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Statistical multifragmentation: Comparison of two quite successful models
NUCLEAR
Nuclear Physics A567 (1994) 317-328 North-Holland
PHYSICS
A
Statistical multifragmentati~n: comparison of two quite successful models D.H.E.
Gross a, K. Sneppen
b
a Hahn-neither-Z~titut, Bereich Physik, Glienickerstr.100, f4109 Berlin, Germany and Fachbereich Physik der Freien UniversitiitBerlin. Berlin, Germany c The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Received 15 June 1993 Abstract We compare the outcome of the Copenhagen and the Berlin statistical multifragmentation models. The two models both start from the assumption of an ensemble of various excited fragments at thermal and chemical equilibrium for an expanded nuclear system at a certain stage, the freeze-out configuration. Details as the volume of this source, the intrinsic excitation of the fragments, and the secondary decays are treated quite differently. Also the technical level of the Monte Carlo sampling differs. Main characteristics such as total charged multiplicity, IMF multiplicity, and total neutron yield are in reasonable agreement. Thus the overall predictions are quite robust against details in the description. Deviations are however seen for high energies (E’ > 6A MeV) where fewer IMF’s, fewer neutrons, and much more (x’s are seen in the Copenhagen model. The isotopic distribution of the very light fragments is more narrow than in the Berlin model.
During the last decade a large effort has been devoted to study the break-up
of hot
nuclei into several fragments. Very early it was conjectured that a hot equilibrated heavy nucleus fissions and at even higher excitation breaks into several larger pieces f 11, it multifragments in contrast to fission or to multiple evaporation of nucleons or cu’s. Within the last year this study has culminated
in the observation
of up to 5 -+ 14 intermediate
mass fragments (Z > 2) formed in nucleus nucleus collisions at 50A MeV Xe+Au [2 ] and 65A MeV Ar + Ag/Br [ 3 1. Because such a big number of IMF’s is difficult to obtain within any sequentially binary decay scenario, it seems that intermediate mass fragments (IMF) are indeed formed in a si~~~t~neous~m~~t~p~e ~is~se~bly of the expan~e~~ot nuclear system. This picture is supported by emulsion studies of central collision Ar+ Ag,fBr at 65A MeV [ 31 where an observed radial expansion velocity of about v % 0.08~ limits the total available time for formation of all the IMF’s to about R/v x 100 fm/c where R is the radius of the disassembling nuclear system. As available data on IMF muItiplicities are well described by assuming that the fragments are thermally as well as chemically equilibrated at the moment of disassembly 0375-9474/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0375-9474(93)E0428-B
318
[4-7]*,
D.H.E. Gross, K. Sneppen / Statistical multifragmentation it is worthwhile
to investigate
different
ways to describe the available
phase
space for the formed nuclear fragments. In the following
we consider
two different
statistical
developed by Gross and coworkers and the Copenhagen
models: The Berlin model
[4]
model [ 5-91 developed by Bon-
dorf and coworkers. Both start from the picture that the fragmentation
is determined
by
the available phase space at a certain freeze-out volume 52. Let us first state the similarities in the basic assumptions of the two models : Sl: At densities
p < po the equilibrium distribution of nuclear matter is not homogeneous. It is in fact consisting of regions of liquid (density /?iiquidz po ) and regions of gas. In both models this distribution is sampled by considering all possible subdivisions of the matter into non overlapping spherical fragments. S2: The probability for each configuration of the fragments is given solely by its available phase space, measured by folding the intrinsic level densities of the fragments and the available phase space for the center of mass motion of the fragments. Population of intrinsically excited states of various fragments are assumed to be independent. S-3: After freeze out each of the formed fragments are allowed to cool by single particle emissions. To state the differences in the two models one has to distinguish between different physical assumptions of insufficiently known inputs into the models and differences in the technical methods used to simplify and speed up the computer simulations. We start with different assumptions about the yet unknown basic physics: Pl: The freeze-out volumes used by the two models are different: The Copenhagen model assumes a multiplicity dependent freeze-out volume that is between 1 -+ 4 times the normal nuclear volume [ 51. Notice that the multiplicity dependent freeze-out volume used in the standard Copenhagen model may not be very realistic for small multiplicities, but nevertheless, an increase in freeze-out volume with excitation energy seems reasonable, or even expected if one believes in BUU simulations of the entrance channel [ 91. Here is a point of inconsistency in the Copenhagen model. Below x 3 times normal volume it is not possible to position the sampled spherical fragments completely within the assumed freeze-out volume. In contrast, the Berlin model assumes that the matter always fragments at a fixed freeze-out volume of about 6 times normal nuclear volume. The distance between the surfaces of neighbouring fragments are about l-3 fm thus comparable to the range of the nuclear force. The constancy of the volume with excitation energy was originally not so much motivated by the physical problem of nuclear fragmentation but more by the desire to work out the generic analogy to other branches of statistical physics. E.g.: Phase transitions are normally signalled by peaks in the specific heat cV(T) at constant volume. It, however, turns out that the available data on nuclear fragmentation at low as well as * In ref. [6] (Nucl. Phys. A470), Fig. 1, the impression may arise that the LTM method of Gross et al. to calculate the quantum partition of a two component system is inaccurate. The contrary is the case: the calculation for a quantum partition gives the crosses in that figure which agree very well with the LTM result, whereas the other curves are for a classical partition.
D.H.E. Gross, K. Sneppen / Statistical multifragmentation
at high excitation
energy can be well described by the model in spite of this assumption
[lO,ll]. P2: The treatment
of the secondary decay of the hot fragments is quite differently.
is more a technical than physical difference, but it reflects the great uncertainty level-density
of the fragments
the level-densities
319
at temperatures
This
about the
of T x 5 MeV or more. At low energies
are fitted by a more or less corrected Bethe formula,
(1) In most cases, the level-density parameter is chosen simply as II = A/8. This formula is obtained from the grand-canonical partition function of a Fermi-gas Z (p, fi ) via double Laplace-backtransform [ 12 ] in saddle-point approximation. For several fragments in equilibrium at a given total excitation-energy we need the folding of several factors p(E). It is evident that this cannot be done with the Bethe formula (1) because of the divergent prefactor. The exponent of the grand-canonical partition sum is -p times the grand-canonical potential J (T, p). The entropy S (E, N) is obtained by the standard thermodynamic relation the saddle point of the inates the level-density. formula ( 1) one obtains
S (E, N) = /3 (E - J - N * ,u ) = 2m which is identical to double-Laplace backtransform P,,u -+ E, N. Normally it domIf one ignores the energy-dependence of the prefactor of p in the familiar result that two nuclei in equilibrium have the same
temperature. The familiar thermodynamic relations like the one above for the potential and the entropy hold for the mean values in the thermodynamic limit (lim,+,m ) where the fluctuations can be ignored. For finite systems the difference between canonical and microcanonical ensembles matters a lot. Even in solid state physics where one often extrapolates simulations of finite systems to cc it is safer to start from finite microcanonical systems than from canonical ones [ 131. Therefore, many authors simply ignore the energy-dependence of the prefactor to avoid the difficulty mentioned above. The Copenhagen model follows this idea and assumes the free energy F(T) for mass A > 4 fragments liquid-gas
to be given in part by the critical temperature
phase transition
FA,z(T) - h,z(T
for infinite
( TC = 16 MeV) of the
nuclear matter (see ref. [ 5 ] ):
= 0) x-LA+16MeV
[($=$)5’4-l]A213,
(2)
thereby counting bulk and surface excitations as independent. For small fragments this may be unrealistic because they are all surface. Moreover, surface vibrations of neighbour fragments might be coupled and not independent degrees of freedom. Fragments with mass A =$ 4 are not allowed for intrinsic excitations. We stress the possible source of mistake in this assumption of too highly excited light IMF’s and completely cold d’s, thereby probably giving too many cy after secondary decay. The total available energy E determines the temperature T = T ({ NA,z} ) and the entropy S = S({ NA,z} 1 of each partition {NA,~} via the thermodynamic relation: S(T)
=
-EdT’
E = F + TS,
(4)
here the free energy is calculated F
( T )
=
C
for each partition NA,Z
FA,Z
( T I
+
F
by: ( { NA,Z,
T}
hslational
(5)
A,Z
Thus the conservation
of energy is guaranteed
on the cost of assigning, for each partition,
the average excitation energy to each available degree of freedom. In order to remedy the influence of fluctuating fragment excitations on the secondary decay chain we set for each fragment f, its internal fragment energy to Er = &z(T) where EB,,
(f) is chosen corresponding
+ ~a”~(~),
to a gaussian
distribution
(61 with variance
C, T2
(where C, = C,(T) = (dE/dT)lr is the internal heat capacity of fragment f) and with the constraint Ef > E/ (T = 0) and total energy conservation. The effect of this internal fragment fluctuations on average observables tends to be negligible, whereas second moments of observables becomes slightly enhanced, In view of the difficulty with the prefactor of the Bethe-formula discussed above and of the fact that the nuclear level-density is not sufficiently known at excitations E/A > 2 MeV - this is especially so for all unstable fragments - the Berlin model uses the following recipe 141: The bound states of the fragments below the lower of the neutron threshold or the proton Coulomb-ba~ier B coUiare taken from the experimental data if available. If not, an empirical level-density formula (Truran-Cameron-HiIf [ 141 or Dilg [ 15 ] ) with pairing-, shell- and deformation-corrections is used. The internal unbound phase space of the fragments is populated by a random number of evaporated nuclei. It is assumed that each nucleon has two possible spin orientations and moves with its classical energy E = p2/2m + V + BcoUl > 0, Bc.,“l resp. inside a potential pocket which is T/ = -50 MeV deep and has the radius R = 1.224A ‘i3 fm . The total number of occupied phase space-cells is proportional to 1/n! [ L2f / (2nti ) ] ’ where n is the total number of evaporated neutrons/protons and Qf is the total volume of all fragments together. These nucleons then leave the fragments after freeze out, as described below. I.e. the higher excited states of the fragments are the product of several unbound s.p. states times the bound states of the evaporation-residue and the entropy is the sum of the entropy of the unbound nucleons and the entropy of the bound residues. In this approximation the internal entropy (level density) of the unbound single-particle states does not change when the fragments are split or combined. Thus any tendency in the Berlin model of the system to fragment does not come from unknown details of the level-density at higher excitations. This has the further advantage that one does not need to sample unstable fragments in the freezeout configuration with mostly unknown binding energies and level densities but only well known stable nuclei as secondary residues. Secondly, fluctuations of the individual excitation energies, which are important for the production of neutron rich very light nuclei, are automatically incorporated. Bound states embedded in the continuum are neglected relative to the ove~helming number of particle unstable states.
321
D.H.E. Gross, K. Sneppen / ~t~tis?icai~ultifra~me~tat~on
Now we list the technical
differences:
Tl: Berlin uses the Metropolis
algorithm
to count the phase space, Copenhagen
the direct sampling method. If there are strong variations are large fluctuations
depending
uses
in the statistical weights, there
on the actual fragment-size
dist~bution
Xt is then often extremely difficult to find the regions in the high dimensional
and positions. phase space
(d M 1000) - configurations - of maximal importance contributing to the observable one wants to calculate. Then the Metropolis algorithm is the only way to drive the sampling systematically there. Moreover, it provides an effective method to calculate the change of entropy which determines the branching ratios. The advantage of Metropolis becomes especially important when one wants to place the fragments in coordinate space in order to account in detail for the avoided volume not available to the center of mass motion of the fragments. This is in general larger than the eigenvolume of the fragments as assumed by the Copenhagen model. At densities of 3 ipo, as applied in both models, the effect of avoided volume is significant, especially close to the onset of fragmentation. T2: Copenhagen model samples only the partition space. An element in this space is a partition {NA,=} where NA,z denotes the number of fragments with mass number A and charge number Z. For each such partition a temperature is assigned, by ignoring fluctuations in internal excitations of the fragments. This is a detour from the microcanonical description to the canonical one. It is further assumed that the total Coulomb-energy is well accounted for by the Wigner-Seitz approximation. This is concerned with the deviations of the Coulomb-energy from its mean (corresponding to a smeared-out charge distribution over the large freeze-out volume) due to local contraction of the charge into fragments with normal density. I.e. the separation of the centers of the fragments is assumed to be the same as in the homogeneous distribution. Any fluctuation away from this is ignored. Moreover, within this approximation a specific fragment has the same Coulomb-energy whether it is at the surface or in the middle of the system. Berlin samples both fluctuations in intrinsic excitations of the fragments tuations fragments
in the mutual
Coulomb-energy
and fluc-
by various actual spatial configurations
of course under the strict constraint
of the conservation
of the
of total energy, mo-
mentum, mass and charge. There are easily fluctuations of up to hundred MeV or so, i.e. much more than the thermal fluctuations. These come mainly from different relative positions of the few heavier fragments (especially at the onset of fragmentation). It matters very much if a larger fragment is at the surface or in the centre of the system, if it touches another heavy one or not. This is important for all fragmentation barriers. In ref. [ 161 such fluctuations connected to positioning a mixture of big and small fragments were not considered. This leads to an interesting effect: Due to the Coulom~repulsion the fragments, especially the larger ones are positioned at the surface of the source, leading to a kind of empty “bubble structure” of the source. T3: The secondary decay after freeze-out is treated differently. Copenhagen uses independent secondary decays of the fragments according to the Weisskopf statistical model [ 17 1, taking into account the emission of protons, deuterons and mass A = 3 particles. As described above, in the Berlin model the secondary evaporation of neutrons and
D.H.E. Gross, K. Sneppen / Statistical multifragmentation
322
16
0”“’ 0
2
4
1
““‘* 6
8
IO
”
12
1
14O
Temperature Fig. 1. Temperature versus excitation energy for fragmentation of a 13’Xe nucleus. Open are calculations with the Berlin code (MMMC = Microcanonical Metropolis Monte Solid dots are for the Copenhagen model with standard multiplicity-dependent freeze-out [ 51. Solid squares are for a modification of the Copenhagen model with a fixed freeze-out of 61/,, which equals that used in the Berlin code.
protons is sampled by populating
the unbound
s.p.-states inside the fragments
squares Carlo). volume volume
at freeze-
out.
Comparisons of the results of the models for various observables: differences,
for the fragmentation
We find the following
of 131Xe*, see Figs. l-4:
( 1) Berlin sees a narrow region of constant
temperature
at low excitations.
In fact at
finer energy steps we see in the Berlin model even a slight backbending of the T(E) curve, signaling a phase-transition [ 41. One also gets a rather broad backbending in the standard Copenhagen model in temperature versus excitation energy. It is interesting that such a plateau or backbending is not obtained in the Copenhagen model if a fixed volume is chosen similarly to the Berlin model. Thus the backbending of T(E) in the standard version of the Copenhagen model is mainly due to the fact that an increased volume favors an increased number of fragments at freeze out, at the cost of the overall temperature. (Ordinary cooling by expansion. ) To understand the nature of the backbending of T(E) in the Berlin model we have also run the Copenhagen code at constant volume (I/ = 6&/o)with completely suppressed intrinsic heating of the fragments at freeze out (“cold fragmentation”). Then one observes
LUKE. Gross, K. Sneppen f Statistical ~~lti~rag~e~tatio~
n .r)
2 8, %
323
50
50
40
40
30
30
20
20
f0
f0
5 4
0
;
Energ y/A 50
_
04
40
40
30
30
20
20
10 .
to
01 0
7
’ 2
*
’ 4
7
’ 6
-
”
8
’
10
7
’
12
I’
14
4
II
16
0
Energ y/A Fig. 2. (a)-(e) Multiplicity of charged particles (Mz ), neutrons (Nn ), H-isotopes (Nz,, ), He-particles (Nz=~), and of fragments with intermediate mass (A&F), (with 2 < Z < 30). Notation as in Fig. 1. The error bars in (b) give the FWHM of the neutron numbers in the Berlin model.
324
D.H.E. Gross, K. Sneppen / Statistical multifragmentation 30
25
25
20
15
f0
5
Energ y/A 16
4 . . . . . . . . _*
(4
16
14
14
12
12
10
10
8 6
6 4 2
0
2
4
6
8
IO
Energ y/A Fig. 2 -continued.
325
D.H.E. Gross, K. Sneppen / Statistical multifragmentation 8 7 -6
6
1
0
0
2
4
6
8
Energ
IO
12
14
16
”
y/A
Fig. 2 -continued.
a clear and significant backbending of T(E) at excitation energies of 2.5 to 5A MeV and at temperatures at about 5-6 MeV. (2) T (Berlin) > T (Copenhagen), except at low excitation energies. This is likely due to the higher level-density (i.e. larger internal phase space) of the small IMFs in the Copenhagen
model.
(3 ) Charged particle multiplicity (4) Neutron
multiplicity
is similar.
is similar for excitations
within the FWHM of the distribu-
tions below E*/A x 4 MeV. Above this there are more evaporated model. (5 ) Isotope-yield
neutrons
in the Berlin
of H- and He- fragments are much broader in the Berlin model than
in the Copenhagen one. In Table 1 this is demonstrated for the H and He isotopes. (6) As seen in Fig. 2e, the MIMF at high energies differs. Due to the higher excitation energies of light IMF’s in the Copenhagen model most of the IMF’s primordially formed will quickly be converted to protons and Q particles, thereby, at E > 10A MeV, leading to much higher QIyields in the Copenhagen than in the Berlin model (see Fig. 2d). To summarize the findings of this chapter: Whereas we see a clear sign of a phasetransition in the Berlin Metropolis calculation (backbending of E(T) ) the transitions from one decay-mode to the other is much more smooth in the constant volume version of the Copenhagen model. For the standard variable volume Copenhagen model the long plateau or backbending of E (T) is connected to cooling due to expansion. The constant
326
D.H.E. Gross, K. Sneppen / Statistical multifragmentation
13fXe charges
three largest
60
* MMMC, ek5 Z I * MMMC,ek5 2.2 * MMMC,ek5 23 * Copenh.,v.v. + Copenh.,v.v. m-Copenh.,v.v. e Copenh.,f.v. * Copenh.,f.v. q Copenh.,f.v.
0
2
4
6
8
IO
12
14
16
50
40
30
0
E/A Fig. 3. Charge of the three biggest fragments. Notation as in Fig. 1.
9
-
-
25
8 -
50
7
75
25
67 73
8498
125
$4
113 133
s\
2‘6
3 2
_
100
n
Copenhagen
0
Berlin
150
#
150 @ 166
\ iaib
1 0 0
‘o-
la9
‘~‘~‘~‘~‘~‘~‘~~ 10 20
30
40
50
60
70
_
80
(Zbound)
Fig. 4. Theoretical excitation energy and masses of the equilibrated sources leading to final fragment-distributions that tit all the fragment correlations measured by the Aladin group [ 18I for 600A MeV Au on different targets. The open circles represent the results by the Berlin model. They are for impact parameters 1, 2, 3, . . ., 10 in sequence.
D.H.E. Gross, 1%Sneppen / Statistical ~ult~~rag~entati~n
327
TABLE 1
Tabulated ratios of light isotopes. The two models differ dramatically. For H isotopes, the Berlin model predicts about f as many tritons as ‘H, whereas Copenhagen gets less than 5% of Z = 1 particles as tritons. Berlin always predicts more than 10 % of the He isotopes are heavier than cy whereas Copenhagen
Ratio
predicts less than one percent of heavy He isotopes E* [MeV]
Copenhagen
Berlin
3H f ‘H
3
0.12/3.6
0.82j2.54
3H,’ ‘H
5
0.14f3.3
0.95/3.3
3H/‘H
7
0.28i4.7
1.1/4.5
3H,“H
10
0.47j7.9
1.9/8.5
3
0.0016/0.38
0.2j2.5
6He / 4He
5
0.0030/ 1.63
0.312.9
6He / 4He
7
0.0058/3.40
0.413.6
‘jHe / 4He
10
0.0210/8.4
0.514.3
6He / 4He
volume Copenhagen model also predicts a backbending E(T) when a Iow intrinsic heat of the fragments is assumed. At high energies there are much more (Y’Sin the Copenhagen model, less neutrons,
and a much narrower isotope yield towards the neutron rich side. Co~~l~~i~n: We have studied two independently developed models aimed both to predict the cluster distributions of nuclear fragments formed out of an expanded system of atomic nuclei. The main, and most important conclusion is that although there are major differences in both amount of intrinsic excitation freeze-out volume, then the overall predictions are simultaneous fragmentation is robust and predicts mass fragments independently on details in further
of the fragments and in the applied fairly similar. Thus the assumption of the formation of many intermediate assumptions for the description of the
available phase space of the expanded Fermi gas. Saying this however, we have to add that important differences remain. Especially it seems that the predicted amounts of rare very light isotopes are signi~cantly
different
between the two models. Also predictions
for the probability of the fission channel at the onset of multifragmentation differences will be evaluated when more experimental data are available. K.. Sneppen is grateful to the Carlsberg Foundation
differs. These
for financial support whereas D.H.E.
Gross is grateful to the Deutsche Forschungsgemeinschaft
for generous support.
References [ 1] D.H.E. Gross, in Topical Meeting on phase space approach to nuclear dynamics, Trieste, 1985, ed. M. di Toro et al. (World Scientific, Singapore, 1986) p. 25 1. [2] D.R. Bowman et al., Phys. Rev. Lett. 67 (1991) 1527 [3] H.W. Barz, J.P. Bondorf, R. Donangelo, F.S. Hansen, B. Jakobsson, L. Karlsson, H. Nifenecker, H. Schulz, F. Schussler, K. Sneppen and K. Soderstrom. Nucl. Phys. A531 ( 1991) 453
328
D.H.E. Gross, K. Sneppen / Statistical multifragmentation
141 D.H.E. Gross, in 4th Nordic Meeting on intermediate and high energy physics (Geilo Sportell, Norway, 198 1) p. 29; D.H.E. Gross, L. Satpathy, Meng Ta-chung and M. Satpathy, Z. Phys. A 309 ( 1982) 41; D.H.E. Gross, Phys. Scripta T5 ( 1983) 2 13: D.H.E. Gross, in High energy nuclear physics, Proc. 6th Balaton Conf. on Nuclear Physics, 1983, ed. J. Eroe, p. 3 11; D.H.E. Gross, in Proc. Int. Conf. on theoretical approaches of heavy ion reaction mechanisms (Paris, 19841, Nucl. Phys. A428 (1984) 313~; D.H.E. Gross and B.H. Sa, Nucl. Phys. A 437 (1985) 643; D.H.E. Gross, in Proc. Winter College on fundamental nuclear physics, ed. K. Dietrich, M. Di Toro and H.J. Mang, ICTP Trieste (World Scientific, Singapore, 1985) p. 1185; D.H.E. Gross and X.Z. Zhang, Phys. Lett. B161 (1985) 47; D.H.E. Gross, Rep. Progr. Phys. 53 ( 1990) 605 [5] J.P. Bondorf, Nucl. Phys. A387 f 1982) 22~; J.P. Bondorf, R. Donangelo, I.N. Mishustin, C. Pethick and K. Sneppen, Phys. Lett. Bl50 (1985) 57; J.P. Bondorf, R. Donangelo, I.N. Mishustin, C. Pethick, H. Schulz and K. Sneppen, Nucl. Phys. A443 (1985) 321; J.P. Bondorf, R. Donangelo, I.N. Mishustin and H. Schulz, Nucl. Phys. A444 (1986) 460; H.W. Barz, J.P. Bondorf, R. Donangelo, I.N. Mishustin and H. Schulz, Nucl. Phys. A448 (1986) 753 [6] K. Sneppen and R. Donangelo, Phys. Rev. C39 (1989) 263; K. Sneppen, Nucl. Phys. A470 ( 1987) 213 [7] H. Barz, J. Bondorf, R. Donangelo, H. Schulz and K. Sneppen, Phys Lett. B191 (1987) 232; H.W. Barz, H. Schulz, J.P. Bondorf, J. Lopez and K. Sneppen, Phys. Lett. B211 (1988) 25; H.W. Barz, J.P. Bondorf, H. Schulz and K. Sneppen, Phys. Lett. 8244 (1990) 161; H.W. Barz, J.P. Bondorf, R. Donangelo, I. Mishustin, H. Schulz and K. Sneppen, Nucl. Phys. A545 (1992) 213~ [8] H.W. Barz, W. Bauer, J.P. Bondorf, AS. Botwina, R. Donangelo, H. Schulz and K. Sneppen, NBI preprint, NBI-92-88 [9] K. Sneppen and L. Vinet, Nucl. Phys. A480 (1988) 342 [lo] D.H.E. Gross, Phys. Lett. B 203 (1988) 26 [ 111 Bao-an Li, A.R. DeAngelis and D.H.E. Gross, Phys. Lett. B303 ( 1992) 225. [ 121 Aa. Bohr and B.R. Mottelson, Nuclear structure (Benjamin, New York, 1969) f 131 R.W. Gerling and A. Huller, Z. Phys. B90 (1993) 207 [ 141 J.W. Truran, A.G.W. Cameron and E. Hilf, Cem Report 70-30 (1970) 215 [ 15 ] W. Dilg, W. Schantl and H. Vonach, Nucl. Phys. A2 17 ( 1973) 269 [ 161 J. Randrup, M. Robinson and K. Sneppen, Phys. Lett. B208 (1988) 25 [ 171 V. Weisskopf, Phys. Rev. 52 ( 1937) 2 11 [ 181 P. Kreutz, thesis GSI-Darmstadt (1992)
Nuclear Physics A567 (1994) 317-328 North-Holland
PHYSICS
A
Statistical multifragmentati~n: comparison of two quite successful models D.H.E.
Gross a, K. Sneppen
b
a Hahn-neither-Z~titut, Bereich Physik, Glienickerstr.100, f4109 Berlin, Germany and Fachbereich Physik der Freien UniversitiitBerlin. Berlin, Germany c The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Received 15 June 1993 Abstract We compare the outcome of the Copenhagen and the Berlin statistical multifragmentation models. The two models both start from the assumption of an ensemble of various excited fragments at thermal and chemical equilibrium for an expanded nuclear system at a certain stage, the freeze-out configuration. Details as the volume of this source, the intrinsic excitation of the fragments, and the secondary decays are treated quite differently. Also the technical level of the Monte Carlo sampling differs. Main characteristics such as total charged multiplicity, IMF multiplicity, and total neutron yield are in reasonable agreement. Thus the overall predictions are quite robust against details in the description. Deviations are however seen for high energies (E’ > 6A MeV) where fewer IMF’s, fewer neutrons, and much more (x’s are seen in the Copenhagen model. The isotopic distribution of the very light fragments is more narrow than in the Berlin model.
During the last decade a large effort has been devoted to study the break-up
of hot
nuclei into several fragments. Very early it was conjectured that a hot equilibrated heavy nucleus fissions and at even higher excitation breaks into several larger pieces f 11, it multifragments in contrast to fission or to multiple evaporation of nucleons or cu’s. Within the last year this study has culminated
in the observation
of up to 5 -+ 14 intermediate
mass fragments (Z > 2) formed in nucleus nucleus collisions at 50A MeV Xe+Au [2 ] and 65A MeV Ar + Ag/Br [ 3 1. Because such a big number of IMF’s is difficult to obtain within any sequentially binary decay scenario, it seems that intermediate mass fragments (IMF) are indeed formed in a si~~~t~neous~m~~t~p~e ~is~se~bly of the expan~e~~ot nuclear system. This picture is supported by emulsion studies of central collision Ar+ Ag,fBr at 65A MeV [ 31 where an observed radial expansion velocity of about v % 0.08~ limits the total available time for formation of all the IMF’s to about R/v x 100 fm/c where R is the radius of the disassembling nuclear system. As available data on IMF muItiplicities are well described by assuming that the fragments are thermally as well as chemically equilibrated at the moment of disassembly 0375-9474/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0375-9474(93)E0428-B
318
[4-7]*,
D.H.E. Gross, K. Sneppen / Statistical multifragmentation it is worthwhile
to investigate
different
ways to describe the available
phase
space for the formed nuclear fragments. In the following
we consider
two different
statistical
developed by Gross and coworkers and the Copenhagen
models: The Berlin model
[4]
model [ 5-91 developed by Bon-
dorf and coworkers. Both start from the picture that the fragmentation
is determined
by
the available phase space at a certain freeze-out volume 52. Let us first state the similarities in the basic assumptions of the two models : Sl: At densities
p < po the equilibrium distribution of nuclear matter is not homogeneous. It is in fact consisting of regions of liquid (density /?iiquidz po ) and regions of gas. In both models this distribution is sampled by considering all possible subdivisions of the matter into non overlapping spherical fragments. S2: The probability for each configuration of the fragments is given solely by its available phase space, measured by folding the intrinsic level densities of the fragments and the available phase space for the center of mass motion of the fragments. Population of intrinsically excited states of various fragments are assumed to be independent. S-3: After freeze out each of the formed fragments are allowed to cool by single particle emissions. To state the differences in the two models one has to distinguish between different physical assumptions of insufficiently known inputs into the models and differences in the technical methods used to simplify and speed up the computer simulations. We start with different assumptions about the yet unknown basic physics: Pl: The freeze-out volumes used by the two models are different: The Copenhagen model assumes a multiplicity dependent freeze-out volume that is between 1 -+ 4 times the normal nuclear volume [ 51. Notice that the multiplicity dependent freeze-out volume used in the standard Copenhagen model may not be very realistic for small multiplicities, but nevertheless, an increase in freeze-out volume with excitation energy seems reasonable, or even expected if one believes in BUU simulations of the entrance channel [ 91. Here is a point of inconsistency in the Copenhagen model. Below x 3 times normal volume it is not possible to position the sampled spherical fragments completely within the assumed freeze-out volume. In contrast, the Berlin model assumes that the matter always fragments at a fixed freeze-out volume of about 6 times normal nuclear volume. The distance between the surfaces of neighbouring fragments are about l-3 fm thus comparable to the range of the nuclear force. The constancy of the volume with excitation energy was originally not so much motivated by the physical problem of nuclear fragmentation but more by the desire to work out the generic analogy to other branches of statistical physics. E.g.: Phase transitions are normally signalled by peaks in the specific heat cV(T) at constant volume. It, however, turns out that the available data on nuclear fragmentation at low as well as * In ref. [6] (Nucl. Phys. A470), Fig. 1, the impression may arise that the LTM method of Gross et al. to calculate the quantum partition of a two component system is inaccurate. The contrary is the case: the calculation for a quantum partition gives the crosses in that figure which agree very well with the LTM result, whereas the other curves are for a classical partition.
D.H.E. Gross, K. Sneppen / Statistical multifragmentation
at high excitation
energy can be well described by the model in spite of this assumption
[lO,ll]. P2: The treatment
of the secondary decay of the hot fragments is quite differently.
is more a technical than physical difference, but it reflects the great uncertainty level-density
of the fragments
the level-densities
319
at temperatures
This
about the
of T x 5 MeV or more. At low energies
are fitted by a more or less corrected Bethe formula,
(1) In most cases, the level-density parameter is chosen simply as II = A/8. This formula is obtained from the grand-canonical partition function of a Fermi-gas Z (p, fi ) via double Laplace-backtransform [ 12 ] in saddle-point approximation. For several fragments in equilibrium at a given total excitation-energy we need the folding of several factors p(E). It is evident that this cannot be done with the Bethe formula (1) because of the divergent prefactor. The exponent of the grand-canonical partition sum is -p times the grand-canonical potential J (T, p). The entropy S (E, N) is obtained by the standard thermodynamic relation the saddle point of the inates the level-density. formula ( 1) one obtains
S (E, N) = /3 (E - J - N * ,u ) = 2m which is identical to double-Laplace backtransform P,,u -+ E, N. Normally it domIf one ignores the energy-dependence of the prefactor of p in the familiar result that two nuclei in equilibrium have the same
temperature. The familiar thermodynamic relations like the one above for the potential and the entropy hold for the mean values in the thermodynamic limit (lim,+,m ) where the fluctuations can be ignored. For finite systems the difference between canonical and microcanonical ensembles matters a lot. Even in solid state physics where one often extrapolates simulations of finite systems to cc it is safer to start from finite microcanonical systems than from canonical ones [ 131. Therefore, many authors simply ignore the energy-dependence of the prefactor to avoid the difficulty mentioned above. The Copenhagen model follows this idea and assumes the free energy F(T) for mass A > 4 fragments liquid-gas
to be given in part by the critical temperature
phase transition
FA,z(T) - h,z(T
for infinite
( TC = 16 MeV) of the
nuclear matter (see ref. [ 5 ] ):
= 0) x-LA+16MeV
[($=$)5’4-l]A213,
(2)
thereby counting bulk and surface excitations as independent. For small fragments this may be unrealistic because they are all surface. Moreover, surface vibrations of neighbour fragments might be coupled and not independent degrees of freedom. Fragments with mass A =$ 4 are not allowed for intrinsic excitations. We stress the possible source of mistake in this assumption of too highly excited light IMF’s and completely cold d’s, thereby probably giving too many cy after secondary decay. The total available energy E determines the temperature T = T ({ NA,z} ) and the entropy S = S({ NA,z} 1 of each partition {NA,~} via the thermodynamic relation: S(T)
=
-EdT’
E = F + TS,
(4)
here the free energy is calculated F
( T )
=
C
for each partition NA,Z
FA,Z
( T I
+
F
by: ( { NA,Z,
T}
hslational
(5)
A,Z
Thus the conservation
of energy is guaranteed
on the cost of assigning, for each partition,
the average excitation energy to each available degree of freedom. In order to remedy the influence of fluctuating fragment excitations on the secondary decay chain we set for each fragment f, its internal fragment energy to Er = &z(T) where EB,,
(f) is chosen corresponding
+ ~a”~(~),
to a gaussian
distribution
(61 with variance
C, T2
(where C, = C,(T) = (dE/dT)lr is the internal heat capacity of fragment f) and with the constraint Ef > E/ (T = 0) and total energy conservation. The effect of this internal fragment fluctuations on average observables tends to be negligible, whereas second moments of observables becomes slightly enhanced, In view of the difficulty with the prefactor of the Bethe-formula discussed above and of the fact that the nuclear level-density is not sufficiently known at excitations E/A > 2 MeV - this is especially so for all unstable fragments - the Berlin model uses the following recipe 141: The bound states of the fragments below the lower of the neutron threshold or the proton Coulomb-ba~ier B coUiare taken from the experimental data if available. If not, an empirical level-density formula (Truran-Cameron-HiIf [ 141 or Dilg [ 15 ] ) with pairing-, shell- and deformation-corrections is used. The internal unbound phase space of the fragments is populated by a random number of evaporated nuclei. It is assumed that each nucleon has two possible spin orientations and moves with its classical energy E = p2/2m + V + BcoUl > 0, Bc.,“l resp. inside a potential pocket which is T/ = -50 MeV deep and has the radius R = 1.224A ‘i3 fm . The total number of occupied phase space-cells is proportional to 1/n! [ L2f / (2nti ) ] ’ where n is the total number of evaporated neutrons/protons and Qf is the total volume of all fragments together. These nucleons then leave the fragments after freeze out, as described below. I.e. the higher excited states of the fragments are the product of several unbound s.p. states times the bound states of the evaporation-residue and the entropy is the sum of the entropy of the unbound nucleons and the entropy of the bound residues. In this approximation the internal entropy (level density) of the unbound single-particle states does not change when the fragments are split or combined. Thus any tendency in the Berlin model of the system to fragment does not come from unknown details of the level-density at higher excitations. This has the further advantage that one does not need to sample unstable fragments in the freezeout configuration with mostly unknown binding energies and level densities but only well known stable nuclei as secondary residues. Secondly, fluctuations of the individual excitation energies, which are important for the production of neutron rich very light nuclei, are automatically incorporated. Bound states embedded in the continuum are neglected relative to the ove~helming number of particle unstable states.
321
D.H.E. Gross, K. Sneppen / ~t~tis?icai~ultifra~me~tat~on
Now we list the technical
differences:
Tl: Berlin uses the Metropolis
algorithm
to count the phase space, Copenhagen
the direct sampling method. If there are strong variations are large fluctuations
depending
uses
in the statistical weights, there
on the actual fragment-size
dist~bution
Xt is then often extremely difficult to find the regions in the high dimensional
and positions. phase space
(d M 1000) - configurations - of maximal importance contributing to the observable one wants to calculate. Then the Metropolis algorithm is the only way to drive the sampling systematically there. Moreover, it provides an effective method to calculate the change of entropy which determines the branching ratios. The advantage of Metropolis becomes especially important when one wants to place the fragments in coordinate space in order to account in detail for the avoided volume not available to the center of mass motion of the fragments. This is in general larger than the eigenvolume of the fragments as assumed by the Copenhagen model. At densities of 3 ipo, as applied in both models, the effect of avoided volume is significant, especially close to the onset of fragmentation. T2: Copenhagen model samples only the partition space. An element in this space is a partition {NA,=} where NA,z denotes the number of fragments with mass number A and charge number Z. For each such partition a temperature is assigned, by ignoring fluctuations in internal excitations of the fragments. This is a detour from the microcanonical description to the canonical one. It is further assumed that the total Coulomb-energy is well accounted for by the Wigner-Seitz approximation. This is concerned with the deviations of the Coulomb-energy from its mean (corresponding to a smeared-out charge distribution over the large freeze-out volume) due to local contraction of the charge into fragments with normal density. I.e. the separation of the centers of the fragments is assumed to be the same as in the homogeneous distribution. Any fluctuation away from this is ignored. Moreover, within this approximation a specific fragment has the same Coulomb-energy whether it is at the surface or in the middle of the system. Berlin samples both fluctuations in intrinsic excitations of the fragments tuations fragments
in the mutual
Coulomb-energy
and fluc-
by various actual spatial configurations
of course under the strict constraint
of the conservation
of the
of total energy, mo-
mentum, mass and charge. There are easily fluctuations of up to hundred MeV or so, i.e. much more than the thermal fluctuations. These come mainly from different relative positions of the few heavier fragments (especially at the onset of fragmentation). It matters very much if a larger fragment is at the surface or in the centre of the system, if it touches another heavy one or not. This is important for all fragmentation barriers. In ref. [ 161 such fluctuations connected to positioning a mixture of big and small fragments were not considered. This leads to an interesting effect: Due to the Coulom~repulsion the fragments, especially the larger ones are positioned at the surface of the source, leading to a kind of empty “bubble structure” of the source. T3: The secondary decay after freeze-out is treated differently. Copenhagen uses independent secondary decays of the fragments according to the Weisskopf statistical model [ 17 1, taking into account the emission of protons, deuterons and mass A = 3 particles. As described above, in the Berlin model the secondary evaporation of neutrons and
D.H.E. Gross, K. Sneppen / Statistical multifragmentation
322
16
0”“’ 0
2
4
1
““‘* 6
8
IO
”
12
1
14O
Temperature Fig. 1. Temperature versus excitation energy for fragmentation of a 13’Xe nucleus. Open are calculations with the Berlin code (MMMC = Microcanonical Metropolis Monte Solid dots are for the Copenhagen model with standard multiplicity-dependent freeze-out [ 51. Solid squares are for a modification of the Copenhagen model with a fixed freeze-out of 61/,, which equals that used in the Berlin code.
protons is sampled by populating
the unbound
s.p.-states inside the fragments
squares Carlo). volume volume
at freeze-
out.
Comparisons of the results of the models for various observables: differences,
for the fragmentation
We find the following
of 131Xe*, see Figs. l-4:
( 1) Berlin sees a narrow region of constant
temperature
at low excitations.
In fact at
finer energy steps we see in the Berlin model even a slight backbending of the T(E) curve, signaling a phase-transition [ 41. One also gets a rather broad backbending in the standard Copenhagen model in temperature versus excitation energy. It is interesting that such a plateau or backbending is not obtained in the Copenhagen model if a fixed volume is chosen similarly to the Berlin model. Thus the backbending of T(E) in the standard version of the Copenhagen model is mainly due to the fact that an increased volume favors an increased number of fragments at freeze out, at the cost of the overall temperature. (Ordinary cooling by expansion. ) To understand the nature of the backbending of T(E) in the Berlin model we have also run the Copenhagen code at constant volume (I/ = 6&/o)with completely suppressed intrinsic heating of the fragments at freeze out (“cold fragmentation”). Then one observes
LUKE. Gross, K. Sneppen f Statistical ~~lti~rag~e~tatio~
n .r)
2 8, %
323
50
50
40
40
30
30
20
20
f0
f0
5 4
0
;
Energ y/A 50
_
04
40
40
30
30
20
20
10 .
to
01 0
7
’ 2
*
’ 4
7
’ 6
-
”
8
’
10
7
’
12
I’
14
4
II
16
0
Energ y/A Fig. 2. (a)-(e) Multiplicity of charged particles (Mz ), neutrons (Nn ), H-isotopes (Nz,, ), He-particles (Nz=~), and of fragments with intermediate mass (A&F), (with 2 < Z < 30). Notation as in Fig. 1. The error bars in (b) give the FWHM of the neutron numbers in the Berlin model.
324
D.H.E. Gross, K. Sneppen / Statistical multifragmentation 30
25
25
20
15
f0
5
Energ y/A 16
4 . . . . . . . . _*
(4
16
14
14
12
12
10
10
8 6
6 4 2
0
2
4
6
8
IO
Energ y/A Fig. 2 -continued.
325
D.H.E. Gross, K. Sneppen / Statistical multifragmentation 8 7 -6
6
1
0
0
2
4
6
8
Energ
IO
12
14
16
”
y/A
Fig. 2 -continued.
a clear and significant backbending of T(E) at excitation energies of 2.5 to 5A MeV and at temperatures at about 5-6 MeV. (2) T (Berlin) > T (Copenhagen), except at low excitation energies. This is likely due to the higher level-density (i.e. larger internal phase space) of the small IMFs in the Copenhagen
model.
(3 ) Charged particle multiplicity (4) Neutron
multiplicity
is similar.
is similar for excitations
within the FWHM of the distribu-
tions below E*/A x 4 MeV. Above this there are more evaporated model. (5 ) Isotope-yield
neutrons
in the Berlin
of H- and He- fragments are much broader in the Berlin model than
in the Copenhagen one. In Table 1 this is demonstrated for the H and He isotopes. (6) As seen in Fig. 2e, the MIMF at high energies differs. Due to the higher excitation energies of light IMF’s in the Copenhagen model most of the IMF’s primordially formed will quickly be converted to protons and Q particles, thereby, at E > 10A MeV, leading to much higher QIyields in the Copenhagen than in the Berlin model (see Fig. 2d). To summarize the findings of this chapter: Whereas we see a clear sign of a phasetransition in the Berlin Metropolis calculation (backbending of E(T) ) the transitions from one decay-mode to the other is much more smooth in the constant volume version of the Copenhagen model. For the standard variable volume Copenhagen model the long plateau or backbending of E (T) is connected to cooling due to expansion. The constant
326
D.H.E. Gross, K. Sneppen / Statistical multifragmentation
13fXe charges
three largest
60
* MMMC, ek5 Z I * MMMC,ek5 2.2 * MMMC,ek5 23 * Copenh.,v.v. + Copenh.,v.v. m-Copenh.,v.v. e Copenh.,f.v. * Copenh.,f.v. q Copenh.,f.v.
0
2
4
6
8
IO
12
14
16
50
40
30
0
E/A Fig. 3. Charge of the three biggest fragments. Notation as in Fig. 1.
9
-
-
25
8 -
50
7
75
25
67 73
8498
125
$4
113 133
s\
2‘6
3 2
_
100
n
Copenhagen
0
Berlin
150
#
150 @ 166
\ iaib
1 0 0
‘o-
la9
‘~‘~‘~‘~‘~‘~‘~~ 10 20
30
40
50
60
70
_
80
(Zbound)
Fig. 4. Theoretical excitation energy and masses of the equilibrated sources leading to final fragment-distributions that tit all the fragment correlations measured by the Aladin group [ 18I for 600A MeV Au on different targets. The open circles represent the results by the Berlin model. They are for impact parameters 1, 2, 3, . . ., 10 in sequence.
D.H.E. Gross, 1%Sneppen / Statistical ~ult~~rag~entati~n
327
TABLE 1
Tabulated ratios of light isotopes. The two models differ dramatically. For H isotopes, the Berlin model predicts about f as many tritons as ‘H, whereas Copenhagen gets less than 5% of Z = 1 particles as tritons. Berlin always predicts more than 10 % of the He isotopes are heavier than cy whereas Copenhagen
Ratio
predicts less than one percent of heavy He isotopes E* [MeV]
Copenhagen
Berlin
3H f ‘H
3
0.12/3.6
0.82j2.54
3H,’ ‘H
5
0.14f3.3
0.95/3.3
3H/‘H
7
0.28i4.7
1.1/4.5
3H,“H
10
0.47j7.9
1.9/8.5
3
0.0016/0.38
0.2j2.5
6He / 4He
5
0.0030/ 1.63
0.312.9
6He / 4He
7
0.0058/3.40
0.413.6
‘jHe / 4He
10
0.0210/8.4
0.514.3
6He / 4He
volume Copenhagen model also predicts a backbending E(T) when a Iow intrinsic heat of the fragments is assumed. At high energies there are much more (Y’Sin the Copenhagen model, less neutrons,
and a much narrower isotope yield towards the neutron rich side. Co~~l~~i~n: We have studied two independently developed models aimed both to predict the cluster distributions of nuclear fragments formed out of an expanded system of atomic nuclei. The main, and most important conclusion is that although there are major differences in both amount of intrinsic excitation freeze-out volume, then the overall predictions are simultaneous fragmentation is robust and predicts mass fragments independently on details in further
of the fragments and in the applied fairly similar. Thus the assumption of the formation of many intermediate assumptions for the description of the
available phase space of the expanded Fermi gas. Saying this however, we have to add that important differences remain. Especially it seems that the predicted amounts of rare very light isotopes are signi~cantly
different
between the two models. Also predictions
for the probability of the fission channel at the onset of multifragmentation differences will be evaluated when more experimental data are available. K.. Sneppen is grateful to the Carlsberg Foundation
differs. These
for financial support whereas D.H.E.
Gross is grateful to the Deutsche Forschungsgemeinschaft
for generous support.
References [ 1] D.H.E. Gross, in Topical Meeting on phase space approach to nuclear dynamics, Trieste, 1985, ed. M. di Toro et al. (World Scientific, Singapore, 1986) p. 25 1. [2] D.R. Bowman et al., Phys. Rev. Lett. 67 (1991) 1527 [3] H.W. Barz, J.P. Bondorf, R. Donangelo, F.S. Hansen, B. Jakobsson, L. Karlsson, H. Nifenecker, H. Schulz, F. Schussler, K. Sneppen and K. Soderstrom. Nucl. Phys. A531 ( 1991) 453
328
D.H.E. Gross, K. Sneppen / Statistical multifragmentation
141 D.H.E. Gross, in 4th Nordic Meeting on intermediate and high energy physics (Geilo Sportell, Norway, 198 1) p. 29; D.H.E. Gross, L. Satpathy, Meng Ta-chung and M. Satpathy, Z. Phys. A 309 ( 1982) 41; D.H.E. Gross, Phys. Scripta T5 ( 1983) 2 13: D.H.E. Gross, in High energy nuclear physics, Proc. 6th Balaton Conf. on Nuclear Physics, 1983, ed. J. Eroe, p. 3 11; D.H.E. Gross, in Proc. Int. Conf. on theoretical approaches of heavy ion reaction mechanisms (Paris, 19841, Nucl. Phys. A428 (1984) 313~; D.H.E. Gross and B.H. Sa, Nucl. Phys. A 437 (1985) 643; D.H.E. Gross, in Proc. Winter College on fundamental nuclear physics, ed. K. Dietrich, M. Di Toro and H.J. Mang, ICTP Trieste (World Scientific, Singapore, 1985) p. 1185; D.H.E. Gross and X.Z. Zhang, Phys. Lett. B161 (1985) 47; D.H.E. Gross, Rep. Progr. Phys. 53 ( 1990) 605 [5] J.P. Bondorf, Nucl. Phys. A387 f 1982) 22~; J.P. Bondorf, R. Donangelo, I.N. Mishustin, C. Pethick and K. Sneppen, Phys. Lett. Bl50 (1985) 57; J.P. Bondorf, R. Donangelo, I.N. Mishustin, C. Pethick, H. Schulz and K. Sneppen, Nucl. Phys. A443 (1985) 321; J.P. Bondorf, R. Donangelo, I.N. Mishustin and H. Schulz, Nucl. Phys. A444 (1986) 460; H.W. Barz, J.P. Bondorf, R. Donangelo, I.N. Mishustin and H. Schulz, Nucl. Phys. A448 (1986) 753 [6] K. Sneppen and R. Donangelo, Phys. Rev. C39 (1989) 263; K. Sneppen, Nucl. Phys. A470 ( 1987) 213 [7] H. Barz, J. Bondorf, R. Donangelo, H. Schulz and K. Sneppen, Phys Lett. B191 (1987) 232; H.W. Barz, H. Schulz, J.P. Bondorf, J. Lopez and K. Sneppen, Phys. Lett. B211 (1988) 25; H.W. Barz, J.P. Bondorf, H. Schulz and K. Sneppen, Phys. Lett. 8244 (1990) 161; H.W. Barz, J.P. Bondorf, R. Donangelo, I. Mishustin, H. Schulz and K. Sneppen, Nucl. Phys. A545 (1992) 213~ [8] H.W. Barz, W. Bauer, J.P. Bondorf, AS. Botwina, R. Donangelo, H. Schulz and K. Sneppen, NBI preprint, NBI-92-88 [9] K. Sneppen and L. Vinet, Nucl. Phys. A480 (1988) 342 [lo] D.H.E. Gross, Phys. Lett. B 203 (1988) 26 [ 111 Bao-an Li, A.R. DeAngelis and D.H.E. Gross, Phys. Lett. B303 ( 1992) 225. [ 121 Aa. Bohr and B.R. Mottelson, Nuclear structure (Benjamin, New York, 1969) f 131 R.W. Gerling and A. Huller, Z. Phys. B90 (1993) 207 [ 141 J.W. Truran, A.G.W. Cameron and E. Hilf, Cem Report 70-30 (1970) 215 [ 15 ] W. Dilg, W. Schantl and H. Vonach, Nucl. Phys. A2 17 ( 1973) 269 [ 161 J. Randrup, M. Robinson and K. Sneppen, Phys. Lett. B208 (1988) 25 [ 171 V. Weisskopf, Phys. Rev. 52 ( 1937) 2 11 [ 181 P. Kreutz, thesis GSI-Darmstadt (1992)