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Microscopic description of heavy-ion collisions in terms of interacting RPA modes
Nuclear 0
Physics
A425 (1984) 152-184
North-Holland
Publishing
Company
MICROSCOPIC
DESCRIPTION
IN TERMS
OF INTERACTING
D. JANSSEN Technische
Universitiit
OF HEAVY-ION
+, M. MILITZER
Dresden, Sektion Received
Physik,
RPA MODES and
R. REIF
8027 Dresden,
31 October
COLLISIONS
Mommsenstr.
13, DDR
1983
Abstract: Starting from the TDHF equation, the equations of motion for the single-particle density matrices refering to projectile and target are derived in an approximative way in a translating and rotating reference frame. An expansion of the fluctuating part of the transformed density in terms of generalized RPA modes in the ph as well as the pp(hh) channel leads to a system of non-linear coupled equations which simultaneously determine the time development of the collective coordmates and momenta of the relative motion, of the Intrinsic angular momentum, and of the amplitudes of coherent surface excitations of the fragments. In a schematic calculatlon the formalism is applied to the angular momentum dissipation in the excitation of damped giant dipole modes.
1. Introduction In heavy-ion collisions with specific bombarding energies below the Fermi energy it is reasonable to expect that the ion-ion interaction is dominated by a collective response of the nuclei with respect to the time-dependent outer field provided by the approaching collision partner. In such a situation the reaction mechanism should be governed by strong mean-field effects, which in a semi-classical manner can be treated satisfactorily in th6 time-dependent Hartree-Fock (TDHF) point of view, the excitation of coherent approximation l, 2). F rom a phisical surface modes (as giant multipole resonances) instead of independent particle-hole pairs could play a decisive role in the approach phase of the reaction, as one can imagine from the analysis of inelastic scattering processes in distant collisions [see e.g. ref. “)I. During the development of the system from the entrance channel to a highly excited compound system such coherent excitations may act as doorway which are subsequently damped by decaying to more complex states, configurations 4). This process leads to a thermalization of the excitation energy, transferred from the relative motion to collective degrees of freedom in the initial stage of the reaction. As was suggested in the framework of linear response
+ Permanent 19, DDR.
address:
Zentralinstitut
fiir Kernforschung 152
Rossendorf,
805 1 Dresden,
PostschlieBfach
153
D. Janssen et al. / Microscopic description
theory 5, for more slowing
down
distant
of the relative
collisions motion
in which
the nuclei
by frictional
keep
forces proceeds
their
identity,
mainly
the
by such
an indirect (one-body) mechanism in which a coherent motion is first excited and then dampled by further interactions. Within this physical picture Broglia, Dasso and Winther6) successfully worked out a phenomenological model for binary deep-inelastic collisions, which for the main reaction characteristics gives results closely related to the TDHF approximation. In this model a nuclear surface-surface interaction, which is empirically fixed from scattering data, couples the relative motion to harmonic surface modes. The relative motion as well as the surface degrees of freedom are treated classically, with the restoring force and mass parameters of the collective vibrations being determined from quanta1 calculations of the nuclear response function. The approach is further improved by taking into account an incoherent particle transfer through the contact zone of both nuclei, which is represented by an additional frictional force. Furthermore, the zero-point fluctuations associated with the low-lying surface modes have been introduced by a set of random initial conditions for integrating the coupled equations of motion. The model has been applied successfully to interpret various deep-inelastic reactions, including angularmomentum-dissipation phenomena ‘). The present paper proposes a description of heavy-ion collisions, which in its basic features represents a microscopic version of the coherent surface excitation model of Broglia, Dasso and Winther. Starting from the TDHF equation for the total single-particle density of the system, and assuming negligible mass transfer as well as small overlap of both nuclear densities during the course of the reaction, we set up approximate equations for the partial single-particle densities, referring to both fragments. The time development of both components is coupled by the common single-particle hamiltonian. According to the simplifications introduced in such a model, its applicability is restricted to a class of more distant processes in which it is possible to distinguish between the fields and nucleons of two nuclei. In such a situation it is meaningful to introduce the collective degrees of freedom associated with the relative motion, which then can be treated formally by a suitable unitary transformation of the partial single-particle densities to translating and rotating reference frames, connected with each nucleus. The parameters of these transformations are taken to be functions of time. Within the intrinsic system, each density is decomposed in a (spherical) stationary field and a part fluctuating in time. For the time-dependent component of the single-particle density one can derive equations of motion, which contain the RPA operator as the dominating term. Therefore, it is reasonable to expand the fluctuating density in terms of the complete, orthonormalized set of generalized RPA solutions’), which have components in the ph as well as the pp(hh) channel. The time development of the expansion coefficients are governed by a set of coupled non-linear equations. The pp(hh) modes with finite frequency are disregarded, so that the basis is restricted to
154
the collective channel and
D. Janssen et al. / Microscopic description
and non-collective RPA modes with finite to relevant spurious modes in the ph and
frequencies in the ph pp(hh) channels. The
spurious mode in the ph channel is associated with the momentum operator. In the intrinsic system this mode, which represents the translation of the nucleus as a whole, is not excited. Its elimination from the RPA spectrum fixes the parameters of the Galilei transformation to the moving reference frame in terms of newtonian equations for the relative motion. In linear approximation, the resulting inertia parameter deviates from the classical reduced mass by terms implying the exchange interaction only. The potential of the force acting on the relative motion consists out of a folding potential and an expression, which provides a coupling to the RPA modes of finite frequency in the ph channel, in some approximation not completely invariant under time-reversal. Assuming a spherical HF basis the spurious mode in the pp(hh) channel is associated with the intrinsic angular momentum of the fragment, so that the time development of its amplitude describes the transfer of angular momentum from the relative motion to intrinsic rotation. The remaining equations for the RPA amplitudes of finite frequency describe coherent vibrations of one fragment within a moving reference frame, influenced by a potential provided by the overlap of the ground-state densities of both nuclei and coupling each particular RPA mode to all other excitations in both nuclei. Introducing statistical aspects, one can find a solution for the amplitudes, which corresponds to a classical damped oscillator in an outer field, shifted in frequency. The resulting set of equations for the relative motion, the transfer of angular momentum and the excitation of surface modes has to be solved simultaneously. So, the main advantage of the present approach is that the collective degrees of freedom associated with the relative motion and the excitation of coherent surface modes can be treated in a unified way by specifying an equation, which has been derived from an universal expansion of the single-particle density in terms of various RPA modes. In sect. 2 the general formalism of the approach is presented, including the unitary transformation to intrinsic reference frames and an expansion of the fluctuating part of the transformed density matrix elements in terms of interacting RPA modes. Dynamical equations for the relative motion, for the transfer of angular momentum, and for the amplitudes of coherent excitations are derived in sect. 3. Based on a schematic model, sect. 4 contains an application of the formalism to the angular momentum dissipation in the excitation of damped dipole modes in various symmetric and asymmetric systems.
2. General formalism We treat a system of N nucleons, interacting via velocity- and spin-independent, translational invariant two-body forces I/(1x - x’j), where X, x’ denote the nucleonic
155
D. Janssen et al. 1 Microscopic description
space
coordinates.
In
TDHF-approximation
described
by a time-dependent
equations
(fi = 1)
The operator h(p) is a single-particle given by (m is the nucleon mass)
density
p2 = p,
q-J= [h(PLP].
0 I,,, = -
the development
single-particle
hamiltonian,
matrix
Trp
of the
system
p(x, x’, t) obeying
= N.
is the
(1)
which in space representation
is
& 6(x-x’)+d(x-x’)
-p(x,
x’)V(Ix
-x/l).
(2)
In order to describe the dynamics of a heavy-ion reaction the solution of eq. (1) has to fullil an initial condition, which corresponds to an entrance channel situation with a definite partition of the total nucleon number N in two fragments N,, N,(N,+ N, = N) with both nuclei localized at a relative position R = R,- R, far outside the interaction region and possessing a relative velocity u = u, - ut, and a corresponding relative momentum p in the c.m. system. Then for the free initial state the suitable solution is a uniform translation p(x,
x’, t) =
exp
(.
z(x - x ’ ) .uy) N
pg”(x
R, - u,t, x’ -R,
p’bo’(x -R,-
- i(x - x’) . r
+ exp
-
-
u,t, x’ -R,
vat)
- u,t)
b>
pk(‘)
of the stationary HF solution calculated in the corresponding
[h(plP’), P:“‘] According decomposed a, b: p(x, x’, equation (1). the interaction
for the ground rest frame, COP= p’o’ k ,
= 0,
PA
states
of target
Tr(pjp’) = N,,
and
k = a, b.
projectile,
(4)
to eq. (3), at the beginning of the reaction the total density p is into two non-overlapping parts, referring to the collision partners t = 0) = pa(x, x’, t = O)+p,(x, x’, t = 0). It follows from the TDHF that the time development of each component during the course of can be described by a set of coupled equations
@a = [h(P, +b
=
+ Pb)r
Pa]+ *%‘a~Pb).
INpa+ Pb), Pbl - w(pa,Pb).
(5)
156
I). Janssen et al. i Microscopic description
These equations define a particular subdivision of p:
P(t) = Pa(t)+Pb(t),
(6)
which is determined by eq, (3) iu the initial channel, for the whole reaction time. The single-particle operator ti. is a complicated functional of the densities pa, pb and the two-particle interaction. Because its trace is connected with the time derivation of the particle number in p,, P,,, &, = Tr@,(t) = Tr ‘Be,
rii, = A$,
(7)
the operator We is responsible for the mutual exchange of particles between the collision partners, which may proceed in the interaction region. The operator 0‘ can be omitted exactly for symmetric systems, for which mass transfer is forbidden because of the conservation of the reflection symmetry of the initial density. For target-projectile combinations, bombarding energies and impact parameters, which in the experiment exhibit a small spread in the mass number of the final fragments only, one can disregard the mass transfer by setting $6’”= 0. Furthermore, for more peripheral collisions the relative motion allows only a small overlap of both densities during the whole course of the reaction, pbpa +p,p, z 0, so that the condition @, + p,J2 = pa + pb can be fulfilled approximately, if the subsidary conditions pk’ =
pk.
Trp,
=
k = a, b,
Nk,
0)
are imposed. One can show, that these equations and the condition ~4. = 0 are quite equivalent. It means that one seeks for approximate solutions of eq. (5) (with tiL^E 0) for pa, P,,, which correspond to determinantal wave functions for the subsystems a, b. So we replace the full TDHF problem formulated in eq. (1) by the following set of coupled equations : ibb = [4P, +PbhPbl P; =
Pbr
Trp, = N,.
(9)
Together with the initial condition (3) it reflects the following physical picture: projectile and target approach each other freely in their HF ground state with given relative velocity and impact parameter. When the HF potential generated by the wave functions in nucleus b starts to overlap with the wave functions in nucleus a it introduces excitations of nucleus a, and vice versa. If the further relative motion proceeds in such a way that the oscillating density dist~butions show only a small overlap, the identity of the collision partners is preserved throughout the reaction and, neglecting the mass transfer, the intrinsic excitation
D. Janssen et al. J Microscopic description
157
can be described as changes of a determinantal wave function for each fragment. Such a picture seems to be sufficient if one is looking for the expection values of single-particle operators in more peripheral collisions of closed-shell nuclei, for which the dominating excitations are rather simple and the width of the final mass distribution is expected to be small. 2.1. INTRINSIC
REFERENCE
FRAMES
In order to separate out the relative motion as well as the intrinsic rotation of the interacting nuclei from the full time development of p, and pi,, the structure of eq. (9) suggests a separate unitary transformation U,, U, into a moving reference frame for each fragment :
r, = Q?JJ,l, The transformation
operators
t-b
=
u,p,u,?
(10)
U,, Vi,,
U, = R;‘exp((*%
U, = R;Lexp(-q%
&)exp(-i$+*),
i)exp(i$.s),
(11)
contain a Galilei transformation as well as a rotation. Because in the cm. system both nuclei are approaching each other with equaf but opposite momenta, the galilean parts of U,, U, are related, and the iotation is expressed by the components of the angular momentum operator Las
According to eqs. (11) and (12) the transformation has to be specified by 12 free parameters q(t), p(t), I, fik(t), y&t), k = a, b, which will be fixed in a convenient manner later on. As a function of time, these parameters describe the translation and rotation of the reference frames with respect to the c.m. system. The transition to the moving coordinate system in eqs. (9) is demonstrated for pa, In order to simplify the notation, the index a is omitted but we emphasize, that in this context p should not be confused with the total density matrix. From ifi = d(U-‘r~~~dt, with the help of the relations
a. X =X+q, +.
( 1
exp q*G
h = - iR(h(cos #?Lz- sin fi(cos yL, -sin yL,)) + &cos y L, +sin yL,) + j$,),
(13)
D. Janssen et al. / Microscopic description
158
one gets
1
\
+ ii + j[L,, r] + &cos yL, + sin yL,, r]
+ k[cos flL, - sin /?(cos yL, -sin yL,), r]
U,
(14)
} which can be solved for i,
- [L,, r]($ + ir cos p) - [L,, r](fi cos y + 6 sin fi sin y) - [A,, r](b sin y - h sin fi cos y).
(15)
In eq. (15) ib has to be replaced by the commutator [h(p+&),
p] = - & [d, U-‘rU]
+ [ bx”
I/U-‘rU,
u-‘r(r]
1 -[U-'r~I/,U-'rU]-[ (16) dx” VU,‘r,U,,
+
U-‘rU
[s
In order to evaluate the various terms of eq. (16) one can utilize the relations
C-4RI = 0, [A,exp(ip*9)]
= -pzexp(ip.f)+2iexp(ip*4)pda;,
(&,)xx’r&, UR\,
’ A,, = r,(D,x,
(17)
&,x’).
The rotation matrix describing the effects of R, is denoted by D,. In this way one ends up with an equation
of motion
if, = [h(r,), r.J - [LX, r&l,
for the density
matrix
ra in the moving
frame:
sin ya- h, sin fi, cos yJ
- [L, rJ(Ba cos Y,+ ka sin B, sin y,) - [L, ra](PB+ k cos 8,)
+
[s 1 vrb,
Ia
-
CWrby
ra]-
W)
159
D. Janssen et al. / Microscopic description
Up to the first term on the r.h.s., one obtains
the TDHF
equation
for the isolated,
unperturbed fragment a. The next three expressions represent Coriolis forces, resulting from a description of the single-particle motion within a rotating coordinate system. Then one has two terms, appearing because of the translation performed in coordinate and momentum space. The interaction between both nuclei is grasped in the remaining terms, consisting of a direct part with
=
6(x - x’) s
XX’ and an exchange
dx” V(lD, lx -Da lx” + qI)r,(x”, x”)
(19)
part with
- ip . T+“l(X-Xf)
( Wrb)xxj = exp
a
b
These interactions receive a complicated time dependence because of the parameters of the transformations U,, vi,, involved in the expressions (19) and (20). An analogous equation can be derived for the component rb.
2.2. DENSITY
FLUCTUATIONS
In the next step each density obeying the stationary fluctuating in time
matrix HF
rk(O),
rk = rip’+ h,(t),
rar rb is subdivided
equation
Tr(Gr,(t))
into a stationary
[h(r’,O)),ri”)] = 0, and
= 0,
a part
k = a, b.
part rk(t)
(21)
The matrix rk has to be traceless, because the particle number is conserved. Introducing eq. (21) in (18) gives an equation of motion for h,(t), which is presented for k = a: dr, = Kc%, + [s(dr,), 6rJ - %?+ 9 + Y. Here, K denotes
the non-hermitean
the interaction
s@,$,,= 6(x-x’)
RPA operator,
= [h(rlp)), Sr,] + [s(Gr,), r:O)],
Sir, and s abbreviates
(22)
s
part of the single-particle
(23) hamiltonian:
dx” V(lx - x”l)br,(x”, x”) - 6r,(x, x’)V(lx - ~‘1).
(24)
160
D. Janssen et al. { Microscopic description
While the contributions
G9and $9 to eq. (22) arise because of the transformation
U,
Q?= [L,, r.J(j, sin y8-ix, sin /3, cos v,) +
CL,,~JtP,cosYa+ 4
sin8,sinY,)+ [L,, r,](& + rjlscm A),
(25)
the last term $’
=
[S
Vrb”’-
Wr’,O’,
1 [s
+ r(O) a + 6r a
VSr, - W&r,
7
rp +Sr
a
I
(27)
expresses the (direct and exchange) interaction between the colliding nuclei. The parts like [f Vri’), r,,+Sr,,] couple the excitations of the fragment k’ to the groundstate HF density of the fragment k, which acts as an outer field. In this sense transitions to excited states occur by a one-body mechanism of excitation. The terms like ~~V~r~,~~~)~~r~,] describe the way, in which the excitations of the fragment k are influenced by the excited fragment k’. 2.3. INTERACTING
RPA MODES
Omitting the index k, the solution of eq. (22) for r, can be sketched as follows: Solving the eigenvalue problem for the RPA operator K, which dominates the equation of motion for 6r, one finds all possible RPA modes 40, with zero and finite frequency w, with components in the ph and pp, hh channels, cp, = {@“, &?, cpt”>.This set 9, can be constructed in such a way, that it can serve as a complete, orthonormalized basis of two-particle wave functions to expand the ~u~tuati~g part 6r of P, which in general is not supposed to be small:
The time-dependent expansion coefficients a,(t) determine the contribution of each collective, non-collective and spurious RPA mode to the change of the density matrix in time, with a substructure according to the various channels, a, = {LZC”, alp, ut”>. From eq. (22) a set of coupled non-linear, first-order differential equations for these RPA amplitudes can be derived, which should be handled further. At first, the structure of the basis 1;~~is discussed in some detail for one subsystem, omitting the index k.
161
D. Janssen et al. / Microscopic description
Because the RPA operator the eigenvalue problem as
K (eq. (23)) is not self-adjoint,
Kc =
one has to formulate
K+Xy = w,x,,.
W/P,.
(29)
The possible solutions to these equations can be characterized in the following way: (i) all frequencies co, are real numbers; (ii) for each solution o,, (py, x, with o, f 0 there exists a second solution oB, cps, xi with w,- = -w,, cpi = cp:, (iii) if the components in the pp and hh channels are taken into account, XC= x: ; the set (p,,, x, formes a complete basis in the space functions. which can be orthonormalized according to
of all two-particle
wave
(30) ph channel:
Because the translational symmetry is broken in the TDHF-initialvalue problem, the momentum operator has non-vanishing matrix elements in the ph channel, giving rise to a spurious RPA mode (0, = 0):
_&
xSPph=fi= These solutions
‘pph=
SP
i
Y[P’,
solution
(31)
of eq. (29) are orthogonal,
Tr(X,“,h+&‘) = Tr(i V@. [r(O), a]) = -Tr(i A further
a].
can be constructed x-ph
=
according
Vr(‘)[@, a]) = 0.
(32)
to
cpph= -i. V[r’O’, a],
f ,
(33)
for which the equations
K+fPh
=
&ph jm
‘P ’
hold. Again,
1 Kcj ph -- _ _ (ps”,”
one has Tr(Xph+(pph) = 0. If one arranges xpph -_ p,A
(34)
im
these solutions
as
cp;” = -i. V[r(O), a],
xpph = f ,
one has the same orthogonality condition for the spurious modes as for phonon factor appearing excitations, Tr(X,ph+(p;h) = Tr(Xsph+(pFh) = 0, and the normalization
162
D. Janssen et ai. / Microscopic
description
in eq. (35) has to be determined from Tr(X;hfrp;h) = Tr(Xih+(pqPh)= 1 leading to V = I/N. All other solutions cpth in the ph channel with finite frequencies o, belong to non-collective and collective modes, which can be interpreted as coherent surface excitations of the nucleus. pi channel: In the pp(hh) channel the spurious mode connected with the particle-number operator is not excited, because the particle number is conserved (Tr(Gr) = 0). If one restricts the consideration to spherical HF ground states, another spurious mode in the pp channel can be created with the angular momentum operator, which has no non-vanishing matrix elements in the ph channel, so that the rotational symmetry is not broken, [r(O), L] = 0: 1,” = L K'
PP (PK
1 ---.-Lx,
ic =
x,y,z.
(36)
'JWLL)
In a representation, in which r (‘I is diagonal, the pp and hh matrix elements of 6r rco)&rtob and G’O%rG (‘) (G “) = 1 -r(O))can be expressed by the ph and hp matrix elements Gco)&rco) and rco%dO’, as follows from the subsidary condition r2 = r, which give 6r = r’0’6r+6rr’0’+Srdr. Multiplying from the left and from the right with the projection operators r(O) and a”‘, respectively, the first two terms disappear because of G(o)r(o)= r(o)g(o) =: 0 and from the last term one gets with 6r6r = Gr(G’“’
+ r”‘)&
o(0)~rrCOl(l
- fl(O)ijrg(O)) = (G(O)Srr(O))(r(O)Gro(O))
- r(0)&r(O)( 1 + r(*)&yJO)) = (r(“)~rG(o))(G(o)sTT(o)).
(37)
Therefore, the RPA amplitudes u~P(~~)in the pp(hh) channel can be expanded in the form uppW) Y
=
+ higher-order EA$E!hyh)a$hullph
terms.
(38)
In lowest order the amplitudes in the pp(hh) channel are quadratic in the amplitudes of the ph channel, so that they are expected to be of minor importance, if one restricts the considerations to small-amplitude motions. Furthermore, if the ph space has large dimension, the random phases of the ph amplitudes guarantee that the sum in eq. (38) remains small because of the destructive coherence of the superposition. But in general, by expanding the single-particle density matrix in terms of generalized RPA modes, the multiple-particle-multiple-hole excitations contained in the TDHF many-body wave function have to be taken into account by the amplitudes in the pp(hh) channel. In this case it is not necessary to restrict 6~ for a small-amplitude limit.
D. Jamsen et al. / Microscopic description
163
Inserting the expansion (28) in the equation of motion (22) for c%, multiplying from the left with x,’ and taking the trace, one can derive first-order, non-linear coupled equations for the RPA amplitudes of the fragment a: ib, = o,a, - %.(r(O))+ ~v(~(o))+ 9”v(#O),Q)
(39) with U,($) = Tr{X,?([L, $](a sin y-dcsin pcos y) + [&, $](B cos y + h sin B cos y)+ [Lz, $f]($ + tftcos /?))I, S&,4) = k+Tr{$[R-IfR, a
$]f
One should keep in mind, that for v = 4 in the first term on the r.h.s. w,a, has to replaced by a,/im because of the special features (34) of the spurious modes’ in the ph channel.
3. Dynamical equations Supposing a spherical HF ground state for both nuclei a, b, the set of equations (39) is handled by eliminating exactly the spurious modes pp, (p4 in the ph channel and imposing the condition up = Tr(‘Gr) = 0, a4 = Tr(26r) = 0 (Tr(~~(‘)) = 0, Tr(fr(“) = 0). These conditions imply, that the expectation values of the operators _? and fi have to vanish, if calculated with the density matrix referring to the moving coordinate systems. This is the case if these reference frames are fixed in the c.m. of the nuclei, so that the parameters q(t) and p(t) take on the physical
D. Janssen et a/. j Microscopic description
164
meaning of the coordinates and the momentum of the relative motion of the fragments, respectively. A specification of eq. (39) for the modes v = p, q leads to equations, which determine the relative motion q(t), p(t) during the collision in terms of conservative mean potentials and excitation of RPA modes in the ph channel. On the other hand, the expectation value of the component
L,=-i
d
( > A
xxdx
K
in the c.m. system is
6%) = TrbQ,) +JW%+J which after transformation function)
(40)
in the intrinsic system takes the form (D is the Wigner
Now, the intrinsic system is chosen in such a way that its z-axis is oriented along the direction of the angular momentum, so that the expectation value for the x- and y-components vanish : QJk
= Tr(L,r,) = 0,
(LJk = Tr(L,r,) = 0.
(42)
The absolute value of the expectation value of fit) gives the magnitude of the intrinsic angular momentum of fragment k: (LJk = Tr(L,r,) E d”(t).
(43)
As a consequence of this choice of the intrinsic reference frame, the amplitudes u(x), uCk) Y at the RPA modes belonging to L,, L, vanish identically, while the RPA amplitude a:)(t) z E@(t) gives the intrinsic angular momentum, finally. From a specification of eq. (39) for the angular momentum modes, equations for the time development of the absolute value I; of the intrinsic angular momentum and its orientation angles CQ,&(yk = 0) can be derived. For the sake of simplicity, in all of these equations resulting for the translational and rotational motion of the fragments the terms quadratic in the amplitudes a, are cancelled. Afterwards, if all pp(hh) components are omitted, one is left with the equations of the amplitudes if the RPA modes of finite frequencies in the ph channel, which have to be solved simultaneously with the equation of motion for the trajectory and the intrinsic rotation. In conclusion, the sum over v in eq. (28) is restricted to the RPA modes with finite frequency in the ph channel and to the relevant spurious modes in the pp(hh) channel.
165
D. Janssen et al. / Macroscopic description
In the following, only the main results of this procedure details of the derivation are given in the appendix A.
are presented, while
3.1. RELATIVE MOTION
For the relative distance q and the relative momentum obtains classical equations of motion 4 = p_x-1,
i,=
p of both nuclei one
(44)
- ii_ vin*> dq
with the inertia parameter .x-i
=: &(A
+p-~.~xdx’V(jx-x’J)Zxsin[~p,(x-x’)]
x r’bo’(x+ q, x’ + q)r’O’(x’, x) + c p - l (i4$&, P ~~(~~,~~)
r~))a, + Wq(Pf, ~~))Q~)~ ) i
dxdx’(x-x’)exp
= i
(
s
-ipa& a Jx-x’)
x $i(DX’, Dx)II/z,(k(x+4),
&(x’+q))*
(45a)
>
Vlx-x’l) (45b)
and the potential Vint =
dxdx’ r(O)(x,x)r~)(x’,
x’)V(lx
-x’
+ ql)
f
-
dxdx’ r(‘)(x’, x)rp)(x -+q, x’ + q) s
(46a) with
(46b)
166
D. Janssen et al. / Microscopic description
~,W,,&)
=
s
dxdx’~,(Dx’,Dw)~,(D,(x+q),D,fn’+q))
-ip
xexp (
. J!-(n -xg)) N,N,
V(jx -x’l),
The quantities a%), cp%’iefer to the subsystem b. In the inertia parameter .X the deviation from the classical reduced mass is caused by the exchange interaction only. The potential Vi,,, consists of two parts: a (conservative) folding potential, depending on the ground-state densities r(O),r$,‘),and a term VcouD which couples the relative motion to the phonon excitations in both nuclei. These equations have to be solved with initial conditions corres~nding to the entrance~hannel situation. 3.2. TRANSFER
OF ANGULAR
MOMENTUM
For the magnitude and the orientation one gets
of the angular momentum of fragment a
(47)
(48)
(49) A change of the intrinsic angular momentum of fragment a is possible because of the interaction excitation of surface modes. 3.3. DAMPED
SURFACE
EXCITATIONS
The remaining equations for the amplitudes of the RPA modes with finite frequency o, in the ph channel referring to fragment a are given by eqs. (39). The first four terms on the r.h.s. represent a harmonic vibration with w, in a translating and rotating reference frame, influenced by the time-dependent outer field, /provided by the overlap of nucleus a with the ground-state density of nucleus b. The Galilei term Y,(r(‘)) and the Coriolis term ‘%,,(r(“)) are vanishing because of the orthogonality of the RPA modes. The outer field Y’“,,(r(‘), r’,“)) can be expressed as
flry(r(0), r’,O))= -
-!!..K T ,o,,
(50)
161
D. Janssen et al. / Microscopic description
As implied
by the following
term in eq. (39) the particular
mode v is coupled
to all
other excitations in both nuclei, with a time dependence of the coupling strength governed by the time dependence of the relative motion. Already the transition to a moving reference frame creates a linear coupling of the various RPA amplitudes within one nucleus given by the expressions WV(cp,), YSV(cp,,).Its strength is determined by matrix elements of the space and momentum operator. For a spherical HF ground-state, the RPA modes can be specified by the angular momentum quantum numbers J, M and an additional quantum number E, v = JME, so that it is possible to derive explicit expressions for the Coriolis term %‘7y((pp)a,,utilizing angular momentum algebra. The main steps of this treatment are given for the x-component (y = 0): (p = J/M’&‘).
Cl”,’ = Tr(X: [&, cp,])h sin pa,,
L,, L_, L, = i(L+ + L_ ), one obtains
Using the shift operators appearing in eq. (51)
[IL,,(PJ'M'd]
= +{J(J’+ +
Inserting
this expression
l)(~~,~,+i~, (52)
in eq. (51) leads to
+J(J’+ to the orthogonality
Cc
for the commutator
(J’+l)J’-M’(M’-l)~J,M<_Ic,J.
CE = *h sin p Tr (xJ+ME (J(J’+
According to
l)J’-M’(M’+
(51)
l)J’-M’(M’+
l)(~~‘~,+i~,
l)J’-M’(M’-
l)cp,,,,_,,,))a,.,.,..
relations
(53)
(30) for the RPA modes, this is equivalent
= $ksinjI(J(J+l)J-M(M-l)a,,_,,
+J(J+ W-M(M+ In this way the equations
for the RPA amplitudes
+$J(J+
l)J-M(M-l)(hsinfl+ifl)aJM_,,
+4,/U+
l)J-M(M+l)(dcsinp-$)a,,+,,
+‘~?%Me((Pp)+ P
lbJM+lE).
“f
JME((P~?
r(b”)))up
+
(54)
get the following
1
YJME(r(o),
form:
$(?b:b’
P
(55)
168
I). Janssen et al. / Microscopic
description
with the indices p, I running over all RPA modes in the corresponding nuclei. One can state, that the Coriolis term couples the mode-JMa only to selected excitations with the same quantum numbers J, E, but differing in the projection quantum number M by k 1. A further source of linear coupling between the RPA amplitudes of nucleus a is the interaction between the excitation of a with the mean field of nucleus b in its ground state, VJcp,,rb*‘). An analogous mechanism connects the time development of aJME with all excitations in nucleus b, YJME(rco), q$“‘). Furthermore, two coupling terms appear which are bilinear in the amplitudes a,,, therefore implying an interaction between different RPA modes in the same nucleus fTr(X: [s(q,), cp,]} as well as in different nuclei ( Yy((pp, cplb))). While the strength of the coupling matrix elements is determined by the ion-ion interaction, and as such strongly depends on the relative distance, the bilinear coupling of RPA modes within the same nucleus is given by the two-body interaction s. Non-linearities of higher order would appear by taking into account the pp(hh) amplitudes to higher than second order in the ph amplitudes (cf. eq. (38)). Such terms would provide a coupling of the ph modes to multiple-particlemultiple-hole configurations. In order to stress the consequencies of the coupling terms between the RPA modes, the outer field V&r(‘), rb(O)), as well as the Coriolis terms, is disregarded for the moment. Then, the system of equations for the RPA amplitudes (55) gets the basic structure (56) For the diagonal elements one assumes 8,” = 0, which can be achieved always by renormalizing (&,, = o,+o,,). The transformation a,(t) = e-1%?,ft) eliminates the time dependence of the non-interacting oscillator modes
(57) In the formal solution of this equation, ii(t) = G(t)a(O), for the propagator
G, a perturbation
G(t) = Texp(
-i /:dr’O(i’)),
expansion with respect to 0 is performed:
(58)
D. Janssen et al. 1 Microscopic description
169
(59) In these integrals the time dependence of 0”’ and 0C2’ is determined by the collective coordinates q(t) and momenta p(t) of the relative motion, only. For lower bombarding energies the relative motion along the trajectory proceeds slowly compared to the period of a spherical vibrational mode, so that it should be a good approximation to keep the quantities 0”’ and O’*’ constant for the time integration in eqs. (59). Furthermore, even if the interactions 0’” and 0C2) are supposed to be small (O(i), O’*’ =: 5,5 K 1) for larger time values the product tt* becomes non-negligible and has to be taken into account. Under these conditions, G(i) and all other odd terms in the perturbation expansion cancel. In Gt2’ the integrand is a superposition of periodic functions which get out of phase for growing T, so that the integrand goes to zero and the upper boundary of the integral can be replaced by infinity [Van Hove limit ‘)I. Furthermore, we assume, that in the summation over /z and A’ the coherent part il’ = 1 gives the dominant contribution. With the ansatz a,(r) = e%-r,r one gets
Explicit expressions for A, and r, can easily be found for limiting cases. (a) tT ez 1. An expansion of the exponential in eq. (60) leads to
= G,,itE,. Because for higher-order
(61)
terms Gt$“) = ((it)“/n!)E:6,, G VP= e’%d
Inserting
this expression
holds, one obtains (62)
VP’
in eq. (58), a comparison
with eq. (61) yields
(63)
170
D. Janssen et al. / Microscopic description
1. The second immediately leads to (b)
tT B-
interaction
term
in eq. (60) goes to zero, which
(64) If one takes into account the force O:‘)(t) as an inhomogeneous term in eq. (56), in a similar way one can find a solution which corresponds to the amplitude of a classical damped oscillator in an outer field, shifted in frequency: f a,(t) = ei(-%+d,)t-r,t
a,(())- i
(
dt’O$)(t’)ei(O”t’-d,t’)+T,t’
s0
.
(65)
)
The damping factor e-rJ for the RPA mode v of fragment a is a consequence of (i) the decay of this mode to a more complex contiguration due to the interaction OC2) within a itself, and (ii) the interaction with the mean field as well as the RPA excitations in fragment b due to the interaction 0”). 3.4. FRICTIONAL
FORCES
Utilizing the expression a,(r) = _
i
’ dt’e(-iW~-r,)(t-t’)OlO)(t’) s 0
=
t&‘e-imv(t-t’)
-_1 s
{cash r,(t - t’) -sinh r,(t - t’)}O\“(t’),
(66)
0
in which the influence of the initial amplitude a,(O) has been neglected, to calculate the potential 9’“C0up of eq. (46a), which determines the coupling of the RPA modes to the relative motion, one finds cash
Y
Y + c+i Y
-
t’){
V,(cp,,
rb”‘)
‘dt’e-. l#(t-t’)cosh rlb)(r - t’){ V,(r(‘), s0
+ C-i
+ c +
r,(t
i
tdt’e-~W~(c-c’)sinhr,(t
-
W,(cp,,
rb’))}
cpib)) - W,(r'"', q',")))
- t’){ V,(cp,, rp’) -
W’(cp,, IL”‘)}
s0 ‘dt’e-. s0
Iwv “I cc-%inh
rtb)(t - t’){ V,(r(“), cp’,“))-
W,(r’“‘,
cpib))).
(67)
Il. Janssen et al. / Microscopic description
171
In first order the last two terms, violating time-reversal invariance, are proportional to the damping width rye In the classical equations of motion (44) they give raise to a frictional component of the force governing the relative motion.
4. Angular momentum dissipation 4.1. EXCITATION OF DIPOLE MODES In order to demonstrate the application of the present approach, the angular momentum dissipation as a function of the impact parameter is treated within a schematic model, utilizing further simplifications, so that the resulting equations to a large extent can be solved analytically. Generally, the exchange interaction has been neglected. Therefore, the equations relevant for the angular momentum transfer do not depend explicitly on the momentum p. But the trajectory q(t) is required, which is taken from calculations within a two-dimensional classical dynamical model lo), including a frictional force of Gross-Kalinowski type ir). Thus, the eqs. (47)-(49) are reduced to
&z/j=&
W
with the sum extended over all RPA modes with w, # 0 in the ph channel. For the mean interaction potential appearing in eqs. (68) a harmonic oscillator (cutted at the interaction radius qo) has been taken:
(69) with the parameters ho, = 41N,i MeV and q. = 1.36(N$+Ng)fm +OS fm for the oscillator frequency and the interaction radius, respectively. The ansatz (69) contains a coupling term, which only allows the excitation of dipole (and in the present context insignificant monopole) modes in the angular momentum transfer, while transitions with higher muttipolarity are excluded. Among the possible dipole transitions the giant dipole resonance should carry the most excitation strength, so that this partidular mode has been taken into account only. Thus, the relevant RPA amplitudes a,, a; have v = JMe with J = 1, M = 0, * 1 and v”= JME, the frequencies uy, CL+depending on the quantum numbers J, E and J, & respectively. For simplicity, the isospin character of the resonance has not been treated explicitly.
172
D. Jamsen et al./ Microscopic description
The RPA equations qp,(x,
x’) = i
I
&td-d’)+tmw,2(x2-x’2) c&(x,x’)
-
dx” (x - x’)w”cp,(x”, x”)Q
-t
(70)
s
with the dipole-coupling strength Q determine the basis elements (py, which are expanded in terms of the eigenfunctions $:, of the harmonic oscillator: (71) The index 1 summarizes the quantum numbers n (number of nodes of the radial wave function). I, m (angular momentum and its projection) of the oscillator state. For simplicity, the spin-orbit coupling has been neglected. From eq. (70) one obtains for the expansion coefficients i2)
(72) In this expression, N,,, denotes the occupation number of the single-particle state n, 1, and _flrdis a strength parameter, which implies the overlap of the radial wave function & in the orbits 2, p: ~
f,~ = 4,/(2l, + 1)W, + 1) (1,01,0110)
r3dr Rn,,,(r)R,,Jr).
(73)
The Clebsch-Gordan coefficient (~~0~~0110) gives the selection rules for parity and angular momentum in the ph excitations. Inserting eq. (69) in eq. (55), modified by the statistical treatment of the interaction terms leads to explicit expressions for the RPA amplitudes in terms of time integrals : (74) The evaluation of the trace yields, e.g. for alle (see appendix B),
D. Janssen et al. J Microscopic description
173
In this equation the 4(t) denote the polar angles of the trajectory q(t). If the z-axis is chosen to be perpendicular to the reaction plane, one has 0 = $r, and oz= 0, /? = 0, because within the classical model the dissipated angular momentum is aligned along this direction. Taking into account the damping of the giant resonance, the eqs. (68) are reduced to a formula for the intrinsic angular momentum L of one fragment along the z-axis,
which can be solved numerically. According to eq. (77) the time dependence of the intrinsic angular momentum carried by one fragment is determined by fi) a strength parameter F, characteristic for the individual structure of the nucleus, (ii) the trajectory specified by the relative distance 4, and the polar angle # in the reaction plane, and (iii) the decay width of the giant dipole resonance. 4.2. NUMERICAL
RESULTS
Numerical calculations have been performed for three symmetric systems with 40Ca, “Zr, ‘**Pb and for the asymmetric systems 40Ca+90Zr, 40Ca+ “*Pb, and 86Kr+208Pb in the energy range of 7-10 MeV/nucleon. The parameters for the isovector giant dipole resonance following from experiments on photoabsorption [see refs. r3, “)I are gathered in table 1 together with the strength parameter F computed according to eq. (76). The position of the resonance in Kr has been fixed in agreement with the empirical relation r3) hog’) = 77.9 MeV x N&l
-e-NJ238)+34.5
MeV x N,$e-NJ23*.
(78)
In order to simulate dissipation mechanisms other than the excitation of collective states which are damped by the coupling to more complicated configuration due to the residual interaction within one nucleus, the width ry as given in table 1 is increased arbitrarily by about 50% in the actual calculation. The corresponding parameter values are presented in column 4 in table 1. Because of the very strict simplifications underlying the calculations one can expect to reproduce only the general tendencies observed ex~rimentalIy in the angular momentum transfer. The numerical results obtained exhibit the following main features.
D. Janssen et al. / Microscopic description
174
TABLE 1 Parameters
‘Ya ‘OZr “‘Pb 86Kr
for numerlcal
calculations
ho, (MeV)
hT:XP (MeV)
W, (MeV)
F
19.0 16.85 13.4 16.8
2 2 2
3 3 3 3
30 92.5 349.1 101.8
For a given impact parameter the time dependence of the z-component of the intrinsic angular momentum L of one fragment represents a damped oscillation with the frequency of the giant dipole resonance. If the interaction time exceeds 1.5 x lo-*‘s, with reasonable parameter values the corresponding lifetime of the excited RPA mode is short enough to achieve a nearly complete damping of the oscillation. Then, the final value of the intrinsic angular momentum determines the fraction of the dissipated angular momentum carried by one fragment. For trajectories with smaller interaction times, at the end of the reaction the intrinsic angular momentum is still in an oscillatory stage. In order to extract a definite dissipated angular momentum, a time average of L has been performed, which simulates the fluctuation of the collective variables, specifying the relative motion around the classical mean value. In such a procedure, a gaussian distribution function has been used with the interaction time given by the classical dynamical model as a mean value and with a standard deviation leading to about 907” of the normalization of the weight function if intergrated over one period of the oscillation. For impact parameters near grazing no well-defined dissipated angular momentum can be derived within the present approach. In fig. 1 the calculated time dependence of the intrinsic fragment spin is E = 8 MeV/nucleon, and 208Pb+20sPb, displayed for the reactions 40Ca+40Ca, E = 7.6 MeV/nucleon, with an angular momentum Li in the entrance channel of Li = 72h and 50h, respectively. Furthermore, it can be seen from the figures, that the final intrinsic angular momentum only weakly depends on the width of the resonance if it is changed within a few MeV. So, the uncertainty in the choice of this parameters does not influence the final results significantly. For symmetric systems in the deep-inelastic region the intrinsic angular momentum L, carried by one fragment in the final channel grows linearly with the entrance angular momentum Li, L, = cLi. The results for the system 20BPb + ‘08Pb are shown in’ fig. 2. This behaviour of L, as a function of Li resembles a sticking situation, but with a slope of the computed straight line well below the value, which one expects from the sticking limit for two touching spheres. The weak angular momentum dissipation predicted within the present approach is explained by the fact, that the oscillator form chosen for the interaction potential restricts the
D. Janssen et al. 1 Microscopic description
175
h 15
10
5
0
-5 %r 15
.
2oepb + 208 Pb , E=1580,8
MeV, Li = 5071
i ‘,
10
5
0 4J I 1 95
-5
I ’ I I :” :
‘1, ,:
”
1P
145 5 /lo-%
Fig. 1. Calculated time dependence of the intrinsic fragment spin L for the reactions %a +‘%a, E = 320 MeV, and 208Pb+*08Pb, E = 1580.8 MeV, with an angular momentum L, in the entrance channel of L, = 72fi and 50h, respectively. The parameters are taken from table 1. In the lower part the dashed curve corresponds to a smaller damping width of fir, = 2 MeV.
to dipole modes only, while higher multipolarities are disregarded. Moreover, the effects of the mass transfer are disregarded completely. According to eq. (77), for symmetric systems the dissipated angular momentum is shared in equal proportions between both nuclei. For the various asymmetric systems investigated within the present approach it was found, that the total dissipated angular momentum is distributed between both fragments according to excitation
D. Janssen et al. 1 Microscopic description
176
L/h
20apb + 208Pb
, E = 1580,8 MeV B
25-
20. 15105-
0
50
I 150
100
200
Fig. 2. The *0sPb+208Pb reaction, E = 1580.8 MeV. intrinsic angular momentum L carried by one fragment m the exit channel as a function of angular momentum L, in the entrance channel: (A) present calculation; (B) from a classical frlctlon model.formulated in ref I’-‘).
the empirical
relation
(cf. table 2)
lSSS2.
(79)
In the classical model one has 6 = -3 for the sticking limit 15). So, in both pictures the heavier fragment carries more angular momentum than the lighter one. In conclusion, one can state that even under the drastic simplifications introduced the present approach allows one to reproduce some of the basic features observed for the angular momentum dissipation in heavy-ion collisions, a phenomenon which turns out to underlying reaction mechanism r6).
be of crucial
The authors would like to thank H. Reinhardt, valuable discussions. The assistance of R. Schmidt trajectory code is gratefully acknowledged.
importance
in
P. Schuck and and J. Teichert
clarifying
the
H. Schulz for in using their
Appendix A In order to illustrate
the derivation
of classical
equations
for the relative
motion,
171
D. Janssen et al. / Microscopic description
TABLE2 Calculated angular momentum dissipation for asymmetric systems [(a) projectile, (b) target, i(f) entrance (exit) channel. AL = @+I!:‘]
‘Wa +‘OZr (E = 8 MeV/nucleon. L,, = 91h, L,, = 141h) 96 101 106
2 2 2
26 27 27
13.0 13.5 13.5
28 29 29
23 23 23
31 41 52 64
21 28 36 43
“‘Kr + “‘“Pb (E = 7 MeV/nucleon. L,, = 46h, I,,, = 254h)
1
72 98 124 150
30 39 50 62
2 2 2
30.0 19.5 25.0 31.0
“t From the classical friction model.
the general equations for the RPA amplitudes for the subsystem a,
64.1)
are specified for the spurious mode connected with the linear momentum, v = p, x, = 8, ‘pp = -i N[r$‘), $1, JV = l/N,, cop = 0. The excitation of this mode would correspond to a finite velocity in the intrinsic system. It has to be eliminated by demanding up = 0, which implies conditions for the time dependence of the parameters of the transformation U. Also for the further spurious modes one has a, = 0, except for the angular momentum mode aL_. But in the latter case the normalization factor V = (Tr(L,L,)j -r goes to zero* for an infinite con~guration space, while the remaining factors like Tr(jI[~Vr~“)- Wrb”‘,LJZY’)) have a finite value. So, also the angular momentum mode does not contribute to the sum, which can then be restricted to the ph modes with w,. # 0. According to these considerations one has from eq. (A.11
+Tr
Vrp) - FVrp), r1p)
fi (
[s
178
D. Janssen et al. / Microscopic description + ‘JWCCL,
dr,]&sin
P, - [Lyr dralba - CL,, dral&, ~0s P,f 1
The parts, in which no interaction terms are involved, can be evaluated explicitly. Some of the traces in eq. (A.2) are vanishing: Tr $. (II
R-l%R,r(o)
1) = Tr([fi,R-l&R]r(o))
= 0,
&R.&l) = 0,
Tr(@.[l?-t
(A-3)
Tr(j?. [R-“S?GR,&]) = Tr([#, R-‘fR]Gr)
= const x Tr(&) = 0,
(A.4)
Tr(${ [L,, Jr]& sin p - [L,, Sr]j - [L,, &]h cos a>), = Tr({[j, L,]hsin~-[$,
Ly]~-[~,Lz]hcosj?~6r)
= 0
(A.5)
(with the help of Tr(@r) = 0). For the other part one obtains $Tr(fi* a
= &
[R,‘fR,,rfP’])
= $Tr(@. a
UD, ’ Tr(jl . [a, ri*)]) = $f
a = !$Trr;O’ a
[UD;ld,r~o)])
Tr([#, &]ri”)) a
zz - iD,p.
(A.61
Inserting this expression in eq. (A.21, one gets
I)
Vr’,O’ - D a Wrp), r(O) a
I) &) ,
Vr(boj- DaWr(bDb,D=cpP
a,
I/@’
p
- D, Wcp~b),r(aO)
(A.7)
with
xx’
=
6(x-x’)
dx” V(lx- x”+ qj)r,(D,x”,
D,x”),
(A.81
179
D. Janssen et al. / Microscopic descrrption
(DaWrb)xxP
=
In the following
~lJDb(x+q),DtJx’+q))V(IX-x’l).
- ip
exp
the evaluation
of an expression
64.9)
of the form
I)
v$b-Daw$b3Da$a will be demonstrated, where $,(x,x’), space representation the first part
$b(~, x’) represent
arbitrary
operators.
In
reads
The arrow leads to
A second
=-
marks
partial
the action
integration
of the differential
operator.
4 first partial
integration
gives
s
dxdx’ V(lx - x’ + ql)ll/b(Dbx’, D,x’) & i,b,(D,:, D,f).
From a further partial to a q-differentiation)
I, =
s
= $
integration
one gets (after converting
dxdx’ & V(l: -x’ + ql)$,(Dbxr,
s
Dbx’)$,(D,x,
dxdx’ v(lx -x’ + ql)ll/b(Dbx’, DbX’)$,(D,X,
the x-differentiation
D,x)
D,x).
(A.lO)
180
D. Janssen et al.
! Microscopic descrrption
Similarly, the exchange interaction part, I, = Tr $[D.wI&.D&J (
= )
dxdx’%6(:-w’)[D,Wi,,D,~,l,,,, s
can be expressed by partial integration according to
+exp
-ip ( x v%.@w’ + 41, &C~ + P)M,aJ,
Using the relation
Q’)
.
D. Janssen et al. { Microscopic description
181
one finds
- ip * +!$-
x +ba(Dax’,D,x) exp
a
=
(x-d,)
V(jx-x’l)
b
dxdx’ ~b(~b(~ +q), &,(x’ + 4 ))~=(~~~‘, D&)
A
dq s xexp
(A.ll)
-ip* i
Altogether, eqs. (A.7~(A.l1) momentum.
leads to the equation
of motion
for the relative
Appendix B In a spherical basis the equations of motion for a, in the ph channel with w, # 0 have the following form: ih, = (CO,-- Mkcos
/?)a,-
~Tr(~:$~~_?(2+09,~) Y
f)J(J+l)j-M(M-l)(hsinB+igfa,,_,, +&/@YijKG(M+
l)(&sinj?-ij3)a,,+1,
(v G JM&). The restriction vanishing :
(B.1)
to dipole excitation
ia loa
=
QvlOE
-
means that only modes with J = 1 are not
Tr(cp~,&noZ(z2 i- Dq)‘)
5
E
+&/5(&sinp+$)a,_,,+(hsin/3-$)a,,,,
ial_,,
= (o,+Ercosj3)a1_,,+~fi(&sin~-$)alo8.
~Tr(~~_,,fmw’(P+Dq)‘) E
(B.2)
182
D. Janssen
et af. j microscopic ~s~r~~tion
Analogous equations exist for the modes with F&O,= -CO,). First one has to calculate TM = Tr(cp:,,$ mw2(~++q)*). With the help of the orthogonality relation Tr(lT;,7;.,,) * 6,,&,. one gets
TM= Trfp :&&tct * q)mw2.
(8.3)
After an expansion of p in terms of oscillator wave functions and xDq in terms of spherical harmonics one obtains
with
81
= - i/$sin(B+fi)eiCt”+“~,
K*
= cos(8 4 /?I,
tB.5)
The next step is the explicit calculation of the six remaining RPA amplitudes a, which are represented in a rotating system: ii, =
c Dycap.
(B-7)
P
The indices V,,Uare characterized
by the projection quantum numbers only:
with
A comparison with eq. (B.l) gives (B. 10)
183
D. Janssen et al. / Microscopic description
This is equivalent
to
= *, 1 Dvaap - 1 D,, $ TV. P B P The solution
of eq. (B.11) is
fs = sf
dr eLwcr1 D,, 2
1 Dvpap = ieeLWJ P The transformation
in the original
a,(r)
0
P
system
leads to
dz eiwer5
ie-l"J
TV.
(B.ll)
(B. 12)
bEI
TV.
(B.13)
0
Note that the start condition a,(O) = 0 is assumed. each particular amplitude is given:
At last the representation
of
f a-,
= i/$X
e-w
dT elwcrqsin(B+ j?)e’(4+“), s0
a o = iXe-1%’
t
dr erwcrqcos(0 + p), s 0 f
a 1=
-
i/$Xe-lWJ
dze’“~*qsin(8+p)e-‘(~+~),
(B. 14)
s 0
with
The relative distance q and the angles 9, C$ of the relative motion as well as the Euler angles CC,p are functions of r. One gets the amplitudes for the modes with E’ from the relation
afM, = (- l)“al_M,.
References 1) J. W. Negele, Rev. Mod. Phys. 54 (1982) 913 2) H. Reinhardt, Nucl. Phys. A390 (1982) 70
(B.15)
184 3) 4) 5) 6)
7) 8) 9) IO) 11) 12) 13) 14) 15) 16)
D. Janssen et al. / Microscopic description R. A Broglia et al., Phys. Lett. 89B (1979) 22 L. Mtinchow, A. Pfitzner and H. Schulz, 2. Phys. A296 (1980) 55 B. C. Sinha and P. J. Siemens, Phys. Lett 65B (1976) 5 R. A. Broglia, C. H. Dasso and A. Wmther, Proc. Int School of Physics “Enrtco Fermi”, Varenna. 1979. ed. R. A. Broglia, R. A. Ricci and C. H. Dasso (North-Holland, Amsterdam, 1981) p. 327 R. A. Broglia et al., Phys. Rev. Lett 43 (1979) 1649 V. G. Zelevmski, Nucl. Phys. A344 (1980) 109 L. van Hove, Physica 21 (1955) 517 R. Schmtdt and J. Teichert, Dubna reports E4-80-527 and E4-80-735 (1980) D. H. E. Gross and H. Kalinowskt, Phys. Lett. 48B (1974) 302 G. E. Brown, Unified theory of nuclear models (North-Holland, Amsterdam, 1964) p. 47 B. L. Berman and S. C. Fultz, Rev. Mod. Phys. 47 (1975) 713 P. F. Bortignon and R A. Broglia. Nucl. Phys. A371 (1981) 405 M. Lefort and C Ngo, RIV. Nuovo Cim. 2 (1979) 1 R. Reif and G. Saupe, Physics of Elementary Parttcles and Atomic Nuclei 14 (1983) 900 [in Russian]
Physics
A425 (1984) 152-184
North-Holland
Publishing
Company
MICROSCOPIC
DESCRIPTION
IN TERMS
OF INTERACTING
D. JANSSEN Technische
Universitiit
OF HEAVY-ION
+, M. MILITZER
Dresden, Sektion Received
Physik,
RPA MODES and
R. REIF
8027 Dresden,
31 October
COLLISIONS
Mommsenstr.
13, DDR
1983
Abstract: Starting from the TDHF equation, the equations of motion for the single-particle density matrices refering to projectile and target are derived in an approximative way in a translating and rotating reference frame. An expansion of the fluctuating part of the transformed density in terms of generalized RPA modes in the ph as well as the pp(hh) channel leads to a system of non-linear coupled equations which simultaneously determine the time development of the collective coordmates and momenta of the relative motion, of the Intrinsic angular momentum, and of the amplitudes of coherent surface excitations of the fragments. In a schematic calculatlon the formalism is applied to the angular momentum dissipation in the excitation of damped giant dipole modes.
1. Introduction In heavy-ion collisions with specific bombarding energies below the Fermi energy it is reasonable to expect that the ion-ion interaction is dominated by a collective response of the nuclei with respect to the time-dependent outer field provided by the approaching collision partner. In such a situation the reaction mechanism should be governed by strong mean-field effects, which in a semi-classical manner can be treated satisfactorily in th6 time-dependent Hartree-Fock (TDHF) point of view, the excitation of coherent approximation l, 2). F rom a phisical surface modes (as giant multipole resonances) instead of independent particle-hole pairs could play a decisive role in the approach phase of the reaction, as one can imagine from the analysis of inelastic scattering processes in distant collisions [see e.g. ref. “)I. During the development of the system from the entrance channel to a highly excited compound system such coherent excitations may act as doorway which are subsequently damped by decaying to more complex states, configurations 4). This process leads to a thermalization of the excitation energy, transferred from the relative motion to collective degrees of freedom in the initial stage of the reaction. As was suggested in the framework of linear response
+ Permanent 19, DDR.
address:
Zentralinstitut
fiir Kernforschung 152
Rossendorf,
805 1 Dresden,
PostschlieBfach
153
D. Janssen et al. / Microscopic description
theory 5, for more slowing
down
distant
of the relative
collisions motion
in which
the nuclei
by frictional
keep
forces proceeds
their
identity,
mainly
the
by such
an indirect (one-body) mechanism in which a coherent motion is first excited and then dampled by further interactions. Within this physical picture Broglia, Dasso and Winther6) successfully worked out a phenomenological model for binary deep-inelastic collisions, which for the main reaction characteristics gives results closely related to the TDHF approximation. In this model a nuclear surface-surface interaction, which is empirically fixed from scattering data, couples the relative motion to harmonic surface modes. The relative motion as well as the surface degrees of freedom are treated classically, with the restoring force and mass parameters of the collective vibrations being determined from quanta1 calculations of the nuclear response function. The approach is further improved by taking into account an incoherent particle transfer through the contact zone of both nuclei, which is represented by an additional frictional force. Furthermore, the zero-point fluctuations associated with the low-lying surface modes have been introduced by a set of random initial conditions for integrating the coupled equations of motion. The model has been applied successfully to interpret various deep-inelastic reactions, including angularmomentum-dissipation phenomena ‘). The present paper proposes a description of heavy-ion collisions, which in its basic features represents a microscopic version of the coherent surface excitation model of Broglia, Dasso and Winther. Starting from the TDHF equation for the total single-particle density of the system, and assuming negligible mass transfer as well as small overlap of both nuclear densities during the course of the reaction, we set up approximate equations for the partial single-particle densities, referring to both fragments. The time development of both components is coupled by the common single-particle hamiltonian. According to the simplifications introduced in such a model, its applicability is restricted to a class of more distant processes in which it is possible to distinguish between the fields and nucleons of two nuclei. In such a situation it is meaningful to introduce the collective degrees of freedom associated with the relative motion, which then can be treated formally by a suitable unitary transformation of the partial single-particle densities to translating and rotating reference frames, connected with each nucleus. The parameters of these transformations are taken to be functions of time. Within the intrinsic system, each density is decomposed in a (spherical) stationary field and a part fluctuating in time. For the time-dependent component of the single-particle density one can derive equations of motion, which contain the RPA operator as the dominating term. Therefore, it is reasonable to expand the fluctuating density in terms of the complete, orthonormalized set of generalized RPA solutions’), which have components in the ph as well as the pp(hh) channel. The time development of the expansion coefficients are governed by a set of coupled non-linear equations. The pp(hh) modes with finite frequency are disregarded, so that the basis is restricted to
154
the collective channel and
D. Janssen et al. / Microscopic description
and non-collective RPA modes with finite to relevant spurious modes in the ph and
frequencies in the ph pp(hh) channels. The
spurious mode in the ph channel is associated with the momentum operator. In the intrinsic system this mode, which represents the translation of the nucleus as a whole, is not excited. Its elimination from the RPA spectrum fixes the parameters of the Galilei transformation to the moving reference frame in terms of newtonian equations for the relative motion. In linear approximation, the resulting inertia parameter deviates from the classical reduced mass by terms implying the exchange interaction only. The potential of the force acting on the relative motion consists out of a folding potential and an expression, which provides a coupling to the RPA modes of finite frequency in the ph channel, in some approximation not completely invariant under time-reversal. Assuming a spherical HF basis the spurious mode in the pp(hh) channel is associated with the intrinsic angular momentum of the fragment, so that the time development of its amplitude describes the transfer of angular momentum from the relative motion to intrinsic rotation. The remaining equations for the RPA amplitudes of finite frequency describe coherent vibrations of one fragment within a moving reference frame, influenced by a potential provided by the overlap of the ground-state densities of both nuclei and coupling each particular RPA mode to all other excitations in both nuclei. Introducing statistical aspects, one can find a solution for the amplitudes, which corresponds to a classical damped oscillator in an outer field, shifted in frequency. The resulting set of equations for the relative motion, the transfer of angular momentum and the excitation of surface modes has to be solved simultaneously. So, the main advantage of the present approach is that the collective degrees of freedom associated with the relative motion and the excitation of coherent surface modes can be treated in a unified way by specifying an equation, which has been derived from an universal expansion of the single-particle density in terms of various RPA modes. In sect. 2 the general formalism of the approach is presented, including the unitary transformation to intrinsic reference frames and an expansion of the fluctuating part of the transformed density matrix elements in terms of interacting RPA modes. Dynamical equations for the relative motion, for the transfer of angular momentum, and for the amplitudes of coherent excitations are derived in sect. 3. Based on a schematic model, sect. 4 contains an application of the formalism to the angular momentum dissipation in the excitation of damped dipole modes in various symmetric and asymmetric systems.
2. General formalism We treat a system of N nucleons, interacting via velocity- and spin-independent, translational invariant two-body forces I/(1x - x’j), where X, x’ denote the nucleonic
155
D. Janssen et al. 1 Microscopic description
space
coordinates.
In
TDHF-approximation
described
by a time-dependent
equations
(fi = 1)
The operator h(p) is a single-particle given by (m is the nucleon mass)
density
p2 = p,
q-J= [h(PLP].
0 I,,, = -
the development
single-particle
hamiltonian,
matrix
Trp
of the
system
p(x, x’, t) obeying
= N.
is the
(1)
which in space representation
is
& 6(x-x’)+d(x-x’)
-p(x,
x’)V(Ix
-x/l).
(2)
In order to describe the dynamics of a heavy-ion reaction the solution of eq. (1) has to fullil an initial condition, which corresponds to an entrance channel situation with a definite partition of the total nucleon number N in two fragments N,, N,(N,+ N, = N) with both nuclei localized at a relative position R = R,- R, far outside the interaction region and possessing a relative velocity u = u, - ut, and a corresponding relative momentum p in the c.m. system. Then for the free initial state the suitable solution is a uniform translation p(x,
x’, t) =
exp
(.
z(x - x ’ ) .uy) N
pg”(x
R, - u,t, x’ -R,
p’bo’(x -R,-
- i(x - x’) . r
+ exp
-
-
u,t, x’ -R,
vat)
- u,t)
b>
pk(‘)
of the stationary HF solution calculated in the corresponding
[h(plP’), P:“‘] According decomposed a, b: p(x, x’, equation (1). the interaction
for the ground rest frame, COP= p’o’ k ,
= 0,
PA
states
of target
Tr(pjp’) = N,,
and
k = a, b.
projectile,
(4)
to eq. (3), at the beginning of the reaction the total density p is into two non-overlapping parts, referring to the collision partners t = 0) = pa(x, x’, t = O)+p,(x, x’, t = 0). It follows from the TDHF that the time development of each component during the course of can be described by a set of coupled equations
@a = [h(P, +b
=
+ Pb)r
Pa]+ *%‘a~Pb).
INpa+ Pb), Pbl - w(pa,Pb).
(5)
156
I). Janssen et al. i Microscopic description
These equations define a particular subdivision of p:
P(t) = Pa(t)+Pb(t),
(6)
which is determined by eq, (3) iu the initial channel, for the whole reaction time. The single-particle operator ti. is a complicated functional of the densities pa, pb and the two-particle interaction. Because its trace is connected with the time derivation of the particle number in p,, P,,, &, = Tr@,(t) = Tr ‘Be,
rii, = A$,
(7)
the operator We is responsible for the mutual exchange of particles between the collision partners, which may proceed in the interaction region. The operator 0‘ can be omitted exactly for symmetric systems, for which mass transfer is forbidden because of the conservation of the reflection symmetry of the initial density. For target-projectile combinations, bombarding energies and impact parameters, which in the experiment exhibit a small spread in the mass number of the final fragments only, one can disregard the mass transfer by setting $6’”= 0. Furthermore, for more peripheral collisions the relative motion allows only a small overlap of both densities during the whole course of the reaction, pbpa +p,p, z 0, so that the condition @, + p,J2 = pa + pb can be fulfilled approximately, if the subsidary conditions pk’ =
pk.
Trp,
=
k = a, b,
Nk,
0)
are imposed. One can show, that these equations and the condition ~4. = 0 are quite equivalent. It means that one seeks for approximate solutions of eq. (5) (with tiL^E 0) for pa, P,,, which correspond to determinantal wave functions for the subsystems a, b. So we replace the full TDHF problem formulated in eq. (1) by the following set of coupled equations : ibb = [4P, +PbhPbl P; =
Pbr
Trp, = N,.
(9)
Together with the initial condition (3) it reflects the following physical picture: projectile and target approach each other freely in their HF ground state with given relative velocity and impact parameter. When the HF potential generated by the wave functions in nucleus b starts to overlap with the wave functions in nucleus a it introduces excitations of nucleus a, and vice versa. If the further relative motion proceeds in such a way that the oscillating density dist~butions show only a small overlap, the identity of the collision partners is preserved throughout the reaction and, neglecting the mass transfer, the intrinsic excitation
D. Janssen et al. J Microscopic description
157
can be described as changes of a determinantal wave function for each fragment. Such a picture seems to be sufficient if one is looking for the expection values of single-particle operators in more peripheral collisions of closed-shell nuclei, for which the dominating excitations are rather simple and the width of the final mass distribution is expected to be small. 2.1. INTRINSIC
REFERENCE
FRAMES
In order to separate out the relative motion as well as the intrinsic rotation of the interacting nuclei from the full time development of p, and pi,, the structure of eq. (9) suggests a separate unitary transformation U,, U, into a moving reference frame for each fragment :
r, = Q?JJ,l, The transformation
operators
t-b
=
u,p,u,?
(10)
U,, Vi,,
U, = R;‘exp((*%
U, = R;Lexp(-q%
&)exp(-i$+*),
i)exp(i$.s),
(11)
contain a Galilei transformation as well as a rotation. Because in the cm. system both nuclei are approaching each other with equaf but opposite momenta, the galilean parts of U,, U, are related, and the iotation is expressed by the components of the angular momentum operator Las
According to eqs. (11) and (12) the transformation has to be specified by 12 free parameters q(t), p(t), I, fik(t), y&t), k = a, b, which will be fixed in a convenient manner later on. As a function of time, these parameters describe the translation and rotation of the reference frames with respect to the c.m. system. The transition to the moving coordinate system in eqs. (9) is demonstrated for pa, In order to simplify the notation, the index a is omitted but we emphasize, that in this context p should not be confused with the total density matrix. From ifi = d(U-‘r~~~dt, with the help of the relations
a. X =X+q, +.
( 1
exp q*G
h = - iR(h(cos #?Lz- sin fi(cos yL, -sin yL,)) + &cos y L, +sin yL,) + j$,),
(13)
D. Janssen et al. / Microscopic description
158
one gets
1
\
+ ii + j[L,, r] + &cos yL, + sin yL,, r]
+ k[cos flL, - sin /?(cos yL, -sin yL,), r]
U,
(14)
} which can be solved for i,
- [L,, r]($ + ir cos p) - [L,, r](fi cos y + 6 sin fi sin y) - [A,, r](b sin y - h sin fi cos y).
(15)
In eq. (15) ib has to be replaced by the commutator [h(p+&),
p] = - & [d, U-‘rU]
+ [ bx”
I/U-‘rU,
u-‘r(r]
1 -[U-'r~I/,U-'rU]-[ (16) dx” VU,‘r,U,,
+
U-‘rU
[s
In order to evaluate the various terms of eq. (16) one can utilize the relations
C-4RI = 0, [A,exp(ip*9)]
= -pzexp(ip.f)+2iexp(ip*4)pda;,
(&,)xx’r&, UR\,
’ A,, = r,(D,x,
(17)
&,x’).
The rotation matrix describing the effects of R, is denoted by D,. In this way one ends up with an equation
of motion
if, = [h(r,), r.J - [LX, r&l,
for the density
matrix
ra in the moving
frame:
sin ya- h, sin fi, cos yJ
- [L, rJ(Ba cos Y,+ ka sin B, sin y,) - [L, ra](PB+ k cos 8,)
+
[s 1 vrb,
Ia
-
CWrby
ra]-
W)
159
D. Janssen et al. / Microscopic description
Up to the first term on the r.h.s., one obtains
the TDHF
equation
for the isolated,
unperturbed fragment a. The next three expressions represent Coriolis forces, resulting from a description of the single-particle motion within a rotating coordinate system. Then one has two terms, appearing because of the translation performed in coordinate and momentum space. The interaction between both nuclei is grasped in the remaining terms, consisting of a direct part with
=
6(x - x’) s
XX’ and an exchange
dx” V(lD, lx -Da lx” + qI)r,(x”, x”)
(19)
part with
- ip . T+“l(X-Xf)
( Wrb)xxj = exp
a
b
These interactions receive a complicated time dependence because of the parameters of the transformations U,, vi,, involved in the expressions (19) and (20). An analogous equation can be derived for the component rb.
2.2. DENSITY
FLUCTUATIONS
In the next step each density obeying the stationary fluctuating in time
matrix HF
rk(O),
rk = rip’+ h,(t),
rar rb is subdivided
equation
Tr(Gr,(t))
into a stationary
[h(r’,O)),ri”)] = 0, and
= 0,
a part
k = a, b.
part rk(t)
(21)
The matrix rk has to be traceless, because the particle number is conserved. Introducing eq. (21) in (18) gives an equation of motion for h,(t), which is presented for k = a: dr, = Kc%, + [s(dr,), 6rJ - %?+ 9 + Y. Here, K denotes
the non-hermitean
the interaction
s@,$,,= 6(x-x’)
RPA operator,
= [h(rlp)), Sr,] + [s(Gr,), r:O)],
Sir, and s abbreviates
(22)
s
part of the single-particle
(23) hamiltonian:
dx” V(lx - x”l)br,(x”, x”) - 6r,(x, x’)V(lx - ~‘1).
(24)
160
D. Janssen et al. { Microscopic description
While the contributions
G9and $9 to eq. (22) arise because of the transformation
U,
Q?= [L,, r.J(j, sin y8-ix, sin /3, cos v,) +
CL,,~JtP,cosYa+ 4
sin8,sinY,)+ [L,, r,](& + rjlscm A),
(25)
the last term $’
=
[S
Vrb”’-
Wr’,O’,
1 [s
+ r(O) a + 6r a
VSr, - W&r,
7
rp +Sr
a
I
(27)
expresses the (direct and exchange) interaction between the colliding nuclei. The parts like [f Vri’), r,,+Sr,,] couple the excitations of the fragment k’ to the groundstate HF density of the fragment k, which acts as an outer field. In this sense transitions to excited states occur by a one-body mechanism of excitation. The terms like ~~V~r~,~~~)~~r~,] describe the way, in which the excitations of the fragment k are influenced by the excited fragment k’. 2.3. INTERACTING
RPA MODES
Omitting the index k, the solution of eq. (22) for r, can be sketched as follows: Solving the eigenvalue problem for the RPA operator K, which dominates the equation of motion for 6r, one finds all possible RPA modes 40, with zero and finite frequency w, with components in the ph and pp, hh channels, cp, = {@“, &?, cpt”>.This set 9, can be constructed in such a way, that it can serve as a complete, orthonormalized basis of two-particle wave functions to expand the ~u~tuati~g part 6r of P, which in general is not supposed to be small:
The time-dependent expansion coefficients a,(t) determine the contribution of each collective, non-collective and spurious RPA mode to the change of the density matrix in time, with a substructure according to the various channels, a, = {LZC”, alp, ut”>. From eq. (22) a set of coupled non-linear, first-order differential equations for these RPA amplitudes can be derived, which should be handled further. At first, the structure of the basis 1;~~is discussed in some detail for one subsystem, omitting the index k.
161
D. Janssen et al. / Microscopic description
Because the RPA operator the eigenvalue problem as
K (eq. (23)) is not self-adjoint,
Kc =
one has to formulate
K+Xy = w,x,,.
W/P,.
(29)
The possible solutions to these equations can be characterized in the following way: (i) all frequencies co, are real numbers; (ii) for each solution o,, (py, x, with o, f 0 there exists a second solution oB, cps, xi with w,- = -w,, cpi = cp:, (iii) if the components in the pp and hh channels are taken into account, XC= x: ; the set (p,,, x, formes a complete basis in the space functions. which can be orthonormalized according to
of all two-particle
wave
(30) ph channel:
Because the translational symmetry is broken in the TDHF-initialvalue problem, the momentum operator has non-vanishing matrix elements in the ph channel, giving rise to a spurious RPA mode (0, = 0):
_&
xSPph=fi= These solutions
‘pph=
SP
i
Y[P’,
solution
(31)
of eq. (29) are orthogonal,
Tr(X,“,h+&‘) = Tr(i V@. [r(O), a]) = -Tr(i A further
a].
can be constructed x-ph
=
according
Vr(‘)[@, a]) = 0.
(32)
to
cpph= -i. V[r’O’, a],
f ,
(33)
for which the equations
K+fPh
=
&ph jm
‘P ’
hold. Again,
1 Kcj ph -- _ _ (ps”,”
one has Tr(Xph+(pph) = 0. If one arranges xpph -_ p,A
(34)
im
these solutions
as
cp;” = -i. V[r(O), a],
xpph = f ,
one has the same orthogonality condition for the spurious modes as for phonon factor appearing excitations, Tr(X,ph+(p;h) = Tr(Xsph+(pFh) = 0, and the normalization
162
D. Janssen et ai. / Microscopic
description
in eq. (35) has to be determined from Tr(X;hfrp;h) = Tr(Xih+(pqPh)= 1 leading to V = I/N. All other solutions cpth in the ph channel with finite frequencies o, belong to non-collective and collective modes, which can be interpreted as coherent surface excitations of the nucleus. pi channel: In the pp(hh) channel the spurious mode connected with the particle-number operator is not excited, because the particle number is conserved (Tr(Gr) = 0). If one restricts the consideration to spherical HF ground states, another spurious mode in the pp channel can be created with the angular momentum operator, which has no non-vanishing matrix elements in the ph channel, so that the rotational symmetry is not broken, [r(O), L] = 0: 1,” = L K'
PP (PK
1 ---.-Lx,
ic =
x,y,z.
(36)
'JWLL)
In a representation, in which r (‘I is diagonal, the pp and hh matrix elements of 6r rco)&rtob and G’O%rG (‘) (G “) = 1 -r(O))can be expressed by the ph and hp matrix elements Gco)&rco) and rco%dO’, as follows from the subsidary condition r2 = r, which give 6r = r’0’6r+6rr’0’+Srdr. Multiplying from the left and from the right with the projection operators r(O) and a”‘, respectively, the first two terms disappear because of G(o)r(o)= r(o)g(o) =: 0 and from the last term one gets with 6r6r = Gr(G’“’
+ r”‘)&
o(0)~rrCOl(l
- fl(O)ijrg(O)) = (G(O)Srr(O))(r(O)Gro(O))
- r(0)&r(O)( 1 + r(*)&yJO)) = (r(“)~rG(o))(G(o)sTT(o)).
(37)
Therefore, the RPA amplitudes u~P(~~)in the pp(hh) channel can be expanded in the form uppW) Y
=
+ higher-order EA$E!hyh)a$hullph
terms.
(38)
In lowest order the amplitudes in the pp(hh) channel are quadratic in the amplitudes of the ph channel, so that they are expected to be of minor importance, if one restricts the considerations to small-amplitude motions. Furthermore, if the ph space has large dimension, the random phases of the ph amplitudes guarantee that the sum in eq. (38) remains small because of the destructive coherence of the superposition. But in general, by expanding the single-particle density matrix in terms of generalized RPA modes, the multiple-particle-multiple-hole excitations contained in the TDHF many-body wave function have to be taken into account by the amplitudes in the pp(hh) channel. In this case it is not necessary to restrict 6~ for a small-amplitude limit.
D. Jamsen et al. / Microscopic description
163
Inserting the expansion (28) in the equation of motion (22) for c%, multiplying from the left with x,’ and taking the trace, one can derive first-order, non-linear coupled equations for the RPA amplitudes of the fragment a: ib, = o,a, - %.(r(O))+ ~v(~(o))+ 9”v(#O),Q)
(39) with U,($) = Tr{X,?([L, $](a sin y-dcsin pcos y) + [&, $](B cos y + h sin B cos y)+ [Lz, $f]($ + tftcos /?))I, S&,4) = k+Tr{$[R-IfR, a
$]f
One should keep in mind, that for v = 4 in the first term on the r.h.s. w,a, has to replaced by a,/im because of the special features (34) of the spurious modes’ in the ph channel.
3. Dynamical equations Supposing a spherical HF ground state for both nuclei a, b, the set of equations (39) is handled by eliminating exactly the spurious modes pp, (p4 in the ph channel and imposing the condition up = Tr(‘Gr) = 0, a4 = Tr(26r) = 0 (Tr(~~(‘)) = 0, Tr(fr(“) = 0). These conditions imply, that the expectation values of the operators _? and fi have to vanish, if calculated with the density matrix referring to the moving coordinate systems. This is the case if these reference frames are fixed in the c.m. of the nuclei, so that the parameters q(t) and p(t) take on the physical
D. Janssen et a/. j Microscopic description
164
meaning of the coordinates and the momentum of the relative motion of the fragments, respectively. A specification of eq. (39) for the modes v = p, q leads to equations, which determine the relative motion q(t), p(t) during the collision in terms of conservative mean potentials and excitation of RPA modes in the ph channel. On the other hand, the expectation value of the component
L,=-i
d
( > A
xxdx
K
in the c.m. system is
6%) = TrbQ,) +JW%+J which after transformation function)
(40)
in the intrinsic system takes the form (D is the Wigner
Now, the intrinsic system is chosen in such a way that its z-axis is oriented along the direction of the angular momentum, so that the expectation value for the x- and y-components vanish : QJk
= Tr(L,r,) = 0,
(LJk = Tr(L,r,) = 0.
(42)
The absolute value of the expectation value of fit) gives the magnitude of the intrinsic angular momentum of fragment k: (LJk = Tr(L,r,) E d”(t).
(43)
As a consequence of this choice of the intrinsic reference frame, the amplitudes u(x), uCk) Y at the RPA modes belonging to L,, L, vanish identically, while the RPA amplitude a:)(t) z E@(t) gives the intrinsic angular momentum, finally. From a specification of eq. (39) for the angular momentum modes, equations for the time development of the absolute value I; of the intrinsic angular momentum and its orientation angles CQ,&(yk = 0) can be derived. For the sake of simplicity, in all of these equations resulting for the translational and rotational motion of the fragments the terms quadratic in the amplitudes a, are cancelled. Afterwards, if all pp(hh) components are omitted, one is left with the equations of the amplitudes if the RPA modes of finite frequencies in the ph channel, which have to be solved simultaneously with the equation of motion for the trajectory and the intrinsic rotation. In conclusion, the sum over v in eq. (28) is restricted to the RPA modes with finite frequency in the ph channel and to the relevant spurious modes in the pp(hh) channel.
165
D. Janssen et al. / Macroscopic description
In the following, only the main results of this procedure details of the derivation are given in the appendix A.
are presented, while
3.1. RELATIVE MOTION
For the relative distance q and the relative momentum obtains classical equations of motion 4 = p_x-1,
i,=
p of both nuclei one
(44)
- ii_ vin*> dq
with the inertia parameter .x-i
=: &(A
+p-~.~xdx’V(jx-x’J)Zxsin[~p,(x-x’)]
x r’bo’(x+ q, x’ + q)r’O’(x’, x) + c p - l (i4$&, P ~~(~~,~~)
r~))a, + Wq(Pf, ~~))Q~)~ ) i
dxdx’(x-x’)exp
= i
(
s
-ipa& a Jx-x’)
x $i(DX’, Dx)II/z,(k(x+4),
&(x’+q))*
(45a)
>
Vlx-x’l) (45b)
and the potential Vint =
dxdx’ r(O)(x,x)r~)(x’,
x’)V(lx
-x’
+ ql)
f
-
dxdx’ r(‘)(x’, x)rp)(x -+q, x’ + q) s
(46a) with
(46b)
166
D. Janssen et al. / Microscopic description
~,W,,&)
=
s
dxdx’~,(Dx’,Dw)~,(D,(x+q),D,fn’+q))
-ip
xexp (
. J!-(n -xg)) N,N,
V(jx -x’l),
The quantities a%), cp%’iefer to the subsystem b. In the inertia parameter .X the deviation from the classical reduced mass is caused by the exchange interaction only. The potential Vi,,, consists of two parts: a (conservative) folding potential, depending on the ground-state densities r(O),r$,‘),and a term VcouD which couples the relative motion to the phonon excitations in both nuclei. These equations have to be solved with initial conditions corres~nding to the entrance~hannel situation. 3.2. TRANSFER
OF ANGULAR
MOMENTUM
For the magnitude and the orientation one gets
of the angular momentum of fragment a
(47)
(48)
(49) A change of the intrinsic angular momentum of fragment a is possible because of the interaction excitation of surface modes. 3.3. DAMPED
SURFACE
EXCITATIONS
The remaining equations for the amplitudes of the RPA modes with finite frequency o, in the ph channel referring to fragment a are given by eqs. (39). The first four terms on the r.h.s. represent a harmonic vibration with w, in a translating and rotating reference frame, influenced by the time-dependent outer field, /provided by the overlap of nucleus a with the ground-state density of nucleus b. The Galilei term Y,(r(‘)) and the Coriolis term ‘%,,(r(“)) are vanishing because of the orthogonality of the RPA modes. The outer field Y’“,,(r(‘), r’,“)) can be expressed as
flry(r(0), r’,O))= -
-!!..K T ,o,,
(50)
161
D. Janssen et al. / Microscopic description
As implied
by the following
term in eq. (39) the particular
mode v is coupled
to all
other excitations in both nuclei, with a time dependence of the coupling strength governed by the time dependence of the relative motion. Already the transition to a moving reference frame creates a linear coupling of the various RPA amplitudes within one nucleus given by the expressions WV(cp,), YSV(cp,,).Its strength is determined by matrix elements of the space and momentum operator. For a spherical HF ground-state, the RPA modes can be specified by the angular momentum quantum numbers J, M and an additional quantum number E, v = JME, so that it is possible to derive explicit expressions for the Coriolis term %‘7y((pp)a,,utilizing angular momentum algebra. The main steps of this treatment are given for the x-component (y = 0): (p = J/M’&‘).
Cl”,’ = Tr(X: [&, cp,])h sin pa,,
L,, L_, L, = i(L+ + L_ ), one obtains
Using the shift operators appearing in eq. (51)
[IL,,(PJ'M'd]
= +{J(J’+ +
Inserting
this expression
l)(~~,~,+i~, (52)
in eq. (51) leads to
+J(J’+ to the orthogonality
Cc
for the commutator
(J’+l)J’-M’(M’-l)~J,M<_Ic,J.
CE = *h sin p Tr (xJ+ME (J(J’+
According to
l)J’-M’(M’+
(51)
l)J’-M’(M’+
l)(~~‘~,+i~,
l)J’-M’(M’-
l)cp,,,,_,,,))a,.,.,..
relations
(53)
(30) for the RPA modes, this is equivalent
= $ksinjI(J(J+l)J-M(M-l)a,,_,,
+J(J+ W-M(M+ In this way the equations
for the RPA amplitudes
+$J(J+
l)J-M(M-l)(hsinfl+ifl)aJM_,,
+4,/U+
l)J-M(M+l)(dcsinp-$)a,,+,,
+‘~?%Me((Pp)+ P
lbJM+lE).
“f
JME((P~?
r(b”)))up
+
(54)
get the following
1
YJME(r(o),
form:
$(?b:b’
P
(55)
168
I). Janssen et al. / Microscopic
description
with the indices p, I running over all RPA modes in the corresponding nuclei. One can state, that the Coriolis term couples the mode-JMa only to selected excitations with the same quantum numbers J, E, but differing in the projection quantum number M by k 1. A further source of linear coupling between the RPA amplitudes of nucleus a is the interaction between the excitation of a with the mean field of nucleus b in its ground state, VJcp,,rb*‘). An analogous mechanism connects the time development of aJME with all excitations in nucleus b, YJME(rco), q$“‘). Furthermore, two coupling terms appear which are bilinear in the amplitudes a,,, therefore implying an interaction between different RPA modes in the same nucleus fTr(X: [s(q,), cp,]} as well as in different nuclei ( Yy((pp, cplb))). While the strength of the coupling matrix elements is determined by the ion-ion interaction, and as such strongly depends on the relative distance, the bilinear coupling of RPA modes within the same nucleus is given by the two-body interaction s. Non-linearities of higher order would appear by taking into account the pp(hh) amplitudes to higher than second order in the ph amplitudes (cf. eq. (38)). Such terms would provide a coupling of the ph modes to multiple-particlemultiple-hole configurations. In order to stress the consequencies of the coupling terms between the RPA modes, the outer field V&r(‘), rb(O)), as well as the Coriolis terms, is disregarded for the moment. Then, the system of equations for the RPA amplitudes (55) gets the basic structure (56) For the diagonal elements one assumes 8,” = 0, which can be achieved always by renormalizing (&,, = o,+o,,). The transformation a,(t) = e-1%?,ft) eliminates the time dependence of the non-interacting oscillator modes
(57) In the formal solution of this equation, ii(t) = G(t)a(O), for the propagator
G, a perturbation
G(t) = Texp(
-i /:dr’O(i’)),
expansion with respect to 0 is performed:
(58)
D. Janssen et al. 1 Microscopic description
169
(59) In these integrals the time dependence of 0”’ and 0C2’ is determined by the collective coordinates q(t) and momenta p(t) of the relative motion, only. For lower bombarding energies the relative motion along the trajectory proceeds slowly compared to the period of a spherical vibrational mode, so that it should be a good approximation to keep the quantities 0”’ and O’*’ constant for the time integration in eqs. (59). Furthermore, even if the interactions 0’” and 0C2) are supposed to be small (O(i), O’*’ =: 5,5 K 1) for larger time values the product tt* becomes non-negligible and has to be taken into account. Under these conditions, G(i) and all other odd terms in the perturbation expansion cancel. In Gt2’ the integrand is a superposition of periodic functions which get out of phase for growing T, so that the integrand goes to zero and the upper boundary of the integral can be replaced by infinity [Van Hove limit ‘)I. Furthermore, we assume, that in the summation over /z and A’ the coherent part il’ = 1 gives the dominant contribution. With the ansatz a,(r) = e%-r,r one gets
Explicit expressions for A, and r, can easily be found for limiting cases. (a) tT ez 1. An expansion of the exponential in eq. (60) leads to
= G,,itE,. Because for higher-order
(61)
terms Gt$“) = ((it)“/n!)E:6,, G VP= e’%d
Inserting
this expression
holds, one obtains (62)
VP’
in eq. (58), a comparison
with eq. (61) yields
(63)
170
D. Janssen et al. / Microscopic description
1. The second immediately leads to (b)
tT B-
interaction
term
in eq. (60) goes to zero, which
(64) If one takes into account the force O:‘)(t) as an inhomogeneous term in eq. (56), in a similar way one can find a solution which corresponds to the amplitude of a classical damped oscillator in an outer field, shifted in frequency: f a,(t) = ei(-%+d,)t-r,t
a,(())- i
(
dt’O$)(t’)ei(O”t’-d,t’)+T,t’
s0
.
(65)
)
The damping factor e-rJ for the RPA mode v of fragment a is a consequence of (i) the decay of this mode to a more complex contiguration due to the interaction OC2) within a itself, and (ii) the interaction with the mean field as well as the RPA excitations in fragment b due to the interaction 0”). 3.4. FRICTIONAL
FORCES
Utilizing the expression a,(r) = _
i
’ dt’e(-iW~-r,)(t-t’)OlO)(t’) s 0
=
t&‘e-imv(t-t’)
-_1 s
{cash r,(t - t’) -sinh r,(t - t’)}O\“(t’),
(66)
0
in which the influence of the initial amplitude a,(O) has been neglected, to calculate the potential 9’“C0up of eq. (46a), which determines the coupling of the RPA modes to the relative motion, one finds cash
Y
Y + c+i Y
-
t’){
V,(cp,,
rb”‘)
‘dt’e-. l#(t-t’)cosh rlb)(r - t’){ V,(r(‘), s0
+ C-i
+ c +
r,(t
i
tdt’e-~W~(c-c’)sinhr,(t
-
W,(cp,,
rb’))}
cpib)) - W,(r'"', q',")))
- t’){ V,(cp,, rp’) -
W’(cp,, IL”‘)}
s0 ‘dt’e-. s0
Iwv “I cc-%inh
rtb)(t - t’){ V,(r(“), cp’,“))-
W,(r’“‘,
cpib))).
(67)
Il. Janssen et al. / Microscopic description
171
In first order the last two terms, violating time-reversal invariance, are proportional to the damping width rye In the classical equations of motion (44) they give raise to a frictional component of the force governing the relative motion.
4. Angular momentum dissipation 4.1. EXCITATION OF DIPOLE MODES In order to demonstrate the application of the present approach, the angular momentum dissipation as a function of the impact parameter is treated within a schematic model, utilizing further simplifications, so that the resulting equations to a large extent can be solved analytically. Generally, the exchange interaction has been neglected. Therefore, the equations relevant for the angular momentum transfer do not depend explicitly on the momentum p. But the trajectory q(t) is required, which is taken from calculations within a two-dimensional classical dynamical model lo), including a frictional force of Gross-Kalinowski type ir). Thus, the eqs. (47)-(49) are reduced to
&z/j=&
W
with the sum extended over all RPA modes with w, # 0 in the ph channel. For the mean interaction potential appearing in eqs. (68) a harmonic oscillator (cutted at the interaction radius qo) has been taken:
(69) with the parameters ho, = 41N,i MeV and q. = 1.36(N$+Ng)fm +OS fm for the oscillator frequency and the interaction radius, respectively. The ansatz (69) contains a coupling term, which only allows the excitation of dipole (and in the present context insignificant monopole) modes in the angular momentum transfer, while transitions with higher muttipolarity are excluded. Among the possible dipole transitions the giant dipole resonance should carry the most excitation strength, so that this partidular mode has been taken into account only. Thus, the relevant RPA amplitudes a,, a; have v = JMe with J = 1, M = 0, * 1 and v”= JME, the frequencies uy, CL+depending on the quantum numbers J, E and J, & respectively. For simplicity, the isospin character of the resonance has not been treated explicitly.
172
D. Jamsen et al./ Microscopic description
The RPA equations qp,(x,
x’) = i
I
&td-d’)+tmw,2(x2-x’2) c&(x,x’)
-
dx” (x - x’)w”cp,(x”, x”)Q
-t
(70)
s
with the dipole-coupling strength Q determine the basis elements (py, which are expanded in terms of the eigenfunctions $:, of the harmonic oscillator: (71) The index 1 summarizes the quantum numbers n (number of nodes of the radial wave function). I, m (angular momentum and its projection) of the oscillator state. For simplicity, the spin-orbit coupling has been neglected. From eq. (70) one obtains for the expansion coefficients i2)
(72) In this expression, N,,, denotes the occupation number of the single-particle state n, 1, and _flrdis a strength parameter, which implies the overlap of the radial wave function & in the orbits 2, p: ~
f,~ = 4,/(2l, + 1)W, + 1) (1,01,0110)
r3dr Rn,,,(r)R,,Jr).
(73)
The Clebsch-Gordan coefficient (~~0~~0110) gives the selection rules for parity and angular momentum in the ph excitations. Inserting eq. (69) in eq. (55), modified by the statistical treatment of the interaction terms leads to explicit expressions for the RPA amplitudes in terms of time integrals : (74) The evaluation of the trace yields, e.g. for alle (see appendix B),
D. Janssen et al. J Microscopic description
173
In this equation the 4(t) denote the polar angles of the trajectory q(t). If the z-axis is chosen to be perpendicular to the reaction plane, one has 0 = $r, and oz= 0, /? = 0, because within the classical model the dissipated angular momentum is aligned along this direction. Taking into account the damping of the giant resonance, the eqs. (68) are reduced to a formula for the intrinsic angular momentum L of one fragment along the z-axis,
which can be solved numerically. According to eq. (77) the time dependence of the intrinsic angular momentum carried by one fragment is determined by fi) a strength parameter F, characteristic for the individual structure of the nucleus, (ii) the trajectory specified by the relative distance 4, and the polar angle # in the reaction plane, and (iii) the decay width of the giant dipole resonance. 4.2. NUMERICAL
RESULTS
Numerical calculations have been performed for three symmetric systems with 40Ca, “Zr, ‘**Pb and for the asymmetric systems 40Ca+90Zr, 40Ca+ “*Pb, and 86Kr+208Pb in the energy range of 7-10 MeV/nucleon. The parameters for the isovector giant dipole resonance following from experiments on photoabsorption [see refs. r3, “)I are gathered in table 1 together with the strength parameter F computed according to eq. (76). The position of the resonance in Kr has been fixed in agreement with the empirical relation r3) hog’) = 77.9 MeV x N&l
-e-NJ238)+34.5
MeV x N,$e-NJ23*.
(78)
In order to simulate dissipation mechanisms other than the excitation of collective states which are damped by the coupling to more complicated configuration due to the residual interaction within one nucleus, the width ry as given in table 1 is increased arbitrarily by about 50% in the actual calculation. The corresponding parameter values are presented in column 4 in table 1. Because of the very strict simplifications underlying the calculations one can expect to reproduce only the general tendencies observed ex~rimentalIy in the angular momentum transfer. The numerical results obtained exhibit the following main features.
D. Janssen et al. / Microscopic description
174
TABLE 1 Parameters
‘Ya ‘OZr “‘Pb 86Kr
for numerlcal
calculations
ho, (MeV)
hT:XP (MeV)
W, (MeV)
F
19.0 16.85 13.4 16.8
2 2 2
3 3 3 3
30 92.5 349.1 101.8
For a given impact parameter the time dependence of the z-component of the intrinsic angular momentum L of one fragment represents a damped oscillation with the frequency of the giant dipole resonance. If the interaction time exceeds 1.5 x lo-*‘s, with reasonable parameter values the corresponding lifetime of the excited RPA mode is short enough to achieve a nearly complete damping of the oscillation. Then, the final value of the intrinsic angular momentum determines the fraction of the dissipated angular momentum carried by one fragment. For trajectories with smaller interaction times, at the end of the reaction the intrinsic angular momentum is still in an oscillatory stage. In order to extract a definite dissipated angular momentum, a time average of L has been performed, which simulates the fluctuation of the collective variables, specifying the relative motion around the classical mean value. In such a procedure, a gaussian distribution function has been used with the interaction time given by the classical dynamical model as a mean value and with a standard deviation leading to about 907” of the normalization of the weight function if intergrated over one period of the oscillation. For impact parameters near grazing no well-defined dissipated angular momentum can be derived within the present approach. In fig. 1 the calculated time dependence of the intrinsic fragment spin is E = 8 MeV/nucleon, and 208Pb+20sPb, displayed for the reactions 40Ca+40Ca, E = 7.6 MeV/nucleon, with an angular momentum Li in the entrance channel of Li = 72h and 50h, respectively. Furthermore, it can be seen from the figures, that the final intrinsic angular momentum only weakly depends on the width of the resonance if it is changed within a few MeV. So, the uncertainty in the choice of this parameters does not influence the final results significantly. For symmetric systems in the deep-inelastic region the intrinsic angular momentum L, carried by one fragment in the final channel grows linearly with the entrance angular momentum Li, L, = cLi. The results for the system 20BPb + ‘08Pb are shown in’ fig. 2. This behaviour of L, as a function of Li resembles a sticking situation, but with a slope of the computed straight line well below the value, which one expects from the sticking limit for two touching spheres. The weak angular momentum dissipation predicted within the present approach is explained by the fact, that the oscillator form chosen for the interaction potential restricts the
D. Janssen et al. 1 Microscopic description
175
h 15
10
5
0
-5 %r 15
.
2oepb + 208 Pb , E=1580,8
MeV, Li = 5071
i ‘,
10
5
0 4J I 1 95
-5
I ’ I I :” :
‘1, ,:
”
1P
145 5 /lo-%
Fig. 1. Calculated time dependence of the intrinsic fragment spin L for the reactions %a +‘%a, E = 320 MeV, and 208Pb+*08Pb, E = 1580.8 MeV, with an angular momentum L, in the entrance channel of L, = 72fi and 50h, respectively. The parameters are taken from table 1. In the lower part the dashed curve corresponds to a smaller damping width of fir, = 2 MeV.
to dipole modes only, while higher multipolarities are disregarded. Moreover, the effects of the mass transfer are disregarded completely. According to eq. (77), for symmetric systems the dissipated angular momentum is shared in equal proportions between both nuclei. For the various asymmetric systems investigated within the present approach it was found, that the total dissipated angular momentum is distributed between both fragments according to excitation
D. Janssen et al. 1 Microscopic description
176
L/h
20apb + 208Pb
, E = 1580,8 MeV B
25-
20. 15105-
0
50
I 150
100
200
Fig. 2. The *0sPb+208Pb reaction, E = 1580.8 MeV. intrinsic angular momentum L carried by one fragment m the exit channel as a function of angular momentum L, in the entrance channel: (A) present calculation; (B) from a classical frlctlon model.formulated in ref I’-‘).
the empirical
relation
(cf. table 2)
lSSS2.
(79)
In the classical model one has 6 = -3 for the sticking limit 15). So, in both pictures the heavier fragment carries more angular momentum than the lighter one. In conclusion, one can state that even under the drastic simplifications introduced the present approach allows one to reproduce some of the basic features observed for the angular momentum dissipation in heavy-ion collisions, a phenomenon which turns out to underlying reaction mechanism r6).
be of crucial
The authors would like to thank H. Reinhardt, valuable discussions. The assistance of R. Schmidt trajectory code is gratefully acknowledged.
importance
in
P. Schuck and and J. Teichert
clarifying
the
H. Schulz for in using their
Appendix A In order to illustrate
the derivation
of classical
equations
for the relative
motion,
171
D. Janssen et al. / Microscopic description
TABLE2 Calculated angular momentum dissipation for asymmetric systems [(a) projectile, (b) target, i(f) entrance (exit) channel. AL = @+I!:‘]
‘Wa +‘OZr (E = 8 MeV/nucleon. L,, = 91h, L,, = 141h) 96 101 106
2 2 2
26 27 27
13.0 13.5 13.5
28 29 29
23 23 23
31 41 52 64
21 28 36 43
“‘Kr + “‘“Pb (E = 7 MeV/nucleon. L,, = 46h, I,,, = 254h)
1
72 98 124 150
30 39 50 62
2 2 2
30.0 19.5 25.0 31.0
“t From the classical friction model.
the general equations for the RPA amplitudes for the subsystem a,
64.1)
are specified for the spurious mode connected with the linear momentum, v = p, x, = 8, ‘pp = -i N[r$‘), $1, JV = l/N,, cop = 0. The excitation of this mode would correspond to a finite velocity in the intrinsic system. It has to be eliminated by demanding up = 0, which implies conditions for the time dependence of the parameters of the transformation U. Also for the further spurious modes one has a, = 0, except for the angular momentum mode aL_. But in the latter case the normalization factor V = (Tr(L,L,)j -r goes to zero* for an infinite con~guration space, while the remaining factors like Tr(jI[~Vr~“)- Wrb”‘,LJZY’)) have a finite value. So, also the angular momentum mode does not contribute to the sum, which can then be restricted to the ph modes with w,. # 0. According to these considerations one has from eq. (A.11
+Tr
Vrp) - FVrp), r1p)
fi (
[s
178
D. Janssen et al. / Microscopic description + ‘JWCCL,
dr,]&sin
P, - [Lyr dralba - CL,, dral&, ~0s P,f 1
The parts, in which no interaction terms are involved, can be evaluated explicitly. Some of the traces in eq. (A.2) are vanishing: Tr $. (II
R-l%R,r(o)
1) = Tr([fi,R-l&R]r(o))
= 0,
&R.&l) = 0,
Tr(@.[l?-t
(A-3)
Tr(j?. [R-“S?GR,&]) = Tr([#, R-‘fR]Gr)
= const x Tr(&) = 0,
(A.4)
Tr(${ [L,, Jr]& sin p - [L,, Sr]j - [L,, &]h cos a>), = Tr({[j, L,]hsin~-[$,
Ly]~-[~,Lz]hcosj?~6r)
= 0
(A.5)
(with the help of Tr(@r) = 0). For the other part one obtains $Tr(fi* a
= &
[R,‘fR,,rfP’])
= $Tr(@. a
UD, ’ Tr(jl . [a, ri*)]) = $f
a = !$Trr;O’ a
[UD;ld,r~o)])
Tr([#, &]ri”)) a
zz - iD,p.
(A.61
Inserting this expression in eq. (A.21, one gets
I)
Vr’,O’ - D a Wrp), r(O) a
I) &) ,
Vr(boj- DaWr(bDb,D=cpP
a,
I/@’
p
- D, Wcp~b),r(aO)
(A.7)
with
xx’
=
6(x-x’)
dx” V(lx- x”+ qj)r,(D,x”,
D,x”),
(A.81
179
D. Janssen et al. / Microscopic descrrption
(DaWrb)xxP
=
In the following
~lJDb(x+q),DtJx’+q))V(IX-x’l).
- ip
exp
the evaluation
of an expression
64.9)
of the form
I)
v$b-Daw$b3Da$a will be demonstrated, where $,(x,x’), space representation the first part
$b(~, x’) represent
arbitrary
operators.
In
reads
The arrow leads to
A second
=-
marks
partial
the action
integration
of the differential
operator.
4 first partial
integration
gives
s
dxdx’ V(lx - x’ + ql)ll/b(Dbx’, D,x’) & i,b,(D,:, D,f).
From a further partial to a q-differentiation)
I, =
s
= $
integration
one gets (after converting
dxdx’ & V(l: -x’ + ql)$,(Dbxr,
s
Dbx’)$,(D,x,
dxdx’ v(lx -x’ + ql)ll/b(Dbx’, DbX’)$,(D,X,
the x-differentiation
D,x)
D,x).
(A.lO)
180
D. Janssen et al.
! Microscopic descrrption
Similarly, the exchange interaction part, I, = Tr $[D.wI&.D&J (
= )
dxdx’%6(:-w’)[D,Wi,,D,~,l,,,, s
can be expressed by partial integration according to
+exp
-ip ( x v%.@w’ + 41, &C~ + P)M,aJ,
Using the relation
Q’)
.
D. Janssen et al. { Microscopic description
181
one finds
- ip * +!$-
x +ba(Dax’,D,x) exp
a
=
(x-d,)
V(jx-x’l)
b
dxdx’ ~b(~b(~ +q), &,(x’ + 4 ))~=(~~~‘, D&)
A
dq s xexp
(A.ll)
-ip* i
Altogether, eqs. (A.7~(A.l1) momentum.
leads to the equation
of motion
for the relative
Appendix B In a spherical basis the equations of motion for a, in the ph channel with w, # 0 have the following form: ih, = (CO,-- Mkcos
/?)a,-
~Tr(~:$~~_?(2+09,~) Y
f)J(J+l)j-M(M-l)(hsinB+igfa,,_,, +&/@YijKG(M+
l)(&sinj?-ij3)a,,+1,
(v G JM&). The restriction vanishing :
(B.1)
to dipole excitation
ia loa
=
QvlOE
-
means that only modes with J = 1 are not
Tr(cp~,&noZ(z2 i- Dq)‘)
5
E
+&/5(&sinp+$)a,_,,+(hsin/3-$)a,,,,
ial_,,
= (o,+Ercosj3)a1_,,+~fi(&sin~-$)alo8.
~Tr(~~_,,fmw’(P+Dq)‘) E
(B.2)
182
D. Janssen
et af. j microscopic ~s~r~~tion
Analogous equations exist for the modes with F&O,= -CO,). First one has to calculate TM = Tr(cp:,,$ mw2(~++q)*). With the help of the orthogonality relation Tr(lT;,7;.,,) * 6,,&,. one gets
TM= Trfp :&&tct * q)mw2.
(8.3)
After an expansion of p in terms of oscillator wave functions and xDq in terms of spherical harmonics one obtains
with
81
= - i/$sin(B+fi)eiCt”+“~,
K*
= cos(8 4 /?I,
tB.5)
The next step is the explicit calculation of the six remaining RPA amplitudes a, which are represented in a rotating system: ii, =
c Dycap.
(B-7)
P
The indices V,,Uare characterized
by the projection quantum numbers only:
with
A comparison with eq. (B.l) gives (B. 10)
183
D. Janssen et al. / Microscopic description
This is equivalent
to
= *, 1 Dvaap - 1 D,, $ TV. P B P The solution
of eq. (B.11) is
fs = sf
dr eLwcr1 D,, 2
1 Dvpap = ieeLWJ P The transformation
in the original
a,(r)
0
P
system
leads to
dz eiwer5
ie-l"J
TV.
(B.ll)
(B. 12)
bEI
TV.
(B.13)
0
Note that the start condition a,(O) = 0 is assumed. each particular amplitude is given:
At last the representation
of
f a-,
= i/$X
e-w
dT elwcrqsin(B+ j?)e’(4+“), s0
a o = iXe-1%’
t
dr erwcrqcos(0 + p), s 0 f
a 1=
-
i/$Xe-lWJ
dze’“~*qsin(8+p)e-‘(~+~),
(B. 14)
s 0
with
The relative distance q and the angles 9, C$ of the relative motion as well as the Euler angles CC,p are functions of r. One gets the amplitudes for the modes with E’ from the relation
afM, = (- l)“al_M,.
References 1) J. W. Negele, Rev. Mod. Phys. 54 (1982) 913 2) H. Reinhardt, Nucl. Phys. A390 (1982) 70
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