Casimir effects in monatomically thin insulators polarizable perpendicularly: nonretarded approximation

A flat monatomically thin insulating sheet is modelled initially as a square lattice and then as an amorphous distribution of harmonic oscillators, polarizable only perpendicularly to the sheet. In an approximation neglecting dissipation and retardation, we calculate the polarizability X(ω,k) per unit area as a function of the frequency ω and surface-parallel wave-vector k of an externally applied electric field. To find the underlying so-called local fields one must first replace the familiar three-dimensional Lorenz–Lorentz accounts of dielectric functions with their well-established and very different two-dimensional analogues. Image fields are given by weighted integrals over X(0,k); the poles of X(ω,k) identify the normal-mode frequencies ω¯(k)?>. The Hamiltonian version of the theory is quantized via the normal modes; from it we determine the van der Waals interaction of the sheet with a nearby atom, and between two dynamically identical parallel sheets.


Background and motivation
In theories of Casimir effects, reflecting layers of nominally infinitesimal thickness play at least three different roles. (i) For a spherical shell taken from the start as perfectly reflecting, Boyer (1968) found that the Casimir stress on it is directed outward, a paradoxical conclusion that has occasioned a vast literature. (ii) Regarding many of their electric properties, giant carbon molecules like C 60 can be modelled quite well as monatomically thin plasma sheets (Barton and Eberlein 1991), with parameters fitted to a single base plane in graphite as discussed by Fetter (1973Fetter ( , 1974. On extending this model to couple such two-dimensional (2D) plasmas to the quantized Maxwell field, one finds that the expression reported by Boyer is a mathematically fascinating but physically far-subdominant part of the energy of spherical plasma shells (Barton 2004a(Barton , 2004b(Barton , 2004c, as is its analogue for indefinitely extended plane plasma sheets (Barton 2005a(Barton , 2005b). (iii) Very different treatments of 2D electron gases via Dirac-like equations underlie the fast-burgeoning theories of graphene layers (see e.g. Katsnelson 2012), which at low frequencies behave in ways totally unlike the 2D plasmas considered under (ii). Latterly, attention has been focussed also on their van der Waals and Casimir interactions with perfect and imperfect metals (references in section 5.3).
Here, however, we shall be concerned with a different generalization of thin-sheet Casimir physics, from 2D plasmas mimicking conductors to 2D insulators, where the central problem is how to take proper account of the local-field effects which dominate their electromagnetic response functions. One prompt for trying to spell out the elements of the theory is a recent paper (Parashar et al 2012, cited as PMSS; see also Milton et al 2013), whose conclusions depend in part on an assertion that sheets polarizable only perpendicularly leave electromagnetic fields wholly unaffected. On examining simple oscillator models of monatomically thin sheets it emerges that this is false: our main purpose is to study the polarizability of such sheets, and some of the physical properties that the polarizability governs.
We shall consider only polarization perpendicular to the sheet, because the theory of parallel polarization, though it features more intricate local fields, is not basically different from the theory of plasma sheets: one merely adds a tangential restoring force to the equation of motion of the charged fluid modelling the plasma. Technically speaking, the fields due to parallel polarization can be ascribed equally well to the surface-charge density produced by its surface divergence, just as for a plasma; whereas perpendicular polarization is unique in producing, as we shall see, discontinuities in the fields but no charge densities whatever.
The sheet is taken to be flat 1 and to extend indefinitely in the (x, y) plane; it is modelled as monatomically thin, and its constituent atoms as dynamically identical linear simple-harmonic oscillators perpendicular to the sheet. The oscillator frequency, internal mass, momentum, coordinate and Hamiltonian are , m, p, ζ and h 0 = p 2 /2m + m 2 ζ 2 /2. The atomic (i.e. the individual oscillator) polarizability 2 is and each oscillator is coupled to electric fields only in the dipole approximation: Recall that for atoms (unlike oscillators) α is generally of the order of (atomic radius) 3 . The scenario where → ∞ at fixed finite α, i.e. where (ω) = α for all ω, we call the nondispersive limit. Dissipation is disregarded; so is retardation 3 , which will be taken into account in a future paper, cited here as II. Conventionally, the nonretarded approximation is identified as the formal limit c → ∞; here it applies provided all physically pertinent distances are far below c/ , and also far below c/ω for all important frequencies ω. We expect, and shall find, that the model makes sense only subject to certain stability conditions of the kind familiar in systems of coupled oscillators. For instance, the normal modes of just two such dipole-dipole-coupled oscillators a distance a apart have squared frequencies 2 1 ∓ α/a 3 , indicating instability unless a > α 1/3 . As regards the response functions and normal modes of the sheet, our basic reasoning is classical, except insofar as it assumes that stable atoms with given (ω) exist.
We shall start by determining the response of the sheet to an externally applied field at this stage ω and k are merely mathematical parameters chosen independently, regardless of how (2) might be realized in practice. Hats denote unit vectors. Generally we shall suppress the factor exp(−iωt) common to all the fields in question. When ω 2 / 2 is very close to 1, i.e. near resonance, one must, in the denominator of , allow for the natural width = α 4 /3c 3 of the oscillator by replacing ω → ω − i /2. As an evidently relativistic correction this is, in principle, beyond the scope of the nonretarded approximation: it will feature in II, but here we disregard it, excluding near-resonance behaviour 4 from our remit. For reference we quote the potential ψ and the field e = −∇ψ at (s, z) due to a polarized atom modelled as a point dipoleẑq 0 placed at the origin: The part of e proportional toẑq 0 δ(s)δ(z) has been dropped: it is ineffective because it cannot contribute to the interaction between the atoms constituting the sheet, and because its source cannot polarize itself. The calculations will demonstrate that physically reasonable models of minimally thin sheets can indeed be polarized in this way. The contrary assertions by PMSS are invalidated by a hidden inadequacy of their initial Ansatz (2a), which for our scenario and in our units would assign to them a dielectric function ε = 1 + 4πλ eẑẑ δ(z). One can see the inadequacy by taking δ(z) = 0 outside and δ(z) = 1/d inside a sheet of thickness d (as do PMSS), and calculating the polarizationẑP produced by a finite externally applied fieldẑE app3 . Then the total perpendicular dipole moment per unit area is Q = Pd = E app3 λ e d/ d + 4πλ e , whence lim d→0 Q = 0. In fact it is well known that the dielectric responses of effectively 2D bodies cannot be described by means of dielectric functions of anything like the kind assigned to three-dimensional (3D) ones. For instance, in 2D the familiar Lorenz-Lorentz (Clausius-Mossotti) theories of 3D local fields must be re-designed root and branch: the more recent history of this and of related problems 5 can be sampled say through the work of Philpott and Sherman (1975), Lee and Bagchi (1980), Christiansen et al (1998) and Ryazanov and Tishchenko (2006).

Preview
Section 2 calculates the polarizability X (k,ω) of a simple square lattice. The prime object is to introduce 2D local fields and the associated stability criteria in a setting and for a model that are long-established, and unproblematic as long as the atomic diameter is well below the lattice spacing a. The second equation of (7) gives the result in terms of the function f (k) defined over the first Brillouin zone by (8). The poles of X identify the normal-mode frequenciesω(k) via (11).
Section 3.1 defines our model of an amorphous sheet. It takes the atoms to be distributed at random, except that no two can come closer than a crucial minimum distance b. The polarizability X (k, ω) is given by (25) in terms of the strength parameter µ ≡ πnα/b, and of the function L(κ = kb) defined and then approximated in (21)-(23). A plausibility argument anticipates a Debye cutoff k < k D , supported in section 3.4 by a count of normal modes. The squared eigenfrequencies are given by (39). Meanwhile, section 3.2 explicates how, once X is known, the expressions for potential and field off the sheet can be matched across it. Section 3.3 determines the potential (36) due to the polarization produced by a point charge e at a distance ζ from the sheet; remarkably, when ζ b the charge experiences the same force (38) as it would from a point dipole at the image position −ζ .
Section 4 uses the polarizability and the normal modes from section 3 to reformulate the theory in canonical terms: the Hamiltonian H 0 of the sheet is identified in section 4.1 (which summarizes itself), and quantized in section 4.2. Equation (55) expresses H 0 in terms of the familiar normal-mode creation and annihilation operators (54), linked to the primary canonical operators q(k), p(k) through (52) and (53). Applications require H 0 , plus the operator ψ(s, z) for the potential off the sheet as expressed in terms of the q(k) by (56).
Section 5 illustrates applications. Section 5.1 calculates the van der Waals potential U (z), equation (58), between the sheet and a ground-state atom at a distance z. When z b this reduces to the relatively simple form (59), falling proportionally to 1/z 4 . In the nondispersive limit it reduces further, and somewhat surprisingly, to the quasi-classical expression U nd = −µb 0 d 2 0 /4 [1 + 2µ] z 4 , where d is the z-component of the atomic dipole-moment operator, and 0 d 2 0 its mean-square value in the ground-state. Section 5.2 finds the electrostatic contribution β(µ, κ D ) to the cohesive energy of the sheet per unit area: β is positive (repulsive) for small κ D , but turns negative for the values of µ and κ D most likely to be of interest. Finally, section 5.3 considers two identical sheets separated by a distance L = bλ. The normal-mode frequenciesω(κ, λ) are given by (68), and the exact van der Waals potential u(µ, κ D , λ) per unit area by (69). For λ 1 it reduces to u −(3π/32)h n 2 α 2 /(1 + 2µ) 3/2 L 4 , as indicated by (70).
Finally, section 6 summarizes the main results, and outlines some points needing to be kept in mind when considering their implications.

Square lattice
We study a square lattice with lattice constant a, so that n ≡(number of atoms)/(unit area)= 1/a 2 . It is acted on by an applied fieldẑẼ app (k) exp(ik · s), producing (dipole moment)/(unit area) =ẑQ(s). Defining polarizability/(unit area) ≡ X , one has 6 Our object is to determine X . The local field experienced by an atom at the Fourier transform of the field due to all the other atoms. Thus It must be kept in mind that E self (s) is defined only on the sheet, and has physical significance only at positions occupied by an atom: it is not the field E pol (s, z) = −∇ψ pol (s, z) perceived elsewhere as due to the polarization of the sheet. In particular, the next section will spell out that the response functions governing ψ pol cannot be read directly from G or from its analogues, but must be determined in their own right. Meanwhile, the central problem is to find G. Define It proves convenient to introduce a dimensionless function f (κ) such that where the prime excludes (only) ν 1 = 0 = ν 2 . Define The mathematics associated with Z and with the function f has a distinguished history dating back at least as far as Hardy (1919) 7 : the challenge is to express slowly converging double sums in terms of far more easily manageable ordinary sums. For the physics in a modern context see Ambjorn and Wolfram (1983), and the appendix to Christiansen et al (1998); for mathematical sophistication and extensive references see McPhedran et al (2004McPhedran et al ( , 2007. For long waves (κ 1) Thus at the zone centre f (0, 0) = Z . At the edge-centres and at the corners, asymptotics supplemented with some numerics yield f (π, 0) −0.935 and f (π, π) −2.646. Figure 1(b) from Christiansen et al (1998) indicates that f is negative all along the edges, minimal at the corners and zero on just one closed locus surrounding the centre. Normal modes have nonzero Q in absence of an applied field, whence they are signalled by the poles of X . Thus the dispersion relation for their frequenciesω reads For long waves, to first order in ak = κ, Remarkably, the frequency drops with rising wave-number. The minimal value of f at the zone corners shows that If a < 1.38α 1/3 , the sheet would be unstable, presumably in the sense of polarizing spontaneously 8 ; compare this with the instability condition a < α 1/3 we have noted for just two atoms.

Generalities
Our model of such a sheet takes the atoms to be distributed over the plane at random, with (mean number)/(unit area) = n, and makes a rudimentary allowance for pair correlations by imposing a minimum interatomic distance b. Accordingly, the field experienced by each atom, called the local field E loc , is the applied field plus the field E self =ẑE self due to the polarized sheet treated as continuous, but with a circular hole of radius b centred on the atom in question. (The main reason why hydrodynamic models of plasma sheets are so much easier to handle than insulators is precisely that their electrons are delocalized, whence the effective fields they experience are not rapidly varying, and are indeed the same as the macroscopic fields perceived outside.) As to orders of magnitude, one expects We consider applied fields (2); theñ and the equation of state reads with X (k) a polarizability per unit area. Next, on the same pattern as for the lattice, we define a function G(k) such that To determine G, we combine the z-components of the fields at (s, 0) due to the dipoleŝ zQ exp(ik · s )d 2 s at (s , 0): On scaling ks = x and redefining we have where the H i are Struve functions. Asymptotically L(κ 1) = 2 π cos(κ + π/4) 2 κ 3/2 − 1185 64κ 7/2 + · · · + sin(κ + π/4) 21 4κ 5/2 − 42735 512κ 9/2 + · · · .
The function L(κ) is plotted in figure 1: it turns negative at κ 0 1.63, with a first (and absolute) minimum L min (κ m 2.6) −0.24. It will prove convenient to introduce a dimensionless coupling constant then the polarizability reads Plausibly, µ attains its maximal value M when α = b 3 /8 is the static polarizability of a perfectly conducting sphere with radius b/2. For orientation, square and hexagonal lattices (labelled sq and hx respectively) with nearest-neighbour distance b have Roughly speaking, µ encodes the strength of the response of an element of the sheet to the field it experiences, while L encodes the effect on this response of variation parallel to the sheet. As to G, we repeat the caution voiced in the preceding section just below equation (6), and complement it by spelling out the physically pertinent potential ψ pol off the sheet. We do this from first principles, i.e. directly from Coulomb's law, in order to dispell any (though misconceived) reservations about admitting delta-singularities ab initio. Thus, in virtue of the first equation of (3), and then changing the integration variable to σ = s −s, Accordingly Finally, in view of the minimum separation b, and anticipating the Debye cutoff presently to be motivated for normal modes, we assume that the polarization cannot support Fourier componentsQ(k) with arbitrarily large k; in other words we impose the condition expecting κ D to be of order unity. Correspondingly, all integrands under d 2 k . . . are understood to carry a step-function factor θ (k D − k) or θ(κ D − κ), which will not be shown explicitly. Improvement on these prescriptions would require microscopic study of the insulating sheets in question: meanwhile, the Debye cutoffs are best regarded as part of the definition of our crude but hopefully serviceable model.

Matching conditions
In one sense, the delta-function part of (29) is just a place holder: off the sheet one can simply discard it, while a microscopic description on the sheet requires the full theory featuring E self and L. Nevertheless the equation of state governing the physics everywhere else can be encapsulated into the standard macroscopic conditions for matching potential ψ and total electric field E = −∇ψ across the sheet. On defining discontinuities by and similarly for other fields, the matching conditions read Though (32) follows with perfect generality from Gauss's law, here we have derived (33) only via the electrostatic representation E = −∇ψ, unwarranted beyond the nonretarded regime. However, in fact it continues to apply subject to Maxwell's equations, which entail also that E 3 is the only Maxwell-field component having a δ(z) singularity on the surface. The proof is straightforward but tedious, and is given in II.
Finally it may bear repeating that in view of (32) and of disc [ψ] = − 0+ 0− dz E 3 , nonzero polarization Q automatically entails a singularity 4π Q(s)δ(z) in E 3 : the connection is explicated text-book fashion say by Barton (1989).

Image potential
To illustrate the uses of the theory in section 3.1, we calculate the image potential and the image force generated by a point charge e fixed to the right of the sheet at (0, ζ ). (Since this is a zerofrequency scenario, the result remains unaffected by retardation: one need merely envisage the Coulomb gauge, and note that photons do not then couple to e.) We start from the applied potential The z-component of the applied field experienced by the sheet is Since it is time-independent one has ω = 0, whence substitution into (28) and integration over the polar angle then lead to The integral is awkward because of the factor [. . .] and of the finite upper limit. However, if at least one of ζ , s, |z| is much larger than b, then ψ pol is dominated by contributions from small k, and can be approximated by replacing L(kb) → L(0) = 2, and extending Remarkably, on the right of the sheet this is the potential due to a point dipole 2µbe/(1 + 2µ) at the image position, a distance ζ behind the sheet; on the left of the sheet, it is the potential due to a point dipole −2µbe/(1 + 2µ) coinciding with the point charge.
The force on the point charge is the exact expression follows trivially on adapting (36), and the approximation from (37). It might seem paradoxical that ψ pol fails to vanish as ζ → 0, when the charge sits on the sheet, and cannot therefore polarize it in the first place. In fact ψ pol (s, z; 0) is zero while lim ζ →0± ψ pol (s, z; ζ ) are not: the approach to the limit is nonuniform, with the zero sitting at the point of discontinuity between peaks of opposite signs to its left and right.

Normal modes
The normal-mode frequenciesω(k) are signalled by the poles of X , whence the dispersion relation readsω 2 / 2 = 1 + µL, allowing the response function (25) to be re-expressed as For long waves, to first order in κ = kb Here tooω/ drops as k rises 9 . For comparison, the analogous equation (12) for the square lattice can be written as ( To ensure that normal modes do not outnumber the pertinent degrees of freedom of the atoms, one must impose a Debye cutoff For our two simple lattices where the κ D are the values prescribed by (42) for an amorphous sheet, given the surface densities n in (26). By (39) the sheet remains stable if µ |L min | < 1, as is likely since |L min | 0.24 while (26) suggests µ M 1. Otherwise translation invariance would presumably be broken by a standing wave of spontaneous antiferroelectric polarization 10 , with the lowest wavenumber for which 1+ µL vanishes. Finally, we anticipate retardation corrections to the extent of noting that our surface-bound normal modes are strictly stable only ifω(k) < ck; otherwise they decay by emitting radiation with frequencyω(k) and wave-vector K ≡ (k,ẑk 3 ), wherek 3 ≡ ω 2 (k)/c 2 − k 2 . Maxwellian theory turns these modes into narrow resonances of the TM reflection and transmission amplitudes. Plasma sheets behave differently, because, without a mechanical restoring force, i.e. lacking a finite frequency parameter like , resonances are replaced by threshold singularities at zero frequency.

Canonical theory
We continue to disregard relativistic and retardation effects, and thus the quantized Maxwell field, which will be considered in paper II. Meanwhile, the interaction energy dH ext of a surface element with an externally generated applied field, and the electrostatic interaction energy dH es of two surface elements with each other, are

Hamiltonian
The canonical 2D fields are p(s), having dimensions [M T −1 ], and dimensionless q(s) normed so that Q(s) = ϕq(s); the coefficient ϕ will be identified presently. Then the Hamiltonian for a single isolated sheet is where θ is the Heaviside step-function, and the last term is the Coulomb self-energy of the sheet. We fit H 0 to our model from section 3, i.e. to n oscillators per unit area, each with internal displacement ζ , charge e, and static polarizability α = e 2 /m 2 , by equating their internal (nonelectrostatic) potential energies. This shows that (m 2 q 2 /2 = nm 2 ζ 2 /2) ⇒ (q = √ nζ ); Hamilton's equations reaḋ Define Fourier transforms 11 p(s) where classically the superscript + indicates the complex and quantally the Hermitean conjugate. It is straightforward to verify that with L from (21)-(23). Accordingly, the Fourier representation diagonalizes H 0 , as by translation invariance it must; and it identifies the normal-mode frequenciesω through tallying with (39).

Quantization
The theory is quantized by imposing the equal-time commutation rules They are satisfied by introducing annihilation and creation operators in standard fashion: Then Recall that the potential ψ generated by Q outside the sheet is odd in z, whence E = −∇ ψ is odd and E z = −∂ψ/∂z is even:

van der Waals interaction between the sheet and an atom
We consider the van der Waals interaction energy U (z) between the sheet and an isotropic atom in its ground state, fixed at a distance z. Write the atomic energy eigenstates and eigenvalues as | j and j , ground state j = 0, with j0 = j − 0 ; the atomic dipole-moment operator as d; and, by hindsight, define The interaction Hamiltonian is H int = −E·d, with E from (57). In view of the first equation of (53) and of (46), and in an obvious shorthand, second-order perturbation theory yields 12 The integral is messy, on account of its finite upper limit and of the awkward variation ofω via L(kb). We evaluate it only for z b: then it is dominated by kz 1, so that ω(k) ω(0) = √ 1 + 2µ and This may be contrasted with the long-distance but still nonretarded van der Waals potential of an atom near a plasma sheet, which falls proportionally to 1/z 3 . In the nondispersive limithω(k) =h √ 1 + µL(κ) → ∞ one has [ j0 +hω(k)] →hω(k). Then closure over the atomic states reduces (58), paradoxically, to the prima facie classical expression

Cohesive energy
The electrostatic contribution to the ground-state cohesive energy per unit area is the second term within the braces subtracts the energy that the oscillators would have if they were infinitely far apart. For stable systems the radicand is positive. Since L can change sign, the sign of F is a matter for calculation.
For weak coupling we approximate by keeping only the first term in the Taylor series for {. . .}: not all that different from the somewhat cavalier estimates (64).

van der Waals interaction between two sheets
Call the sheets 1 and 2, a distance L apart. Call the normal-mode frequenciesω(k, L), so thatω(k) =ω(k, ∞). From the second equation of (44), their interaction Hamiltonian may be written as Combining W with the q + q part of H (1) 0 + H (2) 0 as given by (50, 51), we see that thē ω 2 (k, L), are the eigenvalues of the potential-energy matrix [. . .] in

Thus 13ω
conveniently scaled to The subscripts specify the sign in (68), which is the same as the parity of q with respect to the median plane.
Because of the minus sign in the radicand ofω − (κ, λ), at small λ the two-sheet system is at greater risk from instability than a single sheet. In fact the limit λ → 0 of (68) appears to make no physical sense. For instance, one might plausibly expect that at zero separation the two sheets, each with surface number-density n, merge into a single sheet with surface density 2n; but thenω + (κ, L = 0; n) would reduce toω(κ; 2n) (as happens for plasma sheets), whereas comparing (68) with (39) one sees that it does not. Nor is it clear whether taking λ < 1 is compatible with the minimum dipole-dipole distance b assumed for single sheets. In a preliminary exploration it seems premature to pursue such questions at length. Here we merely observe that with the values of µ and κ D for square or for hexagonal lattices, though λ = 0 would admit instability, λ 1 does not; fudge the issue by imposing the condition λ > 1; and continue to assume stability without further comment.
The van der Waals potential energy per unit area is This integral too is messy, for much the same reasons as the one in (58) This may be compared with u ∼ 1/L 5/2 for two plasma sheets (Bordag 2006) 14 , and with u ∼ 1/L 3 for two graphene sheets (Drosdoff et al 2012, Klimchitskaya andMostepanenko 2013).

Summary
To resume, our main results are as follows. Recall that they apply in the van der Waals regime, disregarding retardation from the start. (i) From monatomically thin sheets polarizable only along their normal, externally applied fields evoke detectable responses of the same kind as they do from macroscopic parallel-sided slabs. (ii) However, their polarizability must be calculated using ideas and methods specific to 2D systems: in particular, to such sheets the familiar 3D concepts of dielectric functions and Lorenz-Lorentz relations do not apply, and provide no sort of handle (cf point (xi) below). (iii) The classic example of the simple-square 2D lattice in the point-dipole approximation is readily adapted to a rough model of an amorphous sheet, which yields results in almost closed form. (iv) It is of the essence that the polarizability exhibits spatial dispersion, i.e. that it is a function not only of the frequency ω (via the underlying atomic polarizability), but also of the wave-vector k parallel to the sheet. It determines (v) the image field evoked by an external charge; (vi) the frequencies of the normal modes; and, after quantization, (vii) the electrostatic contribution to the cohesive energy; plus (viii) the van der Waals attraction between a sheet and a nearby atom, and between two parallel sheets.
In considering the implications, one needs to exercise some circumspection.
(ix) Our object has been to try and clarify some traditional and quite basic questions about the Casimir-type physics of minimally thin surfaces, rather than to derive accurate expressions for realistic systems. Specifically, the dipole approximation, while it minimizes technical complications, is unlikely to prove wholly adequate at the densities in question (except perhaps for some very dilute adsorbed monolayers): at separations of no more than a few atomic units, (a) higher multipoles become competitive, and (b) the wave-functions of electrons in different atoms may overlap non-negligibly. For a critique on such lines, see e.g. Kempa et al (2005).
(x) Real sheets admit parallel as well as perpendicular polarization. The former can be either curl-free or divergence free, making three types of normal mode for given k, and the effects (v)-(viii) listed above are governed jointly by all three.
(xi) As already stressed in section 1.1, our truly 2D sheets have little in common with parallel-sided slabs containing large numbers N of lattice planes, not even with slabs whose overall thickness is small on any macroscopic scale. The central problem for such slabs is to find out how N governs departures from the local field appropriate to bulk material, and thereby position-dependent departures from the Lorenz-Lorentz formula for the dielectric function. This is much the same as the problem of the variation of the local field near the surface of a halfspace. It can be resolved only by calculation: surprisingly perhaps, essentially bulk conditions turn out to prevail already a very few ( 10) lattice planes away from the nearest surface. (Clear accounts are given by Mahan andObermair 1969 andMahan 1972.) Thus our 2D formulae apply when N = 1; macroscopic (continuum) electrostatics should apply at least roughly when N is well above say 10; and numerics are needed in between.