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Kinetic equation for collective modes of a Fermi system with free surface
Nuclear Physics A562 (1993) 41-60 North-Holland
NUCLEAR PHYSICS A
Kinetic equation for collective modes of a Fermi system with free surface V. Abrosimov ‘TV,M. Di Toro a, V. Strutinsky a,b.t ’ Dipartimento di Fisica, Uniuersitci di Catania and INFN, Laboratorio
Nazionale del Sud, Catania, Italy
b Institute for Nuclear Research, 252028 Kiev, Ukraine Received (Revised
7 October 1991 8 March 1993)
Abstract A semiclassical approach for studying collective modes of finite Fermi systems is proposed on the basis of the Landau-Vlasov kinetic equation. In the theory the effective surface is explicitly used. The surface-motion equation is deduced. The problem is reduced to solving a linearized kinetic equation with mirror-reflection conditions at the free-moving effective surface. An equation for eigenfrequencies is obtained when the collision integral and the residual quasiparticle interaction in the bulk of the system are neglected. The eigenfrequencies for oscillations with multipolarities L > 0 are complex. A simple analytical solution is found for monopole eigenfrequencies. The properties of the first monopole mode agree with the ones derived in the corresponding RPA approximation.
1. Introduction
A semiclassical description of nuclear collective motion on the basis of the kinetic equation with a self-consistent field (the Landau-Vlasov kinetic equation) has been the subject of many studies [1,2]. In this picture an open question is the possibility of using explicitly some macroscopic properties of complex nuclei based on the fact that a lot of collective phenomena in nuclei, particularly giant resonances, are observed in processes with small momentum and energy transfers relative to the Fermi momentum and energy. Recently, to study the smooth background in the giant resonance region a free-moving effective surface was used in an approach within the framework of the Vlasov kinetic equation [3]. The found surface response function coincides with the classical limit of the quantum response function derived in the RPA approximation [4]. The essential point of using a free effective surface as a macroscopic collective variable is that it makes it possible to use the same simple residual quasiparticle interaction as in the case of
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42
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the infinite Fermi liquid [5], thus avoiding ambiguous extrapolations in the diffuse edge region. Our aim is to formulate the Landau-Vlasov problem for small vibrations around a spherical equilibrium shape with boundary conditions appropriate for a moving effective surface. In sect. 2 we consider the linearized Landau-Vlasov equation for a system with a central equilibrium potential. In sect. 3 the Landau-Vlasov equation is reduced to the one for a system with a free moving surface. To meet this end we introduce collective variables describing local displacement of the effective surface and then we derive an equation of motion for the surface itself. In sect. 4 an equation for eigenfrequencies of our system is derived and some general properties of solutions, particularly for monopole modes, are discussed. Finally, our study is concluded in sect. 5. Appendix A contains a deduction of the surface equation of motion and in Appendix B the equation for eigenfrequencies is derived.
2. Landau-Vlasov equation for a spherical Fermi system We consider a Fermi liquid bound within a free spherical surface R = const. We assume that the quasiparticle phase-space distribution n(r, p, t) is governed by the Landau-Vlasov kinetic equation for zero temperature [6],
=I[n(r,
(2.1)
r, t)].
Here U(r, p, t) is the self-consistent potential. This is related to the distribution function n(r, p, t) by a self-consistency condition
qr,
P,
where
t>
dn(r’,
1
= ,/dr’
Jr
dp’ FT(r, P; r’, P’)
is the quasiparticle u(r, p, t) is determined as F(r,
p; r’, p’)
effective
P’,
(2.4
,
interaction.
Wr’, u(r,p,t)=b+$jdr’dp’S(r,p;r’,p’)
t)
Jr,
P’,
ap,
The velocity
t)
.
(2.3)
The term in the r.h.s. of Eq. (2.1) is the collision integral. In the present study, we are interested in a small deviation of the distribution 6n(r, p, t) from an equilibrium n,(r, p), i.e. we assume that n(r,
P,
t)
=n,(r,
r-1 +Sn(r,
p,
t).
(2.4)
A. Abrosimou et al. / Kinetic equation
43
If we neglect terms of order fin*, we get the linearized equation for C&G, p, t),
dt -
&j(r,
P, t) + ae”(r, P) @jfl(r, P, t)
dWr3
ar
8P
xw(~,
p> a%+., dr
-
p, t> + asu(r,
P)
Pi f) ap
$l(f-T
p1 t>
dr
8P
dP
P) dWr,
ar
=Jb(r,
P, t>l, (2.5)
where .f[Sn(r, p, t)] is the linearized colIision integral. Here, in accordance with (2.41, we introduce U(r, P, t) = Uo(r, P) +6t’(r,
P, t),
(2.6)
where NJ(r, p, t> is the deviation of i.J(r, p, t> from the equilibrium potential Uo(r, p), related to Sn(r, p, t). The self-consistent quasiparticle hamiltonian is then 2 Eo(f,
P)
=
&
+
u,(r,
(2.7)
P)
and the equilibrium distribution dg,(r,
P) an,(r,
8IJ
P)
tir
It&,
%(r, -
dr
Below, the equilibrium distribution alone, n,(r,
p> satisfies the equation
P) %(r,
p)
ap
=
0.
(2.8)
n,(r, p> is assumed to be a function of &,
P) = nO(E(rY P>).
p)
(2.9)
The distribution n&r, p) and the self-consistent field U,(r, p) are sharply changing functions of the radial variable r near R. Therefore, the semiclassical equation (2.5) is expected not to be valid in the diffuse surface region of the system [4,7]. In order to derive the equation of surface motion we will use Eq. (2.5) also in the edge region of the system. However, we shaIl see that the result obtained does not depend on the surface properties of the equilibrium distribution n,(r, pf and it can be used for studying systems with sharply changing edge density as in the case of nuclei. For simplicity we shall assume below that the quasiparticle effective interaction in eq. (2.5) does not depend on the velocity, F(r,
p; r’, p’) =F(r,
r’)
(2.10)
Then using eqs. (2.21, (2.4) and (2.6) we find that U(r,
t) = Uo(r)
+W(r,
t),
(2.11)
A. Abrosimou et al. / Kinetic equation
44
where dr’ dp’ F(r,
6U(r, t) = ;,
r’)&r(r’,
(2.12)
p’, t).
We shall assume also that the equilibrium potential is central U,(r)
(2.13)
= &Jr)*
Now we simplify eq. (2.5) by changing variables (r, p) to a new set of variables (r, E, I, (Y,p, r> as proposed in ref. ES], (2.14)
(r, P> --) (r, E, f, a, P, Y).
The new variables are quasiparticle energy E = ~a, particle angular momentum I = /r xp 1, radius r and Euler angles (a, p, y). The Euler angles are defined by the rotation of the laboratory frame (x, y, z) to (x’, y’, z’> with i’ along I and 9’ along r. Among the new variables, E, E, cy, /3 are constants of the motion. One can eliminate in Eq. (2.9, written in new variables, the derivative with respect to y by means of an expansion in g-functions [8]. Distributions of quasiparticles with positive radial velocities Gn+(r, E, 1, CX,p, y, t) and with negative ones 6~(r, E, E, (Y,p, y, t) are considered separately. In terms of the new variables the linearized Landau-Vlasov equation is
(2.15) where we have used
SU(r, t) =
c
YLN(;7r
(2.17)
Z.,M,N
C
-z”[(f+(r,E, 1, t>)fM+(f-(r,
E,
1, f))FM]
L,M,N
(2.18)
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45
Here the radial quasiparticle velocity is (2.19)
u(r,E,1)= [(2/m)(c-P/(2mrZ))y2.
represent displacements of the local energy for the functions (f *(r, E, I, t))$ quasiparticles with angular momentum 1 at the distance r from the centre of the system. They describe Fermi surface deformations. It can be shown that the subscripts LA4 of the functions&, determine the multipolarity of the density distribution.
3. System with a free surface We isolate in eq. (2.15) terms which are proportional to dU,/dr dominant at the system’s edge by seeking a solution of the form
6n(r,
P,
t) =
duo(r)
2
Here we introduce
drW+,
the local displacement
and therefore
I
cp,t) .
(3.1)
SR(6,cp, t>of the effective surface
R(6,cp, t>from its equilibrium position 6R(fi, cp, t) =R(fi, rp, t> -R=
c ~R,,(~)Y,M(~, cp>-
(3.2)
L,M
In the new variables it can be written as
aR(a,
(9, t> =
c
&&)L,(h
~+%iv(~, P, Y))*.
(3.3)
L,MJ’
Now taking into account Eqs. (3.1), (2.16) and (3.3) we find 6n(r,
E, 1, a, P, Y, t>
x (q&(%
P7
?I)*.
(3.4)
A. Abrosimou
46
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According to Eq. (3.4) the deviation of the self-consistent potential W’,,(r, t) (see Eq. (2.17)) is divided in a bulk component 8GLM(r, t) and a surface quantity SUL’$(r > l) , &!.I,,( r, t) = Sii,,(
r, t) + SUL$ r, t).
(3.5)
In our approximation the colhsion integral in the r.h.s. of Eq. (2.15) depends only on the bulk component of the distribution function. In fact, using eq. (3.4) and (2.9) we obtain
22
wl(r)
I
--+R,,(r)
1=~R,,(r)-$?[““(E)] =o.
(34
to dCi,/dr
can be neglected.
By means of Eqs. (3.4)-(3.6) one can write Eq. (2.15) as
Now in the bulk region (Y < RI terms proportional Then we obtain
r
(3.8)
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47
In the diffuse edge region at r = R we keep only terms proportional to dU,,/dr. After some transformations one obtains the following equation for the effective surface motion, see appendix A:
= -2p(R,
E, W&,&
fa)&R,,(t)).
(3.9)
The essential point is that Eq. (3.9) does not contain quantities which depend on the equilibrium distribution ~JE(T, p)). Eq. (3.9) has a simple physical meaning: it is the condition of elastic reflection of quasiparticles from the moving surface, set in the frame moving with the surface (the mirror-reflection boundary condition). The same boundary condition was employed in deriving the energy dissipation rate in a deformed nucleus [9] and more recently in studying the nuclear surface response [3]. From eq. (3.9) we can obtain a boundary condition for the radial component of the quasiparticle density current (the macroscopic equation of surface motion). It reads
L,LM(b4r=R=mpo;(%A4w). Here, the quasiparticle
density current is determined
(3.10)
as (3.11)
and the equilibrium quasiparticle density is po =
-$2ri(Zm)‘j21mdr
E”%,(E).
If the distribution ~z,&E)is the Thomas-Fermi value of pO, see e.g. ref. [lo], 1 4?r Po=gTP
;
(3.12) one then we obtain the well-known
(3.13)
A boundary condition (3.10) corresponds to the one used in liquid-drop studies of nuclear vibrations [ll] and recently in liquid-particle approaches [5]. However, the use of an effective-surface equation (3.2) violates the self-consistency condition (2.12) for NJ, because additional (“external”) collective degrees of freedom are introduced. To obtain a closed system of equations one then
A. Abrosimov et al. / Kinetic equation
48
demands that the moving effective surface should be free. The pressure acting on the system’s surface is determined by the momentum flux tensor &ik defined as, see e.g. ref. [12], (3.14) Non-diagonal components L@,+, z%~, should vanish at the free surface. Indeed using the surface equation of motion it can be shown that at the moving effective surface
We also require that the additional pressure caused by the surface tension should equal the normal component @Jr, t) at r =R(6, cp, t>. Then we find
@L,h&?~)lr=RWd =a( f. -
l)( L + 2)RP26R.,(
t).
(3.16)
Here, u is the surface tension parameter (for a nuclear surface u = 1 MeV/fm*). Eq. (3.16) can be obtained from the kinetic equation (2.15) in the same way as it was done for semi-infinite systems [4]. We shall use Eq. (3.16) with a phenomenological value of C, which is a well-determined quantity. This equation is the same as in refs. [11,51. Introducing the new variables we obtain a system of two equations for (f+(r, E, I, t)),N,, see Eq. (2.15). To find the solutions, we should have another boundary condition in addition to Eq. (3.9). We will require that at the inner radial turning point r1 continuity should hold, (3.17) where at r,, u(rl, E, I> = 0. Using Eq. (2.19) and the fact that U&r> is approximately constant for r
1/(2m~)“*.
(3.18)
The problem is reduced to finding a solution to the kinetic equations (3.8) which should satisfy the boundary conditions Eq. (3.9) at the free surface and eq. (3.17) at the turning point r,. The “self-consistency” condition Eq. (3.16) must also be fulfilled, and it determines the eigenfrequencies of our system. Important is that we can use in Eq. (3.8), for the bulk region, the same simple effective interaction as in the infinite Fermi liquid. In the diffuse edge region where the effective
A. Abrosimou et al. / Kinetic equation
49
interaction is complicated we use boundary conditions for the distribution function at the free moving surface, Eq. (3.9).
4. Eigenfrequencies Now we perform the Fourier transformation with respect to time. We put
(f’(r,
E,
I,
u)):~=/~ dt --m
of Eqs. (38, (3.91, (3.16) and (3.17)
exp(iwt)(f*(r,
E, I, t))IM
(4.1)
and
(44 To ensure convergence of these integrals we assume that (f*(y, E, 1, t)>,,, = 0 and 6R,,(t) = 0 for t < r, and suppose that w has a small positive imaginary part in. Eq. (3.8) can be written then as
=r[(Ptr,
E,
1,
w));3;,+ (f-(r,
E,
1,
m))rM],r1
(4.3)
Solutions of these equations must satisfy the boundary conditions
I(
P(r,
E, 1, w)),“,-
(f--(r,
= iZwp( R, E, E)Y,&x,
E, 1, “,):,]I,=, $r)6RLM(
w)
(4.4)
and k P(r,
E, 1, o))FM -
(f (r,
In addition, the “self-consistency”
E, I, D,)1l’J /_,
= 0,
(4.5)
condition
%r,LM (r,o)lr_R=~(L-1)(L+2)R-26RLM(W)
(4.6)
50
A. Abrosimov
et al. / Kinetic equation
must be fulfilled. To establish some general properties of solutions of Eqs. (4.3)-(4.6) we consider first eq. (4.3) without the scattering term. From Eqs. (4.3)-(4.5) we find the following integral equations:
exp[ *i(
m( r,
= sin[+(wT(R,
E,
1) - W( r,
E, I) -NT(R,
E,
Z))]
E, 1))]
y,,($,
Xsin[ +( wT( r’, E, 1) - NT( r’, l, I)) - (oT(
+sin[i(oT(R,
E, 1) -NT(R,
fr)
r’, 6, 4 - NY(r’3
E, l))]/rdr’-$(8~LM(r’,
ET I))]
w))
r1
Xexp[ Ti(m(r’,
E, 1) -Ny(r’,
Here the same quantities are introduced
7(r7 6, I) = jrdr’+ r1
1 E 1) ) 7
E, l))]
1
.
(4.7)
as in ref. [8],
(4.8)
(4.9) The quantities T( R, E, I) = 27( R, E, I)
(4.10)
T(R,
(4.11)
and E, 1) = 2y(R,
E, I)
are, respectively, the radial and angular Because of the condition (4.6) solutions w. In this paper we discuss a solution system of quasiparticles (corresponding
periods for each quasiparticle orbit (E, 1). to Eq. (4.7) exist only at certain values of to the simplified problem of a free-gas to F = 0) in the nuclear volume. Putting
A. Abrosimov et al. / Kinetic equation
51
r, O) = 0 one can find explicit expressions for into Eq. (4.7) 8cLMLM( Using then Eq. (4.6) we obtain an equation for eigenfrequen(f’(r, E, 1, wN&. ties, see appendix B, that can be written as
-6
x
I0
(1 - h2)li2
idA A
=Cr(L-l)(L+2)R-‘. S-(n+Narccos(h)/7r)/(l-A2)1’2 (4.12)
Here we have introduced
the dimensionless quantities
s = wR/n-v,,
(4.13)
A = l/Rmv,,
(4.14)
and
where vr is the Fermi velocity. In the derivation of eq. (4.12) we have used for dn,/de the Thomas-Fermi-like approximation dn,/dE
= -6( E - or)
Positions of the uncorrelated mined, see appendix B, as sin[+(wT--NT)]
(4.15) independent-particle
resonances are explicitly deter-
=O,
(4.16)
where
T(A)
= 231
- A2)1’2,
r(A)
= 2 arccos( A).
(4.17)
It can be seen from Eq. (4.12) that all modes with L > 0 are damped. We remark that the decay of these collective modes may correspond to a one-body damping mechanism because only this mechanism is included in our present approximation.
52
A. Abrosimov
Eq. (4.12) can be transformed modes CL = 0). It reads
et al. / Kinetic equation
to a simple analytical form for the monopole
(4.18) Here s, = s/n
(4.19)
and s, + 1 w(sn) = +sn In -s _ I +i?T~s,O(~s,~ - 1). n
11
It can be seen from Eq. (4.20) that eigenfrequencies determine solutions of eq. (4.18) which correspond quantity (4.20) in powers of s,, w(sn)=sn(sn+$,3+$s,5+
(4.20) with s < 1 should be real. To to s < 1 one can expand the
. ..).
(4.21)
Using Eqs. (4.18) and (4.21) we find that up to terms of the order of s*,
$-*pOEFS2 + P()EF
u =
(4.22)
-
R
and
(4.23) By means of (4.13) we obtain the eigenfrequency l/7-
.
(4.24)
We can verify that for standard values of the parameters: (T= 1 MeV/fm*, pa = 0.17 fme3, lF = 40 MeV, m = 1.04 MeV (1O-22 s)2/fm3, R = 1.12~ll’~ fm the dimensionless frequency s(i) = 0.7 < 1 and tiw(‘) = 114A-‘/3 MeV. Solving Eq. (4.18) numerically we obtain so) = 0.63 and ho(‘) = 103A-‘/3 MeV. Our result is in reasonable agreement with the systematic behaviour of the excitation energy as a function of the mass number for isoscalar monopole resonances [10,13]. When making the comparison one should keep in mind that the present calculation does not include a residual interaction in the bulk of the
A. Abrosimov et al. / Kinetic equation
53
system. Indeed the agreement looks particularly good for heavy nuclei, where the sharp surface approximation is more realistic. In a sense this result represents a further justification of the use of surface peaked form factors for effective separable interactions [ 14,151.
5. Conclusions The Landau-Vlasov equation is reduced to a simple kinetic equation for a system with a free moving effective surface. In this equation the same quasiparticle interaction as in the infinite Fermi liquid can be used. Introducing the moving effective surface we include in a macroscopic way a complicated effective interaction in the surface region. Resonance frequencies in a system of free quasiparticles bound within the effective moving surface of the nucleus, which was determined in a self-consistent way, have the following features. All eigenfrequencies for isoscalar density vibrations with multipolarity L > 0 are complex quantities. This is because the independent-particle resonance frequencies, see Eqs. (4.16) and (4.171, form a continuum from zero to infinity. The collective modes are embedded among the single-particle ones and therefore are Landau damped. Taking into account some results of numerical calculations performed within the semiclassical RPA, see refs. [1,15], one may expect that the Landau damping will be small for resonances with low multipolarity (L = 2, 3) also in our semiclassical model. However, the Landau damping will be growing with increasing multipolarity, as we can clearly see from Eq. (4.12). This is in agreement with fully RPA results [13,19]. The monopole modes are also damped except for the lowest monopole frequency which is real. This comes from the fact that there is a gap in the monopole independent-particle frequency spectrum, which actually starts from oL$, = rv,/R, as we see from Eq. (4.16). On the other hand, the moving-surface assumption corresponds to taking into account an attractive effective interaction in the surface region which leads to a first collective monopole mode with frequency (4.241, smaller than wL$,. This result also agrees with the one obtained in RPA for heavy nuclei [13,18,19], where the “fragmentation width” (Landau damping) of the lowest monopole resonance is small [18,19] or wholly absent [13], depending on the used isoscalar interaction. We finally remark that the linearized Vlasov equation is solved without making any scaling approximation. For the first monopole mode an analytical expression can be found for the distribution function 6&r, p, t). It would be of interest to investigate properties of this monopole mode, especially the widths, in some detail. Particularly interesting is the appearing of the predicted absence of collisionless (Landau) damping for the monopole giant resonance. Since the escape widths seem experimentally quite small [16], the only possibility to understand the
54
A. Abrosimov
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observed total width is to consider more carefully the spreading contribution coming from the collision term. This can easily arise from momentum-space distortions in &z(r, p, t> for the monopole mode. Some work is in progress on this problem. The authors express their gratitude to Professor D.M. Brink for discussions. V.A. and M.D.T. are still shocked by the sudden death of Prof. V. Strutinsky and will always remember the great human and scientific pleasure of working with Vilen. Appendix
A
To derive the equation of surface motion (3.9) we take the sum of Eq. (2.1.5), and to isolate explicitly all terms which are proportional to dU,(r)/dr we use the relation
a
m
ay=
p(r,
(A.11
E, Z)
where p(r, E, 1) = mu(r, E, 1) is the radial momentum. After these transformations one finds
=2p[(f+(r,E, 1, t>)L+(f-(r,
E,
1, t))L].
By using (3.4) this equation can be written in the form
a
z
+ iN
I2 + --
mr3
a ap
(A4
55
A. Abrosimou et al. / Kinetic equation
r, x Km+-, 3%-)GUg)(
t)
=24(f+(r,E, 1,t));M+(f-(r, E,&o.
(A.3)
E,
In the surface region the dominating terms are proportional have at r N R
, $Q&,(t)
to dU,/dr.
Then we
=0.
- SU~~(R, t)
(A.4) Now to transform the surface potential &!$$(R, t) in Eq. (A.4) we use the self-consistency condition (2.2) for the equilibrium potential U,. By using Eq. (2.2) and the assumption (2.11) this condition can be written in the form
-duo = / dr
dr'
rr2Fl(r,
r’)
Here the effective interaction
%(r, r’) =
C jjda
dPo(4 Sf(r,
=
r’>
at I = 1 is determined
dR’ .B(r, r’)Y,,(a)y,;(n’).
m
The equilibrium density distribution p,(r)
W)
dr’
p,,(r)
as
(A.6)
is given by
$/dpno(e(ry ~‘1)
=--
1 1,
E)
nO(Eo)’
(A-7)
A. Abrosimov et al. / Kinetic equation
56
By using Eqs. (AS) and (2.2) we express SU#R,
SU,(&R,
t) =
z =
SR,,(
r
t) in the form
t)
R
+ I a’dr’
Cz(FL( R, r’) - .F,( R, Y’))
dPo(
dr,
r’)
6R,,(
t). (A.81
After integration of Eq. (A.4) over p and taking into account Eq. (A.8 we find
x I aRdr’ r”(
r’) -Fi(R,
r’))
ddr’) dr,
GR,,ft)
1 .
(A.91
In this equation the term
FL{
2 reff
R, r’) - Fl( R, r’) N I__ ( L,R)2 ’
(A.10)
where r,, is the effective radius of the quasiparticle interaction [17]. Therefore, the integral containing this term can be neglected in Eq. (A.9) and Eq. (3.9) is obtained.
Appendix
B
To derive the equation for eigenfrequences (4.12) we should find the multipole rr,LM(~, w) at the surface (r = RI, see Eq. (4.6). component of the normal stress L% The starting point is the normal component @_(T, 0). It is determined from (3.14) as
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57
Here, tr is the radial quasiparticle velocity, p, = mu, and the quantities S(r, p, w> and 6U(r, W) are defined in the same way as (4.1). The expression (B.1) can be written as
The integrals over the coordinates and momenta (r, p) in (B.2) can be transformed into integrals over the new variables (r, E, 1, QI,& y), see sect. 2, by calculating the jacobian of the transformation. Moreover, we can write the multipole expansion of the S-function as
In (13.3) we exploit the transformation
properties of the spherical harmonics
Then, by using the expansion in &$-functions (2.16) for Gii(r, p, w) and the orthogonality of S-functions, we find the normal stress (B.2) in terms of new variables. It writes as
where
x i
N= -L
YL>($T$#+(b
E,
1, w$*
A. Abrosimou
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Here, p(r, E, 1) = mu(r, E, I), where u(r, E, I) is defined by (2.19). The maximum value of quasiparticle angular momentum I,,, is determined by the normalization condition
(B-7)
A = j-_/dr dp n,,( E(T, P)). By using (4.15) we obtain 1max= PFR,
(B*8)
where pF is the Fermi momentum. Putting in the integral equation (4.7) ijgL,,(r, w) = 0 we find the explicit expression for (f*(r, E, 1, WI),“,. At r = R we have
(f+(R,
6, I, w,):~=
exp[+$(wT(R, sin[i(wT(R,
x YLN(b, $+P(
E, 1) -NT(R, E, I) -NT(R, R,
E, /))I E, l))]
(B.9)
E, l)SR,,(w),
where the quantities T(R, E, I) and T(R, E, f) are determined by (4.10) and (4.11). Now putting (B.9) in (B.6) and using (4.15) and (B.8) we get at r = R @+f(R7
w)
1 (4rr)2 = --pwSR$w) h3 2L+l
x/p”RdlI(p(R, 0
,g,\Y,“(;“,
l))*
cot[;(wT(R,
$7r)12
E, I) -NT(R,
E, l))],
(B.lO)
where P(R> I) =P$
T(R,
- (VP~)~]~‘*,
1) =2-+i/p,R)‘]“*,
I’( R, 1) = 2 arccos(E/p,R).
(B.11)
(B.12)
(B.13)
A. Abrosimov et al. / Kinetic equation
By exploiting the dimensionless frequency s and angular momentum and (4.14), the normal stress (B.lO) can be written as
&M(R,
59
A, see (4.13)
4,rr2 s> = -6--2L + 1 EFPoS X
Ltdh ~(l-A*)
cot[rr(l-AZ)1’2,s-Narccos
A]. (B.14)
Here p. is determined expansion for cot 2 cot
2=
2
by (3.13) and Er =pg/2m.
By using in (B.14) the pole
(z-mr-l,
(B.15)
n= -Cc we find the final expression for P,,,LM(R, s). It reads 47 @v,&R,
s) = -6~
2L + 1 EFPoS (1 -A’)“*
x
I0
rdh A
(B.16) s-(n+Narccos(A)/rr)/(l-A*)l’*’
By inserting this result into eq. (4.6) the equation for eigenfrequencies obtained.
(4.12) is
References [l] M. Di Toro, Part. Nucl. 22 (1991) 385 [2] P. Schuck, R.W. Hasse, J. Jaenicke, C. Gregoire, B. Remaud, F. Sebille and E. Surand, Prog. Part. Nucl. Phys. 22 (1989) 181 [3] V.I. Abrosimov and J. Randrup, Nucl. Phys. A449 (1986) 446; A489 (1988) 412 [4] Yu.B. Ivanov, Nucl. Phys. A365 (1981) 301 [S] V. Strutinsky, A. Magner and V. Denisov, Z. Phys. A315 (1984) 301 [6] L.D. Landau, Sov. Phys. JETP 3 (1956) 920; 5 (19.57) 101 [7] A. Magner, Yad. Phys. 45 (1987) 374 [8] D.M. Brink, A. Dellafiore and M. Di Toro, Nucl. Phys. A456 (1986) 205 [9] J. Blocki, Y. Boneh, J. Nix, J. Randrup, U. Robel, A.J. Sierk and W.J. Swiatecki, Ann. of Phys. 113 (1978) 330 [lo] P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980) [ll] A. Bohr and B.R. Mottelson, Nuclear structure, vol. 2 (Benjamin-Cummings, Menlo Park, CA, 1976) [12] E.M. Lifschitz and L.P. Pitajevsky, Physical kinetics (Pergamon, Oxford, 1981)
60 [13] [14] [15] [16] [17] 1181 [19]
A. Abrosimou et al. / Kinetic equation T.S. Dumitrescu, C.H. Dasso, F.E. Serr and T. Suzuki, J. of G.F. Bertsch and S. Esbensen, Phys. Lett. B161 (1985) 248, G.F. Burgio and M. Di Toro, Nucl. Phys. A476 (1988) 189 A. Van der Woude, Nucl. Phys. A519 (1990) 17c E.E. Saperstein, S.A. Fayans and V.A. Khodel, Sov. J. Part. G.F. Bertsch and S.F. Tsai, Phys. Rep. 18 (1975) 125 J. Wambach, Proc. Int. School on nuclear structure, ed. V.
Phys. Cl2 (1986) 349 and references therein
Nucl. 9 (1978) 221 Soloviev (Alushta,
USSR,
1985) p. 83
NUCLEAR PHYSICS A
Kinetic equation for collective modes of a Fermi system with free surface V. Abrosimov ‘TV,M. Di Toro a, V. Strutinsky a,b.t ’ Dipartimento di Fisica, Uniuersitci di Catania and INFN, Laboratorio
Nazionale del Sud, Catania, Italy
b Institute for Nuclear Research, 252028 Kiev, Ukraine Received (Revised
7 October 1991 8 March 1993)
Abstract A semiclassical approach for studying collective modes of finite Fermi systems is proposed on the basis of the Landau-Vlasov kinetic equation. In the theory the effective surface is explicitly used. The surface-motion equation is deduced. The problem is reduced to solving a linearized kinetic equation with mirror-reflection conditions at the free-moving effective surface. An equation for eigenfrequencies is obtained when the collision integral and the residual quasiparticle interaction in the bulk of the system are neglected. The eigenfrequencies for oscillations with multipolarities L > 0 are complex. A simple analytical solution is found for monopole eigenfrequencies. The properties of the first monopole mode agree with the ones derived in the corresponding RPA approximation.
1. Introduction
A semiclassical description of nuclear collective motion on the basis of the kinetic equation with a self-consistent field (the Landau-Vlasov kinetic equation) has been the subject of many studies [1,2]. In this picture an open question is the possibility of using explicitly some macroscopic properties of complex nuclei based on the fact that a lot of collective phenomena in nuclei, particularly giant resonances, are observed in processes with small momentum and energy transfers relative to the Fermi momentum and energy. Recently, to study the smooth background in the giant resonance region a free-moving effective surface was used in an approach within the framework of the Vlasov kinetic equation [3]. The found surface response function coincides with the classical limit of the quantum response function derived in the RPA approximation [4]. The essential point of using a free effective surface as a macroscopic collective variable is that it makes it possible to use the same simple residual quasiparticle interaction as in the case of
’ Deceased
on June 28. 1993.
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42
A. Abrosimou
et al. / Kinetic equation
the infinite Fermi liquid [5], thus avoiding ambiguous extrapolations in the diffuse edge region. Our aim is to formulate the Landau-Vlasov problem for small vibrations around a spherical equilibrium shape with boundary conditions appropriate for a moving effective surface. In sect. 2 we consider the linearized Landau-Vlasov equation for a system with a central equilibrium potential. In sect. 3 the Landau-Vlasov equation is reduced to the one for a system with a free moving surface. To meet this end we introduce collective variables describing local displacement of the effective surface and then we derive an equation of motion for the surface itself. In sect. 4 an equation for eigenfrequencies of our system is derived and some general properties of solutions, particularly for monopole modes, are discussed. Finally, our study is concluded in sect. 5. Appendix A contains a deduction of the surface equation of motion and in Appendix B the equation for eigenfrequencies is derived.
2. Landau-Vlasov equation for a spherical Fermi system We consider a Fermi liquid bound within a free spherical surface R = const. We assume that the quasiparticle phase-space distribution n(r, p, t) is governed by the Landau-Vlasov kinetic equation for zero temperature [6],
=I[n(r,
(2.1)
r, t)].
Here U(r, p, t) is the self-consistent potential. This is related to the distribution function n(r, p, t) by a self-consistency condition
qr,
P,
where
t>
dn(r’,
1
= ,/dr’
Jr
dp’ FT(r, P; r’, P’)
is the quasiparticle u(r, p, t) is determined as F(r,
p; r’, p’)
effective
P’,
(2.4
,
interaction.
Wr’, u(r,p,t)=b+$jdr’dp’S(r,p;r’,p’)
t)
Jr,
P’,
ap,
The velocity
t)
.
(2.3)
The term in the r.h.s. of Eq. (2.1) is the collision integral. In the present study, we are interested in a small deviation of the distribution 6n(r, p, t) from an equilibrium n,(r, p), i.e. we assume that n(r,
P,
t)
=n,(r,
r-1 +Sn(r,
p,
t).
(2.4)
A. Abrosimou et al. / Kinetic equation
43
If we neglect terms of order fin*, we get the linearized equation for C&G, p, t),
dt -
&j(r,
P, t) + ae”(r, P) @jfl(r, P, t)
dWr3
ar
8P
xw(~,
p> a%+., dr
-
p, t> + asu(r,
P)
Pi f) ap
$l(f-T
p1 t>
dr
8P
dP
P) dWr,
ar
=Jb(r,
P, t>l, (2.5)
where .f[Sn(r, p, t)] is the linearized colIision integral. Here, in accordance with (2.41, we introduce U(r, P, t) = Uo(r, P) +6t’(r,
P, t),
(2.6)
where NJ(r, p, t> is the deviation of i.J(r, p, t> from the equilibrium potential Uo(r, p), related to Sn(r, p, t). The self-consistent quasiparticle hamiltonian is then 2 Eo(f,
P)
=
&
+
u,(r,
(2.7)
P)
and the equilibrium distribution dg,(r,
P) an,(r,
8IJ
P)
tir
It&,
%(r, -
dr
Below, the equilibrium distribution alone, n,(r,
p> satisfies the equation
P) %(r,
p)
ap
=
0.
(2.8)
n,(r, p> is assumed to be a function of &,
P) = nO(E(rY P>).
p)
(2.9)
The distribution n&r, p) and the self-consistent field U,(r, p) are sharply changing functions of the radial variable r near R. Therefore, the semiclassical equation (2.5) is expected not to be valid in the diffuse surface region of the system [4,7]. In order to derive the equation of surface motion we will use Eq. (2.5) also in the edge region of the system. However, we shaIl see that the result obtained does not depend on the surface properties of the equilibrium distribution n,(r, pf and it can be used for studying systems with sharply changing edge density as in the case of nuclei. For simplicity we shall assume below that the quasiparticle effective interaction in eq. (2.5) does not depend on the velocity, F(r,
p; r’, p’) =F(r,
r’)
(2.10)
Then using eqs. (2.21, (2.4) and (2.6) we find that U(r,
t) = Uo(r)
+W(r,
t),
(2.11)
A. Abrosimou et al. / Kinetic equation
44
where dr’ dp’ F(r,
6U(r, t) = ;,
r’)&r(r’,
(2.12)
p’, t).
We shall assume also that the equilibrium potential is central U,(r)
(2.13)
= &Jr)*
Now we simplify eq. (2.5) by changing variables (r, p) to a new set of variables (r, E, I, (Y,p, r> as proposed in ref. ES], (2.14)
(r, P> --) (r, E, f, a, P, Y).
The new variables are quasiparticle energy E = ~a, particle angular momentum I = /r xp 1, radius r and Euler angles (a, p, y). The Euler angles are defined by the rotation of the laboratory frame (x, y, z) to (x’, y’, z’> with i’ along I and 9’ along r. Among the new variables, E, E, cy, /3 are constants of the motion. One can eliminate in Eq. (2.9, written in new variables, the derivative with respect to y by means of an expansion in g-functions [8]. Distributions of quasiparticles with positive radial velocities Gn+(r, E, 1, CX,p, y, t) and with negative ones 6~(r, E, E, (Y,p, y, t) are considered separately. In terms of the new variables the linearized Landau-Vlasov equation is
(2.15) where we have used
SU(r, t) =
c
YLN(;7r
(2.17)
Z.,M,N
C
-z”[(f+(r,E, 1, t>)fM+(f-(r,
E,
1, f))FM]
L,M,N
(2.18)
A. Abrosimov
et al. / Kinetic equation
45
Here the radial quasiparticle velocity is (2.19)
u(r,E,1)= [(2/m)(c-P/(2mrZ))y2.
represent displacements of the local energy for the functions (f *(r, E, I, t))$ quasiparticles with angular momentum 1 at the distance r from the centre of the system. They describe Fermi surface deformations. It can be shown that the subscripts LA4 of the functions
3. System with a free surface We isolate in eq. (2.15) terms which are proportional to dU,/dr dominant at the system’s edge by seeking a solution of the form
6n(r,
P,
t) =
duo(r)
2
Here we introduce
drW+,
the local displacement
and therefore
I
cp,t) .
(3.1)
SR(6,cp, t>of the effective surface
R(6,cp, t>from its equilibrium position 6R(fi, cp, t) =R(fi, rp, t> -R=
c ~R,,(~)Y,M(~, cp>-
(3.2)
L,M
In the new variables it can be written as
aR(a,
(9, t> =
c
&&)L,(h
~+%iv(~, P, Y))*.
(3.3)
L,MJ’
Now taking into account Eqs. (3.1), (2.16) and (3.3) we find 6n(r,
E, 1, a, P, Y, t>
x (q&(%
P7
?I)*.
(3.4)
A. Abrosimou
46
et at. / Kinetic equation
According to Eq. (3.4) the deviation of the self-consistent potential W’,,(r, t) (see Eq. (2.17)) is divided in a bulk component 8GLM(r, t) and a surface quantity SUL’$(r > l) , &!.I,,( r, t) = Sii,,(
r, t) + SUL$ r, t).
(3.5)
In our approximation the colhsion integral in the r.h.s. of Eq. (2.15) depends only on the bulk component of the distribution function. In fact, using eq. (3.4) and (2.9) we obtain
22
wl(r)
I
--+R,,(r)
1=~R,,(r)-$?[““(E)] =o.
(34
to dCi,/dr
can be neglected.
By means of Eqs. (3.4)-(3.6) one can write Eq. (2.15) as
Now in the bulk region (Y < RI terms proportional Then we obtain
r
(3.8)
A. Abrosimov et al. / Kinetic equation
47
In the diffuse edge region at r = R we keep only terms proportional to dU,,/dr. After some transformations one obtains the following equation for the effective surface motion, see appendix A:
= -2p(R,
E, W&,&
fa)&R,,(t)).
(3.9)
The essential point is that Eq. (3.9) does not contain quantities which depend on the equilibrium distribution ~JE(T, p)). Eq. (3.9) has a simple physical meaning: it is the condition of elastic reflection of quasiparticles from the moving surface, set in the frame moving with the surface (the mirror-reflection boundary condition). The same boundary condition was employed in deriving the energy dissipation rate in a deformed nucleus [9] and more recently in studying the nuclear surface response [3]. From eq. (3.9) we can obtain a boundary condition for the radial component of the quasiparticle density current (the macroscopic equation of surface motion). It reads
L,LM(b4r=R=mpo;(%A4w). Here, the quasiparticle
density current is determined
(3.10)
as (3.11)
and the equilibrium quasiparticle density is po =
-$2ri(Zm)‘j21mdr
E”%,(E).
If the distribution ~z,&E)is the Thomas-Fermi value of pO, see e.g. ref. [lo], 1 4?r Po=gTP
;
(3.12) one then we obtain the well-known
(3.13)
A boundary condition (3.10) corresponds to the one used in liquid-drop studies of nuclear vibrations [ll] and recently in liquid-particle approaches [5]. However, the use of an effective-surface equation (3.2) violates the self-consistency condition (2.12) for NJ, because additional (“external”) collective degrees of freedom are introduced. To obtain a closed system of equations one then
A. Abrosimov et al. / Kinetic equation
48
demands that the moving effective surface should be free. The pressure acting on the system’s surface is determined by the momentum flux tensor &ik defined as, see e.g. ref. [12], (3.14) Non-diagonal components L@,+, z%~, should vanish at the free surface. Indeed using the surface equation of motion it can be shown that at the moving effective surface
We also require that the additional pressure caused by the surface tension should equal the normal component @Jr, t) at r =R(6, cp, t>. Then we find
@L,h&?~)lr=RWd =a( f. -
l)( L + 2)RP26R.,(
t).
(3.16)
Here, u is the surface tension parameter (for a nuclear surface u = 1 MeV/fm*). Eq. (3.16) can be obtained from the kinetic equation (2.15) in the same way as it was done for semi-infinite systems [4]. We shall use Eq. (3.16) with a phenomenological value of C, which is a well-determined quantity. This equation is the same as in refs. [11,51. Introducing the new variables we obtain a system of two equations for (f+(r, E, I, t)),N,, see Eq. (2.15). To find the solutions, we should have another boundary condition in addition to Eq. (3.9). We will require that at the inner radial turning point r1 continuity should hold, (3.17) where at r,, u(rl, E, I> = 0. Using Eq. (2.19) and the fact that U&r> is approximately constant for r
1/(2m~)“*.
(3.18)
The problem is reduced to finding a solution to the kinetic equations (3.8) which should satisfy the boundary conditions Eq. (3.9) at the free surface and eq. (3.17) at the turning point r,. The “self-consistency” condition Eq. (3.16) must also be fulfilled, and it determines the eigenfrequencies of our system. Important is that we can use in Eq. (3.8), for the bulk region, the same simple effective interaction as in the infinite Fermi liquid. In the diffuse edge region where the effective
A. Abrosimou et al. / Kinetic equation
49
interaction is complicated we use boundary conditions for the distribution function at the free moving surface, Eq. (3.9).
4. Eigenfrequencies Now we perform the Fourier transformation with respect to time. We put
(f’(r,
E,
I,
u)):~=/~ dt --m
of Eqs. (38, (3.91, (3.16) and (3.17)
exp(iwt)(f*(r,
E, I, t))IM
(4.1)
and
(44 To ensure convergence of these integrals we assume that (f*(y, E, 1, t)>,,, = 0 and 6R,,(t) = 0 for t < r, and suppose that w has a small positive imaginary part in. Eq. (3.8) can be written then as
=r[(Ptr,
E,
1,
w));3;,+ (f-(r,
E,
1,
m))rM],r1
(4.3)
Solutions of these equations must satisfy the boundary conditions
I(
P(r,
E, 1, w)),“,-
(f--(r,
= iZwp( R, E, E)Y,&x,
E, 1, “,):,]I,=, $r)6RLM(
w)
(4.4)
and k P(r,
E, 1, o))FM -
(f (r,
In addition, the “self-consistency”
E, I, D,)1l’J /_,
= 0,
(4.5)
condition
%r,LM (r,o)lr_R=~(L-1)(L+2)R-26RLM(W)
(4.6)
50
A. Abrosimov
et al. / Kinetic equation
must be fulfilled. To establish some general properties of solutions of Eqs. (4.3)-(4.6) we consider first eq. (4.3) without the scattering term. From Eqs. (4.3)-(4.5) we find the following integral equations:
exp[ *i(
m( r,
= sin[+(wT(R,
E,
1) - W( r,
E, I) -NT(R,
E,
Z))]
E, 1))]
y,,($,
Xsin[ +( wT( r’, E, 1) - NT( r’, l, I)) - (oT(
+sin[i(oT(R,
E, 1) -NT(R,
fr)
r’, 6, 4 - NY(r’3
E, l))]/rdr’-$(8~LM(r’,
ET I))]
w))
r1
Xexp[ Ti(m(r’,
E, 1) -Ny(r’,
Here the same quantities are introduced
7(r7 6, I) = jrdr’+ r1
1 E 1) ) 7
E, l))]
1
.
(4.7)
as in ref. [8],
(4.8)
(4.9) The quantities T( R, E, I) = 27( R, E, I)
(4.10)
T(R,
(4.11)
and E, 1) = 2y(R,
E, I)
are, respectively, the radial and angular Because of the condition (4.6) solutions w. In this paper we discuss a solution system of quasiparticles (corresponding
periods for each quasiparticle orbit (E, 1). to Eq. (4.7) exist only at certain values of to the simplified problem of a free-gas to F = 0) in the nuclear volume. Putting
A. Abrosimov et al. / Kinetic equation
51
r, O) = 0 one can find explicit expressions for into Eq. (4.7) 8cLMLM( Using then Eq. (4.6) we obtain an equation for eigenfrequen(f’(r, E, 1, wN&. ties, see appendix B, that can be written as
-6
x
I0
(1 - h2)li2
idA A
=Cr(L-l)(L+2)R-‘. S-(n+Narccos(h)/7r)/(l-A2)1’2 (4.12)
Here we have introduced
the dimensionless quantities
s = wR/n-v,,
(4.13)
A = l/Rmv,,
(4.14)
and
where vr is the Fermi velocity. In the derivation of eq. (4.12) we have used for dn,/de the Thomas-Fermi-like approximation dn,/dE
= -6( E - or)
Positions of the uncorrelated mined, see appendix B, as sin[+(wT--NT)]
(4.15) independent-particle
resonances are explicitly deter-
=O,
(4.16)
where
T(A)
= 231
- A2)1’2,
r(A)
= 2 arccos( A).
(4.17)
It can be seen from Eq. (4.12) that all modes with L > 0 are damped. We remark that the decay of these collective modes may correspond to a one-body damping mechanism because only this mechanism is included in our present approximation.
52
A. Abrosimov
Eq. (4.12) can be transformed modes CL = 0). It reads
et al. / Kinetic equation
to a simple analytical form for the monopole
(4.18) Here s, = s/n
(4.19)
and s, + 1 w(sn) = +sn In -s _ I +i?T~s,O(~s,~ - 1). n
11
It can be seen from Eq. (4.20) that eigenfrequencies determine solutions of eq. (4.18) which correspond quantity (4.20) in powers of s,, w(sn)=sn(sn+$,3+$s,5+
(4.20) with s < 1 should be real. To to s < 1 one can expand the
. ..).
(4.21)
Using Eqs. (4.18) and (4.21) we find that up to terms of the order of s*,
$-*pOEFS2 + P()EF
u =
(4.22)
-
R
and
(4.23) By means of (4.13) we obtain the eigenfrequency l/7-
.
(4.24)
We can verify that for standard values of the parameters: (T= 1 MeV/fm*, pa = 0.17 fme3, lF = 40 MeV, m = 1.04 MeV (1O-22 s)2/fm3, R = 1.12~ll’~ fm the dimensionless frequency s(i) = 0.7 < 1 and tiw(‘) = 114A-‘/3 MeV. Solving Eq. (4.18) numerically we obtain so) = 0.63 and ho(‘) = 103A-‘/3 MeV. Our result is in reasonable agreement with the systematic behaviour of the excitation energy as a function of the mass number for isoscalar monopole resonances [10,13]. When making the comparison one should keep in mind that the present calculation does not include a residual interaction in the bulk of the
A. Abrosimov et al. / Kinetic equation
53
system. Indeed the agreement looks particularly good for heavy nuclei, where the sharp surface approximation is more realistic. In a sense this result represents a further justification of the use of surface peaked form factors for effective separable interactions [ 14,151.
5. Conclusions The Landau-Vlasov equation is reduced to a simple kinetic equation for a system with a free moving effective surface. In this equation the same quasiparticle interaction as in the infinite Fermi liquid can be used. Introducing the moving effective surface we include in a macroscopic way a complicated effective interaction in the surface region. Resonance frequencies in a system of free quasiparticles bound within the effective moving surface of the nucleus, which was determined in a self-consistent way, have the following features. All eigenfrequencies for isoscalar density vibrations with multipolarity L > 0 are complex quantities. This is because the independent-particle resonance frequencies, see Eqs. (4.16) and (4.171, form a continuum from zero to infinity. The collective modes are embedded among the single-particle ones and therefore are Landau damped. Taking into account some results of numerical calculations performed within the semiclassical RPA, see refs. [1,15], one may expect that the Landau damping will be small for resonances with low multipolarity (L = 2, 3) also in our semiclassical model. However, the Landau damping will be growing with increasing multipolarity, as we can clearly see from Eq. (4.12). This is in agreement with fully RPA results [13,19]. The monopole modes are also damped except for the lowest monopole frequency which is real. This comes from the fact that there is a gap in the monopole independent-particle frequency spectrum, which actually starts from oL$, = rv,/R, as we see from Eq. (4.16). On the other hand, the moving-surface assumption corresponds to taking into account an attractive effective interaction in the surface region which leads to a first collective monopole mode with frequency (4.241, smaller than wL$,. This result also agrees with the one obtained in RPA for heavy nuclei [13,18,19], where the “fragmentation width” (Landau damping) of the lowest monopole resonance is small [18,19] or wholly absent [13], depending on the used isoscalar interaction. We finally remark that the linearized Vlasov equation is solved without making any scaling approximation. For the first monopole mode an analytical expression can be found for the distribution function 6&r, p, t). It would be of interest to investigate properties of this monopole mode, especially the widths, in some detail. Particularly interesting is the appearing of the predicted absence of collisionless (Landau) damping for the monopole giant resonance. Since the escape widths seem experimentally quite small [16], the only possibility to understand the
54
A. Abrosimov
et al. / Kinetic equation
observed total width is to consider more carefully the spreading contribution coming from the collision term. This can easily arise from momentum-space distortions in &z(r, p, t> for the monopole mode. Some work is in progress on this problem. The authors express their gratitude to Professor D.M. Brink for discussions. V.A. and M.D.T. are still shocked by the sudden death of Prof. V. Strutinsky and will always remember the great human and scientific pleasure of working with Vilen. Appendix
A
To derive the equation of surface motion (3.9) we take the sum of Eq. (2.1.5), and to isolate explicitly all terms which are proportional to dU,(r)/dr we use the relation
a
m
ay=
p(r,
(A.11
E, Z)
where p(r, E, 1) = mu(r, E, 1) is the radial momentum. After these transformations one finds
=2p[(f+(r,E, 1, t>)L+(f-(r,
E,
1, t))L].
By using (3.4) this equation can be written in the form
a
z
+ iN
I2 + --
mr3
a ap
(A4
55
A. Abrosimou et al. / Kinetic equation
r, x Km+-, 3%-)GUg)(
t)
=24(f+(r,E, 1,t));M+(f-(r, E,&o.
(A.3)
E,
In the surface region the dominating terms are proportional have at r N R
, $Q&,(t)
to dU,/dr.
Then we
=0.
- SU~~(R, t)
(A.4) Now to transform the surface potential &!$$(R, t) in Eq. (A.4) we use the self-consistency condition (2.2) for the equilibrium potential U,. By using Eq. (2.2) and the assumption (2.11) this condition can be written in the form
-duo = / dr
dr'
rr2Fl(r,
r’)
Here the effective interaction
%(r, r’) =
C jjda
dPo(4 Sf(r,
=
r’>
at I = 1 is determined
dR’ .B(r, r’)Y,,(a)y,;(n’).
m
The equilibrium density distribution p,(r)
W)
dr’
p,,(r)
as
(A.6)
is given by
$/dpno(e(ry ~‘1)
=--
1 1,
E)
nO(Eo)’
(A-7)
A. Abrosimov et al. / Kinetic equation
56
By using Eqs. (AS) and (2.2) we express SU#R,
SU,(&R,
t) =
z =
SR,,(
r
t) in the form
t)
R
+ I a’dr’
Cz(FL( R, r’) - .F,( R, Y’))
dPo(
dr,
r’)
6R,,(
t). (A.81
After integration of Eq. (A.4) over p and taking into account Eq. (A.8 we find
x I aRdr’ r”(
r’) -Fi(R,
r’))
ddr’) dr,
GR,,ft)
1 .
(A.91
In this equation the term
FL{
2 reff
R, r’) - Fl( R, r’) N I__ ( L,R)2 ’
(A.10)
where r,, is the effective radius of the quasiparticle interaction [17]. Therefore, the integral containing this term can be neglected in Eq. (A.9) and Eq. (3.9) is obtained.
Appendix
B
To derive the equation for eigenfrequences (4.12) we should find the multipole rr,LM(~, w) at the surface (r = RI, see Eq. (4.6). component of the normal stress L% The starting point is the normal component @_(T, 0). It is determined from (3.14) as
A. Abrosimou et al. / Kinetic equation
57
Here, tr is the radial quasiparticle velocity, p, = mu, and the quantities S(r, p, w> and 6U(r, W) are defined in the same way as (4.1). The expression (B.1) can be written as
The integrals over the coordinates and momenta (r, p) in (B.2) can be transformed into integrals over the new variables (r, E, 1, QI,& y), see sect. 2, by calculating the jacobian of the transformation. Moreover, we can write the multipole expansion of the S-function as
In (13.3) we exploit the transformation
properties of the spherical harmonics
Then, by using the expansion in &$-functions (2.16) for Gii(r, p, w) and the orthogonality of S-functions, we find the normal stress (B.2) in terms of new variables. It writes as
where
x i
N= -L
YL>($T$#+(b
E,
1, w$*
A. Abrosimou
58
et al. / Kinetic equation
Here, p(r, E, 1) = mu(r, E, I), where u(r, E, I) is defined by (2.19). The maximum value of quasiparticle angular momentum I,,, is determined by the normalization condition
(B-7)
A = j-_/dr dp n,,( E(T, P)). By using (4.15) we obtain 1max= PFR,
(B*8)
where pF is the Fermi momentum. Putting in the integral equation (4.7) ijgL,,(r, w) = 0 we find the explicit expression for (f*(r, E, 1, WI),“,. At r = R we have
(f+(R,
6, I, w,):~=
exp[+$(wT(R, sin[i(wT(R,
x YLN(b, $+P(
E, 1) -NT(R, E, I) -NT(R, R,
E, /))I E, l))]
(B.9)
E, l)SR,,(w),
where the quantities T(R, E, I) and T(R, E, f) are determined by (4.10) and (4.11). Now putting (B.9) in (B.6) and using (4.15) and (B.8) we get at r = R @+f(R7
w)
1 (4rr)2 = --pwSR$w) h3 2L+l
x/p”RdlI(p(R, 0
,g,\Y,“(;“,
l))*
cot[;(wT(R,
$7r)12
E, I) -NT(R,
E, l))],
(B.lO)
where P(R> I) =P$
T(R,
- (VP~)~]~‘*,
1) =2-+i/p,R)‘]“*,
I’( R, 1) = 2 arccos(E/p,R).
(B.11)
(B.12)
(B.13)
A. Abrosimov et al. / Kinetic equation
By exploiting the dimensionless frequency s and angular momentum and (4.14), the normal stress (B.lO) can be written as
&M(R,
59
A, see (4.13)
4,rr2 s> = -6--2L + 1 EFPoS X
Ltdh ~(l-A*)
cot[rr(l-AZ)1’2,s-Narccos
A]. (B.14)
Here p. is determined expansion for cot 2 cot
2=
2
by (3.13) and Er =pg/2m.
By using in (B.14) the pole
(z-mr-l,
(B.15)
n= -Cc we find the final expression for P,,,LM(R, s). It reads 47 @v,&R,
s) = -6~
2L + 1 EFPoS (1 -A’)“*
x
I0
rdh A
(B.16) s-(n+Narccos(A)/rr)/(l-A*)l’*’
By inserting this result into eq. (4.6) the equation for eigenfrequencies obtained.
(4.12) is
References [l] M. Di Toro, Part. Nucl. 22 (1991) 385 [2] P. Schuck, R.W. Hasse, J. Jaenicke, C. Gregoire, B. Remaud, F. Sebille and E. Surand, Prog. Part. Nucl. Phys. 22 (1989) 181 [3] V.I. Abrosimov and J. Randrup, Nucl. Phys. A449 (1986) 446; A489 (1988) 412 [4] Yu.B. Ivanov, Nucl. Phys. A365 (1981) 301 [S] V. Strutinsky, A. Magner and V. Denisov, Z. Phys. A315 (1984) 301 [6] L.D. Landau, Sov. Phys. JETP 3 (1956) 920; 5 (19.57) 101 [7] A. Magner, Yad. Phys. 45 (1987) 374 [8] D.M. Brink, A. Dellafiore and M. Di Toro, Nucl. Phys. A456 (1986) 205 [9] J. Blocki, Y. Boneh, J. Nix, J. Randrup, U. Robel, A.J. Sierk and W.J. Swiatecki, Ann. of Phys. 113 (1978) 330 [lo] P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980) [ll] A. Bohr and B.R. Mottelson, Nuclear structure, vol. 2 (Benjamin-Cummings, Menlo Park, CA, 1976) [12] E.M. Lifschitz and L.P. Pitajevsky, Physical kinetics (Pergamon, Oxford, 1981)
60 [13] [14] [15] [16] [17] 1181 [19]
A. Abrosimou et al. / Kinetic equation T.S. Dumitrescu, C.H. Dasso, F.E. Serr and T. Suzuki, J. of G.F. Bertsch and S. Esbensen, Phys. Lett. B161 (1985) 248, G.F. Burgio and M. Di Toro, Nucl. Phys. A476 (1988) 189 A. Van der Woude, Nucl. Phys. A519 (1990) 17c E.E. Saperstein, S.A. Fayans and V.A. Khodel, Sov. J. Part. G.F. Bertsch and S.F. Tsai, Phys. Rep. 18 (1975) 125 J. Wambach, Proc. Int. School on nuclear structure, ed. V.
Phys. Cl2 (1986) 349 and references therein
Nucl. 9 (1978) 221 Soloviev (Alushta,
USSR,
1985) p. 83