A simple macroscopic derivation is given of the non-retarded Van der Waals interaction between two semi-infinite dielectric media.Neutral macroscopic bodies, when placed at a distance of the order of 1000 A, attract each other as a result of the Van der Waals forces between their atoms. Lifshitz [1] has developed a macroscopic theory for this phenomenon by introducing fluctuating terms in the Maxwell equations for a dielectric medium. For the interaction energy per unit area at T = 0 between two semi-infinite media a distance d apart IAfshitz finds in the case that d << k (;~ is the principal absorption wave length of the material); ; 1 Starting from the Drude-Lorentz model of an atom (harmonic oscillator model) Renne and Nijboer [2] gave an atomistic derivation of the non-retarded interaction between one atom and a semi-infinite medium by summing two-particle, three-particle, etc., interactions. Their derivation can be extended to the case of two semi-infinite media and for arbitrary atoms [3]. In this note we will give a simple alternative macroscopic derivation of eq. (1), which can easily be generalized to more complicated configurations.Consider two semi-infinite dielectric media a distance d apart. We look for those solutions of the equations of electrostatics: div D = 0, curl/~ = 0, which oscillate harmonically in time. It will be obvious that these equations can be satisfied by: 1°: curl E = O, ~(¢o) = 0 (longitudinal bulk modes); 20: div D = 0, e(c0) = co (transverse bulk modes); 30: div~ = 0, curl E = 0 (surface modes). At T = 0 the interaction between the media may be found from the sum of the zero point energies ~k ½~wk of all modes evaluated at a distance d minus the corresponding sum for infinite d. This procedure, due to Casimir [4], who applied it to the case of ideal conductors in vacuo, can be justified in our case by considering a macroscopic Hamiltonian of the system which is equivalent to a sum of harmonic oscillator Hamiltonians with the frequencies ¢o k as eigenfrequencies. A similar Hamiltonian was used by Hopfield [5]. Now the frequencies of the bulk modes 1 ° and 2 ° are evidently independent of the distance d. Hence for the evaluation of the Van der Waals interaction only the surface modes 3 ° need be considered.Introducing the electrostatic potential ~ (x,y,z,t) by E = -grad ~ we have to solve the equation V2~ = 0, subject to the boundary conditions q~ and ~(w)a~/Sz continuous at z = 0 and z = d (the z-axis is chosen perpendicular to the plane-parallel boundaries of the media).Assuming ¢ to be of the form

SynopsisA s h o r t a n d s t r a i g h t f o r w a r d m e t h o d for t h e conversion of slowly converging lattice sums into expressions with good convergence is presented. As an illustration a wellk n o w n expression for t h e M a d e l u n g c o n s t a n t is rederived. T h e m e t h o d is t h e n applied to two general types of lattice sums, t h e latter of which has not been t r e a t e d before. § 1. Introduction. Lattice sums, i.e. summations over the sites of an infinite perfect lattice of some potential energy function or force interaction, occur in m a n y branches of crystal physics. The appearance of lattice sums in calculations of the lattice energy of ionic crystals and in considerations on the stability of the various lattice types are the best known and also oldest examples. Other cases where the evaluation of these sums is of some importance are investigations of the electromagnetic, optical, or elastic properties of crystals.A difficulty met with in the evaluation of lattice sums always has been the question of their convergence. It presents itself under two aspects, firstly the convergence of the series as such and secondly the rapidity of convergence, the latter question being mainly of practical importance. It is a well-known fact that interactions decreasing slowly with distance (e.g. the Coulomb interaction) can give rise to lattice sums which exhibit only a conditional convergence. Since in this case the sum of the series is not uniquely determined, physical arguments will have to be invoked in order to arrive at physically meaningful answers. But even in the case of unconditional convergence, the actual rapidity of convergence is generally so poor that the series as they stand, are not very useful for computations.In the existing literature on lattice sums a great deal of attention has been devoted to these problems. The classic examples in this connection are the treatments of M a d e l u n g 1) .) (for the Madelung-constant) and of E w a l d s) (for more general types of lattice sums), in which the application of a summation recipe, based on physical arguments, and the transformation of the lattice sum into a more rapidly converging form are closely connected. Still *) An equivalent method has been developed by O r n s t e i n and Z e r n i k e o).--3 0 9 . -

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