signature of exotic nuclear shapes from imf-imf correlations

N U C L E A R PHYSICS A

ELSEVIER Nuclear Physics A 589 (1995) 489-504

Signature of exotic nuclear shapes from IMF-IMF correlations

Subrata Pal a, S.K. Samaddar a, J.N. De b a Saha Institute of Nuclear Physics, I/AE Bidhannagar, Calcutta-700064, India b Variable Energy Cyclotron Centre, 1/AF, Bidhannagar, Calcutta-700064, India

Received 23 January 1995; revised 30 March 1995

Abstract

Fragmentation of exotic nuclear shapes like toroids and bubbles predicted to be formed in intermediate energy heavy ion central collisions within the framework of BUU type calculations is studied in a statistical model with microcanonical simulation of prompt multifragmentation. The relatively smaller interfragment Coulomb interaction energy in the exotic shapes results in an enhanced production of intermediate mass fragments (IMFs) compared to that in the compact spherical configuration. The differences in the spatial geometry of the fragmenting shapes manifested in the inclusive charge distributions are further projected sharply in the IMF-IMF correlations. We exploit the correlation function to delineate the different fragmenting configurations.

1. Introduct ion

Intense efforts have been directed in recent years to the study of the production and decay of hot and compressed nuclear systems in intermediate energy heavy ion reactions.

A large number of intermediate mass fragments (IMF: 3 ~< Z <~ 20) are found to be

produced from such excited nuclear systems. This is commonly referred to as nuclear

multifragmentation. To unravel the underlying reaction mechanism leading to muitifrag-

ment final states, a large number of experiments have recently been performed [ 1-7]

and various theoretical models [8-16] constructed. The models in use can in general

be divided in two groups, dynamical and statistical. One of the most commonly used dynamical models is the Boltzmann-Uehling-Uhlenbeck (BUU) model [ 8] where the trajectory of each nucleon is governed by the ensemble averaged motion of all the

other nucleons with inclusion of two-body collisions. As a result, the correlations and fluctuations are taken into account only partially through binary collisions and therefore

the model cannot be trusted beyond one-body observables. The microscopic quantum

0375-9474/95/$09.50 (~ 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9474(95)00130- 1

490 S. Pal et al. /Nuclear Physics A 589 (1995) 489-504

molecular dynamics (QMD) calculations [9] overcome this limitation, but it is not yet fully geared to describe fragment formation. In statistical models on the other hand, one neglects the dynamical evolution in the early stage of the reaction but presupposes certain source characteristics of the hot nuclear complex. Then it is found to be quite successful in explaining several observables pertaining to multifragmentation. The basic assumption in all statistical models is that prior to fragmentation, the system is in thermodynamic equilibrium. These models are broadly classified into two groups a) sequential binary decay (SBD) [ 10-12] and b) one-step prompt multifragmentation (PM) [ 13-16]. At relatively lower bombarding energies, fragmentation is believed to proceed via sequential binary decay, whereas at higher energies, multifragmentation can possibly be described as a one-step break-up process. Inspite of substantial efforts, study of the extracted physical observables however fail to give an unambiguous delineation of these two processes till now. This warrants more exclusive experimental information to discriminate between the different underlying reaction mechanisms.

Recent simulations of heavy ion central collisions in the BUU model have predicted the formation of exotic nuclear shapes like toroids or bubbles prior to multifragmentation [ 17-19]. The existence of such possible exotic nuclear shapes was put forward long ago by Wheeler [20] and by Siemens and Bethe [21] and their instability was later studied in detail by Wong [22] in macroscopic and microscopic models. The nuclear equation of state (EOS) is found to play an important role in the production of the particular exotic nuclear configuration (toroid for stiff and bubble for soft EOS) [ 19] and thus determination of these shapes through suitable observables may throw some light on the nature of the EOS. As the BUU model is inadequate to describe fragmentation phenomena, geometric as well as statistical models have been employed to get a possible signature of these exotic fragmenting shapes.

Based on the geometric percolation model, Phair et al. [23] have observed enhanced IMF production from exotic shapes relative to that from spherical configuration. Within the microcanonical statistical PM model, it has been found [24] that at a given excitation energy and freeze-out density, the inclusive charge distributions for the three configurations are significantly different. Substantially enhanced IMF production from exotic shapes and significant forward peaking of the folding angle distribution between two heaviest fragments as compared to the spherical configuration were predicted to be the possible signature of the exotic fragmenting shapes.

Recently, Glasmacher et al. [25] exploited the IMF-IMF correlation functions to identify different fragmenting nuclear shapes. Not much noticeable difference in the correlation functions to delineate between different configurations was found in the calculations. They employed the same power law distribution for the fragment charges pertaining to various fragmenting shapes. In Ref. [24], it has however been stressed that the mass distributions for various shapes are different and they play an important role in bringing out the difference in the different physical observables. With this in mind, in this paper, we study the IMF-IMF correlation functions for different fragmenting shapes using the relevant mass distributions obtained from microcanonical statistical PM model.

Contrary to proton-proton correlation function [26-33] where the quantum statistics

s. Pal et al./Nuclear Physics A 589 (1995) 489-504 491

and the mutual interaction between the coincident proton pair are important, for IMF- IMF correlation function [34-39], the quantum effects have minor role to play and their mutual interaction as well as interaction with the neighbouring fragments govern the structure of the correlation function. Moreover the IMFs carry on the average large momentum which induces dynamical correlations in the source arising out of energy and momentum conservation. It is therefore imperative to employ the appropriate mass or charge distribution for various fragmenting shapes.

The paper is organised as follows. In Section 2, we describe the theoretical framework used. Results and discussions are presented in Section 3. The summary and conclusions are given in Section 4.

2. The model

The basic assumption inherent in the statistical model of prompt multifragmentation is that prior to break-up, the system is in thermal and chemical equilibrium so that the probability of finding any particular fragment within the freeze-out volume (treated as a parameter) is determined by the phase-space available to it. In this section, in the first part an outline of the model employed to describe multifragmentation is given; in the second part, the calculation of the IMF-IMF correlation function is discussed.

2.1. Prompt multifragmentation model

The probability pi 's for the production of various fragments (A i , Zi) consistent with the total charge, mass, energy and momentum of the decaying system (A, Z) are calculated in the grand canonical model (GCM) which is then used to generate microcanonical events [40]. The multiplicity of the ith fragment in the GCM is given

by

( mai .~3/2

× exp [-13(B - Bi + Vi - ftnNi -- f t p Z i ) ] , (1)

where .(2 is the freeze-out volume, m the nucleon mass, 13 the inverse of temperature T in the GCM, Ni and Zi the neutron and proton number of the fragment and //,n and #p are the neutron and proton chemical potentials. The average Coulomb interaction of the ith fragment with the rest of the fragments is given by V/, B and Bi are, respectively, the binding energies of the decaying system (A, Z) and of the ith fragment. The internal partition function ~bi (13) of the ith fragment which is excited but below particle emission

threshold is taken as

~2

I dE*pi(e*) e -~t* , (2) ,/,i(13)

492 S. Pal et al./Nuclear Physics A 589 (1995) 489-504

with the density of states pi(e*) given by a Fermi-gas type expression. Here el is the lowest excited state and e2 is the particle decay threshold of the ith fragment. The multiplicity (.o i is sensitive to the proper choice of the average interaction V/. Usually, it has been calculated in the complementary fragment approximation [41,42] given by

f exp[ - l~Ui (R) ] Ui(R) d3R Vi = f exp[ - f lUi( R ) ] d3R ' (3)

where Ui(R) is the interaction energy of the ith fragment with its complementary at a separation R. It is given by

Zi( Z - Zi )e 2 Ui(R) - (4)

R

The integration in Eq. (3) is over the whole freeze-out volume, the volume of the complementary fragment excluded. In our calculation, we take V/given by Eq. (3) as a starting value, but refine it iteratively as discussed later. The temperature and the freeze- out volume are two parameters and the chemical potentials/.tp and/zn are determined iteratively from conservation of charge and baryon number. The probability Pi is given by

o.) i Pi - ~ i °)i" ( 5 )

We simulate a microcanonical event using the isotopic distribution obtained in the GCM in a similar way as followed by Fai and Randrup [ 13]. The probability pi's for i = 1 to N, where N is the total number of isotopes of various elements that are produced in GCM are employed to generate a microcanonical event. A random number x E [0, 1] is generated. The nth isotope (An,Zn) is a member of the event if x lies between Pn-1 and Pn where P~ n 1 ~< n = Ei=l Pi with ~< N. This process is continued till the total charge and mass of the disassembling nucleus is exhausted. The fragments so generated are randomly placed in a non-overlapping configuration within the freeze-out volume for the particular nuclear shape concerned. From energy conservation, one gets

Z g* + 1 Z ~ii - ~-~ Bi + 3 ~[~= EGCM, (6)

where the average energy ECCM of the grand canonical ensemble is

EGCM = Z O.)i(•* ) ..~ 1 Zo . ) iVi__ Zcoini . .}_ 3 Z Z o ) i . (7)

The terms on the left-hand side of Eq. (6) correspond to excitation energy, Coulomb interaction energy, binding energy and kinetic energy of all the N fragments at an effective temperature T for a microcanonical event while those on the right-hand side of Eq. (7) correspond to those in the GCM. Eqs. (6) and (7) enable us to obtain the effective temperature T which is then employed to determine the initial momenta of the fragments according to Maxwell's distribution law. The momenta of the fragments are then boosted in the overall centre of mass frame and finally normalised to the total kinetic energy -~NT of the event.

S. Pal et aL/Nuclear Physics A 589 (1995) 489-504 493

It is expected that the interaction V/ used in the GCM would be equal to the one

obtained from microcanonical ensemble averaging. We calculate the ensemble-averaged interaction V/'s by generating a large number ( ~ 10 5) of microcanonical events and recalculate the GCM multiplicities using these new interactions. The process is repeated till in two successive iterations, V/'s are nearly the same. Due to finite number of events, there would be statistical fluctuations and these introduce an uncertainty, typically ,-~ 1%

in the V/s. The asymptotic configuration is obtained by allowing the fragments to evolve under mutual Coulomb repulsion.

2.2. IMF-IMF correlation function

The dependence of IMF-IMF correlation function on the reduced velocity (Vrea) of the IMF pair is given by

Vrel Vred- ~ , (8)

where Z1 and Z2 are the charges of the two IMFs and Vrel is the asymptotic relative velocity. The use of reduced velocity has the advantage that the IMF-IMF correlation functions for different IMF pairs would display a similar dependence on/)red [ 34,35 ]. In these correlation functions, quantum effects are expected to be negligible but energy and momentum conservation seem to play a very important role. The Coulomb interaction

of the two IMFs with one another, and with the source which coupled with energy- momentum conservation map the correlations inside the source become dominant factors.

The correlation function thus contains information about the spatial distribution of the fragments in the source at freeze-out.

Using the classical approximation of the Koonin-Pratt formula for intermediate mass

fragments [34,35], the IMF-IMF correlation function may be written as

Y12(Vred) (9) 1 + R (Vred) = ~ Yback ( Vred)'

where Yl2(Vred) is the coincidence yield and Yback(Vred) is the "background yield" constructed by event-mixing technique. The background yield is calculated by random selection of each of the fragment pair from different events. It may be pointed out that in many studies of two-particle correlation function, an a priori arbitrary normalisation

of the correlation function is used so that R(vrea) goes 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 functions R(vrea) may not converge asymptotically to zero [39]. So we have not introduced any normalisation of the correlation functions.

Additional insight on different break-up configurations may be obtained by employing directional cuts on the two-fragment correlation functions [25]. It has been observed [35,38] that the Coulomb interaction of the coincident IMF pair with the residual

system may be reflected on the directional correlation functions. The directional correlation functions are constructed by employing cuts on the angle ~b = cos-I [ i p . Vredl/Pvre d ]

494 s. Pal et al./Nuclear Physics A 589 (1995) 489-504

between the reduced velocity Vred and the total momentum P = Pl + P2 of the two coincident IMFs with momenta pj and P2. In this paper, the longitudinal and transverse

correlation functions are calculated by taking 0 ° ~< ~blong ~< 40 ° and 75 ° ~< ~Ptrans ~ 90 °, respectively.

3. Results and discussions

Recent BUU type calculations for the reactions 93Nb+93Nb[ 18] and 92Mo+92Mo[ 19]

at a beam energy of E/A ~ 60 MeV/nucleon predict the formation of exotic shapes in central collisions. For this reaction, the residual system after pre-equilibrium emission of nucleons is found to have mass number A0 ~ 150 in the promptly emitted particle

model [43] with a temperature T ~ 6.5 MeV, which is close to the limiting temperature [44,45] of the system. To look for possible signature for the exotic fragmenting shapes, we have therefore performed calculations for the representative system 15°Sm

having a temperature T = 6.5 MeV. This corresponds to per nucleon excitation energy of ~ 4.0 MeV for the fragmenting system. For all the shapes concerned, we have ini-

tially taken the freeze-out density p to be po/6, where p0 is the normal nuclear matter density. Such low density of the evolved exotic shapes is indeed observed in recent

microscopic calculations [ 19] and also is in conformity with that employed in the statistical model [46] for multifragmentation from spherical freeze-out configuration. For

the exotic shapes, a quantity of prime importance is the aspect ratio which for toroid is defined as [19,22] a~ = R / d whereas for bubble, a = R1/R2. Here R is measured

from the centre of the torus to the centre of the circular meridian, d is the radius of the meridian and R1 and R2 are the inner and outer radii of the bubble, respectively.

In Fig. 1, the inclusive charge distributions for the three (sphere, bubble and toroid)

fragmenting configurations are displayed. The inner radius for both the exotic shapes is taken to be 5 fm. This corresponds to an aspect ratio a = 1.91 for the toroid and ce = 0.43 for the bubble with a freeze-out density of po/6. The top panel corresponds to the calculation in the complementary fragment prescription [Eq. (3)] whereas the bottom

panel refers to the calculation with "self-consistent" interaction as described earlier. In the figure, the solid circles, open circles and the crosses correspond to results for sphere, bubble and toroid, respectively. The effect of inclusion of self-consistency is not insignificant on the charge distribution, and is most marked for the bubble configuration. As is evident from a comparison of the two panels, it is found that self-consistency enhances IMF production for the spherical shape whereas IMF yield remains practically unaltered for the two exotic shapes. From both the calculations, it is clear that exotic shapes produce larger number of IMFs relative to the spherical one. This is attributed to the lesser Coulomb interaction energy for the exotic shapes. Since self-consistency is expected to be a better prescription for the interaction, all the calculations reported below are performed with the self-consistent interaction.

The multiplicity distributions P(NIMF) for intermediate mass fragments for the three fragmenting configurations are shown in Fig. 2. The symbols used in Fig. 1 retain the

S. Pal et al. /Nuclear Physics A 589 (1995) 489-504

101 , , ,

-"S, "%

16 s

5 ~ 1 o °

16'

16 2

I

k

16 3

I I

A , , , , d ~ *

° oOoOo o ° I I ~ I I

0 70

XX~ x 0

I; 2'o s'o ¢o 6'o Z

495

Fig. 1. The charge distributions from 15°Sm for the three fragmenting configurations at T = 6.5 MeV and p = po/6. The solid circles, open circles and the crosses correspond to spherical, bubble and toroidal shapes, respectively. The top panel refers to calculations in the complementary fragment model; the bottom panel refers to those with self-consistent interactions (for details, see text). All the subsequent figures refer to self-consistent calculations.

same meaning in this and the subsequent figure. For the spherical and bubble shapes,

the NIMF distributions show broad shoulders for low NIMV with maxima at NIMF ~ I for

sphere and at NIMF ~ 7 for bubble, respectively. For the toroid, the NIMF distribution

exhibits a nearly gaussian shape peaking at NIMF ~ 7. It can be seen from the figure

that events with large number of IMFs are significantly less for the sphere as compared

to the bubble or torus.

The IMF kinetic energy spectra in the source frame are presented in Fig. 3. The

shapes of the spectra are rather similar. We find that the Coulomb peaks are shifted

a little towards lower energy for the exotic shapes due to lesser Coulomb interaction

there. The relatively harder IMF energy spectra |br the sphere shows a larger effective

temperature T [see Eqs. (6) and (7 ) ] which is reminiscent of the fact that fewer

fragments are produced for the spherical configuration.

In Fig. 4, the angle-integrated I M F - I M F correlation functions for the three fragment-

ing configurations are displayed with the source parameters unaltered. For the sphere,

the correlation function shows a pronounced peak at /)red ~ 0.016C and decreases to a

value less than 1.0 at large reduced velocities. The maximum in the correlation function


I 0 0 , , ,

z- 15 = E

I(~ 3

x ° O Q ~ 0 x

• 0 ~ O X 0 0 0 0 0 •

X • 0 X

X o o

x

I I

NIMF

o x

0

o

i I T 8 10 12 14

Fig. 2. The IMF (3 ~< Z ~< 20) multiplicity distributions with the source parameters the same as in Fig. 1. The symbols have the same meaning as in the previous figure.

is reduced for the bubble configuration while for the toroid, the correlation function

increases monotonically and becomes rather flat at large reduced velocities. For all the three break-up shapes considered here, the correlation functions exhibit pronounced an-

ticorrelations at small ©red- This is a manifestation of the final state Coulomb repulsion

of the two coincident IMFs which prevent them from having similar velocities resulting

in the Coulomb hole at small Vred. A careful look at the charge and IMF multiplicity

distributions presented in Figs. 1 and 2 provides an explanation of the shapes of the cor-

relation functions. For the spherical break-up configuration, the shape is relatively more

10° I . . . . .

X XxX

lo ~ o ..- ' . , . ,~?Xxx x %,. " i ~ X x

ILl " • XXx

.~ 1(3 2

1(3 3 x x x x u ~::~o~. ~ XXxxX/xxx 1

2o 60 8o ,20 E (MeV)

Fig. 3. The 1MF kinetic energy spectra for the three fragmenting configurations. The symbols and the source parameters are the same as in Fig. 1.

S. Pal et al./Nuclear Physics A 589 (1995) 489-504 497

I I I I I I I

e=eo/6 2.0 T=6.5 M

" 0 1.5

tY +

~ - - Toroid (~.=1.911

0.0 ,~ - - - L " ~ 2 ' ~ " I I I I I I

S 10 lS 20 2~ 30 3S

Vred(lO "3 C)

Fig. 4. The IMF-IMF angle-integrated correlation functions for the three fragmenting shapes at T = 6.5 MeV and p = po/6.

compact, the average multiplicity of IMFs is rather small and a very massive fragment is present in the final state [24]. The correlation function in this case is mostly governed by the final state interaction between the coincident IMF pair; the massive fragment

mainly boosts the centre of mass motion of the pair. In addition, the fluctuation in the relative separation between the IMF pair is less due to compactness of the shape. All

these effects lead to a pronounced maximum in the correlation function. For the bubble configuration, the number of produced fragments is large and therefore the interaction of the IMF pair with the source dilutes the role of the mutual interaction between the

IMF pair [39] on the correlation function resulting in a less pronounced peak. The fluctuation of the separation between the IMF pair within the source is larger due to

less compact shape which flattens the peak further. In case of toroid, since the number of fragments is even greater and the shape is more noncompact, the peak washes out leading to a nearly flat correlation function beyond the Coulomb hole.

In Fig. 5, the longitudinal (0 ° ~< ¢, ~< 40°), transverse (75 ° ~< ¢, ~< 90 °) and angle- integrated (0 ° ~< ~p ~< 90 °) IMF-IMF correlation functions are presented for the three break-up configurations, keeping the source parameters same as before. In all the cases, it is observed that the transverse correlation functions are more pronounced than the longitudinal ones, the differences being more marked for the spherical and bubble shapes. In addition, the widths of the Coulomb hole at small Ured are somewhat narrower for the longitudinal correlation functions compared to the transverse correlation functions. Since these two correlation functions display almost similar sensitivity towards different fragmenting shapes, it appears that additional information is difficult to be obtained from

a study of the considered directional cuts in the correlation functions. To illustrate the sensitivity of the angle-integrated two fragment correlation functions

4 9 8 S. Pal et al. /Nuclear Physics A 589 (1995) 489-504

2.0

1.5

1.0

0.5

0.0

I I I I I I I

t ~

Sphere ~ " ' . ~= (?o/6

: i : /

. : / .- ¢

; " -" I I I I I I

Bubble . - . . . . . . . 1.5 " - " - - . -

'>~ 0<.=0.43

1.0 "J" :'1 "'"'--.. : / I "",..

: t " . .

• / " ' " ' " " ' " ' " ' " " " """ " " "

0-5 ,..j . : /

0 . 0 - I I I I I

1.0~ Toroid . . . ~ oK= 1.91 . . ~ o . . . . . . . ,

. . ~ ' ~ - - o-~Y~: 9o- d / • •

. " t • •

. . . . . . o'~ ~<~.o" 0-0 ~ " ' I I I I l

5 tO 15 ZO z5 30 35

Vre d (10 -3 C )

Fig. 5. Comparison of the longitudinal (0 ° ~< ~ .<.< 40°) , transverse (75 ° ~< ~p ~< 90 °) and the angle-integrated (0 ° ~< ~b ~< 90 °) correlation functions for the three shapes at T = 6.5 MeV and p = po/6.

on the aspect ratio of the exotic shapes, calculations have been repeated for the fixed freeze-out density of po/6 with aspect ratio a = 0.26 and 0.57 for the bubble and ot = 1.51 and 2.37 for the toroid. This corresponds to an inner radius Rl = 3 and 7 fm, respectively for either of the exotic shapes. The effect of the variation of ot for bubble and toroid is displayed in Fig. 6, Increasing the aspect ratio makes the shapes more exotic and it widens the difference in the charge distributions from that of the sphere presented in Fig. 1. An increased c~ corresponds to an increased average separation between the IMFs at freeze-out, whereas with decreasing or, the bubble configuration tends towards a sphere. This is reflected from the comparison of the correlation functions presented in Figs. 4 and 6 exhibiting how the pronounced peak is gradually suppressed with increasing oe, from ~ = 0 (sphere) to 0.57. With further increase in o~, the shapes of the correlation functions for the bubble look more like those of the toroid. For the

S. Pal et al./Nuclear Physics A 589 (1995) 489-504 499

, , ; . - - - . . , , , ,

i i "%,% Bubble / - 0 = Col6

x T=6.5 MeV t.S / ~

I . 0 /// ... ~" ~ ~ .

S / . . "

, f ............. ~,= 0.57 .;7J t.l. I I I I I + O.O

1.0 Toroid . ~

...~ ----- ~= 1.51

0 . 5 "~,/ - - oC= 1 .91 ............. o < = 2 . 3 7 " ' "

0.0 ~ l I l i i i S 10 15 2o 25 30 3s

Vre d ( I0 -3 c )

Fig. 6. Dependence of the angle-integrated correlation functions on the aspect ratio o~ for the bubble and

toroidal shapes at T = 6.5 MeV and p = po/6.

toroid, the shape is very noncompact and the variation in a in the range studied here

does not change the shape of the correlation functions appreciably. However for larger

a, since the initial separation between the fragments is larger resulting in a reduced

Coulomb interaction between them, a narrower Coulomb hole in the correlation function

is observed. We found that the charge distributions for the toroid with a = 1.51 and the

bubble with the largest aspect ratio (a = 0.57) studied are almost identical and those

from the other two toroidal shapes are not very different. The near similarity of the

overall shapes of the correlation functions for the aforesaid break-up geometries appears

to be a direct consequence of the similarity in their charge distributions which is also

corroborated in a recent calculation in Ref. [25]. The charge distribution in the PM model is sensitive to the freeze-out density. We

therefore explore the dependence of the angle-integrated correlation functions on the

freeze-out density by repeating the calculations at p = po/4 and po/8. To isolate the

effect of freeze-out density, we have kept the aspect ratio fixed at ce = 0.43 for the bubble

and a = 1.91 for the toroid. This corresponds to an inner radius of 4.4 fm at p = po/4 and 5.5 fm at p = po/8 for both the exotic shapes. The variation of the correlation

functions for the three configurations with the freeze-out density is displayed in Fig. 7. With change in freeze-out density, the charge distributions are modified, so also the mutual Coulomb interactions among the fragments. Both these affect the correlation

function. With increase in freeze-out density, lesser number of fragments are generated,

the Coulomb interaction also becomes stronger and one expects the correlation functions to become more sharply peaked. Comparison of Figs. 4 and 7 does not belie this expectation. The effect is maximum for the bubble shape and minimum for the toroidal


2.0

1.5

1.0

"V

o.s l>

tY

+ 0.0

1.0

O.S

0.0 S I0 15 20 25

Vre d ( 10 -3 c )

I I I I I I I

e = e./~ , ' ~ " , , . T = 6.5 MeV / y ",,, \ ~

~ - ~ - Bubble(o~=0.43) _ J ............ Toroid (o(,=1.91)

J ; : 1 i i

!

30 35

Fig. 7. Dependence of the angle-integrated correlation functions on the freeze-out density. The temperature T is 6.5 MeV and the aspect ratio is kept fixed at a = 0.43 for the bubble and ~ = 1.91 for the toroid.

shape. Increasing the freeze-out density from po/6 to po/4 does not affect the correlation function for the spherical shape appreciably. We then also find that the difference in

the correlation functions between the sphere and the bubble narrows down considerably. We further note that at the low freeze-out density po/8, the correlation functions for

the bubble and the toroidal configurations are practically identical at large Vred; however the Coulomb hole at small Ured is wider for the comparatively more compact bubble configuration. For relatively low freeze-out densities, it is observed that the differences in the charge distributions in general narrow down for various fragmenting shapes resulting in nearly identical correlation functions.

To see the effect of excitation energy of the disassembling system on the physical observables, we have repeated the calculations at a lower temperature of 5.0 MeV and also at a higher temperature of 8.0 MeV, keeping the other source parameters same as those used for Fig. 4. At lower temperatures, the differences between the charge distributions from the three fragmenting shapes and the resulting correlation functions are more pronounced, whereas at higher temperatures, namely at 8.0 MeV, the differences between the respective charge distributions and the correlation functions are nearly washed away. At this very high excitation a large number of small fragments are produced and the charge distributions or correlation functions resemble as those coming from a very extended source. The memory of the fragmenting configuration is however contained only in the narrower Coulomb hole for the exotic shapes, particularly for the

S. Pal et al . /Nuclear Physics A 589 (1995) 489-504 501

I I I I I I I

e=e,/6 T=8.0 MeV

1.o

"' ' 7 . . . . . Bubb le 1~,=0.43) .." , ' / . . . . . . . . . Torold [o(=1.91)

." /

." t

0 - 0 - ' ~ ' : ' : ~ J ' ' I I I 5 10 15 20 25 3tO 35

Vred( lO -3 c )

Fig. 8. Angle-integrated correlation functions for l~)Sm at T = 8.0 MeV and p = t90/6.

toroid. This is displayed in Fig. 8. Recently, substantial collective flow has been observed in central heavy ion colli-

sions [6]. The collective flow velocity superimposed on the thermal motion of the disassembled fragments reflects the strong compression of the interacting nuclei at the

early stage of the collision with subsequent expansion. Microscopic BUU calculations have indicated [47] that the formation and lifetime of the exotic shapes are mainly determined by the total compressional energy stored in the system. To assess the sensitivity of collective flow on the IMF-IMF correlation functions, we have performed calculations with a flow energy of 1.0 MeV/nucleon. This choice of the value of the flow energy is guided by the results reported in Ref. [ 19]. The velocity field of the collective flow

is assumed to be spherically symmetric and the radial flow velocity of a fragment is assumed to be proportional to its distance from the centre of mass of the system [40]. All the other source parameters are the same as reported in connection with Fig. 4. The

resulting correlation functions are shown in Fig. 9. Comparing the results with Fig. 4 where flow velocity is absent, we find that the radial flow increases the relative velocities of the IMFs causing an overall shift of the correlation functions towards higher Ured with widening of Coulomb holes as well [48]. In addition, the peaks of the correlation functions are suppressed a little. Calculations have been repeated with an increased flow energy of 5.0 MeV/nucleon. The shapes of the correlation functions are not changed significantly; the only noticeable effect is an overall shift of the correlation functions

towards higher /:red.


I I I I

e = C o / 6 ~ Eflow/A=l MeV

1.0 ""

+ /S"" .... ---- Sphere o.s .,"]/ - . . . . e u b b i e C . ~ = o . ~ 3 )

. . : / ~ .. Toroid (0t=1.91) i - -

0.0 ="'~:J~ i 10 20 30 40

Vred (10 -3 c)

Fig. 9. Angle-integrated correlation functions at T = 6.5 MeV and p = p0/6 with inclusion of flow energy of 1.0 MeV / n u c l eo n .

4. Summary and conclusions

For delineation of exotic nuclear shapes formed in intermediate energy heavy ion central collisions, we have calculated certain observables with more emphasis on IMF-IMF

correlation functions from multifragmentation of spherical, bubble and toroidal shapes in a microcanonical statistical model. The correlation functions are quite sensitive to the excitation energy, freeze-out volume and the shape of the fragmenting complex and therefore may serve as an important tool to discern the different source characteristics.

In general, the more exotic the shape, the larger are the number of fragments produced. The nature of the charge distributions affects the correlation function rather sensitively. At first, the calculations have been performed for the system 15°Sm at a temperature

of 6.5 MeV, close to the limiting temperature of the system and at a freeze-out density of po/6, in consonance with that obtained in recent microscopic calculations. The fragmenting shapes can easily be discerned then from the correlation functions with a pronounced peak for the sphere, and a much suppressed peak for the bubble. For the toroid, the correlation function does not exhibit any peak, but reaches a plateau at high

reduced velocities beyond the Coulomb hole. For the same freeze-out density, the aspect ratios may however be different for the exotic shapes and we find that for a small aspect ratio, a bubble is difficult to be distinguished from a sphere whereas for large aspect ratio, the bubble and torus display nearly the same correlation functions. The correlation function for the toroidal shape is almost independent of the aspect ratio. If the freeze-out density is decreased further and the temperature of the system is taken beyond the limiting temperature, the mass distributions for the different fragmenting shapes as well as the different IMF-IMF correlation functions become more and more alike, resembling more like that arising from a toroidal configuration. The correlation function for the toroidal shape exhibits a narrower Coulomb hole at the freeze-out density as low as po/8 or the temperature as high as 8.0 MeV. However, as the width of

S. Pal et al. /Nuclear Physics A 589 (1995) 489-504 503

the Coulomb hole at small Vred depends on the a priori unknown characteristics of the emitting system, a clear distinction between the different source shapes by means of

such correlation functions becomes practically impossible. The inclusion of collective flow energy shifts the peaks of the correlation functions towards higher Vred and the peaks are also reduced as higher fragment velocities effectively dilute the Coulomb interaction. The longitudinal and transverse correlation functions display almost identical behaviour as the angle-integrated ones for all the shapes and therefore it is difficult to ex- tract additional information from their study regarding the geometry of the fragmenting configurations.

To sum up, we find that the IMF-IMF correlation functions that depend on the relative location and final state interaction between the emerging fragments display sufficient

sensitivity to the parameters of the break-up configuration, namely the freeze-out volume, the temperature and the fragmenting geometry. These parameters are not a priori known

with certainty. The question arises whether the shape of the correlation function can

provide some information about the fragmentation parameters, more specifically, about the break-up geometry. It is clear from our calculations that if the shape of the corre-

lation function exhibits a peak, a toroidal fragmenting geometry is ruled out. If further information is available from supplementary microscopic calculations on the freeze-out volume and the temperature [ 18,19], distinction between spherical and toroidal shapes may become possible without much difficulty provided the temperature is not too high

compared to the limiting temperature and the freeze-out density not too low. A bubble with low aspect ratio or a bubble with large aspect ratio make finer difference in the

correlation functions with those obtained, respectively, from a sphere or a toroid. These subtle difference in source geometries appear to be difficult to ascertain from the ex-- perimentally measured IMF-IMF correlation functions. Possible admixture of different

exotic shapes arising from different events may further complicate the situation. The simulations performed in Ref. [ 19] indicate however a small mixing ratio which is

expected to have not much of an influence on the correlation functions.

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signature of exotic nuclear shapes from imf-imf correlations

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