- Email: [email protected]
Dynamical structure factor of an electron liquid
0038-1098181/241285--04502.00/0
Solid k a t e Communications, Vol. 38, pp. 1285-1288. Pergamon Press Ltd. 1981. Printed in Great Britain.
DYNAMICAL STRUCTURE FACTOR OF AN ELECTRON LIQUID K. Awa, H. Yasuhara* and T. Asahi Department of Physics, Hokkaido University, Sapporo 060, Japan
(Received 22 January 1981 by Y. Toyozawa) Inelastic X-ray scattering experiments of the dynamical structure factor carried out at Bell Laboratories for various metals are interpreted. The existence of a plasmon-like peak and a broad peak in the intermediate momentum region is ascribed to the striking damping effect of oneelectron states originating from virtual plasmon emission under the influence of strong short-range correlations at metallic densities. An anomalous dispersion of the plasmon around the cut-off wave-number observed in electron scattering experiments is also interpreted. INELASTIC X-RAY SCATTERING EXPERIMENTS [ 1] carried out at Bell Laboratories have revealed very interesting features in the dynamical structure factor S(q, co) of several metals such as Be, Li, graphite,and A1. A double-peak or one-peak-and-one-shoulder structure in S(q, ~) has been observed in the intermediate momentum region PF (or qe) ~ q ~< 2pF where PF is the Fermi momentum and qe is the cut-off momentum. It is commonly believed that the above experimental spectra reflect correlation effects in a free electron liquid, not band structure effects. A theoretical interpretation of the above experiments has been the subject of controversy [2, 4 - 7 ] . Mukhopadhyay et al. have obtained a spectral shape which has some resemblance to the observed one resorting to a rather unnatural modification of Vashishta and Singwi's dielectric function [3]. But their calculation is complicated; its physical implication is obscure. Several authors [4-6] have attempted to apply to this problem Mori's memory function formalism, without much success. A satisfactory solution has not yet been obtained in spite of such great efforts. But the problem is important, considering the fact that the experiments are expected to reflect some unclarified dynamical effects of electron correlations common to a wide variety of different metals. In the present communication we give a clear theoretical interpretation of the spectral structure in the experimental S(q, co) on the basis of the diagrammatic method of many-body perturbation theory. An extensive knowledge as to static aspects of electron correlations at metallic densities has been accumulated since the first successful attempt by Singwi and coworkers [8]. A better understanding [9-12] of a shortdistance behaviour of the pair distribution function g (r)
has been achieved. According to standard diagrammatic method, one of the authors (H.Y.) [9] has given a reasonable expression for g(r) which satisfies g ( r ) ~ 0 at small distances and for any density. It has been clarified there that the particle-particle ladder vertex plays an essential role in the adequate'description of short-range correlations important at metallic densities. The dynamical structure factor, S(q, co) is directly related to the imaginary part of the inverse dielectric function, Im [1/e(q, co)].
S(q, co) = -- h/v(q)rr " Im {1]e(q, co)},
(1)
e(q, co) = 1 + v(q)rr(q, co),
(2)
where v(q) is the Coulomb interaction and n(q, co) is the proper polarization function. Generally speaking from the diagrammatic point of view, the spectrum of S(q, co) or Im [I/e(q, co)] consists of not only one-pair excitations, but also higher order excitations such as two-pair excitation and one-pair-and-one-plasmon excitation, etc. A faithful perturbation calculation of e(q, co) or the proper polarization function including such higher order excitations is exceedingly difficult. Instead, we shall make our way resorting to some physical considerations. The formal expression for the proper polarization function ;r(q, co) is given as, rr(q, ~) = - 2 j"
de (2rri)
x f zra-~G(P' e)G(.p + q, e + co) A(p, e; q, co) t )
(3) =
-- 2 (
J
de
dp
(-~--) ~ ~
{a(p,e)a(p+q,e+ co)
+ G(p, e)G(p + q, e + co)
Permanent address: College of Arts and Sciences, Tohoku University, Sendai, Japan. 1285
1286
DYNAMICAL STRUCTURE FACTOR OF AN ELECTRON LIQUID ×
de'
dp' ~.,
x a ( p ' , e ' ) a ( p ' + q,
. , ,
co)
r + co) + ...) .
rr(q, co) (4)
Here, G(p, e) denotes the one-particle Green function. The functions A(p, e'; q, ~o) and I(p, e; p', e'; q, co) are the proper vertex function and the irreducible particlehole interaction, respectively. Let us first split the Green function into two parts: one is a coherent part which originates from the contribution of a quasi-particle's (or a quasi-hole's) pole (e = Ep) on the analytically continued plane. The other is an incoherent part from configurations of several elementary excitations.
a (p, e) =
zp
e- e(p)- zCp,~)
+ ai.c (p, e),
(5)
where zp is the renormalization constant of a quasiparticle or a quasi-hole. For the moment let us put A = 1 in equation (3). A sharp frequency dependence of rr(q, w) does arise from a convolution integral of two coherent parts of Green functions. On the other hand, the remaining integrals including the incoherent part Ginc (P, e) are not expected to give a significant frequency dependence; they appear as a broad background. At first sight one may suppose that the magnitude of the coherent contribution to 7r(q, co) is considerably reduced owing to the appearance of renormalization constants. In a related problem of manybody effects on the optical absorption strength in the interband region for alkali metals, it has, however, been indicated that the above renormalization effect is overcome by inclusion of the proper vertex function A (p, e; q, ¢o); the vertex correction is partially cancelled by the effect ofzp. The net effect on the strength of onepair absorption amounts roughly to the vertex correction calculated in the statically screened interaction [ 13, 14]. Assuming that the above conclusion is also valid for the present problem, we shall obtain an approximate form for 7r(q, ~o). Considering, in advance, the cancellation between vertex and self-energy corrections, we put zp = 1 and neglect the dynamically interacting part of the irreducible particle-hole interaction. That is, the manybody corrections to 7r(q, co) are effectively represented by an energy width of the quasi-particle or the quasihole coming from the imaginary part of the self-energy correction and those parts of the vertex correction which can substantially be regarded as static interactions. The energy shift of the quasi-particle or the quasi-hole coming from the real part of the self-energy correction is omitted, since it is almost independent of the magnitude of the quasi-particle's momentum or the quasi-hole's [151. The resulting expression for rr(q, co) is given as,
=
1
Vol. 38,,No. 12
(O)(q, co) + I(q)~'CO)(q, co),
(6)
where ~tO)(q, co) is the free polarization function with the modification of the energy width. The function I" (q) in equation (6) denotes an approximate expression for the irreducible particle-hole interaction. A coherent contribution to the spectral shape of S(q, co) originates from such a part of higher order excitations as makes up the energy width of the quasi-particle or the quasi-hole; the remaining part of higher order excitations plays only a background-like role in the spectral shape. The above approximation may be termed the quasi-one-pair excitation approximation. The function I'(q) represents the local field correction to the Hartree field. The statically screened interaction is still strong at short distances. The short-range correlation due to such a strong part of the interaction implies that at metallic densities those interactions whose momentum transfers are larger than the order of hPv are substantially weakened, compared with the case of the RPA. Taking into account the short-range correlation important at metallic densities, we shall approximate I(q) as,
"]"(q) = -- a(q)v(q) = ~ (I(.p, p'; q) -- v(q))pp, + ½ (l(p, p'; q) -- v (q) -- I(p, p'; p -- p' + q))pp, (7)
where G(q) denotes the local field factor and I(p, p'; q) is the particle-particle ladder vertex which is the solution of the integral equation: dk l(p,p';q) = v(q)+ f
v(q -- k)
(__1--f ( p + k)) (1 -- f ( p ' -- k)) I" p'; k). x e(p)-- e(p + k ) + e ( p ' ) - - e ( p ' - - k ) (p' (S) Here, ( • • • )pp, denotes an averaged value over p and p' within Fermi spheres. The functions e(p) and f ( p ) a r e the free electron energy and the Fermi distribution function at zero temperature, respectively. The first term on the right-hand side of equation (7) is the local field correction due to spin-antiparallel correlation which is adequately represented by the ladder vertex; note that the first order term in the Coulomb interaction is excluded there since we treat the proper polarization function. An approximate expression of the averaged value (/(p, p'; q))pp, obtained by one of us (H.Y.) [9] is used for q/> PF where the local field correction due to the strong Coulomb repulsion plays an essential role. For q < PF an extrapolation of the above approximate form is made so as to reproduce the value at q = PF and its derivative;the limiting value of G (q)v (q) at q = 0 is chosen to reproduce the compressibility in the HartreeFock approximation. The second term is the one due to
Vol. a8, No. 1.9_
DYNAMICAL STRUCTURE FACTOR OF AN ELECTRON LIQUID
spin-parallel correlation; the direct process is accompanied with the exchange counterpart. We here estimate it in first order approximation, bemuse the higher order effects of the Coulomb interaction upon spin-parallel correlation probably are of little significance. An averaged value of v (-- p + p' -- q) over p and p' is chosen to be 47re2](q2 + p2F), as Hubbard first did [ 16]. We shall approximate the energy width F (p) using the imaginary part of the self-energy expression estimated at the unperturbed pole, i.e. ~'2 [P, efp)]. The energy width F(p) is proportional to (p - - p F ) 2 in the immediate neighbourhood o f p r . As p goes away from PF it does not continue to increase in such a manner. The energy width F(p) increases suddenly at the immediate vicinity o f p = PF + qc where a damping channel due to plasmon emission opens [15]. For the quantitative estimation of F ( p ) at metallic densities it is necessary to include the local field correction arising from the Coulomb repulsion as well as the exchange effect, which is not allowed for in the RPA's expression for F(p). The resulting expression for F(p) is written as,
rfp) = f
-- 0(eF --
dq v(q) [1 -
e(p
--
q))},
G(q)]
[1 -
C(q)]
0
I
2
3
4
1287
5
6
7
Fig. 1. Intensity of theoretical S(q, co) by the present calculation is shown as a function of ~[er. for various values ofq[pF = 1.2, 1.6, 1.8, 2.0, and 2.4.
(9)
where eF is the Fermi energy and 0 (x) = 1 for x I> 0, zero otherwise. The dielectric function in equation (9) is the Hubbard form with the local field factor G(q) of equation (7). In equation (9) a different type of local field factor C(q) other than G(q) appears. It is the one due to spin-antiparallel correlation alone, i.e. - - C(q)v (q) = q(p, p'; q))pp, -- v (q). For the description of the spectral structure in S(q, co) in the intermediate momentum region one needs to evaluate two functions ]'(q) and F(p) in a consistent way. Numerical calculation of S(q, co) has been carefully performed for the electron density r, = 2.0 appropriate for A1 (r, - 2.0) and roughly for Be (rs = 1.88), over the momentum range 0
0
I
2
3
4
5
6
7
wl~ F
Fig. 2. Intensity of experimental S(q, co) for Be is shown as a function of co/eF for various values of q/Pe = 1.13, 1.46, 1.76, and 2.10. by the RPA is 0.73PF. Thus, the magnitude ofqe is reduced by the local field correction. This reduction effect is more pronounced as r, increases. (2) About at q[PF = 0.8, there occurs a crossing over of the plasmon-like peak and the one due to indi. vidual excitations. For 0.9 < q[p~, < 1.5, a considerably
1288
DYNAMICAL STRUCTURE FACTOR OF AN ELECTRON LIQUID
sharp plasmon-hke peak accompanied with a welldeveloped shoulder can be seen. (3) For 1.5 < q/PF < 2.0, a broad peak appears on the hi~,h energy side of the sharp plasmon-like peak. As q increases, the strength of the broad peak gradually develops relative to that of the sharper one. (4) About at q/PF = 2.0, a switching over of the two strengths occurs. For q/PF > 2.0, the strength of the sharper peak becomes weaker and fades away as q increases. But even at q/PF = 2.5, a weak shoulder as the continuation of the plasmon-like peak can be discerned. (5) As for dispersion of the plasmon and plasmonlike peak, its position, starting from the plasma frequency hoop1/ee = 1.33 at q = 0, reaches its maximum value h¢O[eF = 1.9 at about q[PF = 0.8. For 0.9 < q/PF < 1.2, negative dispersion can be seen. For 1.2 < q/PF, it shows almost no dispersion, located at hc~/eF = 1.7. (6) There appears a high energy tail outside the usual one-pair region. Our calculated spectra orS(q, co), as a whole, are shifted, by a considerable amount, to the low energy side, compared with the RPA's one, in the intermediate momentum region. This is, for the most part, due to the local field correction. Inclusion of the energy width shifts the spectra furthermore to the low energy side. The distinct two peaks and the welldeveloped shoulder shown in Fig. 1 could not possibly be washed out, even if convoluted with an experimental resolution function of width 5 eV. Electron scattering experiments [ 17, 18] of S(q, w) for A1 have also been extended well beyond qc. It has been observed there that just before qe the plasmon dispersion curve bends appreciably from the RPS's one, and its continuation in the continuum shows very small dispersion up to about 1.4PF, though the one-peak-andone-shoulder structure is not found. Our calculation also reproduces excellently the above observations around qe. (See Fig. 3.) In conclusion, the existence of the two peaks in the intermediate momentum region as well as an anomalous dispersion of the plasmon around qc is ascribed to the striking damping effect of one-electron states originating from virtual plasmon emission under the influence of strong short-range correlations at metallic densities.
Acknowledgements - We would like to thank Professors H. Fukuyama and H. Takayama for their interest in this work and useful discussions. Thanks are also due to Dr K. Wada for his encouraging discussions. One of us (H.Y.) is thankful to Professors S. Ichimaru and M. Watabe for many stimulating discussions. Numerical calculations were made by Hitac M-180 at Hokkaido University Computing Center.
Vol. 38, ~0. 12
. . . . R,, .... P.,~,..,o/~/ .///
6C
r, .2.0
/'
4G
,~ 3
a
j"
I I
[
50 20
P" O
I
",de" 2
f
q /PF
Fig. 3. Calculated positions of the plasmon peak and its continuation in the continuum together with those of individual excitation peak are shown as a function of q/PF. The open circles denote the experimental results for A1 by Batson et al. and the crosses those by Zacharias. REFERENCES i. 2. 3. 4. 5. 6. 7. 8. 9. 10. I1. 12. 13. 14. 15. 16. 17. 18.
P.M. Platzman & P. Eisenberger, Phys. Rev. Lett. 33,152 (1974). G. Mukhopadhyay, R.K. Kalia & K.S. Singwi, Phys. Rev. Lett. 34, 950 (1975). P. Vashishta & K.S. Singwi, Phys. Rev. B6, 875 (1972). G. Muldlopadhyay & A. Sj61ander, Phys. Rev. B17, 3589 (1978). H. De Raedt & B. De Raedt, Phys. Rev. BI8, 2039 (1978). F. Yoshida, S. Takeno & H. Yasuhara, Prog. Theor. Phys. 64, 40 (1980). G. Barnea, J. Phys. C12, L268 (1979). K.S. Singwi, M.P. Tosi, R.H. Land & A. SjOlander, Phys. Rev. 176,589 (1968). H. Yasuhara, J. Phys. Soc. Japan 36, 361 (1974). K. Awa & T. Asahi, J'. Phys. Soc. Japan 48, 757 (1980). B.B. Hede&J.P. Carbotte, Can. J. Phys. 50, 1756 (1972). D.N. Lowy & G.E. Brown, Phys. Rev. B12,2138 (1975). L.W. Beeferman & H. Ehrenreich, Phys. Rev. B2, 364 (1970). M. Watabe & H. Yasuhara, Phys. Lett. 34A, 295 (1971). L. Hedin & S. Lundqvist, Solid State Physics (Edited by F. Seitz, D. Turnbull & H. Ehrenreich), p. 1.23, Academic Press, New York (1969). J. Hubbard, Proc. Roy. Soc. London, Ser.A 243, 336 (1957). P. Zacharias, J. Phys. FS, 645 (1975). P.E. Batson, C.H. Chen & J. Silcox, Phys. Rev. Lett. 37,937 (1976).
Note added in proof. Very recently we got to know the paper by G.D. Priftis, J. Boviatsis and A. Vradis (Phyx Lett. 68A, 482 (1978)), whose X-ray scattering experiments have revealed the existence of the double-peak structure in S(q, co) of Li (r s = 3.22) for q > 2PF. Their experimental results are in good agreement with our theoretical ones for r s = 3.0.
Solid k a t e Communications, Vol. 38, pp. 1285-1288. Pergamon Press Ltd. 1981. Printed in Great Britain.
DYNAMICAL STRUCTURE FACTOR OF AN ELECTRON LIQUID K. Awa, H. Yasuhara* and T. Asahi Department of Physics, Hokkaido University, Sapporo 060, Japan
(Received 22 January 1981 by Y. Toyozawa) Inelastic X-ray scattering experiments of the dynamical structure factor carried out at Bell Laboratories for various metals are interpreted. The existence of a plasmon-like peak and a broad peak in the intermediate momentum region is ascribed to the striking damping effect of oneelectron states originating from virtual plasmon emission under the influence of strong short-range correlations at metallic densities. An anomalous dispersion of the plasmon around the cut-off wave-number observed in electron scattering experiments is also interpreted. INELASTIC X-RAY SCATTERING EXPERIMENTS [ 1] carried out at Bell Laboratories have revealed very interesting features in the dynamical structure factor S(q, co) of several metals such as Be, Li, graphite,and A1. A double-peak or one-peak-and-one-shoulder structure in S(q, ~) has been observed in the intermediate momentum region PF (or qe) ~ q ~< 2pF where PF is the Fermi momentum and qe is the cut-off momentum. It is commonly believed that the above experimental spectra reflect correlation effects in a free electron liquid, not band structure effects. A theoretical interpretation of the above experiments has been the subject of controversy [2, 4 - 7 ] . Mukhopadhyay et al. have obtained a spectral shape which has some resemblance to the observed one resorting to a rather unnatural modification of Vashishta and Singwi's dielectric function [3]. But their calculation is complicated; its physical implication is obscure. Several authors [4-6] have attempted to apply to this problem Mori's memory function formalism, without much success. A satisfactory solution has not yet been obtained in spite of such great efforts. But the problem is important, considering the fact that the experiments are expected to reflect some unclarified dynamical effects of electron correlations common to a wide variety of different metals. In the present communication we give a clear theoretical interpretation of the spectral structure in the experimental S(q, co) on the basis of the diagrammatic method of many-body perturbation theory. An extensive knowledge as to static aspects of electron correlations at metallic densities has been accumulated since the first successful attempt by Singwi and coworkers [8]. A better understanding [9-12] of a shortdistance behaviour of the pair distribution function g (r)
has been achieved. According to standard diagrammatic method, one of the authors (H.Y.) [9] has given a reasonable expression for g(r) which satisfies g ( r ) ~ 0 at small distances and for any density. It has been clarified there that the particle-particle ladder vertex plays an essential role in the adequate'description of short-range correlations important at metallic densities. The dynamical structure factor, S(q, co) is directly related to the imaginary part of the inverse dielectric function, Im [1/e(q, co)].
S(q, co) = -- h/v(q)rr " Im {1]e(q, co)},
(1)
e(q, co) = 1 + v(q)rr(q, co),
(2)
where v(q) is the Coulomb interaction and n(q, co) is the proper polarization function. Generally speaking from the diagrammatic point of view, the spectrum of S(q, co) or Im [I/e(q, co)] consists of not only one-pair excitations, but also higher order excitations such as two-pair excitation and one-pair-and-one-plasmon excitation, etc. A faithful perturbation calculation of e(q, co) or the proper polarization function including such higher order excitations is exceedingly difficult. Instead, we shall make our way resorting to some physical considerations. The formal expression for the proper polarization function ;r(q, co) is given as, rr(q, ~) = - 2 j"
de (2rri)
x f zra-~G(P' e)G(.p + q, e + co) A(p, e; q, co) t )
(3) =
-- 2 (
J
de
dp
(-~--) ~ ~
{a(p,e)a(p+q,e+ co)
+ G(p, e)G(p + q, e + co)
Permanent address: College of Arts and Sciences, Tohoku University, Sendai, Japan. 1285
1286
DYNAMICAL STRUCTURE FACTOR OF AN ELECTRON LIQUID ×
de'
dp' ~.,
x a ( p ' , e ' ) a ( p ' + q,
. , ,
co)
r + co) + ...) .
rr(q, co) (4)
Here, G(p, e) denotes the one-particle Green function. The functions A(p, e'; q, ~o) and I(p, e; p', e'; q, co) are the proper vertex function and the irreducible particlehole interaction, respectively. Let us first split the Green function into two parts: one is a coherent part which originates from the contribution of a quasi-particle's (or a quasi-hole's) pole (e = Ep) on the analytically continued plane. The other is an incoherent part from configurations of several elementary excitations.
a (p, e) =
zp
e- e(p)- zCp,~)
+ ai.c (p, e),
(5)
where zp is the renormalization constant of a quasiparticle or a quasi-hole. For the moment let us put A = 1 in equation (3). A sharp frequency dependence of rr(q, w) does arise from a convolution integral of two coherent parts of Green functions. On the other hand, the remaining integrals including the incoherent part Ginc (P, e) are not expected to give a significant frequency dependence; they appear as a broad background. At first sight one may suppose that the magnitude of the coherent contribution to 7r(q, co) is considerably reduced owing to the appearance of renormalization constants. In a related problem of manybody effects on the optical absorption strength in the interband region for alkali metals, it has, however, been indicated that the above renormalization effect is overcome by inclusion of the proper vertex function A (p, e; q, ¢o); the vertex correction is partially cancelled by the effect ofzp. The net effect on the strength of onepair absorption amounts roughly to the vertex correction calculated in the statically screened interaction [ 13, 14]. Assuming that the above conclusion is also valid for the present problem, we shall obtain an approximate form for 7r(q, ~o). Considering, in advance, the cancellation between vertex and self-energy corrections, we put zp = 1 and neglect the dynamically interacting part of the irreducible particle-hole interaction. That is, the manybody corrections to 7r(q, co) are effectively represented by an energy width of the quasi-particle or the quasihole coming from the imaginary part of the self-energy correction and those parts of the vertex correction which can substantially be regarded as static interactions. The energy shift of the quasi-particle or the quasi-hole coming from the real part of the self-energy correction is omitted, since it is almost independent of the magnitude of the quasi-particle's momentum or the quasi-hole's [151. The resulting expression for rr(q, co) is given as,
=
1
Vol. 38,,No. 12
(O)(q, co) + I(q)~'CO)(q, co),
(6)
where ~tO)(q, co) is the free polarization function with the modification of the energy width. The function I" (q) in equation (6) denotes an approximate expression for the irreducible particle-hole interaction. A coherent contribution to the spectral shape of S(q, co) originates from such a part of higher order excitations as makes up the energy width of the quasi-particle or the quasi-hole; the remaining part of higher order excitations plays only a background-like role in the spectral shape. The above approximation may be termed the quasi-one-pair excitation approximation. The function I'(q) represents the local field correction to the Hartree field. The statically screened interaction is still strong at short distances. The short-range correlation due to such a strong part of the interaction implies that at metallic densities those interactions whose momentum transfers are larger than the order of hPv are substantially weakened, compared with the case of the RPA. Taking into account the short-range correlation important at metallic densities, we shall approximate I(q) as,
"]"(q) = -- a(q)v(q) = ~ (I(.p, p'; q) -- v(q))pp, + ½ (l(p, p'; q) -- v (q) -- I(p, p'; p -- p' + q))pp, (7)
where G(q) denotes the local field factor and I(p, p'; q) is the particle-particle ladder vertex which is the solution of the integral equation: dk l(p,p';q) = v(q)+ f
v(q -- k)
(__1--f ( p + k)) (1 -- f ( p ' -- k)) I" p'; k). x e(p)-- e(p + k ) + e ( p ' ) - - e ( p ' - - k ) (p' (S) Here, ( • • • )pp, denotes an averaged value over p and p' within Fermi spheres. The functions e(p) and f ( p ) a r e the free electron energy and the Fermi distribution function at zero temperature, respectively. The first term on the right-hand side of equation (7) is the local field correction due to spin-antiparallel correlation which is adequately represented by the ladder vertex; note that the first order term in the Coulomb interaction is excluded there since we treat the proper polarization function. An approximate expression of the averaged value (/(p, p'; q))pp, obtained by one of us (H.Y.) [9] is used for q/> PF where the local field correction due to the strong Coulomb repulsion plays an essential role. For q < PF an extrapolation of the above approximate form is made so as to reproduce the value at q = PF and its derivative;the limiting value of G (q)v (q) at q = 0 is chosen to reproduce the compressibility in the HartreeFock approximation. The second term is the one due to
Vol. a8, No. 1.9_
DYNAMICAL STRUCTURE FACTOR OF AN ELECTRON LIQUID
spin-parallel correlation; the direct process is accompanied with the exchange counterpart. We here estimate it in first order approximation, bemuse the higher order effects of the Coulomb interaction upon spin-parallel correlation probably are of little significance. An averaged value of v (-- p + p' -- q) over p and p' is chosen to be 47re2](q2 + p2F), as Hubbard first did [ 16]. We shall approximate the energy width F (p) using the imaginary part of the self-energy expression estimated at the unperturbed pole, i.e. ~'2 [P, efp)]. The energy width F(p) is proportional to (p - - p F ) 2 in the immediate neighbourhood o f p r . As p goes away from PF it does not continue to increase in such a manner. The energy width F(p) increases suddenly at the immediate vicinity o f p = PF + qc where a damping channel due to plasmon emission opens [15]. For the quantitative estimation of F ( p ) at metallic densities it is necessary to include the local field correction arising from the Coulomb repulsion as well as the exchange effect, which is not allowed for in the RPA's expression for F(p). The resulting expression for F(p) is written as,
rfp) = f
-- 0(eF --
dq v(q) [1 -
e(p
--
q))},
G(q)]
[1 -
C(q)]
0
I
2
3
4
1287
5
6
7
Fig. 1. Intensity of theoretical S(q, co) by the present calculation is shown as a function of ~[er. for various values ofq[pF = 1.2, 1.6, 1.8, 2.0, and 2.4.
(9)
where eF is the Fermi energy and 0 (x) = 1 for x I> 0, zero otherwise. The dielectric function in equation (9) is the Hubbard form with the local field factor G(q) of equation (7). In equation (9) a different type of local field factor C(q) other than G(q) appears. It is the one due to spin-antiparallel correlation alone, i.e. - - C(q)v (q) = q(p, p'; q))pp, -- v (q). For the description of the spectral structure in S(q, co) in the intermediate momentum region one needs to evaluate two functions ]'(q) and F(p) in a consistent way. Numerical calculation of S(q, co) has been carefully performed for the electron density r, = 2.0 appropriate for A1 (r, - 2.0) and roughly for Be (rs = 1.88), over the momentum range 0
0
I
2
3
4
5
6
7
wl~ F
Fig. 2. Intensity of experimental S(q, co) for Be is shown as a function of co/eF for various values of q/Pe = 1.13, 1.46, 1.76, and 2.10. by the RPA is 0.73PF. Thus, the magnitude ofqe is reduced by the local field correction. This reduction effect is more pronounced as r, increases. (2) About at q[PF = 0.8, there occurs a crossing over of the plasmon-like peak and the one due to indi. vidual excitations. For 0.9 < q[p~, < 1.5, a considerably
1288
DYNAMICAL STRUCTURE FACTOR OF AN ELECTRON LIQUID
sharp plasmon-hke peak accompanied with a welldeveloped shoulder can be seen. (3) For 1.5 < q/PF < 2.0, a broad peak appears on the hi~,h energy side of the sharp plasmon-like peak. As q increases, the strength of the broad peak gradually develops relative to that of the sharper one. (4) About at q/PF = 2.0, a switching over of the two strengths occurs. For q/PF > 2.0, the strength of the sharper peak becomes weaker and fades away as q increases. But even at q/PF = 2.5, a weak shoulder as the continuation of the plasmon-like peak can be discerned. (5) As for dispersion of the plasmon and plasmonlike peak, its position, starting from the plasma frequency hoop1/ee = 1.33 at q = 0, reaches its maximum value h¢O[eF = 1.9 at about q[PF = 0.8. For 0.9 < q/PF < 1.2, negative dispersion can be seen. For 1.2 < q/PF, it shows almost no dispersion, located at hc~/eF = 1.7. (6) There appears a high energy tail outside the usual one-pair region. Our calculated spectra orS(q, co), as a whole, are shifted, by a considerable amount, to the low energy side, compared with the RPA's one, in the intermediate momentum region. This is, for the most part, due to the local field correction. Inclusion of the energy width shifts the spectra furthermore to the low energy side. The distinct two peaks and the welldeveloped shoulder shown in Fig. 1 could not possibly be washed out, even if convoluted with an experimental resolution function of width 5 eV. Electron scattering experiments [ 17, 18] of S(q, w) for A1 have also been extended well beyond qc. It has been observed there that just before qe the plasmon dispersion curve bends appreciably from the RPS's one, and its continuation in the continuum shows very small dispersion up to about 1.4PF, though the one-peak-andone-shoulder structure is not found. Our calculation also reproduces excellently the above observations around qe. (See Fig. 3.) In conclusion, the existence of the two peaks in the intermediate momentum region as well as an anomalous dispersion of the plasmon around qc is ascribed to the striking damping effect of one-electron states originating from virtual plasmon emission under the influence of strong short-range correlations at metallic densities.
Acknowledgements - We would like to thank Professors H. Fukuyama and H. Takayama for their interest in this work and useful discussions. Thanks are also due to Dr K. Wada for his encouraging discussions. One of us (H.Y.) is thankful to Professors S. Ichimaru and M. Watabe for many stimulating discussions. Numerical calculations were made by Hitac M-180 at Hokkaido University Computing Center.
Vol. 38, ~0. 12
. . . . R,, .... P.,~,..,o/~/ .///
6C
r, .2.0
/'
4G
,~ 3
a
j"
I I
[
50 20
P" O
I
",de" 2
f
q /PF
Fig. 3. Calculated positions of the plasmon peak and its continuation in the continuum together with those of individual excitation peak are shown as a function of q/PF. The open circles denote the experimental results for A1 by Batson et al. and the crosses those by Zacharias. REFERENCES i. 2. 3. 4. 5. 6. 7. 8. 9. 10. I1. 12. 13. 14. 15. 16. 17. 18.
P.M. Platzman & P. Eisenberger, Phys. Rev. Lett. 33,152 (1974). G. Mukhopadhyay, R.K. Kalia & K.S. Singwi, Phys. Rev. Lett. 34, 950 (1975). P. Vashishta & K.S. Singwi, Phys. Rev. B6, 875 (1972). G. Muldlopadhyay & A. Sj61ander, Phys. Rev. B17, 3589 (1978). H. De Raedt & B. De Raedt, Phys. Rev. BI8, 2039 (1978). F. Yoshida, S. Takeno & H. Yasuhara, Prog. Theor. Phys. 64, 40 (1980). G. Barnea, J. Phys. C12, L268 (1979). K.S. Singwi, M.P. Tosi, R.H. Land & A. SjOlander, Phys. Rev. 176,589 (1968). H. Yasuhara, J. Phys. Soc. Japan 36, 361 (1974). K. Awa & T. Asahi, J'. Phys. Soc. Japan 48, 757 (1980). B.B. Hede&J.P. Carbotte, Can. J. Phys. 50, 1756 (1972). D.N. Lowy & G.E. Brown, Phys. Rev. B12,2138 (1975). L.W. Beeferman & H. Ehrenreich, Phys. Rev. B2, 364 (1970). M. Watabe & H. Yasuhara, Phys. Lett. 34A, 295 (1971). L. Hedin & S. Lundqvist, Solid State Physics (Edited by F. Seitz, D. Turnbull & H. Ehrenreich), p. 1.23, Academic Press, New York (1969). J. Hubbard, Proc. Roy. Soc. London, Ser.A 243, 336 (1957). P. Zacharias, J. Phys. FS, 645 (1975). P.E. Batson, C.H. Chen & J. Silcox, Phys. Rev. Lett. 37,937 (1976).
Note added in proof. Very recently we got to know the paper by G.D. Priftis, J. Boviatsis and A. Vradis (Phyx Lett. 68A, 482 (1978)), whose X-ray scattering experiments have revealed the existence of the double-peak structure in S(q, co) of Li (r s = 3.22) for q > 2PF. Their experimental results are in good agreement with our theoretical ones for r s = 3.0.