- Email: [email protected]
Damping of nuclear collective motion in the extended mean-field theory
Nuclear Physiw A370 (1981) 317-328 © North-Holland Publishing Company
DAMPING OF NUCLEAR COLLECTIVE MOTION IN THE EXTENDED MEAN-FIELD THEORY S.
AYIK
Gesellschaft fûr Schwerionenforschung mbH (GSI), D-6100 Darmstadt. FR Germany Received 18 May 1981 (Revised 18 June 1981) Ahehact: The dissipation mechanism in slow nuclear collective motion is studied in the frame of the extended mean-field theory . The collective motion is treated explicitly by employing a travelling single-particle representation in the semi-classical approximation. The rate of change of the collective kinetic energy is determined by: (i) one-body dissipation, which reflects uncorrelated particle-hole excitations as a result of the collisions of particles with the mean field, (ü) two-body dissipation, which consists of simultaneous 2 particle-2 hole excitations via direct coupling of the residual two-body interactions, and (iii) potential dissipation due to the redistribution of the single-particle energies as a result of the random two-body collisions. In contrast to the first two processes the potential dissipation exhibits memory effects due to the large values of the local equilibration times.
1. Introduction Recently, there have been attempts to extend the mean-field theory by including two-body collisions caused by the residual interactions t-s ) . The action of two-body collisions randomizes the single-particle momentum distribution and consequently drives the system towards thermal equilibrium 6''). An additional energy dissipation is brought in by two-body collisions on top of the one-body dissipation contained in standard time-dependent Hartree-Fuck theory . However, within the fully microscopic description of the extended mean-field theory, it is difficult to understand the dissipation and fluctuation mechanism of the collective motion . In order to achieve a better understanding of the dissipation process, it seems that, in spite of losing the detailed microscopic description, the collective motion should be explicitly treated in the frame of the extended mean-field theory . In this case, it appears that dissipative forces are acting on the collective motion, in addition to the force resulting from the mean field. From the structure of the dissipative forces, the different mechanisms contributing to the dissipation then become transparent. The appearance of dissipative forces is a general property of statistical theories e), and it is connected with fluctuating forces acting on the part of the system which is considered explicitly . In the extended mean-field theory, the dissipative forces acting on the collective motion are connected with the mean-field fluctuations, 317
31 8
S. Ayik / Damping of collccHve motion
which are imposed by the fluctuating component of the one-body density matrix as a result of the random behaviour of the residual interactions . We identify the collective kinetic energy of the system by employing the travelling single-particle representation in the semi-classical approximation, and calculate the rate of change of the collective kinetic energy using the extended mean-field equations and taking into account both the thermalization and the mean-field fluctuations . As a result of this calculation it turns out that the damping of the collective energy is simultaneously governed by three different mechanisms : (i) the dissipation of the dynamical potential energy by redistributions of the single-particle energies via two-body collisions 9'1 ~, (ü) the one-body dissipation due to the random collisions of the particles with the mean field 11 ), and (iü) the two-body dissipation due to the direct coupling of the single-particle motion via the two-body residual interactions. In sect . 2, a reminder of the extended mean-field equations is given. The rate of change of the collective kinetic energy is evaluated and the relations with existing theories are discussed in sect. 3 . The results are summarized in sect . 4 . 2. Extended mean-field equations In a recent work 1) we have studied the extension of the mean-field theory by including the residual interactions. Here we give a short summary and recáll some of the basic equations. For details see ref. 1). In the extended mean-field theory the one-particle density matrix~p~'~(t)=p(t) is determined from i
at
P(t) _ [~(t), P(t) ]- iK(P(t)) ,
(2 .1)
where ~fl(t) denotes the mean-field hamiltonian and is defined as z
~fC(t)=2m ~ +Trvp(t)= 2m ~ z +v(t),
(2 .2)
via the two-body interactions v. The second term on the r.h .s . of (2 .1) is usually referred to as the collision term, which, in general, is non-local in time and a complicated functional of p(t) . In the limit of long mean-free path (weak-coupling) the collision term can be simplified to a large extent by neglecting the memory effects. Within the Markov approximation K(p(t)) is given by [see eq . (3 .6) in ref. 1)], K(p(t)) =
m
J0
dT Tr [V(t), [V(t), [VI(t, T), páa~ (t)]] ,
(2 .3)
where pö~ (t) is the anti-symmetrized product of four one-particle density matrices : a
Póa ~(t)=~ ~ P(rk. rk ; t) ~ k~l
(2 .4)
S. Ayik / Damping of collective motion
31 9
In (2 .3) the trace is taken over all the coordinates except one, so that K(p(t)) is a one-body operator . The residual interactions are the traceless part of the two-body interaction v and denoted by V(t) = v - v(t) and VI(t, z) is the residual interaction in the interaction representation, v~(t,
T)
= 8(T)V(t)8+(T) ,
(2 .5)
with the modified mean-field propagator g(T), which is given by [ref . 1), eq. (3 .7)] g(T) =
exp {-iT(~(C(t) - il'(t))} .
(2 .6)
Here, the first factor is the mean-field propagator and the second factor is the modification arising from the residual interaction which determines the decay of the propagator . In relation to the Markov approximation, it should be pointed out that the collision term (2 .3) essentially involves two different characteristic times. These are the duration time, Ta-~-lí/d, of a two-body collision and the decay time, Ta ^- fi/T, of the propagator which corresponds to the mean-free path of the nucleons . In low-energy heavy-ion collisions, rA is larger than the nucleon transit time R/vF (R is the nuclear dimension, va the Fermi velocity) and it varies from several -zz x 10 -21 s to a few x 10 s as temperature increases from T = 0 to T = 2-3 MeV lZ)] . [refs. 1 °The duration time of a two-body collision is determined by the energy width d, which describes the strength distributions of the màtrix elements of the residual interactions as a function of the off-shell energy . The on-shell matrix elements V~a, .,~a, with w = e~ + ep - e,. - ea = 0, have the largest strength, and the strength of the matrix elements decreases as ~~~ increases. From a numerical calculation of the matrix elements performed by employing the harmonic oscillator is) wave functions we estimate that for a heavy nucleus d ~ h~o ^~ 7 MeV (wo is the oscillator frequency), which gives -rv --10-ZZ s, and indicates that the weak-coupling limit (TdK Ta) is a good approximation in low-energy nuclear reactions . In the weak-coupling limit rc is the smallest characteristic time in the collision term and introduces an implicit cut-off to the T-integration in (2 .3) . In this case memory effects can be ignored (Markov approximation) by neglecting the variations in ~fC(t), l'(t) and p (t) during a time interval of order z -- . Te, and consequently, the more complicated expression for the collision term takes the form of (3 .2). The solution of (2 .1) for the density matrix is usually considered in the timedependent Hartree-Fork representation'-3 ), which, however, is not very well suited for a separation of the collective and the intrinsic energies of the system. Instead we introduce the travelling representation, which approximately provides an identification of the collective kinetic energy in the semi-classical limit. The travelling single-particle representation is defined by the eigenstates of the self-consistent cranking hamiltonian, ~(t) _ ~l(t) - i a/ar, (~(e(r)-t a/ar)I+~~(t)) = Éa(r)I ~Ga(t)) .
(2 .7)
In the semi-classical approximation, the time dependence of the mean-field is
320
S. Ayik / Damping of collccdve motion
determined by a set of collective parameters q(t), ~fC(t) _ ~C(q(t)) (here we consider one such parameter), which may be specified by the expectation values of some deformation operators q, q (t) _ (q"") . In this case, the time dependence of the travelling wave functions and the energies are specified by q(t) and all higher order derivatives 4(t),4(t), . . ., i.e . ~~Ga(t))=~~G~(q4 ~ ~ ~)) and Éa(t)=êa(g4 ~ ~ ~) . In the limit of slow collective motion, neglecting the dependence of the wave functions and the energies on q(t) and higher order derivatives, and consequently replacing a/ar by 4 8/aq, (2 .7) can be written as [cf. eqs. (3 .6) and (3 .7)] : (~(q) - i4 ala4)I~G~(g4))=Éa(49)~~G~(44)) ~
(2 .8)
We expand the one-particle density matrix in terms of the complete set of the travelling states, ~á(t))=-~((r~(t)), P(t) _ ~ e-l+te(t)lá(t))Pa~(t)(ß(t)I ~a
e 1n~(r)
~
(2 .9)
with n~ (t) = f` dt' ë~ (t') . Theoretical approaches up to now have neglected the off-diagonal elements, pap (t) (a ~ ß), and derived a master equation for the diagonal z _s elements, pa~(t)=pa(t), which correspond to the occupation probabilities ) . The master equation reads [see eq . (3 .10) in ref. 1)], á
P~(t)=-2ra(t)Pa(t)+~ Wa9.rs(t)Pr(t)Pa(t)P~(t)Pß(t) ~
(2 .10)
where p-~ (t) = 1 - pa (t), W~a ..~ (t) is the transition probabilities given in terms of the residual interactions and l'a (t) _ ~ ~ Waa.Ya (t)Pa(t)Py (t)Pa (t)
(2 .11)
is the decay width of the particle state ßá(t)) into all possible 2plh configurations . The expectation value of a one-body operator 0 can then approximatély be evaluated as (O)=Tr ~p(t)=~ (á(t)IO~a(t))P~(t) ~
(2 .12)
which reflects the neglect of the off-diagonal elements paa (t) _ (Sp (t))aa and consequently the neglect of the mean-field fluctuations . In this contribution we want to stress that the approximation (2 .12) should be refined in order to include the effects of the mean-field fluctuations connected with the off-diagonal elements of the density matrix . The off-diagonal part, Sp(t), of the density matrix, which may be considered as a fluctuating quantity with zero mean, introduces a fluctuating component into the mean field, Sv(t)=Tr v8p(t), according to (2 .2). In this case the single-particle wave functions determined by (2 .7) have a fluctuating component ~Sá(t))acSv(t), which is proportional to Sv(t) in the lowest order. The quadratic
S. Ayik / Damping of collective motion
32 1
contributions in Sp(t) to the expectation value (2 .12) do not vanish, and may in fact grow in time . For large mean-field fluctuations (or off-diagonal elements) (2 .12) will then no longer be a good approximation to the expectation values . One way to take account of the off-diagonal elements is to calculate the expectaia) tion values in the spirit of the time-dependent RPA or to calculate the density matrix in the path integral approach' s). This corresponds to the inclusion of quantal fluctuations of the mean field via virtual phonon excitations. In contrast to these approaches, we are here mostly interested in the statistical fluctuations of the mean field. Although it is difficult to calculate the off-diagonal elements, p~a (t), explicitly, they are in principle contained in the equation of motion (2 .1). Instead of calculating the expectation values directly, we prefer to calculate the rate of change of the expectation values, d(0)/dt, using the extended mean-field equation (2 .1). In this way, the effects of the thermalization and the mean-field fluctuations on the expectation values are included implicitly . 3. Damping of collective energy A differential equation for the rate of change of the expectation value of a one-body operator ~(t), which might explicitly depend on time, can be derived by taking time derivatives on both sides of (2.12) : d
dr
(~(r))=Tr~(t)
e
ar
p(t)+Tra~(r) P(r)~ ar
(3 .1)
Using (2 .1) and the cyclic invariance of a trace, (3 .1) can be written as
- ~ dT J0
QL~,
v(r)l~ s(T) v(r)g+(T)~)o ,
(3 .2)
with H the total hamiltonian. Here we introduce the averages (A)°, defined by (A)o ~TrApö ~(t),
(3 .3)
where k is the rank of the operator, A, and pö ~ (t) is the uncorrelated k-particle density matrix defined similar to (2 .4). We want to apply eq . (3 .2) in order to calculate the rate of change of the collective kinetic energy . For this aim we have to define the operator corresponding to the collective kinetic energy of the system . In the semi-classical approximation, the operator corresponding to the collective kinetic energy may be identified with the help of the travelling representation (2 .8) in the following way. We consider the quasi-static single-particle wave functions, defined by
32 2
S. Ayik / Damping of collective motion
and introduce the operator T of infinitesimal displacement of the quasi-static states, ~« (t)) s ~~~ (t)), as ~~tx (t)) _ +i
at
~tx (t)) .
(3 .5)
In the semi-classical approximation the time dependence of the mean field is determined by a set of parameters q(t) [see eq . (2 .8)] and consequently the quasi-static wave functions and energies depend parametrically on time via q(t) . In this case, the infinitesimal displacement operator goes as T = iq(t) a/aq . The travelling wave functions and the energies may be given, approximately, in terms of the quasi-static representation 1°'16'") :
c~ (t) = t;~ (q)
- ZP z(a (q)IM-1
Ia (q)) .
(3 .7)
with
Here the parameter p(t) is identified as the wnjugate momentum and it is related to q(t) via the condition (3 .9) which can be evaluated using (3 .6) and considering the expansion
in powers of p. The expectation value of the first term vanishes due to time-reversal symmetry and the higher order terms do not contribute . Hence, (3 .9) becomes 4(t) =P(t)M-1(q) ,
(3 .11)
M-' (q) = E ~~ (q)IM-1 Ia (q))P~ (t) ~
(3 .12)
with the mass parameter
Employing the approximate expression (3 .6) for the travelling wave functions and assuming that [R~, i~]=1, ~ = i a/aq, it can be seen that the expectation value T(t) of ß` over the travelling states, T(t) _ ~ (ä (t)I ~I~(t))Pa (t) , a
(3 .13)
T(t) = PZM-1 (q) ,
(3 .14)
gives twice the collective kinetic energy l')
which indicates that the operator of the collective kinetic energy may be identified
S. Ayik / Damping of collective motion
32 3
with Tom, and the rate of change of the collective kinetic energy may be calculated using (3 .2) with ~ = ß: It should be stressed that the mean-field propagator g(T) in the collision term of (3 .2) is not diagonal in the travelling representation (3 .6), due to the fact that the travelling states are not eigenstates of ~l(t). In order to take the off-diagonal contributions into account we expand the collision term in powers of the off-diagonal part of the mean-field hamiltonian, ~off(t) = ~ ~á(t))~Qa(t)(ß(t)I , ~~a using 8(T)V(r)B+(T)=8a(r){V(t)-iT[~, V(t) ]+ . . ~}8á (T) "
(3 .15)
Here, ~(lo,~ is replaced by ~, since according to (2 .8) their off-diagonal elements are equivalent, and gd(T) is the propagator corresponding to the diagonal part of the hamiltonian, ~fCa(t) = E~ ~á(t))éea (t)(á (t)~, and is defined via (2 .6), with ~(t) replaced by ~rCa(t). Then the equation for T(t) _ (1) up to the second order in fi (or collective velocity 4) becomes T(t) - (afilat)o +( -i[fi, H])odr J
ao dT([[~,
V(t)] . 8a(T) V(r)8á (T)])o
+~~ dT lT( [[l s V(t)]r gd(T)[~~ V(t)]ód (T)])0 0
"
(3 .16)
The terms on the r.h .s. can be evaluated explicitly in the travelling representation (3 .6) by noting that ! = 4P is acting on the wave functions only and by observing l') (~n(t)I [~, H]I ~n(r)) =-_
aq
=-'áq
(4'n(q)I
e-tPQH
e~°dl~n(q))
(~n(q)I{H+iPzM-1}I~R(q)),
(3 .17)
(~~(t)~[P, ~(r)]~~m(r)> =- =~q(~~(q)I~(r)I~m(q))+i(~n(q)I av/agl~m(q)>, (3 .18) where ~~n(q)) and ~~m(q)j are Slater determinants of the quasi-static single-particle states ~a(q)) . The second line of (3 .17) follows the expansion in powers of p; the higher order terms do not contribute, the term linear in p vanishes due to the time-reversal symmetry and only the mass contribution remains. It is important to note that although V(t) = v - v(t) is a traceless two-body operator, i.e . Tr V(t)p(t) _ 0, this is not true for [~, V(t)], and as can be seen from. the second term on the
32 4
S. Ayik / Damping of collective motion
r.h.s. of (3 .18) it has a non-vanishing one-body part. Collecting the terms together (3.16) becomes t)+dU(q, t)l-LQl(t)+~z(t)l , dt ~pz~2M(q)~=-4a4 ~U(q, where, in the first term U(q, t) is the potential energy (intrinsic energy) U(q, t)=~ (txl - o2i2mla)Pa(t)+i ~ vas.~a(q)Pa(t)Pa(t) , or U(q, t)=E Ea(q)Pa(t) -i E vas.ap(q)P~(t)Pa(t) ,
(3.19)
(3.20)
and dU(q, t) is a correction to the potential energy, due to the virtual transitions imposed by the residual interactions dU(q,t)=á~lVaa .~s(q)Iz
tv~9 .raz Po(t)PB(t)PY(t)Pa(t), ~ ap.ra + raBvó
(3.21)
with p,,(t) = 1 - py (t), arising from the third term on the r.h.s. of (3.16) . The quantities ~1(t) and ~2(t) are given by a a Ql(t) = 4Z E I(~ I aq Y(q)Iß)IZ ~ ~~Pa(t) 2 , awaa waa + r~ß z a ~ z 21'~a,,2 ~Pa(t)Pa(t)PY(t)Pa(t), Qz(t)=á4 z ~ I V~a.ya(q)I aq aw~a.~ ~~t= .,~a+I'~tl,~a
a
(3.22)
(3.23)
with raa (t)=ra ft)+rß (t), r~a,~(t>=r~(t>+rß(t)+r,,(t)+ra(t>
waa(t)=~re~(t)-~a(t),
w~a.~(t) =~C~(t)+~ea(t)-~e,,(t)-~ea(t),
where TQ (t) and ~1~, (t) = taQ (q) +zpz(a IM-11~ ) are the width and the energies of the single-particle states . According to (3.19), the rate of change of the collective kinetic energy is determined by two different contributions. The first term describes the reversible changes in the potential energy (U + d U), which is reversible in the sense that, if the occupation probabilities are held constant, the energy flows back into the kinetic energy. On the other hand, the second term (~1 + ~1z) determines the irreversible changes of the collective kinetic energy as a result of the coupling of the particle motion with the fluctuating mean field. The damping mechanism due to the random collisions of the particles with the mean field, which results from the uncorrelated particle-hole excitations around the Fermi surface, is reflected by the quantity Ql(t) in (3.19) . The ~1(t) dissipation is equivalent to the rate of energy dissipation formula of one-body dissipation theory "'18'19), and it may be attributed to the shape fluctuations of the mean field. The quantity l~z(t) in (3.19) describes the dissipation as a result of the simultaneous 2p2h excitations
S. Ayik / Damping of collective motion
325
via direct coupling of the residual interactions and is referred to as the two-body dissipation . In contrast to ~1i(t), which is a surface effect, Qz(t) describes a volume dissipation and it may be considered as arising from the local fluctuations of the mean field. The rates ~i and ~1s correspond to one-body and two-body friction forces, which are proportional to the collective velocity 4. In the limit of vanishing single-particle widths, T~, -~ 0, the friction coefficients [the coefficients of 4z in (3 .22) and (3 .23)] do not vanish in general, due to the crossings of the energy levels as a function of q. This is in disagreement with the result obtained in ref. z° ) where the friction forces proportional to 4 vanishes in the weak-coupling limit. The result of ref. 2°) may be due to the fact that the crossings (or near crossings) of the many-body adiabatic levels are not taken into consideration. The dissipation mechanism contained in (3 .19) is further complicated by the fact that the force, F(q, t) =-aU(q, t)/aq, evolves in a time irreversible manner as a result of the time dependence of the occupation probabilities [cf. master eq . (2 .10)]. The irreversible behaviour of F(q, t) means that the energy stored in the dynamical potential can be dissipated into the intrinsic degrees of freedom by redistributions of the occupa9' tion probabilities . °). This dissipation mechanism can be understood better by explicitly considering the coupling between the rate equation (3 .19) and the master equation (2 .10) . Using the master equation, an equation for the force can be derived. If the mean field is practically constant in the nuclear interior, the force on the collective variable is mainly determined by the q-dependence of the singleparticle energies . Consequently, neglecting the second term on the r.h .s . of (3 .20), the force may be approximated by
ea(q)P~(t) r)=-~aq Denoting the average value of a2 e~(q)/eq Z by s"(q), we have F(q,
r) dtF(q, r)+4E"(q)=-2I'(t)[F(q, -Fw(q, t)J,
(3 .24)
where the r.h .s. is obtained from the master equation, replacing l'Q (t) by an average decay width I'(t) and approximating the gain term by its equilibrium value. This approximation is equivalent to a relaxation ansatz with the local equilibrium time, T,«(t) = 2/l'(t) [ref . 6'1~] . Assuming F«,(q, t) is a smooth function of time ; the solution of (3 .24) can be given as F(t) =- J~dr'x(t, °
t')4(r~)e"(t')+F~a(r)[1-x(t, 0)J+F(0)x(r, 0),
(3 .25)
with X(t, t~)=exP {-(t-t~)lT~a(t)} ~
(3 .26)
The local equilibration times in low-energy dissipatioe heavy-ion collisions are
32 6
S. Ayik l Damping of collective motion
estimated using (2 .11) within a Ferrai gas model in ref. t° ). According to these estimates, T,oo(t) is very large during the initial phase of the collision, T,oo(t) ^~ -zt several x 10 s, and it becomes considerably smaller during the strongly damped phase, r,~(t) ^- 5 x 10 -zz s. Therefore, during the initial phase, the occupation probabilities remain unchanged for a long time specified by t « T,o~(t) and consequently (3 .25) acts as a conservative force,
F(q)=-~aq
Ea(q)P~(0),
with no explicit time dependence . The energy dissipation during this time interval is mainly determined by the uncorrelated ph excitations and the correlated 2p2h excitations as described by ~1t(t) and ~Z(t), respectively, see (3 .22) and (3 .23) . For later times F(q, t) contributes to the damping via dissipation of the dynamical potential energy which is stored during the initial phase. The potential dissipation exhibits memory effects due to the large value of the local equilibration times. This gives a non-local friction force, the first term on the r.h.s . of (3 .25), as pointed out in a recent work of Nörenberg t° ), in contrast to the local friction forces contained in Qt+QZ. Various dissipation mechanisms described by (3 .19) are illustrated in fig. 1 . Fermi ~ level
collective varinble Fig . 1 . (a) Uncorrelated ph excitations (one-body dissipation due to collisions of particles with the mean field . (b) Correlated 2p2h excitations (two-body dissipation) via the residual interactions . (c) Potential dissipation due to redistributions of the intrinsic energies via two-body collisions .
4. Conclusions In the extended mean-field theory the inclusion of the residual interactions modifies the motion of the particles in two ways . (i) The action of two-body collisions randomizes the single-particle momentum distributions and hence drives the system
S. Ayík / Damping of collective motion
32 7
towards the thermal equilibrium. (ü) The residual interactions introduce a fluctuating component to the mean field via the off-diagonal elements of the density matrix . In the present work, in order to bring out these effects, the collective motion is treated explicitly by employing the travelling single-particle representation, which allows an identification of the collective variables in the semi-classical approximation . The rate of change of the collective energy is calculated using the extended mean-field equations. For the rate of energy dissipation a simple expression is obtained [see eq . (3 .19)], d e [PZ/2M(q) ]= -4 aq U(q, t)-Ql(t)-~z(t), dr
(4.1)
in which dU is neglected and U, !~ 1 and ~z are given by (3 .20), (3 .22) and (3 .23), respectively . The quantity Ql(t) is the rate of energy dissipation due to the ph excitations around the Fermi surface and is essentially equivalent to the rate calculated in the one-body dissipation theory 18'19) (fig . 1) . On the other hand, Qz(t) describes the dissipation as a result of the direct coupling of the single-particle states via the residual interactions (2p2h excitations) and it is identified as the two-body dissipation . The two-body dissipation is expected to be small in lowenergy heavy-ion collisions and may be neglected. The first term on the r.h.s . of (4 .1) determines the rate of change of the potential energy U(q, t), which also contributes to the dissipation, due to the fact that at every action of random two-body collisions some part of the correlated intrinsic energy is converted into uncorrelated energy (heat) . The potential dissipation has memory effects as a result of the large value of the local equilibration times and consequently appears as a non-local friction force 1°), see eq . (3 .25) . Numerical calculations of the one-body and the two-body friction coefficients based on (3 .22) and (3 .23) are presently in progress . The remaining problem to be studied is connected with the fluctuations of the collective variables. There is some evidence that the random two-body collisions are not responsible for the large dispersions in the dissipatioe heavy-ion collisions . The present work however indicates that the mean-field fluctuations should be included, in order to achieve a better understanding of the fluctuations of the collective variables. This point will be studied in a forthcoming paper. The author is grateful to W. Nörenberg and W. Casing for many valuable discussions and for a critical reading of the manuscript.
References 1) S. Ayik, Z. Phys. A298 (1980) 83 2) C.Y. Wong and H.F. Tang, Phys. Rev. Lett . 40 (1978) 1070 3) H. Onland and R. Schaoffer, Z. Phys . A290 (1979) 191
32 8 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
S. Ayik / Damping of collective motion G. Mantzouranis and H.C. Pauli, Z. Phys . A281 (1977) 165 P. Grangé, H.A . Weidenmüller and G. Wolschin, preprint MPI-H-1981-V8 H.S . Köhler, Nucl. Phys . A343 (1980) 315 P. Grangé, J. Richert, G. Wolachin and H.A . Weidenmüller, Nucl. Phys. A3S6 (1981) 260 H.A . Weidenmüller, in Progress in particle and nuclear physics, vol. 3, ed . D.H . Wilkinson (Pergamon, Oxford, 1980) W. Nörenberg and C. Riedel, Z. Phys . A290 (1979) 385 W. Nörenberg, Proc . Int. Workshop IX on gross properties of nuclei and nuclear excitations, Hirschegg, Austria (1981) ; Phys . Lett ., in press S.E . Koorvin, Proc . Int. School of Physics, Enrico Ferrai, Varenna, Italy (1979) P. Morel and P. Nozières, Phys . Rev. 126 (1962) 1909 G. Mantzouranis and H.C . Pauli, Phys . Rev. C20 (1980) 1550 P.G . Rembard, K. Gceke and K.Y. Casson, Z. Phys . A295 (1980) 45 H. Reinhardt, Nucl. Phys. A346 (1980) 1 H.C . Pauli and L. Whets, Z. Phys . A277 (1976) 83 F. Villara, Nucl . Phys . A285 (1977) 269 S.E. Koorvin, R.L. Hatch and J. Randrop, Nucl . Phys. A283 (1977) 87 H. Hofmarvn and P.J . Siemens, Nucl . Phys . A257 (1976) 165; A27S (1977) 464 M.C . Names and H.A. Weidenmüller, preprint MPI-H-1981-V13
DAMPING OF NUCLEAR COLLECTIVE MOTION IN THE EXTENDED MEAN-FIELD THEORY S.
AYIK
Gesellschaft fûr Schwerionenforschung mbH (GSI), D-6100 Darmstadt. FR Germany Received 18 May 1981 (Revised 18 June 1981) Ahehact: The dissipation mechanism in slow nuclear collective motion is studied in the frame of the extended mean-field theory . The collective motion is treated explicitly by employing a travelling single-particle representation in the semi-classical approximation. The rate of change of the collective kinetic energy is determined by: (i) one-body dissipation, which reflects uncorrelated particle-hole excitations as a result of the collisions of particles with the mean field, (ü) two-body dissipation, which consists of simultaneous 2 particle-2 hole excitations via direct coupling of the residual two-body interactions, and (iii) potential dissipation due to the redistribution of the single-particle energies as a result of the random two-body collisions. In contrast to the first two processes the potential dissipation exhibits memory effects due to the large values of the local equilibration times.
1. Introduction Recently, there have been attempts to extend the mean-field theory by including two-body collisions caused by the residual interactions t-s ) . The action of two-body collisions randomizes the single-particle momentum distribution and consequently drives the system towards thermal equilibrium 6''). An additional energy dissipation is brought in by two-body collisions on top of the one-body dissipation contained in standard time-dependent Hartree-Fuck theory . However, within the fully microscopic description of the extended mean-field theory, it is difficult to understand the dissipation and fluctuation mechanism of the collective motion . In order to achieve a better understanding of the dissipation process, it seems that, in spite of losing the detailed microscopic description, the collective motion should be explicitly treated in the frame of the extended mean-field theory . In this case, it appears that dissipative forces are acting on the collective motion, in addition to the force resulting from the mean field. From the structure of the dissipative forces, the different mechanisms contributing to the dissipation then become transparent. The appearance of dissipative forces is a general property of statistical theories e), and it is connected with fluctuating forces acting on the part of the system which is considered explicitly . In the extended mean-field theory, the dissipative forces acting on the collective motion are connected with the mean-field fluctuations, 317
31 8
S. Ayik / Damping of collccHve motion
which are imposed by the fluctuating component of the one-body density matrix as a result of the random behaviour of the residual interactions . We identify the collective kinetic energy of the system by employing the travelling single-particle representation in the semi-classical approximation, and calculate the rate of change of the collective kinetic energy using the extended mean-field equations and taking into account both the thermalization and the mean-field fluctuations . As a result of this calculation it turns out that the damping of the collective energy is simultaneously governed by three different mechanisms : (i) the dissipation of the dynamical potential energy by redistributions of the single-particle energies via two-body collisions 9'1 ~, (ü) the one-body dissipation due to the random collisions of the particles with the mean field 11 ), and (iü) the two-body dissipation due to the direct coupling of the single-particle motion via the two-body residual interactions. In sect . 2, a reminder of the extended mean-field equations is given. The rate of change of the collective kinetic energy is evaluated and the relations with existing theories are discussed in sect. 3 . The results are summarized in sect . 4 . 2. Extended mean-field equations In a recent work 1) we have studied the extension of the mean-field theory by including the residual interactions. Here we give a short summary and recáll some of the basic equations. For details see ref. 1). In the extended mean-field theory the one-particle density matrix~p~'~(t)=p(t) is determined from i
at
P(t) _ [~(t), P(t) ]- iK(P(t)) ,
(2 .1)
where ~fl(t) denotes the mean-field hamiltonian and is defined as z
~fC(t)=2m ~ +Trvp(t)= 2m ~ z +v(t),
(2 .2)
via the two-body interactions v. The second term on the r.h .s . of (2 .1) is usually referred to as the collision term, which, in general, is non-local in time and a complicated functional of p(t) . In the limit of long mean-free path (weak-coupling) the collision term can be simplified to a large extent by neglecting the memory effects. Within the Markov approximation K(p(t)) is given by [see eq . (3 .6) in ref. 1)], K(p(t)) =
m
J0
dT Tr [V(t), [V(t), [VI(t, T), páa~ (t)]] ,
(2 .3)
where pö~ (t) is the anti-symmetrized product of four one-particle density matrices : a
Póa ~(t)=~ ~ P(rk. rk ; t) ~ k~l
(2 .4)
S. Ayik / Damping of collective motion
31 9
In (2 .3) the trace is taken over all the coordinates except one, so that K(p(t)) is a one-body operator . The residual interactions are the traceless part of the two-body interaction v and denoted by V(t) = v - v(t) and VI(t, z) is the residual interaction in the interaction representation, v~(t,
T)
= 8(T)V(t)8+(T) ,
(2 .5)
with the modified mean-field propagator g(T), which is given by [ref . 1), eq. (3 .7)] g(T) =
exp {-iT(~(C(t) - il'(t))} .
(2 .6)
Here, the first factor is the mean-field propagator and the second factor is the modification arising from the residual interaction which determines the decay of the propagator . In relation to the Markov approximation, it should be pointed out that the collision term (2 .3) essentially involves two different characteristic times. These are the duration time, Ta-~-lí/d, of a two-body collision and the decay time, Ta ^- fi/T, of the propagator which corresponds to the mean-free path of the nucleons . In low-energy heavy-ion collisions, rA is larger than the nucleon transit time R/vF (R is the nuclear dimension, va the Fermi velocity) and it varies from several -zz x 10 -21 s to a few x 10 s as temperature increases from T = 0 to T = 2-3 MeV lZ)] . [refs. 1 °The duration time of a two-body collision is determined by the energy width d, which describes the strength distributions of the màtrix elements of the residual interactions as a function of the off-shell energy . The on-shell matrix elements V~a, .,~a, with w = e~ + ep - e,. - ea = 0, have the largest strength, and the strength of the matrix elements decreases as ~~~ increases. From a numerical calculation of the matrix elements performed by employing the harmonic oscillator is) wave functions we estimate that for a heavy nucleus d ~ h~o ^~ 7 MeV (wo is the oscillator frequency), which gives -rv --10-ZZ s, and indicates that the weak-coupling limit (TdK Ta) is a good approximation in low-energy nuclear reactions . In the weak-coupling limit rc is the smallest characteristic time in the collision term and introduces an implicit cut-off to the T-integration in (2 .3) . In this case memory effects can be ignored (Markov approximation) by neglecting the variations in ~fC(t), l'(t) and p (t) during a time interval of order z -- . Te, and consequently, the more complicated expression for the collision term takes the form of (3 .2). The solution of (2 .1) for the density matrix is usually considered in the timedependent Hartree-Fork representation'-3 ), which, however, is not very well suited for a separation of the collective and the intrinsic energies of the system. Instead we introduce the travelling representation, which approximately provides an identification of the collective kinetic energy in the semi-classical limit. The travelling single-particle representation is defined by the eigenstates of the self-consistent cranking hamiltonian, ~(t) _ ~l(t) - i a/ar, (~(e(r)-t a/ar)I+~~(t)) = Éa(r)I ~Ga(t)) .
(2 .7)
In the semi-classical approximation, the time dependence of the mean-field is
320
S. Ayik / Damping of collccdve motion
determined by a set of collective parameters q(t), ~fC(t) _ ~C(q(t)) (here we consider one such parameter), which may be specified by the expectation values of some deformation operators q, q (t) _ (q"") . In this case, the time dependence of the travelling wave functions and the energies are specified by q(t) and all higher order derivatives 4(t),4(t), . . ., i.e . ~~Ga(t))=~~G~(q4 ~ ~ ~)) and Éa(t)=êa(g4 ~ ~ ~) . In the limit of slow collective motion, neglecting the dependence of the wave functions and the energies on q(t) and higher order derivatives, and consequently replacing a/ar by 4 8/aq, (2 .7) can be written as [cf. eqs. (3 .6) and (3 .7)] : (~(q) - i4 ala4)I~G~(g4))=Éa(49)~~G~(44)) ~
(2 .8)
We expand the one-particle density matrix in terms of the complete set of the travelling states, ~á(t))=-~((r~(t)), P(t) _ ~ e-l+te(t)lá(t))Pa~(t)(ß(t)I ~a
e 1n~(r)
~
(2 .9)
with n~ (t) = f` dt' ë~ (t') . Theoretical approaches up to now have neglected the off-diagonal elements, pap (t) (a ~ ß), and derived a master equation for the diagonal z _s elements, pa~(t)=pa(t), which correspond to the occupation probabilities ) . The master equation reads [see eq . (3 .10) in ref. 1)], á
P~(t)=-2ra(t)Pa(t)+~ Wa9.rs(t)Pr(t)Pa(t)P~(t)Pß(t) ~
(2 .10)
where p-~ (t) = 1 - pa (t), W~a ..~ (t) is the transition probabilities given in terms of the residual interactions and l'a (t) _ ~ ~ Waa.Ya (t)Pa(t)Py (t)Pa (t)
(2 .11)
is the decay width of the particle state ßá(t)) into all possible 2plh configurations . The expectation value of a one-body operator 0 can then approximatély be evaluated as (O)=Tr ~p(t)=~ (á(t)IO~a(t))P~(t) ~
(2 .12)
which reflects the neglect of the off-diagonal elements paa (t) _ (Sp (t))aa and consequently the neglect of the mean-field fluctuations . In this contribution we want to stress that the approximation (2 .12) should be refined in order to include the effects of the mean-field fluctuations connected with the off-diagonal elements of the density matrix . The off-diagonal part, Sp(t), of the density matrix, which may be considered as a fluctuating quantity with zero mean, introduces a fluctuating component into the mean field, Sv(t)=Tr v8p(t), according to (2 .2). In this case the single-particle wave functions determined by (2 .7) have a fluctuating component ~Sá(t))acSv(t), which is proportional to Sv(t) in the lowest order. The quadratic
S. Ayik / Damping of collective motion
32 1
contributions in Sp(t) to the expectation value (2 .12) do not vanish, and may in fact grow in time . For large mean-field fluctuations (or off-diagonal elements) (2 .12) will then no longer be a good approximation to the expectation values . One way to take account of the off-diagonal elements is to calculate the expectaia) tion values in the spirit of the time-dependent RPA or to calculate the density matrix in the path integral approach' s). This corresponds to the inclusion of quantal fluctuations of the mean field via virtual phonon excitations. In contrast to these approaches, we are here mostly interested in the statistical fluctuations of the mean field. Although it is difficult to calculate the off-diagonal elements, p~a (t), explicitly, they are in principle contained in the equation of motion (2 .1). Instead of calculating the expectation values directly, we prefer to calculate the rate of change of the expectation values, d(0)/dt, using the extended mean-field equation (2 .1). In this way, the effects of the thermalization and the mean-field fluctuations on the expectation values are included implicitly . 3. Damping of collective energy A differential equation for the rate of change of the expectation value of a one-body operator ~(t), which might explicitly depend on time, can be derived by taking time derivatives on both sides of (2.12) : d
dr
(~(r))=Tr~(t)
e
ar
p(t)+Tra~(r) P(r)~ ar
(3 .1)
Using (2 .1) and the cyclic invariance of a trace, (3 .1) can be written as
- ~ dT J0
QL~,
v(r)l~ s(T) v(r)g+(T)~)o ,
(3 .2)
with H the total hamiltonian. Here we introduce the averages (A)°, defined by (A)o ~TrApö ~(t),
(3 .3)
where k is the rank of the operator, A, and pö ~ (t) is the uncorrelated k-particle density matrix defined similar to (2 .4). We want to apply eq . (3 .2) in order to calculate the rate of change of the collective kinetic energy . For this aim we have to define the operator corresponding to the collective kinetic energy of the system . In the semi-classical approximation, the operator corresponding to the collective kinetic energy may be identified with the help of the travelling representation (2 .8) in the following way. We consider the quasi-static single-particle wave functions, defined by
32 2
S. Ayik / Damping of collective motion
and introduce the operator T of infinitesimal displacement of the quasi-static states, ~« (t)) s ~~~ (t)), as ~~tx (t)) _ +i
at
~tx (t)) .
(3 .5)
In the semi-classical approximation the time dependence of the mean field is determined by a set of parameters q(t) [see eq . (2 .8)] and consequently the quasi-static wave functions and energies depend parametrically on time via q(t) . In this case, the infinitesimal displacement operator goes as T = iq(t) a/aq . The travelling wave functions and the energies may be given, approximately, in terms of the quasi-static representation 1°'16'") :
c~ (t) = t;~ (q)
- ZP z(a (q)IM-1
Ia (q)) .
(3 .7)
with
Here the parameter p(t) is identified as the wnjugate momentum and it is related to q(t) via the condition (3 .9) which can be evaluated using (3 .6) and considering the expansion
in powers of p. The expectation value of the first term vanishes due to time-reversal symmetry and the higher order terms do not contribute . Hence, (3 .9) becomes 4(t) =P(t)M-1(q) ,
(3 .11)
M-' (q) = E ~~ (q)IM-1 Ia (q))P~ (t) ~
(3 .12)
with the mass parameter
Employing the approximate expression (3 .6) for the travelling wave functions and assuming that [R~, i~]=1, ~ = i a/aq, it can be seen that the expectation value T(t) of ß` over the travelling states, T(t) _ ~ (ä (t)I ~I~(t))Pa (t) , a
(3 .13)
T(t) = PZM-1 (q) ,
(3 .14)
gives twice the collective kinetic energy l')
which indicates that the operator of the collective kinetic energy may be identified
S. Ayik / Damping of collective motion
32 3
with Tom, and the rate of change of the collective kinetic energy may be calculated using (3 .2) with ~ = ß: It should be stressed that the mean-field propagator g(T) in the collision term of (3 .2) is not diagonal in the travelling representation (3 .6), due to the fact that the travelling states are not eigenstates of ~l(t). In order to take the off-diagonal contributions into account we expand the collision term in powers of the off-diagonal part of the mean-field hamiltonian, ~off(t) = ~ ~á(t))~Qa(t)(ß(t)I , ~~a using 8(T)V(r)B+(T)=8a(r){V(t)-iT[~, V(t) ]+ . . ~}8á (T) "
(3 .15)
Here, ~(lo,~ is replaced by ~, since according to (2 .8) their off-diagonal elements are equivalent, and gd(T) is the propagator corresponding to the diagonal part of the hamiltonian, ~fCa(t) = E~ ~á(t))éea (t)(á (t)~, and is defined via (2 .6), with ~(t) replaced by ~rCa(t). Then the equation for T(t) _ (1) up to the second order in fi (or collective velocity 4) becomes T(t) - (afilat)o +( -i[fi, H])odr J
ao dT([[~,
V(t)] . 8a(T) V(r)8á (T)])o
+~~ dT lT( [[l s V(t)]r gd(T)[~~ V(t)]ód (T)])0 0
"
(3 .16)
The terms on the r.h .s. can be evaluated explicitly in the travelling representation (3 .6) by noting that ! = 4P is acting on the wave functions only and by observing l') (~n(t)I [~, H]I ~n(r)) =-_
aq
=-'áq
(4'n(q)I
e-tPQH
e~°dl~n(q))
(~n(q)I{H+iPzM-1}I~R(q)),
(3 .17)
(~~(t)~[P, ~(r)]~~m(r)> =- =~q(~~(q)I~(r)I~m(q))+i(~n(q)I av/agl~m(q)>, (3 .18) where ~~n(q)) and ~~m(q)j are Slater determinants of the quasi-static single-particle states ~a(q)) . The second line of (3 .17) follows the expansion in powers of p; the higher order terms do not contribute, the term linear in p vanishes due to the time-reversal symmetry and only the mass contribution remains. It is important to note that although V(t) = v - v(t) is a traceless two-body operator, i.e . Tr V(t)p(t) _ 0, this is not true for [~, V(t)], and as can be seen from. the second term on the
32 4
S. Ayik / Damping of collective motion
r.h.s. of (3 .18) it has a non-vanishing one-body part. Collecting the terms together (3.16) becomes t)+dU(q, t)l-LQl(t)+~z(t)l , dt ~pz~2M(q)~=-4a4 ~U(q, where, in the first term U(q, t) is the potential energy (intrinsic energy) U(q, t)=~ (txl - o2i2mla)Pa(t)+i ~ vas.~a(q)Pa(t)Pa(t) , or U(q, t)=E Ea(q)Pa(t) -i E vas.ap(q)P~(t)Pa(t) ,
(3.19)
(3.20)
and dU(q, t) is a correction to the potential energy, due to the virtual transitions imposed by the residual interactions dU(q,t)=á~lVaa .~s(q)Iz
tv~9 .raz Po(t)PB(t)PY(t)Pa(t), ~ ap.ra + raBvó
(3.21)
with p,,(t) = 1 - py (t), arising from the third term on the r.h.s. of (3.16) . The quantities ~1(t) and ~2(t) are given by a a Ql(t) = 4Z E I(~ I aq Y(q)Iß)IZ ~ ~~Pa(t) 2 , awaa waa + r~ß z a ~ z 21'~a,,2 ~Pa(t)Pa(t)PY(t)Pa(t), Qz(t)=á4 z ~ I V~a.ya(q)I aq aw~a.~ ~~t= .,~a+I'~tl,~a
a
(3.22)
(3.23)
with raa (t)=ra ft)+rß (t), r~a,~(t>=r~(t>+rß(t)+r,,(t)+ra(t>
waa(t)=~re~(t)-~a(t),
w~a.~(t) =~C~(t)+~ea(t)-~e,,(t)-~ea(t),
where TQ (t) and ~1~, (t) = taQ (q) +zpz(a IM-11~ ) are the width and the energies of the single-particle states . According to (3.19), the rate of change of the collective kinetic energy is determined by two different contributions. The first term describes the reversible changes in the potential energy (U + d U), which is reversible in the sense that, if the occupation probabilities are held constant, the energy flows back into the kinetic energy. On the other hand, the second term (~1 + ~1z) determines the irreversible changes of the collective kinetic energy as a result of the coupling of the particle motion with the fluctuating mean field. The damping mechanism due to the random collisions of the particles with the mean field, which results from the uncorrelated particle-hole excitations around the Fermi surface, is reflected by the quantity Ql(t) in (3.19) . The ~1(t) dissipation is equivalent to the rate of energy dissipation formula of one-body dissipation theory "'18'19), and it may be attributed to the shape fluctuations of the mean field. The quantity l~z(t) in (3.19) describes the dissipation as a result of the simultaneous 2p2h excitations
S. Ayik / Damping of collective motion
325
via direct coupling of the residual interactions and is referred to as the two-body dissipation . In contrast to ~1i(t), which is a surface effect, Qz(t) describes a volume dissipation and it may be considered as arising from the local fluctuations of the mean field. The rates ~i and ~1s correspond to one-body and two-body friction forces, which are proportional to the collective velocity 4. In the limit of vanishing single-particle widths, T~, -~ 0, the friction coefficients [the coefficients of 4z in (3 .22) and (3 .23)] do not vanish in general, due to the crossings of the energy levels as a function of q. This is in disagreement with the result obtained in ref. z° ) where the friction forces proportional to 4 vanishes in the weak-coupling limit. The result of ref. 2°) may be due to the fact that the crossings (or near crossings) of the many-body adiabatic levels are not taken into consideration. The dissipation mechanism contained in (3 .19) is further complicated by the fact that the force, F(q, t) =-aU(q, t)/aq, evolves in a time irreversible manner as a result of the time dependence of the occupation probabilities [cf. master eq . (2 .10)]. The irreversible behaviour of F(q, t) means that the energy stored in the dynamical potential can be dissipated into the intrinsic degrees of freedom by redistributions of the occupa9' tion probabilities . °). This dissipation mechanism can be understood better by explicitly considering the coupling between the rate equation (3 .19) and the master equation (2 .10) . Using the master equation, an equation for the force can be derived. If the mean field is practically constant in the nuclear interior, the force on the collective variable is mainly determined by the q-dependence of the singleparticle energies . Consequently, neglecting the second term on the r.h .s . of (3 .20), the force may be approximated by
ea(q)P~(t) r)=-~aq Denoting the average value of a2 e~(q)/eq Z by s"(q), we have F(q,
r) dtF(q, r)+4E"(q)=-2I'(t)[F(q, -Fw(q, t)J,
(3 .24)
where the r.h .s. is obtained from the master equation, replacing l'Q (t) by an average decay width I'(t) and approximating the gain term by its equilibrium value. This approximation is equivalent to a relaxation ansatz with the local equilibrium time, T,«(t) = 2/l'(t) [ref . 6'1~] . Assuming F«,(q, t) is a smooth function of time ; the solution of (3 .24) can be given as F(t) =- J~dr'x(t, °
t')4(r~)e"(t')+F~a(r)[1-x(t, 0)J+F(0)x(r, 0),
(3 .25)
with X(t, t~)=exP {-(t-t~)lT~a(t)} ~
(3 .26)
The local equilibration times in low-energy dissipatioe heavy-ion collisions are
32 6
S. Ayik l Damping of collective motion
estimated using (2 .11) within a Ferrai gas model in ref. t° ). According to these estimates, T,oo(t) is very large during the initial phase of the collision, T,oo(t) ^~ -zt several x 10 s, and it becomes considerably smaller during the strongly damped phase, r,~(t) ^- 5 x 10 -zz s. Therefore, during the initial phase, the occupation probabilities remain unchanged for a long time specified by t « T,o~(t) and consequently (3 .25) acts as a conservative force,
F(q)=-~aq
Ea(q)P~(0),
with no explicit time dependence . The energy dissipation during this time interval is mainly determined by the uncorrelated ph excitations and the correlated 2p2h excitations as described by ~1t(t) and ~Z(t), respectively, see (3 .22) and (3 .23) . For later times F(q, t) contributes to the damping via dissipation of the dynamical potential energy which is stored during the initial phase. The potential dissipation exhibits memory effects due to the large value of the local equilibration times. This gives a non-local friction force, the first term on the r.h.s . of (3 .25), as pointed out in a recent work of Nörenberg t° ), in contrast to the local friction forces contained in Qt+QZ. Various dissipation mechanisms described by (3 .19) are illustrated in fig. 1 . Fermi ~ level
collective varinble Fig . 1 . (a) Uncorrelated ph excitations (one-body dissipation due to collisions of particles with the mean field . (b) Correlated 2p2h excitations (two-body dissipation) via the residual interactions . (c) Potential dissipation due to redistributions of the intrinsic energies via two-body collisions .
4. Conclusions In the extended mean-field theory the inclusion of the residual interactions modifies the motion of the particles in two ways . (i) The action of two-body collisions randomizes the single-particle momentum distributions and hence drives the system
S. Ayík / Damping of collective motion
32 7
towards the thermal equilibrium. (ü) The residual interactions introduce a fluctuating component to the mean field via the off-diagonal elements of the density matrix . In the present work, in order to bring out these effects, the collective motion is treated explicitly by employing the travelling single-particle representation, which allows an identification of the collective variables in the semi-classical approximation . The rate of change of the collective energy is calculated using the extended mean-field equations. For the rate of energy dissipation a simple expression is obtained [see eq . (3 .19)], d e [PZ/2M(q) ]= -4 aq U(q, t)-Ql(t)-~z(t), dr
(4.1)
in which dU is neglected and U, !~ 1 and ~z are given by (3 .20), (3 .22) and (3 .23), respectively . The quantity Ql(t) is the rate of energy dissipation due to the ph excitations around the Fermi surface and is essentially equivalent to the rate calculated in the one-body dissipation theory 18'19) (fig . 1) . On the other hand, Qz(t) describes the dissipation as a result of the direct coupling of the single-particle states via the residual interactions (2p2h excitations) and it is identified as the two-body dissipation . The two-body dissipation is expected to be small in lowenergy heavy-ion collisions and may be neglected. The first term on the r.h.s . of (4 .1) determines the rate of change of the potential energy U(q, t), which also contributes to the dissipation, due to the fact that at every action of random two-body collisions some part of the correlated intrinsic energy is converted into uncorrelated energy (heat) . The potential dissipation has memory effects as a result of the large value of the local equilibration times and consequently appears as a non-local friction force 1°), see eq . (3 .25) . Numerical calculations of the one-body and the two-body friction coefficients based on (3 .22) and (3 .23) are presently in progress . The remaining problem to be studied is connected with the fluctuations of the collective variables. There is some evidence that the random two-body collisions are not responsible for the large dispersions in the dissipatioe heavy-ion collisions . The present work however indicates that the mean-field fluctuations should be included, in order to achieve a better understanding of the fluctuations of the collective variables. This point will be studied in a forthcoming paper. The author is grateful to W. Nörenberg and W. Casing for many valuable discussions and for a critical reading of the manuscript.
References 1) S. Ayik, Z. Phys. A298 (1980) 83 2) C.Y. Wong and H.F. Tang, Phys. Rev. Lett . 40 (1978) 1070 3) H. Onland and R. Schaoffer, Z. Phys . A290 (1979) 191
32 8 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
S. Ayik / Damping of collective motion G. Mantzouranis and H.C. Pauli, Z. Phys . A281 (1977) 165 P. Grangé, H.A . Weidenmüller and G. Wolschin, preprint MPI-H-1981-V8 H.S . Köhler, Nucl. Phys . A343 (1980) 315 P. Grangé, J. Richert, G. Wolachin and H.A . Weidenmüller, Nucl. Phys. A3S6 (1981) 260 H.A . Weidenmüller, in Progress in particle and nuclear physics, vol. 3, ed . D.H . Wilkinson (Pergamon, Oxford, 1980) W. Nörenberg and C. Riedel, Z. Phys . A290 (1979) 385 W. Nörenberg, Proc . Int. Workshop IX on gross properties of nuclei and nuclear excitations, Hirschegg, Austria (1981) ; Phys . Lett ., in press S.E . Koorvin, Proc . Int. School of Physics, Enrico Ferrai, Varenna, Italy (1979) P. Morel and P. Nozières, Phys . Rev. 126 (1962) 1909 G. Mantzouranis and H.C . Pauli, Phys . Rev. C20 (1980) 1550 P.G . Rembard, K. Gceke and K.Y. Casson, Z. Phys . A295 (1980) 45 H. Reinhardt, Nucl. Phys. A346 (1980) 1 H.C . Pauli and L. Whets, Z. Phys . A277 (1976) 83 F. Villara, Nucl . Phys . A285 (1977) 269 S.E. Koorvin, R.L. Hatch and J. Randrop, Nucl . Phys. A283 (1977) 87 H. Hofmarvn and P.J . Siemens, Nucl . Phys . A257 (1976) 165; A27S (1977) 464 M.C . Names and H.A. Weidenmüller, preprint MPI-H-1981-V13