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Monopole modes in a finite Fermi system with diffuse reflection boundary conditions
Nuclear Physics A571 (1994) 117-131 North-Holland
NUCLEAR PHYSICS A
Monopo~c modes in a finite Fermi system with diffuse reflection boundary conditions V.M. Kolomietz a, A.G. Magner a, V.M. Strutinsky a8b,S.M. Vydrug-Vlasenko a Institute for Nuclear Research, Kiev, Ukraine
a
b National Institute for Nuclear Physics, LNS, Catania, Italy
Received 22 March 1993 (Revised 13 September 1993)
Abstract At semi-classical response function for a finite system of Fermi particles on which acts a spherical external perturbation is obtained assuming a boundary condition with diffuse reflection which takes into account particle correlation interaction within the nuclear surface region.
1. Introduction
The problem of coherent collective modes is one of the most important in the theory of complex nuclei. Significant advances were made within theoretical approaches based upon the so-called Landau-Vlasov equation which describes a system of Fermi particles interacting with each other via perturbation of the common self-consistent field. Unfortunately, in the application to finite nuclear systems only numerical solutions are generally possible and it is difficult sometimes to view the physics behind it. Periodic perturbations, spread over the volume or concentrated at the surface, excite compression-like collective motion of particles. It is of certain interest in this respect to study quasi-stationa~ modes (resonances) which may appear in the response to such perturbations in a system of free particles whose motion is restricted in one or more dimensions [1,2]. The dynamics of the density distribution in the surface region plays an important role in the formation of such collective modes: the surface is one of the most flexible degrees of freedom in heavy nuclei. However, it is just in this region of the nucleus that the Landau-Vlasov equation fails. In the surface region, indeed, the density gradient is so large that particle momenta of the order of the nucleon’s Fermi momentum pr are involved, which hinders the quasi-particle concept of the Landau-Fe~i-Iiquid theory. The difficult may be overcome, however, by introducing a certain dynamic effective sharp surface as a dynamical variable. Formally, 03759474/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0375-9474(93)E0513-8
KM. Kolomietz et al. / Monopole modes
118
such a surface is conveniently defined as an envelope of maximal-density gradient points. The unrealistic task of solving realistic equations of microscopic dynamics in the surface region can be replaced by imposing a certain set of boundary conditions at the effective surface on quantities determined for the interior region where the density gradient is small [3-51. The boundary conditions state that the normal-to-surface component of the velocity of the nuclear matter must equal the normal velocity of the surface itself and the normal component of particle-pressure tensor must equilibrate the excess force due to the surface tension. The boundary conditions relate the volume dynamics to the surface shift. The momentum distribution of particles reflected from the surface must also be specified. As an alternative to the widely used model of elastic reflection from the wall of the moving potential well one may consider the so-called diffuse reflection condition [6] which assumes a uniform distribution of momenta of reflected particles. This may be a more realistic approach because the wavelength of quasiparticles, &r, colliding with the effective nuclear surface, is of the order of reflecting surface nonhomogeneities and equals the internucleonic distance rO in our case, i.e. h/p,r, - 1, see ref. [6]. As was noted above, the quasiparticle concept of the Landau Fermi-liquid theory fails in the surface layer: a particle arising in this layer interacts with several partners rather than the mean potential, so that the direction of the momentum of the outgoing particle may differ significantly from the mirror reflection value. Inside the nuclear volume the quasiparticle interaction is known to affect the dynamics. It created the zero-sound collective modes and is an important factor which must be included in a realistic theory. However, some resonance-response properties are predetermined by not always trivial features of the ensemble of particles of the finite system interacting with the dynamical surface, which makes it worthwhile to consider such features separately. The purpose of the present paper is to study these features by making use of the diffuse boundary conditions.
2. Boundary condition of diffise reflection Let us consider the linearized collisionless Landau-Vlasov
equation
Here f,, is the equilibrium distribution function, which is assumed to be of the form of a step, p and m are the nucleon’s momentum and mass, and Sf is the dynamical component of the distribution function, so that f
=fe, + Sf*
KM. Kdomietz et al. / MompIe
In Eq. (1) SV is the variation of the self-consistent external force acting on the system has the form V,,=h(t)q(r)
+c.c.,
-S(e -eF)v(r,
119
potential. The potential of the
A(f) =A, e-‘a’,
where o contains an infinitely ing on V, at c = - ~0.We shall and that the perturbation does tion with frequency w one can Sf=
modes
(3)
small positive imaginary part of adiabaticly switchassume that the mean field is spherically symmetric not break this ~met~. For a periodic perturbawrite then in the nuclear interior
6) e-‘“‘+C.C.
(4)
where e is the energy of the particle, eF =p$‘2m, and pr is the Fermi momentum. The quantity &, 61, which has to be found, at a given point r depends on two variables only: namely, the radius r = I r 1 and angle 6 between vectors r and p (I p 1 =pF). As was noted in the introduction we shall consider below the boundary conditions of the diffuse reflection. By the definition of the diffuse reflection [6], particles are isotropically reflected from the wall in the coordinate system moving with the wall. At cos 8 < 0 the distribution function (2) near the wall takes the form f(COS
i+)l r=R(t)=f,,(e
-P&(f)(COS
6 +
c)),
(5)
where U(t) is the surface velocity,
~
=
U,(w) em’“”+
C.C.
U,, stands for the complex amplitude yet to be found. The linearized boundary ~ndition for the distribution function (2) and (41 can be written as v( r, cos 4)
r=R,
= -pFUO(COS4 t C),
(7)
cos 6>0
where R, is the radius of the equilibrium distribution. The constant C is determined from the condition that the mean radial velocity of particles at the surface should equal the velocity of the surface itself,
(8) where peq is an equilibrium particle density in the interior. The boundary condition (7) is not sufficient, however, for determining
unam-
KM. Kolomietz et al. / Monopole modes
120
biguously the distribution function, because the amplitude lJ, remains yet unknown. A condition balancing the forces acting upon the surface must now be used, but a specific formulation of this condition requires a more detailed knowledge of the perturbation.
3. Response to a volume-distributed As a characteristic consider q(r)
example
external field
of a volume-distributed
external
=r2.
potential
we
(9)
The dynamical balance of forces at the surface is represented 13-51 S17,,( r=R=8P,
by the equation
(10)
Here SII,,. is the radial component of the momentum flux tensor,
(11) and SPvp is the pressure tensor SP,,=2/-----
d3p
P,P,
(2Trh)3
m
Sf(r,fi,
t>*
(12)
In (10) the quantity SP, = -
$SR(t) 0
(13)
is the excess pressure caused by the surface tension, u is the surface-tension coefficient and SR = R(t) - R,. In the derivation of (10) the consistency between variations of the averaged potential and the particle density in the surface region is taken into account 151.
4. Solution of the Landau-Vlasov
equation
We now want to solve Eq. (1) for the volume part of the distribution function Sf, see Eq. (4), and substitute this solution into the boundary conditions (7) and (10). Below we shall neglect the variation of the self-consistent potential in the
KM. Kolomietz
et al. / Monopole modes
interior assuming SV= 0 for r < R(t). Introducing r$=
$0, 6,
q = isin
121
coordinates
6,
(14)
0
one writes (1) as
where a = oR,/v,
= o/f2
(16)
is a dimensionless quantity. Here the parameter 0 = 2av,/L
R is introduced
defined as
= vF/Ro,
(17)
being approximately the frequency of the periodic motion along the shortest periodic paths in the spherical well which determine the amplitude of the shell structure energy interval [7-91,
L = 2rR,,
(18)
equal to the perimeter length of the well which approximately determines the mean length of such paths. The solution to (16) satisfying the boundary condition of diffuse reflection (7) can be written as 2hoRo w(n) eza5+ 7
(19)
where w(q) = exp(ia\ll_)[{m
(20)
-D(a)],
where
B,(a) + f
D(a)= (Bl(a)
-
;)
The quantities B,(a) are determined B n+l =&[e2’“-(n+l)B,,],
(21)
.
by the recurrence B,,=-&(eZ’“-1).
relationships
(22)
KM. Kolomietz
122
et al. / Monopole modes
Substituting (19) into the second boundary condition (10) we obtain the amplitude of velocity of the effective surface u = 0
yo% 4(1- fcs) _ 1 aPE
(
3aZ(a)
(23)
1’
where Z(a)
= -2i{B,(a)
+ f-
[&(a)
+ f]D(a)}
- &Cs.
(24)
Here, C, = bs/e,41i3,
b, = 4rr&,
(24a)
where u is the surface-tension parameter of the mass formula, R, = roA’13 and A is particle number. For realistic nuclear values of b, the dimensionless parameter C, = 0.03. The function (24) contains information about the system’s response.
5. Response function
The response function determined
for the external field (3) can be written as
mm S(w) =
i/d3rq(r)Q(r, m>, 0
where 6p(r, w) is the Fourier transform density,
(25) of the dynamical, or transition particle
6p( r, t) = 6p( r, cd) eerof + c.c..
It is convenient to represent terms, 6p(r,
t) =t?p”“(r,
t)-
this quantity as the sum of “volume” and “surface”
PO(‘) Peq
dPo(r) - - dr
6R(t)*
(26)
In this expression PO(r) coincides with the equilibrium density peq in the nuclear interior far from the edge region and it turns into zero outside the nucleus. In the surface region p. decreases sharply from its maximal asymptotic value in the
V.M. Kolomietz et al. / Monopole modes
123
nucleus to zero outside. The quantity SR(t) is the shift of the radius of the effective surface and Spv%, t) is the relatively small dynamical part of the density in the interior. Here we use the same sharp effective surface approximation to the calculation of the dynamical particle density and, in particular, p,(r) as in ref. [12]. It is defined as the zero p-moment of the distribution function Sj” determined above, +vol(r,
r) =
Spvo’(r, w) em’@’+ cx.
(27)
Here,
x = cos 9, y = r/R, and I/(X, y) is the amplitude (19) expressed in these new variables. The response function &o) can be calculated now by means of (25) with 6p as given by Eqs. (26j-428). Substituting z4{, q), see Eq. (191, into (28) and neglecting in this volume term the width of the diffuse layer in comparison with R,, we obtain 8 3&(a)
(29)
For a z+ 1
3(w>
2 3-s __i $(I h2WZ i
- &,)“,
where
3, = ~h2~R&h2.
(311
As is seen from Eq. (301, for large w the real part of L?(W) decreases as l/w’. The coefficient 3, in Eq. (30) coincides with the estimated value for the model-independent energy-weighted sum for monopole excitations, as it should [ll]. For small frequencies o K L? one obtains
(32)
Vhf Kolomietz et al. / Monopole modes
124
where terms containing C, were omitted for brevity. From Eq. (32) we find [lO,ll] the adiabatic stiffness parameter C,, related to the variation of the mean-square radius ( r2> by external field (91, Cad=
-[aye)]-‘=g& 0
and the corresponding B,,=
-
(33)
mass parameter
dTJ20) [‘w(o)]-“= &.
(34)
Here 5%” is the real part of L@‘.The adiabatic frequency mean-square root radius is estimated therefore as
of oscillations of the
In what follows we shall be interested also in the imaginary part L%“‘(O)of the response function which determines the dissipative properties of the system. Separating real and imaginary parts in Z(a) in Eq. (24) one finds
(36)
R"(o) = -
where X(a) = Re Z(a),Y(a)= -Zm Z(u).From Eqs. (20, (22) and (24) one gets X(u)=
--&,+2B;-2B;D~-2(B;cf)D”,
(37)
Y(u) = 2B; + ; - 2(B; + $)D’ + 2B;D", where prime and double primes are correspondingly Here, D'=
(38) real and imaginary parts.
(B;++)(B;-;)+B;B; (39) ’
(B;- $+B;2 ,,,=(B;-$)B;-(B;+f)B;' (B;-;)'+B;* and the following recurrence B;,,=B;-
(40)
’
relationships
are used:
cos 2u n+l n+l -B'" B;,l= _~ + -B' 2u 2u 2u Ia,
?I'
(41)
KM. Koiomietzet al. / Mono&e modes
125
where sin 2a B;,=2a,
cos 2a B;;=-2a+2a.
1
(42)
6. Energy absorption rate To determine stationary and quasi-stationary states it is convenient to consider the rate of energy absorption from the external field averaged over the period T = 2r/w. In order to facilitate comparison with other models, particularly with the slab geometry problem [l], we define the energy absorption rate per unit of the surface area as
(43) where A, is real. Substitution of (36) into (43) results in
aE -=_ dt The dependence of the absorption rate d E/dt on the frequency w is iflustrated in Fig. 1 where the parameter C, = 0. In the lower part of the figure are shown the positions of the complex roots ak of the characteristic equation Z(a,)
=0
(45)
which determine the frequencies of maximal absorption. Condition (45) can also be obtained directly from the boundary conditions (7),(10), not relating properties of quasi-stationa~ states to the present of the external field. Thus, in line with the general theory of the response function 1131 it can be concluded that there exist spherical quasi-stationary modes in the system of Fermi particles interacting with a moving spherical surface which give rise to a resonant behavior of absorption. The first root a, of Eq. (45) is found at Re a, = 2, which corresponds to the resonance energy ho, approximately MeV in a heavy nucleus.
(46) equal to 2hQ = 15
126
VIM. Kolomietz
et al. / Monopole modes
I
,A
0.0 0
I
a=w/Q
Re a Fig. 1. Energy adsorption rate dE/dt for a volume-extended monopole perturbing potential, see Eq. (441, versus the dimensionless parameter a = o / 0 (in units of AR,,Ai /pF). Shown below are roots of the characteristic equation (45) which determine the positions and widths of the resonances.
All roots of eq. (45) have finite imaginary parts and thus describe quasi-stationary (damped) periodic modes. The widths of the resonances are evaluated as
ay(a)
ax(a)
a(~ a) ‘a(Re
a)
a=Reka’
(47)
The ratio of the derivatives in the brackets is of the order of unity and one can estimate then Yk
=
(48)
-27ma,.
For the lowest resonance gives the width Aho, = y,hR = @IL
in the energy dependence
of the response function it
(49)
With increasing k the distance of poles uk from the real axis is increasing and so do the widths of the resonances. One can also determine the contributions of the resonances to the energyweighted sum rule, (50)
KM. Kolomietzet al. / Monopolemodes
127
For the first resonance it is nearly 100%. Higher resonances are depressed due to the rapid decrease of the dissipative response function (36) for larger 6.1.Such a distribution of strength among resonances explains why the adiabatic frequency wad, see Eq. (39, is so close to wr.
7. Response
to surface external
compression
Now we consider the case of a perturbation V,, localized at the edge of the density distribution. For small-amplitude surface vibrations such an external force can be simulated by means of Eq. (3) with
4(r) =
aPo(r) 7
i
1
r=R
+
(51)
The width d of the surface-density layer is small in comparison with R, and, following ref. [5], it can then be shown that the external force (3) with q(r) as in Eq. (51) results in the following condition of the balance of forces acting on a free surface:
(52)
8rr,,lr=Ro=8Pm+Pea.
The second term on the right-hand side is the effective surface pressure, peti = - /fdr
%xt= PO emrot + CC.,
po( r) 7
where PO= -&a/2/3
(54)
Here,
(55) see ref. [12], and up to a small fi2-correction it coincides with the surface-tension coefficient in Eq. (13) assuming that the quantity p is the same as the coefficient in the (Vpj2-term in the energy-density expression [ll]. We now transform (11, taking into account that VIJr, t> as determined by Eqs. (3) and (51) is zero in the interior region. With Eq. (14) it gives
VX Kolomietzet al. / Monopolemodes
128
for r
(58)
’
where 7’3
(59)
4PeqPF9
where Z(a) is the same as in Eq. (24). Substituting (26) into (25) and using (57), we get Sp”‘( r = R,)
L%‘(O) = -47rR;
(60)
2kl PlZq
where 3mUo
6pvo’( r) = 7
F
pqjoldx
W(T)
COS(~YX)
(61)
and l=z
VP,,
(62)
0
is a small dimensionless quantity of the order of d/R,, d being the width of the density edge. In the first order in 5 one can neglect the “volume” contribution in the response function to obtain approximately G 9(o)
= --
aZ( a) ’
where
(64)
lCM. Kolomietz
et al. / Monopole modes
The poles of this response function are determined istic equation (45). As small frequencies o +Cfi(a -=c1) we have
5(1-
fc,)’
In the opposite limit of large frequencies becomes
-$(gCs +G cos 2a)
as roots of the same character-
ia
a2
129
- 200(1-
ic,)’
(a x=- 1) the quantity
-
(65) L%‘(O)(Eq. (63))
(66)
Generally, the imaginary part of the response function is s”(w)
=
22
Y(a)
a X”(a) + Y”(a)
’
(67)
Note that this quantity now turns into zero as o-l, which means that for the surface-localized external field the convergency is weaker than in the case of the volume-extended potential (9). Substituting Eq. (67) into (43) gives ZC dt
17 = 18 X2(a)
Y(a) + Y’(u)
(68)
where (dE/dt), is the energy dissipation rate per unit of surface areas in the semi-infinite matter [l],
The quantity PO was defined in Eq. (54). It can be noted that the expression (68) for (E/d) can also be obtained directly by calculating the amount of work performed by the external force acting on the surface per unit of time, =P,J dt
t)U( t) = 2P, Re U,.
Again, after substitution of (58) into (70) we get (68). The quantity (68) measured in units of (dE/dt), is shown in Fig. 2 as a function of the dimensionless parameter a. For simplicity, we neglected small corrections
130
VTM. Kolomietz
0
et al. / Monopole modes
El
4
12
Re a Fig. 2. Energy absorption rate v for a spherically symmetric perturbation acting on the surface, see Eq. (661, in units of the wall-formula value (69). Roots of the characteristic equation (45) are the same as in Fig. 1 and are included for comparison with positions of resonances.
due to finite surface tension (C, = 0). The characteristic equation (45) is the same for both types of external fields, and the positions and widths of the maxima in dE/dt are the same. For a B 1 the quantity (70) turns into the wall formula [14]
(71) This limit may correspond to the case of semi-infinite matter (the system’s size parameter R, = COand 62 = 0) or of rapid perturbation in a finite-size system (0 Z=-n>.
8. Conclusion
In this paper we show that in a finite system of Fermi particles constrained within a dynamical spherical surface, quasi-stationary macroscopic modes may arise with a frequency close to that at which nuclear giant-monopole resonances are known to appear. In our mode the boundary conditions of diffuse reflection from the effective sharp surface are set. The diffusive reflection boundary conditions may be found as more appropriate because it accounts for the complexity of physical conditions in the region of large gradients of the density near the nuclear
KM. Kolomietz et al. / Monopole modes
131
edge where the picture of potential motion fails for the bulk of the nucleons. The lowest resonance found almost exhausts the energy-weighted sum. The position of this resonance appears to be close to the one determined as a pure stationary state in ref. [2] where the analogous problem was considered for the case of the mirror reflection boundary condition. So, the width of the lowest resonance in Figs. 1 and 2 is caused by surface correlation forces taken into account by means of the boundary conditions of the diffuse reflection. The origin of quasi-stationary modes can be related to the periodicity of particle motion in the closed volume of the potential well. The widths of the resonances are determined by specific distributions of velocities of the participating particles. For the model system considered here the widths are - partly or entirely - due to many-body interaction in the system’s surface region, which is phenomenologically included in the model in the form of a boundary condition of diffuse reflection. These widths are not contributed by the two-body Fermi-liquid viscosity [U-17] in the nuclear volume because we neglected the collision integral in the LandauVlasov equation.
References [l] [2] [3] [4] [S] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17]
V.M. Strutinsky, S. Vydrug-Vlasenko and A. Magner, Z. Phys. A327 (1987) 267 V.I. Abrosimov, M. Di Toro and V.M. Strutinsky, Nucl. Phys. AS62 (1993) 41 V.M. Strutinsky, A.G. Magner and M. Brack, Z. Phys. A318 (1984) 205 Yu.B. Ivanov, Nucl. Phys. A322 (1985) 633 A.G. Magner and V.M. Strutinsky, Z. Phys. A322 (1985) 633 I.L. Bekarevich and I.M. Khalatnikov, Sov. Phys. JETP 12 (1961) 1187 R. Balian and C. Bloch, Ann. of Phys. 64 (1971) 271 V.M. Strutinsky, in Proc. Mazurian School on Nuclear Physics, Mykolaiky Poland, 1974; Nucleonica 20 (1975) 679 V.M. Strutinsky and A.G. Magner, Sov. J. Part. Nucl. 7 (19771 138 B.K. Jennings and A.D. Jackson, Phys. Rep. 7 (1980) 141 V.M. Kolomietz, Local density approach for atomic and nuclear physics (Naukova Dumka, Kiev, 1990) [in Russian] V.M. Strutinsky, A.G. Magner and V.Yu. Denisov, Z. Phys. A322 (1985) 149 A. Isihara, Statistical physics (Academic Press, New York, 1971) J. Blocki, Y. Boneh, J.R. Nix, J. Randrup, M. Robel, A.J. Sierk and W.J. Swiatecki, Ann. of Phys. 113 (1978) 330 A.A. Abrikosov and I.M. Khalatnikov, Rep. Prog. Phys. 22 (1959) 329 G.F. Bertsch, P.F. Bortignon and R.A. Broglia, Rep. Mod. Phys. 55 (1983) 329 V.M. Kolomietz, A.G. Magner and V.A. Plujko, Z. Phys. A345 (1993) 131, 137
NUCLEAR PHYSICS A
Monopo~c modes in a finite Fermi system with diffuse reflection boundary conditions V.M. Kolomietz a, A.G. Magner a, V.M. Strutinsky a8b,S.M. Vydrug-Vlasenko a Institute for Nuclear Research, Kiev, Ukraine
a
b National Institute for Nuclear Physics, LNS, Catania, Italy
Received 22 March 1993 (Revised 13 September 1993)
Abstract At semi-classical response function for a finite system of Fermi particles on which acts a spherical external perturbation is obtained assuming a boundary condition with diffuse reflection which takes into account particle correlation interaction within the nuclear surface region.
1. Introduction
The problem of coherent collective modes is one of the most important in the theory of complex nuclei. Significant advances were made within theoretical approaches based upon the so-called Landau-Vlasov equation which describes a system of Fermi particles interacting with each other via perturbation of the common self-consistent field. Unfortunately, in the application to finite nuclear systems only numerical solutions are generally possible and it is difficult sometimes to view the physics behind it. Periodic perturbations, spread over the volume or concentrated at the surface, excite compression-like collective motion of particles. It is of certain interest in this respect to study quasi-stationa~ modes (resonances) which may appear in the response to such perturbations in a system of free particles whose motion is restricted in one or more dimensions [1,2]. The dynamics of the density distribution in the surface region plays an important role in the formation of such collective modes: the surface is one of the most flexible degrees of freedom in heavy nuclei. However, it is just in this region of the nucleus that the Landau-Vlasov equation fails. In the surface region, indeed, the density gradient is so large that particle momenta of the order of the nucleon’s Fermi momentum pr are involved, which hinders the quasi-particle concept of the Landau-Fe~i-Iiquid theory. The difficult may be overcome, however, by introducing a certain dynamic effective sharp surface as a dynamical variable. Formally, 03759474/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0375-9474(93)E0513-8
KM. Kolomietz et al. / Monopole modes
118
such a surface is conveniently defined as an envelope of maximal-density gradient points. The unrealistic task of solving realistic equations of microscopic dynamics in the surface region can be replaced by imposing a certain set of boundary conditions at the effective surface on quantities determined for the interior region where the density gradient is small [3-51. The boundary conditions state that the normal-to-surface component of the velocity of the nuclear matter must equal the normal velocity of the surface itself and the normal component of particle-pressure tensor must equilibrate the excess force due to the surface tension. The boundary conditions relate the volume dynamics to the surface shift. The momentum distribution of particles reflected from the surface must also be specified. As an alternative to the widely used model of elastic reflection from the wall of the moving potential well one may consider the so-called diffuse reflection condition [6] which assumes a uniform distribution of momenta of reflected particles. This may be a more realistic approach because the wavelength of quasiparticles, &r, colliding with the effective nuclear surface, is of the order of reflecting surface nonhomogeneities and equals the internucleonic distance rO in our case, i.e. h/p,r, - 1, see ref. [6]. As was noted above, the quasiparticle concept of the Landau Fermi-liquid theory fails in the surface layer: a particle arising in this layer interacts with several partners rather than the mean potential, so that the direction of the momentum of the outgoing particle may differ significantly from the mirror reflection value. Inside the nuclear volume the quasiparticle interaction is known to affect the dynamics. It created the zero-sound collective modes and is an important factor which must be included in a realistic theory. However, some resonance-response properties are predetermined by not always trivial features of the ensemble of particles of the finite system interacting with the dynamical surface, which makes it worthwhile to consider such features separately. The purpose of the present paper is to study these features by making use of the diffuse boundary conditions.
2. Boundary condition of diffise reflection Let us consider the linearized collisionless Landau-Vlasov
equation
Here f,, is the equilibrium distribution function, which is assumed to be of the form of a step, p and m are the nucleon’s momentum and mass, and Sf is the dynamical component of the distribution function, so that f
=fe, + Sf*
KM. Kdomietz et al. / MompIe
In Eq. (1) SV is the variation of the self-consistent external force acting on the system has the form V,,=h(t)q(r)
+c.c.,
-S(e -eF)v(r,
119
potential. The potential of the
A(f) =A, e-‘a’,
where o contains an infinitely ing on V, at c = - ~0.We shall and that the perturbation does tion with frequency w one can Sf=
modes
(3)
small positive imaginary part of adiabaticly switchassume that the mean field is spherically symmetric not break this ~met~. For a periodic perturbawrite then in the nuclear interior
6) e-‘“‘+C.C.
(4)
where e is the energy of the particle, eF =p$‘2m, and pr is the Fermi momentum. The quantity &, 61, which has to be found, at a given point r depends on two variables only: namely, the radius r = I r 1 and angle 6 between vectors r and p (I p 1 =pF). As was noted in the introduction we shall consider below the boundary conditions of the diffuse reflection. By the definition of the diffuse reflection [6], particles are isotropically reflected from the wall in the coordinate system moving with the wall. At cos 8 < 0 the distribution function (2) near the wall takes the form f(COS
i+)l r=R(t)=f,,(e
-P&(f)(COS
6 +
c)),
(5)
where U(t) is the surface velocity,
~
=
U,(w) em’“”+
C.C.
U,, stands for the complex amplitude yet to be found. The linearized boundary ~ndition for the distribution function (2) and (41 can be written as v( r, cos 4)
r=R,
= -pFUO(COS4 t C),
(7)
cos 6>0
where R, is the radius of the equilibrium distribution. The constant C is determined from the condition that the mean radial velocity of particles at the surface should equal the velocity of the surface itself,
(8) where peq is an equilibrium particle density in the interior. The boundary condition (7) is not sufficient, however, for determining
unam-
KM. Kolomietz et al. / Monopole modes
120
biguously the distribution function, because the amplitude lJ, remains yet unknown. A condition balancing the forces acting upon the surface must now be used, but a specific formulation of this condition requires a more detailed knowledge of the perturbation.
3. Response to a volume-distributed As a characteristic consider q(r)
example
external field
of a volume-distributed
external
=r2.
potential
we
(9)
The dynamical balance of forces at the surface is represented 13-51 S17,,( r=R=8P,
by the equation
(10)
Here SII,,. is the radial component of the momentum flux tensor,
(11) and SPvp is the pressure tensor SP,,=2/-----
d3p
P,P,
(2Trh)3
m
Sf(r,fi,
t>*
(12)
In (10) the quantity SP, = -
$SR(t) 0
(13)
is the excess pressure caused by the surface tension, u is the surface-tension coefficient and SR = R(t) - R,. In the derivation of (10) the consistency between variations of the averaged potential and the particle density in the surface region is taken into account 151.
4. Solution of the Landau-Vlasov
equation
We now want to solve Eq. (1) for the volume part of the distribution function Sf, see Eq. (4), and substitute this solution into the boundary conditions (7) and (10). Below we shall neglect the variation of the self-consistent potential in the
KM. Kolomietz
et al. / Monopole modes
interior assuming SV= 0 for r < R(t). Introducing r$=
$0, 6,
q = isin
121
coordinates
6,
(14)
0
one writes (1) as
where a = oR,/v,
= o/f2
(16)
is a dimensionless quantity. Here the parameter 0 = 2av,/L
R is introduced
defined as
= vF/Ro,
(17)
being approximately the frequency of the periodic motion along the shortest periodic paths in the spherical well which determine the amplitude of the shell structure energy interval [7-91,
L = 2rR,,
(18)
equal to the perimeter length of the well which approximately determines the mean length of such paths. The solution to (16) satisfying the boundary condition of diffuse reflection (7) can be written as 2hoRo w(n) eza5+ 7
(19)
where w(q) = exp(ia\ll_)[{m
(20)
-D(a)],
where
B,(a) + f
D(a)= (Bl(a)
-
;)
The quantities B,(a) are determined B n+l =&[e2’“-(n+l)B,,],
(21)
.
by the recurrence B,,=-&(eZ’“-1).
relationships
(22)
KM. Kolomietz
122
et al. / Monopole modes
Substituting (19) into the second boundary condition (10) we obtain the amplitude of velocity of the effective surface u = 0
yo% 4(1- fcs) _ 1 aPE
(
3aZ(a)
(23)
1’
where Z(a)
= -2i{B,(a)
+ f-
[&(a)
+ f]D(a)}
- &Cs.
(24)
Here, C, = bs/e,41i3,
b, = 4rr&,
(24a)
where u is the surface-tension parameter of the mass formula, R, = roA’13 and A is particle number. For realistic nuclear values of b, the dimensionless parameter C, = 0.03. The function (24) contains information about the system’s response.
5. Response function
The response function determined
for the external field (3) can be written as
mm S(w) =
i/d3rq(r)Q(r, m>, 0
where 6p(r, w) is the Fourier transform density,
(25) of the dynamical, or transition particle
6p( r, t) = 6p( r, cd) eerof + c.c..
It is convenient to represent terms, 6p(r,
t) =t?p”“(r,
t)-
this quantity as the sum of “volume” and “surface”
PO(‘) Peq
dPo(r) - - dr
6R(t)*
(26)
In this expression PO(r) coincides with the equilibrium density peq in the nuclear interior far from the edge region and it turns into zero outside the nucleus. In the surface region p. decreases sharply from its maximal asymptotic value in the
V.M. Kolomietz et al. / Monopole modes
123
nucleus to zero outside. The quantity SR(t) is the shift of the radius of the effective surface and Spv%, t) is the relatively small dynamical part of the density in the interior. Here we use the same sharp effective surface approximation to the calculation of the dynamical particle density and, in particular, p,(r) as in ref. [12]. It is defined as the zero p-moment of the distribution function Sj” determined above, +vol(r,
r) =
Spvo’(r, w) em’@’+ cx.
(27)
Here,
x = cos 9, y = r/R, and I/(X, y) is the amplitude (19) expressed in these new variables. The response function &o) can be calculated now by means of (25) with 6p as given by Eqs. (26j-428). Substituting z4{, q), see Eq. (191, into (28) and neglecting in this volume term the width of the diffuse layer in comparison with R,, we obtain 8 3&(a)
(29)
For a z+ 1
3(w>
2 3-s __i $(I h2WZ i
- &,)“,
where
3, = ~h2~R&h2.
(311
As is seen from Eq. (301, for large w the real part of L?(W) decreases as l/w’. The coefficient 3, in Eq. (30) coincides with the estimated value for the model-independent energy-weighted sum for monopole excitations, as it should [ll]. For small frequencies o K L? one obtains
(32)
Vhf Kolomietz et al. / Monopole modes
124
where terms containing C, were omitted for brevity. From Eq. (32) we find [lO,ll] the adiabatic stiffness parameter C,, related to the variation of the mean-square radius ( r2> by external field (91, Cad=
-[aye)]-‘=g& 0
and the corresponding B,,=
-
(33)
mass parameter
dTJ20) [‘w(o)]-“= &.
(34)
Here 5%” is the real part of L@‘.The adiabatic frequency mean-square root radius is estimated therefore as
of oscillations of the
In what follows we shall be interested also in the imaginary part L%“‘(O)of the response function which determines the dissipative properties of the system. Separating real and imaginary parts in Z(a) in Eq. (24) one finds
(36)
R"(o) = -
where X(a) = Re Z(a),Y(a)= -Zm Z(u).From Eqs. (20, (22) and (24) one gets X(u)=
--&,+2B;-2B;D~-2(B;cf)D”,
(37)
Y(u) = 2B; + ; - 2(B; + $)D’ + 2B;D", where prime and double primes are correspondingly Here, D'=
(38) real and imaginary parts.
(B;++)(B;-;)+B;B; (39) ’
(B;- $+B;2 ,,,=(B;-$)B;-(B;+f)B;' (B;-;)'+B;* and the following recurrence B;,,=B;-
(40)
’
relationships
are used:
cos 2u n+l n+l -B'" B;,l= _~ + -B' 2u 2u 2u Ia,
?I'
(41)
KM. Koiomietzet al. / Mono&e modes
125
where sin 2a B;,=2a,
cos 2a B;;=-2a+2a.
1
(42)
6. Energy absorption rate To determine stationary and quasi-stationary states it is convenient to consider the rate of energy absorption from the external field averaged over the period T = 2r/w. In order to facilitate comparison with other models, particularly with the slab geometry problem [l], we define the energy absorption rate per unit of the surface area as
(43) where A, is real. Substitution of (36) into (43) results in
aE -=_ dt The dependence of the absorption rate d E/dt on the frequency w is iflustrated in Fig. 1 where the parameter C, = 0. In the lower part of the figure are shown the positions of the complex roots ak of the characteristic equation Z(a,)
=0
(45)
which determine the frequencies of maximal absorption. Condition (45) can also be obtained directly from the boundary conditions (7),(10), not relating properties of quasi-stationa~ states to the present of the external field. Thus, in line with the general theory of the response function 1131 it can be concluded that there exist spherical quasi-stationary modes in the system of Fermi particles interacting with a moving spherical surface which give rise to a resonant behavior of absorption. The first root a, of Eq. (45) is found at Re a, = 2, which corresponds to the resonance energy ho, approximately MeV in a heavy nucleus.
(46) equal to 2hQ = 15
126
VIM. Kolomietz
et al. / Monopole modes
I
,A
0.0 0
I
a=w/Q
Re a Fig. 1. Energy adsorption rate dE/dt for a volume-extended monopole perturbing potential, see Eq. (441, versus the dimensionless parameter a = o / 0 (in units of AR,,Ai /pF). Shown below are roots of the characteristic equation (45) which determine the positions and widths of the resonances.
All roots of eq. (45) have finite imaginary parts and thus describe quasi-stationary (damped) periodic modes. The widths of the resonances are evaluated as
ay(a)
ax(a)
a(~ a) ‘a(Re
a)
a=Reka’
(47)
The ratio of the derivatives in the brackets is of the order of unity and one can estimate then Yk
=
(48)
-27ma,.
For the lowest resonance gives the width Aho, = y,hR = @IL
in the energy dependence
of the response function it
(49)
With increasing k the distance of poles uk from the real axis is increasing and so do the widths of the resonances. One can also determine the contributions of the resonances to the energyweighted sum rule, (50)
KM. Kolomietzet al. / Monopolemodes
127
For the first resonance it is nearly 100%. Higher resonances are depressed due to the rapid decrease of the dissipative response function (36) for larger 6.1.Such a distribution of strength among resonances explains why the adiabatic frequency wad, see Eq. (39, is so close to wr.
7. Response
to surface external
compression
Now we consider the case of a perturbation V,, localized at the edge of the density distribution. For small-amplitude surface vibrations such an external force can be simulated by means of Eq. (3) with
4(r) =
aPo(r) 7
i
1
r=R
+
(51)
The width d of the surface-density layer is small in comparison with R, and, following ref. [5], it can then be shown that the external force (3) with q(r) as in Eq. (51) results in the following condition of the balance of forces acting on a free surface:
(52)
8rr,,lr=Ro=8Pm+Pea.
The second term on the right-hand side is the effective surface pressure, peti = - /fdr
%xt= PO emrot + CC.,
po( r) 7
where PO= -&a/2/3
(54)
Here,
(55) see ref. [12], and up to a small fi2-correction it coincides with the surface-tension coefficient in Eq. (13) assuming that the quantity p is the same as the coefficient in the (Vpj2-term in the energy-density expression [ll]. We now transform (11, taking into account that VIJr, t> as determined by Eqs. (3) and (51) is zero in the interior region. With Eq. (14) it gives
VX Kolomietzet al. / Monopolemodes
128
for r
(58)
’
where 7’3
(59)
4PeqPF9
where Z(a) is the same as in Eq. (24). Substituting (26) into (25) and using (57), we get Sp”‘( r = R,)
L%‘(O) = -47rR;
(60)
2kl PlZq
where 3mUo
6pvo’( r) = 7
F
pqjoldx
W(T)
COS(~YX)
(61)
and l=z
VP,,
(62)
0
is a small dimensionless quantity of the order of d/R,, d being the width of the density edge. In the first order in 5 one can neglect the “volume” contribution in the response function to obtain approximately G 9(o)
= --
aZ( a) ’
where
(64)
lCM. Kolomietz
et al. / Monopole modes
The poles of this response function are determined istic equation (45). As small frequencies o +Cfi(a -=c1) we have
5(1-
fc,)’
In the opposite limit of large frequencies becomes
-$(gCs +G cos 2a)
as roots of the same character-
ia
a2
129
- 200(1-
ic,)’
(a x=- 1) the quantity
-
(65) L%‘(O)(Eq. (63))
(66)
Generally, the imaginary part of the response function is s”(w)
=
22
Y(a)
a X”(a) + Y”(a)
’
(67)
Note that this quantity now turns into zero as o-l, which means that for the surface-localized external field the convergency is weaker than in the case of the volume-extended potential (9). Substituting Eq. (67) into (43) gives ZC dt
17 = 18 X2(a)
Y(a) + Y’(u)
(68)
where (dE/dt), is the energy dissipation rate per unit of surface areas in the semi-infinite matter [l],
The quantity PO was defined in Eq. (54). It can be noted that the expression (68) for (E/d) can also be obtained directly by calculating the amount of work performed by the external force acting on the surface per unit of time, =P,J dt
t)U( t) = 2P, Re U,.
Again, after substitution of (58) into (70) we get (68). The quantity (68) measured in units of (dE/dt), is shown in Fig. 2 as a function of the dimensionless parameter a. For simplicity, we neglected small corrections
130
VTM. Kolomietz
0
et al. / Monopole modes
El
4
12
Re a Fig. 2. Energy absorption rate v for a spherically symmetric perturbation acting on the surface, see Eq. (661, in units of the wall-formula value (69). Roots of the characteristic equation (45) are the same as in Fig. 1 and are included for comparison with positions of resonances.
due to finite surface tension (C, = 0). The characteristic equation (45) is the same for both types of external fields, and the positions and widths of the maxima in dE/dt are the same. For a B 1 the quantity (70) turns into the wall formula [14]
(71) This limit may correspond to the case of semi-infinite matter (the system’s size parameter R, = COand 62 = 0) or of rapid perturbation in a finite-size system (0 Z=-n>.
8. Conclusion
In this paper we show that in a finite system of Fermi particles constrained within a dynamical spherical surface, quasi-stationary macroscopic modes may arise with a frequency close to that at which nuclear giant-monopole resonances are known to appear. In our mode the boundary conditions of diffuse reflection from the effective sharp surface are set. The diffusive reflection boundary conditions may be found as more appropriate because it accounts for the complexity of physical conditions in the region of large gradients of the density near the nuclear
KM. Kolomietz et al. / Monopole modes
131
edge where the picture of potential motion fails for the bulk of the nucleons. The lowest resonance found almost exhausts the energy-weighted sum. The position of this resonance appears to be close to the one determined as a pure stationary state in ref. [2] where the analogous problem was considered for the case of the mirror reflection boundary condition. So, the width of the lowest resonance in Figs. 1 and 2 is caused by surface correlation forces taken into account by means of the boundary conditions of the diffuse reflection. The origin of quasi-stationary modes can be related to the periodicity of particle motion in the closed volume of the potential well. The widths of the resonances are determined by specific distributions of velocities of the participating particles. For the model system considered here the widths are - partly or entirely - due to many-body interaction in the system’s surface region, which is phenomenologically included in the model in the form of a boundary condition of diffuse reflection. These widths are not contributed by the two-body Fermi-liquid viscosity [U-17] in the nuclear volume because we neglected the collision integral in the LandauVlasov equation.
References [l] [2] [3] [4] [S] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17]
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