linear response functions for an inhomogeneous electron gas in the hydrodynamic approximation

PHYSICAL REVIE%' B VOLUME 24, NUMBER 10 15 NOVEMBER 1981

Linear response functions for an inhomogeneous electron gasin the hydrodynamic approximation

Frank J. CrowneUS Army Electronics Research and Development Command, Harry Diamond Labs, Adelphi, Maryland

See-Chen YingI'Ayslcs Department, Brolon UnlUerslty, I'roUIdence, Rhode Island

(Received 10 October 1980)

The current-current response functions for one-component inhomogeneous plasmas areevaluated in two geometries: the infinite half-space and the sphere. Retardation effectsand dispersion to order k are included by constructing Green's functions at the plasma-vacuum interface for the hydrodynamic plasma equations coupled to the Maxwell equa-tions. The surface-plasmon dispersion relation is found to appear as a pole of thecurrent-current response, and the residue at this pole is identified as the appropriate cou-

pling constant for interactions of surface plasmons with weak external probes. Contact ismade with Feibelman's theory of linear response for the infinite half-space.

I. INTRODUCTION

In this paper the linear-response functions for aone-component dispersive plasma will be calculatedin the hydrodynamic approximation for two inho-mogeneous geometries: the infinite half-space andthe sphere. As originally derived by Van Hove,these response functions can, in turn, be related tothe nonlocal dielectric constants of the plasma, andthereby to various correlation functions which arisein the theory of inelastic scattering. The calcula-tions described herein will include retardation ef-fects, so that the response of the plasma to externalprobes will involve radiative decay of the appropri-ate elementary excitations where such radiation isnot forbidden by symmetry.

An early attempt to calculate response functionsfor inhomogeneous plasmas was made by Feibel-man, ' who used a normal-mode analysis to evalu-ate the nonretarded response function for an infin-ite half-space. Feibelman assumed that the plasmawas nondispersive; at the end of his paper hepo11lts out certain problems which arisc whendispersion is taken into account. His calculationwas essentially classical; a quantum approach wasused somewhat earlier by Crowell and Ritchie forthe spherical geometry. Crowell and Ritchie did in-clude retardation in a perturbative fashion byevaluating the transition rate for emission of a pho-ton by a plasma mode, using the golden rule. Itcan be shown that this procedure is equivalent toexpanding the classical radiative decay rate in the

parameter

COp Vo

where ro is the radius of the sphere, ~z the plasma

frequency, and c the velocity of light. A calcula-tion which included all the effects of retardationwas done by Fujimoto and Komaki, who exam-

ined the response of a metal sphere to a fast elec-tron within the hydrodynamic formalism. Bychoosing a specific form for their probe (a fast elec-

tron), however, they sidestepped evaluating aresponse function, and found the radiated fields

directly. Some formal results concerning the hy-

drodynamic response function with retardation ef-

fects were also reported by Eguiluz and Quinn, al-

though explicit expressions were not given.Because these response functions are ultimately

to be used as input to quantum problems such asscatterixlg, 1t pays to evaluate them in isolationrather than in the confines of a specific pxoblem; in

particular, the density-density response functiongiven here is directly related to the cross sectionfor Raman scattering by a surface plasmon, andhence can be used in analyzing the "surface-enhanced" Raman scattering experiments whichhave recently attracted so much interest in theliterature. This paper, however will address thecalculation of these functions without regard totheir eventual usc; the information they contain—coupling constants, normal-mode dispersion rela-

tions, etc.—will be utilized in future publications.

5455 1981 The American Physical Society

FRANK J. CRO%NE AND SEE-CHEN YING

II. BASIC FORMALISM

The linearized equations of hydrodynamics for aone-component plasma are

Bn

Bt+no%. v =0,

g;(r, co}=IGJ(r, r ',co)S'J'"(r;co)d r',

using a dyadic Green's function G which satisfies

the equation

N 2 2

V +e G; — 1 ——B BkGk.S

2 'J 2

P E=—4me(n —no),

V' 8=0,1 BBVXE= ——c Bt

1 BE 4m.V XH= — — noev,

c Bt c

(4)

Np

1 5;15(r—r ) . (11)

Now, if the system is translation invariant, Eq.(11}is easily solved in Fourier space by writing

3

G,t.(r, r ';co)= I G,J.(k,co)e'""2n. 3

where —e and m are the charge and mass of a sin-

gle carrier (here an electron), n and no the per-turbed and unperturbed electron charge densities, vtheir hydrodynamic velocity, and h the fluid

enthalpy per unit volume of the electron gas. Asusual, a positive background of infinitely-massiveparticles at density no is assumed to be present.The fields E, H, and 8 are electromagnetic fields,while E'x is a prescribed external field. Within alocal approximation to the electron gas thermo-

dynamics,

h(r, t)=ms n(r, t),intoducing a compressibility ms . If a harmonicperturbation

Eex( ~t} icyiEex(—~}

1s app11ed, Eqs. (1)—(8) ca11 11e combined 111to asingle equation for the electric field E(r;co) induced

by E'"(r;co):

V' +e g' — 1 — V('f. 8') = 8""(r,co),C C C

which gives

k;kj —(eco Ic )5;JG; (k,co)= ——1

k —EN /C

(12)k —6N /S

The quantity G (k, co) is one type of plasmaresponse function, although not the one of real in-terest here; specifically, it is the electric fieldresponse to a 6-function external field. It is clearthat G has poles at

k =e(co)co Ic, k =e(co)co Is

which imply the usual dispersion relations

N =Np+C k, N =Np+S k

for transverse and longitudinal plasmons. Accord-ing to Van Hove, it is now possible to relate theresidues of G at these poles to a (somewhat unin-teresting) correlation function (E;(r,t)EJ(r ', t'))k„.Of greater interest, however, is the function

JJ(r, r ';co)=(j;(r,t)j;(r ', t'))~,

and

&N E @ex &N Eex4m

'4m

are the usual plasma dielectric constant and plasmafrequency. Equation (9) can be formally invertedto give

i.e., the current-current correlation function, whichis directly related to the system normal modes ( thesubscript co denotes a time Fourier transform) andother many-body properties such as scattering am-plitudes.

Now, Kubo has shown that the response of asystem to a probe is related to the average (thermalor ground state) of a certain commutator, i.e., aquantum-mechanical expression; this is the caseeven for a classically observed response like a sys-

LINEAR RESPONSE FUNCTIONS FOR AN INHOMOGENEOUS. . . 5457

tern dielectric constant. Normally one derives mi-croscopic expressions from the Kubo formalismwhich generalize the better-known macroscopicresults; in this paper, however, one starts from amacroscopic set of equations and works "back-wards, " i.e., the classical equation (9) will be usedto evaluate the quantity

([j;(r,t),j;(r ', t')])

which is quantum mechanical. The physical con-tent of this procedure is clear: The system responseis bound up with its free oscillations. If these os-cillations are quantized, it can be shown' that theclassical and quantum-mechanical responses coin-cide. Thus, the coupling of "plasmons" in an elec-tron gas to an external probe can be evaluated bycalculating a classical response function for thecorresponding classical plasma.

III. RESPONSE FUNCTIONS FOR NON-TRANSLATION-INVARIANT SYSTEMS

A. General formalism

A response function for a plasma with sharp boundaries will, in general, consist of two terms, one due tothe bulk material and one due to surface charges at the boundaries. In order to evaluate these terms an aux-iliary Green's function I is introduced which satisfies Eq. (11):

22 2 Q)pV +a' I,z(r, r';co) 1———

z 8;BkI kj(r, r ',co)= 5&5(r —r ')C

but whose behavior at the boundary is for the moment unspecified. Introducing the notation

D,J(r, co) = 8 8 67 S+~ 2 ~Ij — 1 —2Brk Ark q2 q2 Br; 8rj

Eqs. (13) and (10) can be written in the form

Q)~Dtk(r, co)I'kj(r, r';co)= 5;~5(r —r ),Ij (14)

2~P -ex

Dtk(r, ,co) 8'k(r, co) = 8' "(r,co) .2

One now introduces a generalization of Green's theorem and writes

8'J(r, co)DJk(r, co)I k;(r, r ', co) —I J, (r, r ';co)DJk(r, co)$'k(r, co)= P~, [S',I'],r

where the functional

s+ 1 ——~

I ~;(r, r ',co) 8'k(r, co)

is referred to as the "bilinear concomitant" of the dyadic operator D Using Eqs. (.14) and (15), Eq. (16) isnow integrated over the plasma volume V to get

2 2COp COp

N';(r, co)= n~PJ, [l', I ]dS'+ I 1;(r ', r;co)8'„'"(r ',co)d3r',

where S( V) is the surface of V and n is an outwardnormal to S(V).

As in standard diffraction theory, it is now pos-sible to introduce "formal" boundary conditions for

I which aBow the boundary values given for 8' toenter the problem correctly. This procedure resem-

bles the standard treatments of Dirichlet and Neu-mann boundary-value problems in electrostatics.

5458 FRANK J. CROWNE AND SEE-CHEN YING

As opposed to the "formal" conditions on I, thereare standard physical boundary conditions for thehydrodynaimc approach:

(I) Continuity of tangential I'(r, co), r ES(V).(II) Continuity of tangential A (r, co), r ES(V)

[4 =(ico/4m)H)](III) n j(r.,co)=0, r ES(V).

The proper choice for the I conditions will be evi-dent once the structure of P has been elucidated.

The boundary condition (III) has been somewhatcontroversial in the past; while plausible, it is hardto justify microscopically. In Sec. IV, it will befound that (III) can be justified from a rathersurprising standpoint; for now, it is enough to note

FIG. 1. Geometry for infinite half-space.

that these conditions are standard ones fornorinal-mode calculations such as Bennctt s. Thisin turn implies that precisely those modes deter-mined by conditions (I)—(III) will appear in theIcsponsc functions.

B. The infinite half-space

In Fig. 1 the usual half-space geometry is illustrated. The vectors r and r ' are decomposed into com-ponents parallel and perpendicular to the surface: r =-(z,R), r '=(z', R'). The volume V is the half spacez y 0, the surface the R plane plus a large hemisphere enclosing V. In the R plane, dS=d R, n =(0,0, —1),and Eq. (17) becomes

2——— g';(r, co)= f d R'P~ [8',I ~] . + f, d vTJ;(1 greco)@J (r ~co) ~

2

P~[$', I ]=8'J(z,R;co) I J;(z,R;z', R', co) 1 ———2

g', (z, R;co) I J;.(z, R;z', R';co)

2—I"J., (z,R;z', R';co) — 8'J(z, R;co)+ 1 ——I ~(z, R;z', R';co) — g'J(z, R;co) . (18)

At thisgoint, one recalls that the Maxwell boundary conditions on 8', as well as the plasma current con-dition n j (r,co)=0, r CS(V), do not involve first derivatives of 5', although these derivatives appear in(18). They can be removed, however, if it is required that

I",J(r, r ';co) =0, r ES(V) or r 'E S(V)

in which case one finds

2

g';(z, R;co)=2

2

d R ' 8'i(z', R';co)- , I J,(z', R',z,R;co)—1 ——2

8', (z', R', co) -- — I J,-(z', R',z,R;co)

(19)

One now observes that thc systcII1 is translation 1QVRriaQt in Rny dlicctioIl paiallcl to the surface, and intio-duces the Fourier transform along the surface. Then

I,J(z,R;z', R', co) = f I J(z,z';K, co)e' ' 'd K/(2m)

8' (z R co)= f 8' (z'K co)e''

d K/(2m)

and Eq. (19) becomes, in Fourier space,

I.lwah. R RESIONsE FUNcnONS FOR AN mHoMOGENEOUS. . .

2

g,.(z;k,~)= 1 ——, g', (z', K,~)B,' I;;(z',z;K,~), ,— , I J, (z', z;K, co) 8';(z', K,a))

BZ

+ dz'I'i, (z', z;K,co) 8'J"(z',K,co),0

where

It is now expedient to introduce some notation: a 3-vector a wiH be written as a column vector,

a,

where A is the component of a parallel to the surface and a, is its z component. In this notation,

—iKBr

BZ

Any tensor quantity like I wiH likewise be decomposed

I II Io

where I is a "surface tensor, " I and I are row and column "surface vectors, " and I"o is a "scalar."The problem of determining I;J is now easily solved. Details will be left to Appendix A; it is found that

I „(z,z', k~) =6,,'.(z —z', k,~)+I,', (z,z', k,~)+rq(z, z';k, ~)+ I,',~(,z z'k, ~),where 6 is simply the translation-invariant result, Eq. (12},back-Fourier-transformed with respect to k, .Expliritly,

6 (z —z', K,co)= — ——1

e 1 —KKC2 e

—S ~z —z'I

"TaiK-Clz

where a dyadic notation is introduced here,

and

KKy x y

1 01

O 1

while

are "decay constants" assoriated with the free oscillations of the plasma. To see this, one notes that I hasbranch points in o~ corresponding to P=O, Q=0. The cuts which these points produce are defined by

co =co&+c (K +k, }, co =co&+s (K +k, ),


where k, is arbitrary. These equations are, of course, the bulk dispersion relations. The other terms in I,

~p ~ lI (z,z';K, co)= ——1

2

e 1 —KK i KPC2

—iKPe

—P(z+z')

2P

i K—(e ' e—~')~P Pr ~(z,z;K,~)= — ——1PQ E'—E —Pz ~ —Qze —~eP

KK i K—Q e—Q(~+*')

r(z, ';K, )= ——1iK 2Q

2

iK'(e "—e-~') e"'—c-ie-&'

P

arise from the surface boundary conditions. It is clear that

I; (z,z';K, co)= I ';(z,z', K,co)

which reflects the self-adjointness of the operator D together with the boundary conditions chosen here. Us-ing this expression, we can write the electric field as

8';(z;K,co)=MJ(z)8'i(z =0;K,co)+ ri(zz';Kco)S'i "(z',K,co)dz'0

(20)

where

8'(z &0;K,co)= 8'(z=0;K, co)e ',where

2

pp ——E- CO

le '+ 60 (e ' e+)KK—T 5 (e '—e ~')iKP

'(e * e+)—i K Q 6 ( Ee '+—PQe +)

with b,o—Pg E. At z =—0, M—(z) reduces to the identity matrix, while I vanishes; thus, the boundary

values for 8' have been isolated in the first term of (20).It is now convenient to introduce the physical boundary conditions into the problem as stated in Sec.

III A. Recall that these conditions are also used in the free-oscillation problem, where they characterize aset of bulk and surface normal modes; it is therefore expected that these same modes will appear as singular-ities in the response functions, while the residues at these singularities measure the coupling strengths of themodes to an external probe. If interest is confined to perturbations which vanish outside the plasma, the in-

duced electric field outside is given by

makes 8' a solution to the vacuum wave equation. After much algebra, it is found that the boundary values

for 8' are given by

8';(z =0;K,co )= — A ,&(z')8'J'"(z';K, co)dz',0

where

2 —Pz1 [(P—Q — )e '+ e +]KK (e ' —e +)iK+ ~ Pp e Ppe

5"(z')=(1—e)(Pge ' Ee + )iKT— E'Q(, e g. )

and

&=Q(cpa+ P) (1 e)E——

LINEAR RESPONSE FUNCTIONS FOR AN INHOMOGENEOUS. . .

This can be combined with Eq. (20) to get the field 8' as a function of 5"". The result will not be givenhere, since the interesting quantity is the current induced by 8"",which is given by

This is found to take the form

J',.(z;K,co) = ——1 Jq(z, z', K,co) Ã'J."(z',K,co)dz',g y y 0 fj

where

e1 0J(z,z';K, co)= 5(z —z')+(1 —e)J (z,z';K, co)+J (z,z';K, co)+J (z,z';K, co)

2

e 1 —KK iKC BZ —I'

) z —z'I8.KT ~ K2 2P

P2Po —P1

1 KKy Pp +P K2 K

i K~P Ke

—I'(,s+z')

KITBz —Q I~ —~'I8

—KKr iKQ

iK—Q —Q7 2

e—Q(z+z' j

where J is written as a dyadic and

g(z; K,co)=i KP/K iK(1/Q)

8 —1

8

One now notes the following properties of J:(1) J is symmetric in z and z . As will be discussed in Sec. IV, this is a nontrivial consequence of our

choice ofboundary

conditions.(2) The terms J' and J~ are singular at P =0 and Q =0, having the same branch points as I, again re-

flecting the bulk excitations of the plasma.(3}As shown by Crowell and Ritchie, ' the current j,„associated with a surface plasmon in the type of

system discussed here is given by

j,p(z;K, co) =Hog(z;K, co(K)),

where Ao is an arbitrary constant and co(K) is given by the equation

Q(epo+P}=( I e}E—

in this notation. Therefore, one can write

J =Ko j,~(z;K,co) j ~~(z', K,co),coz —'co (K)

where Ko is a coupling constant. J is therefore associated with that term in the eigenfunction expansion of

5462 FRANK J. CROWNE AND SEE-CHEN YING

J which corresponds to the surface mode.(4) There is a fictitious singularity at I'+pa ——0 in the response function; this singularity was also reported

by Kliewer and Fuchs' in connection with reflection of P-polarized waves from a surface and does not cor-respond to any real co(K).

(5) The continuity equation allows one to calculate the density-density response from the current-currentresponse: Setting

N(r, t; r ', t') = ([n( rt), n(r ', t')]),where n is the number-density operator, gives

, ([n(r, t),n(r, t')]) = —2 ([j (r, t)j ~(r ', t')]) .

It is found that

.-+ -+N(~, ~t I} I /K'(a —R )—F01(j j )N( I, K )

(2n }2 2'where

I

N(z, ';K, )= 8( )8( ') 5( —')+—( — )4me s' s'

It would be interesting to compare this result to the one derived by Feibelman; however, Feibelman actuallycalculated the "potential-potential" response function,

4(r, t; r ', t') = ([P(r, t);P(r', t')] )8(t t')—which in the limit c~ ao is related to N by

(4me ) N(r, t;r ', t')= 8

a Bra4&(r, t;r ', t') .

Br& Br~

The function 4(r, t; r ', t') for this system is given in Appendix 8, and is shown there to reduce toFeibelman's expression as s~ 0.

C. The sphere

Figure 2 illustrates the spherical geometry; standard spherical coordinates are clearly in order here, so onewrites r =(r,Q), r =(r', 0') and dS =ref Q„n =r, where ro is the radius of the sphere. Analogous to thesurface Fourier transform used in Sec. III8 are the usual decompositions into spherical harmonics; sincevectors are involved here, however, one must invoke the formalism of vector spherical harmonics. In this

paper, extensive use is made of the notation used in a paper by Ford and Werner'; one writes the (tensor)bulk Green's function 6 (r, r ';co) in the dyadic form

G'(r r ',~)=—g I q'dq[Alm(r, q)gl'm'(q)AI (r ',q)+BI (r,q)gl'm'(q)B1 (r ', q}lm

+Ci (r;q}g&' '(q)Ci (r ',q)],

where the functions A, B,C are discussed in Appendix C, and

2 2

EN —Cg CN —Sg

LINEAR RESPONSE FUNCTIONS FOR AN INHOMOQENEOUS. . .

The q integration can be done, but requires a choice of boundary conditions at infinity; the usual outgoingspherical wave condition is appropriate, so that 6 corresponds to electromagnetic radiation by a 5-functionsource. Note that this choice wi11 affect the self-adjointness of 6, since time-reversal invariance no longerholds.

Equation (17) becomes

—I';(r ', r;ro), g~(r ',~)+ 1 ——2 [rz 1 J;(r ', r;co}] ~}'k(r ';co)

Br c ~rk, I~ 'I =~o

and once more one chooses I to satisfy the boundary condition

Again, details are relegated to Appendix 0; proceeding exactly as in Sec. III 8, it is straightforward to arriveat the expression

j(r;co)= f J(r, r', ro) (}t""(r',ra)d r'.The spherical harmonic decomposition allows us to write

j ~ (r,co}=f JI (r, r ',co) ÃI'"(r', ro)d r'.The expression for JI~ is rather complex; if it is broken into two terms,

then

2

JI (r, r ';a))= ip 2(1 —e)[AI+—(r;p)A)~(r ',p)8(r r')+A} (r p)A—I+' (r ',p)8(r' —r)]

C

where

—iQ 2 [BIm (r Q)B/~(r ';Q)8(r —r')+B~ (r;Q)BI~ (r ';Q)8(&' —&)1S

2

ip 2(—1 —e}[C}+(r;p)CI (r ';p)8(r —r')+C& (r;p)CI+ (r ';p)8(r' r)],

C

p2 (~2 ~2) Q2 (2 2)

is the "bulk" contribution to the current, and the functions A'+', 8'+', and C'+' are defined in Appendix C.As for J, let

M~(r)=AI (r;p), M2(r)=B}~(r,Q), M3(r)=C~~(r;p) .

Then

J,(r, r';co)=M~(r):- ~Mp(r'),

where summation is implied over the repeated indices (which are not vector indices for A, B, or C), and


Z11 Z12 0

Z21 Z22 0

0 0 Z33

with

Z, l———(1 e)A—p 'I yl(coro/c)[yl(prp)gl(Qrp) pl(p—ro)J'i (Qro)]

I'o

&yl—(Pro)[yl(coro«)gl(Qro) yl(—co"o«j)i (Q"o)1 j

Zl2 ——Z2l —— ——1 Ap —3[1(1+I)] yl(corpIC),1 1 1 1/2

ro

Z22= Ao I(1 &)yl(—Qro)yl(coro«)gl(pro) "l (—Qro)[fl(pro)yi(cpro~c) &gl(pro—)apl(coro~c)] JC tQ (1)'

S I'

Z33 ——~(1 F)8p '[h—l' '(corp Ic

)fl(pro�)

(pl(piro l—cj)l(pr p)],ro

where jl(x) and hl "(x) are the usual sphericalBessel function, ji (x) and hl'" (x) their derivatives,

and

so="l (coro«)fl(pro) gl(piro—«j~i(pro)(1) k

Note that the equation

Ap =yl (piro IC )f (1 E)gl (pro )gl(Q—ro )

fl(pro Vi (p"o)1—

+egl (Pr 0 )(Pl (~r o «Vi ( Qr o»

s(v)

FIG. 2. Spherical geometry.

can be shown to be equivalent to the dispersion re-lation derived by Crowell and Ritchie as ro~ ~.

The response function Jl~ is seen to have the fol-

lowing features.(1) It is symmetric under interchange of the vec-

tors r, r '. Note, however, that it is not Hermitian,due to the radiation condition which was imposedat the beginning of the calculation.

(2) In the c~ co limit, Jl has no branch-cutstructure, just poles at the system's free-oscillationfrequencies. This is an expected result for a trulyfinite system, in which the only possible normalmodes are standing waves for which the "propaga-tion vectors" are quantized by the systemgeometry. As I o~ ao it can be shown that onesuch pole will split on' and eventually give rise tothe half-space surface-plasmon pole, while the oth-er poles merge into the branch cuts described inSec. III 8. Note that for a very large sphere, anexperiment whose duration is less than the time awave takes to tranverse the sphere and reflect offthe other side will not see those discrete poles, butwill instead see a quasicontinuum part of the spec-trum.

(3) The coupling strength of the "surface-


plasmon" pole is a function of r0, and for r0 g 00

partakes of the "bulk" oscillator strength to a re-markable degree; this suggests that the distinctionbetween "bulk" and "surface" excitation is not avery useful one until r0 is quite large.

The density-density response function for thesphere now takes the form

([n(r, t),n(r ', t')])„=g Ft~ (Qo)nt (r, r';co)lm

where 00 is the solid angle associated with the vec-tor

~

r —r '~, r and r' are magnitudes, and the sub-

script m denotes the time-Fourier transform. ThecoeAicients nl~ are given by

(s~V +co —co~ )n =cop 2V +ex .

4me

Introducing the usual spatial (three-dimensional}Fourier transform gives

n,„(k, co)=—k 2 2

4&e QP —6) —s kp

In the standard theory of the electron gas, the f-sum rule in particular j~ commonly invoked forth1s purpose; 1n this scctlon of thc paper, thc appli-cability of the f-sum rule to the infinite half-spaceunder study here will be discussed.

For a translation-invariant, nonretarded (c~ 00 }plasma, the charge density n(r, co) induced by anexternal potential y,„(r,co) is given by the equation

21 p 1

nt~ (r, r';co) = 5(r—r')+ gt(r—,r')4me s r

nt Jt(gr V—t(g'}

and so the density-density response is

k2

N( k, co)4&e QP —6) —s kp

In this notation, the f-sum rule takes the form

(21)

g, (r, r') =tg'[J, (gr')It, '"(Qr}«r' —r)Im codcoN(k, co)= — k n ,o0 2' (22)

+At'"(Qr') jt(gr)8(r —r')],

(1—e)yt(gro) —Htht"'(Qr )91

(1 e)gt(gro) —Htj/ (Qro)—

fry ro) mt(~ro«}l=gib ro) rt(coro«}

where m is the electron mass and n0 the electrondensity; it is easily shown that the X given by Eq.(21) satisfies this equation. Upon reflection onesees that this should be so, since the f-sum rule isa direct consequence of the conservation of electroncharge, i.e., the continuity equation (1) of Sec. II.

The inhomogeneous form of Eq. (22) has beengiven by Bagchi':

f co dco ImN(z, z', K,co)0

One again notes that nl splits into a "bulk" and a"surface" part; the quantity ql is still singular atthe "surface-plasmon" frequency. It is found thatthese singularities occur for complex co, however,

indicating that radiation damping is present. Anexperiment such as Raman scattering will, there-

fore, give linewidths associated with radiative de-

cay; these widths are found to be quite small, how-

cvcr.

E +, no(z)5(z —z')32m 9zi3z

which is seen to reduce to (22) when

N(z, z', K,co) =N(z —z', K,co)

(23)

IV. THE f-SUM RULE FOR THE INFINITEHALF-SPACE

When one undertakes to calculate physically in-

teresting quantities relating to a many-body sys-

tem, the use of approximations is unavoidable; forthis reason, sum rules —which are exact conse-quences of the many-body physics —provide a use-

ful measure of how good the approx1mations are.

after a Fourier transforIn. Bagchi's calculation wasmicroscopic, however, and never invoked a sharpsurface; hence, in the calculations presented here itis important to verify that (23) is indeed satisfied.In order to do this, it is useful to ask a relatedquestion: a typical calculation of this kind yieldsinformation about the normal modes of a system.It is well known that the half-space problem ad-mits both surface plasmons and bulk plasmons. In

5466 FRANK J. CRO%NE AND SEE-CHEN YING

gf.(z)f„*(z')=5(z -z'P. (24)

this situation, it is natural to ask about complete-ness: treating the normal modes f„as eigenfunc-

tions, is it true that

tailed nature of the f„. If the f„are density eigen-

functions, then

f„(z;K,co„(K})f„*(z';K,co„(K) )N(z, z';K, co)= —~„(K)

The answer to this question depends on the de- Putting this expression into (23) gives

g f„(z;K,co„)f„'(z';K,co„)=— 2a'

K +, no(z)5(z —z')2m BZBZ

and it is thus clear that the density eigenfunctions are in fact overcomplete —they can produce functionsmore singular than 5(z —z').

At this point, it should be noted that the basic equations plus boundary conditions are indeed one step re-

moved from the density Auctuations; in fact, the fundamental object calculated thus far is the current-current

response. The question of when Eq. (24) holds can now be translated to a question about current eigenfunc-tions: Suppose that f„ is a current eigenfunction. Then the current-current response can be written

f„'(z;K,a)„}fJ (z;K,co„)J' (Jzz'; Kco) =top g 2 ~

8(z)8(z'),N —Q)z

where (24) becomes

(25)

g f„'(z;K,co„)fJ (z', K,co„)=515(z—z')6(z)8(z'), (2

Now, it is shown in Appendix E that the function J is related to the current-current response via the equa-

tion

JJ(r, r ', to) = ([j;(r,t),jj(r ', t')] }„.

([n(r, t},n(r ', t')] }„=— Jtj(r, r ', to),1 8 8Bl'g BI'J

or, after the surface Fourier transform,T

gT

N(z, z';K, co)= iKyz4me

J—iK+z ', J(ti,z'z, Kco),

BZ

where z,z ' are unit vectors perpendicular to the surface. Then

Taking two divergences of this equation and using the continuity equation gives

(27)

tN

Im coN(z, z';K, co)dho= — iK+z4me' BZ

—iK+z '-, codco ImJ,J.(z,z';K, co)BZ'

Now, if (26) holds,

I,

2 I

ai K.+Z4' BZ

JiK+z '—

, g f„'(z;K,co„)f~j (z';K, ro„) .BZ

I to de ImX(z, z';K, a) )=—2 i J

p m'

iK+z i K+z ' — 5"5(z —z')6(z)8(z')4~e

' —SC'+, 5(z —z )e(z)e(z')2 BZOZ

J."+, 5(z —z')n, e(z)2m BZBZ'


using

8(z)e(z')5(z —z') =e(z)&(z —z'),

and so Bagchi's form of the f-sum rule is seen tobe satisfied. Similar arguments can be used in thecase of thc spherical geometry.

The f-sum rule is thus seen to be a direct conse-quence of the completeness relation (26) for thecurrent eigenfunctions. At this point, one can ex-

amine the Green's functions calculated in this pa-

per, and note that they are all symmetric in z andz' (or r and r'). This symmetry allows one to in-

voke what is caHed Mercer's theorem, ' which

states that the eigenfunctions of a. symmetric kernelare indeed complete. The f-sum rule is thus arather arcane consequence of the symmetry of thecurrent-current response, which in turn is deter-mined by the choice of boundary conditions. Be-cause these boundary conditions involve bothcurrent and electric field, which are related nonlo-

caHy to one another via Eq. (21), it is not at all

clear a priori that this boundary-value problemleads to symmetric Green's function; indeed, thef-sum rule will not be satisfied, in general, forevery choice of boundary conditions. Since manyquestions of principle have been raised about theapplicability of the boundary condition of vanish-

ing normal current density, the results found hereshed some light on the success of the hydrodynam-ic formalism ln predicting surface-plasmonbehavior using this condition.

electron-gas enthalpy predicted by Kohn andHohenberg. ' As these corrections introducehigher derivatives into the problem, the responsecalculation will require additional boundary condi-tions whose characteristics are not obvious; theconsiderations outlined herein offer considerableguidance in choosing such boundary conditions. Ina future publication, the problem of two-componentplasmas will be dealt with, with the intention ofexamining the hydrodynamic behavior of electron-hole droplets.

ACKNOWLEDGMENT

One of the authors (S. Y.) acknowledges supportby the Material Sciences Laboratory at BrownUniversity, funded by thc National Sclcncc Foun-dation.

APPENDIX A: INF~INITE HALF-SPACE—EVALUATION OF GREEN'S FUNCTIONS

Onc bcglns by writing

I;J(z,z'; K,co) =GJ (z,z';K, co)+ I",'J (z,z'; K,co),

where I ' is a solution to the homogeneous equa-tion

82 2—

z E+e2

I—';J(z,z';K, co)BZ C

V. CONCLUSION

Thc rcsponsc functloI18 calculated ln this 'paper

incorporate all the efFects of retardation into their

structure, and as such afFord an opportunity for de-

tailed examination of these CAccts in a dispersive

inhomogeneous plasma. Use of the formalism ofhydrodynamics gives functions which automatical-

ly satisfy the f-sum rule, thereby avoiding difficul-

ties associated with perturbation theory. Thc func-

tions exhibit a surprising amount of symmetry; the

connection between this symmetry and the conser-vation laws via the self-adjointness property of the

boundary conditions has been remarked upon. Itwould be interesting to identify a class of boundary

conditions for which this problem remains self-

adjoint; this class would in some sense "generalize"the hydrodynamic boundary conditions. Themethods used ln this paper can easily bc cxtcndcdto more complex modeling of the electron-gas ther-

modynamics, i.e., the "gradient" corrections to the

i K. -

BZ

82

Z

is a differential operator, written ln tensor form inaccordance with the notation discussed in the mainbody of the text; thus,

/E~ —,EEy8

6 is the translation-invariant response function[Eq. {12],back-Fourier-analyzed perpendicular tothe surface; explicitly,

~ =~~+c'(K'+k,')

~ =~~+s'(E'+k,')

%herc k 18 Rny QQIDbcr froID zcI0 to infinity.The solution F to Eq. (Al) which 1s appropriate

to R nonradiat]tng surface charge 18

&'(z,z', K,a)) =I'"(z';K,co)e

+&' '(z';K, co)e

Now, I"' must behave hke a tensor both in the fullythree-dimensional space Rnd thc t%'0-diITlcnsional

subspace spanned by thc surface; the only tmo-dimcnsional vector in thc probleID 18 K, Rnd so thconly two-dimensional tensors are 1 and KKT.Hence, r"' and I""'must tave the forms

Ao(z')1+3 t(z')KK" iKB2(z')

iKTB) (z') C(z')

D(z')KKT iKE2(z')r'"z'K~ =

iK E~(z') F{z")

Viewed as a function of frequency ~, G has

branch points at

which correspond to

(iK, —P).F'"=0,+

lo. K=0 s

Qo.

tK .o. 0

Furthermore, direct substitution of (A2) into (Al)shovvs that

Rnd

icspcctivcly; thc cuts refjcct thc existence of R con-

tinuum of bulk excitations of the form Thcsc relations RI10%v onc to %rite

I'(z,z';K, co)=[Ho+A —)K ]iKI'

pA2iK

~

EC

I' h

—iKe Pz+—

using dyadic notation in thc second teim, and suPPrcssing thc dcPcndcncc of ~s and ~; on ~ .In order to solve for the A; ~d 8;, one uses

P(O,z', K,co) =G (O,z';K, co)+1"(O,z';K, co)=0 „ (A3)

6 (O,z';K, ra)= ———1

e H 1+(6—H)KK (PH —Qb, )iK2

(PH Qh)i —KT

LINEAR RESPONSE FUNCTIONS FOR AN INHOMOGENEOUS. . . 5469

where

H= e-~', a= e-&z',2P '

2Q

the unknowns 2; and 8; can be found by equating the (scalar) coefficients of l, KK, and i K T.he result

takes the form given in the text.Once I is known, Eq. (20) can be combined with the boundary conditions of Sec. III A; the Maxwell

equations give

2- 2-m=Vx 8', , j =, @' —Vx(Vx@').

C C C

If 8' and A are now written

8'(z;K, co) =8'~(z;K, co)

A (z;K,co)=8', (z;K,co) A g(z;K, co)

then at z=0,

A ~(0;K,co)

A, (0;K,co)

Po — crK—KPQ —K

+iK 0.

. P' —Z2

PQ E—0

w'~(0;K, co)M(0;z';K, co) @'"(z';%co)dz',

8', (0;K,co)

where

~(0,z';K, ~)= V x I (z,z', K,~) ~,=,2

—E 0C2

0

—lPo K

0

p, 0 —iQoKe e

0 0

1

PQ Ic. —0 0

2

e (e ' —e &' )crKKT (Q P)(K e ' —PQe ~' )ioK- .C2

If the perturbation 8""is now chosen to vanish outside the solid, then the vacuum response takes the form

8'i"(0;K,co)

8',""(0;K,co )

2co

po ——E—2

insures that 8' ' is a solution to the vacuum wave equation. Boundary condition l of See. ILIA, plus theassumption that the static dielectric constant 6O= I gives

T

S'I"(0;K,co) S'I(0;K,co)

S',""(0;K,co) S',(0;K,co)(A4)

Similarly, it is found that outside the plasma,

P'j "(0;K,co)

~"-(0K )

—icrK S'z"(0;K,co)

0 S',""(0;K,co)I

Boundary collclltloll (II) liow says to cquRtc A~I~ on tllc two sldcs of tllc surface, glvlllg tllc followlllg cqllR-

tion (suppressing dependence on K,co):

I'2 —E2@ocr O'I~'(0) —icrKS',"'(0)= Pcr S' —(I0) — — (crK)[K ~ S'I(0)]—i (crK) S', (0)

P E — PQ E—+ I Nl(0, z';J, co) S""(z',K,co)dz', (A5)

wlllcll, with Eq. (A4), glvcs Rll cqllatlon fol' S j slid S' (0) 111 terms of S ". A second equation like thisone can now be derived from condition (III):

j,(0;K,co)=0 .

This equation (eventually) gives

(1 e) coS' (0)= — B (z'K co) S""(z'K co)dz' (A6)

9 T( Kz, )=co(iKT(PQ "e' &'e + ),&—Q(e

&=(eyo+P)Q —(1—&)X

Note that the quantity PQ Eno longer a—ppears in any denominators; its presence there was due to the"formal" boundary condition (A3), and it should drop out in the end. Equation (A6) can now be inserted

lllto (A5) to glvc Rll cqllRtlofl for S I(0) 111 tc11Ils of S (z;K,co); uslllg tllcsc cxpl'csslolls 111 Eq. (20) of tllctext yields a homogeneous relation between S'(z;K,co) and S""(z;K,co).

APPENDIX 8: POTENTIAL-POTENTIAL RESPONSE FUNCTIONAND FEISRLMAN*S CALCULATION

For c—+ ao, Eq. (1)—(8) of Sec. II simplify considerably; introducing the electrostatic and velocity poten-tials

a single equation for q inside the plasma obtains

(s V +co co~)V y=4Ireco~n'"=co~—V qr'".

24 LINEAR RESPONSE FUNCTIONS FOR AN INHOMOGENEOUS. . . 5471

In this form, cp can be formally evaluated:

((o(r,N)= f F(r, r ';N)n'"(r ';N)d r',

and the function I" is clearly related to

(I)(r, r ',N)= f e'"" ' ([p(r, t),y(r, t')])B(t t')—d(t t')—.

Again going to Fourier space,

2

f @( i.K )iK (R —R')

(2ir)

Using the methods described in the text, it is found that

C (z,z';K, N) =C,(z,z')e( —z)e( —z')+e, (z,z')e( —z)e(z')+ e,(z,z )e(z)e( —z')

+e,(z,')e(.)e(z ),with

C&)(z,z') =— 22~e Q K(+ )E'42(z,z')= — e + ——1+—e ' e ',4me ~ 1 Q

2 E

4~e43(z,z') =— e

—Qz 1 ~ Q —Kz eEz'2 E

2Np

44(z, z') = —4rreN —

N& 2Qe Ql —'I l+2

N E 2

e—Q(z+z') N

(e (Qz+ Kz') +e—(Qz +K—z) )'

K)z —z') 2 ++—Qe —K(z~z')2E

where

Q2=I(. 2 ——(N2 —Nz), &„= lim &=(2N Nz)Q N—~& . —s C~ ee

Now, Feibelman calculates the quantity 4(z,z; kN) in his paper; to recover his result, the limit s ~ 0 mustfirst be taken. For small s,

„—=—(2N —N~ )(N~ —N )~ ~ a), Q (Np —N ) ~ ~—~,2 2 2 2 1/2 ~ 2 2 1/2

s s

so that

Writing

2 2K(z+z')

2N —N&

me

2N —N&

2 N2(

t)

2)re i K(z —z')2ZZ 2 2e

2N —N&

2~—Np

2 —x(~+z']2N —N2 2'

gives

2 2 2 2N N& N& CQ&

(N —N )(2N —N ) N —N 2N —N2 2 2 2 2 2 2 2Jp P

S472 FRANK J. CROWNE AND SEE-CHEN YING 24

(e—K [2 [e Ic[—z'[ )+2' —

N&

which is Feibelman's result.

2

( —K~

z —2'~

—E(z+2') $

2 2 8 e 7

CO —COP

APPENDIX C: SPHERICAL-HARMONICFORMALISM AND WAVE-EQUATION

SOLUTIONSFollowing Ford and Werner, the following vector

functions are introduced:

mY

and that

l2l+1

1/2 ' 1/2l+1X1+1 X3,2l+1

Yi, i+i(Q) = [(l +1)(2l +1)]X"+'Vr "+"Y-, (Q),

Yii(Q)= i[l(—l+1)] '~ rx VYi (Q),

(C1)

J Xi.XJdQ=5J .

Now, Ford and Werner introduce the following setof functions in order to solve the wave equation in

spherical coordinates:

Yii i(Q)=[1(21+1)] ' 'r " "Vr'Y (Q) A, (r;q)= l21+1

1/2

ji+i(qr»l I+ 1(Q)

Xi(Q) =r Yi~ (Q), (C4)

(C3)

where r is the usual radius vector, 0 is its angularposition, and r =

~

r~

. These functions in turn al-

low one to introduce a second orthogonal set:

1/2l+1

2l +1' 1/2

l+1Bi (r;q) =

2l+1 ji+i(qr}Yi,i+i«}

X2(Q) =Yi i(Q),

X3(Q)= r"x Yi i—(Q)

(C5)

(C6)

l

21+1

' 1/2

ji,(qr)Yi i,(Q},

Then it is easily shown thatr 1/2l+1 - . l+

' 1/2

X3,

Ci (r;q)=ji(qr)Yi((Q) .

These vector functions have the following useful

properties:

V' AI~ ——0, V Cl~=0p VXBl

VX(VXAi )=q Ai~, VX(VXCi )=q Ci~, V(V Bi )=—q Bi~ .

Clearly these functions allow one to solve a varietyof vector equations, in particular Eq. (9) in thetext. For the purpose of fitting boundary condi-tions, the functions X; are more convenient than

+ +

the Y~,' so, the functions A, 8, and C are moreusefully expressed thus:

Ai~(r;q) = —gi(qr)X&(Q) —ifi(qr)X3(Q),

Bi ( r;q) =J'i (qr)Xi(Q)+igi(qr)X3(Q),

Ci (r;q)= ji(qr)X2(Q),

where

gi(qr) =—[l (I +1)]'~ji(qr),1

qr

1fi(qr) =j/ (qr}+ ji(qr) . —

gr

(C7)

A,'+'(r;q) =2l+1

„

1/2

hi+i (qr}Yi i+i(Q)

1/2

hi i (qr)Yi i i(Q),l+1+

In this paper, the outgoing-wave conditions re-quire one to introduce the following functions inaddition to A, B, and C:


' 1/2

BI (r q)=(+) . I + l (1) m

2t+l hi+ & (q")Yc,I+ &(&)

1/2

with

s

hi i(qr)YI ( i(A),

CI~ ( r; q) =hi (q&)Yli (0) .

These are "traveling-wave" analogs to the"standing-wave" function (C4) —(C6). Written in

terms of the X;, they become

AI+ (r;q)= —yr(qr)Xi(Q) —ill(qr)X3(Q),

8'I~ '( r:q) =bc" (qr)X, (Q)+ iy((qr)X3(Q),

CI+ '( r;q) =hi "(qr)X2(Q),

where

y, (qr) = [1(1+—I )]' 'hl "(qr),qr

and the functions with adjoint signs are those in

which the spherical harmonics are complex conju-gated but not the Hankel functions, since a changefrom hI"' to hI' ' destroys the radiation condition.

SNow, the term I I is assumed to take the form

I I (r, r ',co)=MooCi (r;p)CI (r ';p)

+(Ai (r;p), Bi (r;Q))r

AI~(r;p)X 8, (rg)

using a somewhat cumbersome matrix notation(note that the terms in I I are still dyadics). Thenthe condition

f'I(qi)=hi ' (qr)+ hj "(qr)—gr

I i (r, r 'co

+ ~l ( r, r ',~)I

APPENDIX D CALCULATIONS FORSPHERICAL RESPONSE FUNCTION

The function I;J is decomposed, as in the half-

space problem, into two parts:

I;1(r,r ',co) =G;1 (r, r ';co)+I";J(r,r ';co)

allows one to evaluate the Mpj For reference theyare given here:

[yI (p"o )gI ( Q"o )lp

&p

qc(pro)ji —(Qro)],

G ( r, r ';co ) = g Gi~ ( r, r '; co ),lm

I' (r, r';co)= gl ~ (r, r', co) .Im

It is found that, letting r =fr f, r'=

fr '

f,

G (r, r ';co) = —e(r —r')a (r, r ')

—9(r' —r )2''(r, r ),where (in dyadic notation)

~(r, r ')=ipAIm (r;p)A1~(r ';p)

+ipCi+'(r;p)CI (r;p)2-

+iQ—BI+ (r;Q)BI (r ', Q),s

~'(r, r ')=ipAI (r;p)Ai (r;p)

+ipCI (r,p)C,'+' (r ',p)2

+ig —,Bi (r;Q)BI'"(r ';Q),s

—,[yi(Q&o Vi(gro)iQ c I

s2

gc(Q&o)h—i"' (Q&o)]

~2i = [yI(pro)fI(pro)Lp

0'p

O(pro)cPI(pr—o) l

iQ c~~22=

2 [yl(gro)gl(pro)~~p S'

fi(pro)hi' (Qr—o)f

where

&o=O(pro)gI(Q"o) fi(pro Vi (Q&o)—is analogous to the Ap which appeared in the half-space problem, ' again, imposition of physical boun-dary conditions should cause the denominators in

Np to disappear in the final answer.


%ith the evaluation of the M; one has the "for-mal" Green's function I I~. At this point the func-

tions X; and a new notation can be introduced: Ifthe electric field is written

@'=g @'Im

lm

This decomposition permits the evaluation of suchangular integrals as

f dQ'8';(r ) I IJ.(r, r ')

and eventually an analog to Eq. (20) can bederived:

a8'T(r)=, I'zT{r,r')3r'

+ f p'r' dr'[I zT(r, r')8"z"(r')+I r„(r,r')&'„"(r')], (D1)

8'„(r)=,I'„z(r,r')8r'

(D2)

5'p(r) = (D3)

+r«r')= —,O'I~p '[fib r)~12 gl(Qr)-It'22]JI(Qr')+gl(Qr)IIi"'(Qr')I

I'pg(r r')=I' 2Q I&p —'[gl(pr)~12 il' Q(r~)—22]il Q('r—) gl(Qr)III' '(Qr')j

Hcavy llsc of Bcsscl-function ldclltl'tlcs reveal tllat Eq. (Dl) —(D3) rcdllcc to t11c ldclltlty

(rp)= F (rp)

when r =ro, just as in the half-space expression.Now, the electric field outside the sphere is given by

-(+) -. -(+) -.N I (rp)=e1AI I'; +e2CI I",

C C

which is an outgoing wave. This expression evaluated at r =ro gives the boundary conditions

@'T(rp)= I'e1yl(perp/c), @—'„(rp)= e,yl(corp/c—), 8'p(rp)=e2III "(corp/c) .

Similarly, one calculates the current and magnetic field and applies the boundary conditions as in the half-space situation; this rather lengthy procedure results in the expressions given in the text.

APPENDIX E: RELATION BET%'EEN GREEN'S FUNCTIONSAND TRUE RESPONSE FUNCTIONS

The objects calculated in this paper are all of the form

i (r;co)= f G ~(r, r;cp)S'Il"(r ',cp)d r' .


To see how these relate to the usual Kubo expressions, we consider a harmonic perturbation

E'"(r,t)=E'"( )

Then the interaction Hamiltonian is

H (t)= — j(r', t) A'"(r', t)d r'= e '"' j(r', t) E'"(r')d r'.rC lN

Now, Kubo's formula gives the induced current

( j (r, t))= —f dt'B(t t')(—[H (t'), j (r, t)]),

the angular brackets denoting an average. Then

(j ~(r, t)) = f d3r'Ett" (r ') f e '~'B(t —t')([j ~(r ', t'),j~(r, t)])

where

e '"' f d (r ')E~"Jott(r, r ',co),le

J tt(r, r ',co)= — e'"" ''B(t —t')([j (r, t),jtt(r ', t')])d(t t')—is the usual current-current response function. Now, since

@ex(~s) tco Eex(~g)4~

it is clear that

(j ~( r, t) ) = — e '"' f J~tt(r, r ';co)$'p"( r ')d r'N

or

(j (r,co))= — J p(r, r ';co)S'p"(r')d r',CO

and so

G~tt(r, r ',co)= — J~~(r, r ';co) .CO

If we define the density-density response as

N(r, r ';co)= —f e'"" "B(t—t')([n(r, t),n(r ', t')])d(t t'), —fi

then the continuity equation implies that

J~tt(r, r ';co)=co e N(r, r ',co),Br Br~

and so

N(r, r ',co)= — G ~(r, r ';co),a a4me2 Br 0r~

as stated in the text.


&P. J. Feibelman, Phys. Rev. 8 5, 2463 (1972)2J. Crowell and R. Ritchie, Phys. Rev. 172, 436 (1968).3F. Crowne, Ph.D. thesis, Brown Umversity, 1974 (un-

published).4F. Fujimoto and K. Komaki, J. Phys, Soc. Jpn. 25,

1679 (1968).sA. Eguiluz and J. J. Quinn, Phys. Rev. 8 14, 1346

(1976).6See, e.g., N. A. Krall and A. %. Trivelpiece, I'nneiples

of Plasma Physics (McGraw-Hill, New York, 1973),Chap. I.

7S. C. Ying, Nuovo Cimento 238, 270 (1974).3L. Van Hove, Phys. Rev. 95, 249 (1954).9R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).toSee, e g , C..K. ittel, Quantum Theory of Solids (Wiley,

New York, 1963), pp. 42—46."J.Bennett, Phys. Rev. B 1, 203 (1970).~2J. Crowell and R. Ritchie, J. Opt. Soc. Am. 60, 794

(1970).~3K. I,. Kliewer and R. Fuchs, Phys. Rec. 172, 607

(1968).~40. Ford and S. %'erner, Phys. Rev. 8 8, 3072 (1973).'5F. Crowne, Ph.D. thesis, Brown University, 1974 (un-

published).'sA. 8agchi, Phys. Rev. 8 +5, 3060 (19/6).'7R. Courant and D. Hilbert, Methods of Mathematical

Physics, (Interscience, New York, 1953), Vol. I, pp.138—140.

~~%. Kohn and P. Hohenberg, Phys. Rev. 136, 864(1964).

linear response functions for an inhomogeneous electron gas in the hydrodynamic approximation

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