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QUANTAL VERSUS CLASSICAL PHASE-SPACE DYNAMICS OF HEAVY-ION REACTIONS
W. Cassing
To cite this version:
W. Cassing. QUANTAL VERSUS CLASSICAL PHASE-SPACE DYNAMICS OF HEAVY- ION REACTIONS. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-185-C2-194.
�10.1051/jphyscol:1987227�. �jpa-00226493�
JOURNAL DE PHYSIQUE
Colloque C2, supplBment au n o 6, Tome 48, juin 1987
QUANTAL VERSUS CLASSICAL PHASE-SPACE DYNAMICS OF HEAVY-ION REACTIONS'~'
W. CASSING
Institut fiir Theoretische Physik, Universitst Giessen, 0-6300 Giessen, F.R.G
Abstract - The Vlasov-Uehling-Uhlenbcck equation is derived from coupled equations of motion for the one-body density matrix and the two-body correlation function as obtained from the density-matrix hierarchy. The respective approximations introduced for the classical collision term are related with energy and momentum conservation in single nucleon-nucleon collisions and found numerically to be approximately valid for hard scattering processes in case of intermediate energy heavy-ion reactions. A related second order approximation in the nucleon-nucleon interaction, however, turns out not to be justified for colliding nuclear systems in the Fermi energy domain.
I - Introduction
Intermediate energy nucleus-nucleus collisions from roughly 15 MeV/u to 150 MeV/u provide experimental probes for the competition between mean-field dynamics /l/, dominant at lower energies, and successive nucleon-nucleon collisions /2/ known from the relativistic regime. It thus appears natural to include mean-field effects in cascade simulations of heavy-ion collisions / 3 - 6 / at inkermediate energies which have been interpreted as solutions of the Vlasov-Uehling- Uhlenbeck equation (VUU) for the one-body phase-space distribution f(r,k;t), i.e.
Equation (1.1) originally has been proposed for low density fermion systems /7-9/ and not necessarily applies to nucleus-nucleus collisions which are expected to provide information on nonequilibrium dynamics of strongly interacting quantal many-body systems at high density.
At present, it is neither known what to implement for the mean-field potential U(r,k;t) properly nor what to use for do/dR in the r.h.s. of l . Energy conservation and momentum conservation in single nucleon-nucleon collisions are not expected to hold for finite quantum systems due to the uncertainty relation nor the Markov approximation in time adopted in (1.1).
(l)~upported by BMFT and GSI Darmstadt
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987227
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On the other hand, using parametrizations for U(r,k;t) and do/dR, experimental data on particle emission 6 pion /10/ and photon production /11/ appear to be well understood in the limit of mean-field dynamics accomplished by random (on-shell) nucleon-nucleon scattering. One thus would like to bridge the gap between more fundamental quantal many-body theories /12-19/ and the classical limit adopted in (1.1).
I1 - Derivation of the Vlasov-Uehling-Uhlenbeck Equation
The quantal description of the nuclear many-body system is based on coupled equations of motion for the one-body density matrix p
and the two-body correlation function cp(12,1'2';t) defined by
where p , denotes the two-body density matrix, i.e.
as obtained from the density matrix hierarchy /17/. In eqs.(2.1) to (2.3) t(i) and v(ij) denote kinetic energy and the free nucleon- nucleon interaction, respectively, while the index i refers to coordinate r(i), spin a(i) and isospin ~ ( i ) of particle i. The limits (2.3a) to (2.3f) correspond to different classes of two-body theories which are all compatible with total energy conservation /20/.
In the limit (2.3b), neglecting additional terms of order v.c2, the equation of motion for the two-body correlation function can be integrated analytically /20-22/. Adopting c,(+ - - ) = 0 , inserting the result in (2.1) and performing a Wigner transformation
one ends up with (R=(r+r')/2)
I(R,k;t) = -ig/fiz S d3s exp(-iks) S d3r2
(2n)-13 SSS S d3qld3ql ' d k s 2 d 3 q , ' S dw [v(R+s/2-r, )-v(R-s/2-rz ) ]
exp( iql r1 +iq,r2 -iql ' r1 -iq2 ' r2 -iwt)
* . f S S ~ d ~ s ~ d ~ s ~ ' d 3 s 2 d 3 s Z 'SSoo t dt' exp(-iq,sl-iq,s,+iq~ s1 '+iq2 s2 +iotl )
{Q - .fi/2M(ql 2+qz '-q, l Z - q z l ) + ~ E $ - ~ ~ S S ~ ~ S ~ ~ ~ S ~ ~ { ( ( V ( S ~ - S ~ ) - V ( S ~ ~ - ~ ~ ~ ) ~
(~(slsl' ;t' )p(s,szl ; t l )-p(s,s2' ; t l )p(s2sl' ; t l )/g)S3(s3) - ( ~ ( s l - s 3 ) - ~ ( s l ' - s 3 ' ) ) ~ ( ~ I s l ' ; t ' ) ~ ( ~ 3 5 2 ' ; t ' ) p ( ~ ~ S 3 ' ; t ' )
- (v(s2-53)--?(S,'-S3' ))p(s2sz1 ; t t )p(s3sL1 ; t l )p(slSjl ; t l )
+ (v(sx-s3)-v(sz'-S3' ))P(s&' ;t' )p(53S1' it' )Q(szS~' ; t o )/g
+ ( ~ ( ~ 2 - s 3 ) - v ( s l ' - s 3 ' ) ) p ( ~ 3 ~ 2 ' ; t ' ) ~ ( ~ 1 S 3 ' ; t ' ) ~ ( ~ 2 S l ' ; t o / g ) $
S3(53-53' 1 . (2.5)
For reasons of simplicity the elementary nucleon-nucleon interaction v
in (2.5) is assumed to be averaged over spin and isospin and to depend on /r,-r,l only due to translational and rotational invariance. The factor g=4 accounts for spin and isospin degeneracy as in /20/.
Neglecting the exchange term in the 1.h.s. of (2.5), introducing a time-dependent Hartree-mean-field by
and expanding U(R+s/Z;t) and U(R-s/2;t) up to second order in s one directly obtains the 1.h.s. of (1.1) from the 1.h.s. of (2.5) /23,24/
known as the Vlasov equation for Icoll(t)=O.
11.1 - TDHF Versus Vlasov Dynamics
Equation (2.1) together with (2.3a) is equivalent to TDHF 1 In order to compare the quantal time evolution of the one-body phase- space distribution (2.4) with the corresponding result from the 1.h.s.
of (1.1) central collisions of 4 0 ~ a + 4 0 ~ a have been analysed in the intermediate energy regime /22,25/ within three-dimensional TDHF /26/.
The result for f(rl,,k,l;t), which corresponds to (2.4) integrated over degrees of freedom perpendicular to the beam direction, is displayed in Fig. 1 at 20 MeV/u within the various limits also including the full numerical simulation of (1.1) (VUU /3-6/). The Vlasov equation is seen to reproduce roughly the TDHF results, especially the open phase space at k,,*O for strongly overlapping nuclei. This open phase space gets partly filled up during the course of reaction when including residual nucleon-nucleon collisions (r.h.5. of Fig. 1) and proves to be vital in case of energetic photon or pion production by independent nucleon-nucleon collisions /11,22,27/. High momentum components of the nuclear phase- space distribution, which also have to be attributed to energetic nucleons (cf. /25/), however, are missed in the limit (1.1).
11.2 - Momentum Conservation in Single Nucleon-Nucleon Collisions Since the 1.h.s. of (1.1) and (2.5) describes independent single- particle motion in phase space and turns out to be roughly similar, we concentrate on the collision terms in (1.1) and (2.5), respectively.
Introducing the Fourier transforms v(q) = S d3r ~ ( r ) exp(-iqr)
and
P(q1q';t) = SS d3rd3r' exp(-iqr+iqlr') p(rrv;t) the r.h.s. of (2.5) can be cast into the form /20/
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TDHF Vkasov VUU
... ... . . .
Fig.1 The one-body phase-space distribution f(r,,,k,,;t) for a central collision of 4 0 C a + 4 0 ~ a at 20 MeV/u according to TDHF, the Vlasov equation (1.h.s. of (1.1)) as well as the Vlasov- Uehling-Uhlenbeck equation (1.1) within cluster plots. The intensity of the dots increases linearly with f(r,,,kl,;t) while the time is given in units of fm/c
I(R,k;t) = -g/fi2 SSSS d 3 q l d 3 q 1 8 d 3 q 2 d 3 q 2
*v(q2-qz') ex~(i(q~+q,-q,'-q,')R)
*Cs3((ql+ql'+qz-q2')/2-k)-&3((ql+qli-q2+q2')/2-k)}
S-= t dt' exp(-i.K/2M(q12+q2 2-ql I 2-q2 ' )(t-t' ))*C(~H)-~ S d3q ~ ( q )
' C~(ql-q>ql' it' )~(qz+q*qz' ;t' )-p(q1-q1q2' ; t l )p(q2+q,qlt it' )/g +p(qlJq2'-q;tq lql'+q;t8 )/g-p(ql ,q1 l+q;ta )p(q, , % S - q ; t a ) l +(2n) SS d3q d J Q v(q)
* C { ~ ( q i ~ q 1 ' + q ; t ' ) - ~ ( q l - q t q t ' ; t ' ) ) * p ( 2 Q - q 2 ' ,qqr ;t9)p(qz,2Q-9-92, it') +{p(% 3q2'+q;t')-p(q2-q,q2 ;t')j*p(2Q-q,' ,ql ;t1)p(q1 ,2Q-q-qll;t') +{~(q~-qsqz';t' )-p(ql 1 q 2 2 + q ; t 9 )>*p(2Q-q18 ,q18 it' )p(q2,2Q-9-91' ;tS)/g +{p(qz-q,ql' it' )-P(% ,ql'+q;t' )}*~(2Q-q~' ,q2' it' )p(ql ,2Q-q-qZ1 ;tq)/gl)
(2.9) which so far does not involve further approximations.
Performing three-dimensional TDHF calculations for central collisions of 4 0 ~ a + 4 0 ~ a at e.g. 80 MeV/u the phase-space distribution in momentum space (Fig. 2) is seen to be roughly diagonal in q and q' with a width Aq/fm = 0.15-0.3 for q=(O,O,q? and ql=(O,O,q'). When givingup resolution in coordinate space, 1.e. averaging the quantal Wigner function over a volume AV of about 50 fm3 in the dinuclear reaction zone /20/, i.e.
the density matrix may be approximated by
in analogy to the infinite nuclear matter limit where (2.11) holds exactly. Since the momentum transfer q in (2.9) is fixed by (2.11) one introduces a momentum mismatch of order Aq/fm 2 0.15-0.3 by (2.11) which can be neglected in case of hard nucleon-nucleon scattering.
Since v in (2.9) denotes the free nucleon-nucleon interaction its antisymmetrized product in momentum space averaged over spin and isospin may approximately be replaced by the .experimental free nucleon-nucleon differential cross section du/dR /28/ in the Born approximation, i.e.
Using (2.11) and (2.12) the collision term (2.9) reduces to Iav(k;t) = gi'~~/(4n~M~) SSS d 3 q 2 d 3 q 1 1 d 3 q 2 I-= t dt'
2 * ~ 0 s C K / 2 M ( k ~ + q ~ ~ - q ~ ,-qz (t-tt ))-K3 (k+q2-ql -q2 * 1-do/dQ {faV(ql' ; t q )fav(q2' ;tl)?(k;t')r(qz;tt)
with f=(l-fav)
and resembles very closely the Uehling-Uhlenbeck limit (1.1) apart from the additional time integration. One should note, however, that the use of the Born approximation (2.12) is only consistent when defining the mean field U(r;t) in (1.1) by (2.6). This should be kept in mind especially when defining U(r,k;t) via a G-matrix /l/ instead of the free interaction v or using Skyrme type parametrizations /3-6,'.
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Fig.2 The4$ne-body density matrix p(q,qU ;t) for a central collision of C a + 4 n ~ a at 80 Mev/u for q=(O,O,q) and q1=(O,O,q' ) in isometric representation. The time is given in units of fm/c 11.3 - Energy Conservation in Individual Nucleon-Nucleon Collisions The next step in approaching the limit (1.1) is to assume that fav(k;t') is a smooth function of time and can be replaced by fav(k;t) in (2.13). The integral over dt' then gives 21-145
*
&-function in energy and (2.13) readsIav(k;t) = g fi3/(2n3M2 ) S S S d3q2d3ql 'd3q2 ' do/dQ
* S 3 ( k + q 2 - q l l - q Z 1 ) S ( K Z / 2 M ( k 2 + q 2 2 - q 1 1 2 - q 2 2 2 ) ) Cfav(s,' ;t)faV(qZa ;t)j(k;t)f(q2 it)
which is identical to the r.h.s. of (1.1) (with g=4) except for the fact that the quantum mechanical version does not involve the Wigner function itself according to the uncertainty principle and strictly only holds in the infinite nuclear matter limit.
In order to obtain a clear picture on the validity of (2.14) the calculations of Danielewicz /29/ have to be repeated for finite nuclei at intermediate energies, i.e. one has to compare directly the solutions of (2.13) with those of (2.14). Since the infinite nuclear matter limit yields a rather poor approximation for f(k) prior to two-body collisions in the intermediate energy regime the initial conditions for fav(k;t=O) are determined from TDHF calculations /25/.
The average collision terms (2.13) and (2.14) determine the change in time of fav(k;t), i.e.
provided that the gradient of the mean field U(r;t) is close to zero in the reaction zone and energy and momentum of the nucleons follow
the free relation
The latter restrictions could be shown in /25/ to hold sufficiently well in case of intermediate energy nucleus-nucleus collisions.
The actual comparison is performed for the quadrupole moments of the average phase-space distributions
and displayed in Fig. 3 for a heavy-ion reaction at 40 MeV/u. The solution corresponding to the on-shell collision limit (2.14) is given by the full line while the off-shell result from (2.13) is shown by the full dots. The vertical error bars show the range of values obtained for Q,(t) due to different sets of random momenta used in the Monte Carlo integration for (2.13). Similar to the relativistic case /29/ one obtains a retardation of equilibration by off-shell scattering which, however, is only a minor effect on the 15% to 20%
level at intermediate energies. The same picture arizes at all bombarding endrgies from 20 MeV/u to 140 MeV/u indicating that memory effects play a minor role with respect to equilibration though fav(k;t) is a rapidly changing function of time.
Fig. 3 The quadrupole moment Q? (t) in momentum space for a heavy-lon collision at 40 MeV/u according to
solutions of (2.13) for the average o phase-space distribution in the di- t: II
nuclear reaction zone (full dots). The result for Q,(t) from the on- 1 0"
shell limit (2.14) is shown by the 2 0.2 - full line. The vertical error bars g
denote the uncertainty in Q2(t) ac- cording to (2.13) which results from different sets of Monte-Carlo ensem- bles used in the integration over the momentum variables
0 5 10 15 20 25
t (fm/c)
I11 - Probing Higher Order Terms in the Nucleon-Nucleon Interaction In Section I1 the quantal two-body theory has been investigated in the limit (2.3b) for the two-body correlation function which implies a restriction to second order terms in the nucleon-nucleon interaction v(ij) and should be regarded as the quantal off-shell version of the classical VUU approach 1.1). It is not clear, however, if this restriction of the many-body theory holds for high density fermion systems where interactions between a nucleon and a correlated pair of nucleons (2.3e) might become important. In order to explore convergence properties of the many-body theory, equations (2.3a) to (2.3e) are investigated numerically for a one-dimensional problem representing a+a collisions at 80 MeV/u /30/.
In order to integrate (2.1) and (2.3) in time the initial conditions for p and c, as well as the interaction v(x,-X,) have to be specified.
For the present explorative investigation each nucleus is determined by stationary Hartree-Fock with v(x,-X,) given by the sum of a short range repulsive (al=0.4 fm) and a long range attractive (a,=1.5 fm)
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Fig.4 The phase-space distribution f(x,k;t) according to equations (2.3a), (2.3b), (2.3d), and (2.3e) for a one-dimensional problem modelling a+a collisions at 80 MeV/u. The intensity of the dots increases linearly from 0.1 to 1.0 while the time is given in units of 10-'~.5
gaussian interaction. Contrary to three-dimensional problems, where a G-matrix has to be introduced in the mean-field theory in order to achieve nuclear stability /l/, the one-dimensional problem saturates for more elementary interactions due to a favourable scaling of the kinetic energy per nucleon with the Fermi momentum. The single -
particle wave functions (as evaluated from Hartree-Fock) then are boosted with the appropriate relative momentum per nucleon and p(x,xl';t=Oj is determined in the usual way /l/ while c,(t=O)=O is assumed initially.
The actual computations are performed on a grid of size C-15fm, 15fml with 91 meshpoints using conventional Runge-Kutta techniques and exploiting all symmetries of c,(12,112';t) with respect to exchange of particles and complex conjugation. Conservation laws concerning particle number, linear momentum and energy are controlled numerically at each time step and found to be violated by at most a few percent due to the finite gridsize in coordinate space.
The results for the Wigner function f(x,k;t) are shown in Fig. 4 for the limits (2.3a), (2.3b), (2.3d), and (2.3ej in terms of cluster plots. The different times are indicated in units of 1 0 - ~ ~ s while the intensity of the dots increases linearly from 0.1 to 1.0. The results for ( 2 . 3 ~ ) are not shown explicitely since f(x,k;t) turns out to be close to the limit (2.3b). In all cases the system a+a is found to be transparent at the bombarding energy considered, however, with different asymptotic kinetic energy defined by
= lim gfi2/2M S dx j(x;tI2/p(xx;t)
*c011 *.*m
(3.1) with
changes from 144 MeV in case of (2.3a) to 55,65,85, and 100 MeV for (2.3b), (2.3c), (2.3d), and (2.3e), respectively, indicating a considerable change in stopping power for the different approximations since the total energy available in the center-of-mass system is only 160 MeV. The low stopping pgwer in the one-body limit (2.3a) is known from various TDHF calculations for one- and three-dimensional systems while a drastic increase in energy and momentum transfer is also observed in simulations based on the VUU equation (1.1) /31/. This is not surprising since terms of second order in v(ij? ought be attributed to nucleon-nucleon collisions in the classical limit.
However, higher order terms in the two-body interaction, accounted for by [v,c,] in (2.3d), change the time evolution of the system considerably. In classical terms, the two-body collision rate decreases due to higher order terms in the interaction. This becomes even more apparent when considering interactions between a nucleon and a correlated pair of nucleons additionally with all exchange terms in (2.3e). Apparently, expansions in terms of the nucleon-nucleon interaction do not seem to converge in case of violent heavy-ion collisions which represent a strongly interacting many-body system far from equilibrium.
IV - Summary
The quantal description of Yntermediate-energy heavy-ion reactions has been based on coupled equations of motion for the one-body density matrix and the two-body correlation function ((2.1) and (2.3)) as obtained from the density-matrix hierarchy. Restricting to second order in the nucleon-nucleon interaction the Vlasov-Uehling-Uhlenbeck equation (1.1) is derived for a phase-space distribution averaged over the dinuclear reaction zone. The approximations introduced are related with momentum and energy conservation in sirgle nucleon-nucleon
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collisions and found numerically to be approximately valid in case of hard nucleon-nucleon scattering. The restriction to second order processes in the interaction v(ij), however, is found not to hold for intermediate-energy heavy-ion reactions indicating that the nuclear many-body system may be far from the simple limit adopted in (1.1).
The author acknowledges valuable discussions with G.F. Bertsch, P.
Buck, H. Feldmeier, U. Mosel, K. Niita, A. Pfitzner, C. Toepffer ,and S.J. Wang.
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