I 2 4 6 8 10 12 14 16

NIMF

Fig. 3. The IMF (3 ~< Z ~< 20) multiplicity distributions P(NIMF) from the decay of i97Au in sequential fission. The symbols retain the same meaning as in Fig. 1.

mass fragments. The charged particle multiplicity distribution is peaked at Nc ~ 23 (Fig. 2) and the P(NIMP) distribution (Fig. 3) is gaussian peaking at NIMF ~ 7. At a higher excitation energy of 9 MeV/nucleon, it is evident from the charge distribution that heavier fragments disappear. Large numbers of charged particles are produced with

the maximum of the P (Nc) distribution occurring at Nc ~ 40. However, the most prob-

able value of IMF at this energy is only one unit higher compared to E*/A = 6 MeV.

This is quite understandable. It is now known that with an increase in bombarding energy (excitation energy), the IMF production shows a "rise and fall" pattern [7]. For this particular system, we have found that the maximum number of IMF of N 8.3 is produced at an excitation energy per particle of ,-, 8 MeV. The IMFs produced at

E*/A = 9 MeV correspond to the falling part of the IMF production pattern. In Fig. 4, the ensemble averaged values for the time evolution of various quantities

from the break-up of 197Au in SF are displayed at E*/A = 6 MeV. The top panel of

this figure presents the dynamical evolution of the average number of IMFs (NIMF), the average number of charged particles (Nc), and the average value of the root mean square radius of the disassembling system (Rrms), as a function of time. The bottom panel shows the ensemble averaged values for the time evolution of the charge of the first, second and third largest fragment in an event. It is observed that (Nc) saturates at a time of ,-~ 100 fm/c. The IMFs are produced in the very early stage of the reaction when the sources are still very hot and it saturates at an earlier time of ,-~ 50 fm/c compared to (Nc}. At early times when the IMF production rate is quite high, the root mean square radius (Rrms) of the fragmenting system increases rapidly with time. When IMF production practically ceases, the radius shows a constant expansion rate indicating as if the fragments are moving outwards in an effectively force-free region. The constant radial expansion rate is found to be practically the same as that of the average value of the asymptotic fragment velocities. From the bottom panel of Fig. 4, we find that the


E

E i_

v

A

U

Z v

Z v

A

W

v

25 i

2O

15

10 . / "

.'/

0 I

70'

60

50

30

20

100~" I . . . . . .

0

/ /

/

/ / . . . "

/ . . ' ' "

I I I I I

/ /

/

. . . . (NIMF: (Nc) _

. . . . . . ¢ ,R~r .s >

I I I I

- - (Zmax>

- - -(Z2>

. . . . . (Z3>

I I I I I I

20 40 60 80 100 120 140 160

t ( f m / c )

Fig. 4. The ensemble averaged values of NIMF, NC and the root mean square radius Rrms (top panel) as a function of time. The bottom panel exhibits the ensemble averaged values of the largest three charges as a function of time from the decay of 197Au at an excitation energy of 6 MeV/nucleon in sequential fission.

first break-up of the initial excited source is nearly symmetric. With time, they further disintegrate and the charges of the first three heaviest fragments reach their asymptotic

values at t ~ 60 f m / c when IMF production has already ceased and (Nc) has nearly

reached the saturation value. It may be noted that all the three heaviest fragments are IMFs in most of the events. Almost all these features are qualitatively retained in the other two excitation energies studied in this paper.

In the remainder of the paper, we will investigate the sensitivity of IMF-IMF correlation functions to the three different decay mechanisms with emphasis on an excitation energy of 6 MeV/nucleon which is close to the limiting temperature of the decaying system [44,46]. For this end, we will first describe in detail the features exhibited by the correlation functions from decay through sequential fission.

In Fig. 5, the angle-integrated IMF-IMF correlation functions for different windows

on the events containing a certain number of produced IMFs (NxMF = 2, 4, etc.) are displayed. The correlation functions show a pronounced peak around Vred ~-- 0.017c which gradually dissolves with an increasing number of IMFs. The correlation function considering all events containing all possible number of IMFs is also shown. For this case, (NIMF) ,~ 7 which explains the overall position of this correlation function relative to the other functions. All the correlation functions exhibit pronounced anticorrelations at small u=d. This is a manifestation of the final-state Coulomb repulsion of the two co-


t -

r Y

Jr

2.0

1. - , - -

1 . 0 -

0.5-

0.0 0

I I

SF

I 5 10

I I I I t I

- - oil IMFs .-...

" ...... 2 IMFs ." "..

" 4 I M F s

!. ' . "......

F I I " " " * ' " ' " * " " " . . . . . .

:!

E * / A -=6 MeV

I I I I 15 20 25 30

Vred (I0 -3 c )

i I

35 40

Fig. 5. The angle-integrated IMF-IMF correlation functions from the decay of 197Au at E*/A = 6 MeV by sequential fission (SF). The solid line refers to calculations in which events with all possible number of IMFs in the final state are included. The other curves represent calculations in which events with fixed number of IMFs in the final state (as given in the legend) arc considered.

incident IMFs which prevent them from having similar velocities resulting in a Coulomb

hole at small Vred. For large Vred, the correlation functions are significantly lower than unity for events with a smaller number of IMFs and gradually tend to unity for events

with larger NIMF. In order to understand the shapes of the correlation functions, we first refer to only

two-IMF events. From the bottom panel of Fig. 4, we find that the first break-up of the

initial excited source is near-symmetric, resulting in two heavy fragments moving back to back which are therefore highly correlated. The two IMFs might originate from these

two sources and because of the small time interval in which the IMFs are produced would contain the imprint of this correlation. If the two IMFs come from a single source, the correlation is even stronger. All these effects result in a pronounced peak in the correlation function for two-IMF events. In case of events with more IMFs, there are multiple sources of IMF production; the presence of a large number of charged fragments dilutes the role of mutual interaction between the coincident IMF pair on the correlation function, leading to a gradual suppression of the peaks. The correlation functions from PM and SE exhibit a similar behaviour with varying number of IMFs.

It would be of interest to compare the correlation functions for the multifragment yield originating from different decay mechanisms. This is depicted in Fig. 6 from the break-up of 197Au at E*/A = 6 MeV. The correlation functions displayed here and hereafter comprise all events with all possible numbers of IMFs. It is observed that the angle-integrated correlation function for SF shows a peak which is suppressed for PM and attenuated further for SE. From the procedure employed to generate the freeze-out state in PM from the SF channel, the freeze-out volume for the allowed most compact configuration averaged over all the events is found to be about three times the initial


I i I i I i i 1 . 5 - E * / A =6 M e V

-... - - . . .

,~ 1.O-

cr SF

4- :i PM 0.5- .iSI...i . . . . . . . . SE

.: i I

0.0 I _ ~ I i I I 0 5 10 15 20 25 30 35 40

Vred (10 -3 c)

Fig. 6. The angle-integrated correlation functions from the decay of 197Au at E*/A = 6 MeV by sequential fission (SF), prompt multifragmentation (PM) and sequential evaporation (SE).

volume of 197Au. The corresponding average freeze-out radius is then (gf) ~ 10.0 fm.

Since in PM the fragments originate from the freeze-out volume simultaneously, the

correlation inside the source plays a very crucial role. The Coulomb repulsion exerted

simultaneously by the charged particles situated in between an IMF pair forces the pair to large reduced velocities [36,38]. As a result the mutual interaction of the IMF pair becomes less important leading to a dilution of the correlation peak. The correlation function from sequential fission is comparatively more peaked mainly because of the following three reasons. (i) The IMF production is complete in a very short time ~ 50 fm /c . Since IMF production is sequential, there are possibilities of early IMF emission. The IMF pair then becomes highly time-correlated. The effective volume in which the IMF pair is produced may also be comparatively small (corroborated from the evolution

of the rms radius shown in Fig. 4) as a result of which the mutual interaction between the fragments in the IMF pair mainly governs the correlation function. (ii) The number of charged particles produced in the early stages of IMF emission in SF is small in contrast

to that in prompt multifragmentation where all the charged particles are produced at the same time. The smaller number of charged particles causes less distortion on the mutual interaction of the coincident IMF pair. The light charged particles produced at later

stages in SF are expected to have a minor influence on the mutual interaction and hence on the correlation. (iii) The possibility of an enhancement of the peak of the correlation function might also arise for the following reason. At relatively later stages of IMF emission, it may so happen that two IMFs are generated from the binary decay of a heavier fragment. I f the two IMFs do not undergo any further particle emission, a very strong correlation is expected between them. In fact, the significant enhancement in the a a correlation function observed in the reaction 160 -3 t-27 AI at 140 MeV [47] has been indicated as a preponderance of events where preformed 8Be decays into a particles.


2.0 I I I I I I I I

o 4 ~,,(4o 1.5--

1.0 _; ~'- . ~ -

- ,4 0.0 ~ , [ i i I I I

+-..- 75°,,( ~ 4 90° , ~

1.0 - -' "~ i J''''i i I

0.5 ~:" I"

ii. f

/.'" t I 0.0 I .~ I 5 10 15 20 25 30 35 40

Vred ( 10-3 c)

Fig. 7. The longitudinal (0 ° ~ ~ ~ 40 °) (top panel) and transverse (75 ° ~ ~ ~ 90 °) (bottom panel) correlation functions from the decay of 197Au at E*/A = 6 MeV by SF, PM and SE.

However, in SF where a large number of different IMFs are produced from different sources, this effect may be diluted.

In case of the decay of a nucleus by sequential evaporation, in Fig. 6, we do not see

much of a structure in the correlation function at E * / A = 6 MeV. Here a large number of fragments emitted sequentially in random directions from a single source does not

have much room for building up correlation. To study the dependence of the correlation function on the ordering of fragment emission in SE, calculations were also performed for the random emission of fragment species (corresponding to those generated in SF)

instead of using the decay widths of Eq. (1). The correlation function was found to be almost insensitive to this ordering of emission in SE. It may, however, be pointed out that in some experiments in intermediate energy heavy ion collisions [30,32,33], peaks in IMF-IMF correlation functions have been observed.

The angle-integrated (0 ° ~< ~b ~< 90 °) IMF-IMF correlation functions probe the volume of the phase-space distribution of emitted particles with little sensitivity to its shape. Information on the shape of the phase-space distribution of fragments may possibly be gained by employing directional cuts on the correlation functions. The longitudinal

(0 ° ~< ~b ~< 40 °) and transverse (75 ° ~< ~ ~< 90 °) correlation functions are exhibited in Fig. 7 for the three decay mechanisms. In comparison to the angle-integrated correlation functions presented in Fig. 6, an enhanced difference in the longitudinal correlation


function is observed (top panel of Fig. 7) for the sequential fission model with the other

two decay mechanisms. On the other hand, the distinction between transverse correlation functions (bottom panel of Fig. 7) in SF and PM is nearly washed out. Simultaneous investigation of the angle-integrated, longitudinal and transverse correlation functions reveals that the correlation function is more sensitive to the directional cut for the SF model as compared to the other two models and therefore may serve as a tool for possible delineation of the reaction mechanisms.

Two-fragment correlation functions are sensitive to the strength of the final-state Coulomb interaction between the fragments in the coincident pair as well as to their

interaction with the source, which in turn depend on the time scale of fragment emission in the sequential binary decay model. The simple estimate of the lifetimes used in

Eq. (8) is not very accurate; we have therefore repeated the calculation with lifetimes

0.1, 25 and 50 times the lifetime r given by Eq. (8) in the sequential binary decay model. The sensitivity of the angle-integrated correlation functions on the lifetime of

particle emission is illustrated in Fig. 8 from the decay of 197Au at E*/A = 6 MeV in

SF (top panel) and SE (central panel). A longer emission time corresponds to a larger initial separation between the fragments, resulting in a weaker Coulomb interaction between them. This is manifested in a gradual suppression of the peak with a narrower Coulomb hole in the correlation function with increasing lifetimes.

To explore the sensitivity of the angle-integrated correlation functions on the freeze-out density in PM, we have performed calculations with freeze-out densities of p = po/6 and p = p0/lO, where P0 is the normal nuclear density. These correlation functions along with that obtained for the allowed most compact configuration (with (p) = po/3 as mentioned earlier) are displayed in the bottom panel of Fig. 8. With increasing freeze-

out volume, the Coulomb interaction gets diluted, which is reflected in the somewhat suppressed peaks for the expanded volume. In this context it may be mentioned that

the directional correlation functions display an almost similar sensitivity as the angle- integrated ones for varying lifetimes and freeze-out densities.

To see the effect of angular momentum on the correlation functions using the same isotopic yield (obtained from SF) for the three models, the calculations have been

repeated with an angular momentum of J = 75h deposited in the fragmenting nucleus. Inclusion of the angular momentum in the decay widths has been done following the prescription of Charity et al. [ 15]. The angle-integrated IMF-IMF correlation functions with and without angular momentum are compared for all the three models in Fig. 9. The

heights of the peaks of the correlation functions are found to be somewhat suppressed in all the three models when the angular momentum is included; the shifts of the

peaks are, however, nearly the same. The suppression in the peaks is a reflection of the somewhat larger number of IMFs produced, which dilutes the role of mutual Coulomb interaction between the IMF pair. The effects on the directional correlation functions are

also similar. To assess the sensitivity of the angle-integrated IMF-IMF correlation function on the

excitation energy of the fragmenting system for different decay scenarios, we have also calculated the correlation functions for break-up at E*/A = 3 and 9 MeV. These are


I I ! I I I I i

1.5 - S F .. e

1.0- / • ¢ ~ .........

h ." ! / ! :

0.5- .,, [- .1~ II .,

, I s .."

0.~ - ' F ~ ' i I I I I I I m

S E

- - 0.s- 7 / / - - - - i i -~ 1 , ; , ' / ~ ! .o ~ J z . ; . j . . . . . . so I

PM , . , . ~ 1.0 - , : ' - - .-..-..--

': - 9 ~ <e) =

o . s - ,,.../ . . . . . . e = eo/S - , , : . .y ---- ~ = eo?lO

0.0 _~-'..'.~-~/t i i I i -' i I

5 10 15 20 25 30 35 40

Vred (10 -3 c)

Fig. 8. Dependence of the angle-integrated correlation functions from the decay of 197Au at E*/A = 6 MeV on the lifetime z of particle emission in SF (top panel) and SE (central panel). The dependence on the freeze-out density is displayed in the bottom panel for the same reaction in the PM model. For details, see text.

shown in Fig. 10. At the high excitation energy of 9 MeV/nucleon, a large number of small fragments are produced within a very short time. The differences in correlation functions for SF and PM then narrow down considerably; the correlation functions

in all the models are nearly flat beyond the Coulomb hole. This may be understood from the fact that at this high excitation the number of emitted charged fragments is very large; they provide a strong residual interaction which dilutes the effect of the mutual interaction of the coincident IMF pair. The memory of rapid particle emission and thereby smaller spatial extent of the decaying system in SF compared to PM is, however, contained in the wider Coulomb hole for SF. In addition to a large number of charged particles generated within a very short time, the effect of isotropic emission from a single relatively heavy charged fragment in SE causes a significant enhancement in the velocities of the IMFs. This is reflected in the correlation function which tends to a value greater than unity at large urea.

For the break-up at the lowest excitation energy of 3 MeV/nucleon studied in this paper, it is interesting to observe that the angle-integrated correlation function for PM exhibits a peak which is more pronounced than that for SF. Since a lower temperature


1 , 5 I I I I I I I I_

% " ....

P 1.0 " " ......

rr

+ .F~ - - - - - P M •

" - 0 3 - 1i~/~I . . . . . . . SE -

$ 7 0.0 ~ ~,#Jr I I I I I I I

0 5 10 15 20 2 5 30 3 5 40

Vre d (10 -3 c }

Fig. 9. The angle-integrated correlation functions from the decay of [97Au at E*/A = 6 MeV with ( J = 75h) and without angular momentum in SE PM and SE.

leads to smaller emission rates and therefore to a larger apparent source dimension,

the IMFs originating in SF and SE have a reduced mutual Coulomb interaction. In

contrast, the correlation function in PM is mostly governed by the relatively stronger

2 . 5 I I I I I i I I

E"IA=3 M V_. - - S F 2 . 0 - - / \

/ ~ . \ - P M

1 . 5 - / . . . . . . . . SE

-- I y x . . . . . . . . . . . . . _ 1.0 / ... -.. ....

> o.s rY .../

4 . 0 . 0 , , I : : I i

E * / A = 9 M e V 1.0 - . ~ - - r . ~ "-" ~- - - - -- -----=---

S 0 . 5 - /

I / 0,0 _ L ~ I I I I I I

0 5 10 15 20 25 30 35 40

V r e d (10 -3 c }

Fig. 10. The angle-integrated correlation functions from the decay of 197Au at excitation energies per nucleon of 3 MeV (top panel) and 9 MeV (bottom panel) in SF, PM and SE.

172 s. Pal~Nuclear Physics A 594 (1995) 156-174

final-state interaction between the coincident IMF pair emanating from a comparatively smaller source size (freeze-out volume). This results in a stronger correlation in PM. The directional correlation functions, however, do not provide any additional insight into the break-up mechanism at these excitation energies.

4. Summary and conclusions

In this paper, we have investigated the correlation functions for intermediate mass fragments for a possible delineation between different multifragment production mechanisms in energetic heavy ion collisions. At low excitation energies, it is expected that sequential binary decay (SF or SE) is the source of multifragment production. At very high excitations, intuitively it is also expected that the hot nuclear system would decay very promptly, possibly by single-step prompt multifragmentation. At excitations close to the limiting temperature when large-scale instabilities start growing up in the hot system, knowledge of the decay mechanism is still obscure. To have a better understanding of the mechanism of nuclear disassembly, we have therefore employed models of prompt multifragmentation, sequential fission and sequential evaporation (with the same fragment yield obtained from the decay of 197Au in the SF model) at low (E*/A = 3 MeV), intermediate (6 MeV) and high (9 MeV) excitation energies and exploited the IMF-IMF correlation functions to discern the reaction mechanisms.

At E*/A = 3 MeV, the angle-integrated correlation functions for all three decay mechanisms show pronounced peaks. At the very high excitation energy of 9 MeV/nucleon, the correlation functions do not exhibit any peaked structure, but reach a plateau at high reduced velocities beyond the Coulomb hole. From the correlation functions, it is thus seen that a distinction between the reaction mechanisms at high excitation energies is practically impossible; at low excitation energies, the kurtoses in the correlation functions are different but their overall shapes are nearly the same and therefore here also, unambiguous delineation of the reaction mechanism is difficult. The directional cuts on the correlation functions at these excitation energies do not provide additional information on the break-up mechanism.

At the excitation energy of 6 MeV/nucleon, the angle-integrated correlation functions in the SF and PM models show almost similar peaked structures whereas the correlation function in the SE model is almost fiat beyond the Coulomb hole. Thus the SE model can be easily distinguished from the SF and PM models. The peakedness of the correlation function in the SF model is, however, quite sensitive to the directional cuts whereas it is almost insensitive to these cuts for the correlation functions in the PM and SE model. A simultaneous study of the angle-integrated, longitudinal and transverse correlation functions thus provide a possible delineation of the reaction mechanisms at this excitation energy window. A considerable variation in the emission lifetimes ~- or in the freeze-out volume as well as the inclusion of angular momentum deposited in the fragmenting system do not strongly affect the shapes of the correlation functions in the different models, and thus the conclusions arrived at remain unaltered.


Acknowledgements

The a u t h o r g ra te fu l ly a c k n o w l e d g e s he lpfu l d i scuss ions wi th S.K. S a m a d d a r and

J.N. De.

References

[ 1] R. Trockei et al., Phys. Rev. Lett. 59 (1987) 2844. 12] D.R. Bowman et al., Phys. Lett. B 189 (1987) 282. 131 G. Klotz-Engmann et al., Nucl. Phys. A 499 (1989) 392. [4] Y.D. Kim et al., Phys. Rev. Lett. 63 (1989) 494. [5] D. Cebra et al., Phys. Rev. Lett. 64 (1990) 2246. [6] Y. Blumenfeld et ai., Phys. Rev. Lett. 66 (1991) 576. 171 C.A. Ogilvie et al., Phys. Rev. Lett. 67 (1991) 1214. 18] K. Hagel et ai., Phys. Rev. Lett. 68 (1992) 2141. [9] G.E Bertsch, H. Kruse and S. DasGupta, Phys. Rev. C 29 (1984) 673;

L. Vinet, C. Gr6goire, P. Schuck, B. Remaud and E S6bille, Nucl. Phys. A 468 (1987) 321; G.E Bertsch and S. DasGupta, Phys. Reports 160 (1988) 189.

1101 G. Peilert, H. St6cker, W. Greiner, A. Rosenhauer, A. Bohnet and J. Aichelin, Phys. Rev. C 39 (1989) 1402; J. Aichelin, Phys. Reports 202 (1991) 233.

[11] S.E. Koonin and J. Randrup, Nucl. Phys. A 356 (1981) 223; A 474 (1987) 173; G. Fai and J. Randrup, Nucl. Phys. A 381 (1982) 557; A 404 (1983) 551.

[ 121 J.P. Bondorf, R. Donangelo, I.N. Mishustin, C.J. Pethick, H. Schultz and K. Sneppen, Nucl. Phys. A 443 (1985) 321; J.P. Bondorf, R. Donangelo, I.N. Mishustin and H. Schultz, Nucl. Phys. A 444 (1985) 460.

1131 Sa Ban-Hao and D.H.E. Gross, Nucl. Phys. A 437 (1985) 643; D.H.E. Gross, Rep. Prog. Phys. 53 (1990) 605.

1141 W.A. Friedman and W.G. Lynch, Phys. Rev. C 28 (1983) 16; W.A. Friedman, Phys. Rev. Lett. 60 (1988) 2125; Phys. Rev. C 42 (1990) 667.

1151 R.J. Charity et al., Nucl. Phys. A 483 (1988) 371. [ 161 C. Barbagallo, J. Richert and P. Wagner, Z. Phys. A 324 (1986) 97;

J. Richert and P. Wagner, Nucl. Phys. A 517 (1990) 399. [ 171 J.A. L6pez and J. Randrup, Nucl. Phys. A 491 (1989) 477. [18] W. Gawlikowicz and K. Grotowski, Nucl. Phys. A 551 (1993) 73. [19] S.E. Koonin, Phys. Lett. B 70 (1977) 43. [20] S. Pratt and M.B. Tsang, Phys. Rev. C 36 (1987) 2390. [21] T.C. Awes et al., Phys. Rev. Lett. 61 (1988) 2665. 122] W.G. Gong, W. Bauer, C.K. Gelbke and S. Pratt, Phys. Rev. C 43 (1991) 781. [231 W.G. Gong et al., Phys. Rev. C 43 (1991) 1804. [24] M.A. Lisa, W.G. Gong, C.K. Gelbke and W.G. Lynch, Phys. Rev. C 44 (1991) 2865. [25] W. Bauer, C.K. Gelbke and S. Pratt, Ann. Rev. Nucl. Part. Sci. 42 (1992) 77. 1261 M.A. Lisa et al., Phys. Rev. Lett. 71 (1993) 2863. [271 D.O. Handzy et al., Phys. Rev. C 50 (1994) 858. [28] Y.D. Kim et al., Phys. Rev. C 45 (1992) 338. 129] Y.D. Kim, R.T. de Souza, C.K. Gelbke, W.G. Gong and S. Pratt, Phys. Rev. C 45 (1992) 387. 1301 D.R. Bowman et al., Phys. Rev. Lett. 70 (1993) 3534. I311 E. Bauge et al., Phys. Rev. Lett. 70 (1993) 3705. 1321 T.C. Sangster et al., Phys. Rev. C 47 (1993) R2457. I331 B. K~impfer et al., Phys. Rev. C 48 (1993) R955. [341 T. Glasmacher et al., Phys. Rev. C 50 (1994) 952. 1351 D. Fox et al., Phys. Rev. C 50 (1994) 2424. 1361 O. Schapiro, A.R. DeAngelis and D.H.E. Gross, Nucl. Phys. A 568 (1994) 333;

O. Schapiro and D.H.E. Gross, Nucl. Phys. A 573 (1994) 143; A 576 (1994) 428.


[37] T.C. Sangster et al., Phys. Rev. C 51 (1995) 1280. [38] S. Pal, S.K. Samaddar and J.N. De, Nucl. Phys. A 589 (1995) 489. [39] W.J. Swiatecki, Aust. J. Phys. 36 (1983) 641. [40] S. Pal, S.K. Samaddar, A. Das and J.N. De, Nucl. Phys. A 586 (1995) 466. [41] S. Pal, S.K. Samaddar and J.N. De, Nucl. Phys. A 591 (1995) 719. [42] C.M. Perey and EG. Perey, At. Data Nucl. Data Tables 17 (1976) 1. [43] J. Blocki, J. Randrup, W.J. Swiatecki and C.F. Tsang, Ann. Phys. 105 (1977) 427. [441 S. Levit and P. Bonche, Nucl. Phys. A 437 (1985) 426. [45] J. Richert and P. Wagner, Z. Phys. A 341 (1992) 171. [46] D. Bandyopadhyay, C. Samanta, S.K. Samaddar and J.N. De, Nuci. Phys. A 511 (1990) 1. [47] P.A. DeYoung et al., Phys. Rev. C 39 (1989) 128.

distinction between multifragmentation mechanisms from imf-imf correlation functions

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