On the many-body Van der Waals forces between macroscopic bodies

Abstract

The non-retarded Van der Waals interaction of a neutral atom with a semi-infinite perfect crystal is analyzed through the coupling of the atom to the electronic polarization waves of the crystal. First, we derive a closed form, valid to all order in d -1 (where d is the distance separating the atom from the crystal surface), for the exact interaction energy of the system. Second, by means of a renormalization procedure, we extract from the exact energy, the first non-vanishing order (and the only significant one, macroscopically) which depends on d -3. A numerically tractable formula results which includes the complete N-body contribution to this order. As a function of the “refractivity” z = 4 πNα 0 of the crystal, the d -3 energy possesses a singularity at z = z 0 where z 0 is any pole of the static dielectric constant ε 0( z). For the case of cubic crystals a singularity occurs at z 0 = 3, a situation known as the “ 4π 3 polarizability catastrophe” in the theory of polar dielectrics. This feature clearly indicates the limited validity of the perturbational treatment of the problem. Finally, a comparison is made between our results and those recently obtained by Renne and Nijboer. Also, the necessary modifications to allow for retardation effects are discussed briefly.