Some surface effects in the hydrodynamic model metalstotuvach.free.fr/Articles/barton79.pdfSome...

Rep. Prog. Phys., Vol. 42, 1979. Printed in Great Britain

Some surface effects in the hydrodynamic model sf metals

G BARTON

School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, Sussex BNI 9QH, UK

‘All too often is heard the alibi that since the theory itself is only approximate, the mathematics need be no better. In truth the opposite follows. Granted that the model represents but a part of nature, we are to find what such an ideal picture implies. A result strictly derived serves to test the model; a false result proves nothing but the failure of the theorist. T o call an error by a sweeter name does not correct it. The oversimplification or extension afforded by the model is not error: the model, if well made, shows at least how the universe might behave, but logical errors bring us no closer to the reality of any universe.’

Truesdell and Toupin (1960)

Abstract

Heuristic guidance to image and van der Waals potentials and to parts of the surface energy CT should be furnished by the continuum-hydrodynamic model for metallic electrons, with assigned plasma frequency wp and sound velocity p, with an impenetrable barrier at the surface, and neglecting relativistic retardation. But such uses of the model when p>O have been obscured in recent years by errors about: (i) the orthogonality relation between normal modes; (ii) the Coulomb potential, outside the medium, due to continuum (‘bulk’) modes; and (iii) the enumeration of modes, and the likely dependence of the wavenumber cutoff on system size. This review aims mainly to disencumber the proper heuristic functions of the model from such misconceptions, avoidable in the light of the classic papers by Bloch (1933)) Jensen (1937) and Samoilovich (1945).

The boundary conditions and linearised differential equations are established without cutoff; they determine the normal modes and orthogonality relations. The model is quantised through its normal modes, and the equal-time commutation rules are discussed. The equations in Fourier space are found; it is argued that a cutoff, if required, should be imposed on the Hamiltonian in this representation and before diagonalisation, and the consequences are explored. With such a cutoff, surface though not bulk modes become dispersive even when p= 0. The formalism is applied briefly to image potentials, and in more detail to the attraction between two half-spaces; the role of bulk modes (when p>O) is stressed; the asymptotics are discussed at long and short distances.

Finally (T is calculated through the zero-point energy, strictly according to the 00344885/79/060963 + 50 $05.00 0 1979 The Institute of Physics

65

964 G Barton

model, and in the simplest case p= 0, which suffices to exhibit the crucial arguments. Magnitude and sign both depend on just how the cutoff Q is imposed. Defining (wavenumber parallel to surface) k, one has (surface-plasmon frequency) &(k) = wp[l - 7-1 cos-1 (k /Q)]1 /2 , and U = (27)-2Jd2k~(k) . If Q varies with system size L so that the number of modes is strictly proportional to volume, then ~ ( k ) = o~(k) = @[!&(k) - up] ; if Q is strictly independent of L, then o(k) = U&) = Bti[Qs(k) - 3wp/4]. The popular formula ~ ( k ) = UIII(~) = &%[fZs(k) - &up] is usually found by an incorrect enumeration of modes, adducing transverse modes which in metals (as opposed to insulators) do not exist, though formulae akin to a111 can be obtained by suitably defined cutoffs in the ‘long-wavelength approximation’ Qs(k)+ Rs(0). But in fact a111 relates more naturally not to U but to the energy needed to separate two half-spaces initially in contact; in the model the separation energy differs from U because the barriers at the two surfaces affect the energy even when they touch. Neglect of this difference explains why in certain limits some calculations starting from many-body theory obtain 0111 rather than UI or UII.

This review was received in August 1978.

Some surface eflects in the hydrodynamic model of metals 965

Contents

1. Introduction . 2. The model without cutoff in a half-space: normal modes, quantisation and

image potentials . 2.1. Formulation of the model . 2.2. Normal modes . 2.3. Quantisation . 2.4. Orthogonality and equal-time commutators 2.5. The non-dispersive limit . 2.6. Status of the boundary condition 2.7. Length scales and relation to relativistic calculations 2.8. Image potentials. .

3. Fourier-space integral equations and cutoff . 3.1. Integral equations in Fourier space . 3.2. Cutoff . 3.3. Non-dispersive limit . 3.4. The decoupling fallacy .

4.1. Introduction . 4.2. Bulk-mode phase shifts and frequencies in a single half-space I

4.3. The normal modes of two separated half-spaces 4.4. The van der Waals potential 4.5. Non-dispersive limit ,

4.6. The squared-frequency sum rule for separated half-spaces

5.1. Introduction . 5.2. Ideal gases . 5.3. Surface energy of the plasmon gas with mode-conserving cutoff 5.4. Other prescriptions for the cutoff . Acknowledgments . Appendix. Surface energy and surface correlation energy with /3 > 0 References .

.

4. van der Waals attraction between half-spaces .

. .

. 5 . Surface energy .

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1002 1004 1007 1012 1012 1014

Some surface efects in the hydrodynamic model of metals 967

1. Introduction

It seems exceptionally hard to devise, for the electron gas near a metal surface, a theory which is both realistic and yet reasonably transparent; witness the demand for simple models which has not noticeably diminished even though rigorous treatments have been developing for some time (see, for instance, Lang 1973, Budd and Vannimenus 1973, Kohn and Lang 1975, Vannimenus and Budd 1977). While such treatments are undoubtedly essential for even a qualitative understanding of the surface energy, one expects intuitively the one widely exploited model, namely the hydrodynamic (HD) modelt, to be a valid heuristic guide to those surface effects which, like image and van der Waals potentials, should be dominated by long-range (Le. collective) processes. Unfortunately, the (limited) insight afforded by the model has become obscured by a profusion of mistaken and of easily misunderstood statements in the published literature, especially about the enumeration of the normal modes and about the effects of spatial dispersion on the contributions of bulk modes to surface effects. (Discrepancies both tacit and explicit between many papers show that not all can be correct.) This confusion has arisen because the model by itself tends to be treated by needlessly advanced methods apt to invite misuse, and when embedded in more realistic calculations its own simple but non-trivial physics is insufficiently respected. Moreover, when the model is unambiguously defined and its predictions are evaluated correctly, some appear paradoxical at first sight, and further confusion arises because calculations of the surface energy of the model, if performed erroneously rather than correctly, can lead to results agreeing better, rather than worse, with the properties of real metals.

Here we hope to rectify matters by discussing the HD model in its own right and in its own terms, carefully, but as simply as possible. The model should then be better able to serve either of its main purposes, namely to act as a rough heuristic guide to long-range effects, or to illustrate certain limited aspects of more realistic and therefore much more complicated calculations. Accordingly the present review differs in flavour from most others in this journal; it is narrowly focused on the HD model, regarded partly as an exercise in mathematical physics, and it does not attempt to give a balanced account of the physics of real metal surfaces. It is further restricted to the three effects already mentioned (namely image potential, van der Waals potential and surface energy, excluding in particular light reflection and energy losses by fast charged particles), to systems at zero temperature, and to a wholly non-relativistic treatment (unretarded Coulomb forces). Apart from some seminal early papers, practically all the work which will concern us has appeared

f Some nomenclature: by the HD model we mean the model defined in 92; it treats the electrons as a continuous fluid. Its normal modes subdivide into bulk modes whose amplitudes extend throughout the fluid, and surface modes whose amplitudes decay exponentially towards the interior. The quanta of these modes are called plasmons. The hydrodynamic pressure of the fluid is responsible for the spatial dispersion, i.e. for the variation of the mode frequencies and of the effective dielectric function with wavenumber. (By contrast, the dielectric function does vary with frequency even when pressure effects are ignored.) Pressure and spatial dispersion vanish together in the non-dispersive limit. Dissipation, i.e. ohmic conduction and plasmon damping, is neglected, but the plasma frequency tends to infinity in the perfect- conductor limit, so called by analogy because the fluid then excludes electric fields completely.

968 G Barton

since about 1970. As a rule we start each topic with the appropriate calculation for the model, in enough detail to allow comparison with other work, which we regard as of the essence because of the widespread discrepancies already noted; but there will be only brief asides about comparisons with real metals. The references do not aim at completeness; they have been chosen mainly to facilitate access, to help avoid certain standard pitfalls, or because they themselves contain useful reviews of or references to other work.

The hydrodynamic model was introduced by Bloch (1933) as a generalisation of the hydrostatic Thomas-Fermi theory, and applied to the energy lost by fast charged particles to heavy atoms, initially without trying to calculate the normal modes of the electron gas directly from the model. Jensen (1937), simplifying further, considered an electron gas enclosed in a rigid spherical container. He effectively neglected the central potential but allowed for an inert overall-neutralising background charge, and pointed out that this treatment extends naturally to small but macroscopic metallic particles. He determined explicitly those normal modes which have non-zero dipole strength, elucidating the essential qualitative features of bulk and surface modes in the HD model; read 40 years later, his paper can still dispel misconceptions created by intervening publications.

As is well known, our present understanding of bulk plasma oscillations stems from the work of Bohm and Pines in the early 1950s, which anchored it firmly to first principles, i.e. to the quantum theory of the interacting electron gas. Exceptionally lucid accounts are given, for instance, by Raimes (1957), Pines (1964), Fetter and Walecka (1971) and Doniach and Sondheimer (1974). The classical-continuum model is given a textbook treatment by Jackson (1975). These ideas are confirmed dramatically by measurements of the energy lost by fast charged particles traversing metal foils, much of which can be accounted for by the excitation of discrete plasmons. However, similar studies reveal just as directly that, in addition to bulk plasmons, bounded metals also support surface modes. The modern theory of these starts from the prediction by Ritchie (1957) that surface plasmons would be excited in such experimentst (see also Ferrell 1958). Here we shall not discuss applications to energy losses further. For a more recent and clear theoretical treatment see Feibelman et a l (1972); for experimental aspects see Raether (1965) and Lucas and ,”$ (1972). A brief history of surface plasmons more generally is given by Ritchie (1973).

The theory from first principles is much harder for surface modes than for bulk modes. It remains the object of research, and we shall not document it systematically, though we shall, later, give references where results from the HD model can illuminate aspects of such more ambitious work. In particular, the damping of surface plasmons, which here we neglect, is governed by mechanisms more effective and more intricate than the Landau damping of bulk plasmons: see for instance the early and especially clear papers by Ritchie and Marusak (1966), Wagner (1966), Feibelman (1968) and Heger and Wagner (1971) (the disagreement between the last two papers is

t Basically, the earliest modern papers dealt with the HD model, neglecting the effects of retardation on the normal modes of the medium. This makes sense because the medium is active, in that it supports longitudinal oscillations representing degrees of freedom additional to the transverse oscillations supported by the (relativistic) Maxwell field. Remarkably, the problem of how light is reflected from the surface of such an active medium was not satis- factorily solved until the allied papers by Sauter (1967), Forstmann (1967) and Sturm (1968), completing an argument begun by Fresnel in 1832, missed by Maxwell, and for passive media settled by Helmholtz and Lorentz. For the beginning of the story see Whittalter (1951).

Some surface ezects in the hydrodynamic model of metals 969

instructive). Another topic we shall not pursue is the generation of surface plasmons by light incident on imperfectly flat metal surfaces (see for instance Elson and Ritchie (1971); by contrast, light incident on perfectly flat mirrors is dealt with by the papers mentioned in the last footnote).

Sections 2 and 3 formulate the HD model. They are self-contained and detailed enough to underpin the applications, and to allow a coherent discussion of previous work. Section 2 alone should just suffice as an introduction to those problems which do not absolutely require a cutoff, namely the coupling of an external ion or molecule to a half-space (summarised briefly in $2.8), and the attraction between two half- spaces ($4). But $S on the surface energy, which does require a cutoff, exploits most of the ideas from $$2, 3 and 4.

2. The model without cutoff in a half-space: normal modes, quantisation and image potentials

2.1. Fovmulation of the model

Our model aspires to resemble an electron gas as regards the latter’s collective (long-range) aspects but not its particle aspects?. In this section we formulate it without imposing a cutoff. It is then taken to consist of a strictly continuous fluid governed by differential equations. (In $3 we shall re-express these as integral equations in Fourier space, in which form they allow a cutoff, if needed, to be imposed in a sensible fashion, and the physics to be modified accordingly.) The fluid has mass density m(n + An), charge density e(n + An), plus an immobile, uniform, overall- neutralising background charge distribution of charge density - en, both extending throughout the half-space z 6 0. All equations will be linearised in the deviation An from the equilibrium value n, or equivalently in the displacement !(r) of the fluid from equilibrium:

An= -n div 5. (2.1)

The deviation AP of the hydrodynamic pressure from its equilibrium value is

where /3 would be the sound velocity if the medium were neutral. It is introduced here as a parameter of the model, responsible for dispersion. Two alternative values are commonly considered; each stems solely from the kinetic energy of the degenerate electron gas, neglecting exchange and correlation energies. In terms of the Fermi velocity VF = Q p / m , the value /32 = 3 ~ ~ 2 1 5 is appropriate when high frequencies play the leading role (of the order of the plasma frequency), so that the Fermi gas fails to reach internal equilibrium in one oscillation period; while /32= v$/3 is appropriate at low frequencies or for static screening. This distinction was discussed by Bloch (1934) and is clearly set out by Jackson (1975). We chose the larger value, since high-frequency aspects will be stressed more often.

t Experts might call it the ‘hydrodynamic (or classical) approximation to the jellium model with impenetrable-barrier boundary conditions’, where ‘classical’ emphasises that the model lacks an analogue to the effects of quantum-mechanical interference on electron waves reflected from the barrier. I t has also been called the ‘dielectric approximation’.

970 G Barton

We neglect relativistic retardation, taking the limit c+co from the outset (see 92.7). Then the electric field inside the plasma is governed by Poisson’s equation?

E = -grad @, V2@ = 4rrne div E,. (2.3) The force per unit volume is (neE-grad A P ) , and for normal modes having the time dependence exp (- iQt) the equation of motion reads

- QZmE, = - grad (e@ - mp2 div E,). (2.4) Accordingly we can without loss of generality derive g from a displacement potential Y:

and chose to work with Y as our basic variablef. Substituting (2.5) into (2.4) we can immediately integrate the result once (the integration constant having no physical significance) and obtain

E, = -grad Y ( 2 . 5 )

cI> = - (.lie)( 0 2 + p2V2)Y (x < 0) . (2.6) Acting on both sides with V2 and using (2.3) we find the basic differential equation for the plasma normal modes:

V2( Q 2 - wp2 + /32V2)Y = 0 ( 2 . 7 )

wp2 = 4rne2/nz. (2.8)

V2@ = 0 (x > 0). ( 2 * 9)

where the plasma frequency wp is defined by

Outside the plasma, 0 obeys Laplace’s equation

It remains to specify the boundary conditions. At infinity these rule out exponential increase. At the surface we adopt the only condition which is natural in the HD model, namely that the normal component of displacement (and hence of velocity) vanish there:

f z = o = ay/ax at z=O-. (2.10(a)) This is just the condition introduced explicitly by Sauter (1967). By virtue of (2.4) and (2.3) it is equivalent to the mixed condition

8/8x(e@ + m,B2V2Y) = 0 at Z = O - . (2*10(b)) Equations (2.10) allow the volume charge-density to remain non-zero at the surface; but because of the finite hydrodynamic pressure no true (singular) surface charge- density can arise, whence the conditions on @ are

@ and a@/& continuous across z = 0. (2.11)

Though (2.7) is a local differential equation for Y alone, we stress that it cannot be solved directly as such, for lack of enough local boundary conditions involving

.i. We take the dielectric constants both of the immobile background and of the exterior as unity. Kornyshev et al (1977) have discussed other cases, where the image potential can become repulsive at small distances.

$ This is equivalent and marginally preferable to the more traditional velocity-potential ; but it is incomparably simpler than working directly with the density fluctuation - div 5, as is often done in order to preserve formal analogies with the methods of many-body theory.

Some surface effects in the hydrodynamic model of metals 971

only Y and a finite number of its derivatives. But enough non-local boundary conditions on Y" do result if @(Y) in (2.10(b)) is expressed by Coulomb's law as Jdy' I Y - r' I -lneV'ZY(v').

Writing r=(p, x), with p a two-component vector parallel to the surface (and k, below, likewise), translational invariance parallel to the surface allows the individual normal modes to be written in complex notation as

Y= exp (ik. p)+(x), @=exp (ik.p)+(z). (2.12)

Inside the plasma, (2.6) and (2.7) then become, respectively,

( - k* + d'/d.')( Q2 - u p 2 + pzkz - p 2 d'/d.*)$ = 0 (.GO) (2.14)

while outside the plasma we have, from (2.9),

C$ = constant x exp ( -ax) (x > 0). (2.15)

Finally, the energy H (not yet a Hamiltonian) is the sum of kinetic energy (with density gnmtz), hydrodynamic compressional energy (with density .I,nmpZ(div E,)2),

and electrical energy (expressible through a density -$ne@ div E,), so that

H = J?., dZppSO_, dx[$nm(VY)2fgnmp2(V")2+ane@V2'~]. (2.16(a))

From now on we do not indicate these integration limits explicitly, and denote volume integrals over the interior, like those in (2.16(a)), simply by Jdr. An equivalent expression? for H in terms of Y alone, and hence non-local in form, is

W = dv[$nm( V Y ) z + Jnmp2(V2Y)z] + 4 VzY(r)V'2Y(r'). (2.16(6)) s The model ignores, by way of definition, all energies associated directly with the immobile background or with the surface barrier (though the boundary conditions enforced by the latter, once it is erected, do play a central role). We shall adhere to this strictly even when considering cleavages of the medium in $5; this limitation must be remembered before exploiting predictions of the model as guides to expecta- tions about real metals.

2.2. Normal modes

Equations (2.10)-(2.15) entail that for each value of k there exists one discrete (normalisable) 'surface' mode having R2 < (wp2+ p2k2), plus a continuum of 'bulk' modes (subject only to &function normalisation) having Q 2 2 (wp2+ p2k2). We proceed to determine the normal-mode amplitudes; their orthogonality properties will be discussed in $2.4.

I- Yet another form of the energy results if in (2.6) we replace ClZY'by - and then substitute it for @ in (2.16(a)) ; but because it involves the second time derivative, this is useless as a Hamiltonian and hence useless in quantum mechanics.

972 G Barton

2.2.1. Surface modes. Ruling out exponential increase as z-+co, the only+ solution of (2.14) and (2.10(a)) when R2<(wp2+/32k2) is

$ k ( Z ) = N k [ p S exp ( k z ) - k exp ( P S Z ) ] (2.17)

where Nk is a norming constant to be chosen later, and where, after setting R2 = Rsz(k), p s is defined by

Rs2(k) = u p 2 + p2k2 - p2ps2. (2.18)

The eigenvalue !& is determined, via ps, by using (2.17) together with the equations (2.1 l), (2.13) and (2.15) for 4. One obtains straightforwardly

2 W p s - wp2(ps + k) = 0 and eventually$

(2.19)

Rs2(k) = 3[wp2+ P2k2+ pk(2wp2 + p2k2)1/2] (2 20(a))

(2*20(b)) (2.21)

pps = *[ - /3k + (2wp2 + p2k2)lq

4k = - Nk(m/e) exp ( - kz)(Qs2ps - up") (Z20).

For x < 0, q3k is given by (2.13). Notice that (i) Rs(k= 0) = w p / d 2 ; (ii) QS has a term linear in k for small k (in contrast as we shall see to the bulk-mode frequencies); and (iii) RS increases monotonically with k.

As far as this writer knows, the dispersion relation (2.20(u)) was first written down explicitly by Ritchie and Wilems (1969), though it is implicit in expressions given by Ritchie (1963) for a finite slab; it is also obtainable as the non-relativistic limit of a far more involved expression given implicitly by Sauter (1967) and discussed lucidly and at some length by Sturm (1968).

2.2.2. Bulk modes. In a truly semi-infinite plasma it is clear that all R2> (wp2+P2k2)

are eigenvalues. In 994 and 5 we shall be concerned with somewhat subtle size- dependent deviations from this statement, but here our only task is to find the (&function normable) normal-mode amplitudes. They will be labelled by a continuous index p in addition to k. Accordingly, we set RZ= Q B ~ , where

RB2(k, p ) 3 u p 2 + p w + p 2 p 2 (2.22)

defining p to be "negative. One finds straightforwardly

(2.23)

t That only one surface mode occurs is a specific consequence of the very idealised surface profile adopted in our model. With more elaborate profiles there might be more (Bennett 1970, Eguiluz et al 1975), though as far as this writer knows no experiment suggests more. By contrast, the one mode we are about to find exists for very general reasons (see for instance Feibelman 1971).

1 Equations (2.18) and (2.19) by themselves have another solution, namely p s = k, 0 s = up. However, substitution soon reveals that the corresponding expressions for 6 violate the continuity conditions (except when P2k2 = wp2/4, when this extra solution accidentally coincides with the true surface mode). Coupled differential equations frequently generate such mirages which satisfy one of the derived uncoupled equations but not the original equations themselves.


where M k p is a norming constant to be chosen later. A useful alternative to (2 .23) is

$ k p = A k p COS [pz- s ( k p ) ] + B k p exp (Kx) ( 2 . 2 5 )

where the values of the constants A and B are irrelevant, and where the phase shift 8 is defined by

the principal branch of the arctangent being meant. Note that 8 is non-negative, and

6(k, 0) = in-, S(k, 00) = 0. ( 2 * 26(b))

2.2.3. Theye w e no tvnnsvevse modes. Equation (2.4) probably makes it obvious that in our model all motion (i.e. every mode with non-zero frequency) is longitudinal?. Nevertheless we sketch a simple explicit argument, since the modes are sometimes miscounted, as we shall see in $94.3 and 5.4. The proof proceeds by showing that even if (2.5) is initially replaced by the apparently more general expression

5 = -grad Y* - curl A

the curl can be reabsorbed by a gauge transformation on ‘1’. As before, ‘3’ and A are non-zero only for z<O, and we chose a gauge where div A is strictly zero. Then At(O-)=0 in order to avoid a S(z)-proportional singularity of div A on the surface. By taking the divergence of (2.4) we deduce for Y the same equation (2.7) as before, and by taking the curl we find V 2 A =O. For the normal-mode amplitudes this, together with div A = 0 = A,(O -), leads to

A = ( - k y , h z , O ) f e x p ( i k . p + k z )

where f is a norming constant, while Y! is written as in (2.12). But now g can be expressed as

5 = -grad [exp (ik . p)( 4 - ikf rxp (kz ) )]

so that the argument from this point on proceeds exactly as it would from (2.5) and (2.12), with the combination $‘= $-ikfexp (kz) playing the role which $ played before.

Accordingly, there exist no modes other than those found in $92.2.1 and 2.2.2 above. This argument is independent of the value of and continues to hold in the non-dispersive limit /?+O discussed in 92.5.

2.3. Quantisation

The model is quantised by the standard elementary method of rewriting the displacement potential Y as a Hermitean field operator in terms of the complex normal modes introduced above :

( 2 . 2 7 )

where ‘HC’ stands for Hermitean conjugate, and where the coefficients ak and U k p

Y = J dk exp (ik. p)ak$k(z)+ J dkJ dp exp (ik. p)akp$kp(z) + HC

.i- Transverse motions do occur if relativistic retardation is allowed for, or if, as in many models of insulators, the medium is taken to contain oscillators with restoring forces that are not purely electrostatic. The normal-mode structure of such systems, necessarily much more elaborate than ours, is described by Ruppin and Englman (1970), though some further effort is needed to apply their results to our model. Schram (1975) gives an especially clear discussion of a relatively simple- model for insulators which reduces to ours in a transparently obtainable limit. Lucas and SunjiC (1972) also enumerate the modes correctly.

974 G Barton

are annihilation operators required to obey the usual commutation rules?, e.g.

[Uk, U&’+] = 6(k’ - k )

[Ukp , uk’p’+] = 6(k’- k)6(p’ - p). (2.28)

Finally we chose the norming constants in (2.17) and (2.23) so that the expression (2.16(a)) for the energy turns into the Hamiltonian operator

H = fdkQS(k)(ak+uk + i) + JdkjdpQB(k, p)(ulcp+akp + 4). (2,29)

We chose units that are Gaussian and such that R = 1, and adopt the convention that integrals over p run from 0 to + 00 unless otherwise stated, while integrals over both components of k run from - CO to + CO. The zero-point energies in H will play no role until $$4 and 5 .

After some algebra one finds

(2.30)

Then, by virtue of (2.21), (2.19) and (2.24), the electric potential outside the plasma becomes, and this is one of our central results,

4- HC (2 B 0). For z < 0, @ is given by (2.6) and (2.27).

canonical form in terms of the operators defined by Two other expressions for the Hamiltonian prove useful later. The first is the

Pkll = iSIu1/2( - a-kp + akp+) (2.33)

and similarly for the surface modes (without the index p and replacing QB by as); this yields

x k p = ( a k p + a-kp+)/(2QB1/2),

[ x k p , pk’p’] =i6(k-k’)S(p-pf)

H=Sdk($[j4k+p1,4- Qsz(k) xk+xk)

+ J dkS dp( Pkp’ p k p + QB ( A, P)&p”Xkp). (2.34)

The corresponding expansion of CD for z 2 0 is obtained from (2.32) by replacing ak+2Zls1izXk, U&p-+2ZlBli2Xkp and omitting the addend (+ HC).

The second alternative form of H results from expanding Y (and Y) not in terms of the exact modes for a half-space as in (2.27), but in terms of the complete set of functions exp (ik. p) cos (pz) which, though they satisfy the actual boundary condition (2,10(a)), would, if extended throughout all space, coincide with the

f We discuss in $2.4 why one cannot conveniently follow a procedure more tempting at first sight, namely to adopt ng and nm4 as canonically conjugate vector-density operators.

Some surface efsects in the hydrodynamic model of metals 975

x-symmetric modes appropriate to an infinite rather than a semi-infinite medium. We shall call these the cosine modes (C-modes) in order to distinguish them from the exact modes for the half-space. It proves convenient to adopt canonical expansion coefficients analogous to the operators in (2.33) :

d p exp (ik.p) cospxXkpc - 1”’ (2.35) (x3nm(kZ + p2)

d p exp ( - ik. p) cospxPkpc

[Xkp’, Pk’p‘’] = i8(k - k’)8(p - p ’ ) (2.37) where the normalisations have been chosen so that after substituting from (2.35)- (2.37) into (2,16(b)) some manipulation yields

dp[a[FDkpcf pkpC 4- aB2(k, p)xkpcfxkpc]

Section (3.1) will demonstrate that (2.38) is unitary-equivalent to (2.34). For an infinite medium only the first integral would be present in (2.38), identifying QB (equation (2.22)) as the plasmon dispersion relation in such a system, and the C- modes as its normal modes. In (2.38) the two terms involving wp2 embody the Coulomb energy; the second of these results from the circumstance that in (2.16(b)) the x and x’ integrations extend from - -CO only to 0. Accordingly it embodies effects of the surface (but not all such effects, as we shall see in §§52 and 5.3).

2.4. Orthogonality and equal-time commutators

The orthogonality relations between normal-mode amplitudes were found by Bloch (1933)Jy. For brevity we label the modes with a single index X (i.e. X=k or X=kp) and define, e.g. for bulk modes, gA= g k p = -grad Ykp= -grad [#kp(Y) exp (ik. p)]. In the following argument, integrals run over the interior (x< 0). When we integrate by parts, the surface integrals vanish by virtue of (2,lO(a)). Using (2.6) we start from

1 - ,BZf du(V2‘€rA*)( V2YA~) = 1 dv( V2YAs) - + QA!2YAt (: dr - @A* + QA’YA+ (V’YA.). =s (: 1

Since the difference between the second and third expressions vanishes, one has

f du (: [( V2YA*)@,! - OA*( V2YAf)] + [!JAZZ( V2YA*)YA! - QA2YA*( V2YAt)] = 0. 1 But the first pair of terms on the left vanishes. To see this, recall that the charge density is neV*Y, define R=(v-u’), and note that

QA(u) = J du’ne(V’WA~(u’))/R.

-t. This paper is often ignored, sometimes with unfortunate consequences which we shall see in $3.4.

976 G Barton

Therefore the first term may be written as

(ne2jm)JS dv dv’( V~Y,*(v>)( V’2YA@’))/R

the second term likewise, and the two cancel identically. Finally, after an integration by parts in each of the two remaining terms, one finds Bloch’s result

Since two modes differing in k are automatically orthogonal under integration with respect to p, we can disregard accidental degeneracy between such modes, and express the orthogonality condition in the form

The equal-time commutation rules (ETCR) are obtained most conveniently? from (2.35)-(2,37). Defining, subject always to x, x’ 6 0:

R E r - v ’ = ( p - p ’ , x-x’), # E v - 7 ’ E ( p - p’, x+x’)

(i.e. F’ is the mirror image of r’ in the surface), one finds

[Y(r), rI(v’)] = -(i /4~)(1/R+ l/a) (2.40)

which implies the local ETCR

[V2Y(v), rI(v’)] = [Y(r), V‘2I’I(v’)] = i8(v-r’) (2.41)

once we note that S(R) = 0, because x and - x’ cannot both lie within the medium. One obtains full formal agreement with the standard ETCR appropriate to a medium filling all space on realising that ( l jR+ l/a) is expressible as (-47r/VZ)S(R), whence (2.40) implies

[<i(v>, nm(j(v’)] = i(a,af/VZ)8(r- r’). (2 -42)

However, this has only symbolic value, because in a half-space the solution of Poisson’s equation written as V-ZS(R) is not uniquely determined until explicit use is made of the boundary conditions at the surface.

The formal ETCR (2.42), corresponding to the orthogonality conditions (2,39), is precisely what one expects for a purely longitudinal vector field E= -grad Y, i.e. for one derivable from a potential. Its attendant complications arise because for such a field the three components of 5 are not dynamically independent degrees of freedom, even though it is 5 rather than Y which is observable. This is why it would have been inadvisable to begin by attempting to impose canonical commutation rules directly on the densities 5 and nm&. The situation is familiar from the standard quantisation of the Maxwell field in the Coulomb gauge (see, for example, Power 1964, Bjorken and Drell 1965).

f Relying on the unitary equivalence to be demonstrated in $3.1.

Some surface efects in the hydrodynamic model of metals 977

2.5. The non-dispersive limit One of the features which saves our model from triviality is that for p > 0 its

dielectric response is 'non-local', in the sense that the electric displacement at a point Y depends on the electric field in a finite neighbourhood of r and not only at r itself. Correspondingly the dielectric function commonly assigned to the medium

K ) = 1 - wp2 (U2 - p q - 1

depends on wavenumber as well as on frequency+. As pointed out in passing by Gerlach (1969) and then forcefully by Heinrichs (1975a), this removes the system from the category covered by Lifshitz's fluctuation theory, at least in the latter's native form (Lifshitz 1955, Dzyaloshinskii et al 1961). In the limit P-+O some calculations indeed simplify; but it can impede the correct enumeration and classi- fication of the normal modes, and even if taken is often better postponed to a late stage. (Eguiluz (1978) discusses the additional non-trivial problems encountered when the limit P - t O is taken for a slab of finite width rather than for an infinite half-space.)

For surface modes, (2.20(a)) shows that Qs= cop/@ for all K . Next, one finds from (2.20(b)) and (2 .30) that ps-+co, Nrc->O, but (Nkps)-,(w,mnKn242/2)-1/2; hence, by (2,17), +k-t(Nrcps) exp (Ax). But this result is deceptive as regards the charge density, which becomes singular in the limit:

pk(z) exp (ik. p)-neV2[+k exp (ik. p)] = -nekNk(--K2+ps2) exp (psz+ik.p)

+ - mk(Nkps )ps exp ( p s ~ + ik. p)

-+ -(KwP/16n3\/2)1/26(z) exp (ik. e) (2.43)

where it is clear from the context that the 6 function is wholly within the medium (10- dz6(x) = 1). Finally, (2.21) yields + k ( x > O)+ - (wp2m,Qe)(Nkps) exp (-Ax).

For bulk modes, (2.22) shows that Q B = w ~ for all1 k and p ; (2.31) gives Mkp2-+p2[4n3mnwp(K2 + p 2 ) 2 ] - 1 ; then by (2 2 3 ) the charge density remains non- singular and, by (2.24), 4 k p ( x > 0) = 0; in other words, electric potentials and electric fields of bulk modes vanish outside the medium in the limit P = O . We shall refer to this as the decoupling of bulk plasmons from the exterior. (Note that both $kp and 8$kP/2x now vanish on the boundary.)

Remarkably, in the limit P+O (and only in this limit) the orthogonality condition (2.39(b)) can be rewritten in the very different form

J dr@,"V2'?',. = 0 for X # X ' when p=O. (2.44)

.f- Here K is a three-dimensional wavenumber which in our applications will enter as (P+ p2) . The limit p=O, where the K dependence of E is lost, is sometimes called the 'local approximation', but elsewhere in this review we avoid this description, since even for p>O the system obeys the local differential equations of $2.1 and the local ETCR of $2.4. Therefore, we reserve 'non-locality' to describe the consequences of the wavenumber cutoff introduced in $3.2 below. The pole of EHD at o2 = can be thought of as a vestigial representation of particle-hole excitations in the neutral Fermi gas, which the HD model evidently treats as phonons (see also $5.2) . The non-dispersive limit is also known as the 'high-frequency' approximation (somewhat misleadingly from our point of view) because for given K the term p2K2 in the denominator of EHD becomes negligible as o2 -+ CO.

$' Thus, for both surface and bulk modes the group velocity vanishes, so that energy cannot propagate. Conversely, for p > 0 energy can propagate both along the surface and into the interior.

978 G Barton

In other words the electric potential of one mode is orthogonal to the charge density of another. The equivalence of (2.44) to (2.39(b)) is suggested by (2.6), which shows that, for P = O , and YA become proportional, so that YA* in (2.39(b)) may be replaced by Qn*, Because we have seen that V2Y develops a surface singularity in the limit P+O, one might have reservations about this argument; but (2.44) can also be verified directly in the limit P = O , as follows. (i) Modes with different k are obviously orthogonal under Jdp. For given k, (ii) the surface mode is orthogonal to all bulk modes, because the charge density of the former is proportional to 6(x), while the potentials of the latter all vanish at the surface, as we have seen. Finally, (iii) for two bulk modes, no singularities occur, so that the above argument through the proportionality between 6, and Y is unexceptionable.

Perhaps it is worth noting that straightforward integrations with the explicit normal-mode amplitudes (2.17) and (2.23) bear out (2.39), and also bear out the breakdown of (2.44) when P > 0. To summarise, the orthogonality condition may be expressed either as (2.39) or (2.44) when P= 0, but when ,B > 0 it generalises only in the form (2.39).

The Hamiltonian in any of its forms is of course unaffected by the limit P-+O except as it governs the values of the Oh. But the external potential (2.32) reduces

( ~ 2 0 , ,8=0). (2.45) to

6, = - J dk exp (ik. p - Kx)ak( wP/4.rrd2 k)1/2 + I-IC

Section 3.4 will discuss the probable technical roots of the two widespread and allied misconceptions that bulk modes decouple from the exterior and that the orthogonality rule (2.44) applies for all values of P (and not only for P = O ) . Historically they may be rooted in an unkind double bluff by the model. On the one hand, the general rule (for P > O ) is clearly that all modes have external potentials. On the other hand, in the special case ( P = O ) , the burden of proof is reversed, because now it is only some special (surface) modes, distinguished by a singular surface charge-density, which have such potentials, while the (infinitely more numerous) bulk modes have none. Hence what one might want to explain in this special case is why a singular surface charge-density is needed to allow external potentials to survive in the limit. This point must have been clearly understood by Jensen (1937), who provides just such a qualitative explanation, which we shall not repeat in detail. I t starts by noting from ( 2 . 6 ) that for P = 0, @ becomes proportional to Y ; then argues from the boundary conditions (2.10) that a,@ must vanish at x = 0 - unless the charge density, proportional to V W , has a singularity there ; and finally that by Gauss's theorem the vanishing of a*@ at 0 - implies complete decoupling.

2.6. Status of the boundary condition

The condition &(O - ) = 0 represents an impenetrable barrier precisely at the edge of the background charge distribution, and is plainly unrealistic. We adopt it (without further apologies beyond this subsection) because it yields a tractable problem and has a distinguished history as a first approximationj-. One basic limitation of the model is that it involves only a single length scale ,8/wp, which is not very different from those governing other physical effects that are neglected. First amongst these is the quantum-mechanical interference between the electron waves incident on and reflected from the surface, governed by the Fermi wavelength &-I. This entails that the surface barrier is effectively relocated at a finite distance outside the edge of the background charge distribution. The implications for the surface

t For further comments on some of the complications in the real world, see 55.1.


energy will be mentioned in $5.2.2; those for image potentials have been explored by Heinrichs (1973~). Next, even within the HD model, it would be more realistic to allow the fluid to leak out beyond the background distribution until restrained by the resulting electric forces, at distances of the order of the static Thomas-Fermi screening length. Moreover, not only is the background charge density (ionic lattice) granular rather than continuous, but it is itself likely to relax near the surface, so that a more gradual decrease of n should be allowed for; both these effects are likely to be governed by the lattice spacing. In such cases, given an expression for pressure as a function of density trustworthy beyond linearisation around the bulk equilibrium value, one would first determine the equilibrium density profile of the surface layer, and would then attempt to set up normal modes by linearising the equations of motion around this profile. (For better founded attempts in this direction, see for example Zaremba and Kohn (1976, 1977).)

The only feature of the model which is probably immune to appreciable corrections due to such effects is the value of as at Iz = 0. For an electron gas, Feibelman (1 971) has shown that under quite general conditions any self-consistent treatment results in Qs(Iz= 0) = wp/2/2, independently of surface details; similar conclusions are reached by Harris and Griffin (1971). Beyond this, however, corrections to the model would have at least two crucial qualitative effects: they would modify the surface-plasmon dispersion relation (2.20(a)), and would alter the coefficient of the annihilation operators in the expansion (2.32) of at. Clearly it would be incon- sistent to allow for only one of these two modifications; in the present review of the HD model we adopt neither. For all these reasons, the short-distance expressions we shall obtain are not to be trusted in detail; they would be appreciably different in any more realistic treatment of the boundary.

2.7. Length scales and relation to relativistic calculations

When the velocity of light is taken as finite rather than infinite, the HD model contains two characteristic length scalesf, namely c/wp and p/wp. Evidently c/wpg/3/wp; hence, as discussed by Sturm (1968) and Buhl (1976), distances of physical interest, like the distance x of a molecule from the surface, or the wavelength h of a surface plasmon, can be classified into three qualitatively different regimes§,

From a mathematical point of view, the most distinguishing peculiarity of the I : z<P/w,; 11: /3/wp<x<c/wp; 111: c/wp<x.

Not surprisingly there are strong experimental and theoretical indications that (2.20(a)) overestimates the dispersion. For an early discussion in terms of the HD model, see Bennett (1970). For recent references, see Duke et a2 (1975) and Forstmann and Stenschke (1978).

$ If a nearby external molecule, say, has excitation frequencies E very different from up, then a third length scale c / E enters. The consequences are readily allowed for.

$This can create a verbal trap. In calculations which treat the electromagnetic field relativistically (i.e. by Maxwell’s equations) but set j3 = 0 (e.g. Babiker and Barton 1976), one calls region I11 ‘long-distance’ and region I1 ‘short-distance’. This approximation is unable to distinguish region I from region 11, so that the properties which in fact are peculiar to region I are lost. By contrast, here we let c --f CO from the outset. Then region I11 and the properties peculiar to it are lost, I1 is called ‘long-distance’, and I ‘short-distance’. Thus the ‘long-distance’ regime here coincides with the ‘short-distance’ regime of the relativistic treatment with p = O ; we have spelled this out to avoid confusion. As pointed out in $2.6, for real metals the predictions of the HD model for short distances must be taken with a pinch of salt.

66

980 G Barton

HD model with simultaneously finite c < CO and p> 0 is that the boundary conditions then unavoidably couple longitudinal (curl E= 0) and transverse (div E= 0) motions, even though the differential equations in an unbounded medium would allow them to remain uncoupled. (This contrasts with the non-relativistic limit which allows no transverse motions even in the presence of a boundary, as we saw in $2.2.3. For a general discussion of such problems see Morse and Feshbach (1953)) For bulk modes such coupling merely modifies the amplitudes near the surface, but for surface modes it exerts a crucial influence on the dispersion relation, leading to sharply contrasted behaviour in the two limiting cases. This contrast, perhaps the most remarkable physical feature of the relativistic HD model, has been discussed in detail by Sturm (1968) and again by Forstmann and Stenschke (1978). For ,8 > 0 and c = CO,

we have seen in $2.2.1 that the surface modes are pure longitudinal, with frequencies Qs which approach o p / 2 / 2 from above as k+O, i.e. as the wavelength h-tco. But as h increases, it inevitably enters region I11 where our non-relativistic model breaks down. Detailed calculation shows that under the influence of the surface-induced coupling the character of the mode gradually changes from pure longitudinal at h = 0 to pure transverse at h = CO. In the transverse-dominated region the unretarded Coulomb forces (responsible for the dispersion relation (2.20(a))) must of course be replaced by properly retarded and accordingly less effective interactions. I t then turns out that in the other limit (p= 0, c < CO) the dispersion relation-/- becomes

so that Rs drops to zero as k+O and approaches wp12/2 from below as k - t c o . As shown by Sturm (1968), the non-relativistic formula (Z.ZO(a)) applies when h is in regions I or 11; the relativistic formula (2.46) applies in region 111; and there is a remarkably fast transition between them near ckNoP/2/2 , This softening of Qs (Qs-tO as X - t c o ) is clearly governed jointly by relativistic retardation and by the surface-induced longitudinal-transverse mixing; witness the fact that in an unbounded medium (i.e. with retardation but without a surface) the bulk modes, which remain pure longitudinal at all wavelengths, retain a non-zero frequency wp even for h+co (Jackson 1975).

2.8. Image potentials

An external charge q fixed at (p=O, x) couples to a half-space through the Hamiltonian H' = q@(O, x), with CD the operator (2.32). The leading (second-order) perturbation V of the ground-state energy combines contributions Vs and V B from virtual intermediate states with, respectively, one surface or one bulk plasmon excited.

.f. The result is ascribed to E A Stern by Ferrell (1958). With hindsight, today's reader could extract it from the classic paper by Sommerfeld (1909) on the propagation of radio waves over a Aat conducting Earth. (For an updated version see Stratton (1941).) To do this one must ascribe to the medium the real dielectric function E = 1 - wp2/u2, whereas Som- merfeld was concerned primarily with the effects of ohmic conduction, i.e. with the imaginary part of E . The relevant general result of Sommerfeld is that the response of the medium to a sender with frequency w has a pole at wavenumber s= [k12k$ (kiz + k22)-1]1/2, where ki = w / c and k2= ~ l / Z k l are the wavenumbers in Vacuo and in the medium, respectively. By setting s = k l , using the above form of E , and solving for U, one obtains (2.46).

Some surface eflects in the hydrodynamic model of metals 98 1

One finds straightforwardly (with Qs Qs(k) and QB = ! 2 ~ ( k , p)) :

Vs = - J dkl ( k I H' IO) 1 '/i& (2.47)

As expected, V B vanishes when /3= 0, but not otherwise. After a change of integration variables, VS and VB combine into the expression first obtained by Newns (1965) in an entirely classical calculation :

V = VS+ VB= -(q2Uwp/2p)f(p) (2.45)

As x+m, this gives V N ( - ~ ~ / ~ X + ~ ~ ~ / ~ O J ~ X ~ + . . . ); the leading term, i.e. the perfect-conductor image potential, comes wholly from Vs, while Vs and V B share comparably in the term of 0 ( x p 2 ) . By contrast, Vs and VB are strictly comparable as z+O. For instance, Vs(0) = - (q2wp/3p)2-l/2 and VB(O) = - (q2wp/3/3)( 1 - 2 - 9 .

An external fixed neutral molecule with electric dipole-moment operator d couples to a half-space through the Hamiltonian H " = d . V@. The perturbation U = Us+ UB again subdivides as indicated. Let the unperturbed molecular states and energies be I xn ) and En, with n = 0 the state whose shift is being calculated (not necessarily the ground state). Exploiting the azimuthal symmetry of the k integrations, and defining Eno= En - Eo,

f(P) = .To" dY exp ( - PY>[(Y2 + 1) l j2 - YI2, p = 2wpx/p.

dlno2= 8 I ( xn l dll I XO) I 2 + I (Xn I d , I XO) ( 2 (2.50)

where the suffices 1 1 (I_) denote components parallel (normal) to the surface, one finds :

(2.51) w 4 m

us= - 4 2 d P 2 0 2 1 dkk2 exp (-2kx) -

lis = - 2 2 A n 0 2 1 dkk3 exp ( - 2kx)

-

0 ( G O -t- Qs)Qs(wp2+ 2Qs2) 1%

77 0 12

(2.52)

Each term in these sums clearly depends on two dimensionless variables en,, E EnO/wp and PE 2wpx//3. Hence a systematic treatment of the asymptotics (p+co and p+O) and of the perfect-conductor limit (wp+co) becomes very elaborate and will be given elsewhere (Plaskett and Barton 1979). Qualitatively speaking, the contributions from surface and from bulk plasmons compare as they did in the potential V.

When vanishing denominators are encountered, the Cauchy principal-value prescription is understood. But if the molecule has an excitation very close to the

982 G Barton

limiting surface-plasmon frequency wp/ 1/2, then interesting resonance shifts and splittings can arise which are not covered by the discussion below (see Mavroyannis and Hutchinson 1977, Barton 1978).

Again, Us and UB can be combined into a relatively compact resultant. T o this end one uses the identity+:

(2.53)

Then thep integration in Ug becomes a simple contour integral, which when evaluated combines naturally with Us to yield

U = - (2/rr)(wpip)3 ~ , 0 2 ~ ( E n 0 , p) n

Here we have changed the integration variables to x = f l u p and y = /3k/wp. Remark- ably, (2.54) is the rationalised version of a result obtained by Mahanty and Paranjape (1977) using response-function methods :

(2.55) where y, related by analytic continuation to the function p s of $2.2.1, is given by /3y(k, f ) = (wp2+ /32k2+ f2)112, and where the dynamic molecular polarisability IT, evaluated at pure imaginary frequencies w = if, is

U = - (wp2/477)j0" dflT(if)J; dkk2 exp ( - 2kx)(P2y2+ /3%y - 1, 2 P 2)-1

II(if) = 2 d,o2Eno (E,02 + f 2 ) - 1 . n

For large p, but for any value of E, expansion of the y integrand in (2.54) (apart from the exponential) in ascending powers of y yields Mahanty and Paranjape's result :

By contrast, it is surprisingly difficult to obtain a systematic approximation for small p. The two leading terms turn out to be (J S Plaskett 1978 private communication)

(2.57) Substituting (2.57) into (2.54) we can perform the sum over molecular states by using closure in the first term and the Thomas-Reiche-Kuhn sum rule in the second; this yields, for a single-electron atom (with me = electron mass)

F( E, pu.30) % [rr/16p + ( 4 6 ) log p+ (terms remaining finite as p+O)].

U % [ - ( w p 2 / 1 6 P 2 ~ ) ( X O I ( ~ ~ , , 2 + ~ ~ 2 > 1 X O )

- (up2e2/37rP3me) log (2wpx/P)+(terms remaining finite as x+O)]. (2.58) Several provisos must be remembered when interpreting these results. Thus,

the dipole approximation H " breaks down at distances x comparable to the molecular

t Circumspection is needed when calculating the shifts of excited states, where some of the En0 are negative. For negative E,o (2.53) obviously fails. Consequently, in (2.54) and (2.56) one cannot in such cases simply set E = - I E I. Instead, one must carefully continue these expressions analytically to negative E ; but it turns out eventually that (2.57) and (2.58) below do remain valid. For details see Plaskett and Barton (1979).


diameter; however, except for very large molecules, this is below the distance p/wp where our model cannot in any case be trusted quantitatively. Furthermore, at very small distances where the calculated potential becomes comparable to typical molecular excitations, the perturbation treatment obviously breaks down, and would have to be improved along the lines first discussed by Antoniewicz (1974) for the analogous problem with perfect conductors (up+ CO)?.

Finally, for excited molecular states not only the energy but also the lifetime vary with distance from the surface. This is a much-discussed phenomenon which we do not review in detail, especially because it is closely linked with ordinary spon- taneous decay, which in turn is a relativistic effect governed wholly by the coupling to the transverse Maxwell field. For a recent review of experimental results and of semiclassical theory, see Chance et aZ( l978) . A lucid theoretical account based on the HD model without dispersion (,B=O) is given by Philpott (1975). There are especially marked effects when the light frequency is close to wp/1 /2 . This writer knows of no systematic account of molecular decay embodying both relativity and finite dispersion, nor of any non-relativistic treatment (with C+CO but p #O). At small distances the problem is complicated further by the possibility of direct energy transfer from the molecule to the dissipative mechanisms responsible for ohmic losses in the medium (see for example Lyuboshitz 1977, Persson 1978).

3. Fourier-space integral equations and cutoff

In $2 the HD model was described by differential equations. To compare it with the many-body theory of the electron gas, it must be reformulated in terms of (integral) equations in Fourier space, starting with the expansions over cosine modes, equations (2.35)-(2.38). This is done in $3.1. It is also a prerequisite for imposing a cutoff in a physically sensible manner, a very delicate problem motivated and discussed in $3.2 (though after that we shall proceed without a cutoff until dealing with the surface energy in $5).

3.1, Integral equations in Fourier space

Consider the diagonalisation of the Hamiltonian ( 2 . 3 8 ) subject to the commutation rules (2 .37) . The double integral couples C-modes having a common value of k but all values of p (it couples all Fourier cosine coefficients for the 2;-dependence). Without such coupling H would be a sum of Hamiltonians for mutually independent harmonic oscillators, i.e. it would effectively be diagonal already. Therefore H is effectively diagonalised by diagonalising the matrices of the real symmetric quadratic forms involving the X k p C for given k, by means of a unitary transformation acting on the indicesf p , and leaving form-invariant the terms in H which contain the PC,

+ For more realistic attempts to deal with adsorbed atoms too close to the surface for a simple perturbation treatment, see Zaremba and Kohn (1976, 1977) in the p = O limit; Meixner and Antoniewicz (1976); Linder and Kromhout (1976). For a discussion of experimental data see Le Roy (1976) and Mehl and Schaich (1977). For a recent discussion of plasmon effects on x-ray emission from adsorbed atoms, and for references, see Hussain and Newns (1978) and Gunnarsson and Schonhammer (1978).

1 In a slab of finite width (see $5.3) p would be a discrete index; but here it is continuous, which leads to integral rather than matrix equations. The transition from a finite slab to an infinite half-space is surprisingly delicate (see Eguiluz 1978).

984 G Barton

I t is important that such a transformation corresponds to (i.e. is brought about by) a canonical transformation in Hilbert space, under which the complete set of operators ( x k p c , P k p C } transforms into the alternative complete set consisting jointly of the {Xk, Pk) and the { x k p , P k p } in (2.33)-(2,34), while the Hamiltonian (2.38) transforms into (2.34).

T o discuss the diagonalisation process conveniently, we shall in this section reserve the labels (kp) for the C-modes, label the exact eigenmodes (to be discovered) by kpL, and write their frequencies as QkP (eventually we shall set p = S for the surface mode, and let p range from 0 to CO for the (exact) bulk modes). Define also

g(k, p ) = [ k w p 2 / 7 r ( k 2 + p 2 ) ] 1 / 2 .

x k p = j d P X k p c < k P I )k,c

(3 a 1)

(3 *2>

( 3 . 3 )

Then the transformation being looked for can be written as

and the eigenvalue equation for the QkP2 becomes

rQka2 - - p2(k2 -k p2)1<&' I ) k p = - g(k , p)j dp'g(k p')<kp' I ) k p

Recall also the definition (2.22). Equation (3.3) has the familiar structure of the Lippmann-Schwinger equation in scattering theory ; moreover it is easily soluble because it is separable. Evidently it has a continuous spectrum of eigenvalues Qkf12 > (wp2+p2k2), for which only the eigenfunctions remain to be calculated. The analogy to scattering theory reminds one of two very important facts about functions in the continuous spectrum: (i) they are subject only to &function normalisation, i.e. they are not normalisable (square-integrable) in the elementary sense; (ii) if equation ( 3 . 3 ) is rewritten in configuration space as an integral equation (rather than as the differential equations of 92), then these are necessarily inhomo- geneous, with the inhomogeneity representing a wave 'incident from infinity'. By contrast, an eigenvalue equation for the discrete spectrum with properly normalisable eigenfunctions is obtained in standard fashion by dividing both sides of ( 3 . 3 ) by (Qkp2- Q B ~ ) , multiplying byg(k, p ) , and integrating over p . After cancellation of the common factor Jdpg(k, p)(kp I ) k p this yields

1 = - J dPg2(k, p ) [QkP2 - Q B ~ ( ~ , p)]-'. (3 -4) The integration is elementary and yields the expected unique solution R k p 2 = &*(A) given by (2.20(a)). It must be stressed that for functions of the continuous spectrum (Le. for bulk modes), division by (Qka2- Q$) is not allowed, and that these normal modes do not satisfy (3.4).

Considering (3.3) as a Lippmann-Schwinger equation, we have dealt with it so far by treating the surface-induced interaction on its right-hand side as the 'potential', and Q ~ 2 ( k , p ) as the zero-order Hamiltonian. Alternatively, by analogy with many- body-theory calculations of correlation energies, one can take the term in wp2 over to the right, and treat the new right-hand side as the potential. This then embodies all the terms derived from the Coulomb energy in (2.16(b)) and proportional to wp2, while the new zero-order Hamiltonian is just /32(k2+ $2). Using (3. l), ( 3 . 3 ) is accordingly rewritten as


which is much harder to handle because the new potential is singular and non- separable.

Equation (3.5) leads to two others that are often encountered; we stress that both arise after effectively dividing by [axl? - /32(k2 +p2)], whence it is dangerous to rely on them when discussing bulk modes. For easier comparison with the literature we drop the suffix from Qkp2, define an effective dielectric function for the HD model

€HD(a, k, p ) 35 1 - U p 2 (a2 - p Z k 2 - p2P2)-’

( Q 2 - p2k2 - P ‘ L p W p I ) k , k ( k , PI = F k p ( P )

(kp I ) k , k @ , PI = f k p ( P ) .

(3.6)

(3 7 ( 4 ) (3 7(b ) )

and two new amplitudes by

Then one finds straightforwardly?

Finally, we re-express the discrete-eigenvalue equation (3 .4) in terms of EHD :

The integral over the first term yields 8, whence one obtains

Equations like (3.8)-(3,10) are widespread in approaches to the bounded electron gas through the random-phase approximation (RPA), which is known to describe the unbounded electron gas in the high-density limit. They will be found in many of the papers quoted in later sections. In place of EHD as above such approaches generally introduce the bulk dielectric function (Lindhard function) ERPA of the RPA, to which EHD can then be regarded as a further crude approximation. Ritchie and Marusak (1966) used (3.10) with EHD+ ERPA to discuss the surface-plasmon dispersion relation and damping (see also Wagner 1966). illthough equations (3.8)-(3.10) so modified can thus be arrived at in an apparently more realistic context, their precise relation to a well-defined underlying Hamiltonian is then less clear than in the HI) model.

3.2. Cutofjc

The HD model is meant to illuminate the collective behaviour of an electron gas;

t Equations ( 3 . 5 ) and (3.8) seem ambiguous for the special value p = 0, when the singularity of the 6 function coincides with the lower limit of the p‘ integration. This is resolved by reference back to the unambiguous parent equation (3.3). Alternatively, one could abandon the convention (see 92.2.2) that p is non-negative. Noting that EHD and the non-singular parts of the kernels are all symmetric functions of p, one would define F andf to be symmetric in p, extend the p’ integrals from - CO to + CO, and at the same time halve the non-singular kernel, i.e. replace rZ/v(k2 + p 2 ) by k/27i(kZ+p2)).

986 G Barton

therefore ultimately some kind of wavenumber cutoff Q must become essential, because, obviously, no collective wave can propagate with a wavelength less than the mean interelectron spacing YO, conventionally defined by 4nr03/3 = n-1. Never- theless the strict continuum model without cutoff (i.e. with Q+w) yields sensible answers to some questions, and after this subsection we shall revert to it throughout $4, reintroducing finite Q only in $5 where it becomes indispensable for discussing the surface energy. When a cutoff does become indispensable, it turns out that its consequences depend qualitatively on the precise manner in which it is imposed; though from the mathematical point of view this is a matter of defining the model, from the point of view of any guidance which the results may afford, it is essential to avoid arbitrariness and to motivate the definitions physically as far as possible. There are two especially important points. The first concerns comparability between surface and bulk modes, and the second the size-dependence of Q in finite systems. We comment on the second point, briefly, later in this subsection, and again in $5.

The first point can be appreciated best by asking? how cutoffs should be imposed, consistently, on two integrals contributing (most likely with opposite signs) to the same effect, say a van der Waals potential or a surface energy, one integral three- dimensional and running over bulk modes, i.e. over I& andp, and the other two-dimensional and running over surface modes, i.e. only over k . T o resolve the question one must recall that a cutoff is of itself a source of non-locality, causing the response of the medium at a point to depend on the forces throughout a region having linear dimensions of the order of 9-1. This non-locality in turn affects both the disturbance caused by a surface and the dispersion relation for surface plasmons. This writer believes that the only physically sensible choice is to impose the cutoff on the Hamil- tonian at a stage when it can still be done without tacitly or explicitly assigning a special role to the direction normal to the surface, and before, not after, the exact bulk and surface modes have been identified. In other words, we impose the cutoff not on expressions like (2.27) or (2.29), but on the expansions in terms of C-modes, equations (2.35)-(2.38). In C-mode expansions we restrict all integrations to the region

k2+ pa 6 Q 2 . (3.11)

C-modes violating (3.11) are simply excluded from the Hilbert space of the model. The Fourier-space commutation rules (2.37) are retained; then the 6 functions in the configuration space ETCR (2.41) and (2.42) are smeared over distances of the order of Q-1, the precise details of this smearing being irrelevant.

Of course it is essential for the physics that the cutoff be imposed on the normal modes and not be allowed, instead, to soften the singularity of the underlying Coulomb interaction R-1 at R=O or as R-tco. The cutoff is chosen as sharp rather than gradual because this makes it easy to handle and to appreciate conceptually, though for applications a gradual cutoff might be more realistic; (it would also involve another adjustable parameter).

Though Q is clearly of the order of magnitude of YO-1, we must settle on a precise (if provisional) prescription. There are two obvious choices. The first is inspired by the Debye theory of specific heats, and defines Q so that the number of degrees of freedom (Le. of C-modes) per unit volume becomes equal to the number n of

.1- This question has caused exceptional acrimony: see for instance Phillips (1975) and Kohn and Lang (1975).


electrons per unit volume. (We chose n rather than 3n because only the longitudinal component of the displacement constitutes an independent degree of freedom, as discussed in $2.3 and near the end of $2.4.) Allowing for spin degeneracy this would entailt

Of course the essential feature here is not the precise equality, but the proportionality to QBT with a proportionality constant of the order of unity.

The other obvious choice of cutoff stems from the random-phase approximation in the standard theory of the (unbounded) dense electron gas (e.g. Raimes 1957, Pines 1964, Fetter and Walecka 1971, Doniach and Sondheimer 1974). There, a reasonable separation between collective and single-particle aspects is achieved if plasmon wavenumbers are limited by

Q N wp/vF - 0.903 ~ ~ - 1 1 2 au. (3.13)

This is best thought of as a variational parameter optimised for an unbounded medium, though it happens also to be the wavenumber where the plasmon and the particle-hole excitations merge in energy, and where the Landau damping of bulk plasmons becomes effective. For definiteness (and by hindsight) we adopt the cutoff (3.13) from here on.

Though (3.12) and (3.13) vary differently with density, neither cutoff introduces a radically new length scale into the problem. Nevertheless no cut-off integral over normal modes can diverge, so that even those zero-distance limits of image potentials remain finite, which without a cutoff were in $2.8 found to diverge. This is yet another reason why the small-distance behaviour in the model cannot be trusted in detailf.

More important, we see from (3.12) and (3.13) that plausible cutoffs depend on the Fermi momentum QF, Now in $5 we shall meet the well-known fact that QF has a size-dependent component which is crucial for surface energies; therefore it is reasonable to anticipate that similar size-dependent corrections may also become important in Q.

It remains to explore the effects of a cutoff on the integral equations in $3.1. Those p-integrations where EHD either does not appear explicitly, or appears only in the combination (EH= - 1) or ( E H D - ~ - l), must now be evaluated with the upper limit (Q2-rZ2)1/2; this is obvious from their provenance and makes it clear that EHD must now be defined by

cutoff = 2113Qp N 2-42 rS-l au. (3.12)

EHD 3 1 - Up2 ( Q2 - p2k2 - p2P2)-' for (It2 +p2) Q Q2

EHDG 1 for ( k 2 + $ 2 ) > 92. (3.14)

This is expected physically: since C-modes violating (3.11) cannot be excited, at such wavenumbers the medium behaves like the vacuum. (By contrast, when ERPA

t For later use, note the following formulae in atomic units, where f i = m = e2 = 1, so that the Bohr radius aB=@/me2=1, and where we define m / a B = Y s (TS ranges from 2.17 in A1 to 5.78 in Cs). wp=31/2~s-3i2; Q~=(9.rr/4)1/3rs-1-1.92 vS-1; wp/Ep= [16(4/9?r)113/3.rr]1/2 rs1i2 -0.94 rs1i2; and with p2= 3 v ~ ~ / 5 , j2Q~2/wp2=(g.rr/4)4/3 5-1 Y S - ~ -2.71 Y S - ~ . Also, wPQ$= 3112 (9n/4)213 Ts-712-6*38 78-712, and with Q defined by (3.13), p2Q2/wp2= 3/5,

r Note also that the non-dispersive limit cannot be a quantitatively reliable approximation. For instance, by (3.13) the ratio of maximum to minimum bulk plasmon frequencies is (1 + p2Q2/wpz)li2=(8/5)1/2- 1.27, which is not very close to unity.

wpQ2= 33/2(4/9~)213 rs-512 N 1.41 ~ s - 5 1 ~ .

988 G Barton

is used as in some electron-gas calculations, the integrals need not be cut off explicitly, since ERPA effectively contains a built-in cutoff near Q.)

The surface-plasmon dispersion relation is determined by the elementary integral (3.4) with or without a cutoff. But with finite Q, Qs approaches QB as k+Q:

Qs2(Q) = u p 2 + p2p. (3 * 15)

This can be seen directly from (3 -4): as k+Q, the integration region shrinks to zero and the integral can continue to equal - 1 only if the denominator vanishes. Alternatively, the Hamiltonian (2.38) shows directly that at k = Q the effective coupling term vanishes altogether, so that in particular it cannot then cause a discrete state to form below the continuum+. Instead of quoting the full dispersion relation in tedious detail, we record only its non-dispersive limit:

(3.16)

Notice that with a cutoff !& (unlike QB) becomes dispersive even when P = O . Of course, the cutoff modifies not only the dispersion relation but also the normal-

mode amplitude, and in particular the external potential, causing it to differ from (2.32).

The result (3.14) can create a trap; for instance, if one forgets it, one might evaluate Ritchie and Marusak’s integrals with an upper limit ( Q z - k 2 ) 1 / 2 , which would be correct in (3.9) but is wrong in (3.10). Amusingly, for p=O this would lead to

Rs2= w&1+ (2174 cos-1 (k /Q)] -1

instead of (3.16); these expressions differ in detail though they agree at their end points k = O and k = Q . When Griffin and Kranz (1977) discuss how a cutoff in the HD model affects the relative contributions to the surface energy, from bulk and surface plasmons, they appear to have been caught in just this trap (see their equation (2) and the remarks between their equations (7) and (8)). Accordingly their quantitative results need revision.

3.3. Non-dispersive limit

In this limit, with or without cutoff, the integral eigenvalue-equation (3.3) continues to apply. Though all the bulk modes now have the unique frequency up, they remain ‘continuum’ modes in the mathematical sense of being subject only to &function normalisationr . However, since the zero-order Hamiltonian on the left now ceases to depend on p , one can make some very simple statements about the normal modes, which will prove useful in $5.3. Evidently, the surface-mode amplitude (kp 1)ks is just that superposition of C-modes which is proportional to g(k , p ) , and the bulk-mode amplitudes (kp I )kpp are consequently any such superpositions orthogonal to g (i.e. for bulk modes the integral on the right of (3.3) vanishes); if one wishes to construct them explicitly, one must of course ensure that they are also orthogonal to each other. Lastly one can show without much difficulty that even with a cutoff the exterior potentials of the bulk modes vanish when P = O ; the exterior potential @ and the surface-mode contribution to the charge density

-1- As far as this writer knows, this junction of the surface and bulk-plasmon branches was first exhibited clearly by Grifin et a2 (1974).

1 The infinite integrals responsible for the 6 functions are the underlying configuration- space integrals over the infinite half-space, and not of course the wavenumber integrals over p , which for finite Q run over only a finite range.


n e V V are now given by

+ HC + (bulk-mode contributions) (3.17)

(3.18)

3.4. The decoupling fallacy

The area under review includes two especially foggy topics, identifiable through manifest contradictions in the literature. Those relating to surface energies will be discussed in $ 5 ; here we try to exorcise the double fallacy that, even when P # O , bulk plasmons continue to decouple from the exterior and the orthogonality rule (2.44) continues to hold. This is easier to dispel because it is unaffected by any arbitrariness related to cutoffs, is unambiguously stated, and is known from $92.4 and 2.5 to be simply wrong. Hence it would be redundant to locate the specific errors in every paper promulgating the fallacy. Historically, one induce- ment to error may have been the original demonstration by van Kampen et al (1968) that for /3=0 the entire van der Waals potential W between two half-spaces stems from the surface plasmons, which are then non-dispersive. This might tempt one to jump to the conclusion that the only effect of finite dispersion would be to change the surface-plasmon contribution through its influence on Rs(k), whereas 55 will show that P # O also elicits a contribution from bulk plasmons. These contributions were initially omitted by Heinrichs (1973a), perhaps for want of a clear statement, while using response functions to calculate W (Heinrichs 1975a), that a dispersive half-space has normal modes not decaying exponentially towards the interior. Remarkably, the clear verbal assertion that only surface plasmons contribute seems to have forestalled the recognition, in the mathematics, of the branch cut supplying the contribution of bulk plasmons; the erratum corrects the mathematical error but does not clearly cancel the verbal statement?.

Another influential paper, by Feibelman et al (1972, to be referred to as FDB), may also have been instrumental in innocently eliciting error. In the body of their paper FDB consider, elegantly and at some length, the non-dispersive case (‘high-frequency approximation’), explicitly basing their argument on the orthogonality condition (2.44). In their appendix they explain that in the presence of dispersion this must be replaced by what is effectively Bloch’s general condition (2.39). However, they do not spell out in complete detail how this replacement and dispersion generally affect the charge densities and potentials (which we discussed in S2.5)f. An incomplete appreciation of FDB’s appendix is one possible origin of the unqualified assertion by Inglesfield and Wikborg (1975) that bulk plasmons do not

+ There exists also an ambitious calculation of W by Buhl (1976), which aims to allow, simultaneously, for the effects of relativistic retardation, dispersion and finite temperature. In the appropriate limit the results are stated to agree with those of Heinrichs (1973a, before correction). If so, the argument must contain an error equivalent to his, though the present writer has been unable to locate it because of the complications of the formalism. See also the footnote commenting on equation (4.35).

3 (i) Because FDB start from standard response-function theory, their field variable is not a displacement- or velocity-potential, but the density fluctuation neV2Y. This makes it harder for them to discuss the boundary conditions (and thereby the orthogonality rule) because, as we saw in $2.1, there is no intuitively transparent and simple prescription for imposing a boundary condition directly on the density. (Moreover, V V can be worse afflicted than Y by Gibb’s phenomenon.)

(ii) In the light of the remarks made between equations (3.3) and (3.6), it is not clear to this writer whether FDB’s treatment is immune to mathematical ambiguities stemming from their use of effectively homogeneous integral equations even for continuum (bulk) modes. I t would be interesting to investigate this question rigorously.

990 G Barton

contribute to the van der Waals potential?. Unfortunately, this misconception also runs right through the long analysis by Wikborg and Inglesfield (1977), reflected especially in the adoption of the non-generalisable orthogonality rule (2.44). This makes the interesting technical achievements of their analysis hard to disentangle.

Finally, the expansion (2.32) of the external potential itself sets a trap through the analytic properties of the expansion coefficients regarded as functions of frequency (as or C ~ B respectively). Here it may help to consider for definiteness the matrix elements encountered when using the Golden Rule to calculate the decay into plasmons of an external molecule having excitation energies E. Matrix elements for such decays contain the expansion coefficients of (2.32) but with Rs and L ~ B replaced by E. When interpreting the result for decay into bulk plasmons with given k, confusion may arise from the fact the squared matrix element contains in its denominator the factor [(2E 2 - wp2)z- 4 P 2 P E 21, which vanishes when E 2 = Qs2(k) ; in other words, the decay rate into bulk plasmons, regarded as a function of energy E, has a pole at the surface-plasmon energy. Though this analytic structure has an analogue in the wavefunctions of scattering theory when bound states exist, it is perhaps unsurprising that in the present context and in calculations based on analyticity properties it should have invited misattributions of external coupling effects as between surface and bulk plasmons. This appears to have happened for instance in the discussion by Fuse and Ichimaru (1975).

Insofar as the above comments concern the propagation of errors, they are only surmises based on published papers and on their references. But with respect to the HD model the errors themselves are just that; they have been spelled out and located in order to make it easier for the true results to be distinguished from the false where they tangle. Fortunately, correct treatments of these points are also available in the literature, starting with the original results of Bloch (1933) on the orthogonality rule and of Jensen (1937) on decoupling. Section 2.8 has shown that Newns’ (1969) classical result for the image potential is also due in part to bulk plasmons. Accounts of the van der Waals potential which visibly embody the bulk- plasmon contributions have been given by Harris and Griffin (1975) (their ‘mixed modes’ are our ‘bulk modes’), and by Griffin and Kranz (1977) (who however make a technical slip in handling the cutoff: see the last paragraph of $3.2 above). They start from the RPA approximation to the dielectric function (see $53.1 and 3.2 above) but reveal the results for the HD model in the appropriate limit. After correction along the lines of his erratum, Heinrich’s (1975a) calculation leads to the same expressions.

4. van der Waals attraction between half-spaces

4.1. Introduction

Consider two similar infinite half-spaces of the kind discussed in $2, with parallel surfaces at x= &a, respectively, i.e. separated by a distance d=2a . We define 2 W ( d ) as the energy of attraction between them per unit area, so that W is the energy per unit area for just one surface, normed through W(co)=O. Wwill be determined by calculating how the total zero-point energy (ZPE) varies with d. Relativistic retardation is neglected, and no cutoff is imposed, so that the system is governed by the differential equations and the boundary conditions introduced in $2. We shall use them to find the exact frequencies and normal modes of the joint system, which is easier than allowing for the mutual coupling between the two half-spaces as an explicit perturbation.

Besides recording the correct result for W, we aim to show in an elementary way how the continuum modes, whose individual contributions vanish inversely with the size L of the system, nevertheless conspire to produce a non-zero overall

J E Inglesfield (1976 private communication) has pointed out, and a careful reading of the paper confirms, that this statement, though used interpretatively, does not actually enter the quantitative analysis, which proceeds through a contour deformation automatically including all contributions.

Some surface eflects in the hydrodynamic model of metals 99 1

contribution to W; in more highbrow treatments such effects enter through changes in the density of states, and experience shows that they are easily forgotten. (Our approach to continuum modes is the one followed by simple proofs of Levinson's theorem (see for instance Weinberg 1965).) These contributions call for a first look at the thermodynamic limit L+ CO. We shall also prove a sum rule for the model, new in this context, showing that the sum of the squared frequencies of all normal modes is independent of d.

Previous theoretical work on W has already been mentioned at the end of $3.4. As far as this writer knows the only measurement with metals is an early and not fully conclusive one by Sparnaay (1958) in the relativistically-retarded (Casimir) regime. All other work has used dielectrics, and is accounted for by Lifshitz's theory for materials with dielectric functions independent of wavenumber (see Israelachvili 1972, Israelachvili and Tabor 1973). For a thorough review of non-dispersive theory, and for a critique of the foundations of Lifshitz's method, see the exceptionally scholarly dissertation by Schram (1975).

Section 4.2 takes a preparatory look at bulk-mode phase shifts and frequencies in a single half-space; $4.3 determines the exact normal modes of two separated half-spaces; $4.4 evaluates W ( d ) ; $4.5 considers the non-dispersive limit; and $4.6 proves the squared-frequency sum rule and comments on its implications.

4.2. Bulk-mode phase shifts and frequencies in a single half-space

We start from the bulk normal-modes of the displacement potential in a single infinite half-space as obtained in equations (2.25)-(2.26).

In order to enumerate them conveniently and to identify their frequencies as functions of k and p , we define a fictitious boundary at x = - L , and require Z/J to have vanishing slope there. (Any other boundary condition would serve equally well.) The limit L-t CO is to be taken at the end of the calculation. For large enough L, the exponential in (2.25) is negligible at x = - L and the allowed values of p are determined by

PnL + 6(k , P,) = nT, n=1 ,2 , 3 , . , , . (4.1)

The value n=O is forbidden because p is defined non-negative while 6 is positive. Though in principle (2.26(a)) and (4.1) constitute a transcendental equation for the pn, it will become clear in the course of the argument that in the limit L+CO this equation need not formally be solved. The mode eigenfrequencies Qkn are given by Rkn2 = (wp2 + P2k2+ P'pn'), It will further become clear that Q is required only to order L-1; accordingly we rewrite Q as below, defining in the process a new summation (eventually integration) variable q by

q nir/L. Thus

whence to order L-1

Expressions like the second term on the right will be called frequency shifts.

992 G Barton

Finally, to leading order in L-1, the summations over n which we shall encounter can be turned into integrals over n, and in view of (4.2) into integrals over q, by the standard equivalence

E . . . +(L/.rr)Jdq.. . . (4 * 4) n

(Recall that the perhaps more familiar formula Zn+(L/2n)Jdq applies to running waves admitting both signs of q, while here we have standing waves with q defined non-negative.) In $5, when we come effectively to count the modes themselves, we shall need the next-order corrections, but in the present section (4.4) suffices.

4.3. The normal modes of two separated half-spaces

Consider the system of two half-spaces described in $4.1. Fictitious boundaries, like those in $4.2, are introduced at z= 5 (L+ a). The normal modes are again of the form (2.12) ; the equations to be solved are (2.13) and (2.14) inside each plasma, i.e. for I X I 2 a, and Laplace’s equation in the gap, i.e. for [ X I < a:

( - k2 + d’/d.*)+ = 0. (4 * 5 )

The boundary conditions at the surfaces (z= k a) are that + and d+/dz are continuous and d#/dx = 0. We rule out exponential increase as I x I -+ CO.

By symmetry, solutions are either even or odd in z (suffices + and - re- spective1y)t. In the gap, ++ and +- are proportional to cosh kx and sinh kz, respectively. Clearly it is sufficient to consider solutions for $ only in, say, the left-hand half-space. T o each pair of labels, k and p, indexing a mode of a single half-space, there now correspond two modes, $kP+ This doubling is as expected since we are now dealing with twice the total depth of plasma (2L instead of L).

Again, solutions subdivide into bulk and surface modes according to !3 2 (wp2+ p2k2). The calculations are straightforward and we quote only the results, The absolute normalisations of the amplitudes are now irrelevant.

For surface modes we set &2= (wP2+p2k2--p2ps*2) in the light of (2. IS). Then in the left-hand half-space

#M = PS, exp [k(z + a)] - k exp [PS*(X + a)] (4 * 6)

and the boundary conditions eventually yield the eigenvalue equations

which give Heinrich’s (1973a, 1975a) results

Qs+2(d) = *,@[cQ~+ k2+ k(k2+ 2a(T2)1/2] (4.8)

where we introduce the abbreviations

Note also “+yo) = 2012, a(-2(0) = 0. (4. lo)

f Solutions even (odd) in the potentials are odd (even) in the displacements and velocities.


Naturally one has (compare to (2.20)) Qs+( C O ) = Qs-(CO) = Qs(k). (4.11)

More surprisingly perhaps

and Rs+2(0) = u p 2 + p2k2 = QB2(k , 0) (4.12)

Qs-*(O) = i[p"k2 + pk(P2k2+ 4 w p y ] (4.13)

so that in the non-dispersive limit

QS+(O) = u p , Qs-(O) = 0 when p = 0. (4.14)

Bulk modes exist for all Qza (up2+ p2k2) . In the light of $2.2.2 one finds in the

(4.15) left-hand half-space :

$kqlt={COS [ q ( x + a ) - & ( k , q, a) ]+C+exp [ k ( x + a ) ] )

where the constants Ci are irrelevant, and where

(4.16)

Note that, as a+w, &+S as defined in (2.26(a)); and the asymptotics

6+(n = 0) = 0, &.(a = 0 ) = tan-1 [ k 0 ~ 2 / q Q ~ 2 ( k , q)]. (4.17)

By the same argument as in $4.2, we see in the light of (4.3) that because of the finite (rather than infinite) separation between the half-spaces, the frequency of each even (odd) bulk mode is shifted by the a-dependent amount

(4.18)

defined so as to vanish when a+ CO. This is our central result. We shall see in $5 that misinterpretations of the results (4.12) and (4.14) can thoroughly

prejudice and confuse discussions of the surface energy. The result ! k ( d =0, /3=O)=O has been explained (Schmit and Lucas 1972) by saying that this is a transverse mode which, as such, lacks a restoring force, and has zero frequency for that reason. That this explanation is false can be seen by mode-counting and continuity. 'rhus, (4.8)-(4.9) show that 0 s - vanishes only when both d = 0 and /3 = 0. Away from either or both these limits S- is a manifestly ordinary, longitudinal, finite-frequency mode ; indeed, in the HD model all finite- frequency modes are longitudinal, as was shown in $2.2.3 whose argument extends Tadily from one half-space to two (see also the correct enumeration of modes by Lucas and Sunji6 1972). The real reason why Rs-(d =0, p = O ) vanishes stems from the fact that in this mode the motions are symmetric in z , i.e. towards the surface barrier in one half-space and away from it in the other. Hence, a t zero separation, and for p = O , the G(z)-proportional surface charge built up on one side is precisely neutralised by that on the other side, and the restoring forces vanish because for p=O surface modes have no other charge densities, as shown in $2.5.

The result %+(d =0) = Qu(k, p = O ) has been explained by saying that, for given k, S+ is really the one bulk mode which propagates parallel to the surface (with p = 0) and remains unaffected by the boundary conditions. That this explanation is misleading is best seen from a mode-count based on the Hamiltonian in $4.5, conveniently performed in the special case /3 = O and at finite separation, the limits d -+O and d -+ 00 being covered by continuity. Thus, the real reason for the result is that in any mode with this symmetry, the motions in the z direction are antisymmetric in z. When the two half-spaces touch, this antisymmetry conspires with the mirror reflection enforced by the boundary conditions to produce precisely the

994 G Barton

same pattern of motion as would result in an undivided infinite space without any barriers at all. Hence, in the language of 452 and 3, the surface-induced coupling between cosine modes having this symmetry vanishes when the two half-spaces touch, whence it would indeed be correct to say that Si is the lowest cosine mode, and is unaffected by the surfaces. But if the exact eigenmodes of the coupled half-spaces are expressed as superpositions of the exact eigenmodes of the individual half-spaces (rather than as superpositions of their cosine modes), then the correct description is that, for d =0, the lowest bulk mode is accidentally degenerate with the surface mode of this symmetry, while remaining orthogonal to it.

4.4. The van der Waals potential

The energy W defined in $4.1 is evaluated as half the total ZPE of the system ($2 for each mode) at separation d, less its value in the limit d-tco. Separating the contributionst of surface and bulk modes, noting that Ck = (2~)-2Jdk, and using (4.8), (4.11), (4.18) and (4.4) we have

W = Ws+ (4.19)

Ws = &(2~)-2J dk(Q,q+ + as- - 2Qs) (4.20(4)

WB= - i ( 2 ~ ) - ~ J dkJ dq[B2q/fiB(k, q)](8++ 6--28). (4 * 20(b) )

Different values of k contribute independently, and we introduce the notation,

W(d) = (2~)-2J dkW(d, k) . (4.21)

The q integration in (4,20(b)) can be done in closed form, but the result is too cumber- some to be worth quoting. Instead, we confine ourselves to specifying the asymptotic behaviour of W as d+O and as d+ CO.

As d=Za+co, equations (4.9), (4.8) and (4.16) show that the W(d, k ) vanish exponentially fast. Hence the k integrations are dominated by the region of low k, and we obtain asymptotic expressions by expanding the integrands (apart from the exponentials exp ( - 2ka)) in ascending powers of h, and integrating term by term. Changing the integration variable to x=ka, and keeping only the two leading terms, we obtain straightforwardly

for any quantities like W, Ws and WB:

Ws= WS(O)+ Ws(l)+ . . . (4.22)

(4.23)

(4.24)

where now 01*2 = a2[ 1 _+ exp ( - 2x)]. The integrals in (4.23)-(4.24) are pure numbers,

j- When an observable quantity like W is subdivided into 'contributions' from different kinds of modes, the subdivision is always inspired by some convenient mathematics, and is never enforced directly by the physics. For instance, in $2.8 we subdivide in a way suggested by the formalism of second-order perturbation theory, while in $94 and 5 we shall subdivide in a way suggested by expansions in terms of exact normal modes. Nevertheless, these particular subdivisions have been adopted as almost universal conventions. By contrast, the overall influence of finite dispersion on the various potentials is a genuine physical effect independent of any particular method of calculation.


but unfortunately they must be evaluated by yet again expandingt the integrands, say in powers of exp (-2%). Keeping two terms in WS@) one reproduces Heinrich's (1973a, 1975a) results for no cutoff; these, and the results obtained by evaluating the integrals numerically, are as follows:

(4-25)

(4.26)

WB is calculated by the same prescription applied to the k integration, We confine ourselves to the leading term, which is of order d-3, and hence comparable to Ws(1). Here it is a great help that the phase shifts 6* and 6 are already of order k. After considerable manipulation one obtains

(4,27)

Expansion in powers of exp ( - 4%) and integration yields m

Finally, numerical evaluation yields

P 0.227- - 0.719 x 10-3 WBN- 32n-2d3 d3

to be compared with (4.26). Combining (4.25), (4.26) and (4.30),

(4.29)

(4-30)

(4.31)

As by now we expect, the leading long-distance term, of order d-2, comes entirely

As regards short distances, Ws and WB both remain finite at d = 0. For Ws, from Ws, while Ws and WB share comparably in the term of order d-3.

we obtain from (4.20(a)) and (4.9)-(4.10):

(4.32)

Much more labour is needed to evaluate WB(O), but from (4.20(b)) and (4.16)- (4.17) one finds eventually

(4 a 33)

t The nth-order terms in (Ws + WB) are just the nth-order perturbation contribution when the coupling between the half-spaces is treated as a perturbation.

67

996 G Barton

Note that WB(O) and Ws(0) have opposite signs, and that there is considerable cancellation :

W(O)= Ws(O)-t W~(O)=(wp3/96n/3~)( -3n/2+8.\/2-7)- -(wp3//32)(1*32 x 10-3)

- 3.1 10-3~,-5/2 au.

(4.34)

Though the total van der Waals energy 2W(O) is finite, the attractive force F = - 8(2W)/a(2a) = - aW/aa diverges logarithmically as a+O. Keeping only terms which diverge in this limit, one finds after some manipulation? the explicit formulae

(4.35)

4.5. Nola-dispersive limit In the limit p = O , bulk plasmons decouple, so that WB vanishes; the surface-

mode frequencies (4.8) reduce to

!&*2 = wD2[1 I exp ( -Ad)] and one finds

(4.36)

W = (wp/d28.rrdZ)J,’ dxx[(l +e-”)l/2+(1- e-z)1/2--2]. (4.37(a))

W now varies like d-2 for ail values of d, and diverges as d+O.

Lifshitz’s theory by van Kampen et nl (1968), Schram (1975) shows that (4.37(a)) is equivalent to the expression derived from

where .(U) = (1 - wp2/co2) is the non-dispersive dielectric function (3.6). By expanding the square roots in (4,37(a)) he also derives the fast-converging expansion

2W = - (wP/d28.rr.\/2) czn/n2 co

n-1

c p = (2p) ! [(2p - l)(p !)222P]-1 (4 37(c))

whose first term is often found as a ‘leading approximation’. Using Lifshitz’s theory, Teodorovich (1978) has considered the dragging (fric-

tional) force in a tangential direction between two half-spaces in relative motion with velocity a,, at a constant separation d, and with their surfaces remaining parallel. Since (4.37) gives a normal force per unit area of F , - R ~ ~ d - 3 , one expects on dimensional grounds that to leading order in a, the frictional force F,, should be proportional

.f. In the paper already discussed in 93.4, Buhl (1976) gives the coefficient of (wp4//33) x log (wpa//3) in F as 0.006. According to him this should agree with Heinrich’s result Fs, though in ( 4 . 3 5 ) we have 1/8x-0.040. These manipulations are so involved that an independent check is desirable.


to uhd-4. Remarkably, this is independent of up. Teodorovich obtains an expression in terms of the static dielectric function ~ ( 0 ) of the medium, namely

If this is applied naively to a metal having E ( w ) = ( ~ - wP2/w2) and E(O)+CO, one obtains F : , = - v(h/128~2d4)(~4/15).

Finally, for comparison in 55.5.3 with other work, we record an expression for W(0) obtainable through (4.21) by making a so-called long-wavelength approximation (LWA) and (contrary to the arguments of 53.3 above) imposing a cutoff as follows: (i) Ignoring the circumstance that a cutoff makes surface modes dispersive even when /3 == 0 (see for instance equation ( 3 . 1 6 ) ) , approximate !&(d = 0) and Qs by their K-independent values given in (4.12) and (4.13), obtaining

W(d, k)+*(wp+ 0 - 2w,/d2)= - *WP(d2 - 1). (4 38(a)) (With a proper handling of the cutoff, it turns out that (4.38(a)) is actually correct as k+O, whence the designation of the LWA.) (ii) Substitute (4.38(a)) into (4.21) and perforrn the k integration subject to the cutoff K2< Q2; this yields

W~,w*(d = 0, /3= 0 ) = - ( d 2 - l ) w p Q 2 / 1 6 ~

N - 8.2 x ~ O - ~ W , Q ~ N - 12 x 10-3rS-5/2 au. (4 ' 38(b)) By contrast, if a cutoff is imposed on the two half-spaces along the lines followed

in $3 for a single half-space, then a short calculation starting from (3.17) and ( 3 . 1 8 ) leads to

Qs*2(d, ,8=O)=wp2{1-~-1 COS-~(K/Q)[~ Texp ( - A d ) ] )

whence, evaluating the integrals numerically,

W(d=O, ,8=0)= -(wpQ2/Sn)J; dxx{l+ [ 1 - ( 2 / ~ ) ~ 0 ~ - 1 ~ ] 1 / 2 - 2 ( 1 - T - ~ c o s - ~ x ) ~ / ~ }

N - 0.80 x 10-3wpQ2- - 1.1 x 10-3 rS-5/2 au. (4 ' 38(c)) Note how much smaller the propcrly cut-off result (4.38(c)) is than the LWA (4.38(b)).

4.6. The squared-jrequency sum rule for separated half-spaces

We outline the proof of the sum rule stated in $4.1. The plasma frequencies w P ~ , R of the two half-spaces L and R may be different. One starts from the Hamiltonian

€€=HI,+ HR + Hint. Here, HL, R are, respectively, the Hamiltonians for the L and the R half-spaces in isolation, each conveniently given by (2.34) with the appropriate value of up; and Hint is the interaction between them :

Hint = - 3s drR(DLne div ER - 3s drr,(DRne div g ~ . (4.40) Each integral runs over the interior of one half-space; OL, R is the potential due to the indicated half-space beyond its surface, given by an expression like (2.32) but in terms of the operators X rather than a, as explained just below equation (2.34); and the factors -ne div 5 are the charge densities. The following features are important: (i) Because (DL, R depend only on the X but not on the P, Hint is

(4 * 39)

998 G Barton

bilinear in the XL and XR. (ii) Only modes with equal values of k are coupled by Hint, so that H separates into a sum of terms H(k) , each referring to a given k .

For simplicity we drop the index k and, for given k, label the normal modes of each separate half-space schematically by discrete indices n( v) for the L(R) half- spaces (each index runs both over the surface and over the bull: modes). Then H ( k ) assumes the form

H ( k ) = 1 (gPr~+~nfQI,n~X,+X,)f ($~ ,+~v+QR,2X,+X,) n V

+ 4 Cf,”(d)(X,+X, + X,X,+) (4.41)

where only the coefficients f,, depend on the separation d. By the same kind of argument as in $3.1, H(k) is effectively diagonalised by diagonalising the matrix of the quadratic form involving the X , and X,. The elements of the resulting diagonal matrix are the squared frequencies Qkr2 of the exact normal modes for the two half-spaces jointly. The crucial point is that such a transformation cannot change the trace of the quadratic form; since the coupling term involving the fn, naturally lacks diagonal elements, we have

n v

(4.42)

whose right-hand side is manifestly independent of d, which completes the proof. The sum rule is useful chiefly as a consistency check; for instance it shows at

once that the effects of bulk-mode frequency shifts cannot be ignored (unless ,8= 0). Thus, the squared frequencies of the surface modes are found from (4.8)-(4. lo), and those of the bulk modes from (4.18). Summing over bulk modes by the prescription (4,4), we can accordingly state the sum rule in the form

(Qs+2+ Qs-2-2Qs’)- ( Z , ~ ’ / T ) J ~ dqq(6++ S--ZS)=O. (4.43)

The surface-mode contribution by itself, i.e. the first term on the left, is plainly non-zero unless ,8= 0, whereas (4.36) shows that for ,8= 0 it does vanish. We will not evaluate the integral in (4.43) explicitly, which is a non-trivial exercise; but it does indeed cancel the first term. However, note that to obtain the requisite cancellation one must, as indicated in (4.43), integrate over all q for given k. Hence at this stage there is no physically sensible way to impose a cutoff while respecting the sum rule. In order to do so one would have to start again with the cosine modes and the methods of $3.2.

5. Surface energy fi sempre bene il sospettare un poco in questo mondo (da Ponte 1790).

5.1. lntrodzrction

T o identify the energy U per unit surface area of a fluid, one must first express the total energy E of a large but finite system, having volume V and surface area A, in the form (Vu+Au+ . . . ), and then take the thermodynamic limit of infinite V and A, where the bulk energy density U, and U, remain finite. For simplicity we discuss a flat-faced slab of finite width L in the ,z direction, but already of infinite


extent in the x and y directions, and consider the contribution E(L) to its energy from a portion having unit cross-sectional area. Then the thermodynamic limit is L + 00, and U is identified through the expansion

E(L)=Lu+2a+ . . a .

The L-independent term is twice a because the slab has two faces. Contributions to E which vanish as L+co will be dropped without further comment. However, we shall see that before the limit is taken most methods yield E in the form

E(L) = Lu(L) + 2 4 5 ) + * . . . Then the L-1-proportional part of the quasi-intensive variable

u(L)=(u+u1L-l+ . . . ) must be retained, since it furnishes a finite (L-independent) contribution to a = (U( co) + h.1). Another mathematically equivalent method is to consider a slab of width 2L, and then two slabs, say each of width L, obtained from the first by cleavage and separation, a process which creates two new surfaces. Then U is identified through?

2 a = lim [2E(L)-E(2L)]. (5-3) L-+m

The surface energy of a real metal is governed by many factors, including (i) the effect of finite instead of infinite volume on the kinetic energy density even of an ideal Fermi gas confined by impenetrable barriers; (ii) further corrections to this if the barriers are positioned beyond the edge of the neutralising background, as they should be in order to ensure strict overall charge neutrality; (iii) changes in the exchange and correlation energy near the surface of a non-ideal gas similarly confined, subdivided into (a ) long-range and (b) short-range parts; (iv) for permeable barriers, the effects on (i)-(iii) of electron spillage beyond the surface; and (v) relaxation of the positive ionic lattice near the surface. Some of these factors favour a positive and others a negative a; several are comparably important; and their effects interlock, i.e. they are not simply additive. Hence one would not expect a realistic account to be simple, nor the final result to be easily predictable from a few basic principles. For a clear pioneer- ing discussion see Samoilovich (1945); for a review of early attempts, and for a thorough account of the beginning of systematic modern work on metal surfaces generally, begun by Lang and Kohn (1970), see Lang (1973). Measured values are conveniently summarised by the two preceding references, and by Schmit and Lucas (1972)I.

Although we shall see that it is not physically reasonable to compare the values of a given by the HD model directly with experiment, we note for the record that the measured values are all positive, and for several simple metals are roughly described by the formula proposed by Schmit and Lucas (1972):

(5.4) t It is fortunate that the two definitions agree, because in the HD model (5.3) implies

two new impenetrable barriers with the ensuing boundary conditions. Hence the energies 2E (L) and E (2L) in (5.3) are eigenvalues of Hamiltonians in manifestly different Hilbert spaces. The point is that the limit L+CO in any case involves comparison between non- trivially diflerent Hilbert spaces. By contrast, we shall see in 95.4.3 that certain other prescriptions for identifying a can and do clash with those above, and cannot therefore be accepted even if plausible at first sight.

$ Many measurements seem to date back quite far, and the scatter of some (Handbook of Chemistry and Physics 1972) inspires reservations.

1000 G Barton

(The atomic unit for surface energy is e2/a$= 1.56 x 106 erg cm-2.) By contrast, without an artificial surface barrier, the Thomas-Fermi approximation, which should apply as rs+O, gives a leading term proportional to r , -9/2 (Samoilovich 1945); the cutoff given in equation (3.13) suggests the quasi-dimensional estimate wpQ2m

1.4 rS-5I2 au, while the cutoff in (3.12) would suggest wpQ2- 10 ~ ~ - 7 1 2 au. Precisely because the problem is so complex, it offers scope for a simple model em-

bodying only some of the contributory factors, and intended to elucidate the signs and orders of magnitude of their effects which in more realistic treatments are more easily obscured. In this spirit and without further apologies we aim to calculate the absolute surface energy according to the HD model, and to dispel some misconceptions which have become attached to it, vitiating even the limited diagnostic uses which it could otherwise serve. We start from the Hamiltonian (2.38) adapted to a finite slab; for clarity we shall work mostly in the non-dispersive limit /3 = 0, which is not an essential restriction but allows all calculations to be performed transparently and in closed form. The problem is now mathematically well-defined, but two points must be borne in mind. First, the forces and energies associated with the barrier and the background charge do not enter at all, except indirectly through the boundary conditions (hence for instance the virial theorems for Coulomb forces are not directly applicable). Second, a cutoff Q is essential for convergence, and a depends crucially on how Q varies with L. On dimensional grounds we expect

Such L-dependence is well known in the closely allied Fermi momentum, but as far as this writer knows it has not been discussed explicitly for plasmons. From the point of view of the model the cutoff is in principle a matter for definition, but some precise choice must be made, and it is better that it should be made explicitly and that the physical motivation for it should be stated. The need for a precise definition is best evidenced as follows. We shall see later that a, like W in 94, can be expressed as an integral over independent contributions with different wavenumbers k :

a= (27~-2f dko(k). (5.6) Now in the non-dispersive limit there are at least three reasonable or popular candi- dates for a(k). Expressed in terms of the surface-plasmon frequency Rs they are

OI(K) = 3(Qs - Up)

UII(k) = +(as - &Jp) (11) ( P = O ) (5 *7)

(1) ( P = O >

aIII(k) &( QS - & u p ) (111) ( P = O ) .

(I), (11), and expressions more or less similar to (111) can all be validated by a suitably chosen Q(L), although some have occasionally been obtained by incorrect reasoning instead, and we shall try later to disentangle their status. Further confusion flows from the fact that some treatments ignore the k-dependence of Rs in (3.16) induced by the cutoff (even in the non-dispersive limit), and effectively propose (11) or (111) with Qs(k) replaced by Qs(0). Clearly it is necessary to impose the cutoff on some well-defined Hamiltonian, and to leave no room for arbitrariness by choosing cutoffs independently for surface and for bulk modes.

As we shall see, (I) results if the cutoff is defined so that the number&” of normal modes is strictly unchanged by the cleavage process described earlier, or if, equivalently, is strictly proportional to L ; this is the prescription which this writer con-

Some surface eflects zf.1 tlze hydrodyiaaiiiic model of metals 1001

siders the most plausible, and it will be adopted in the subsequent discussion unless otherwise stated. Then U is manifestly negative?. The alternative (11) results if the cutoff is taken as independent of L ; then the sign of cr depends critically on how 0 s varies with K . Ways of obtaining (111), which gives positive cr, will be mentioned later.

Another physically artificial but mathematically unambiguous consequence of the boundary conditions in the HD model is that two initially disjoint half-spaces brought into contact (Le. with separation d reduced to zero) are not equivalent to an undivided infinite medium, since the boundary conditions continue to affect the energy. In other words, to establish a free surface, one must first cleave the medium (enforce the boundary conditions) at zero separation, and then separate the two parts. For unit area of each surface, the first step costs a cleavage energy E,1 and the second the vdW energy - W ( 0 ) discussed in 54:

Ecl has no physical relevance apart from (5 . 8), and (5 8) is indeed the best way to calculate it. Heinrichs (1973a, 1975a) has argued strongly on physical grounds that, as regards the surface energy of real metals, the HD model affords more plausible guidance through - W(0) (see equations (4.38)) than through U. Here, however, we proceed to discuss the well-defined node1 surface-energy U, This artificial feature of the model, that U # - W(O), can be a salutary reminder that even if the density of a system has at every point its undisturbed bulk value, this alone does not by any means guarantee the absence of all interfacial energies. Consequently any formalism which implies such a guarantee should be approached with caution, a point that will become important in 95.4.31.

The rest of this section is laid out as follows. For simplicity the discussion is almost entirely confined to the non-dispersive limit. Section 5.2 calculates the surface energy of the ideal (neutral) fluid and of an ideal Fermi gas; it introduces the modc- counting techniques4 and the size-dependence of the cutoff which are crucial to the argument. Section 5.3 calculates the surface energy of the charged fluid with a strictly mode-conserving cutoff, obtaining the negative result UI (equation (5 .7)).

-f This could have been foreseen from arguments given by Samoilovich (1945), who proves that o is negative in any system whose mechanical (i.e. non-electrical) stress tensor is isotropic (as is the pressure in the HD model), and where deformations involving the creation of extra surface area are carried out strictly at constant volume (which is the prescription effectively mimicked in the HD model by the mode-conserving prescription for the cutoff). The positive result for U in an ideal Fermi gas results from anisotropy enforced near the surface by quantum- mechanical interference effects lacking any counterpart in the HD model.

t. Note the paradox that the HD model, even when it yields a negative surface energy (as with a mode-conserving cutoff leading to formula (I)), nevertheless combines this with a van der Waals force which remains attractive down to zero separation. Taken literally this would imply that the bulk material tends to fragment spontaneously while the fragments nevertheless remain in contact; the true ultimate scale of subdivision would then be dictated by physical effects not built into the model. This unrealistic combination of features stems from the assumed strict impenetrability of the surface barriers, which prevents any overlap between the ‘electrons’ belonging to initially distinct half-spaces even when the half-spaces touch; the standard variational argument which shows that the van der Waals force is attractive hinges on this absence of overlap.

§ These are elementary applications of ideas whose more sophisticated developments are described by Balian and Bloch (1970).

1002 G Barton

Section 5.4 reviews the consequence of other prescriptions for the cutoff, and some common pitfalls and snares which beset such calculations.

5.2. Ideal gases

We consider first the surface energy of an ideal (neutral) fluid (phonon gas) with the cutoff Q so defined that the total number of modes is strictly conserved on cleavage. This will introduce much of the formalism needed for dealing with the charged fluid (plasmon gas) in 95.3. For comparison, and because of their relevance to real metals, we also outline briefly the corresponding results for an ideal Fermi gas; the Fermi momentum QF then plays a role analogous to the cutoff Q for a continuous fluid. Throughout, the system is confined to a slab with surfaces situated at x = - L and x =0, where the fluid obeys the boundary conditions familiar from 92, whereas the fermion wavefunctions vanish.

5.2.1. Ideal phonon gas. By a phonon gas we understand the HD model with a neutral fluid, i.e. in the limit wp+O. Within the slab defined above the normal-mode amplitudes and frequencies are-f.

YiCr- exp (ik. p) cos ( r n ~ / L ) (5 a 9(4)

(5 * 9@)) MO(k, Y ) = p(k2 + YW/L2)1/2

where the index r runs over all integers greater than or equal to 0. A sum over such modes assumes the form

For summands independent of the direction of k this becomes (2n)-lfdkkCva 0.

The total number of modes X and the cutoff Q are connected by

(5 , l o )

where 8 is the step function, O[x>O]=l, 8[x<O]=O. The effective upper limit of the integral is clearly Q, and of the sum, the integer R given by K ( k , L)= [L(Q2-k2)1/2/77]. In this context [ . . . ] specifies the integer part. We write

(5.11)

I n the limit L+co the quantity 7 fluctuates rapidly. Therefore we replace it by its average over a uniform distribution of 7 over its permitted range; later we shall similarly replace higher powers of 7, setting

77+h +Q.. . (5 * 12)

and call this process '77-averaging'. Accordingly, since C = ( R + l), we have

(5.13)

Substitution into (5.10) and integration yields

Jlr = LQ3/6n2+ Qz/Sn. (5.14)

-i. The superscript on no distinguishes it from the eigenfrequencies of the charged fluid.

Some surface effects in the laydrodynanzic model of metals 1003

Recall that Q is still a function of L. Our prescription requires N to be strictly proportional to L, i.e. on substituting (5.5) into (5.14) we require the L-independent term on the right to vanish. This gives immediately (with NIL = n)

Q(L)=Qm+ 6Q=(6&~)1’3- 7/4L. (5.15)

In (5.14), and in similar expressions, the explicitly L-proportional terms are determined directly by the standard density of states in an unbounded system; it is the extra terms which need careful handling.

The addend 3 on the right of (5.13) has an immediate physical significance best seen if, contrary to our present prescription, we consider the cutoff Q fixed and independent of L. Since the sum on the left of (5.13) simply counts the normal modes with given k, we see that this number then consists of an L-proportional part (the first term), which as such is unchanged on cleavage, plus an extra mode, i.e. an extra 8 mode for each surface. In other words, when Q is independent of L, 8 of a mode is created per unit area for each new surface created; we shall exploit this interpretation later. (Inglesfield (1977) has rediscovered these quarter-modes and discussed some of their implications, apparently taking for granted the special choice of cutoff from which they stem,)

We are now in a position to calculate the surface energy. The total ZPE is given

As L+CO the sum may be approximated by the Euler-Maclaurin formula (see for instance Hardy 1949, Abramowitz and Stegun 1965, Mathews and Walker 1965):

R Qo(k, r ) = drSZO(k, Y) + &[LIO(k, 0) i- QO(h, R)] -+ . . . (5.17)

where the neglected terms all involve derivatives of L20 with respect to Y, and give contributions to EO which vanish as L-t CO. To evaluate EO one performs the following steps which are elementary but moderately tedious. (i) Evaluate the integral in (5.17); (ii) express R in terms of L, Q and 7 as in (5.11); (iii) expand the result in powers of 7, dropping terms which vanish as L+ 00 ; (iv) 7-average as in (5.12). The result is

r:O

Finally, (v) substitute into (5-16) and integrate over k . Actually it is far easier to exploit the remark just below equation (5.15), and to obtain the coefficient of L in the explicitly L-proportional term of EO by a direct three-dimensional integration carried out isotropically, as if calculating the energy density in an unbounded system, The essential point in obtaining (5.18) was to identify the term, a@, not explicitly proportional to L. In this way one finds

Eo= LPQ4/16~2+ PQ3124x. (5.19)

Finally we substitute (5. IS), which gives

EO= LPQm4/16~2- PQm3/48~ (5.20)

1004 G Barton

whence, by (5.1) or (5.3), and using (3.13) for Qa, g o = -PQm3196x- -3.6 x 10-3rS-512 au. (5.21)

Notice that without the L-1-proportional correction to 9, the negative result ( 5 -21) would be replaced by the positive value PQZ3/48n.

5.2.2. Ideal Fermigas. Here the normal modes and energies are:

yk, r - exp (ik. p) sin ( rx z /L ) , ~ ( k , Y ) = (k2+ rZn2/L2)/2m. (5.22) Sums over Y now exclude r = 0 and run over Y 2 1, and the total number of electrons N F is naturally unchanged on cleavage. The arguments of $5.2.1 are readily adapted to this situation, but the exclusion of the terms with Y = 0 reverses the sign of the effects. Allowing for the spin-degeneracy factor one finds in an obvious notation

(5.23)

( 5 .24)

(5.25) The result (5.25) dates back to Brager and Schuchowitzky (1946). One refinement

is to relocate each barrier a distance A outside the edge of the background charge distribution, choosing A so as to ensure overall charge neutrality as we11 as an exactly vanishing charge density deep inside the gas when L+ CO. On dimensional grounds one must have A=a/QFZ, and it turns out that a=3n/8. It is straightforward to repeat the above calculation with the distance between the barriers increased to ( L + 2a/QFm). Then (5.23) is replaced by QF = [ Q F ~ + ( x / 4 - 2a/3)/L] = Q F ~ and (5 .25) by

o ~ = ( Q ~ ~ 4 / 3 2 n m ) ( 1 - 3 2 a / 1 5 n ) = Q ~ ~ 4 / 1 6 0 n i ~ 1 ~ 0 . 0 2 7 rS-4 au. (5.26)

For a self-contained discussion of this result see Sugiyama (19GO), and for more recent references see 1,ang (1973, especially the Appendix). The expression (5.26) shows that leakage of the electrons beyond the surface decreases a. Leakage controlled by a different mechanism could in principle lead to a larger effective value of a, and thus to a negative surface energy.

Unfortunately, the opposite signs of (5.21) and (5.25) or (5 .26) show that as regards the surface energy of neutral gases the continuum-m model is an inadequate guide to Fermi systems (recall the first footnote to p977). Therefore the most one can hope for from the model are indications as to the so-called correlation part of LP,

namely the difference between a for a charged and for a neutral system. This is the chief reason why from here on we concentrate mainly on the non-dispersive limit p=O. In this limit (5.20) and (5.21) remind us that the neutral phonon gas has zero energy and a fortiori zero surface energy. Then the entire surface energy stems from correlations, which is indeed how the indications of the HD model have usually been interpreted.

5.3. Surface energy of the plasmon gas with mode-conser.z;ing cutoff

The surface energy of the neutral fluid proved easy to calculate because the frequency of every mode is uniquely and simply given by the (three-dimensional)


wavenumber. The Hamiltonian (2.38) for a half-space already shows that in the charged fluid this simplicity is destroyed by the surface-dependent coupling between the cosine modes. Only the bulk energy-density U for given Q (analogous to the first term in (5.19)) remains unaffected by this complication, being given straightforwardly by

where (see footnotes to p987):

In the non-dispersive limit ( 5 ,27) reduces to

T o calculate the surface energy we start by expanding the displacement potential in a slab in terms of the discrete set of cosine modes, norming them so that the surface-independent part of the Hamiltonian assumes the standard canonical form, and so that without a cutoff the same local commutation rule (2.41) would be obeyed as in an unbounded system. As analogues of (2.35) and (2.36) this yields

Y? = f dk C exp (ik. p) cos (prz)[Lx2nm(K2+ pr2)(1 + S , O ) ] - ~ / ~ XkrC

n = f dk r

exp ( - ik. p) cos (prz)[nm/L4x2(K2+ pr2)(l + ~3~o)]1 /2[FPk~C (5.30) r

p,=mlL.

Next, these expansions are substituted into the expression (2.16(b)) where the z and x' integrations now run between - L and 0. Because the system is mirror- symmetric about z = - gL, the C-modes with even (odd) Y couple only to each other. The integrals are evaluated as before, with the additional purely numerical approximation that terms proportional to exp ( - KL) are dropped, which is justified since in the limit L-+ CO they become exponentially small compared to all the terms retained. (These terms are discussed fully by Griffin and Zaremba (1973).) One obtains the analogue of (2.38)

H=H++H-

where in H+ (H-) the sums run only over even (odd) values of Y and Y', and where we now impose the cutoff (k2+pr2)<Q2, which is to be understood throughout the following. The eigenvalue equations for the squared frequencies (analogous to

1006 G Barton

r’ (Y, Y‘ both even or both odd). (5 .32(a))

With the sums not yet replaced by integrals, i.e. with L still finite, we can derive for all modes an equation analogous to (3.4):

When modes are to be counted, the discrete nature of the sums in (5 .32) is of the essence. But anticipating the limit L-t CO, it is clear that (5 ,32(a)) will then admit a continuum of bulk modes with !2 2 QB(k,p=O); and in order to calculate the frequency Qs,t of the one even and one odd surface mode depressed below Q B ( K , 0), it is an adequate numerical approximation (for large but still finite L ) to replace the sum in (5 ,32 (b ) ) by an integral. Then the F equations coincide, since

( 5 . 33) r even r odd

and we recover the same equation (3 .4) as in a half-space?. For non-zero values of /3 it would be impossibly awkward to proceed as in $5.2,

and in practice one would need to borrow the techniques of response-function theory. This is done briefly in the appendix, mainly to facilitate comparison with more elaborate calculations. But when ,#= 0, the calculation can fortunately be completed by elementary methods, exploiting the fact that, as we have seen, Q* then has the same spectrum in a slab as in a half-space. In particular, is given by the simple formula (3 .16 ) ; all bulk modes have the frequency up; the total number of modes for fixed k is given by (R+ 1); and mode-counting and 7-averaging, which are unaffected by the value of ,#, proceed exactly as in $5.2.1.

Since there are two surface modes, the number of bulk modes is (R+ 1-2); accordingly the total ZPE E can be written down immediately:

E =(2~) -2Jdk&[wp(R+ 1)+2(!&-wp)]~El+E2 (5 .34 )

subdividing E by hindsight. Note that E1 is just dwp times the total number of modes; with the cutoff defined to conserve modes, E1 is strictly proportional to L, i.e. it lacks an L-independent part and does not contribute to D. By contrast, E2 lacks an L-proportional part, so that 20 is given by Ez, where Q may be replaced

f- In the limit L -> CO the two surfaces of the slab effectively decouple from each other; although in principle the two discrete modes are still symmetric and antisymmetric respectively, in practice the amplitudes of the unsymmetrised functions overlap negligibly, and each surface may be regarded as having its own surface mode, identical in all relevant respects to that in a half-space.


by Q,,,. equation (5 .7) :

Using (3,16) for Qs and (3.13) for Q we obtain the negative result (I) of

a ~ = $ E ~ = ( Z T ) - ~ J dk$(Qs-wp)= ( ~ p Q 2 / 4 ~ ) f i dx[(l- n-1 COS-’ x)’12- 11%

- 5.49 10-3w,~2

N - 7.74 x 10-3 ~ ~ - 5 1 2 au. (5.35)

The integral Jidxx( 1 - r -1 cos-1 x)1/2- 0-432 has been evaluated numerically. If, ad hoc, we were to adopt the so-called long-wavelength approximation (LWA) by replacing Q;zs(k) by Qs(0) = w p / d Z , then (5.35) would be replaced by

N - 16.4 x 10-3 r,-5/* ail. (5.36)

5.4. Other prescriptions for the cutoff

5.4.1. Introduction. Section 5.3 showed that a mode-conserving cutoff leads to formula (I) of equation (5.7). We end this review by considering the provenance of the rival formulae (11) and (111); these often turn up in calculations using the response-function formalism and the random-phase approximation (RPA) of many- body theory, and yielding a as an integral over the dielectric function ERPA. We shall not try to analyse such calculations in technical detail, but note that all reduce to the HD model in a suitable limit where ERPA is replaced by EHD, as discussed in $3 (and investigated in detail by Eguiluz (1978)); hence they can be partially tested in this limit. Recall that in such many-body calculations there is no need to impose a cutoff by hand. As regards plasmon contributions they have an effective built-in cutoff near Q = w p / v ~ , equation (3.13). It is obvious of course that the response- function formalism as such can add nothing new to the physics; its proper role is to elucidate succinctly the implications of a Hamiltonian.

We shall assess calculations of a by two criteria. The more searching is their result for a itself. The less searching is their result for the integrand o(k = 0), equation (5.6), in the limit k+O, which we shall call the long-wavelength approximation (LWA). One is interested in the second criterion because it is in this limit that many- body calculations are most apt to exhibit collective aspects. Though we shall some times indicate results for

these contain no reliable information about the surface energies entailed by any well-defined Hamiltonians, and are quoted mainly to facilitate comparisons with other work. For simplicity we continue to confine ourselves to the non-dispersive limit, though many of the references quoted go beyond this,

5.4.2. Formula (IQ. If the cutoff is taken as independent of L, then, as explained in the paragraph following equation (5.15), the total number of modes (and hence of bulk modes) with given k is augmented by for unit area of each new surface. The ZPE of these extra modes augments the integrand o(k) by $up, whence (5.35)

1008 G Barton

and (5 -36) are replaced by

In the LWA, one would obtain the very different result:

(5 * 38)

(5.39)

N - 2.4 x 10-3 ~ ~ - 5 1 2 au.

The remarkable factor # (or equivalently -a) was found by Griffin et al (1974), and appears also in the work of Wikborg (1974) and Inglesfield and Wikborg (1974). Its role is particularly clear in the paper by Griffin and Kranz (1977)t.

We conclude that the result (11) is correct if the cutoff is chosen to be strictly independent of L.

5.4.3. Formula (111). In the literature formula (111) of equation (5.7) generally appears in the LWA

(5.40(a)) leading to

0111, LWA(k) = $(wp/Z/Z - wp/2)

UIII, LWA’ (2r)-21 dki(wp/.\/2 -

N 12 x 10-3 ~ ~ - 5 1 2 au

contrasting sharply, and as to sign, with 01, LWA, equation (5.36). (Equation (5.40(b)) but with a cutoff Q smaller by a factor 2-ll2 is the formula proposed by Schmit and Lucas (1972).) I n fact (5.40) coincide with the LWA to the quite distinct quantity - WL’WA (d =0) given by (4.38). Here we recall Heinrich’s argument outlined

just below equation (5.8). Note however that this coincidence holds only in the LWA; when a111 is evaluated with correctly k-dependent Qs, one finds

q11=(wpQz/4n)J; dx[(l- 7-1 COS-^ x)’/’- B]x N 0.014 wpQ2

N 20 x 10-3 ~ ~ - 5 1 2 au (5.41)

while - W ( d = O ) is then given by (4,38(c)), which is quite different from 0111. Amusingly, 0111 emerges in the LWA if the cutoff is made to vary with L so that

the sum of the squared frequencies is independent of I , (i.e. is unaltered on cleavage).

-1 Their function D is essentially the same as ours. In the non-dispersive limit the first term of the integrand in their equation (1) gives the contribution -$wp characteristic of an L-independent cutoff, while the second term of the integrand by itself would lead to UI.

Some surface ejfects in the hydrodynamic model of metals 1009

This is best seen by adapting the argument linking (5.34) to (5,35), but summing the squared frequencies rather than the ZPE of the modes. Suppose that for each k a fraction t of a mode is created per unit area of each new surface, by an appropriate change of cutoff; then the constraint is

( Q S Z ( 0 ) - wp2 + twp2) = 0, hence t = 4 and the extra + mode leads directly to the energy balance

i-(Qs(o) - wp + tup) = ~III, L W A ( ~ ) .

Unfortunately, the result is peculiar to the LWA; without this approximation the same constraint reads

( 2 5 ~ ~ 2 s dk(Qs2(k) - wp2) + wp2A.M = 0

where AM is the total number of new modes created with each new surface, The consequent energy balance yields

= (2~)-2Jdk4(Qs(k> - ! & ~ ( K ) / o J ~ ) (5.42)

= 011

where the last equality stems from the remarkable coincidence that the k integral over Qs2(K)/wp has the same value as the k integral over a constant integrand 3 4 4 . Thus, for ,B = 0, the squared-frequency constraint when implemented exactly rather than in the LWA leads not to 0111 but to 011. Some interest attaches to it because it has formal similarities to the constraint that the HD model should obey the so-called .f sum-rule; whether it does has caused some controversy (Inglesfield and Wikborg 1974, 1978, Griffin and Harris 1976) which can be resolved either way by choosing the cutoff appropriately?.

Table 1 compares the numerical results discussed above, not as an invitation to compare these with experiment, but to show how sensitive they are to assumptions about the cutoff and to the long-wavelength approximation.

Table 1. The surface energy in (atomic units) x 10-3vs-5/2, for the hydrodynamic model in the non-dispersive limit p = O , and a cutoff Q = W ~ / V F . (The empirically quite successful Schmit-Lucas formula corresponds to 0111, LWA evaluated with the cutoff Q= w p / 2 / 2 VF, which would halve all the table entries.)

Squared-frequency- 01 011 0111 conserving - W(0)

Exact -7 .8 6 . 2 20 6 . 2 LWA - 16 -2.4 12 12

1.1 12

-f These references show that in the HD model, with its abruptly terminated background charge density, the f sum-rule can be implemented only by some very delicate handling of singular surface terms, which it is difficult to relate clearly to a well-defined underlying Hamiltonian. In the quantum mechanics of particles, the f sum-rule is essentially a device for counting degrees of freedom. Hence this writer believes that in the HD model its spirit is best observed by choosing a mode-conserving cutoff; but if it is to be enforced according to the letter, then one runs into the surface singularities just mentioned, and finds in any case that the validity of any such rule must depend on the precise definition of the cutoff.

1010 G Barton

The empirically quite successful formula (111) was first proposed by Schmit and Lucas (1972), in the long-wavelength approximation, and with the cutoff up/2/2 VF rather than u ~ / z ~ F . They were led to (111) by an argument, often repeated since, conserving modes by intention but not in effect, which was mentioned already in 44.3. Briefly, it asserts that on cutting a slab into two and separating the halves, two surface modes are created, but that only one bulk mode disappears, the balance being made up by the further disappearance of one transverse mode which had zero frequency and whose loss therefore liberates no ZPE. If correct, this would immediately yield the energy balance 24k) = &(2& - up - 0), i.e. formula (111). But the argument is wrong, since as we saw in $2 and especially in $2.2.3, there exist no transverse modes. We have seen that in a mode-conserving hydrodynamic model formula (111) is also ruled out a pviori by Samoilovich’s (1945) proof that U must be negative.

I t is ironic that the original argument for formula (111) stems from a simple miscount of normal modes, because fierce controversies have been pursued over the secondary defect that, as originally proposed, it embodied no convincing prescription for cutting off the integrals over bulk plasmons and over surface plasmons in a consistent manner. Feibelman (1973) has shown that if these two cutoffs are to be chosen by plausibility arguments, then equally or even more plausibly one can be led to formula (I) and to a negative value of U. (For more about the controversy, and for references, see Phillips (1975) and Kohn and Lang (1975).) Although this secondary defect can be remedied, as we have seen, by imposing the cutoff on the cosine modes, and when remedied as in (5.41) predicts a positive value for U, formula (111) cannot be salvaged in any mode-conserving approach.

I t is all the more remarkable that formula (111) in the long-wavelength approximation has appeared so often as the output of much more sophisticated many-body calculations, like those by Peuckert (1971), Craig (1972) (after correction by Barrera and Duke 1976), Mukhopadhyay and Lundqvist (1975) and Barrera and Duke (1976, hereafter referred to as BD). In most cases the reason is difficult to pin down because by the time the HD limit is taken the surface contribution to the total energy is usually identified by plausibility arguments rather than explicitly from a Hamiltonian, and because the possible L-dependence of the Fermi momentum or of the cutoff is ignored?.

However, (111) appears to be a characteristic consequence of the often tacit assumption or approximation that interfacial energies in an infinite system would automatically vanish if the background density, and therefore the local plasma frequency and dielectric function, had the same value at every point. As we shall see this is suggested by analysis of the exceptionally clear paper by BD which we take as representative of those just cited (see also the closely related paper by Barrera and Gerlach (1974)). Since the assumption amounts to neglecting the cleavage energy in ( 5 . S), it is not surprising in the light of Heinrich’s argument that such calculations should produce the result UIII which, in the LWA, coincides with - W(0).

BD consider two half-spaces in contact across the interface z = 0, with non-dispersive dielectric functions €1, ~(n) = 1 - uB1, 22 (Cl + i8)-2 for z 5 0 respectively (8 determines the signs of the imaginary parts needed below, and is allowed to vanish at the end of the calculation). The response function of the system is defined by the functional derivative x ( r ’ , Y, t‘ - t ) = 8 p ( r ’ , t ’ ) / 8 p e x t ( r , t ) , where p is the charge density induced by the perturbing external density p e x t . After taking Fourier transforms with respect to x, y and t , BD use equations equivalent to those of the HD model without a cutoff and find (in our notation)

~- - ( z ‘ , z, k, Cl) = 8(z) exp (Kz‘) (5.43)

a+-(%’, z, k, 0) = 8(z) exp ( - k Iz‘ I )

where the first and second suffices specify whether z and z‘, respectively, are positive or negative.

j- By contrast, the papers cited in $5.4.2 show that formula (111) is not an automatic consequence of sophistication per se.

Some surface eflects in the hydrodynamic model of metals

Finally, the total energy of the system is expressed in the standard form

E = ( 2 ~ ) - 2 1 dkE (k)

E (k) = 1 1: d Q j m dz Im a(z , z , k, Cl). T -a0

101 1

( 5 .44)

We have anticipated slightly by replacing a coupling-constant integration by the factor of two which it would eventually yield with the dielectric functions adopted here.

BD assert that the surface energy U is obtained by replacing a in (5.44) with the average of the S(z)-proportional parts of the diagonal elements a+.+ and a-- of a. (The average is suggested by a somewhat delicate prescription for handling coincident singularities of 6 functions and step functions.) This leads to

(5.45)

Setting €1 = 1, € 2 = E = 1 - wp2/Qz, one finds straightforwardly that U B D ( ~ ) = UIII, L W A ( ~ ) . However, the energy UBD thus defined differs from the surface energy U identified, in the

manner of 95.1, so as to ensure that for each value of k the bulk energy is strictly proportional to the volume of the medium. For the simple case si = 1, E Z = E , such an argument when based on (5.44) would run as follows. (i) Determine um(z’, 2, k , Cl), the response function as it would be in a truly infinite medium (i.e. with EI= E Z = E everywhere). This gives the standard result

am(z’, z , k , Q)= S(z’-z)(l - l / + (5.46)

(ii) Notice that a++ vanishes when ei=l, because in the z>O half-space there is then no medium, and therefore no induced charge density. (iii) Notice that for the same reason the S(z)-proportional part of a-- lies effectively within the z < 0 half-space. (iv) Accordingly, the integral in (5.44) runs only from - CO to 0 and involves only Im U-.... (v) Far inside the half-space CL- +amg. Hence we identify the surface energy by subtracting from E (equation ( 5 .44)) the strictly volume-proportional quantity obtained by replacing its integrand by Im “(2, z, k, Q), while still extending the integration only over the interior, i.e. over z<O. This eives

(5 .47)

which leads straightforwardly to u(k) = UI, L W A ( ~ ) . From a mathematical point of view neither of the above arguments based on (5.44) is

rigorous, for two reasons common to both. First, U is not a well-defined quantity without a cutoff, whereas the cutoff above is imposed only on the final integral over k rather than on an initially well-defined Hamiltonian. This is also why in both cases one obtains only the long-wavelength approximation. Second, U should be identified by approaching the thermodynamic limit through systems of indefinitely increasing finite size, whereas the argument above attempts to deal directly with a semi-infinite system. Hence by itself the second argument, in favour of (5.47), is certainly not conclusive; but the first argument, in favour of (5.454, is certainly not correct unless suitable further assumptions are adopted, because (5.45) asserts that the interfacial energy vanishes when the two media are the same ( el= a= e) , or in other words that the cleavage energy discussed earlier is zero, whereas we know that in the HD model for instance this is not so.

On the other hand, from an intuitive point of view, the argument leading to (5 .45 ) and the argument leading to (5 .47) are both plausible. The fact that in the HD model and in the ‘dielectric approximation’ they reach different conclusions, forcing one to choose between them, reflects the possibly paradoxical feature that U and - W(0) differ. However, this feature reminds one that in general the energy is a complicated functional of the background density n, sensitive to all derivatives of n, and is not fully determined by integrals over local functions of n itself. If it were so determined, or rather if one could neglect all effects involving derivatives of n, then BD’s argument makes it plausible that UBD= UIII might result at least in the long-wavelength approximation. (In fact, once an approach along lines like these is adopted, it becomes very difficult to allow properly for effects governed by gradients of n, especially in models including or implying an impenetrable surface barrier. The difficulties are apparent

68

1012 G Barton

for instance from the work of Shy-Yih Wang and Rasolt (1976).) In the HD model the dis- continuous step in the background density introduces singularities into the relevant integrands, and it seems difficult in practice to extract the correct results from such singularities through response-function formalisms without explicit guidance from some model Hamiltonian, The important moral is not that in the HD model (111) is wrong, since it might be right in other equally or more plausible models ; but rather that the above plausibility argument leading to (111), and other arguments to the same effect, are not conclusive, and that even in other models where (111) may be correct in fact, it still needs to be validated by further assumptions or arguments which do not apply to the HD model. But in the HD model the most persuasive (mode-conserving) prescription for the cutoff suggests that (111) represents not the surface energy o (which in this model is negative), but an approximation to the separation energy - W(0). That - W(0) is positive is, of course, a general variational theorem in the absence of overlaps, which are eliminated from the MD model by the surface barriers; and the validity of this theorem does not depend on adjustable cutoffs.

Acknowledgments

It is a pleasure to thank Professor A Griffin for early advice and criticism, and Dr J S Plaskett for his derivation of equation (2.57), and especially for his insistence on the respect due to the continuous spectrum in integral equations.

Appendix. Surface energy and surface correlation energy with p > 0

We calculate U with a mode-conserving cutoff as in $5.3; other prescriptions for the cutoff can be handled similarly. We shall also consider briefly gcorr = (U - uo), the change which occurs in the surface energy of a neutral fluid when the Coulomb forces are turned on. This is perhaps the more interesting quantity, because we saw in 45.2 that 00 itself is altogether different from the corresponding (and physically relevant) surface energy UF of an ideal Fermi gas. (For p=O, u0 vanishes, and 0 coincides with uc,rr.)

It is convenient to write u=uC+Au (Al . 1)

where uc is the result calculated for cosine modes, i.e. as if the final term in the Hamiltonian (5.31) and the right-hand side of (5,32(a)) were dropped, and Au corrects for these terms. & is calculated just as was 4, but substituting i 2 ~ ( k , p,) for W ( k , p r ) . This is a straightforward exercise and yields, in the notation of (5,6),

( A l . 2)

where we have anticipated the fact that the two explicitly Q-dependent terms give the same end-result in the integration over k, which yields the negative quantity

uc=wPQ 2 - ( 1 + p)1/2/327 + [( 1 + (2)3/2 - 1]/487(2} ( A l . 3)

t E PQlwp. (Al .4)

In (Al.2) and ( A l . 3) the first (negative) terms stem from the size-dependent cor-


rection SQ= -n/4L to the cutoff, explained in 95.2.1; with a size-independent cutoff these terms would be absent. In this notation, 00 (equation (5 -21)) becomes

00 = - 0pQ2[/96~. (Al . 5 )

I t remains to calculate A u = (AD++ Au.-), which subdivides into independent contributions from even and odd modes. As we shall show,

( A l . 6) ho*(k) -= (Zn-i)-I dQ@ d - log D+(k, Q) $ d!2

where D* are defined by (5 ,32 (b ) ) , and the contour encircles the positive real Cl axis in the positive direction. For given k , the functions D have zeros at the exact eigen- mode frequencies, and poles at the C-mode frequencies QB. Thus d log DjdQ has poles with residues + 1 at the cormer and - 1 at the latter, whence (Al . 6) is indeed the required difference of zero-point energies (one factor of 4 allows for the two surfaces of the slab). Finally, since Au is evidently a pure surface term (lacking 1,-proportional parts), we may safely take the limit I,+ CO. Then the sums in D, are replaced by integrals according to (5 .33) , D+ and D- coincide, and we have

Ao(k) = (Zni)-I$ dCl$Q iR log D(K, a) (Al .7)

In the last step we have relied on the arguments of 93.1, There are several standard alternative forms for Ao(k). The prescription

(2~ i ) - l$dQ . . . can be replaced by - 71-1s: dQ Im . . . , followed by an integration by parts where the integrated term vanishes because as Q+ CO one has Cl log D - R-1. This yields

Ao(k) = (Zn)-lf: dQ Im log D(k, R). ( A l . 9)

Alternatively, the contour in (-41.7) may be deformed to run along the imaginary axis, again followed by an integration by parts. Writing Q=iy, this yields

which shows explicitly that Ao(k), and hence Au, like crc, is negative. A short calculation confirms that A o reduces to a1 in the limit ( = O , and shows that for ( ~ 1 it behaves like -(wPQ2/32n[2) (log [+ constant)t. Hence for very large ( it becomes neligible compared to the other components of ocorr, namely (uc - 00) - wpQ2/64n[.

-f Though the limit [= /3&/wp -+ CO is academic (physical values of 5 being of the order of unity), it is amusing to consider it formally from the point of view wp = (4nne2/m)l/2 +. 0 at fixed ,13 and Q. Then ocort. is the change in U due to the charge density ne regarded as a weak perturbation. The result is:

UcorrN (wp2Q3/64~P)E1 - ( 2 w d P Q ) log (BQ/wp)l

and the logarithmic singularity shows that in the HD model, just as in the many-body theory of the electron gas, the long range of the Coulomb interaction prevents one from expressing its consequences as power series in ne.

1014 G Barton

However, at f = O (i.e. at p= 0), where 00 vanishes, crc, Acr and gcorr are all negative, and though gcorr must turn positive with increasing f , this happens only at quite unphysically large f . By contrast, with an L-independent cutoff, ocorr, like cr, is always positive. In other words, for reasonable values of f , finite dispersion makes no difference as regards the signs of cr or crcorr, though it does affect their magnitudes.

Finally, our expressions for cr may be compared with others written in terms of a dielectric function E , by setting E-+ EHD. For instance, Griffin and Kranz (1977), adopting a size-independent cutoff, obtain (in our notation)

a log ~ ( 0 , k, O)+log 277

Their second term corresponds precisely to our Ao(k), while the term 4 log E corresponds to the second term of crc (equation (Al.2)), as expected with such a cutoff. If UGK represents ucorr rather than U, it is not clear why it appears to contain no equivalent at all to the counterterm - 00.

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