Screening effects on plasmon damping in an electron liquid

Physica B 183 (1993) 293-302 North-Holland

Screening effects on plasmon damping in an electron liquid Martina E. Bachlechner, Znstitut fiir Theoretische

Physik,

Helga M. Biihm and Andreas Schinner Johannes

Kepler

Universitiit,

Linz,

Austria

Received 18 February 1992 Revised 10 April and 31 August 1992

Plasmon damping in the three-dimensional homogeneous electron gas is investigated order perturbation theory, concentrating on the effects of static and dynamic screening. theoretical approaches leading to comparable results, especially in the metallic-density interaction, however, significantly improves the results of our theory towards a better

1. Introduction

In the region of small values of the momentum transfer q the full width at half maximum of the plasmon peak in an electron gas is given by A&

=fiwp Im e(q, up) =fLo,b( q/k,)2+ 0(q4).

(1)

wp denotes the plasmon frequency

for q + 0, i.e. 0: = 4nne2/m, with n being the electron density and m the electron mass. Plasmon dispersion can be neglected for the evaluation of the q*-coefficient b (see ref. [l]). In simple metals this coefficient is determined mainly by the decay of the plasmon into two electron-hole pairs, which can be investigated for the homogeneous electron gas within secondorder perturbation theory for the density-density response function [l-3]. More refined approaches have been presented, but only two of those yield good results, i.e. the works of Utsumi and Ichimaru [4], and of Aravind et al. [5]. It is seen that including higher orders of the electronelectron interaction is of great importance. To Correspondence

the Physik, Austria.

too: M.E. Bachlechner, Institut fur TheoretisJohannes Kepler Universitiit, A-4040 Linz,

within the formalism of secondWe have found several different regime. Using a spin-dependent agreement with the experiments.

investigate these effects of screening is the main objective of this paper, which is organized as follows. In section 2 we present the well-known second-order expression for Im E, generalized to the case of arbitrary effective electron-electron interactions. From this point of view we investigate various theoretical approaches to the problems of static and dynamic screening in this field. A further generalization of our basic damping formula is given in section 3 where we allow the effective potential to be spin-dependent, too. The analytic and numerical results are summarized in section 4. Two facts are especially noteworthy. First, the theories investigated here which are using dynamic screening were not found to produce results being significantly superior to those based on a static effective interaction. Second, it appears to be of great importance not to neglect the spin-dependence of the effective potential. Finally, we end this paper with a short conclusion in section 5. The following conventions are used. Energies and momenta are measured in units of 2~~ (Ed denotes the Fermi energy) and k, (the Fermi momentum), respectively. Consequently, so, 0 means the dimensionless plasmon frequency, i.e. R2 = (4ar,/37r) with (Y= (4/9~)“~ and rs being the usual density parameter.

0921-4526/93/$06.00 @ 1993 - Elsevier Science Publishers B.V. All rights reserved

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et al. I Screening effects on plasmon damping in an electron liquid

The contributions of polarizability lead to nonvanishing imaginary of the function s(q, 0) are denoted by czpairs in the following.

2. Effects of screening As it is seen from the results of previous works pure second-order perturbation theory overestimates the effect of plasmon decay by the order of one magnitude with respect to the experiments [6-121. Since in a real metal the plasmon half width is increased by lattice effects, too, the results of a realistic pure electron gas model should instead provide a lower boundary to the measured data. Obviously, these problems arise from neglecting higher-order perturbation contributions, which would result in an effective reduction of the electron-electron interaction. There are two principal ways to build such an effect into the theory, which we will call ‘static’ and ‘dynamic’ screening. (i) A both simple and promising way is to substitute a ‘renormalized’ (screened) effective potential for the bare Coulomb interaction into the second-order formula. In the following we shall concentrate on this method, and, as a further simplification, restrict ourselves to the static (o-0) limit of the investigated potentials. This involves some sort of time-scale separation idea, where we assume that the relevant dynamics of the plasmon originates from the principle structure of the second-order formula, while the potential is taken at its t- CC (‘equilibrium’) value. Although this point of view seems to be principally questionable the comparison with more refined (‘dynamically screened’) theories appears to justify this concept (as will be shown in more detail later on). (ii) A more systematic way of taking into account screening effects is to go beyond secondorder perturbation theory by using diagrammatic techniques or decoupling the equations of motions including higher-order terms. We will refer to this concept as ‘dynamical screening’ within this paper.

Thus, following the concept (i) we have to generalize eqs. (8)-( 11) of ref. [l] to the case of an arbitrary potential up. This is not trivially accomplished since we have to bear in mind that the small q expansion leading to eq. (11) of ref. [l] makes explicit use of the shape of up. In appendix A we show one way of getting the following result for Im e2 pairsby using the formulation of Hasegawa and Watabe [3]:

[l]

m

Im E2pairs(q,~)=~~d~~d3k~d3kf 0

X

nkn~-pnk~n~~+p a(0 + P’P)P2y7 d

(2)

where p’ is an abbreviation for -(k’ - k + p). In this expression the function ti corresponds to the definition of eq. (11) in ref. [l]: &4= vp5P

- &+&P

(3)

with the generalized

functions d”” and ~2’” as

22”” = ; - up +

dex =

s-

ap

;a’, + $

[k2p2

- (kp)2]

+ iapapf + 2 on2 apap, PP

)

(44

P’P’” + 2 2 n ’ (4b)

where the quantity up is connected derivative of the potential via

with the

The general features of eq. (2), i.e. the excitation of a pair from occupied to unoccupied states with conservation of energy and momentum, will certainly appear similarly in the description of the exact pair process. This being the dominant damping mechanism for q + 0 it can be expected that many of the reliable theories for AE1,2 are mainly distinguished by different functions (u,&), which might become quite involved. In order to avoid unreasonable effort, however, more obvious introductions of screening are investigated next.

M. E. Bachlechner

et al. I Screening effects on plasmon damping in an electron liquid

2.1. Static screening

For bare Coulomb interaction (up = 1 lp’) the function JZ~of eq. (3) is, of course, identical to that of eq. (11) in ref. [l]. The most obvious choices for a statically screened interaction then are the Thomas-Fermi potential and the potential u:” defined by RPA

VP

vP

_

= ERPA(

p,

w

=

(6)

0)

where ERPA(P, w =O) = 1+ 9

(7)

C(p/2)

with

These potentials, together with the resulting functions ap from eq. (5)) are listed in table 1. The case of the Thomas-Fermi potential has already been evaluated by Glick and Long [13]. 2.2. Static local-field corrections Further improvements of the ansatz eq. (6) by including a static local field correction G(p) do not necessarily lead to reasonable results as is already seen from the modified Hubbard approximation:

Here n can be calculated from Kimball’s relation [14] while the compressibility K or the third moment Mc3) sum rule determine 5 (see e.g. ref. [El). Neglecting the r,-dependence of n and 5 we obtain the original Hubbard local field with n = 5 = 1. One could use the above G(p) to correct the dielectric function sRPA(p, w = 0) in eq. (6). It is well known, however, that both K and Mc3) become negative in the jellium model (for K roughly at rs = 5). One thus finds that the resulting effective potential becomes attractive for large rs in the region of small p, independent from how we calculate r~. This fact causes an unphysical behavior of the resulting plasmon half width, so that we will not further discuss this approach here. It is especially noteworthy that this problem is not originating from the special choice eq. (8) for the local field, but, instead, is caused by the small-p behavior of this ansatz even when using ‘exact’ expressions for G(p) and E( p, 0), respectively. 2.3. Dynamic screening As the plasmon itself is a dynamical effect, dynamical screening might be expected to be necessary to get a realistic description of its decay probability. Dynamical RPA-decoupling of the density-density function as suggested by Hasegawa and Watabe [3] and Sturm [16] results in the following substitution for the self contributions in eq. (4a): 2

up2 G(p)

=

” ”

295

4p,

pk’

+ p2W

2 pz+.

Table

I2 + $

1

The functions

P(d/aPb, ap = u P

for various

Potential

UD

a,

Coulomb

1 7 P

-2

ThomasFermi Static

RPA

potentials

[k2p2 -

&d21] . (9)

up.

Ninham et al. [12] have simplified the above formula by setting pk’+pZ/2=pk+p212-012.

1

-2-_

7-x

2PZ + (1/2)q:,Pl’(

1 P2 + q:AP/2)

P + qw

-

PZ + 4%

P/2)

P/2)

The modifications of the exchange contributions being rather lengthy we refer to ref. [3] for the corresponding results. It is stressed, however,

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et al. I Screening effects on plasmon damping in an electron liquid

that this approach has shown that self- and exchange-interaction should be screened differently. How this important feature can be implemented in a more elegant way will be demonstrated in section 3. 2.4. Dynamic local-jield corrections An approach which accounts for both dynamical screening and local field corrections has been presented by Dabrowski [17]. He obtains the plasmon lifetime from AE,,, = up *m Y4,

(10)

“J

where the dynamic local field %( q, 0) is fixed by a simple PadC interpolation procedure as suggested by Gross and Kohn [18]. A brief description of this approximation is given in appendix B. Dabrowski uses the static local-field corrections of Vashista-Singwi [19] and Pathak-Vashista [20] in order to cover the whole q range. All plasmon damping investigations, however, are restricted to q G 1. We therefore follow Gross and Kohn in calculating the interpolation parameters ‘exactly’ by deriving them from the Monte Carlo values for the correlation energy [21,22] (cf. eqs. (Al)(A3)). It should be noted that the whole approach is based on the assumption that Kramers-Kronig relations hold for the local-field correction, which, to the best of our knowledge, has not been proven. The dielectric function is known to be possibly nonanalytic [23] (and as a consequence E( q, 0) may take negative values). A nonretarded behavior may as well be expected for %. The analytic and numerical results for the models introduced above will be presented in section 4.

sumptions of our method one is inevitably confronted by two points of criticism. (i) Using a statically screened potential of the form uJ.s(~, o = 0) might not be appropriate for a dynamic phenomenon like the plasmon. (ii) Although th e expression U/E gives the correct effective interaction between an electron and an external point charge, e.g. an ion, the situation becomes significantly different when the source of the potential is also belonging to the system. An additional exchange interaction between two indistinguishable electrons with parallel spins causes the resulting effective potential to be spin-dependent even if the bare Coulomb interaction is not. To avoid problem (i) one has to go beyond our basic concept, which - implicitly or explicitly - means factorizing the three-pair Green’s function on RPA-level or higher. As it is seen from the works of Hasegawa and Watabe [3], Ninham et al. [12] and Dabrowski [17] that these methods do not appear to produce significantly improved results in the regions of our interest (cf. section 4). Consequently, we continue using the static limit of the multipair-excitation effects that are averaged in the renormalized interaction, and concentrate on problem (ii). As a starting point we assume the (effective) potential v, in eq. (2) to be different of the direct (self) and the exchange contributions to the imaginary part of E. Calling these two parts ur and uy, respectively, it is straightforward to obtain a generalization of eq. (2): oz

Im E‘Pairs(q,.)=~~dp~d3~~d3k’ 0

x ncn,_pn,.n,+Ps(n

+ p’p)p%

(11)

where 3. Spin-orientation

dependent

screening

(124 In the previous section we discussed how a renormalized (screened) potential modifies the second-order perturbation approach to the plasmon half width. Keeping in view the basic as-

9’“” = i - a:

+ ;(a”,“)’

+ -$

[k2p2

- (kp)2]

, ( 12b)

M. E. Bachlechner et al. I Screening effects on plasmon damping in an electron liquid V

“p”=

+ U,(P))

;(u,,(P>

vpxo( P, 0) Cl+ 2 P’P” 7’

’ + 1-

(124

*

G"(P)>'

0) (1 - G”(p))

’

“p”= q,(P)

(13)

Obviously, eqs. (11) and (12) reduce to eqs. (2) and (3) for vr = up. The complex problem of how to obtain a realistic model for the effective two-body interaction in an electron liquid has already been studied intensively by several authors: Kukkonen and Overhauser [25] developed a modified Hartree-Fock theory; Kukkonen and Wilkins [24] were using approximate solutions of the BetheSalpeter equations for the four-point scattering function. From the later point of view by Singwi [26] the latter work corresponds to a summation of the so-called ‘improper’ diagrams and their exchange counterparts only, while the method of Kukkonen and Overhauser turns out to be a similar but more phenomenological access to the extensive diagrammatic analysis by Singwi. Thus, we chose the first-principles based theory of Singwi to describe the spin-dependent effective potentials in our eqs. (11) and (12). Following our concept we set o = 0 in the expressions for U,, and U, from eq. (13) in ref. [26]. From its physical meaning the ‘self’-contribution to the plasmon half width contains all excitation processes despite of the fact whether they are forbidden by Pauli’s principle or not. The ‘exchange’ part then reduces this overestimated decay probability by the amount of forbidden processes involving electrons with parallel spin orientation. From this point of view it is also clear that the ‘exchange’ contributions must become just - 1 of the ‘self’ contributions when rs ---, ~0, since in this limit the radius of the Fermi sphere tends to zero, and thus just one half of all excitation processes - those with parallel spins would violate Pauli’s principle. Consequently we write down

v,x’(p,

(144 V

se,ex _ PWP)~“,“‘” ap = seiex VP

297

1

v,x”(P, 0) Cl- G”(P>)'

= Up’ + 1 -

vpxo( p, 0) (1 - G”(p))

v,x”( P, 0) (G”(P))’ + 1+ y,x”(p,

0) G”(P)

1’

(14b)

where, following ref. [26], the local field factors G”(P) = G"(P)

+(G,(P)+ G,(P))

= t(G,(p)

-

7

are given by the parameterized by Pines [27],

G,(P)=

p2 pz+

(15)

G,(P))

G,(P)=

form proposed

p2 pz’

(16)

together with the r,-dependences of qtt and qti. The Lindhard polarizability is denoted by x0(4, 0). Now, having written down the basic relations that are describing spin-dependent screening within our approximation, it is important once again to clearly state the various assumptions and motivations leading to them. The central idea of our method is to replace the Coulomb potential in the second-order results for the dielectric function by an effective electron-electron interaction. Of course, from a purist’s point of view this approach has the unpleasant disadvantage that it is not completely first-principles based, as long as there cannot be given a microscopic proof of it. However, looking at the various steps of diagrammatic analysis (and approximation) carried out by Singwi in order to obtain the result in eq. (14), it becomes quite obvious that such a microscopic justification for our approach would be an extremely difficult, if not

298

M. E. Bachlechner

et al. I Screening effects on plasmon damping in an electron liquid

even impossible task. It clearly lies beyond the scope of our present work. The empirical approach used here is justified, however, by the fact that it is the only theory presently available which yields realistic results throughout the whole r,-range. Nevertheless, there are arguments giving some evidence for the on-principle usefulness of our method. The work of Hasegawa and Watabe [3] has shown that a first-principle extension of the pure perturbational analysis leads to results comparable with our method when using an RPAtype effective interaction. Furthermore, the analysis by Schinner and Bachlechner [35,36] of the self- and distinct correlations in an electron liquid confirms our point of view that the subsequent potential-‘renormalization’ yields reliable results even in the case of a highly nonstatic phenomenon. Finally, as already discussed above, the choice of the spin-averaged potential for the ‘self’ part eq. (14a) and the parallel-spin interaction for the ‘exchange’ contribution eq. (14b) naturally follows from their physical interpretation: Only electrons with parallel spin orientation can produce exchange effects.

4. Analytic and numerical results In this section we are presenting the analytic and numerical results corresponding to the various models for the plasmon decay probability that have been introduced in sections 2 and 3. The numerical evaluation of the appearing seven-fold integrals (cf. eqs. (2) and (11)) uses the Monte Carlo method described in detail in ref. [l]. Furthermore we were able to obtain analytic expressions in the rs + ~0 (i.e. 0 + ~0) limit for all of these models. It is stressed that all the theories presented here involve the Coulomb interaction to infinite order, either explicitly (as e.g. in the case of dynamical RPA-decoupling) or in a global-qualitative way (as e.g. in the case of Thomas-Fermi screening). The approaches are intended to account for ‘true’ two-pair processes and the investigation of the large-r, limit is therefore reasonable. In the r,+O (i.e. 0-+ 0) limit the usual method of Du Bois [28] leads to

analytical results for the bare Coulomb and Thomas-Fermi potential only, while in the case of static RPA and Singwi’s spin-dependent interaction the remaining p-integral has to be done numerically. Additionally, the result following Dabrowski [17] can straightforwardly be expanded analytically in both limits. Table 2 gives a summary of these asymptotic behaviors. In fig. l(a) we show a comparison of the self-contributions to the plasmon damping coefficient b (defined in eq. (1)) as a function of rs for various theoretical approaches. It is seen that, up to the range of metallic densities, the models including screening effects produce quantitatively similar results. Since this picture remains qualitatively the same when exchange effects are included, too, only the large-r, limit can be used as a criterion to decide which approach is superior. It is definitely not to be expected that the plasmon damping should vanish for large couplings as it is the case using Thomas-Fermi or dynamical RPA-screening for rs P 10 (cf. fig. l(a)). Physically this arises from the fact that the screening length vanishes with increasing rs in these theories. Therefore the approaches of refs. [3] and [13] are unsatisfactorily for practical applications, although they have the merit of high transparency. From the physical point of view it appears reasonable that the plasmon half width should tend to infinity as rs -+ 30 (that is because the case of lower densities is corresponding to a more dominant influence of the pair interaction responsible for the entire damping). Thus we Table 2 Plasmon half width coefficient A and exponent B defined AE,,,/(fL*q2/m) = A rf in the r, asymptotic limits.

by

rs+a

r,+O Potential

A

B

A

B

Coulomb Thomas-Fermi Static RPA Dynamic RPA Dabrowski Singwi

0.2297 0.02622 0.02622”’ 0.023”’ 0.2913b’ 0.02622”’

312 312 1.50”’ 1.50”’ 3/4b’ 1.50”’

0.2913 0.0445 0.00419 0.0351 0.1364b’ 0.2913

314 -l/4 3/4 -1.4 1/3h’ 314

‘) Result of a numerical calculation, not an analytic expression. ‘) This asymptotic behavior has not yet been reached in fig. l(b).

M. E. Bachlechner

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Damping

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et al. I Screening effects on plasmon damping in an electron liquid

find the results of the static RPA screening much more reliable than those of the above-mentioned methods. Despite its static nature it is superior to the dynamic screening of Hasegawa and Watabe

Coefficient

t31. In fig. l(b) we present the total contributions (self and exchange) to the plasmon damping coefficient b calculated from the dynamic local field eq. (10) in comparison with our data for bare Coulomb and RPA screened potentials. Although within the metallic-density regime the static RPA and the method following Dabrowski lead to quantitatively similar results, the latter becomes unsatisfactory in the small-r, regime: there the plasmon half width calculated from eq. (10) crosses over the bare Coulomb curve which in our picture would correspond to some sort of ‘negative screening’. Furthermore it is seen from fig. l(b) that all of the screened potentials investigated so far underestimate the plasmon decay probability when compared to experiment. However, using the spin-dependent potential developed by Singwi (cf. section 3) leads to an interesting crossover between the statically RPA-

l E-o3L rS

Plasmon

-

Damping

Coefficient

(b)

bore_<__/------_,--

1 E+OO=

,/’

/*-

,/

Plasmon

”

Halfwidth

,/ ,/’ ,,“ ,/

x

,/’

lE-01: -

x vFeldc

,I stat

0 Theoretical x Experimental

1 E--04-Y 1 E-01

1 E+OO

1 E+Ol

J

1 E+O: 2

rs

Fig. 1. (a) Self part of the plasmon damping coefficient b defined in eq. (1) for various types of screening: unscreened (bare) Coulomb, Thomas-Fermi (TF) potential, static and dynamic RPA (Stat and Dyn, respectively). The dotted line represents dynamical screening using ~(p, 012) (cf. ref. [12]). (b) Plasmon damping coefficient b in the Dabrowskil Gross-Kohn approximation [17] (Dab); perturbational result with and without static RPA-screening (long- and shortdashed, respectively). The crosses denote the experimental values of ref. [6].

1 E-01

1 E+OO

lE+Ol

r

lE+02

1 E+03

S

Fig. 2. Plasmon full width at half maximum versus r, for various theoretical approaches and experimental results. The curves correspond to our present calculations, using bare Coulomb interaction (short-dashed), the static RPA-screened potential from eq. (6) (long-dashed) and the spin-dependent interaction eq. (14) (solid line). The points from bottom to top refer to Utsumi and Ichimaru [4] (without and with lattice effects) and Aravind et al. [S] (theoretical results), Vom Felde [6], Gibbons et al. [8], Kunz [lo] (experimental values for Na), Ninham et al. [12], Krane [7], Kunz [lo], Kloos [9] and Gibbons et al. [8] (experimental values for Al).

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M. E. Bachlechner

et al.

I Screening effects on plasmon damping in an electron liquid

screened behavior (for rs+ 0) and the bare Coulomb result (for r,--) co), as seen in fig. 2. Thus, the resulting curve shows a significantly better agreement with most of the experiments. Since it is the purpose of this work to investigate the plasmon damping effects arising from the electrons’ interaction only, any reliable results have to lie clearly below the experimental data, since e.g. lattice effects or impurities provide further decay channels. From this point of view our theory yields very satisfactory results. Additionally, the remarkable agreement of our data for Al and Na with those calculated by Utsumi and Ichimaru [4] is noteworthy since they were using a completely different theoretical approach.

5. Conclusions The main objective of this paper was a comprehensive investigation of various theoretical approaches to the plasmon half width in a pure electron gas. We have found that the ‘simple’ method of using a statically screened potential within the framework of second-order perturbation theory in many ways produces results comparable to or even superior over theories being significantly more extensive (‘dynamically screened’). Thus, concentrating on the concept of ‘static screening’, taking into account the correct spindependence of the effective potential turned out to be essential. This was accomplished by making use of the effective two-body interaction analyzed by Singwi [26]. Although this method appears to provide a reliable description of the plasmon decay in an electron gas, it is not impossible that another more extensive way of treating higher-order contributions nevertheless might lead to new phenomena. The experiments of Vom Felde et al. [6] on Cs and Rb could be an example for this. future investigations must show However, whether the effects found there are latticeinduced or can be explained within a more sophisticated electron gas analysis, e.g., going beyond linear response theory [29].

Acknowledgements

Among many other acknowledgements the authors wish to thank especially Prof. KS. Singwi: it was the last work we had the pleasure to discuss with him. We thank him for this last time, and we will miss him very much. This work was supported by the “Fonds zur Forderung der Wissenschaftlichen Forschung” (FWF No. P7846-PHY).

Appendix

A

Following Hasegawa and Watabe [3] the density-density response function x( q, o) reads

MC3’( 4)

*(q,w)=~+~ mu2

0

where (( . ; * ) denotes the Fourier transform of the time-retarded Green’s function. Mc3’( q) is the third moment, and the currency operatorj, is given via

iP, = [P,, HI = ~9,.

642)

In the original work of Hasegawa and Watabe [3] temperature-dependent Matsubara functions are used. They are easily transformed, however, into the corresponding expressions for time-ordered Green’s functions. The commutation with the Hamiltonian H = H” + gives, leading order

3 bi!L/~fn

(~)2c:_q,2ck+q,*

= 1 =1

$

[ei,?

Cd

xc> -kq

k

m

* + ‘kCk

9

VI= ; T 7 -j (E (up- up_,>

(A3)

M. E. Bachlechner

et al. I Screening effects on plasmon damping in an

electron

301

liquid

qq+o,o)=-$(1-~)=q2%(), VW (A4) Inserting

these expressions

Im .c(q+O,

0~~)= -u,

=

4Epm_

%(q++O,w)= -q2 3

into eq. (Al) yields

Im X(q-+0,6~~)

u~u~,A,(P> W,(P’, k’)

=q%=

@lb)

P

Here, K and Epot denote the compressibility of the system and its potential energy per particle (in 2&r). These quantities can be straightforwardly obtained from the correlation energy &COTT’ This immediately implies

where

A,(p,k) =

( Pd2 $ +2p2

ap + 2 (P4)(W mw

(AV

with ap =p(dldp)v,lu,

(A@

.

The Green’s function can be evaluated straightforwardly (see e.g. ref. [30]). Tasking its imaginary part gives Im E(q+0,

1

upAq( P, k)[A,(

[2 + rs

(J=b)

$]dWl

Equations (B2) are easily calculated from the fits of the Ceperley-Alder [21] Monte Carlo energies by Vosko et al. [22]. The ansatz Im Y( q, 0) = Cr,wq 2[ +]

5’4

(BW

+&j+]4’3, .-$+763. P, k) + A&-P,

(B3b)

k - P’)]

Finally, averaging over the angles of p as described in appendix A of ref. [l] and in ref. [3] and using dimensionless units leads to eq. (2) in the main text.

Appendix

7

together with the assumption that KramersKronig relations hold for the local-field factor leads to

cop) =

x S(w + PP’) x

+-

’ = I-

x0(4, w) u(q)]1 - G(q, w)lx’(q,

55

XX4,~) I-

B

The following exact asymptotic properties known to hold for the local-field correction e.g. ref. [15]):

Finally, the parameter C = 23&/60 is fixed by using the asymptotic result of Glick and Long [13], although the latter has been derived for weakly coupled systems only. It is to be noted that two types of definitions are common for the local-field correction [31,32], namely

are (cf.

u(q)]1 - ~(4~4lXX4~

w)

0) ’

O34)

differing in that x0 and xy are evaluated by using the free and the fully interacting momentum distribution, respectively. Two distinct asymptotic behaviors result from these definitions, in

M. E. Bachlechner

302

et al. I Screening effects on plasmon damping in an electron liquid

particular a divergence is obtained for G [32] in the limit CJ+ 03: G(q+m,

w) = -2q*

3

+ 3[1-g(O)]. P Wa>

Here, A&, = Ekin - EEin denotes the correction to the true kinetic energy per particle (in 2~~) due to exchange and correlation, a quantity that also enters the limit G( q + 0, ~0) [32-341. Equation (BSa) has to be contrasted with the finite result [31] %(q+a,

w) = $[l -g(O)].

In this work the following view is adopted. In order to avoid divergencies and unphysical modifications of the local-field-corrected potential one should use Niklasson’s definition of the local field factor, i.e. 5% Since, however, xf is not explicitly known, the Lindhard function x0 is used as its approximation. This procedure ensures that basic ‘self-properties’ [35] of the system are not artificially packed into 9( q, w). Then the plasmon damping coefficient b = Im(%) /q* obeys the following asymptotic relations: -314

b(r,+O)

= C 2 (

b(r,-+m)

= Cd 2

1

r:1’4, 1.4522r,1’6,

Wb)

References [II M.E.

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