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Description of nuclear fragment formation in terms of a stochastic nucleation model
Volume 244, number 2
PHYSICS LETTERS B
19 July 1990
Description of nuclear fragment formation in terms of a stochastic nucleation model C.O. D o r s o ~ a n d R. D o n a n g e l o 2 Instituto de Fisica. Facultad de lngenieria, C. C. 30. Montevideo. Uruguay
Received 9 January 1990; revised manuscript received 2 May 1990
Fragment formation is described as a stochastic nucleation process without making explicit assumptions about the degree of equilibration of the nuclear system.
The study o f nuclear multifragmentation has attracted considerable interest both from the experimental and theoretical standpoints. On the theoretical side many different models have been employed to describe this process. These models may be roughly cast into two broad categories: (i) dynamical, in which the time evolution o f the system is followed according to some microscopic description (for example the studies ofAichelin et al. [ 1 ], Boal et al. [ 2 ], and Vinet et al. [ 3 ] ), and (ii) non-dynamical, in which the process is studied without making reference to its time dependence. The latter group involves most of the currently available descriptions [ 4 - 1 0 ] . All these treatments implicitly contain the assumption that some sort o f equilibrium is reached previous to fragmentation. This is done by maximizing entropy, minimizing information or some other process equivalent to these. In the present work we attempt to understand fragmentation from a purely stochastic point of view, without making any assumption about the degree o f equilibration o f the system. Moreover, our model will include only those basic ingredients necessary to describe the nuclear characteristics: n u m b e r of nucleons, density, and the nucleon-nucleon interaction parameters. On leave of absence from Departamento de Fisica, Facultad de Ciencias Exactas, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina. On leave of absence from Instituto de Fisica, Universidade Federal do Rio de Janeiro, 21944 Rio de Janeiro, Brazil.
We consider that after the collision event during which energy was deposited into a system composed o f A nucleons this system expands evolving into the unstable region of the liquid-gas coexistence zone. At this stage fragment formation is driven by the appearance of randomly distributed fluctuations which cause the collapse o f the up to then homogeneous nuclear matter into fragments o f varying size, shape and excitation energy. We mock-up the clusterization stage by the following procedure: (a) The nucleon positions are randomly selected using a spherical uniform probability distribution o f radius R, that corresponds to the break-up density value p,~_ 0.10 fm -3. F r o m the nucleon-nucleon interaction characteristics only the hard core radius Rc ~ 0.5 fm is considered in the generation of these primordial configurations. This means that the available configurational phase space has a volume £2= ( V B - - A ~ ) A, where V B =~1gR 4 t3 and eo=Tzu~o. 16 n 3 (b) We let the system perform a r a n d o m walk in the configuration space volume. Using a MetropolisMonte Carlo calculational scheme we make sure that every region of this space is sampled. An alternative procedure we have also considered here consists in not performing this r a n d o m walk and directly using the primordial configurations generated in (a) (using the r a n d o m placement method). (c) Once the configurations are generated we assume that the non-homogeneities in them are representative of those appearing in an unstable system and proceed to define the formation of fragments associ-
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
165
Volume 244, number 2
PHYSICS LETTERS B
ated to them. In fact, what we are doing is to analyze the possible partitions o f a system o f particles u n d e r the restrictions i m p o s e d b y the non-zero particle size and given container volume and clusterization radius. We define a cluster as a group o f nucleons such t h a t each o f t h e m is closer than a given clusterization distance Rd to some other m e m b e r o f the same subset. In this work we consider R~ as a free p a r a m e t e r a n d study the characteristics o f the fragmentation spect r u m as a function o f it. This t r e a t m e n t implies that the C o u l o m b interaction as well as the kinetic energy effects are implicitly included through the R~ value. We note that we have not explicitly taken into account the F e r m i m o t i o n o f the nucleons in our description o f the clusterization process. We can include it in an effective way through the n u c l e o n nucleon potential. To properly include nucleon m o tion an aggregation m o d e l in phase space would be needed, a n d not only in configuration space. We would like to remark, though, that Ngo et al. [11 ] recently considered this point a n d concluded that the a d d i t i o n o f the F e r m i m o t i o n should not lead to a strong constraint in m o m e n t u m space. For this reason we believe that our present treatment is adequate. The clusters generated through the p r o c e d u r e described above are not spherical. Thus we m a y associate a d e f o r m a t i o n energy to each fragment. In o r d e r to evaluate this energy we consider the following effective interaction potential:
19 July 1990
0.0
/7
-1.0v
-2.0R:t = 1.2 1.0
• Rc~ =
• Rc~ = 0 . 6
-3.0
Ro
Vo=Vo e x p ( - R a R ° ) FD, Ru
Rh~
Vo = 0 ,
Rij >lRcl,
Fig. 1. Nucleon-nucleon potential as function of the clusterization radius R¢1.
14.0 o
:E
v >-
(.9 rr" 10.0 ILl Z LU
(.9 Z rt
6.0
I
I
2.0
40
0
(1)
where Vo is the potential strength, Ra is a constam related to the diffuseness o f the potential a n d D is defined so that Vu = 0 for R i j = Rcb i.e. Rd is t h e effective range o f the potential. This is a simple p a r a m e t r i z a tion o f the functional form o f an i n t e m u c l e o n potential that b o t h saturates a n d has b o u n d states. In fig. 1 we illustrate the shape o f the potential for different values o f the clusterization radius. Using the internucleon potential we d e t e r m i n e the associated ground state binding energy per nucleon through a procedure similar to that e m p l o y e d in ref. [ 12 ]. The result is c o m p a r e d in fig. 2 with the semit66
3.0
Rij(fm)
z
VO=oo ,
2.0
1.0
MASS
NUMBER
80 A
Fig. 2. Binding energy per nucleon as a function of mass number for a system of particles interacting through the potential of eq. (1). e m p i r i c a l mass formula values. N o t e that in neither o f the two calculations C o u l o m b or a s y m m e t r y effects were taken into account. The calculated mass spectra o f the fragments as a function o f Re1 are shown in fig. 3. It is clear that the evolution o f these spectra with increasing R¢~ is quite similar to that f o u n d in the percolation a p p r o a c h for a decreasing value o f the site occupation p a r a m e t e r
Volume 244, n u m b e r 2
10-01
PHYSICS LETTERS B
I0.0-~
b)
1]~
t/') 2.0-
011\ 0.0
50.0
100.0 MASS NUMBER A
10.0~
>" 0.011 -0.001 I
0.0
(21 rr" , 50.0
100.0
MASS NUMBER A
0.0
/.
5010 100.0 MASS NUMBER A
z o
1.6
N ¢¢LtJ 1.2¸
,~ I°0 l c)
:2,1\
19 July 1990
:3 .._1 U 0.00! 0.0
50.'O
100,0
MASS NUMBER A
0.8 0:0
~0
EXCITATION Fig. 3. Average n u m b e r of events as a function o f the mass n u m ber of the fragments for the following values of the clusterization radius Rcl: 1.1 (a), 1.2 (b), 1.3 (c), 1.4 (d).
or in the statistical multifragmentation model for decreasing excitation energies per nucleon. Namely we observe the evolution from exponentially decaying mass spectra all the way to a U-shaped behavior. In particular we observe that for Rd = 1.8 the mass spectra can be parametrized as ( M ) = aA-2.37 which is in basic accordance with Fisher's nucleation theory and with the results o f the percolation model in the low density-vanishing interaction limit. This resemblance has been explored in fig. 4 where we relate the value o f the clusterization radius to the excitation energy per nucleon that leads, according to the statistical multifragmentation model o f ref. [ 10], to the same average multiplicity. The nuclear density was taken to be the same in the two calculations. It as interesting to remark at this point that the present model might also be considered as a configuration space version o f the coalescence model [ 13 ]. Some important differences between the two approaches must be noted, however. In the coalescence model a fragment o f mass number A is a set of A nucleons which m o m e n t a all lie inside a sphere o f radius Po (A) im m o m e n t u m space. The parameters Po (A) are individually adjusted to fit the experimental yields o f fragments of masses A = 2, 3 ..... In the present model only one parameter, Rd, is used to define the production o f all clusters. Then although an increase in Rd is operationally equivalent
810
12:0
ENERGY PER NUCLEON
Fig. 4. Clusterization radius Rel that gives the same value of the m e a n fragment multiplicity as the excitation energy per nucleon E*/Ao in the statistical model of ref. [ 10].
to an increase in the value o f the Po (A) parameters, the connection between them is not straightforward. However, the fact that the Po (A) parameters depend smoothly on A probably means that the coalescence model may be reformulated as a nucleation model in m o m e n t u m space in which only one parameter Pa is needed. In this case a close correspondence should exist between the parameters Rc] and Pd in the two models. In fig. 5 we show the total multiplicity as a function o f R¢l. These results, together with those o f figs. 3 and 4, reflect the fact that in our model a decrease in the value of Rcl, is equivalent to an increase in the value o f D and it thus mocks up a higher mean kinetic energy o f the constituent nucleons. We have checked that these results are insensitive to the choice o f the hard-core radius parameter Rho as long as it remains much smaller than the system's dimensions. Another relevant point is that the results obtained with the use of the Metropolis-Monte Carlo r a n d o m walk method coincide with those obtained with the much simpler r a n d o m placement treatment. This finding is not obvious since, as it is well known, the closely packed configurations that naturally occur at higher densities are highly improbable and consequently are not conveniently studied with the r a n d o m placement method. At high densities the 167
Volume 244, number 2
PHYSICS LETTERS B
The m o d e l also allows to study the effect o f density fluctuations on the fragmentation process. This m a y be done b y purposefully biasing the r a n d o m configurations generated in part ( a ) o f the procedure described in the initial part o f the present work [ 15 ]. This is a feature o f our m o d e l which cannot be incorp o r a t e d by any m o d e l assuming equilibrium a n d which, to our knowledge, has not been included in current percolation models. We would like to a d d that, in contrast to most percolation models, our treatm e n t does not e m p l o y a lattice, so we expect it to lead to better estimates o f the fragment d e f o r m a t i o n energies. Finally, the procedure described in this work does not consider clusters v i a site a n d / o r b o n d percolation p a r a m e t e r s b u t through the m o r e a p p r o p r i ate concept (in nuclear physics) o f the effective interaction range.
40.0.
1
30.0 A
20.0 ,4-'
~E v
10.0
0.0
0.8
°
t2
2.0
Rct Fig. 5. Average total multiplicity as a function of the clusterization radius. M e t r o p o l i s - M o n t e Carlo m e t h o d naturally searches for the small region o f phase space which corresponds to packed configurations. This property makes it most convenient for such situations. In the rarefied case we are dealing with here, the close agreement between the two t r e a t m e n t s indicates that the r a n d o m placement m e t h o d is preferable because o f its simpler structure a n d considerably shorter c o m p u t a tional time requirement. One o f the interesting features o f the present model is that, since there is no restriction o n the geometrical configuration o f the clusters, it takes into account effects due to fragment d e f o r m a t i o n and, in particular, it allows t o easily calculate their excitation energies. The i m p o r t a n c e o f fragment d e f o r m a t i o n has been e m p h a s i z e d by several authors, i.e. Vicentini et al. [ 14 ]. In o r d e r to calculate the excitation energy Of a fragment o f size A we sum the effective interaction energies given by eq. ( 1 ) ,
eo(A)=½ ~ VU, 6
(2)
where i, j c o r r e s p o n d to nucleons belonging to the given cluster, and then subtract the b i n d i n g energy o f a nucleus with the same n u m b e r o f nucleons as the cluster as taken from fig. 2. T h e study o f d e f o r m a t i o n energies on the fragmentation process is presently underway. 168
19 July 1990
C,O.D. thanks the Centro L a t i n o a m e r i c a n o de Fisica for financial support. R.D. acknowledges partial support from the C N P q a n d F I N E P (Brazil).
References [ 1] J. Aichelin, G. Peilert, A. Bohnet, A, Rosenhauer, H. Strcker andW. Greiner, Phys. Rev. C 37 (1989) 2451. [2] D.H. Boal and J.N. Glosh, Phys. Rev. C 38 (1988) 1870. [ 3 ] L. Vinet, C. Gregoire, P. Schuck, B. Remand and F. Sebille, Nucl. Phys. A 468 (1987) 321. [4] J. Aichelin, J. Hiifner and R. Ibarra, Phys. Rev. C 30 (198~4) 107. [ 5 ] J. Randrup and S. Koonin, Nucl. Phys. A 356 ( 1981 ) 223; G. Fai and J. Randrup, Nucl. Phys. A 381 (1982) 557. [6] A.Z. Mekjian, Phys. Rev. C 17 (1978) 1051; C 30 (1984) 851. [7 ] W.A. Friedman and W.G. Lynch, Phys. Rev. C 28 (t983) 16, 950. [8] W. Bauer, D.R. Dean, U. Mosel andU. Post, Phys. Lett. B 150 (1984) 53. [9] Sa Ban-Hao and D.H.E. Gross. NucL Phys. A 437 (1985) 643. [10] J.P. Bondorf et al., Nucl. Phys. A 443 (1985) 321; A 444 (1985) 460. [ 11 ] C. Ngo et al., preprint IPNO/TH 88-770 (November 1988). [ 12] J.K. Lee, J.A. Baker and F.F. Abraham, J. Chem. Phys. 58 (1973) 3166. [ 13] H.H. Gutbrod el al., Phys. Rev. Lett. 37 (1976) 667. [ 14] A. Vicentini, G. Jacucci and V.R. Pandharipande, Phys. Rev. C31 (1985) 1783. [ 15 ] R. Donangelo, C.O. Dorso and H.D. Marta, to be published.
PHYSICS LETTERS B
19 July 1990
Description of nuclear fragment formation in terms of a stochastic nucleation model C.O. D o r s o ~ a n d R. D o n a n g e l o 2 Instituto de Fisica. Facultad de lngenieria, C. C. 30. Montevideo. Uruguay
Received 9 January 1990; revised manuscript received 2 May 1990
Fragment formation is described as a stochastic nucleation process without making explicit assumptions about the degree of equilibration of the nuclear system.
The study o f nuclear multifragmentation has attracted considerable interest both from the experimental and theoretical standpoints. On the theoretical side many different models have been employed to describe this process. These models may be roughly cast into two broad categories: (i) dynamical, in which the time evolution o f the system is followed according to some microscopic description (for example the studies ofAichelin et al. [ 1 ], Boal et al. [ 2 ], and Vinet et al. [ 3 ] ), and (ii) non-dynamical, in which the process is studied without making reference to its time dependence. The latter group involves most of the currently available descriptions [ 4 - 1 0 ] . All these treatments implicitly contain the assumption that some sort o f equilibrium is reached previous to fragmentation. This is done by maximizing entropy, minimizing information or some other process equivalent to these. In the present work we attempt to understand fragmentation from a purely stochastic point of view, without making any assumption about the degree o f equilibration o f the system. Moreover, our model will include only those basic ingredients necessary to describe the nuclear characteristics: n u m b e r of nucleons, density, and the nucleon-nucleon interaction parameters. On leave of absence from Departamento de Fisica, Facultad de Ciencias Exactas, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina. On leave of absence from Instituto de Fisica, Universidade Federal do Rio de Janeiro, 21944 Rio de Janeiro, Brazil.
We consider that after the collision event during which energy was deposited into a system composed o f A nucleons this system expands evolving into the unstable region of the liquid-gas coexistence zone. At this stage fragment formation is driven by the appearance of randomly distributed fluctuations which cause the collapse o f the up to then homogeneous nuclear matter into fragments o f varying size, shape and excitation energy. We mock-up the clusterization stage by the following procedure: (a) The nucleon positions are randomly selected using a spherical uniform probability distribution o f radius R, that corresponds to the break-up density value p,~_ 0.10 fm -3. F r o m the nucleon-nucleon interaction characteristics only the hard core radius Rc ~ 0.5 fm is considered in the generation of these primordial configurations. This means that the available configurational phase space has a volume £2= ( V B - - A ~ ) A, where V B =~1gR 4 t3 and eo=Tzu~o. 16 n 3 (b) We let the system perform a r a n d o m walk in the configuration space volume. Using a MetropolisMonte Carlo calculational scheme we make sure that every region of this space is sampled. An alternative procedure we have also considered here consists in not performing this r a n d o m walk and directly using the primordial configurations generated in (a) (using the r a n d o m placement method). (c) Once the configurations are generated we assume that the non-homogeneities in them are representative of those appearing in an unstable system and proceed to define the formation of fragments associ-
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
165
Volume 244, number 2
PHYSICS LETTERS B
ated to them. In fact, what we are doing is to analyze the possible partitions o f a system o f particles u n d e r the restrictions i m p o s e d b y the non-zero particle size and given container volume and clusterization radius. We define a cluster as a group o f nucleons such t h a t each o f t h e m is closer than a given clusterization distance Rd to some other m e m b e r o f the same subset. In this work we consider R~ as a free p a r a m e t e r a n d study the characteristics o f the fragmentation spect r u m as a function o f it. This t r e a t m e n t implies that the C o u l o m b interaction as well as the kinetic energy effects are implicitly included through the R~ value. We note that we have not explicitly taken into account the F e r m i m o t i o n o f the nucleons in our description o f the clusterization process. We can include it in an effective way through the n u c l e o n nucleon potential. To properly include nucleon m o tion an aggregation m o d e l in phase space would be needed, a n d not only in configuration space. We would like to remark, though, that Ngo et al. [11 ] recently considered this point a n d concluded that the a d d i t i o n o f the F e r m i m o t i o n should not lead to a strong constraint in m o m e n t u m space. For this reason we believe that our present treatment is adequate. The clusters generated through the p r o c e d u r e described above are not spherical. Thus we m a y associate a d e f o r m a t i o n energy to each fragment. In o r d e r to evaluate this energy we consider the following effective interaction potential:
19 July 1990
0.0
/7
-1.0v
-2.0R:t = 1.2 1.0
• Rc~ =
• Rc~ = 0 . 6
-3.0
Ro
Vo=Vo e x p ( - R a R ° ) FD, Ru
Rh~
Vo = 0 ,
Rij >lRcl,
Fig. 1. Nucleon-nucleon potential as function of the clusterization radius R¢1.
14.0 o
:E
v >-
(.9 rr" 10.0 ILl Z LU
(.9 Z rt
6.0
I
I
2.0
40
0
(1)
where Vo is the potential strength, Ra is a constam related to the diffuseness o f the potential a n d D is defined so that Vu = 0 for R i j = Rcb i.e. Rd is t h e effective range o f the potential. This is a simple p a r a m e t r i z a tion o f the functional form o f an i n t e m u c l e o n potential that b o t h saturates a n d has b o u n d states. In fig. 1 we illustrate the shape o f the potential for different values o f the clusterization radius. Using the internucleon potential we d e t e r m i n e the associated ground state binding energy per nucleon through a procedure similar to that e m p l o y e d in ref. [ 12 ]. The result is c o m p a r e d in fig. 2 with the semit66
3.0
Rij(fm)
z
VO=oo ,
2.0
1.0
MASS
NUMBER
80 A
Fig. 2. Binding energy per nucleon as a function of mass number for a system of particles interacting through the potential of eq. (1). e m p i r i c a l mass formula values. N o t e that in neither o f the two calculations C o u l o m b or a s y m m e t r y effects were taken into account. The calculated mass spectra o f the fragments as a function o f Re1 are shown in fig. 3. It is clear that the evolution o f these spectra with increasing R¢~ is quite similar to that f o u n d in the percolation a p p r o a c h for a decreasing value o f the site occupation p a r a m e t e r
Volume 244, n u m b e r 2
10-01
PHYSICS LETTERS B
I0.0-~
b)
1]~
t/') 2.0-
011\ 0.0
50.0
100.0 MASS NUMBER A
10.0~
>" 0.011 -0.001 I
0.0
(21 rr" , 50.0
100.0
MASS NUMBER A
0.0
/.
5010 100.0 MASS NUMBER A
z o
1.6
N ¢¢LtJ 1.2¸
,~ I°0 l c)
:2,1\
19 July 1990
:3 .._1 U 0.00! 0.0
50.'O
100,0
MASS NUMBER A
0.8 0:0
~0
EXCITATION Fig. 3. Average n u m b e r of events as a function o f the mass n u m ber of the fragments for the following values of the clusterization radius Rcl: 1.1 (a), 1.2 (b), 1.3 (c), 1.4 (d).
or in the statistical multifragmentation model for decreasing excitation energies per nucleon. Namely we observe the evolution from exponentially decaying mass spectra all the way to a U-shaped behavior. In particular we observe that for Rd = 1.8 the mass spectra can be parametrized as ( M ) = aA-2.37 which is in basic accordance with Fisher's nucleation theory and with the results o f the percolation model in the low density-vanishing interaction limit. This resemblance has been explored in fig. 4 where we relate the value o f the clusterization radius to the excitation energy per nucleon that leads, according to the statistical multifragmentation model o f ref. [ 10], to the same average multiplicity. The nuclear density was taken to be the same in the two calculations. It as interesting to remark at this point that the present model might also be considered as a configuration space version o f the coalescence model [ 13 ]. Some important differences between the two approaches must be noted, however. In the coalescence model a fragment o f mass number A is a set of A nucleons which m o m e n t a all lie inside a sphere o f radius Po (A) im m o m e n t u m space. The parameters Po (A) are individually adjusted to fit the experimental yields o f fragments of masses A = 2, 3 ..... In the present model only one parameter, Rd, is used to define the production o f all clusters. Then although an increase in Rd is operationally equivalent
810
12:0
ENERGY PER NUCLEON
Fig. 4. Clusterization radius Rel that gives the same value of the m e a n fragment multiplicity as the excitation energy per nucleon E*/Ao in the statistical model of ref. [ 10].
to an increase in the value o f the Po (A) parameters, the connection between them is not straightforward. However, the fact that the Po (A) parameters depend smoothly on A probably means that the coalescence model may be reformulated as a nucleation model in m o m e n t u m space in which only one parameter Pa is needed. In this case a close correspondence should exist between the parameters Rc] and Pd in the two models. In fig. 5 we show the total multiplicity as a function o f R¢l. These results, together with those o f figs. 3 and 4, reflect the fact that in our model a decrease in the value of Rcl, is equivalent to an increase in the value o f D and it thus mocks up a higher mean kinetic energy o f the constituent nucleons. We have checked that these results are insensitive to the choice o f the hard-core radius parameter Rho as long as it remains much smaller than the system's dimensions. Another relevant point is that the results obtained with the use of the Metropolis-Monte Carlo r a n d o m walk method coincide with those obtained with the much simpler r a n d o m placement treatment. This finding is not obvious since, as it is well known, the closely packed configurations that naturally occur at higher densities are highly improbable and consequently are not conveniently studied with the r a n d o m placement method. At high densities the 167
Volume 244, number 2
PHYSICS LETTERS B
The m o d e l also allows to study the effect o f density fluctuations on the fragmentation process. This m a y be done b y purposefully biasing the r a n d o m configurations generated in part ( a ) o f the procedure described in the initial part o f the present work [ 15 ]. This is a feature o f our m o d e l which cannot be incorp o r a t e d by any m o d e l assuming equilibrium a n d which, to our knowledge, has not been included in current percolation models. We would like to a d d that, in contrast to most percolation models, our treatm e n t does not e m p l o y a lattice, so we expect it to lead to better estimates o f the fragment d e f o r m a t i o n energies. Finally, the procedure described in this work does not consider clusters v i a site a n d / o r b o n d percolation p a r a m e t e r s b u t through the m o r e a p p r o p r i ate concept (in nuclear physics) o f the effective interaction range.
40.0.
1
30.0 A
20.0 ,4-'
~E v
10.0
0.0
0.8
°
t2
2.0
Rct Fig. 5. Average total multiplicity as a function of the clusterization radius. M e t r o p o l i s - M o n t e Carlo m e t h o d naturally searches for the small region o f phase space which corresponds to packed configurations. This property makes it most convenient for such situations. In the rarefied case we are dealing with here, the close agreement between the two t r e a t m e n t s indicates that the r a n d o m placement m e t h o d is preferable because o f its simpler structure a n d considerably shorter c o m p u t a tional time requirement. One o f the interesting features o f the present model is that, since there is no restriction o n the geometrical configuration o f the clusters, it takes into account effects due to fragment d e f o r m a t i o n and, in particular, it allows t o easily calculate their excitation energies. The i m p o r t a n c e o f fragment d e f o r m a t i o n has been e m p h a s i z e d by several authors, i.e. Vicentini et al. [ 14 ]. In o r d e r to calculate the excitation energy Of a fragment o f size A we sum the effective interaction energies given by eq. ( 1 ) ,
eo(A)=½ ~ VU, 6
(2)
where i, j c o r r e s p o n d to nucleons belonging to the given cluster, and then subtract the b i n d i n g energy o f a nucleus with the same n u m b e r o f nucleons as the cluster as taken from fig. 2. T h e study o f d e f o r m a t i o n energies on the fragmentation process is presently underway. 168
19 July 1990
C,O.D. thanks the Centro L a t i n o a m e r i c a n o de Fisica for financial support. R.D. acknowledges partial support from the C N P q a n d F I N E P (Brazil).
References [ 1] J. Aichelin, G. Peilert, A. Bohnet, A, Rosenhauer, H. Strcker andW. Greiner, Phys. Rev. C 37 (1989) 2451. [2] D.H. Boal and J.N. Glosh, Phys. Rev. C 38 (1988) 1870. [ 3 ] L. Vinet, C. Gregoire, P. Schuck, B. Remand and F. Sebille, Nucl. Phys. A 468 (1987) 321. [4] J. Aichelin, J. Hiifner and R. Ibarra, Phys. Rev. C 30 (198~4) 107. [ 5 ] J. Randrup and S. Koonin, Nucl. Phys. A 356 ( 1981 ) 223; G. Fai and J. Randrup, Nucl. Phys. A 381 (1982) 557. [6] A.Z. Mekjian, Phys. Rev. C 17 (1978) 1051; C 30 (1984) 851. [7 ] W.A. Friedman and W.G. Lynch, Phys. Rev. C 28 (t983) 16, 950. [8] W. Bauer, D.R. Dean, U. Mosel andU. Post, Phys. Lett. B 150 (1984) 53. [9] Sa Ban-Hao and D.H.E. Gross. NucL Phys. A 437 (1985) 643. [10] J.P. Bondorf et al., Nucl. Phys. A 443 (1985) 321; A 444 (1985) 460. [ 11 ] C. Ngo et al., preprint IPNO/TH 88-770 (November 1988). [ 12] J.K. Lee, J.A. Baker and F.F. Abraham, J. Chem. Phys. 58 (1973) 3166. [ 13] H.H. Gutbrod el al., Phys. Rev. Lett. 37 (1976) 667. [ 14] A. Vicentini, G. Jacucci and V.R. Pandharipande, Phys. Rev. C31 (1985) 1783. [ 15 ] R. Donangelo, C.O. Dorso and H.D. Marta, to be published.