Effect of neighbouring fragments on sequential binary decay

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 591 (1995) 719-737

Effect of neighbouring fragments on sequential binary decay S u b r a t a Pal a, S . K . S a m a d d a r a, J.N. D e b a Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Calcutta 700 064, India h Variable Energy Cyclotron Centre, 1/AF, Bidhannagar, Calcutta 700 064, India Received 16 December 1994; revised 30 March 1995

Abstract

The effective fission barrier of a hot nucleus in nuclear multifragmentation in the sequential binary decay model depends on the configuration of the neighbouring fragments generated in the previous stages of binary decay. Taking this into consideration, various observables related to multifragmentation in intermediate energy nuclear collisions are evaluated in the transition state model. The effect of recombination of the fragments during dynamical evolution of the system is also taken into account. The results of our calculations are confronted with the experimental data from fragmentation in the reaction Au on Cu at 600 MeV/nucleon bombarding energy.

I. Introduction

Intermediate energy heavy ion collisions or high energy proton-induced reactions are associated with the production of a large number of intermediate mass fragments. Commonly referred to as multifragmentation, this phenomenon offers a unique tool to understand the properties of hot nuclear matter. Intense experimental efforts [ l - 1 1 ] in the last decade have been directed to unravel the underlying reaction mechanism and a few theoretical models have been proposed to understand the experimental findings. These models are basically statistical in nature and may be broadly classified in two groups: (a) Prompt multifragmentation (PM) [12-15] and (b) sequential binary decay (SBD) [ 16-18]. Besides, microscopic models based on modified BUU calculations [ 19,20] or on quantum molecular dynamics [ 21,22] are being recently employed to study multifragmentation phenomena. However, these microscopic models are not yet fully geared to explain the experimental observables related to multifragmentation. In prompt multifragmentation, the branching ratios of various channels are dictated by the 0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0375 - 9 4 7 4 ( 9 5 ) 0 0 1 8 9 - l

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S. Pal et al./Nuclear Physics A 591 (1995) 719-737

available phase-space at a fixed freeze-out volume (usually taken to be six to seven times the normal nuclear volume) and the interaction of a fragment with the neighbouring fragments is taken into account in counting the different fragmenting configurations. In SBD, fragmentation proceeds through a sequence of binary decays. There the decay probability or the fragmentation barrier, till now, has always been taken to be that of an isolated nucleus. However, in this model, since fragmentation of the primary hot nucleus proceeds through a series of sequential binary decays, at any stage of the decay chain but for the very first one, a fissioning fragment in the event would always be surrounded by other nuclei created in earlier series of binary decays. These nuclei generated from the previous stages of binary decays would be referred to as "surrounding medium" or "neighbouring nuclei" in the context of fission of a fragment in the event concerned. The fission barrier of a fragmenting nucleus would be modified due to the presence of these neighbouring nuclei. This change in barrier would affect the decay widths and hence would alter the fragment multiplicity distributions. Moreover, during dynamical evolution, particularly at early times, two fragments may come in close proximity and experience the strong nuclear attraction leading to their recombination [23]. Both these effects, namely the effect of the surrounding medium on the fragmentation barrier and recombination, may have a significant influence on the physical observables. The aim of the paper is to make a detailed study of these effects in multifragmentation in the SBD model, which has not been performed till date. In a recent experiment performed by the ALADIN group at GSI [8-11], the fragmentation of Au projectiles after collisions with different targets at a bombarding energy of 600 MeV/nucleon was studied. They measured different charged particle observables as a function of Zbou,a (defined as the sum of the charges of the projectile fragments with Z ~> 2). In particular, they have studied the correlations with Zbound of the mean multiplicity of the intermediate mass fragments (NIMF), the mean largest charged fragment (Zmax), the normalised mean charge variance ()'2) and the two- and three-body asymmetries. We have confronted these experimental charge correlation data with our model predictions and then tried to find out the sensitivity of the medium effect and the effects due to recombination in sequential binary decay. In Section 2, we give a brief outline of the theoretical framework used in this paper. Results and discussions are presented in Section 3. The summary and conclusions are given in Section 4.

2. The model

The transition state model of Swiatecki [24] is employed to calculate the decay probability for the fragmentation of an excited nucleus into two fragments. At each stage of the sequential decay of the fragments, they are assumed to be in thermal equilibrium. The fission barrier height V~B of an isolated hot nucleus of mass A and charge Z and excitation energy E* decaying into two fragments is calculated in the two-sphere approximation. We do not consider angular momentum at any stage of the

S. Pal et al./Nuclear Physics A 591 (1995) 719-737

721

calculation. The barrier height then may be written as (1)

V°(Ts) = Vc + VN + Esep(To, Ts) ,

and the position and height of the barrier are calculated numerically. Here To is the temperature of the fissioning nucleus given by To = V/-~-/a, a being the level density parameter (taken as a = A / I O MeV -1) and Ts is the temperature of the system at the saddle point, which is also the temperature of the daughter nuclei. It is given by (2)

Ts = ~/(E* - V° - K ) / a ,

where K is the kinetic energy of relative motion of the two fragments at the saddle point. It is assumed to follow a thermal distribution P ( K ) ~ x/--Ke -K/rs. The complicated interrelationships among V~B,K and Ts makes a fully consistent determination of Ts very difficult. To simplify the problem, K in Eq. (2) is replaced by its average value 3Ts 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 the kinetic energy is ~-, Ts and E* - V~B is generally much greater than Ts. The average value of K is used only to extract the temperature Ts; its actual value obtained from the thermal distribution is used for trajectory calculations as discussed later. In Eq. (1), Vc is the Coulomb interaction taken to be that between two uniformly charged spheres. The nuclear interaction VN between two fragments of masses A1 and A2 is classified in three groups depending on their masses [23], (i) A1 ~< 4, A2 ~ 4, (ii) Ai ~< 4, A2 > 4 and (iii) AI > 4, A2 > 4. In (i) the nucleon-nucleon interaction is taken to be a gaussian with a range parameter of 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 in the mass range in this class is then calculated by convoluting the nucleon-nucleon interaction with the gaussian density distribution. In (ii), the interfragment interaction is taken to be the real part of the optical potential [25]. In the mass range given by (iii), the proximity interaction [26] of Blocki et al. has been used. The temperature dependent separation energy Esep(T0, Ts) is taken as Esep(T0, Ts) = B(To) - BI (Ts) - B2(Ts) ,

(3)

where B, BI and B2 refer to the temperature dependent binding energies of the parent nucleus and the daughter nuclei. The binding energy, modified to include the temperature dependent surface energy, is given by B ( T ) = 15.677A - 28

(N-Z)

2

18.56o-(T) A 2 / 3

A

+ 0.245 " f f Z z 2 . 2A The surface tension constant o-(T) is taken as [27]

+ 3T

(,-

~j

,

-

Z2 0.717--

A1/3

(4)

(5)

S. Pal et aL/Nuclear Physics A 591 (1995) 719-737

722

with o-(0) = 0 . 9 5 1 7 1 1 - 1 . 7 8 2 6 ( - ~ ) 2 ] .

(6)

The critical temperature Tc is taken to be 16 MeV. As mentioned earlier, the fission barrier of an isolated nucleus gets modified in the presence of neighbouring fragments generated in the earlier stages of the binary division in a decay chain. The fission barrier for a nucleus due to the surrounding medium is then modified as (7)

V8 = V° + AVB,

where V° is the barrier as given by Eq. (1) for an isolated nucleus and zlVB is the correction term taken as / % = u~ + u 2 - u .

(8)

In Eq. (8), U, Uj and U2 are the interactions of the fragmenting nucleus and of the daughter nuclei with the neighbouring fragments. The interactions Ul and U2 are evaluated corresponding to the unperturbed saddle configuration for which the centre of mass of the daughter nuclei and the fragmenting nucleus are the same. The presence of the surrounding medium modifies the saddle configuration of the fissioning nucleus; we have not taken it into account for simplicity. Both the nuclear and Coulomb terms are included in these interactions. Fission can occur in any direction, therefore the interactions UI and U2 are angle dependent and so is the fission barrier. The saddle point temperature Ts is also modified accordingly as Ts = v / ( E * - VB - K ) / a .

(9)

The decay probability of an excited nucleus with mass A and charge Z into two fragments (A1, ZI ) and (A - At, Z - Z1) with relative kinetic energy K at the saddle point is calculated in the transition state model and is given by P(A,Z,E*;At,Z1)

oc exp I 2 v / a ( E * - VB - K ) - 2x/ax/~] .

(10)

For an isolated nucleus, fission is isotropic in its own frame. In the presence of the surrounding medium the barrier is angle dependent and therefore fission is anisotropic. To make the computation tractable, we neglect the anisotropy in choosing the fissioning direction in the Monte Carlo method (as discussed in the next section), but retain the angle dependence in the barrier in calculating the decay probability as given by Eq. (10). The kinetic energy K of the fissioning fragments lying in the range 0 ~< K .<, E* - Ve is generated in a Monte Carlo method obeying the thermal distribution mentioned earlier. This kinetic energy is plugged into Eq. (9) to recalculate the temperature of the daughter nuclei and thus energy conservation is assured. The fragment kinetic energies and hence their velocities are obtained from momentum conservation. The trajectories of the fragments are calculated in the overall centre of mass frame. If the fragments have sufficient excitation energy, they may decay in flight. The lifetimes

S. Pal et al./Nuclear Physics A 591 (1995) 719-737

723

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 too small though the branching ratios in various channels are reasonable. In the absence of any better simple prescription, for the lifetime of emission of a fragment of mass A l from a source at temperature T, we take a simple parametrised form [28] ~-=2 e 13/T e a'/8

(fm/c),

(11)

obtained from a good fit of the available data for the lifetimes of emission of different species. The calculations are performed both with and without recombination. For the latter, the trajectories evolve with Coulomb interaction alone; for the former, nuclear interaction is included. During evolution, the distance between all the fragment pairs are checked and two fragments recombine if the distance between their centres is less than some critical value. This is taken as the sum of the fragment radii R1 + Re calculated with Ri = roA]/3 where the radius parameter r0 is taken as 1.18 fm. The position and momentum of the recombined complex are obtained by reversing the procedure employed in fission. The excitation energy of the fused complex is obtained from energy conservation. The integration of the trajectories is continued in the overall centre of mass frame till the asymptotic region is reached when the interaction energy is very small (~, I MeV) and the excitation energies of all the fragments are below particle emission threshold.

3. Results and discussions

3.1. Model predictions f o r a representative system To study the effect of the neighbouring fragments on the sequential binary decay of a hot nucleus, we have taken 15°Sm as a representative system with an excitation energy of 3.6 MeV/particle. With the choice of level density parameter as a = A / I O MeV -1, the temperature of the nucleus is then 6.0 MeV, close to the limiting temperature of such a system [29]. The effective barrier of a fragmenting nucleus may increase or decrease depending on the direction of fission and the configuration of the neighbouring fragments. To illustrate it by a simple example, we consider the symmetric fission of 4°Ca in the presence of 2°Ne. In the inset of Fig. 1, A and B represent the centres of the daughter nuclei at the saddle configuration with O as the centre of mass of the 4°Ca nucleus; C is the centre of the neighbouring 2°Ne nucleus. The fissioning direction AB makes an angle 0 with OC. The distance OC is chosen in such a way that when the three nuclei are linearly placed (0 = 0°), the minimum surthce separation Smin between the neighbouring fragment and the closest daughter nucleus is 2 fm. The change in barrier AVB given by Eq. (8) is then dependent on the angle 0 and is displayed in Fig. 1 with both Coulomb and nuclear proximity interaction. In this particular illustration, we find that the effective barrier is mostly reduced due to the presence of a neighbouring nucleus. In reality, the

724

S. Pal et al./Nuclear Physics A 591 (1995) 719-737

\ \

\x \ \ \ \ \

-

>~-1 I[ /." /

-3

/: / .." .

. . . . .

C

.................

N

-

C+N

-

."

~

20

I

40

I

I

60

J

I

80

9 (deg.) Fig. 1. The modification of the barrier AV8 (Eq. ( 8 ) ) as a function of the emission angle 0 for symmetric fission of 4~Ca due to the presence of a ZONe fragment. The dotted, dashed and solid lines represent nuclear, Coulomb and Coulomb plus nuclear contributions, respectively. In the inset the fission configuration is shown (for details, see text).

number and configuration of the neighbouring fragments evolve dynamically giving a more complex shade to the barrier of the fissioning nucleus. It may be pointed out that the magnitude of zlV8 is very sensitive to the separation of the parent and daughter nuclei with the neighbouring fragments, especially at small separation. For instance, in the above illustration, if Srnin "~ 0, then AVB for 0 ,-~ 0 ° is as low as ,~ - 2 8 MeV. On the other hand if the fissioning nucleus were in contact with the neighbouring fragment, then for 0 ~ 90 °, the surface separations of the daughter nuclei at the saddle point with the neighbouring fragment are ,,~2 fm; the value of AVB then becomes as a high as ~ + 2 2 MeV. In Fig. 2, we display the ensemble averaged change in barrier for different fragmenting nuclei (depicted by charge Z) arising out of the sequential binary decay of 15°Sin at T = 6.0 MeV, averaged over the initial time (0 ~< t ~< 60 f m / c ) . As discussed earlier, the change in barrier can be both positive (AV+) or negative (AVe) depending on the configurations. In order to have better insight into the role of AVB on fission probability, averaging has been performed for positive and negative AVB separately. To see the effect of mass asymmetry on AVB, calculations have been performed for two asymmetries a = 0.1 and 0.5 where a = A]/(A1 + Az), A] and A2 being the masses of the daughter nuclei. The change in barrier for symmetric fission ( a = 0.5) without recombination is shown by the dotted lines and is larger than those compared to asymmetric fission with o~ = 0.1 (shown by dash-dotted lines). When recombination is taken into account, the absolute magnitude of AVB (shown by the thin solid line and the dashed line for a = 0.5 and 0.1 respectively) decreases. This is predominantly due to the change in

S. Pal et al./Nuclear Physics A 591 (1995) 719-737

725

30

2O

-10 -20 -3

, 0

I 10

,

I 20

,

I 30

~

I &O

,

I 50

I 60

Z

Fig. 2. The change in barrier AVa for various fragments arising out of sequential binary decay of 15°Sm at T = 6.0 MeV averaged over early time (0 ~ t ~ 60 fm/c) and ensemble. The dotted and thin solid lines correspond to symmetric fission ( a = 0.5) without and with recombination whereas the dash-dotted and dashed lines represent those for a = 0.1. The thick solid line corresponds to the barrier VB° (Eq. ( l ) ) for symmetric fission of isolated nuclei produced in the same reaction with the same averaging procedure.

the configuration of the system after recombination. The reduction in the number of fragments has also a somewhat significant role to play. The magnitude of AVB is found to be maximum around Z = Z0/2 where Z0 is the charge of the initial fragmenting nucleus 15°Sm. The ensemble and time averaged barrier V~B for symmetric fission of isolated nuclei arising out of sequential binary decay of 15°Sm at T = 6.0 MeV is also displayed in the figure by the thick solid line which also has a maximum at around Z = Z0/2. It is seen that the magnitude of AV8 can be as high as ,-~ half of V° which should have a very large effect on the fission widths in various channels. However, due to the simultaneous operation of both suppression (AV8 -- positive) and enhancement (zlVB = negative) of fission decay, the overall effect on the fission width is diluted. The generally high value of AVB can be understood from the fact that the temperature of the parent nucleus and the fragments created in the early stages of the decay chains is quite high, the lifetime of different species as obtained from Eq. (11) is quite low and therefore in the time interval concerned (0 ~< t ~< 60 fm/¢), the number of neighbouring fragments is appreciable. Moreover, since the velocities of the fragments are very small (kinetic energy ~ 1.5T), the fragments are still very close to each other in the time interval mentioned and therefore the value of AVB is high. When AV~ is averaged over a time interval 60 ~< t ~< 120 fm/c, we find that AVB is decreased by a factor of more than two as the fragments become relatively more separated. The suppression or enhancement of fission decay does not depend on the magnitudes of zlV+ or AVe- alone, but also on the frequencies (N + or N - ) with which they occur during the dynamical evolution of the system. We define a quantity S = ( N - - N+)/ ( N - + N +), a measure of the relative importance of the frequencies. In Fig. 3, we plot

726

S. Pal et a L I N u c l e a r Physics A 591 (1995) 7 1 9 - 7 3 7

0.6

/ o.~ x ~ /

/o., /i I

~

~ o.s

iit I

\\~\\

0.2

I1 I I II

\\ \\

o.s/

\x

//

" ~\--.

0.1

/ / o.1

\ ' -\, .,,,.

/ /

/ /

\x o.o

/

\\

/ /

\\

i I

\\ \\

-o.1

i I j // \..~//

-0.2

r

I 10

u

I 20

n

I "n j 30 Z

I

I

40

50

J 60

Fig. 3. The asymmetry 6 for frequencies of A V + and AVe- (see text) for the reaction as mentioned in Fig. 2. The dashed and solid lines correspond to calculations without and with recombination respectively. The mass asymmetries a = 0.5 and 0.1 are as indicated.

8 as a function of Z, the charge of the fissioning fragment. The frequencies N + and N are averaged over the same initial time interval (0 ~< t ~< 60 f m / c ) by taking a large number of events. The dashed lines correspond to calculations without recombination and the solid lines are those with the inclusion of recombination, the calculations being performed for a = 0.1 and o~ = 0.5. It turns out from the calculations that AV~ occurs more frequently than AV+ as depicted in the figure. It is difficult to assign any specific reason for this in a complex situation with many fragments; however, for the simple case as illustrated in Fig. 1 involving only one fragment in a planar geometry, it is found that AVe- is much more frequent and there the quantity 8 is ~ 0.6. This feature is possibly to a large extent retained when more fragments are present. It is generally found that with a decrease in the number of neighbouring fragments, 8 tends to increase. This, along with the modified configuration, may be the reason for higher values of 8 with inclusion of recombination when the number of neighbouring fragments is generally less (see the charge particle multiplicity as displayed in the top panel of Fig. 5). The charge distribution from the fragmentation of the system is displayed in Fig. 4. The open circles refer to conventional calculations without recombination or any other medium effect (referred to as model I). The crosses correspond to those with inclusion of the modification of the fission barrier due to the neighbouring fragments (model II), and the results with further inclusion of recombination (model III) are shown by the filled circles. The most visible effect due to the modification of the fission barrier is in the enhanced production of relatively lighter intermediate mass fragments (IMF defined as 3 ~< Z ~< 30) and the suppression of heavier fragments. This can be understood in the

S. Pal et al./Nuclear PhysicsA 591 (1995) 719-737

10 z -

i

i

i

i

i

x

Io io

727

10111-

100d_ 10-' _

23 N

o~

Ooe~ele~XXXx

10-2

Xx x

10-3

o xx

××Xx

°~o Xx x

o

l'o

2'0

3'o

z.~o

-

° °

5~o

z Fig. 4. The charge yield from the decay of i~iSm at an excitation energy of 3.6 MeV/particle in the sequential binary decay. The open circles refer to conventional calculations without any medium effect (model l); the crosses correspond to results with modified barrier (model II) and the filled circles to results with a further inclusion of recombination (model Ill). following way. The fission yield is governed by a factor ~ e -v'/r. Since ]AV~] ~ AVe(see Fig. 2) the yield would be controlled more by AV~ than by AV+. Moreover, the frequencies o f AVe- are generally more than those of AV~ which accentuates the effect o f AVF further. The complicated dependence of AV~ on the charge of the fragmenting nucleus and their asymmetry dependence are reflected in the modified charge distribution. The difference between the crosses and the filled circles is the measure o f the effect of recombination. In a recent calculation i [23] where the effect of recombination has been studied without any barrier modification, it has been found that the yield of heavier fragments is enhanced whereas the production of lighter fragments is suppressed a little. Qualitatively, the same effect is noticeable here. However, the reduction in the yield o f lighter fragments is a little more pronounced, which may be attributed to the environmental effect on the barrier. We thus find that the effects due to barrier modification and recombination act in opposite directions in the charge yield. In Fig. 5, we present the dynamical evolution of the average number of charged particles (Nc) (top panel) and the average number of IMF, (NJMF) (bottom panel) as a function of time. The thin lines in the figures refer to conventional calculations in model I, the dashed lines correspond to those in model II and the thick solid lines refer to those in model III. The barrier modification is seen to increase the charge particle multiplicity whereas recombination reduces it. We find that without recombination (Nc) saturates at a time o f ~ 150 fm/c whereas saturation is reached at a somewhat later time when recombination is included. The IMFs are produced in the very early stage of the reaction when the sources are still very hot The I M F production saturates at a I In Ref. 1231, the temperature dependence of the separation energy tor neutrons was inadveilently not included. The qualitative conclusions drawn in the paper, however, remain unaltered once the said temperature dependence is taken into account. Quantitatively, a reduction by ~ 25% of (Nc) and (NIMF) at the expense of enhanced neutron emission is found.

728

S. Pal et al./NuclearPhysicsA 591 (1995) 719-737

12

i

i

i

r

r

I0

4

I

/

/

/

I

I

I

I

J

200

i

/

/ N/

I00

300

t (fm/c)

Fig. 5. The dynamicalevolution of (Nc) (top panel) and (N1MF) (bottom panel). The thin solid lines, the dashed lines and the thick solid lines correspondto calculations with models I, II and Ill respectively. somewhat earlier time compared to that for (Nc). Modification of the fission barrier due to interaction with the neighbouring fragments increases (NIMF) by a factor of about two which is due to the role of AV~ as discussed earlier in the context of charge yield. With the inclusion of coalescence, the IMF multiplicity decreases but it is still significantly higher than that obtained in model I. In Ref. [23], it is seen that coalescence increases (NIMF) somewhat in the absence of any barrier modification. Recombination increases (NIMF) intrinsically; however, its role on AVn is to dilute the effect of barrier modification (as seen in Fig. 2), thereby reducing IMF multiplicity. The conventional sequential binary decay model (model I) generally underestimates IMF production in nuclear fragmentation, inclusion of the medium effect may partially remove this deficiency. The probability distribution P ( N c ) for charge particle multiplicity Nc is displayed in Fig. 6. The symbols used in this figure are the same as in Fig. 4 and retain the same meaning in subsequent figures. In Fig. 5, we have already seen that the average value of charge particle multiplicity (Nc) is more in model II than in model I which is reflected in the distribution function P(Nc ). With the inclusion of recombination the most probable value of Nc is shifted towards lower values by about two units. In Fig. 7, the probability distribution P(NjMF) for IMF multiplicity NIMF is shown. The most probable value for N1MF agrees with (NIMF) at saturation as depicted in Fig. 5 and the model dependence of the most probable value follows the trend of (NIMF)In Fig. 8, the correlation between charge particle multiplicity Nc and the average number of IMFs produced (NlMF) is displayed. In the conventional model (model I), (N1MF) decreases with increasing Nc as large Nc corresponds to events with a large number of light charged particles. With the inclusion of AVB, as already discussed, the

S. Pal et al.INuclear Physics A 591 (1995) 719-737

729

10 o

e ~

x

10-~ x 0 u

1(3-:'

Q-

x

o

•

x x

o

8

10-3

o

8 I

r

Nc

Fig. 6. The probability distribution of charge particle multiplicity Nc. The symbols have the same meaning as in Fig. 4.

production of IMF is enhanced. From the correlation result, it is seen that IMF enhancement occurs for all values of Nc. With the inclusion of recombination (model III), the light charged particles recombine to produce IMFs, thereby reducing both Nc and NIMF (see Fig. 5). In model II and model III, the near constancy of (NIMF) as a function of Nc results from the intricate role played by AVB and & In Fig. 9, the IMF kinetic energy spectra in the source frame are presented. When barrier modification is included, it is found that the kinetic energy spectra are nearly unchanged except in the low energy domain where the yield is enhanced. This enhanced yield comes from a lesser Coulomb repulsion due to an increased production of lighter IMFs in model II as observed from Fig. 4. When recombination is included, the Coulomb peak is shifted a little towards lower energy due to nuclear force effects. We further find that here the relative yield of high energy IMFs is greater. We attribute this to the possible higher temperature of the secondary sources that are formed from coalescence

I00

o

O

.

8

;

x

161

x o

•

x h.

162

O

0.

163 x o I

I

J

8 NIMF

Fig 7 Same as in Fig 6 for NIMF

730

S. Pal et al./Nuclear Physics A 591 (1995) 719-737

5

I

X X

X

X

X

X

X

X

X

X

3 x_

•

2¸

o

0

O

o o

o

0

0 0

O O

I

I

I

I

Nc

Fig. 8. The correlation of average IMF (3 ~< Z ~< 30) multiplicity /NIMF/ with charge particle multiplicity Nc, Symbols have the same meaning as in Fig. 4.

of the generated fragments. All the calculations mentioned above have been done with lifetimes for fragment emission evaluated from Eq. (11). The lifetimes are, however, not very exact. For a given temperature, the lifetimes determine the nature of the configuration of the whole system at any instant and therefore are expected to influence the environmental effect for sequential binary decay. The medium effect is expected to get diluted with increased lifetime. We have repeated the calculations by a fivefold increase in r as given by Eq. ( 11 ). It is observed that the effects of AVB as well as that of recombination become comparatively weaker. However, as the two effects act oppositely, the overall reduction of the medium effect is not very significant by increasing ~- even by fivefold. To investigate the excitation energy dependence of the medium effect, the calculations have been repeated at an excitation energy of 6.0 MeV/particle. At this higher excitation energy the barrier effect is found to be more significant resulting in a larger number

100

161

xXXXXxx

oOOoo~

z

XOooO

xo c

vjj

o

2 16~ N W

~°o o

• Ooo

10-3

0000

Ox

0

I

20

L0

6'0

8'o

I

100

12o

E (MeV) Fig, 9. The IMF kinetic energy spectra with the symbols having the same meaning as in Fig. 4.

S. Pal et al./Nuclear Physics A 591 (1995) 719-737

731

of IMFs. However, because of the close proximity of the large number of produced fragments recombination effect is also substantially enhanced. Thus the combined effect does not alter the conclusions already arrived at at 3.6 MeV/particle for the observables at the higher excitation energy. 3.2. Confrontation with experimental data

A large body of experimental data [8-11] on charge correlations expressed as a function of Zbound (Zbound is a measure of the violence of the collision; a large Zbound implies peripheral collisions with smaller excitation energy; a small Zboundcorresponds to central collisions with higher excitation) exists for the projectile fragmentation of the Au nucleus incident on C, A1, Cu and Pb targets at 600 MeV/nucleon. The experimental results are found to be insensitive to the choice of the target. This is interpreted as equilibration reached by the projectile spectator prior to fragmentation. In this subsection, we compare our model predictions with the data for the Au+Cu reaction at the aforesaid energy. This comparison allows us to probe the effectiveness of the sequential binary decay model in understanding the fragmentation scenario in energetic heavy ion collisions; it also throws light on the sensitivity of the medium effects on the experimentally measured charge correlations. Comparison of the experimental data with the model predictions, however, becomes difficult as the input parameters such as the size and the excitation energy of the fragmenting projectile spectator are a priori unknown. One way to overcome this problem is to perform a hybrid calculation in which the statistical models are supplemented with the dynamical BUU model. The BUU model [19] describes the first stage of the collision quite effectively and provides an estimate of the size and the amount of excitation energy deposited into the projectile spectator which can then be used as input to our sequential binary decay model. The essential input parameters, namely the source size and the excitation energy per particle for the Au+Cu reaction at 600 MeV/nucleon for different impact parameters are taken from the BUU calculations in Ref. [ 11 ]. Subsequently, these input parameters would be referred to as parameter set I. The observables studied as a function of Zbound are the average IMF multiplicity (NIMF>, the average value of the largest charge (Zmax), asymmetry between the largest two charge fragments a2 defined as Zmax -- Z2 a2 - Zmax + Z2 '

(12)

and the three-body asymmetry a3 given by

a3 =

v/(Zmax -- ( Z ) ) 2 -4- (Z2 - ) 2 "q'-(Z3 - ) 2 V ~ (Z)

(13)

Here Z2 and Z3 are the second and third largest charge fragments and (Z) = Zmox+ Z2 + Z3 3

(14)

732

S. Pal et al./Nuclear Physics A 591 (1995) 719-737

/

/

/ f "~\ \

\\

A z

v 2

"f ,/:

0

0

..... 20

, 40

....

'o

,,o 60

Zbound

Fig. 10. The average IMF multiplicity (NIMF) as a function of abound in the reaction Au+Cu at 600 MeV/nucleon. The squares correspond to experimental data [ll], the dotted, dashed and solid lines refer to calculations in models I, II and Ill respectively with the input parameter set I. The open circles, triangles and filled circles correspond to calculated results in models l, lI and III with parameter set II. We also study the second moment T2 [30], expressed through the normalised variance tre of the charge distribution within an event defined as T2 -- ~

Or2e

+ 1,

(15)

(Z)e being the mean charge in the event; the largest charge in the event is excluded in the calculation of T2. In order to facilitate comparison with the data, the Monte Carlo generated events are filtered by a simple efficiency function e ( Z ) depending on the charge Z of the fragment [ 10,1 l ] , e(Z) =l-2e e(Z)--0,

-z,

Z~>2, Z ~< 1.

(16)

The calculated correlations are compared with the experimental data and are displayed in Figs. 10-12. In all the figures, the experimental data are represented by squares. The dotted, dashed and solid lines refer to calculations with models I, II and III respectively. The correlation of (NIMF) with Zbound shown in Fig. 10 displays the rise and fall pattern of (NIMF) as experimentally observed. However, all the three model calculations underpredict the data for very central and near-central collisions (small Zbound) but overestimate the data for mid-peripheral and peripheral collisions. From the top panel of Fig. l 1, the systematically smaller values of calculated (Zmax) as a function of Zbound show more symmetric exit channels as compared to those obtained from experiments. The smaller calculated values of (a2) and (a3) as shown in Fig. 12 reflect the predominance of symmetric decay. As mentioned earlier in the context of predictions for the decay of ]5°Sm at an excitation energy of 3.6 MeV/particle, comparison of calculated

S. Pal et al./Nuclear Physics A 591 (1995) 719-737

733

60 50

I-~ ."

40

N N/20 /

10

E "'l /

o

1.6 1.5

o •

1.4

0 c~oD

~1.3

-

_ 1.2

~t~

~

[]

El~

El 1.1 1.0

..l..iil~li i ~ l ~ 0

, i 20

, ,

I , , 40

,

, I ~ , 60

Zbound

Fig. I 1. The average value of the largest charge (Zmax) (top panel) and the average value of the normalised charge variance (~'2) (bottom panel) as a function of Zbouod for the reaction Au+Cu at 600 MeV/nucleon. The symbols have the same meaning as in Fig. 10.

results from different models depicted by the dotted, dashed and solid lines in Figs. 1012 shows that the effects of recombination and modification of the fission barrier due to surrounding medium act in opposite directions. The high values of (NIMF) and the symmetric decay channels obtained in our model as compared to the experimental data have their origin in the rather large value of the temperature of the fragmenting source in parameter set I. In fact, the COPENHAGEN simultaneous multifragmentation model [ 13] also predicted enhanced symmetric decay channels with this parameter set in the mid-peripheral region (see Ref. [ 11]) in comparison to the experimental data. The charge correlation data were therefore reanalysed by this group [ 31 ]. It was found that once the source sizes and excitation energies used as input in their statistical model were adjusted so as to reproduce the (NIMF)-Zbound correlation, the other measured charge correlations as a function of Zbound were satisfactorily explained. The source parameters so extracted (hereafter referred to as parameter set II) were consistent with their BUU calculations [ 31 ]. These values are significantly smaller than those deduced in Ref. [ 11 ], especially for central collisions where a limiting excitation energy of 8.0 MeV/particle was obtained in set II. The difference arises primarily due to different definitions employed for the fragmenting source. In Ref. [ 11 ], the source was defined to be composed of all the nucleons within a sphere in coordinate space which had not yet collided while a particular lower bound in density was used in Ref. [31].

734

S. Pal et al./Nuclear Physics A 591 (1995) 719-737

0.B []~

0.6 []

." .'

A

0.~

W. A . J / / 0.2

0.8

A 0,6 [] V

A ""

OJ+ •.,

..''

0.2

0-0

~l

i

i

I . . . .

20

I

40

i

i

i

i I i

,

60

Zbound

Fig. 12. The average value of two-body asymmetry (a2) (top panel) and the average value of three-body asymmetry {a3) (bottom panel) as a function of Z~oundfor the reaction Au+Cu at 600 MeV/nucleon. The

symbols retain the same meaning as in Fig. 10. We have repeated the calculations with the parameter set II as inputs in our model• The charge correlations so obtained as a function of Zbound are displayed also in Figs. 10-12. The open circles correspond to calculations performed in model I, the triangles refer to those from model II while the filled circles are results from model III. The last point in the three models at high Zbound is obtained by an extrapolation of the source parameters reported in Ref. [31]. The fit to the data is strikingly improved with the parameter set II. The average value of )'2 as a function of Zbound is displayed in the bottom panel of Fig. 11. The calculated values of (~2) with parameter set I are quite wide off the experimental data and are not displayed in the figure• The calculated values with parameter set II in all the three models agree well with the experimental data for central and very peripheral collisions, but the predicted values are overestimated for mid-peripheral collisions. The sensitivity of the model predictions is found to be more apparent near the peak value• A peak in (')'2/as a function of excitation energy is believed to be an indication of a phase transition [30]. In the sequential binary decay model, a phase transition does not arise• Our model predictions thus show that the occurrence of a peak in (3/2) is probably not a definitive manifestation of a phase transition in nuclear fragmentation• The GEMINI code [ 17] which is also based on the sequential binary decay model fails to reproduce most of the charge correlations [ 10,11 ]. The model predicts highly asymmetric decay channels with a very low mean multiplicity of IMFs and large values

S. Pal et al./Nuclear Physics A 591 (1995) 719-737

735

of (a2), (a3) and (Zmax). On the contrary, the sequential binary decay model as employed by us is able to reproduce most of the data quite satisfactorily. Such distinction mainly arises from the different fission barriers used in the two calculations. The GEMINI code utilises temperature independent finite range liquid drop model conditional barriers and is quite successful in describing low energy collisions. However, the introduction of temperature dependence in the barriers is necessary in energetic nuclear collisions when the temperature involved may be high. The exit channels are then more symmetric as is evident from our results. In all the calculations till date, the effect of the surrounding medium on fragmentation in the sequential binary decay model has not been included. With the COPENHAGEN input source parameters (set II), it is found that the fit to the experimental data with the conventional SBD model (model I) or its modification with the inclusion of medium effects (model III) is equally fair. The model predictions, however, differ significantly around the mid-peripheral collisions for certain observables, namely in the values of (N1MF)and (Y2). This sensitivity to the model predictions is due to the production of a significant number of charged particles and IMFs in close proximity. This enhances the recombination and influences the conditional barrier affecting binary decay.

4. Summary and conclusions In the sequential binary decay model of nuclear fragmentation, fission barriers of hot decaying nuclei are modified in the presence of neighbouring fragments produced in earlier stages of the decay chain. The effect of such a modification on some physical observables in fragmentation phenomena has been studied in this paper. Close proximity of produced fragments may result in coalescence and this effect has also been included in the present study. Depending on the direction of emission of the fission fragments and the configuration of the surrounding medium, the effective fission barrier may sometimes get enhanced or reduced. From the present study it is found that the reduction of the barrier has a more dominant role, thereby increasing the IMF multiplicity substantially, at the cost of heavier fragments. When the effect of recombination is taken into consideration alone, the recombined sources become hotter and may decay subsequently resulting in a somewhat larger number of IMFs produced [23]. Modification of the barrier increases (NIMF) substantially (model II), but the simultaneous treatment of barrier modification and recombination (model III) does not further increase (NIMF), rather it reduces it compared to model II due to a delicate interplay of the medium effect on the barrier and recombination. These two effects thus go in opposite directions, particularly for charge and IMF yield. The overall IMF yield is still substantially greater compared to that obtained from the conventional model of sequential binary decay (model I). The relatively weak correlation between N c and (NIMF) observed in the conventional SBD model practically vanishes with inclusion of barrier modification (model II) and also with further recombination (model III) for the representative system studied in this paper. The IMF kinetic energy spectra become harder (model III) because of the

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possible higher temperatures o f the secondary sources that are formed from the fusion o f generated fragments. Lifetimes o f the hot sources are expected to influence the physical effects that are studied in this paper. To explore this influence, the lifetime as given by Eq. ( 1 1 ) is increased fivefold and we find that the net effect on the physical observables is not very significant. To test the sensitivity o f the physical observables on the combined effect o f the modification o f the fission barrier due to the surrounding medium and recombination, we have confronted our model predictions with the experimental data in relation to projectile fragmentation from Au-induced reactions on the Cu target at 600 MeV/nucleon. The a priori unknown source parameters are taken from B U U calculations, that span a large range o f temperatures and sizes o f the fragmenting source. I f the input parameters are suitably chosen, it is found that a host o f experimental data can be fairly well explained in the sequential binary decay model, both with or without the inclusion o f medium effects. When the size o f the source is rather small or when the source temperature is not too high (corresponding to a small or large Zbouad for the case studied), the combined effect o f the surrounding medium on the physical observables is not very significant; however, for sources o f sizeable magnitude with moderately high excitation energies (corresponding to mid-peripheral collisions for the experimental data analysed or the model case for the representative system 15°Sm studied) the numbers o f charged particles and IMFs are rather large and the sensitivity o f the charge correlation observables to the effects due to the surrounding medium is enhanced. The medium effect on the barrier induces an anisotropy in the direction o f emission o f the fission fragments which has been neglected here to make the calculations numerically tractable. But the observables studied in this paper are mostly inclusive or semi-inclusive in nature and hence may not be influenced much by this anisotropy because o f ensemble averaging.

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