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Relaxation processes in intermediate-energy heavy-ion collisions
Nuclear Physics A519 (1990) 345c-356c North-Holland
345c
RELAXATION PROCESSES IN INTERMEDIATE-ENERGY HEAVY-ION COLLISIONS M. CUBERO% M. SCHONIIOFEN% B. L. FRIMANa'b and W. NORENBERG a'~ "Gesellschaff flit Schwerionenforschung (GSI), Postfach 110552 D-6100 Darmstadt, West-Germany bInstitut ffir Kernphysik, Technische Hochschule Darmstadt D-6100 Darmstadt, West-Germany Covariant kinetic equations are derived from quantum hadrodynamics (QHD) and applied to heavy-ion collisions at intermediate energies (Ebeam < 2GeV/u). It is shown that medium effects strongly modify the initial equilibration process and the reabsorption of pions and deltas during the final expansion phase.
1. INTRODUCTION AND SUMMARY Heavy-ion collisions up to beam energies Ebe,,m ~ 2 GeV per nucleon (GeV/u) probe nuclear matter at large densities (up to ca. 4 times normal nuclear density) and high excitation energies (up to ca. 200 MeV/u) 1, 2 However, since such densities and excitation energies are reached only in transient states for time intervals of the order 10 -22 s and in volumes of the order 100 fm3, it is necessary to understand the complete dynamicM evolution of the collision in order to extract properties, such as the nuclear equation of state or the dispersion relation of pions in dense and hot nuclear matter from the experimental data. At these intermediate energies of several hundred MeV/u in the center-of-mass system, mesonic degrees of freedom and baryonic excitations become important, whereas the quarks and gluons remain confined within the hadrons. Thus, the appropriate framework for attacking these problems is a relativistic quantum field theory for hadrons. For this exploratory calculation we choose a relatively simple, but still fairly realistic model, quantum hadrodynamics 3 (QHD). This is an effective quantum field theory for baryons and mesons at low momenta (< 1 GeV/c); form factors are introduced to cut off high momentum components. For a semi-realistic description of intermediate-energy heavy-ion collisions, it is sufficient to take into account the 7r-, (7- and w-mesons, the nucleons and the delta-resonance. The latter plays a very important role in these problems, due to its strong coupling to pions and nucleons (cf. section 2). The p-meson must be taken into account, when electromagnetic probes are considered. Starting from the Kadanoff-Baym equations 4 for QHD, we derive, by applying the customary gradient expansion as well as the quasi-particle approximation, semiclassical kinetic equations (Landau-Vlasov equations) for nucleons, deltas and pions. The heavier mesons (¢r and w) are treated in the mean-field approximation. Finally, the collision term is evaluated in the Born approximation. Since the Born approximation does not yield a satisfactory description of nucleon-nucleon 0375-9474/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
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M. Cubero et al. /Intermediate-energy heavy-ion collisions
collisions in a microscopic theory, the interaction should be viewed as an effective parametrization at this stage. Higher-order diagrams are effectively included in the collision term by fitting, in the Born approximation, elastic and inelastic nucleon-nucleon scattering cross sections (cf. section 3). In kinetic models of nucleus-nucleus collisions one either simulates the solution of the kinetic equations for semi-classical nucleons (VUU, BUU, Landau-Vlasov etc.) or one solves Newton's equations of motion for classical nucleons (molecular dynamics) 1, 2.
Although
these models involve drastic approximations, the calculations are still very time consuming. Thus, for computational convenience, a simple form for the nucleon-nucleon cross section is usually assumed. A major difficulty in solving kinetic equations numerically is the high dimensionality of phase space; 6+1 in the classical and 6+2 in the quantum-mechanical case. However, several interesting aspects of nucleus-nucleus collisions can be explored in spatially homogeneous systems. The advantage of such models is that the kinetic equations can be solved explicitly, microscopically evaluated collision ternas can be implemented and medium effects can be studied. Therefore, we model the equilibration process for a heavy-ion collision by considering an extended honmgeneous system in space of two counterstreaming nuclear fluids (of. section 4). The numerical results show that, for a small effective baryon mass (which carries most of the m o m e n t u m dependence of the effective interaction), the equilibration times are strongly reduced. The inelastic scattering of nucleons associated with production of deltas provide an essential contribution to the equilibration process. For the Walecka model, generalized by including deltas, very short equilibration times, on the order 10 =22 s, are obtained. This result supports the notion of a fireball in heavy-ion collisions at beam energies around 1 G e V / u . However, in view of the very strong density dependence of the effective baryon mass in this model, which leads to a strong enhancement of the effective cross sections, this result should probably be regarded with some caution. Some years ago it was suggested that the pion yield in intermediate energy heavy-ion collisions could be used to extract information on the nuclear equation of state, in particular the stiffness at densities above the saturation point s. It was argued that the final ~r + A abundance corresponds to the state of nmximum compression, i.e., that the absorption of pion-like degrees of freedom during the expansion of the fireball is negligible. Later it was pointed out that the pion abundance at m a x i m u m compression depends more strongly on the baryon effective mass than on the stiffness 6. Here we study the evolution of the pion- and delta-abundances during the expansion in detail, by solving the kinetic equations for nucleons, deltas and pions in the homogeneous interior part of the expanding fireball. (cf. section 5) Both the multiplicity of pions at maximum compression and the absorption rate depend strongly on the nucleon effective mass. However, the two effects compensate each other in such a way, that we find a very weak net dependence on the equation of state; if more pions are produced during the compression, more pions are also absorbed during the expansion. Thus, pion multiplicities seem not to be sensitive to the equation of state.
M. Cubero et aL / Intermediate-energy heavy-ion collisions 2.
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KINETIC EQUATIONS IN QttD Q u a n t u m hadrodynamics is an effective, relativistic quantum-field theory, consisting of
baryon fields (~I'N, ~'A) which interact via meson fields (scalar a, vector w and charged pseudoscalar 7?). For the lagrangian density we use a generalized version of the Walecka model 3,
+ ~(i.y.(o.. - -
g..,.)
+ ~(o.oo.o - . ~ )
-
mA
+ g.o)%
v
- u(~)
(1)
1 2 ~ F ~ F "~ + ~rn~w~w ~
+ - -
ig,,/~N%q'N
g~NA~NTff2A~,O"~ g~.NA~A~T | ~NO*'~ - -
where F~,, --- O,,w,, - O,,wt, and m N , m a m , ,mo.,m~ denote the Fuasses of nucleons, deltas, pions, scalar and vector mesons, respectively. The additional relation %~I'~, = 0 projects out the spin-3/2 part of the vector spinor ~'[x- The potential U(cr) denotes non-linear self-coupling terms of the scalar field. In studying the equilibration problem we set U(cr) = 0, while the pion-reabsorption problem is studied for several models with different choices for U(cr). In the non-linear models, the effective mass and the stiffness can be varied independently. Since we consider a system, which initially is far from equilibrium, we need transport equations which allow us to describe the equilibration process. A convenient starting point for the derivation of the semi-classical kinetic equations are the equations of motion of the Green functions, i.e., the Kadanoff-Baym equations 4, 7.
= / d%(z>(~,, ~)co(~, ~) - zo(~,, ~)c>(~, ~))
(3)
where
ia<(~,, ~) = -(~(~)~(~,)),
iac(~,, ~ ) = e(t, - t ~ ) i a > ( ~ , ~ ) + 6(t~ - t,)ia<(~,, ~ ) , ic°(~,, ~ ) = o(~
~"(~,, ~ )
= o(t2
-
-
t,)ic>(~,, ~ ) + o(~,
t,)~>(~,, ~ ) +
o(t~ -
-
t~)ic<(~,, ~2),
t~)r~<(~,, ~ ) + ~(t~ - t~)r~,(~,, ~)
and E ¢'~'>'< denote the different components of the non-equilibrium nucleon self-energy. The above equations apply for spin-½ fermions. Analogous equations apply for bosons and fermions
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M. Cubero et al. / Intermediate-energy heavy-ion collisions
of higher spin. We use the same notation as in 4, which is related to that of 7 in a straightforward manner. Assuming that the system is quasi-classical, i.e., that all quantities vary slowly in space and time, we approximate equations (2) and (3) by the leading terms in the gradient expansion. Furthermore, we assume that the nucleon spectral function is peaked, so that the quasi-particle approximation is valid. The resulting quasi-classical kinetic equation for the phase-space distribution function of the nucleons f ( z , p*) is
((0," + F,~G..) P--~
Tr -iC<(,,p'))
Tr (r.<(~, p*)G>(~, p') - ~>(~, p')C<(~, p*)),
(4)
where p*" -- p " - g~ (w') is the kinetic m o m e n t u m and m~v -- m N - go(a) the nucleon effective mass. Analogous equations are obtained for deltas and pions. A similar kinetic equation for the nucleons has also been derived by Li et al. 8 The real part of the self energy is approximated with the Hartree contribution, evaluated with mean-field coupling constants 3. This approximation, is consistent with the effective nature of the model and with the quasi-particle approximation. The spectral function of the baryons is then proportional to a delta-function i~(p*"p*~ - m'N) , as are the Green functions
s
i a > (~,
~(~; ", ~)~(~;*, ~)~(p;
p*) = 2~ ~
- ~
m;~)(1 - f(~,
~;-))
J
where u(/Y*, s) are the Dirac spinors for nuclear matter in the Hartree approximation. Antinucleons are neglected throughout our work. We calculate the self-energy functions N>,< up to second order in perturbation theory, which corresponds to the Born approximation for the scattering amplitude, i.e., we take the following self-energy diagrams into account:
--Z~B'or
n
~
7T, (Y 1 0 d i
3.
>
-~-
"K 1 Or , Od
.
r
NUCLEON-NUCLEON CROSS SECTIONS The exact solution of equation (4) in the full 6+1 dimensional phase space is in most cases too complicated to be feasible at present. However, if the spatial dependence of f(z,/ff*) is neglected, equation (4) can be solved numerically. The aim of this work is to gain insight into the global properties, like relaxation time-scales, and to study medium effects using a microscopic or semi-microscopic collision term. The simple model with two counter-streaming nuclear matter currents is well suited for addressing these
M. Cubero et al. / Intermediate-energy heavy-ion collisions
349c
questions. Furthermore, it can be utilized as a testing ground for various approximations, which are necessary to make the full calculation feasible. The parameters of the model are the baryon and meson masses as well as the coupling constants. For the masses we choose: mN = 939 MeV, mr, = 1232 MeV, m s = 550 MeV, m,, = 784 MeV and m,~ = 139 MeV. T h e nucleon-meson coupling constants are determined by fitting the free, elastic, differential nucleon-nucleon cross section at a b e a m energy of 1 GeV, while for the ~rNA-coupling constant we take the s t a n d a r d value, determined by the free decay width of the A resonance. In figure I we show the elastic differential cross section for free nucleon-nucleon scattering in our model as a function of the scattering angle in the center-of-mass frame ®c,, at different b e a m energies from 100 MeV to 1 GeV. In order to get a reasonable agreement with the reference curve from 9 one has to introduce form factors F(q2), where q is the m o m e n t u m transfer at the nucleon-meson vertices. For F = A2/(A 2 - q2) we find 10: g~ = 5.74, g~ = 6.00, g,, = 13.45,A~ = 975 MeV, A~ = 706 MeV, h,, = 693 MeV. The angular dependence
..•
'
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I
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"_~.:...
"
~:
,,
- -
",, 5-
"-. "~
-
,
,
_ _ _
...
"'.
"'~
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--
100 300 500 700
MeV MeV MeV MeV
0
0
20
4.0
60
80
0=. Figure 1: Differential elastic nucleon-nucleon cross section as a function of the scattering angle for different b e a m energies. of the differential cross section favours scattering in the forward direction at a b e a m energy of 1 GeV. At smaller energies (100 MeV to 300 MeV) the differential cross section is almost isotropic. The energy dependence of the total, elastic nucleon-nucleon cross section c~a is displayed in figure 2. The d o t t e d curve is from reference 9 and the full line is the elastic cross section obtained in our model. In order to illustrate the density dependence of the cross section in this model, we also show the total cross section obtained for m ~ = 0.56raN.
The energy
dependence of the cross section is not reproduced in the Born approximation. Thus the cross section in our model is too high at low energies and too low at energies above 1 GeV. Note also t h a t the cross section depends strongly on the nucleon effective mass, indicating a strong density dependence of the collision term.
M. Cubero et aL / Intermediate-energy heavy-ion collisions
350c
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20
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'
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I
I-i...o.i
.
~'~"
0 i 0
i
i I Et,, b
i 2 (GeV)
Figure 2: Total elastic cross section for nucleon-nucleon scattering as a function of the b e a m energy.
..-... U
10
E
'4-
5 u
o
.
o
I
200
,
I
t
400
Eta b
I
,
600
I
800
.
I
,
1000 1200
(MeV)
Figure 3: Mean time between two successive nucleon-nucleon collisions for ground state nuclear m a t t e r as a function of the kinetic energy of the incident nucleon. For the m e a n field we employ coupling constants fitted to the ground-state properties of nuclear m a t t e r 3. Since the higher order diagrams contributing to the m e a n field do not coincide with those c o n t r i b u t i n g to the forward scattering amplitude, there is no reason why one, in a n effective theory, should be forced to use the same coupling constants in b o t h cases. By allowing the coupling constants to differ, we a t t e m p t to get a reasonable description of the m e a n field as well as the scattering amplitude.
We note, however, t h a t in one of the
non-linear models the coupling constants are nearly the same as o b t a i n e d by fitting the elastic cross section In order to study the density dependence of the collision 8,
11. term, we have calculated
M. Cubero et aL / Intermediate-energy heavy-ion collisions
351 c
the m e a n collision time for a nucleon incident on nuclear m a t t e r in its ground state, taking only elastic nucleon-nucleon collisions into account. There is empirical information available from optical potential fits to proton-nucleus scattering data. In figure 3 we show the collision time tco 1 = 1/2Ira ( ~ ) obtained in our model as a function of the kinetic energy of the nucleon. One can u n d e r s t a n d the behaviour of the full line in figure 3 (m~v = raN) in terms of three effects. Below Ezab = 150 MeV the Pauli exclusion principle is responsible for the very large collision time. Near the Fermi surface most of the scattering events are forbidden. Between Etab = 200 MeV a n d 500 MeV the m e a n collision time reaches its m i n i m u m ( ~ 5 fm/c). The i m p o r t a n c e of the Pauli principle is decreased a n d the nucleon-nucleon cross section is still fairly isotropic. At higher energies the scattering is governed by the strongly forward peaked elastic cross section. This implies t h a t a lot of finM states are blocked again because they lie within the Fermi sphere. The net result is an increase of tco 1 to 9 fln/c at Ezab = 1 GeV. The curve for m ~ = 0.56
my
shows a less d r a m a t i c energy dependence of the collision time, since
in this case the cross section is b o t h larger a n d more isotropic over the whole energy range (compare figure 2). T h e collision times extracted from the empirical fits of H a m a et al 12 at energies below 400 MeV, where inelastic scattering is negligible, lies between our results for m ) = mN a n d m~r = 0.56/; raN. A detailed comparison is not meaningful at this stage, however, since in our model the total elastic nucleon-nucleon cross section is too large in this energy range. W h e n inelastic collisions are included, a comparison with optical potential d a t a at Elab = 1 GeV can be done. 4.
EQUILIBRATION Let us first consider the equilibration stage of a nucleus-nucleus collision
10. T h e time
evolution of the m o m e n t u m distribution in a symmetric nucleus-nucleus collision with a b e a m energy of I GeV per nucleon, modelled by counter-streaming nuclear m a t t e r , is displayed in figure 4 at different times. Only elastic nucleon-nucleon collisions are included. Here 1 TO -- - ~ 1.8 f m / c 27p0cr~l is the collision time for the free cross section neglecting Pauli blocking. The initiM condition is constructed by boosting two Fermi spheres with v~ = =t=0.6 c. The mass of the nucleons is m~v = m N a n d the influence of the meson fields is neglected. In the first collision the nucleons scatter, due to e n e r g y - m o m e n t u m conservation, preferentially into states within a m o m e n t u m shell, with a diameter equal to the relative m o m e n t u m of the original Fermi ellipsoids. At these energies the cross section is strongly forward peaked and therefore, only m o m e n t u m states near the Fermi ellipsoids will be occupied in the first collisions. T h e r m o d y n a m i c a l equilibrium is reached only after 12 TO. The slope of the equilibrium distribution corresponds to a t e m p e r a t u r e of a b o u t 135 MeV. We have also performed a calculation with mean-field effects included. In this case, the relation between the b e a m energy a n d the initial m o m e n t u m distribution is more complicated, since the two Fermi ellipsoids interact t h r o u g h the m e a n fields 13. Consider two colliding slabs of nuclear m a t t e r .
Before the slabs overlap, there is no interaction between t h e m , a n d the
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M, Cubero et al. / Intermediate-energy heavy-ion collisions
i
i
i
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i
i
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o.
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-5 I
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l
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0
§
~
0
S
P'z (fro-l)
P'z ffTn-l)
Figure 4: The evolution of the m o m e n t u m distribution for m~v 0.02, 0.04, 0.08, 0.16, 0.32 and 0.64.
=
raN.
The contour lines are:
m o m e n t u m distribution in each slab is simply a boosted Fermi ellipsoid. Once the slabs start overlapping, the two Fermi ellipsoids change due to mean-field effects as well as collisions. In constructing the initial state of our model, which corresponds to overlapping slabs, we take only the mean-field effects into account; the effects of collisions are included in the the solution of the kinetic equations. Thus the initial state is obtained by requiring energy, m o m e n t u m and baryon-number conservation in going from non-overlapping to overlapping slabs, not allowing for nucleon-nucleon collisions. This is the analogy of the one-dimensional shock problem in the collisionless regime, where the temperature in the shocked region is replaced by the relative m o m e n t u m of the two Fermi ellipsoids. Because of the cancellation of the spatial components of the vector field and the strong dependence of the effective mass of the nucleons on density, the initial distance between the Fermi ellipsoids is strongly reduced. The mean collision time without Pauli blocking (r0 = 0.8 fin/c) was in this case determined by the effective cross section at the relative m o m e n t u m of the shifted Fermi ellipsoids. This procedure eliminates the spurious energy dependence of the cross section. Still, the relaxation is much faster than in the free Fermi gas calculation. This is due to the increasing isotropy of the cross section at smaller relative momenta. Thus, the Pauli blocking is not as effective as in the previous calculation. However, if plotted against the number of true collisions, including the effect of Pauli blocking, the equilibration times are more equal in the two systems. The production of entropy is completed after roughly three collisions. The inclusion of delta
M. Cubero et al. / Intermediate-energy heavy-ion collisions
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Figure 5: Same as figure 4, but including mean-field effects. excitations reduce the equilibration time in b o t h cases by roughly a factor 2. 5.
R E A B S O R P T I O N OF PIONS AND DELTAS We now t u r n to the expansion stage of a nucleus-nucleus collision, studying in particular
the a b u n d a n c e of pion-like excitations 11. T h e equilibration process, considered in the previous section is assumed to be completed, so t h a t the initial state of the expansion is an extended fireball in t h e r m o d y n a m i c a l equilibrium. In the sense of a Thomas-Fermi a p p r o x i m a t i o n the expansion of the central part of the fireball is modelled by homogeneous m a t t e r with a density decreasing with time. In the absence of collisions, the abundances are not changed during the expansion a n d the entropy remains constant. The characteristic timescale to for the expansion is given by the time it takes a sound wave to travel from the surface to the center of the fireball 14, 15 A solution of the coupled Landau-Vlasov equations for nucleons, deltas a n d pions is shown in fig. 6. T h e initial fireball is assumed to have a baryon density PB = 3pB0 a n d a t e m p e r a t u r e T = 100 MeV. These values are typical for fireballs obtained in the one-dimensional shock a p p r o x i m a t i o n for a central heavy-ion collision with b e a m energies a r o u n d 6 1 G e V / u . The expansion rate is illustrated by the time dependence of the b a r y o n density. T h e effective masses of the nucleon a n d the a-meson a p p r o a c h their free values. The n u m b e r of deltas decreases due to their n a t u r a l decay (A ---, N+~r) as well as the reabsorption process ( N + A ---, N + N ) , while
354c
M. Cubero et al. ' Intermediate-energy heavy-ion collisions
.4 ~5
(a) 2
.IL v
o
(ff 0 -o
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2 ;
l
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(b)
;;-... (~ "'.....
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~ . . . . ...... ~
~
............ ..- ..........
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-
(c)
0.2
.IL v
on 0¢
0.1
C~ 0.0 0
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6
8
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t (fm/c) Figure 6: (a) The densities of baryons, deltas and pions in units of the normal nuclear matter density, (b) effective masses of nucleons and sigmas, and (c) the number baryon of pions, deltas and pions plus deltas per baryon as functions of time. Here the equation of state includes self-interactions of the sigma field (rn~v0 = 0.85 raN, K0 ----400 MeV, cf. ref. 6). the number of pions increases. The sum of deltas and pions remains constant after 4 fm/c. This time is referred to as freeze-out point for the pion multiplicities, the corresponding density is P~reeze ~ 0"5pB0" During the expansion the 7r + A abundance is reduced by approximately 30 %. In figure 7 we illustrate the effect of pion and delta absorption: we show, for different QHD models, the primary 6 and the final 7r + A abundances as functions of the bombarding energy. We observe that in those models, where the abundance of pions and deltas in the initial fireball is high, the reabsorption rate is also high. This effect is particularly drastic for the pure Walecka model. The strong dependence of the primary abundance of pions and deltas on the nuclear equation of state, in particular on the baryon effective mass, is to a large extent
M. Cubero et aL / Intermediate-energy heavy-ion collisions
355c
0.5 0.4. _ m*N0(3)=0.85mN
,," ..,.-" _
_ m N014.)=0.85mN
~,'..."
_
0.3 0.2
i / .,"
.o-
/ / ..'" ~.~'" , ..." .#~"
0.1
,,¢:~-"~
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I
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/
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J
m'N0(l)=O-56mN J
0.4. _
0.3 0.2
...,.,....,
-
.........
0.1 0.0
-
0
..Jr" "~''''1 ~
500
,
I
,
I
1000
1500
ELAB/A p
(MeV)
2000
Figure 7: The number of pions plus deltas as a function of the beam energy for central collisions of equal nuclei. The two upper curves denote the pion-plus-delta abundances in the fireball where 1,3,4 correspond to different nuclear equations of state with (m~v, K0) = (0.56 raN, 540MeV), (0.85 raN, 400MeV) and (0.85 raN, 210MeV), respectively 6. The two lower curves give the pion multiplicities after reabsorption during the expansion and the arrows indicate the loss due to reabsorption. The data points are the observed pion multiplicities t6, 17 compensated by an similar dependence of the absorption rate. Thus the final pion abundance shows almost no dependence on the equation of state. REFERENCES 1) G.F. Bertsch and S. Das Gupta, Phys. Reports 160 (1988) 189 2) See Proc. Int. Workshop on Nuclear Dynamics at Medium and High Energies, Bad Honnef 1988, eds. W. Cussing and U. Mosel, Nucl. Phys. A495 (1989) 3) B. Serot and J. D. Walecka, Adv. Nucl. Phys. vol. 16 eds. J. W. Negele and E. Vogt (Plenum Press, New York, 1986); see also J. D. Walecka, this volume 4) L.P. Kadanoffand G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962)
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M. Cubero et al. / Intermediate-energy heavy-ion collisions
5) R. Stock et al. , Phys. Rev. Left. 49 (1982) 1236; R. Stock, Phys. Reports 135 (1986) 259 6) M. Schhnhofen, M. Cubero, M. Gering, M. Sambataro, It. Feldmeier and W. Nhernberg, Nucl. Phys. A504 (1989) 875 7) L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 (Soy. Phys.-JETP 20 (1965) 1018); E. M. Lifshitz and L. P. Pitaevski~, Physical Kinetics (Pergamon, Oxford, 1981) 8) Q. Li, Q. Wu and C. M. Ko, Phys. Rev. C39 (1989) 849 9) J. Cugnofi, T. Mizutani and J. Vandermeulen, Nucl. Phys. A352 (1981) 505 10) M. Schhnhofen, Doctoral Thesis, Technische Hochschule Darmstadt, 1990 11) M. Cubero, Doctoral Thesis, Technische Hochschule Darmstadt, 1990 12) S. Hama, B.C. Clark, R.E. Kozack, S. Shim, E.D. Cooper, R.L. Mercer and B. Serot, Phys. Rev. C37 (1988) 1111 13) C. IIorowitz, in Proc. XVI International Workshop on Gross Properties of Nuclei and Nuclear Excitations, Hirschegg, Austria, 1988, ed. H. Feldmeier, p. 10 14) H. W. Barz, J. Knoll, B. L. Friman and H. Schulz, Nucl. Phys. A484 (1988) 661 15) G. Baym, B. L. Friman, J. P. Blaizot, M. Soyeur and W. Czy~, Nucl. Phys. A407 (1983) 541 16) J.W. Harris et al., Phys. Rev. Left. 58 (1987) 463 17) J.W. Harris et al., Phys. Left. B153 (1985) 377
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RELAXATION PROCESSES IN INTERMEDIATE-ENERGY HEAVY-ION COLLISIONS M. CUBERO% M. SCHONIIOFEN% B. L. FRIMANa'b and W. NORENBERG a'~ "Gesellschaff flit Schwerionenforschung (GSI), Postfach 110552 D-6100 Darmstadt, West-Germany bInstitut ffir Kernphysik, Technische Hochschule Darmstadt D-6100 Darmstadt, West-Germany Covariant kinetic equations are derived from quantum hadrodynamics (QHD) and applied to heavy-ion collisions at intermediate energies (Ebeam < 2GeV/u). It is shown that medium effects strongly modify the initial equilibration process and the reabsorption of pions and deltas during the final expansion phase.
1. INTRODUCTION AND SUMMARY Heavy-ion collisions up to beam energies Ebe,,m ~ 2 GeV per nucleon (GeV/u) probe nuclear matter at large densities (up to ca. 4 times normal nuclear density) and high excitation energies (up to ca. 200 MeV/u) 1, 2 However, since such densities and excitation energies are reached only in transient states for time intervals of the order 10 -22 s and in volumes of the order 100 fm3, it is necessary to understand the complete dynamicM evolution of the collision in order to extract properties, such as the nuclear equation of state or the dispersion relation of pions in dense and hot nuclear matter from the experimental data. At these intermediate energies of several hundred MeV/u in the center-of-mass system, mesonic degrees of freedom and baryonic excitations become important, whereas the quarks and gluons remain confined within the hadrons. Thus, the appropriate framework for attacking these problems is a relativistic quantum field theory for hadrons. For this exploratory calculation we choose a relatively simple, but still fairly realistic model, quantum hadrodynamics 3 (QHD). This is an effective quantum field theory for baryons and mesons at low momenta (< 1 GeV/c); form factors are introduced to cut off high momentum components. For a semi-realistic description of intermediate-energy heavy-ion collisions, it is sufficient to take into account the 7r-, (7- and w-mesons, the nucleons and the delta-resonance. The latter plays a very important role in these problems, due to its strong coupling to pions and nucleons (cf. section 2). The p-meson must be taken into account, when electromagnetic probes are considered. Starting from the Kadanoff-Baym equations 4 for QHD, we derive, by applying the customary gradient expansion as well as the quasi-particle approximation, semiclassical kinetic equations (Landau-Vlasov equations) for nucleons, deltas and pions. The heavier mesons (¢r and w) are treated in the mean-field approximation. Finally, the collision term is evaluated in the Born approximation. Since the Born approximation does not yield a satisfactory description of nucleon-nucleon 0375-9474/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
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M. Cubero et al. /Intermediate-energy heavy-ion collisions
collisions in a microscopic theory, the interaction should be viewed as an effective parametrization at this stage. Higher-order diagrams are effectively included in the collision term by fitting, in the Born approximation, elastic and inelastic nucleon-nucleon scattering cross sections (cf. section 3). In kinetic models of nucleus-nucleus collisions one either simulates the solution of the kinetic equations for semi-classical nucleons (VUU, BUU, Landau-Vlasov etc.) or one solves Newton's equations of motion for classical nucleons (molecular dynamics) 1, 2.
Although
these models involve drastic approximations, the calculations are still very time consuming. Thus, for computational convenience, a simple form for the nucleon-nucleon cross section is usually assumed. A major difficulty in solving kinetic equations numerically is the high dimensionality of phase space; 6+1 in the classical and 6+2 in the quantum-mechanical case. However, several interesting aspects of nucleus-nucleus collisions can be explored in spatially homogeneous systems. The advantage of such models is that the kinetic equations can be solved explicitly, microscopically evaluated collision ternas can be implemented and medium effects can be studied. Therefore, we model the equilibration process for a heavy-ion collision by considering an extended honmgeneous system in space of two counterstreaming nuclear fluids (of. section 4). The numerical results show that, for a small effective baryon mass (which carries most of the m o m e n t u m dependence of the effective interaction), the equilibration times are strongly reduced. The inelastic scattering of nucleons associated with production of deltas provide an essential contribution to the equilibration process. For the Walecka model, generalized by including deltas, very short equilibration times, on the order 10 =22 s, are obtained. This result supports the notion of a fireball in heavy-ion collisions at beam energies around 1 G e V / u . However, in view of the very strong density dependence of the effective baryon mass in this model, which leads to a strong enhancement of the effective cross sections, this result should probably be regarded with some caution. Some years ago it was suggested that the pion yield in intermediate energy heavy-ion collisions could be used to extract information on the nuclear equation of state, in particular the stiffness at densities above the saturation point s. It was argued that the final ~r + A abundance corresponds to the state of nmximum compression, i.e., that the absorption of pion-like degrees of freedom during the expansion of the fireball is negligible. Later it was pointed out that the pion abundance at m a x i m u m compression depends more strongly on the baryon effective mass than on the stiffness 6. Here we study the evolution of the pion- and delta-abundances during the expansion in detail, by solving the kinetic equations for nucleons, deltas and pions in the homogeneous interior part of the expanding fireball. (cf. section 5) Both the multiplicity of pions at maximum compression and the absorption rate depend strongly on the nucleon effective mass. However, the two effects compensate each other in such a way, that we find a very weak net dependence on the equation of state; if more pions are produced during the compression, more pions are also absorbed during the expansion. Thus, pion multiplicities seem not to be sensitive to the equation of state.
M. Cubero et aL / Intermediate-energy heavy-ion collisions 2.
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KINETIC EQUATIONS IN QttD Q u a n t u m hadrodynamics is an effective, relativistic quantum-field theory, consisting of
baryon fields (~I'N, ~'A) which interact via meson fields (scalar a, vector w and charged pseudoscalar 7?). For the lagrangian density we use a generalized version of the Walecka model 3,
+ ~(i.y.(o.. - -
g..,.)
+ ~(o.oo.o - . ~ )
-
mA
+ g.o)%
v
- u(~)
(1)
1 2 ~ F ~ F "~ + ~rn~w~w ~
+ - -
ig,,/~N%q'N
g~NA~NTff2A~,O"~ g~.NA~A~T | ~NO*'~ - -
where F~,, --- O,,w,, - O,,wt, and m N , m a m , ,mo.,m~ denote the Fuasses of nucleons, deltas, pions, scalar and vector mesons, respectively. The additional relation %~I'~, = 0 projects out the spin-3/2 part of the vector spinor ~'[x- The potential U(cr) denotes non-linear self-coupling terms of the scalar field. In studying the equilibration problem we set U(cr) = 0, while the pion-reabsorption problem is studied for several models with different choices for U(cr). In the non-linear models, the effective mass and the stiffness can be varied independently. Since we consider a system, which initially is far from equilibrium, we need transport equations which allow us to describe the equilibration process. A convenient starting point for the derivation of the semi-classical kinetic equations are the equations of motion of the Green functions, i.e., the Kadanoff-Baym equations 4, 7.
= / d%(z>(~,, ~)co(~, ~) - zo(~,, ~)c>(~, ~))
(3)
where
ia<(~,, ~) = -(~(~)~(~,)),
iac(~,, ~ ) = e(t, - t ~ ) i a > ( ~ , ~ ) + 6(t~ - t,)ia<(~,, ~ ) , ic°(~,, ~ ) = o(~
~"(~,, ~ )
= o(t2
-
-
t,)ic>(~,, ~ ) + o(~,
t,)~>(~,, ~ ) +
o(t~ -
-
t~)ic<(~,, ~2),
t~)r~<(~,, ~ ) + ~(t~ - t~)r~,(~,, ~)
and E ¢'~'>'< denote the different components of the non-equilibrium nucleon self-energy. The above equations apply for spin-½ fermions. Analogous equations apply for bosons and fermions
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M. Cubero et al. / Intermediate-energy heavy-ion collisions
of higher spin. We use the same notation as in 4, which is related to that of 7 in a straightforward manner. Assuming that the system is quasi-classical, i.e., that all quantities vary slowly in space and time, we approximate equations (2) and (3) by the leading terms in the gradient expansion. Furthermore, we assume that the nucleon spectral function is peaked, so that the quasi-particle approximation is valid. The resulting quasi-classical kinetic equation for the phase-space distribution function of the nucleons f ( z , p*) is
((0," + F,~G..) P--~
Tr -iC<(,,p'))
Tr (r.<(~, p*)G>(~, p') - ~>(~, p')C<(~, p*)),
(4)
where p*" -- p " - g~ (w') is the kinetic m o m e n t u m and m~v -- m N - go(a) the nucleon effective mass. Analogous equations are obtained for deltas and pions. A similar kinetic equation for the nucleons has also been derived by Li et al. 8 The real part of the self energy is approximated with the Hartree contribution, evaluated with mean-field coupling constants 3. This approximation, is consistent with the effective nature of the model and with the quasi-particle approximation. The spectral function of the baryons is then proportional to a delta-function i~(p*"p*~ - m'N) , as are the Green functions
s
i a > (~,
~(~; ", ~)~(~;*, ~)~(p;
p*) = 2~ ~
- ~
m;~)(1 - f(~,
~;-))
J
where u(/Y*, s) are the Dirac spinors for nuclear matter in the Hartree approximation. Antinucleons are neglected throughout our work. We calculate the self-energy functions N>,< up to second order in perturbation theory, which corresponds to the Born approximation for the scattering amplitude, i.e., we take the following self-energy diagrams into account:
--Z~B'or
n
~
7T, (Y 1 0 d i
3.
>
-~-
"K 1 Or , Od
.
r
NUCLEON-NUCLEON CROSS SECTIONS The exact solution of equation (4) in the full 6+1 dimensional phase space is in most cases too complicated to be feasible at present. However, if the spatial dependence of f(z,/ff*) is neglected, equation (4) can be solved numerically. The aim of this work is to gain insight into the global properties, like relaxation time-scales, and to study medium effects using a microscopic or semi-microscopic collision term. The simple model with two counter-streaming nuclear matter currents is well suited for addressing these
M. Cubero et al. / Intermediate-energy heavy-ion collisions
349c
questions. Furthermore, it can be utilized as a testing ground for various approximations, which are necessary to make the full calculation feasible. The parameters of the model are the baryon and meson masses as well as the coupling constants. For the masses we choose: mN = 939 MeV, mr, = 1232 MeV, m s = 550 MeV, m,, = 784 MeV and m,~ = 139 MeV. T h e nucleon-meson coupling constants are determined by fitting the free, elastic, differential nucleon-nucleon cross section at a b e a m energy of 1 GeV, while for the ~rNA-coupling constant we take the s t a n d a r d value, determined by the free decay width of the A resonance. In figure I we show the elastic differential cross section for free nucleon-nucleon scattering in our model as a function of the scattering angle in the center-of-mass frame ®c,, at different b e a m energies from 100 MeV to 1 GeV. In order to get a reasonable agreement with the reference curve from 9 one has to introduce form factors F(q2), where q is the m o m e n t u m transfer at the nucleon-meson vertices. For F = A2/(A 2 - q2) we find 10: g~ = 5.74, g~ = 6.00, g,, = 13.45,A~ = 975 MeV, A~ = 706 MeV, h,, = 693 MeV. The angular dependence
..•
'
I
'
I
'
I
'
I
10
"_~.:...
"
~:
,,
- -
",, 5-
"-. "~
-
,
,
_ _ _
...
"'.
"'~
"""
--
100 300 500 700
MeV MeV MeV MeV
0
0
20
4.0
60
80
0=. Figure 1: Differential elastic nucleon-nucleon cross section as a function of the scattering angle for different b e a m energies. of the differential cross section favours scattering in the forward direction at a b e a m energy of 1 GeV. At smaller energies (100 MeV to 300 MeV) the differential cross section is almost isotropic. The energy dependence of the total, elastic nucleon-nucleon cross section c~a is displayed in figure 2. The d o t t e d curve is from reference 9 and the full line is the elastic cross section obtained in our model. In order to illustrate the density dependence of the cross section in this model, we also show the total cross section obtained for m ~ = 0.56raN.
The energy
dependence of the cross section is not reproduced in the Born approximation. Thus the cross section in our model is too high at low energies and too low at energies above 1 GeV. Note also t h a t the cross section depends strongly on the nucleon effective mass, indicating a strong density dependence of the collision term.
M. Cubero et aL / Intermediate-energy heavy-ion collisions
350c
8o
_
20
'
'
"-..
I
I-i...o.i
.
~'~"
0 i 0
i
i I Et,, b
i 2 (GeV)
Figure 2: Total elastic cross section for nucleon-nucleon scattering as a function of the b e a m energy.
..-... U
10
E
'4-
5 u
o
.
o
I
200
,
I
t
400
Eta b
I
,
600
I
800
.
I
,
1000 1200
(MeV)
Figure 3: Mean time between two successive nucleon-nucleon collisions for ground state nuclear m a t t e r as a function of the kinetic energy of the incident nucleon. For the m e a n field we employ coupling constants fitted to the ground-state properties of nuclear m a t t e r 3. Since the higher order diagrams contributing to the m e a n field do not coincide with those c o n t r i b u t i n g to the forward scattering amplitude, there is no reason why one, in a n effective theory, should be forced to use the same coupling constants in b o t h cases. By allowing the coupling constants to differ, we a t t e m p t to get a reasonable description of the m e a n field as well as the scattering amplitude.
We note, however, t h a t in one of the
non-linear models the coupling constants are nearly the same as o b t a i n e d by fitting the elastic cross section In order to study the density dependence of the collision 8,
11. term, we have calculated
M. Cubero et aL / Intermediate-energy heavy-ion collisions
351 c
the m e a n collision time for a nucleon incident on nuclear m a t t e r in its ground state, taking only elastic nucleon-nucleon collisions into account. There is empirical information available from optical potential fits to proton-nucleus scattering data. In figure 3 we show the collision time tco 1 = 1/2Ira ( ~ ) obtained in our model as a function of the kinetic energy of the nucleon. One can u n d e r s t a n d the behaviour of the full line in figure 3 (m~v = raN) in terms of three effects. Below Ezab = 150 MeV the Pauli exclusion principle is responsible for the very large collision time. Near the Fermi surface most of the scattering events are forbidden. Between Etab = 200 MeV a n d 500 MeV the m e a n collision time reaches its m i n i m u m ( ~ 5 fm/c). The i m p o r t a n c e of the Pauli principle is decreased a n d the nucleon-nucleon cross section is still fairly isotropic. At higher energies the scattering is governed by the strongly forward peaked elastic cross section. This implies t h a t a lot of finM states are blocked again because they lie within the Fermi sphere. The net result is an increase of tco 1 to 9 fln/c at Ezab = 1 GeV. The curve for m ~ = 0.56
my
shows a less d r a m a t i c energy dependence of the collision time, since
in this case the cross section is b o t h larger a n d more isotropic over the whole energy range (compare figure 2). T h e collision times extracted from the empirical fits of H a m a et al 12 at energies below 400 MeV, where inelastic scattering is negligible, lies between our results for m ) = mN a n d m~r = 0.56/; raN. A detailed comparison is not meaningful at this stage, however, since in our model the total elastic nucleon-nucleon cross section is too large in this energy range. W h e n inelastic collisions are included, a comparison with optical potential d a t a at Elab = 1 GeV can be done. 4.
EQUILIBRATION Let us first consider the equilibration stage of a nucleus-nucleus collision
10. T h e time
evolution of the m o m e n t u m distribution in a symmetric nucleus-nucleus collision with a b e a m energy of I GeV per nucleon, modelled by counter-streaming nuclear m a t t e r , is displayed in figure 4 at different times. Only elastic nucleon-nucleon collisions are included. Here 1 TO -- - ~ 1.8 f m / c 27p0cr~l is the collision time for the free cross section neglecting Pauli blocking. The initiM condition is constructed by boosting two Fermi spheres with v~ = =t=0.6 c. The mass of the nucleons is m~v = m N a n d the influence of the meson fields is neglected. In the first collision the nucleons scatter, due to e n e r g y - m o m e n t u m conservation, preferentially into states within a m o m e n t u m shell, with a diameter equal to the relative m o m e n t u m of the original Fermi ellipsoids. At these energies the cross section is strongly forward peaked and therefore, only m o m e n t u m states near the Fermi ellipsoids will be occupied in the first collisions. T h e r m o d y n a m i c a l equilibrium is reached only after 12 TO. The slope of the equilibrium distribution corresponds to a t e m p e r a t u r e of a b o u t 135 MeV. We have also performed a calculation with mean-field effects included. In this case, the relation between the b e a m energy a n d the initial m o m e n t u m distribution is more complicated, since the two Fermi ellipsoids interact t h r o u g h the m e a n fields 13. Consider two colliding slabs of nuclear m a t t e r .
Before the slabs overlap, there is no interaction between t h e m , a n d the
352c
M, Cubero et al. / Intermediate-energy heavy-ion collisions
i
i
i
!
i
i
2~o I
E o
ON
o.
I
I
I
I
I
I 8~o
5
E
~
0
o
o x
-5 I
I
!
l
I
I
•"5
0
§
~
0
S
P'z (fro-l)
P'z ffTn-l)
Figure 4: The evolution of the m o m e n t u m distribution for m~v 0.02, 0.04, 0.08, 0.16, 0.32 and 0.64.
=
raN.
The contour lines are:
m o m e n t u m distribution in each slab is simply a boosted Fermi ellipsoid. Once the slabs start overlapping, the two Fermi ellipsoids change due to mean-field effects as well as collisions. In constructing the initial state of our model, which corresponds to overlapping slabs, we take only the mean-field effects into account; the effects of collisions are included in the the solution of the kinetic equations. Thus the initial state is obtained by requiring energy, m o m e n t u m and baryon-number conservation in going from non-overlapping to overlapping slabs, not allowing for nucleon-nucleon collisions. This is the analogy of the one-dimensional shock problem in the collisionless regime, where the temperature in the shocked region is replaced by the relative m o m e n t u m of the two Fermi ellipsoids. Because of the cancellation of the spatial components of the vector field and the strong dependence of the effective mass of the nucleons on density, the initial distance between the Fermi ellipsoids is strongly reduced. The mean collision time without Pauli blocking (r0 = 0.8 fin/c) was in this case determined by the effective cross section at the relative m o m e n t u m of the shifted Fermi ellipsoids. This procedure eliminates the spurious energy dependence of the cross section. Still, the relaxation is much faster than in the free Fermi gas calculation. This is due to the increasing isotropy of the cross section at smaller relative momenta. Thus, the Pauli blocking is not as effective as in the previous calculation. However, if plotted against the number of true collisions, including the effect of Pauli blocking, the equilibration times are more equal in the two systems. The production of entropy is completed after roughly three collisions. The inclusion of delta
M. Cubero et al. / Intermediate-energy heavy-ion collisions
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Figure 5: Same as figure 4, but including mean-field effects. excitations reduce the equilibration time in b o t h cases by roughly a factor 2. 5.
R E A B S O R P T I O N OF PIONS AND DELTAS We now t u r n to the expansion stage of a nucleus-nucleus collision, studying in particular
the a b u n d a n c e of pion-like excitations 11. T h e equilibration process, considered in the previous section is assumed to be completed, so t h a t the initial state of the expansion is an extended fireball in t h e r m o d y n a m i c a l equilibrium. In the sense of a Thomas-Fermi a p p r o x i m a t i o n the expansion of the central part of the fireball is modelled by homogeneous m a t t e r with a density decreasing with time. In the absence of collisions, the abundances are not changed during the expansion a n d the entropy remains constant. The characteristic timescale to for the expansion is given by the time it takes a sound wave to travel from the surface to the center of the fireball 14, 15 A solution of the coupled Landau-Vlasov equations for nucleons, deltas a n d pions is shown in fig. 6. T h e initial fireball is assumed to have a baryon density PB = 3pB0 a n d a t e m p e r a t u r e T = 100 MeV. These values are typical for fireballs obtained in the one-dimensional shock a p p r o x i m a t i o n for a central heavy-ion collision with b e a m energies a r o u n d 6 1 G e V / u . The expansion rate is illustrated by the time dependence of the b a r y o n density. T h e effective masses of the nucleon a n d the a-meson a p p r o a c h their free values. The n u m b e r of deltas decreases due to their n a t u r a l decay (A ---, N+~r) as well as the reabsorption process ( N + A ---, N + N ) , while
354c
M. Cubero et al. ' Intermediate-energy heavy-ion collisions
.4 ~5
(a) 2
.IL v
o
(ff 0 -o
'
2 ;
l
'
I
'
I
'
I
'
I
'
(b)
;;-... (~ "'.....
E-,
~ . . . . ...... ~
~
............ ..- ..........
N
<3
0 0.3
_
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-
(c)
0.2
.IL v
on 0¢
0.1
C~ 0.0 0
2
b,
6
8
10
12
t (fm/c) Figure 6: (a) The densities of baryons, deltas and pions in units of the normal nuclear matter density, (b) effective masses of nucleons and sigmas, and (c) the number baryon of pions, deltas and pions plus deltas per baryon as functions of time. Here the equation of state includes self-interactions of the sigma field (rn~v0 = 0.85 raN, K0 ----400 MeV, cf. ref. 6). the number of pions increases. The sum of deltas and pions remains constant after 4 fm/c. This time is referred to as freeze-out point for the pion multiplicities, the corresponding density is P~reeze ~ 0"5pB0" During the expansion the 7r + A abundance is reduced by approximately 30 %. In figure 7 we illustrate the effect of pion and delta absorption: we show, for different QHD models, the primary 6 and the final 7r + A abundances as functions of the bombarding energy. We observe that in those models, where the abundance of pions and deltas in the initial fireball is high, the reabsorption rate is also high. This effect is particularly drastic for the pure Walecka model. The strong dependence of the primary abundance of pions and deltas on the nuclear equation of state, in particular on the baryon effective mass, is to a large extent
M. Cubero et aL / Intermediate-energy heavy-ion collisions
355c
0.5 0.4. _ m*N0(3)=0.85mN
,," ..,.-" _
_ m N014.)=0.85mN
~,'..."
_
0.3 0.2
i / .,"
.o-
/ / ..'" ~.~'" , ..." .#~"
0.1
,,¢:~-"~
0.0
(b)
I
I
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/
I
J
m'N0(l)=O-56mN J
0.4. _
0.3 0.2
...,.,....,
-
.........
0.1 0.0
-
0
..Jr" "~''''1 ~
500
,
I
,
I
1000
1500
ELAB/A p
(MeV)
2000
Figure 7: The number of pions plus deltas as a function of the beam energy for central collisions of equal nuclei. The two upper curves denote the pion-plus-delta abundances in the fireball where 1,3,4 correspond to different nuclear equations of state with (m~v, K0) = (0.56 raN, 540MeV), (0.85 raN, 400MeV) and (0.85 raN, 210MeV), respectively 6. The two lower curves give the pion multiplicities after reabsorption during the expansion and the arrows indicate the loss due to reabsorption. The data points are the observed pion multiplicities t6, 17 compensated by an similar dependence of the absorption rate. Thus the final pion abundance shows almost no dependence on the equation of state. REFERENCES 1) G.F. Bertsch and S. Das Gupta, Phys. Reports 160 (1988) 189 2) See Proc. Int. Workshop on Nuclear Dynamics at Medium and High Energies, Bad Honnef 1988, eds. W. Cussing and U. Mosel, Nucl. Phys. A495 (1989) 3) B. Serot and J. D. Walecka, Adv. Nucl. Phys. vol. 16 eds. J. W. Negele and E. Vogt (Plenum Press, New York, 1986); see also J. D. Walecka, this volume 4) L.P. Kadanoffand G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962)
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M. Cubero et al. / Intermediate-energy heavy-ion collisions
5) R. Stock et al. , Phys. Rev. Left. 49 (1982) 1236; R. Stock, Phys. Reports 135 (1986) 259 6) M. Schhnhofen, M. Cubero, M. Gering, M. Sambataro, It. Feldmeier and W. Nhernberg, Nucl. Phys. A504 (1989) 875 7) L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 (Soy. Phys.-JETP 20 (1965) 1018); E. M. Lifshitz and L. P. Pitaevski~, Physical Kinetics (Pergamon, Oxford, 1981) 8) Q. Li, Q. Wu and C. M. Ko, Phys. Rev. C39 (1989) 849 9) J. Cugnofi, T. Mizutani and J. Vandermeulen, Nucl. Phys. A352 (1981) 505 10) M. Schhnhofen, Doctoral Thesis, Technische Hochschule Darmstadt, 1990 11) M. Cubero, Doctoral Thesis, Technische Hochschule Darmstadt, 1990 12) S. Hama, B.C. Clark, R.E. Kozack, S. Shim, E.D. Cooper, R.L. Mercer and B. Serot, Phys. Rev. C37 (1988) 1111 13) C. IIorowitz, in Proc. XVI International Workshop on Gross Properties of Nuclei and Nuclear Excitations, Hirschegg, Austria, 1988, ed. H. Feldmeier, p. 10 14) H. W. Barz, J. Knoll, B. L. Friman and H. Schulz, Nucl. Phys. A484 (1988) 661 15) G. Baym, B. L. Friman, J. P. Blaizot, M. Soyeur and W. Czy~, Nucl. Phys. A407 (1983) 541 16) J.W. Harris et al., Phys. Rev. Left. 58 (1987) 463 17) J.W. Harris et al., Phys. Left. B153 (1985) 377