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Coulomb instability in collisions between very heavy nuclei around 30 MeV per nucleon
Physics Letters B 302 (1993) 15-17 North-Holland
PHYSICS LETTERS B
Coulomb instability in collisions between very heavy nuclei around 30 MeV per nucleon B. Borderie a, B. R e m a u d b, M.F. Rivet a and F. Sebille b Institut de Physique Nuclbaire, IN2P3-CNRS, F-91406 Orsay Cedex, France b Laboratoire de Physique Nucl~aire, 2 rue de la Houssini~re, F-44072 Nantes Cedex 03, France
Received 28 July 1992
Simulations, based on the Landau-Vlasov model, of central collisions (b=0-3 fm) between very heavy nuclei around 30 MeV per nucleon reveal the occurrence of the Coulomb instability as already predicted by static calculations of hot nuclei a few years ago. The Coulomb instability shows itself in the formation of unstable bubbles.
During the last few years, experiments p e r f o r m e d with i n t e r m e d i a t e energy heavy-ion projectiles have m a d e possible studies o f the f o r m a t i o n a n d decay o f highly excited nuclei [ 1 ]; there was evidence that such hot c o m p a c t c o m p o s i t e nuclei behave like thermalized nuclei up to initial temperatures o f at least 6 MeV [2,3 ]. On the other hand, a few years ago, theoretical calculations, in which the nucleus is in thermal equilibrium with its vapor, have shown that C o u l o m b repulsion can cause an instability in a hot nucleus at n o r m a l density when its t e m p e r a t u r e is raised bey o n d a certain limiting value [ 4,5 ]; thus an unstable bubble nucleus is expected to be formed. The value o f the t e m p e r a t u r e at which this C o u l o m b instability takes place d e p e n d s on the balance between the Coulomb, the surface a n d the bulk nuclear energy. It is therefore strongly influenced by the equation o f state, the t e m p e r a t u r e d e p e n d e n c e o f the surface tension o f nuclear m a t t e r a n d the charge to mass ratio o f the hot nucleus [ 6 ]. F r o m the confrontation o f these limiting values with those d e d u c e d from experiments, no conclusion can be given at the present time. W i t h recent capabilities o f accelerating very heavy projectiles, one has a possible way to prove or not the existence o f such a C o u l o m b instability; indeed by using projectiles heavier than Xe on heavy targets (from Au to U ) one would be able to produce, a r o u n d 30 M e V per nucleon, very heavy c o m p a c t and thermalized compos-
ite systems (A>~300) with mass to charge ratios a r o u n d 2.5, i.e. with C o u l o m b energy values larger than those deduced for the beta stability line. As a consequence low limiting temperatures for C o u l o m b instability are expected, as presented in fig. 1; these curves have been obtained with an a p p r o x i m a t e calculation [ 7 ] which well agrees, especially for large A values, with the results o f ref. [ 5 ]. Note that the limiting values lie in the range 3-5 MeV. These calculations correspond to a static situation but we know well that d y n a m i c a l aspects can have
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60
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160
BOO 860 A
800
860
l , , ,
400
Fig. 1. Limiting temperature of nuclei as a function of their mass. The dashed line is relative to nuclei along the fl stability line while the full curve refers to composite nuclear systems ( A / Z ~ 2 . 5 ) which can be produced in collisions between very heavy nuclei.
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
15
Volume 302, number 1
PHYSICS LETTERS B
strong consequences in heavy-ion collisions; it therefore appeared to be very fruitful to simulate these collisions between very heavy nuclei. For this purpose semi-classical calculations based on the Landau-Vlasov model [ 8 ], which were shown to reproduce well dynamical aspects of collisions especially around 30 MeV per nucleon, have been performed [2,9]. The Landau-Vlasov equation has been solved with 30 pseudo-particles per nucleon and the mean field was approximated by a simplified Skyrme interaction [ 10 ] yielding an incompressibility modulus K = 200 MeV. The free nucleon-nucleon cross-section was used in the collision term. Evidently the Coulomb interaction was correctly accounted for. We present here the results obtained for the system 1 S S G d + 2 3 s U at 27 and 35 MeV per nucleon, for impact parameters equal to 3 fm. Similar trends were observed for head-on collisions. At both bombarding energies the two in-
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18 March 1993
cident nuclei fuse in an entity with a compact shape produced by a gentle compression phase (P/Po ~ 1.2). Then the composite system expands, and at about 140 f m / c , when it has thermalized by ejecting a large number of preequilibrium nucleons (27 at 27 MeV per nucleon and 40 at 35 MeV per nucleon), it enters a bubble geometry (very low central density). The thermal excitation energy of the composite system at that time lies around 2.0 MeV/nucleon irrespective of the bombarding energy; this value has to be regarded as a minimum one due to the very low density. The evolution of the system can be followed by looking at equal density levels in the three planes (x, z), (y, z) and (x, y), zbeing the beam direction and x the impact parameter axis. An illustration is presented in fig. 2, for one plane, but the two other pictures are completely similar, revealing that indeed a bubble is formed. We must underline that similar
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Fig. 2. t55Gd + 23sU collisions at 35 MeV/nucleon calculated with the Landau-Vlasov equation for an impact parameter b = 3 fm. The beam axis is vertical and the impact parameter axis horizontal. The different colors represent equal density levels in the (x, z) plan e. The system is displayed every 20 fm/c from 0 to 220 fm/c when scanning from top to bottom, and from left to right.
16
Volume 302, number 1
PHYSICS LETTERS B
........ t =
8o.
/ ' ~
c u l a t i o n s are in progress to a n s w e r this q u e s t i o n tentatively [ 17 ]. F o r e x p e r i m e n t a l i s t s the challenge is n o w to search for u n a m b i g u o u s signatures o f such a m u l t i f r a g m e n t a t i o n b y using very powerful m u l t i d e rectors [ 1 8 - 2 0 ].
~t=-.ouo.
0.2 =
----t
160./
~.
t= 2 4 0 . / ~
--.
I ,
. ~ ,,...,
"~,,
T h e a u t h o r s t h a n k D. V a u t h e r i n for v a l u a b l e discussions a n d for p r o v i d i n g us with its c o m p u t e r prog r a m for l i m i t i n g t e m p e r a t u r e .
0.1 Q)
0,0
18 March 1993
,-"-TY" I-"" I . _~"~
. . . . . .
0
5
R(fm)
X-.
1
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10
I
, . . "l~, ;:'==---J
. . . . . .
6
10
15
R(fm)
Fig. 3. Density profiles (R is the radial distance) at different times (in fm/c) extracted from the Landau-Vlasov simulation for the same system as in fig. 2 (left side). At the right side the Coulomb interaction has been turned off. values ofp/po for lighter systems [ 11,12] do n o t induce the f o r m a t i o n o f a b u b b l e . To be m o r e q u a n t i t a t i v e a n d to clearly e m p h a s i z e the role o f the C o u l o m b i n t e r a c t i o n , we have plotted in fig. 3 the d e n s i t y profile o f the system for different times. Values as low as 0. lpo (Do ~ 0.13 f m - 3 with the force used here) are o b s e r v e d in the central part after 160 fm/c for the c o m p l e t e calculation (left side). Conversely, i f we t u r n o f f t h e C o u l o m b interaction, a higher i n i t i a l c o m p r e s s i o n is observed, followed b y m o n o p o l e type oscillations (right side). N o t e that b u b b l e s were also o b t a i n e d in calculations o f very hot o r / a n d highly c o m p r e s s e d nuclei [ 13-15 ] b u t it was argued that such a f u n n y geometry a p p e a r e d because o f the i m p o s e d spherical symmetry. I n fig. 2, we observe well that, after a b o u t 160 f m / c, a m u l t i f r a g m e n t a t i o n process occurs. Evidently, the s i m u l a t i o n discussed above is n o m o r e r e l e v a n t to exp l a i n the f o r m a t i o n o f fragments. However, c a n it indicate w h e t h e r there are surface instabilities [ 16 ] or w h e t h e r the system enters the s p i n o d a l region? Cal-
References
[ 1 ] D. Guerreau, Nuclear matter and heavy-ion collisions, eds. M. Soyeur, H. Flocard, B. Tamain and M. Porneuf (Plenum, New York, 1989)p. 187. [2] D. Jouan et al., Z. Phys. A 340 ( 1991 ) 63. [ 3 ] E. Crema et al., Phys. Left. B 258 ( 1991 ) 266. [4] P. Bonche, S. Levit and D. Vautherin, Nucl. Phys. A 427 (1984) 278. [ 5 ] S. Levit and P. Bonche, Nucl. Phys. A 437 ( 1985 ) 426. [ 6 ] J, Besprosvany and S. Levit, Phys. Lett. B 217 (1989) 1. [ 7 ] D. Vautherin, Lectures NATO Advanced Study Institute on Supernovae (Les Houches, France, August 1990), eds. S. Bludman, R. Mochkovitch and J. Zinn-Justin (NorthHolland, Amsterdam). [ 8 ] C. Gr6goire et al., Nucl. Phys. A 465 (1987) 317. [9] C. Gr6goire et al., Nucl. Phys. A 471 (1987) 399c. [ 10] L. Zamick, Phys. Left. B 45 (1973) 313. [ 11 ] M.F. Rivet et al., Phys. Lett. B 215 ( 1988 ) 55. [ 12 ] E. Suraud et al., Phys. Lett. B 229 (1989) 359. [ 13 ] J. Nemeth et al., Z. Phys. A 320 ( 1985 ) 691. [14] L. Vinet et al., Phys. Lett. B 172 (1986) 17. [ 15 ] D. Vautherin, J. Treiner and M. V6n6roni, Phys. Lett. B 191 (1987) 6. [ 16] L.G. Moretto et al., LBL report 31812 (1992). [ 17 ] F. S6bille et al., to be published. [ 18 ] D.W. Stracener el al., Nucl. Instrum. Methods A 294 (1990) 485. [ 19 ] R.T. de Souza et al., Nucl. Instrum. Methods A 295 (1990) 109. [20] A. Chbihi et al., Proc. first European Biennal Workshop on Nuclear physics (Meg~ve, France, March 1991 ), eds. D. Guinet and R. Pizzi (World Scientific, Singapore ) p. 378.
17
PHYSICS LETTERS B
Coulomb instability in collisions between very heavy nuclei around 30 MeV per nucleon B. Borderie a, B. R e m a u d b, M.F. Rivet a and F. Sebille b Institut de Physique Nuclbaire, IN2P3-CNRS, F-91406 Orsay Cedex, France b Laboratoire de Physique Nucl~aire, 2 rue de la Houssini~re, F-44072 Nantes Cedex 03, France
Received 28 July 1992
Simulations, based on the Landau-Vlasov model, of central collisions (b=0-3 fm) between very heavy nuclei around 30 MeV per nucleon reveal the occurrence of the Coulomb instability as already predicted by static calculations of hot nuclei a few years ago. The Coulomb instability shows itself in the formation of unstable bubbles.
During the last few years, experiments p e r f o r m e d with i n t e r m e d i a t e energy heavy-ion projectiles have m a d e possible studies o f the f o r m a t i o n a n d decay o f highly excited nuclei [ 1 ]; there was evidence that such hot c o m p a c t c o m p o s i t e nuclei behave like thermalized nuclei up to initial temperatures o f at least 6 MeV [2,3 ]. On the other hand, a few years ago, theoretical calculations, in which the nucleus is in thermal equilibrium with its vapor, have shown that C o u l o m b repulsion can cause an instability in a hot nucleus at n o r m a l density when its t e m p e r a t u r e is raised bey o n d a certain limiting value [ 4,5 ]; thus an unstable bubble nucleus is expected to be formed. The value o f the t e m p e r a t u r e at which this C o u l o m b instability takes place d e p e n d s on the balance between the Coulomb, the surface a n d the bulk nuclear energy. It is therefore strongly influenced by the equation o f state, the t e m p e r a t u r e d e p e n d e n c e o f the surface tension o f nuclear m a t t e r a n d the charge to mass ratio o f the hot nucleus [ 6 ]. F r o m the confrontation o f these limiting values with those d e d u c e d from experiments, no conclusion can be given at the present time. W i t h recent capabilities o f accelerating very heavy projectiles, one has a possible way to prove or not the existence o f such a C o u l o m b instability; indeed by using projectiles heavier than Xe on heavy targets (from Au to U ) one would be able to produce, a r o u n d 30 M e V per nucleon, very heavy c o m p a c t and thermalized compos-
ite systems (A>~300) with mass to charge ratios a r o u n d 2.5, i.e. with C o u l o m b energy values larger than those deduced for the beta stability line. As a consequence low limiting temperatures for C o u l o m b instability are expected, as presented in fig. 1; these curves have been obtained with an a p p r o x i m a t e calculation [ 7 ] which well agrees, especially for large A values, with the results o f ref. [ 5 ]. Note that the limiting values lie in the range 3-5 MeV. These calculations correspond to a static situation but we know well that d y n a m i c a l aspects can have
. . . .
I
. . . .
I
. . . .
I
. . . .
!
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
l
. . . .
I
. . . .
16
10
",,,
'|
-
0
l
0
z
- -
-o.o
-
. . . .
I
60
. . . .
a . . . .
100
I
. . . .
160
BOO 860 A
800
860
l , , ,
400
Fig. 1. Limiting temperature of nuclei as a function of their mass. The dashed line is relative to nuclei along the fl stability line while the full curve refers to composite nuclear systems ( A / Z ~ 2 . 5 ) which can be produced in collisions between very heavy nuclei.
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
15
Volume 302, number 1
PHYSICS LETTERS B
strong consequences in heavy-ion collisions; it therefore appeared to be very fruitful to simulate these collisions between very heavy nuclei. For this purpose semi-classical calculations based on the Landau-Vlasov model [ 8 ], which were shown to reproduce well dynamical aspects of collisions especially around 30 MeV per nucleon, have been performed [2,9]. The Landau-Vlasov equation has been solved with 30 pseudo-particles per nucleon and the mean field was approximated by a simplified Skyrme interaction [ 10 ] yielding an incompressibility modulus K = 200 MeV. The free nucleon-nucleon cross-section was used in the collision term. Evidently the Coulomb interaction was correctly accounted for. We present here the results obtained for the system 1 S S G d + 2 3 s U at 27 and 35 MeV per nucleon, for impact parameters equal to 3 fm. Similar trends were observed for head-on collisions. At both bombarding energies the two in-
I
'
'
'
'
I
'
'
"
'
I
'
18 March 1993
cident nuclei fuse in an entity with a compact shape produced by a gentle compression phase (P/Po ~ 1.2). Then the composite system expands, and at about 140 f m / c , when it has thermalized by ejecting a large number of preequilibrium nucleons (27 at 27 MeV per nucleon and 40 at 35 MeV per nucleon), it enters a bubble geometry (very low central density). The thermal excitation energy of the composite system at that time lies around 2.0 MeV/nucleon irrespective of the bombarding energy; this value has to be regarded as a minimum one due to the very low density. The evolution of the system can be followed by looking at equal density levels in the three planes (x, z), (y, z) and (x, y), zbeing the beam direction and x the impact parameter axis. An illustration is presented in fig. 2, for one plane, but the two other pictures are completely similar, revealing that indeed a bubble is formed. We must underline that similar
'
'
'
I
'
m
!
'
'
m*
i
'
i
I
'
|
h
r
~,
|
im
i
" !
r •
"'i
•
*~
~
,i
E
i
i
1, • m
m t.
i q
m-
m
Immm
F
m~
I
q m
•
ms am
mm
q
• ,.ram
E _
. _
|
.
.m
• i
B
E
|m
q
i
.
Fig. 2. t55Gd + 23sU collisions at 35 MeV/nucleon calculated with the Landau-Vlasov equation for an impact parameter b = 3 fm. The beam axis is vertical and the impact parameter axis horizontal. The different colors represent equal density levels in the (x, z) plan e. The system is displayed every 20 fm/c from 0 to 220 fm/c when scanning from top to bottom, and from left to right.
16
Volume 302, number 1
PHYSICS LETTERS B
........ t =
8o.
/ ' ~
c u l a t i o n s are in progress to a n s w e r this q u e s t i o n tentatively [ 17 ]. F o r e x p e r i m e n t a l i s t s the challenge is n o w to search for u n a m b i g u o u s signatures o f such a m u l t i f r a g m e n t a t i o n b y using very powerful m u l t i d e rectors [ 1 8 - 2 0 ].
~t=-.ouo.
0.2 =
----t
160./
~.
t= 2 4 0 . / ~
--.
I ,
. ~ ,,...,
"~,,
T h e a u t h o r s t h a n k D. V a u t h e r i n for v a l u a b l e discussions a n d for p r o v i d i n g us with its c o m p u t e r prog r a m for l i m i t i n g t e m p e r a t u r e .
0.1 Q)
0,0
18 March 1993
,-"-TY" I-"" I . _~"~
. . . . . .
0
5
R(fm)
X-.
1
~"-,...~..~J
10
I
, . . "l~, ;:'==---J
. . . . . .
6
10
15
R(fm)
Fig. 3. Density profiles (R is the radial distance) at different times (in fm/c) extracted from the Landau-Vlasov simulation for the same system as in fig. 2 (left side). At the right side the Coulomb interaction has been turned off. values ofp/po for lighter systems [ 11,12] do n o t induce the f o r m a t i o n o f a b u b b l e . To be m o r e q u a n t i t a t i v e a n d to clearly e m p h a s i z e the role o f the C o u l o m b i n t e r a c t i o n , we have plotted in fig. 3 the d e n s i t y profile o f the system for different times. Values as low as 0. lpo (Do ~ 0.13 f m - 3 with the force used here) are o b s e r v e d in the central part after 160 fm/c for the c o m p l e t e calculation (left side). Conversely, i f we t u r n o f f t h e C o u l o m b interaction, a higher i n i t i a l c o m p r e s s i o n is observed, followed b y m o n o p o l e type oscillations (right side). N o t e that b u b b l e s were also o b t a i n e d in calculations o f very hot o r / a n d highly c o m p r e s s e d nuclei [ 13-15 ] b u t it was argued that such a f u n n y geometry a p p e a r e d because o f the i m p o s e d spherical symmetry. I n fig. 2, we observe well that, after a b o u t 160 f m / c, a m u l t i f r a g m e n t a t i o n process occurs. Evidently, the s i m u l a t i o n discussed above is n o m o r e r e l e v a n t to exp l a i n the f o r m a t i o n o f fragments. However, c a n it indicate w h e t h e r there are surface instabilities [ 16 ] or w h e t h e r the system enters the s p i n o d a l region? Cal-
References
[ 1 ] D. Guerreau, Nuclear matter and heavy-ion collisions, eds. M. Soyeur, H. Flocard, B. Tamain and M. Porneuf (Plenum, New York, 1989)p. 187. [2] D. Jouan et al., Z. Phys. A 340 ( 1991 ) 63. [ 3 ] E. Crema et al., Phys. Left. B 258 ( 1991 ) 266. [4] P. Bonche, S. Levit and D. Vautherin, Nucl. Phys. A 427 (1984) 278. [ 5 ] S. Levit and P. Bonche, Nucl. Phys. A 437 ( 1985 ) 426. [ 6 ] J, Besprosvany and S. Levit, Phys. Lett. B 217 (1989) 1. [ 7 ] D. Vautherin, Lectures NATO Advanced Study Institute on Supernovae (Les Houches, France, August 1990), eds. S. Bludman, R. Mochkovitch and J. Zinn-Justin (NorthHolland, Amsterdam). [ 8 ] C. Gr6goire et al., Nucl. Phys. A 465 (1987) 317. [9] C. Gr6goire et al., Nucl. Phys. A 471 (1987) 399c. [ 10] L. Zamick, Phys. Left. B 45 (1973) 313. [ 11 ] M.F. Rivet et al., Phys. Lett. B 215 ( 1988 ) 55. [ 12 ] E. Suraud et al., Phys. Lett. B 229 (1989) 359. [ 13 ] J. Nemeth et al., Z. Phys. A 320 ( 1985 ) 691. [14] L. Vinet et al., Phys. Lett. B 172 (1986) 17. [ 15 ] D. Vautherin, J. Treiner and M. V6n6roni, Phys. Lett. B 191 (1987) 6. [ 16] L.G. Moretto et al., LBL report 31812 (1992). [ 17 ] F. S6bille et al., to be published. [ 18 ] D.W. Stracener el al., Nucl. Instrum. Methods A 294 (1990) 485. [ 19 ] R.T. de Souza et al., Nucl. Instrum. Methods A 295 (1990) 109. [20] A. Chbihi et al., Proc. first European Biennal Workshop on Nuclear physics (Meg~ve, France, March 1991 ), eds. D. Guinet and R. Pizzi (World Scientific, Singapore ) p. 378.
17