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Static polarizability of metal clusters and the related blue shift of the dipolar surface plasmon
Solid State Communications, Vol. 46, No. 7, pp. 571-574, 1983. Printed in Great Britain.
0038-1098/83 / 190571-04 $03.00[0 Pergamon Press Ltd.
STATIC POLARIZABILITY OF METAL CLUSTERS AND THE RELATED BLUE SHIFT OF THE DIPOLAR SURFACE PLASMON W. Ekardt, D.B. Tran Thoai, F. Frank and W. Schulze Fritz-Haber-Institut der Max-Hanck-Gesellschaft Faradayweg 4 - 6 , D-1000 Berlin 33, West Germany
(Received 26 November 1982 by L. Hedin)
The blue shift of the dipolar surface plasmon of small metal clusters is studied with the use of the sum rule techniques due to Lushnikov et al. The static polarizability needed for the application of this technique is obtained by solving Poisson's equation for the total electrostatic potential of a small metal particle pheed in an external electric field. The screening properties of the metal electrons are described within the modified Thomas-Fermi approximation. The ground state charge density determining the local Thomas-Fermi wavenumber is obtained from a variational estimate of the total energy of the metal electrons. The static polarizability obtained is smaller than the classical value R a (where R is the cluster radius) but larger than the value following from the step density ThomasFermi approximation. Correspondingly, the surface plasmon frequency is blue shifted from its classical value but red shifted from its step-density value. The shift is strongly rs-dependent, where r, is the well known electron gas parameter. 1. INTRODUCTION THE BEHAVIOUR of the collective surface excitations of small metallic particles is still a subject of controversial discussion. Whereas a number of authors predict a blue shift compared to the classical value [1-3] ~o8 = t~p/X/3 with o ~ = 4ne2n/m, other authors predict a red shift [4, 5]. Obviously, the discussion suffers from a lack of detailed knowledge of the electronic density profile near the surface of the cluster. Closely connected with the htter point is that in theories of the dynamical polarizability of small metal clusters usually the infinite barrier model is used [5] which is oversimplifying with respect to the properties of a real surface. For instance, the electrostatic contribution to the work function of a plane surface following from the infinite barrier model is far too small, and the reason for this is simply that the electrons are pushed much too close to the positive back-ground boundary [6]. In 1975 Cini [7] calculated the ionization potential and the electron affinity of small metal clusters in strict analogy to Smith's work on the theory of work functions of plane metal surfaces [8]. In this type of calculation one avoids automatically a super artificial surface barrier because the electron density profile follows from minimizing the surface energy of the particle. The density, in turn, determines a reasonable approximation to the surface barrier confining the electrons within the cluster. Work functions determined in this way are in reasonable agreement with experimental data [7], and 571
we believe that this type of charge density might also be useful for the problem under discussion. Since the static polarizability of a system is a ground state property it is a unique functional of its electronic density [9]. However, as yet nobody knows the exact form of this functional. A possible way out of this difficulty is contained in the work by Rice et ai. [10]. These authors have found that the exact RPApolarizability of small particles (derived from the oneparticle wavefunctions of an inf'mite barrier) may be obtained in a much simpler way with the help of the socalled modified Thomas-Fermi approximation (MTFA). It is only for very small cluster size that the approximation breaks down. Because the exact self-consistent charge density for larger rs-values [(4¢t/3)r~ = 1In] looks very similar to the inf'mite barrier charge density [11 ], it seems to be reasonable to expect that the MTFA works quite well also in the present case. Spurious results obtained at small r s will then probably reflect the breakdown both of the MTFA and of the special form of the variational charge density.
2. STATIC POLARIZABILITY The starting point of our method is a variational determination of the electronic charge density n(r) in the ground state for further use in the MTFA. In strict analogy to the original work by Smith [8] we adopt for this procedure the following family of trial densities
572 n(r)
Vol. 46, No. 7
STATIC POLARIZABILITY OF METAL CLUSTERS 10
=
3
1+
e
08 06
x {[1 --0.5ea(r-R)]O(R --r) + 0.5e-a(r-R)O(r--R)}. (1)
Here, R is the cluster radius, ~ = 1/(41r/3)r~ is the jellium back-ground density, O(x) the unit step function, and the parameter 3 is to be determined by minimizing the total energy E(n). The pre-factor in equation (1) assures correct normalization [ 12]. Because different local approximations are in use for the description of exchange and correlation of the electroncs, we have performed the variational calculation with both the Wigner approximation and the RPA expression [6]. At low rs-values the RPA should yield better results, and for large G-values the Wigner approximation is preferable. Having obtained a reasonable ground state density we solve next the Poisson equation with the use of the MTFA as defined by Rice etal. [10]
A V = KZTFA[n(r)]V,
......
0( r= =2
r== 6
04
O~
0.2
O;
20 40 60 8o 100 R [oB]
20 ~0 0o 8o 10o R [a6]
Fig. 1. Static polarizability a of a small metallic particle, with the radius R (in units of the Bohr radius an). a is shown in units of its classical value acl = R 3. r s is the electron gas parameter, G = (1/(47r/3)n) v3. Continuous line: step-density Thomas-Fermi approximation. Dashed and dashed-dotted lines give the result for a diffuse density, equation (1) of the text. Dashed line: Wigner-type correlation was used for the minimization of the surface energy; dashed-dotted line: same calculation with RPA-correlation energy. For a diffuse boundary, the modified Thomas-Fermi approximation of [10] was used. In the case o f r s = 2, Wigner- and RPA-type correlation result in nearly identical a-values.
(2)
K~ITFA[n(r)] = K~,r(n ) n(_~_r)= [9/(91r/4)2/31
1
W., = e 2 ~ cosl(OlVls)[ 2 = - ½e2 f d3rV(r)VV(r)
n(r)
ff
?1
(3) From the asymptotic solution of this equation for the dipole field
the dipolar polarizability
a = aMeTFA(R,rs)R 3
(5)
is obtained. Here, a~TFA(R, rs) gives the correction factor to the polarizability R 3 of a classical metal particle. If we let the diffuseness parameter/3 be infinitely large the step-density Thomas-Fermi polarizability aeTF(R, re) is recovered. Both results are shown in Fig. 1. 3. SURFACE PLASMON ENERGY As has been shown by Lushnikov et al. [1, 13, 14] the collective frequency of a given type of symmetry (in our case the dipolar excitation) can be obtained from the following sum-rule expressions co---~= W.t W-l
(6)
with I¢-1 = e 2 ~co;tl(0iVIs)12
$
F
1.00E ~
= ½a
(7)
x Vn(r).
(8)
In these expressions, V(r) = r cos 0 is the external potential, acting on the electrons, cos and Is) are the exact frequencies and eigenstates of the system of electrons, and a is the static polarizability of t h . ~ s t e m . The essential point of interest to us is that co" is determined solely by the ground state density n(r) because n(r) determines a, as discussed in Section 2. Whereas the sum rule (6) is exact [13, 14], the identification of co2 with the collective frequency (in our case the dipolar surface plasmon frequency co~) rests on the assumption that nearly all of the total oscillator strength of the system is stored in the collective pole. This assumption may or may not be fulfilled. A glance at the experimental spectra of the light metals or at theoretical results following from the infinite barrier model [4] shows that for the model under discussion this assumption should be fulfilled. On the basis of the present model the dynamical polarizability a(co) of small metal clusters can be shown to be
47#ie2/m a(co) = ~R 3 co# 2_co2 icor."
(9)
Here, I"s is the damping of the collective pole [1, 15, 16]. An expression of exactly this form is tacitely assumed to be valid in the discussion of SERS (surface enhanced Raman scattering) due to the microscopic roughness effect [17].
573
STATIC POLARIZABILITY OF METAL CLUSTERS
Vol. 46, No. 7
i 1&
3
ff
I/.
"l r 10 08
"
~
,
20
'
,
40
~
,
60
I0 ~
80
,
100
R [Oa]
0.8
l
37
.I
38
:.\I
q=2
"1
3.5
2b 10 d0 do ,b0 R (aa]
• \\
3/.
Fig. 2. Surface plasmon frequency ~ , in units of its classical value co~l = COp/~/3. Due to the diffuse boundary, ~s is red-shifted compared to its stepdensity value. However, compared to its classical value, 6o8 is blueshifted in all the three different approximations.
xt \I %
3.3 32 20
From equation (9) we obtain the surface plasmon frequency in a very nice form as c°t'/~/3 r.) co~TFA = ~/a~'rFA(R,
(10)
where otMTFA(R,rs) was defined in Section 2. Equation (10) makes it especially clear that the blue shift e f the surface plasmon frequency is a direct consequence of the lowered static polarizability of the metal cluster. The results are shown in Fig. 2 and compared with the step density result [ 1]. The red shift of LOM T F A compared with the step-density result is a direct consequence of the enlarged polarizability due to a diffuse boundary: Part of the induced dipole moments inside the sphere relax to those outside the sphere, resulting in an enlarged total dipole moment of the cluster. The diffuseness of the surface results in a lowered local plasmon frequency Wp(r) and this in turn results in a red shift compared to the step-density value. There is indeed experimental evidence that the trend calculated here is correct. The step-density result for Ag clusters embedded in an argon matrix is shown by the dotted line in Fig. 3 and compared with the experimental data by Schulze et al. [18]. As it should be, the step-density result is above the experimental result at smaller values o f R . Unfortunately, our method is not directly applicable to a quantitative interpretation of these experiments, mainly because the short-range forces at the Ag-Ar interface are unknown. In a first step, both the Ag-background and the Ar-matrix could be described phenomenologically by two dielectric constants. However, this is not sufficient for the application of our procedure. Because the density profile follows from a delicate balance between the various parts of the total energy functional (kinetic part, electrostatic part and exchange-correlation part), we need to know the effective masses in the interface region. And this is still an unsolved problem. Hence, it might be more useful to investigate clusters o f the light metals in the gas phase. In this situation, our method is applicable, and the remaining lattice effects
40
60
80
100
DIAMETER []~ ]
Fig. 3. Comparison between theory (dashed line) and experiment (dots) for surface plasmons of small silver particles embedded in an argon matrix. For further discussion see text [20]. may be taken into account via pseudo-potential perturbation theory. 4. CONCLUSION The dynamical polarizability of small particles awaits further investigations. A final c___onfirmationof the validity of the identification of ~2 ~ cos2 can only be given by first principle methods. However, the good agreement with experimental data which we obtained also in the case of the collective excitations in atomic Xe and Cs [19] shows the wide applicability of the methods discussed here. The underlying principle seems to be that homogeneous electron gas theory can be applied locally to inhomogeneous systems down to atomic dimensions. Acknowledgements - One of use (D.B.T.T.) would like to thank the Max-Planck-Gesellschaft for a postdoctoral fellowship. We are very much indebted to Profs. H. Gerischer and E. Zeitler for their continuing interest and support. REFERENCES 1. 2. 3. 4. 5.
6. 7. 8. 9.
A.A. Lushnikov & A.J. Simonov, Z. Phys. 270, 17 (1974). R. Ruppin,J. Phys. Chem. Solids 39,233 (1978). L. Genzel, T.P. Martin & U. Kreibig, Z. Phys. B21, 339 (1975). D.M. Wood & N.W. Ashcroft, Phys. Rev. B25, 6255 (1982). M. Cini,J. Opt. Soc. Am. 71,386 (1981); In this work a finite step is also discussed. But the main deficiency remains that a finite-step potential has no structure resembling a real surface. N.D. Lang, Solid State Phys. 28,225 (1973). M. Cini,J. Cat. 37,187 (1975). J.R. Smith,Phys. Rev. 181,522 (1969). A. Zangwill & P. Soven, Phys. R ev. A21,1561 (1980).
574 10.
STATIC POLARIZABILITY OF METAL CLUSTERS
Vol. 46 No. 7
M.J. Rice, W.R. Schneider & S. Stdissler, Phys. 1765 (I 966). Rev. !18,474 (1973). 16. D.B. Tran Thoai & W. Ekardt, Solid State Com11. D.M. Newns, Phys. Rev. BI, 3304 (1970). mun. 41,687 (1982) (and references therein). 12. This charge density is not a very sophisticated one. 17. E. Burstein, C.Y. Chen & S. Lundquist, Light ScatCini [7] has included two more parameters. But tering in Solids, (Edited by J.L. Birman, H.Z. equation (1) is just enough to see what happens. Cummins & K.K. Rebane), p. 479. Plenum, New The linear term included by Cini for making the York (1979). derivative dn/dr vanish at r = 0 is quite small at 18. H. Abe, W. Schulze & B. Tesche, Chem. Phys. 47, larger R-values and is therefore neglected. 95 (1980); and F. Frank & W. Schulze (unpub13. A.A. Lusnikov & D.F. Zaretsky,Nucl. Phys. 66, lished). 35 (1965). 19. W. Ekardt & D.B. Tran Thoai, Physica Scrip ta 26, 14. A.B. Migdal, A.A. Lushnikov & D.F. Zaretsky, 194 (1982). Nucl. Phys. 66,193 (1965). 20. The dielectric constants entering this calculation 15. A. Kawabata & R. Kubo,J. Phys. Soc. Japan 21, were taken from [18].
0038-1098/83 / 190571-04 $03.00[0 Pergamon Press Ltd.
STATIC POLARIZABILITY OF METAL CLUSTERS AND THE RELATED BLUE SHIFT OF THE DIPOLAR SURFACE PLASMON W. Ekardt, D.B. Tran Thoai, F. Frank and W. Schulze Fritz-Haber-Institut der Max-Hanck-Gesellschaft Faradayweg 4 - 6 , D-1000 Berlin 33, West Germany
(Received 26 November 1982 by L. Hedin)
The blue shift of the dipolar surface plasmon of small metal clusters is studied with the use of the sum rule techniques due to Lushnikov et al. The static polarizability needed for the application of this technique is obtained by solving Poisson's equation for the total electrostatic potential of a small metal particle pheed in an external electric field. The screening properties of the metal electrons are described within the modified Thomas-Fermi approximation. The ground state charge density determining the local Thomas-Fermi wavenumber is obtained from a variational estimate of the total energy of the metal electrons. The static polarizability obtained is smaller than the classical value R a (where R is the cluster radius) but larger than the value following from the step density ThomasFermi approximation. Correspondingly, the surface plasmon frequency is blue shifted from its classical value but red shifted from its step-density value. The shift is strongly rs-dependent, where r, is the well known electron gas parameter. 1. INTRODUCTION THE BEHAVIOUR of the collective surface excitations of small metallic particles is still a subject of controversial discussion. Whereas a number of authors predict a blue shift compared to the classical value [1-3] ~o8 = t~p/X/3 with o ~ = 4ne2n/m, other authors predict a red shift [4, 5]. Obviously, the discussion suffers from a lack of detailed knowledge of the electronic density profile near the surface of the cluster. Closely connected with the htter point is that in theories of the dynamical polarizability of small metal clusters usually the infinite barrier model is used [5] which is oversimplifying with respect to the properties of a real surface. For instance, the electrostatic contribution to the work function of a plane surface following from the infinite barrier model is far too small, and the reason for this is simply that the electrons are pushed much too close to the positive back-ground boundary [6]. In 1975 Cini [7] calculated the ionization potential and the electron affinity of small metal clusters in strict analogy to Smith's work on the theory of work functions of plane metal surfaces [8]. In this type of calculation one avoids automatically a super artificial surface barrier because the electron density profile follows from minimizing the surface energy of the particle. The density, in turn, determines a reasonable approximation to the surface barrier confining the electrons within the cluster. Work functions determined in this way are in reasonable agreement with experimental data [7], and 571
we believe that this type of charge density might also be useful for the problem under discussion. Since the static polarizability of a system is a ground state property it is a unique functional of its electronic density [9]. However, as yet nobody knows the exact form of this functional. A possible way out of this difficulty is contained in the work by Rice et ai. [10]. These authors have found that the exact RPApolarizability of small particles (derived from the oneparticle wavefunctions of an inf'mite barrier) may be obtained in a much simpler way with the help of the socalled modified Thomas-Fermi approximation (MTFA). It is only for very small cluster size that the approximation breaks down. Because the exact self-consistent charge density for larger rs-values [(4¢t/3)r~ = 1In] looks very similar to the inf'mite barrier charge density [11 ], it seems to be reasonable to expect that the MTFA works quite well also in the present case. Spurious results obtained at small r s will then probably reflect the breakdown both of the MTFA and of the special form of the variational charge density.
2. STATIC POLARIZABILITY The starting point of our method is a variational determination of the electronic charge density n(r) in the ground state for further use in the MTFA. In strict analogy to the original work by Smith [8] we adopt for this procedure the following family of trial densities
572 n(r)
Vol. 46, No. 7
STATIC POLARIZABILITY OF METAL CLUSTERS 10
=
3
1+
e
08 06
x {[1 --0.5ea(r-R)]O(R --r) + 0.5e-a(r-R)O(r--R)}. (1)
Here, R is the cluster radius, ~ = 1/(41r/3)r~ is the jellium back-ground density, O(x) the unit step function, and the parameter 3 is to be determined by minimizing the total energy E(n). The pre-factor in equation (1) assures correct normalization [ 12]. Because different local approximations are in use for the description of exchange and correlation of the electroncs, we have performed the variational calculation with both the Wigner approximation and the RPA expression [6]. At low rs-values the RPA should yield better results, and for large G-values the Wigner approximation is preferable. Having obtained a reasonable ground state density we solve next the Poisson equation with the use of the MTFA as defined by Rice etal. [10]
A V = KZTFA[n(r)]V,
......
0( r= =2
r== 6
04
O~
0.2
O;
20 40 60 8o 100 R [oB]
20 ~0 0o 8o 10o R [a6]
Fig. 1. Static polarizability a of a small metallic particle, with the radius R (in units of the Bohr radius an). a is shown in units of its classical value acl = R 3. r s is the electron gas parameter, G = (1/(47r/3)n) v3. Continuous line: step-density Thomas-Fermi approximation. Dashed and dashed-dotted lines give the result for a diffuse density, equation (1) of the text. Dashed line: Wigner-type correlation was used for the minimization of the surface energy; dashed-dotted line: same calculation with RPA-correlation energy. For a diffuse boundary, the modified Thomas-Fermi approximation of [10] was used. In the case o f r s = 2, Wigner- and RPA-type correlation result in nearly identical a-values.
(2)
K~ITFA[n(r)] = K~,r(n ) n(_~_r)= [9/(91r/4)2/31
1
W., = e 2 ~ cosl(OlVls)[ 2 = - ½e2 f d3rV(r)VV(r)
n(r)
ff
?1
(3) From the asymptotic solution of this equation for the dipole field
the dipolar polarizability
a = aMeTFA(R,rs)R 3
(5)
is obtained. Here, a~TFA(R, rs) gives the correction factor to the polarizability R 3 of a classical metal particle. If we let the diffuseness parameter/3 be infinitely large the step-density Thomas-Fermi polarizability aeTF(R, re) is recovered. Both results are shown in Fig. 1. 3. SURFACE PLASMON ENERGY As has been shown by Lushnikov et al. [1, 13, 14] the collective frequency of a given type of symmetry (in our case the dipolar excitation) can be obtained from the following sum-rule expressions co---~= W.t W-l
(6)
with I¢-1 = e 2 ~co;tl(0iVIs)12
$
F
1.00E ~
= ½a
(7)
x Vn(r).
(8)
In these expressions, V(r) = r cos 0 is the external potential, acting on the electrons, cos and Is) are the exact frequencies and eigenstates of the system of electrons, and a is the static polarizability of t h . ~ s t e m . The essential point of interest to us is that co" is determined solely by the ground state density n(r) because n(r) determines a, as discussed in Section 2. Whereas the sum rule (6) is exact [13, 14], the identification of co2 with the collective frequency (in our case the dipolar surface plasmon frequency co~) rests on the assumption that nearly all of the total oscillator strength of the system is stored in the collective pole. This assumption may or may not be fulfilled. A glance at the experimental spectra of the light metals or at theoretical results following from the infinite barrier model [4] shows that for the model under discussion this assumption should be fulfilled. On the basis of the present model the dynamical polarizability a(co) of small metal clusters can be shown to be
47#ie2/m a(co) = ~R 3 co# 2_co2 icor."
(9)
Here, I"s is the damping of the collective pole [1, 15, 16]. An expression of exactly this form is tacitely assumed to be valid in the discussion of SERS (surface enhanced Raman scattering) due to the microscopic roughness effect [17].
573
STATIC POLARIZABILITY OF METAL CLUSTERS
Vol. 46, No. 7
i 1&
3
ff
I/.
"l r 10 08
"
~
,
20
'
,
40
~
,
60
I0 ~
80
,
100
R [Oa]
0.8
l
37
.I
38
:.\I
q=2
"1
3.5
2b 10 d0 do ,b0 R (aa]
• \\
3/.
Fig. 2. Surface plasmon frequency ~ , in units of its classical value co~l = COp/~/3. Due to the diffuse boundary, ~s is red-shifted compared to its stepdensity value. However, compared to its classical value, 6o8 is blueshifted in all the three different approximations.
xt \I %
3.3 32 20
From equation (9) we obtain the surface plasmon frequency in a very nice form as c°t'/~/3 r.) co~TFA = ~/a~'rFA(R,
(10)
where otMTFA(R,rs) was defined in Section 2. Equation (10) makes it especially clear that the blue shift e f the surface plasmon frequency is a direct consequence of the lowered static polarizability of the metal cluster. The results are shown in Fig. 2 and compared with the step density result [ 1]. The red shift of LOM T F A compared with the step-density result is a direct consequence of the enlarged polarizability due to a diffuse boundary: Part of the induced dipole moments inside the sphere relax to those outside the sphere, resulting in an enlarged total dipole moment of the cluster. The diffuseness of the surface results in a lowered local plasmon frequency Wp(r) and this in turn results in a red shift compared to the step-density value. There is indeed experimental evidence that the trend calculated here is correct. The step-density result for Ag clusters embedded in an argon matrix is shown by the dotted line in Fig. 3 and compared with the experimental data by Schulze et al. [18]. As it should be, the step-density result is above the experimental result at smaller values o f R . Unfortunately, our method is not directly applicable to a quantitative interpretation of these experiments, mainly because the short-range forces at the Ag-Ar interface are unknown. In a first step, both the Ag-background and the Ar-matrix could be described phenomenologically by two dielectric constants. However, this is not sufficient for the application of our procedure. Because the density profile follows from a delicate balance between the various parts of the total energy functional (kinetic part, electrostatic part and exchange-correlation part), we need to know the effective masses in the interface region. And this is still an unsolved problem. Hence, it might be more useful to investigate clusters o f the light metals in the gas phase. In this situation, our method is applicable, and the remaining lattice effects
40
60
80
100
DIAMETER []~ ]
Fig. 3. Comparison between theory (dashed line) and experiment (dots) for surface plasmons of small silver particles embedded in an argon matrix. For further discussion see text [20]. may be taken into account via pseudo-potential perturbation theory. 4. CONCLUSION The dynamical polarizability of small particles awaits further investigations. A final c___onfirmationof the validity of the identification of ~2 ~ cos2 can only be given by first principle methods. However, the good agreement with experimental data which we obtained also in the case of the collective excitations in atomic Xe and Cs [19] shows the wide applicability of the methods discussed here. The underlying principle seems to be that homogeneous electron gas theory can be applied locally to inhomogeneous systems down to atomic dimensions. Acknowledgements - One of use (D.B.T.T.) would like to thank the Max-Planck-Gesellschaft for a postdoctoral fellowship. We are very much indebted to Profs. H. Gerischer and E. Zeitler for their continuing interest and support. REFERENCES 1. 2. 3. 4. 5.
6. 7. 8. 9.
A.A. Lushnikov & A.J. Simonov, Z. Phys. 270, 17 (1974). R. Ruppin,J. Phys. Chem. Solids 39,233 (1978). L. Genzel, T.P. Martin & U. Kreibig, Z. Phys. B21, 339 (1975). D.M. Wood & N.W. Ashcroft, Phys. Rev. B25, 6255 (1982). M. Cini,J. Opt. Soc. Am. 71,386 (1981); In this work a finite step is also discussed. But the main deficiency remains that a finite-step potential has no structure resembling a real surface. N.D. Lang, Solid State Phys. 28,225 (1973). M. Cini,J. Cat. 37,187 (1975). J.R. Smith,Phys. Rev. 181,522 (1969). A. Zangwill & P. Soven, Phys. R ev. A21,1561 (1980).
574 10.
STATIC POLARIZABILITY OF METAL CLUSTERS
Vol. 46 No. 7
M.J. Rice, W.R. Schneider & S. Stdissler, Phys. 1765 (I 966). Rev. !18,474 (1973). 16. D.B. Tran Thoai & W. Ekardt, Solid State Com11. D.M. Newns, Phys. Rev. BI, 3304 (1970). mun. 41,687 (1982) (and references therein). 12. This charge density is not a very sophisticated one. 17. E. Burstein, C.Y. Chen & S. Lundquist, Light ScatCini [7] has included two more parameters. But tering in Solids, (Edited by J.L. Birman, H.Z. equation (1) is just enough to see what happens. Cummins & K.K. Rebane), p. 479. Plenum, New The linear term included by Cini for making the York (1979). derivative dn/dr vanish at r = 0 is quite small at 18. H. Abe, W. Schulze & B. Tesche, Chem. Phys. 47, larger R-values and is therefore neglected. 95 (1980); and F. Frank & W. Schulze (unpub13. A.A. Lusnikov & D.F. Zaretsky,Nucl. Phys. 66, lished). 35 (1965). 19. W. Ekardt & D.B. Tran Thoai, Physica Scrip ta 26, 14. A.B. Migdal, A.A. Lushnikov & D.F. Zaretsky, 194 (1982). Nucl. Phys. 66,193 (1965). 20. The dielectric constants entering this calculation 15. A. Kawabata & R. Kubo,J. Phys. Soc. Japan 21, were taken from [18].