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Eigenmodes of a compressible liquid drop.

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Ann. Physik 5 (1996) 471-500
Annalen
der Physik
0 Johann Ambrosius Barth 1996
Eigenmodes of a compressible liquid drop
J. Wehner and HJ. Krappe
Hahn-Meitner Institut Berlin, D- 14091 Berlin, Germany
Received 21 December 1995, in revised form 21 February 1996, accepted 6 March 1996
Abstract. The eigenmodes of a non-viscous, compressible liquid drop are investigated. The spectrum is shown to be derivable from two Hermitian eigenvalue problems which are weakly coupled
by a non-Hermitian operator. It is shown that both eigenvalue problems admit an asymptotic, leptodermous expansion. Their contribution to the entropy of the droplet therefore also allows for such
an expansion.
Keywords: Droplet vibrations; Debye model for atomic clusters.
1 Introduction
The free oscillations of a homogeneous, elastic, solid sphere have been studied first
by H. Lamb [l]. Numerical tables and displacement fields were published by Sat0
and Usami [2]. Elastic modes of a solid sphere with fixed surface were treated in a
famous paper by Debye [3]. For an incompressible, ideal liquid sphere Rayleigh first
derived the eigenmodes [3]. The case of a viscous incompressible - charged and uncharged - spherical drop was treated only more than seventy years later by Chandrasekhar [5] and Reid [6]. In this paper we want to study the eigenmodes of a compressible, inviscid drop. In view of applications in statistical mechanics we are interested in the energy-averaged density of these modes as a function of the size and
shape of the drop.
In the next section we consider the eigenmodes of a free spherical drop as well as
those of a drop with fixed surface. Analytical expressions for the average density of eigenmodes in the limits of vanishing compressibility and vanishing surface tension will
be given. We shall then treat these limiting cases for drops of any smooth shape and
show how the general case may be obtained from the two limiting situations. The result
will finally be used to discuss the question whether the entropy of a drop can be expected to possess a leptodermous expansion, i.e. can be written as the sum of terms proportional to the volume, the surface area, and the mean curvature of the drop.
2 Eigenmodes of spherical drops
For an inviscid liquid the velocity field has vanishing curl and can be derived from a
velocity potential 4 which satisfies the wave equation
472
Ann. Physik 5 (1996)
with the sound velocity c related to the compressibility K and mass density pm
through
The stationary solution satisfies the Helmholtz equation
A4 + k 2 4 = 0
with wave number k = o / c [7].
2. I Helmholtz equation with Neumann and Dirichlet boundary conditions
Two types of boundary conditions will be considered, which both lead to a self-adjoint
eigenvalue problem. For a fixed surface we have the Neumann boundary condition
on the surface,
=0
(A)
where the positive normal derivative shall here and in the following be defined to
point inwards. For a sphere with radius R separation of the Helmholtz equation in
spherical coordinates leads to the eigenvalue equation
for the wave numbers kl, corresponding to each partial wave I.
As we will see below the Dirichlet boundary condition
4 =0
corresponds to a free surface in the limit of vanishing surface tension
the eigenvalue equation
"nR)
(B)
on the surface
=0
0.
It leads to
(2)
for the case of the sphere.
2.2 Helmholtz equation with the boundary condition of a free sui$ace
More interesting are the eigenmodes of a droplet with a free surface and finite surface tension. To obtain the boundary condition in this case we have to satisfy on the
surface the geometrical condition
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
473
where n is the displacement field of the surface in the normal direction (pointing inwards) with respect to the equilibrium shape, and the equilibrium condition
p = 2uH,
(C2')
+
where H = 1/2 (R;'
R;') is the mean curvature (with main curvature radii R1
and R2) and p the pressure. In view of the linearized Euler equation (linearized in
the deviations from equilibrium)
pmi,+V(p-p0)
=o
1
we find
4 = - - P, - P o
Pm
where p , is the equilibrium mass density. Therefore with the equilibrium condition
(C2') we obtain
on the surface. In the following we will replace (C2') by ((22).
Representing the surface by
yields [7] to linear order in P,,
r
1
Inserting the multipole expansion for the velocity potential
into the boundary conditions (C) we obtain
and
474
with
Ann. Phvsik 5 (1996)
oln= kl,,c. The
wave numbers kl,, are the solutions of the eigenvalue equation
1
-8,Wl(P) =
P
O(Z
R2Pm
- 1)(Z 2)
+
(3)
for p = Rkl,,. For a free surface there is no restoring force against a shift of the center of mass. Therefore the dipole 1 = 1 is to be excluded in (3).
The solutions of (3) are controlled by the dimensionless quantity
Rc2pm- R
o
OK
(=-----
(4)
which is the ratio between the restoring forces for sound waves and capillary waves.
For liquids is always large compared to one.
To discuss an incompressible fluid, we consider the limit c2 03 with W/n = ckl,,
fixed. Keeping only the leading term in the power series expansion of the left-hand
side of (3) yields the well-known [4] spectrum of capillary-type surface waves
---f
2
Wl
0
==-Z(Z-l)(Z+2),
pmR3
122
.
2.3 Eigenvalue densities
In order to compare the energy-averaged density of the eigenmodes W/n with analytical
results obtained by other methods, it is useful to consider first the averaged density
of the dimensionless quantity
For boundary conditions (A), (B) or (C) the sums in (6) run over all solutions of
Eqs. (l), (2) or (3), respectively. For the width f of the Gaussian averaging function
we will consider two choices: when f is smaller than the distances of neighboring
eigenvalues wln (on the p t scale), a convenient graphical representation of the full
density of eigenmodes is obtained. Figure 1 shows the result for the Neumann
boundary condition (1). The lowest 30 000 eigenvalues (properly accounting for their
(21 1)-fold degeneracy) are considered. A blow-up of the lower part of the figure
is presented in Fig. 2, such that individual eigenvalues - weighted with their multiplicity - are resolved. Also shown in the figures is the result of choosing a sufficiently
large averaging width f : in the scale chosen the averaged density is almost a straight
line.
+
475
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
400
300
n
N
Q
W
N
200
100
0
0
50
100
150
200
250
300
350
p2=(k*R)2
Fig. 1 Averaged eigenvalue density (6) for the sphere with fixed surface. Full lines averaging
width r = 1 , dashed line r = 150.
8000
n
6000
N
Q
W
N
4000
2000
0
0
1000
2000
3000
4000
5000
p2= ( k t R)2
Fig. 2 Blowup of the lower left part of Fig. 1.
This result is not unexpected. For boundary conditions (A) and (B) leading to selfadjoint eigenvalue problems the averaged eigenvalue density has an asymptotic expansion of leptodermous type [8]. In the form given by Balian and Bloch [9] it reads
(7)
'
with the volume V and the surface area S. The upper sign in the second term refers
to boundary condition (B), the lower to (A). For the sphere (7) yields
476
Ann. Physik 5 (1996)
In this expression the leading quadratic term is dominant as can be seen in Fig. 3,
where z(p) is shown for both si ns of the surface term.
The explicit appearance of cc) in the boundary condition (C2) shows that the case
of a free surface does not lead to a self-adjoint eigenvalue problem. Therefore the solutions of the eigenvalue equation (3) do not lead to a set of orthogonal functions in
contrast to the solutions to the eigenvalue equations ( I ) and (2). Condition (2) is
however recovered in the limit cr + 0. We solved (3) with = 156, which corresponds to a sodium cluster with N =10000 atoms, with Wigner-Seitz radius
r, = 2.11,4, connected with the radius R of a spherical drop by R = rSN’l3,sound
~ 22.99 amu, and surface tension
velocity c = 2.5 .JO’ ms-I, atomic mass M N =
Q = 0.0105 eVA[Ill. Insertion of the solutions into (6) yields the upper curve in
Fig. 3. The curve is seen to lie well above the curves corresponding to the self-adjoint eigenvalue problems considered by Balian and Bloch [9].
It should be noted in passing that an intermediate choice of the averaging width r
leads to the well-known shell oscillations of the eigenvalue density superimposed on
a smooth background. For r = 50 Fig. 4 shows the two main oscillation frequencies
leading to a beat phenomenon (supershells [ 12]), which are characteristic for a spherical shape.
Also seen in Fig. 4 are the edge effects and truncation errors at the beginning of
the spectrum and at the upper cutoff, unavoidable in all averaging procedures. Their
range increases with the averaging width r. Alternative averaging procedures turned
out to be even less satisfactory whether they are based on a Lorentzian [9] or on
truncated completeness sums like [ 131
i!
1400
1200
1000
n
Q
v
N
800
600
400
200
0
0
10
20
30
40
50
60
70
p=R*k
Fig. 3 Averaged eigenvalue density for a sphere with different boundary conditions on the surface,
N = 10000. Full line: boundary condition (A), dotted line: boundary condition (B), dashed line:
boundary condition (C).
477
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
1400
1200
1000
n
N
800
Q
W
N
600
400
200
0I I
0
1000
I
I
I
I
2000
3000
4000
5000
I
p2=(k*R)’
Fig. 4 Averaged eigenvalue density for a sphere with fixed boundary and different averaging
widths.
where Pj are Legendre polynomials, SZ is the range of the spectrum to be averaged
(on the p*-scale), and j,,,,, M n/r. We also did not include curvature correction
polynomials in (6) which are often used in nuclear physics in connection with the
Strutinsky procedure [14] since they also increase the edge effects and there is indeed little curvature on the p2-scale.
On the scale of the ordinate in Fig. 4 the edge effects at the beginning of the spectrum are barely visible, although they are of the same order as the surface term in
(8). A more drastic edge effect is seen in Fig. 3. The slight bump in the upper curve
corresponding to boundary condition (C) around p x 13 is due to a lower-end edge
effect because of the peculiar w ’ / ~behavior of the capillary modes for small o as
will be shown below.
2.4 Analysis of the solutions for boundary condition (C)
To further analyze the structure of the eigenvalue problem (3) we plotted in Fig. 5
for each value of I the numerically found solutions and used the same value of
= 156 as considered before. The lowest eigenvalue for each I corresponding to a
mode with no radial node is clearly seen to be separated from the rest of the solutions, which are approximately equally spaced for each 1. This observation suggests
that the lowest modes are generically capillary modes and would be the only surviving modes in the limit of an incompressible fluid. The other modes, with radial
nodes, would be sound waves with the compressibility as restoring force. They
would be the only surviving modes for vanishing surface tension, + 00, and satisfy boundary condition (B) in this limit. On the surface these modes would have maximal amplitudes.
<
478
Ann. Phvsik 5 (1996)
100
I
'
I
80
60
40
20
0
20
10
0
30
p=R*k
Fig. 5 Position of eigenvalues in the @,I)-plane for a sphere with boundary condition (C),
N = 10000. Open circles:
r = 156, squares:
= 156/20, triangles:
r = 5 . 156.
1 .OEl4
8.OE1 3
-
A
I
6.OE1 3
v)
W
3 4.OE1 3
2.OE1 3
O.OEO
0
40
80
120
angular m o m e n t u m
160
t
Fig. 6 Position of eigenvalues in the (w,[)-plane for a sphere with boundary condition (C),
N = 10000. Open circles: c = 2 . 5 . I03m/s, 5 = 156, black squares: incompressible limit, c = 00,
black circles: frequencies of capillary waves with the first two terms of (1 2),
r = 156.
That this picture is in fact correct can be further substantiated by changing the surface tension Q. In Fig. 5 the position of the eigenvalues is also plotted for values of
which are 5 times larger and 20 times smaller than the previously used value. It is
seen that the sound modes are only weakly affected by this change, in contrast to the
capillary modes which shift to smaller values for smaller Q.
The change in 5 may also be interpreted as due to a change in the compressibility,
i.e. in c2 with Q fixed. For that purpose we now plot the eigenvalues olnvs. 1 for
= 156 and = 5 - 156 in Figs. 6 and 7 respectively. Also shown for comparison is
the position of the capillary-mode frequency for an incompressible (5 = 00) fluid.
c
c
479
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
1 .OEl 4
8.OE1 3
-
n
I
6.OE1 3
v)
v
3 4.OE1 3
2.OE1 3
O.OEO I
~
0
"
20
"
'
40
"
60
'
80
~
~
~
~
.
l
.
100 120 140 160 180
angular m o m e n t u m
t
Fig. 7 Position of eigenvalues in the im,I)-plane for a sphere with boundary condition (C),
N = 10000. Open circles: c = 5.59 . 10 d s , 4 = 5 * 156, black squares: incompressible limit,
c = 00.
+
It is remarkable that the summation - with (21 1)-fold multiplicity - of the
rather regularly spaced eigenvalues for each 1 give rise to the bunched eigenvalue
density of the full spherical problem. This is typical for a separable system since one
expects no bunching of eigenvalues in a one-dimensional eigenvalue problem.
In order to show the appropriateness of our picture of the nature of the eigenmodes in the case of a free surface, boundary condition (C), we will expand the velocity potential in this case in a basis consisting of eigensolutions for the boundary
value (B) plus one pure capillary mode
where the knl satisfy (2). Note that this expansion is not overcomplete since all
j/(k,,R) are zero by construction, however 4 is not. Therefore an additional basis
function is needed which does not vanish at the surface. Insertion into the wave
equation yields
where we have dropped the index m on A', B and k. Multiplication by
~ ~ j , ( k / , , r ) ? Rand
- ~ integration over r from 0 to R gives
where the two integrals [ 151
I
I
I
I
480
Ann. Physik 5 (1996)
R
0
have been used. The boundary condition (Cl) yields
and from (C2) follows
-hi
0
=7
(I -
1)(1+ 2)pi .
PmR
Recalling ( 5 ) for w: the two latter equations can be written, after eliminating Rp,,
n
Equations (9) and (10) form a set of harmonic oscillator equations for the coefficients A' and B which are coupled by a nonsymmetric, co-dependent coupling matrix.
This again clearly shows the non-selfadjoint nature of the boundary value problem
(C) and confirms our earlier assumption that the solution consist of capillary and
sound modes coupled weakly to each other.
We shall now treat the coupling by perturbation theory.
Case I) w >> 01, i.e. eigensolutions in the range of sound-wave frequencies. We
eliminate B from (9) and (10) and obtain for the perturbed eigenfrequency w
with the ratio
To lowest order in the coupling matrix elements (11) yields
(mi - w 2 ) ( 0 2- a;)= qinnw2w; .
Expansion of this expression to second order in the ratio
O,/Oln
finally gives
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
or in terms of the control parameter
48 1
< (4)
As we have seen in Fig. 5 the last term is negligibly small for a sodium cluster with
10 000 atoms.
Case 2) o << win, i.e. eigensolutions in the range of the capillary wave frequencies. We eliminate A’ from (9) and (10) and obtain
To fourth order in
o//o,n
or in terms of (
Alternatively we can expand the 1.h.s. of ( 3 ) up to the third term of its power series
in p = o R / c << 1 and we obtain
( 1 - 1)(1+ 2 )
o2
7 = l - (21 + 3)<
+
+
( 1 - q 2 ( 1 + 212(1 5 )
(21 q 2 ( 2 1 5)t2
+
+
’
which is identical with the preceding equation if and only if
We are not aware that these sums over zeros of spherical Bessel functions have been
Written down before.
482
Ann. Physik 5 (1996)
2.5 Density of capillary modes
We now return to the representation of the smooth part of the eigenvalue density corresponding to boundary condition (C).
The eigenvalue density of the capillary modes alone may be represented by the expression
can be obtained from a spline interpolation of the lowest
where the function ocap(l)
solutions of (3) for each 1. This function may then be numerically inverted to obtain
l(o)which is inserted into (13). The same was done for the second eigenvalues for
each 1, for the third eigenvalues and so on. Finally all these contributions to z ( o ) were
summed up and compared with the result of the averaging procedure (6) applied to the
total spectrum. The above-mentioned deficiencies of the Gaussian averaging at the lower edge of the spectrum are clearly seen. To show the sensitivity to a change of the compressibility we included in Fig. 8 eigenvalue densities for values of which are 20 times
smaller and 5 times larger than the one appropriate for a sodium cluster with
N = 10000, this time interpreted as a corresponding change in the sound velocity by
l/mand &,respectively, and with fixed surface tension.
Replacement of integer 1 values by a continuous real variable 1 amounts to an effective averaging over one unit on the I-scale. This is apparently sufficient to wash
out all shell effects.
To obtain an analytical expression to first order in
we inverted (12) to this order in and substituted the result into (13). To perform the necessary algebraic manipulations it is convenient to introduce the abbreviations
<
<-'
<
500
400
n
Q
300
W
N
200
100
0
p=R*k
Fig. 8 Averaged eigenvalue density of capillary modes for a sphere with boundary condition (C),
N = 10000 and different compressibilities. Curves 1) correspond to the actual sound velocity in sodium c = 2 . 5 . 103m/s,curves 2) to c = 5.59. 102m/s,curves 3) to c = 5.59. 103m/s. Dashed lines:
Gaussian average (6) with r = 150, full lines: (13) with numerically inverted functions /(a).
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
r3
483
We first invert (5) i.e.
= f ( l o ) to obtain lo(C). Cardan’s solution of this cubic
up to the sixth order. The result is
equation is expanded in powers of
c-’
with
Writing
and inserting in (12)
yields
c.
as a rational function of The eigenvalue density (13) can be written in terms of the
functions f , g, h and their derivatives with respect to 1. Using (12) to first order in
we obtain
r-’
Insertion of (14) and (15) into this expression allows to write it as a rational function
of The final result is
r.
484
Ann. Physik 5 (1996)
This expression shows the characteristic de endence of the eigenvalue density of the
capillary modes on 0113.The N213 and N' terms suggest a dependence on the surface area and the integral of the mean curvature over the surface, respectively. The
discussion of non-spherical shapes in the next section will show that this is in fact
correct. The coupling to the sound modes through 5-l introduces in addition a term
proportional to o.
P
3 Eigenvalue density for drops of arbitrary shape
For smooth, but not necessarily spherical shapes the smoothed eigenvalue density is
given directly by (7) for boundary conditions (A) and (B). We therefore have to consider only the boundary condition (C) in this section. First the condition (C2) has to
be expressed in terms of the characteristic geometrical properties of the surface to be
considered. We show in the Appendix that in terms of the metric tensor gij and the
second fundamental form of the theory of surfaces L,j we have
2a6H
a
- -n = -(LijLklgikd'+ AB)a,,+
P , 6n
P,
on the surface S,
where the Beltrqmi operator A B is the generalization of the two-dimensional Laplacian
in curved spaces. Introducing the Green function G(7, 7';k 2 )for the Helmholtz equation
with Neumann boundary conditions (A) and a fixed value of k 2 , the potential Q, on the
surface may formally be expressed in terms of its normal derivative on the surface
+(Z) = -
s
G(G,p)an4(/?)d2$
,
fi.
where surface coordinates shall be denoted by Eu' and The boundary condition (C2)
is then written as an eigenvalue problem in two dimensions (for fixed k2 !)
(L2 + A B + dG)a,,+ = 0
on the surface
with
-
Pm
u --02.
a
2
This equation is formally brought into the standard form of a self-adjoint eigenvalue
equation by introducing the function I+Y= G'I2a,+, defined on the surface,
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
485
However, since G depends on k and through w = ck also on w and therefore on d ,
(17) is not a standard Hermitian eigenvalue problem, as we have already seen above
for the spherical shape. The limiting case of an incompressible fluid corresponds to
k = 0. Then G is the Green function of the Laplace equation, and (17) is self-adjoint
and describes pure capillary waves.
The eigenvalue density corresponding to the operator AB has been obtained by Balian and Bloch [16]. The somewhat more complicated operator in (17) was not yet
treated, however. We therefore give a derivation using the same method as Balian
and Bloch [9, 161. We first introduce the (two-dimensional) Green function 6 for the
generalized Helmholtz equation (17), appropriate for a closed surface
+ A ~ ) G - ' /+~ K
(~-'/2(~2
2 )=
~
-d2(6
- 6')
.
In terms of this Green function the density of eigenmodes is
S
where the imaginary part has to be taken first and the limit d + d' afterwards. This
relation holds however only if 6 is the Green function of an Hermitian eigenvalue
problem. In the following we will nevertheless keep the k-dependence of G (up to order k2) and consider it as a fixed external parameter. Later we shall show that even a
perturbation expansion of z ( K ~ )for small k around k = 0 is not possible.
We seek a leptodermous expansion of z up to terms linear in the inverse curvature
radii. Since L2 is proportional to the sum of the inverse squares of the curvature radii, its contribution can be dropped. As shown in [I61 the differeye between A and
A s yields a term proportional to the Gaussian curvature (RlR2)- and can also be
neglected in the order considered here. The Green function 6 therefore satisfies in
this order the equation
(G-'12AG-*12 + x 2 ) 6 = -d2(6
- 6')
.
(19)
In [9] an expansion of the three-dimensional Green function is given, which is
appropriate for our purposes,
with
We need this Green function with both arguments on the surface, more precisely, in
the limit where r' and r" approach the surface from inside.
486
Ann. Physik 5 (1996)
Following the procedure and terminology in [9] Cartesian coordinates in a tangential plane at a point d on the surface are introduced with the z-axis along the inward
pointing surface normal. In the tangential plane two-dimensional Fourier transforms
are introduced. From the three-dimensional Fourier transform of GO(?,7’)
we obtain immediately the two-dimensional transform
with
a=&-,
Rea>O
and
As shown in [9] the normal derivatives of G can be expanded to linear order in the
curvature H and yield
- k2
an,Go (p, z = 0, z’ + Of ) = -2I + H a2
___
4a3
a2 - k2
& , G o ( ~ , z = z ‘ = O ) = H - 4a3
,
Using (22), (24), (25) the Fourier transform of (20) yields in the limit z = 0, z’
up to linear order in H
1
a
--+
O+
H a2 - k2
.
2 a4
Gf$)=-+--
Inserting this result in the Fourier transform of (19) gives for the Fourier transform
of the two-dimensional Green function G
I
G(P) = p2a(l
and for 6 itself
-By)
- K 2 - iy
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
487
where JO is the Bessel function of order zero.
The imaginary part of G is
and
-
1
1 Po
-ImG(a
- CY- + I = 0) = ,
n
271 b ' b o )
where po is the real, positive zero of
b@)=p 2J
p
q -H
--p2
2
+ H-k2 ___
P2 - K 2
2
p 2 - k2
To order k2 we have
and to linear order in H and k2,
Insertion of this result into (26) and integration over the surface yields to the same
order
This is the density of eigenvalues K~ for fixed k2. Especially for an incompressible
drop k2 = 0. Rescaling the density from the K~ to the co scale gives in this limit
which is identical with (16) for (-' = 0 in the spherical case.
This confirms our earlier suggestion that the first two terms in (16) are indeed
Proportional to the surface area and the mean curvature.
488
Ann. Physik 5 (1 996)
To obtain the effect of the coupling to the sound modes one may be tempted to
simply substitute k = o / c in (27). Specializing to the spherical case one does however not obtain the result (16) including the coupling terms. In particular the leading
term in N of the coupling contribution proportional to 5-l does not appear to be derivable from the last term in (27) by any kind of perturbation approach. This shows
that the relation between eigenvalue density and Green function, on which the trace
formula (1 8) is based, is definitely restricted to Hermitian eigenvalue problems. The
structure of (1 6) suggests however that also the terms multiplied with 5-l may be interpreted as being proportional to the surface area and the mean curvature taking into
account that is itself proportional to
We conclude this section by summarizing that the shape dependence of the averaged eigenvalue density of pure capillary modes is given by (28) and that of pure
sound modes by (7). The coupling between these modes was obtained for spherical
shapes in (16) for capillary modes whereas for the sound modes the effect of the coupling was seen to be negligibly small. For non-spherical shapes the effect of the coupling to the eigenvalue density of capillary waves could not be obtained with the
trace formula techniques used here.
4 Thermodynamic applications
Recent treatments of unimolecular evaporation from atomic clusters, in particular
sodium clusters [19], require the knowledge of the level density as a function of the
size and shape of the cluster. The oscillator ansatz for the level density used in [17,
181 does however not reflect the special role played by the surface of a cluster and
its shape. In terms of the entropy S the level density is given by &S . exp(S). To
account for its shape dependence the entropy is assumed in later treatments [19, 201
to admit a leptodermous expansion, i.e. a representation in terms of a bulk contribution proportional to the volume, a surface contribution and, if one wants to continue
this expansion, contributions proportional to higher invariants of the surface. This behavior of the entropy is to be seen in contrast to the binding energy of atomic clusters, which shows, as a function of the number of atoms, striking minima at “magic
numbers”.
4. I Leptodermous expansions of thermodynamic functions
One may therefore ask why the entropy should depend smoothly on the size of the
cluster without magic-number effects. Since the largest contribution to the entropy
comes from the ionic degrees of freedom we will first discuss their behavior. In the
spirit of the Debye theory of the specific heat we may describe the cluster as an elastic, homogeneous medium.
A description in terms of a hydrodynamical theory can certainly not exhaust all
degrees of freedom of a liquid cluster, since quantities like pressure or velocity imply
that each volume element contains sufficiently many atoms to allow the construction
of stable averages. Therefore only the slowest modes, connected with long wave
lengths, can be expected to be described in an hydrodynamical model. In the highfrequency limit on the other hand, a liquid behaves like a solid body and sustains in
particular transversal modes, cf. [21] for a more detailed discussion of the contribu-
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
489
tions of the different dynamical regimes to the thermodynamic functions of liquid alkali metals in the bulk.
In the following we will pursue the rather restricted aim to understand the basis
for a leptodermous expansion of those parts of the entropy which allow a description
in terms of the hydrodynamical continuum model. The actual situation is further idealized by the assumption that material parameters like K and o are independent of the
temperature and the frequency.
In the harmonic approximation the phonon contribution to the canonical partition
sum is
and the free energy becomes
nD
ln(2 sinhPhoi/2)
PFphon= - In Zphon=
i= 1
with the Debye frequency oD= a n D l where U D is the number of degrees of freedom,
which can be treated in the harmonic approximation in a given model. Its size in
classical hydrodynamics will be discussed in the next section.
From (29) follows the specific heat
with the effective number of particles Neyy = no/3, the entropy
nD
S = -&F = C((Bfi~i/2) ~ 0 t h P h ~ i / 2ln(2 sinhphoi/2)) ,
i= 1
and the average excitation energy
We will first discuss the case of a fixed boundary. AS shown in Figs. 1 and 2 the
density of eigenvalues oi can be decomposed into a rapidly fluctuating part and a
smooth part z ( o ) which admits the leptodermous expansion (7) for a compressible,
non-viscous liquid. According to [16] the same is true for an elastic solid body.
Since the individual terms in the sums (29-31) are very smooth functions of oi one
can replace them by integrals, i.e.
(33)
490
Ann. Physik 5 (1996)
introducing the smoothed density of eigenmodes
quency W D by
Z(W)
and defining the Debye fre-
00
z(w)do .
n D = /
(34)
0
Figure 9 shows a comparison between (30) and (33) for various values of the averaging width r for a sodium cluster with Nen = 92, which was, for this numerical
comparison, taken equal to the actual number of atoms in the cluster. It is seen that
c,, is rather insensitive to the averaging interval used to define z(w).
The shape dependence of the thermodynamic functions is due to the explicit shape
dependence of Z ( W ) in the integrand and the implicit one of W D through (34). Using
(7) (with the lower sign) in (34) the shape dependence of OD can be expressed in
terms of the shape functions Bs and Bcury[22] which are the surface area and the
mean curvature in units of the surface area and mean curvature of the sphere of equal
volume, respectively,
2
nD = -pD
972
3
+ 4-1 Bs
2
2
+ -BB,UW
PO
3n
(35)
with
If the quantity
1
,
,
,
,
,
4
,
I
3.OEO -
-
I
I
--
-
I
temperature (K)
Fig. 9 Specific heat per particle for a spherical cluster with N = 92, ( = 156 vs. temperature T .
Full line, dotted and short-dashed lines: boundary condition (A), averaging width in the eigenvalue
density Z ( W ) is r = 150,50,1, respectively. Long-dashed line, dashed-dotted and dash-triple-dotted
lines: boundary condition (C) averaging width r = I50,50, 1, respectively.
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
49 1
the solution of ( 3 5 ) for wD can be expanded in inverse powers of p D
(0)
For nD = 100 one obtains p!) = 11.2 and the correction to the leading term in ( 3 6 )
becomes 10% for the spherical shape (Bs = B,,,, = 1).
The thermodynamic functions F(p) may be expressed in terms of characteristic
functions f(Pho/2) by writing
OD
F(P)=
J z ( o ) f ( P h w / 2 )dm
*
0
For example, for the entropy F = S we have
f ( x ) = xcothx - ln(2sinhx) .
Introducing the incomplete moments
the shape dependence of F(p)can be exhibited more explicitly
2
1
F(p) = t 3 N F 2 ( P h m ~ / 2+) - t2BsN2I3F1( p h w ~ / 2 )
372
2
2
+tBcu,N'13Fo(phw~/2)
372
(37)
with the variable t = 2rs/(phc), which is proportional to the temperature. If only the
leading term of the expansion ( 3 6 ) is used and ~ D / N
is of order 1, the moments F,,
become independent of the size and shape of the system and ( 3 7 ) represents the leptodermous expansion of the thermodynamic functions. In general the moments have
to be expanded around w g ) = ~ p ; ) / ( r ~ N ' Restricting
/~).
the expansion to the leading two orders in N 1 I 3one obtains
492
Ann. Physik 5 (1996)
In this order the Debye theory is therefore seen to lead again to a leptodermous expansion, provided n D / N is shape and size independent.
More realistic for free clusters is the case of a free surface. In this case the pressure on the surface is zero and (29) represents the free enthalpy and therefore (33)
the specific heat cp. Using the set of eigensolutions of (3) we show the entropy for
spherical sodium clusters with the (somewhat academic) assumption N = Nesf =
40,92 and 10000 in Figs. 10, 11 and 12 respectively. To obtain S ( E ) (31) and (32)
were used and p eliminated numerically by spline techniques. Also shown for comparison is the entropy for the same clusters with fixed surface, based on the eigen-
300
x
Q
0
200
I
100
0
energy (ev)
Fig. 10 Entropy for a sodium cluster with N = 40 vs. excitation energy. Full line: boundary condition (A), dashed line: boundary condition (C).
700
600
500
400
300
200
100
0
energy (ev)
Fig. 11 Same as Fig. 10, but for N = 92.
493
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
60000
50000
40000
Q
0
-b
c
a
30000
20000
10000
0
energy (ev)
Fig. 12 Same as Fig. 10, but for N = 10 000.
a
-c
1.5EO
0
.+-
.-
1.OEO
Q
m
5.OE-1
LL-d
_ _ _ _ _ - - --
O.OEO
0
100
_-_-_______________
200
300
400
500
temperature (K)
Fig. 13 Specific heat cp per particle for a sodium cluster with boundary condition (C), N = 40.
Full line: total specific heat, dashed line: contribution of capillary modes, dashed-dotted line: contribution of sound modes.
solutions of (2). The increase of the phase-space volume because of the free surface
is clearly visible. In Figs. 13 and 14 the specific heat cp is broken up into the contributions from the capillary modes and the sound modes. As expected, the dominant
role of the former decreases with increasing size of the cluster until eventually the
volume-type sound modes yield the leading contribution.
The rather steep increase of the contribution of the capillary modes to ,c, for small
temperatures is due to the peculiar 0.1'1~ dependence of the density of eigenvalues of
those modes. We showed in Fig. 9 that the use of Gaussian-averaged eigenvalue densities tends to distort the behavior of cp for T + 0.
494
Ann. Physik 5 (1996)
3.OEO
a
0
2.5EO
2.OEO
1.OEO
Q
v)
5.OE-1
O.OEO
0
100
200
300
400
500
t e m p e ra t u re (K)
Fig. 14 Same as Fig. 13, but for N = 10000.
4.2 Damping of the elastic eigenmodes
The elastic model on which the last five figures are based is rather unrealistic since
below the melting temperature one would not expect hydrodynamical behavior and
above melting only the hydrodynamical modes with sufficiently long wave length
can be expected to be described in the inviscid approximation. An estimate for the
validity of this approximation can be obtained in the following way. In order that the
non-viscous treatment remains valid, the damping time l / y must be large compared
to the oscillation period. In terms of the viscosity v this ratio is given for capillary
waves by [23]
Taking the wave length of a multipole mode of order I on a sphere to be A = 2nR/I
we obtain k = 1/R. With the frequency (5) and p, = 3 M ~ , / ( 4 n r , 3()3 8 ) yields
where terms of the order 1 / 1 have been neglected. Using the viscosity of sodium at
63OOC [24] q = 2 mP as a typical value and other material parameters as given
above, the relation (38) becomes 1 << 0.2Ni13.This shows that even for a cluster with
N = lo6 only the lowest few capillary multipole modes can be treated as non-viscous,
hydrodynamical modes and their contribution to the entropy is of course negligible.
For the sound modes the damping is given by [23]
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
495
c,
with (i= (4/3) v] + where is the volume viscosity. We did not find this parameter for liquid sodium in the literature, but for an estimate it is reasonable to use the
same value for as for q. The condition o / y >> 1 can be rewritten as
c
Assuming generously that all pnl which are smaller than N'l3 contribute to no, we
obtain by integrating (8)
This relation is shown in Fig. 15. Asymptotically n~ = 2 / ( 9 n ) N or N e =~ 0.02N.
4.3 Specific heat of the electron subsystem
It is instructive to compare the contribution of the valence electrons to the thermodynamic functions with that of the phonons. The excitation energy is given in terms of
the inverse temperature /3 and the chemical potential p by P51
a
3001
/
N
Fig. 15 Effective number of degrees of freedom to be treated at most by the inviscid hydrodynamica1 model VS. the particle number of the cluster.
496
Ann. Physik 5 (1996)
in analogy to (32); Z,I(E) is the density of electronic levels in the jellium potential. In
contrast to the rather smooth functionsf(x) in the thermodynamic potentials for boson systems the cofactor of Z,I(E) in this integral emphasizes a neighbourhood of
E = p of width p-’. The thermodynamic properties of Fermion systems are therefore
much more sensitive to shell fluctuations of the level density z,,(E) around the Fermi
energy than those of boson systems. This is particularly apparent for the Fermi gas
where the Sommerfeld expansion of the integral (42) yields the well-known power
series in T 2 for E* starting with
As usual the chemical potential p has been eliminated in favor of
tion
~f
using the rela-
The electronic contribution to cv is small compared to the ionic contribution. For the
two “magic” numbers N = 40 and 92 we compare in Fig. 16 the electronic contribution to cv,calculated strictly in the canonical frame, from [26] with the ionic contribution. It is calculated in the Debye model with the density of eigenvalues (8) and
boundary condition (A). Since the Debye model underestimates the specific heat by
20-25% [22] around the melting point and in the fluid phase, the relative contribution of the electrons is even somewhat smaller than shown in Fig. 16. Moreover, for
3
2
>
0
100
U
0
a,
_c
6
5
4
3
2
10-1
6
5
0
400
800
1200
1600
temperature (K)
Fig. 16 Specific heat c,, per particle vs. temperature T for sodium clusters. Electronic contribution
for N = 40 (full line), N = 92 (dashed line) and phonon contribution in the Debye model, boundary condition (A) (dashed dotted line) the same for N = 40 and N = 92 on this scale.
J. Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
497
non-magic electron numbers the specific heat per electron can even be considerably
smaller than for magic numbers [271.
5 Concluding remarks
We have shown that a compressible liquid with a fixed surface has an energy-averaged eigenmode spectrum as predicted by the Balian-Bloch formula for a rather wide
range of Gaussian averaging widths with only minor edge effects for small w. For
the sphere the eigenvalue density exhibits shell fluctuations around the energy-averaged background with the well-known characteristic two frequencies.
In the case of a free surface capillary modes exist besides the sound modes. They
are weakly coupled to each other with a frequency-dependent non-hermitian coupling
matrix. The eigenvalue equation for the pure capillary modes is formulated for droplets with a smooth, but otherwise arbitrary surface. The energy-averaged density of
eigenmodes is shown to have a leptodermous expansion. The leading term is proportional to the surface area and to
The next term, proportional to the surface integral over the mean curvature, has a w-Il3 dependence on the frequency. Because of
the peculiar, non-analytic behavior for w -+ 0, Gaussian averaging is seen to lead to
serious distortions of the averaged density for small w . For spherical droplets we
showed that also the modification of the eigenvalue density due to the coupling to
the sound modes allows a leptodermous expansion.
The use of these results for hot metallic clusters, though it motivated this investigation, turned out to be of rather limited value because viscosity effects in clusters of
mesoscopic size lead to overdamped collective motion. This model is therefore more
appropriate for droplets of macroscopic size, or for superfluid helium clusters.
A Appendix
we want to represent the operator 6H/6n in terms of the first and second fundamental forms of the theory of curved surfaces. We follow [281 in this appendix. Introducing surface coordinates a l , a 2the surface may be given by the three-dimensional
vector field y(6).We define tangential vectors by
and the unit normal vector
we further define
498
Ann. Physik 5 (1996)
In terms of these quantities the first fundamental form, the metric tensor, is given by
the scalar product
- + - t
gij = Y i ' Y j
(43)
and the second fundamental form by
Lr.l .- - N+ . y .'I. = -N-+
1
.y.J
'
(44)
The mean curvature is written in terms of these forms as
H = -1 g"..
2 ' I
"
(45)
with
gil gr j=dj
.
(46)
We introduce an infinitesimal scalar distortion field &(&) to describe an infinitesimal
shift of the surface y'(&) in the normal direction
si; = &n(G)
With abbreviations
136n
6ni = aai
,
(47)
the variations of y'i and Y'ij can be expressed by the distortion field and its derivatives
From (45)we find
1
6 H = - (dgi' Lij
2
+ g'j6Lij)
and from (46)
6g"grj = -g"6gy
.
With (43)therefore
6g'i = -gik g"6gkI = -g'k g11(yk @[ + ?[si;k)
and in terms of the distortion field 6n
J.
Wehner, H.J. Krappe, Eigenmodes of a compressible liquid drop
g"[(ykfiI + Y&k)6n + fi . ( y k 6nl+
6grj = -gik
499
6nk)] .
The second term in the brackets vanishes since is orthogonal to y'k. For the variation of the second fundamental form one finds from (44)
6L, = k&j + j , &
To obtain 6G we note that Gyi = 0 implies 6Gy'j = -GSy'j and since 6 N can only
have components in the tangential directions
Because
G2 = 1,
ilifii=o,
- + - +
$fii++,~=~
and
From (49)then follows
SL, = ijn,
and
+ k f i o d n - gkl+yiyijdnk
+
500
Ann. Physik 5 (1996)
The functional derivative is therefore
Besides the L2 term this is the Beltrami operator AB on the surface, which generalizes the Laplacian in a covariant way in curved spaces. The second term, written in
local Riemannian coordinates, yields the sum of the squared main curvatures
We are indebted to Prof. B. Miihlschlegel for suggesting this investigation and to Prof. W. Swiatecki
for valuable discussions, in particular concerning the limits of the inviscid liquid-drop model.
References
H. Lamb, Proc. Lond. Math. Soc. 13 (1882) 189
Y. Sato, T. Usami, Geophys. Mag. 31 (1962) 15
P. Debye, Ann. d. Phys. 39 (1912) 789
J.W. Strutt (Baron Rayleigh), Phil. Mag. 14 (1882) 184; The Theory of Sound, (i 364, Dover, New
York 1945
[5] S. Chandrasekhar, Proc. Lond. Math. SOC. 9 (1959) 141
[6] W.H. Reid, Proc. Lond. Math. SOC. 9 (1959) 388; Quart. Appl. Math. 18 (1960) 86
[7] H. Lamb, Hydrodynamics, 6th edition, Dover 1945, 5 275, 5 287
[8] A. Pleijel, Ark. Mat. 2 (1952) 553; Proc. 12th Skand. Mat. Kongr. 1953, p 222
[9] R. Balian, C. Bloch, Ann. of Phys. 60 (1970) 401
[lo] S. Frauendorf, V.V. Pashkevich, Z. Phys. D 26 (1992) 98
[ I I ] Landolt-Bomstein, Zahlenwerte und Funktionen aus Natunvissenschaft und Technik, Neue Sene,
Springer 1967, 11, 5, p. 14
[I21 H. Nishioka, Z. Phys. D 19 (1991) 19
[I31 F.A. Ivanyuk, V.M. Strutinsky, Z. Phys. A 286 (1978) 291
[I41 G.G. Bunatian, V.M. Kolomietz, V.M. Strutinsky, Nucl. Phys. A 188 (1972) 225
[I51 IS.Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, 4th translated edition, Academic 1965, formulae 6.561.5 and 6.521.1
[I61 R. Balian, C. Bloch, Ann. of Phys. 64 (1971) 271
[I71 P.C. Engelking, J. Chem. Phys. 87 (1987) 936
[ 181 P. Frobrich, Phys. Lett. A 202 ( I 995) 99
[I91 P.A. Hervieux, D. Gross, Z. Phys. D 33 (1995) 295
[20] S. Frauendorf, Z. Phys. D 35 (1995) 191
[ZI] C.C. Hsu, in Handbook of Thermodynamic and Transport Properties of Alkali Metals, R.W. Ohse
ed., Blackwell, Oxford, 1985, p. 105
[22] R.W. Hasse, W.D. Myers, Geometrical Relations in Macroscopic Nuclear Physics, Springer, Berlin
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[23] L.D. Landau, E.M. Lifschitz, Hydrodynamik, Akademie Verlag 1971, chaps. 25 and 77
[24] E.E. Shipilrain, K.A. Yakimovich, V.A. Fomin, S.N. Skovoredjko, M.S. Mozgovoi. in Handbook of
Thermodynamic and Transport Properties of Alkali Metals, R.W. Ohse ed., Blackwell Scientific, Oxford, 1985, p. 753
[25] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Saunders College, Philadelphia, I976
1261 M. Brack, 0. Genzken, K. Hansen, Z. Phys. D 21 (1991) 65
[27] V. Pashkevich, private communication
[28] D. Laugwitz, Differentialgeometrie, Teubner, Stuttgart, I968
[I]
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