Finite-size effects and nonadditivity in the van der Waals interaction

Abstract

We obtain analytically the exact nonretarded dispersive interaction energy between an atom and a perfectly conducting disk. We consider the atom in the symmetry axis of the disk and assume that the atom is predominantly polarizable in the direction of this axis. For this situation we discuss the finite-size effects on the corresponding interaction energy. We follow the recent procedure introduced by Eberlein and Zietal together with the old and powerful Sommerfeld's image method for nontrivial geometries. For the sake of clarity we present a detailed discussion of Sommerfeld's image method. Comparing our results for the atom-disk system with those recently obtained for an atom near a conducting plane with a circular aperture, we discuss the nonadditivity of the van der Waals interactions involving an atom and two complementary surfaces. We show that there is a given ratio z/a between the distance z from the atom to the center of the disk (aperture) and the radius of the disk a (aperture) for which nonadditivity effects vanish. Qualitative arguments suggest that this quite unexpected result will occur not only for a circular hole, but for any other symmetric hole.

Image method in the calculation of the van der Waals force between an atom and a conducting surface

“…where f (r) must assume the unit value for points belonging to the surface of the sphere, in order to satisfy equation (54). Using the fact that the Green function must be symmetric 68 by the exchange r ←→ r ′ , and since f (r) = 1 for |r| = R, we are led to the result…”

On membrane interactions and a three-dimensional analog of Riemann surfaces

“…This paper considers a Riemann space consisting of two copies of R 3 connected by a flat circular branch disk and computes the potential generated by a point charge located in the first R 3 . More recently, Riemann spaces have been considered in [32][33][34][35]. It can be shown that Sommerfeld's solution with a straight branch line and Hobson's solution with a circular branch loop can be related by an inversion transformation, see appendix D.…”

Sommerfeld's Image Method in the Calculation of Van Der Waals Forces

“…A relevant example is that of an atom interacting with an infinite plane with a circular aperture, which leads to repulsion as shown by Levin et al [2]. In a recent paper [17], we treated this problem by Sommerfeld's image method and obtained an analytical result for the corresponding interaction energy of this system. Our results agree with those previously obtained by Eberlein and Zietal [5].…”

“…In this connection, we would like to mention particularly the possibility of studying finite size effects and, in some cases, non-additivity effects involving complemetary surfaces. In the latter case, the complementary geometries of an atom-disk system and the system constituted by an atom and a plane with a circular aperture was studied in [ [17]], where a surprising result was found, namely: that there exists a distance between the atom and the center of the aperture for which the non-additivity effects vanish. The authors believe this result may occur in other systems involving complementary surfaces.…”

“…With this method we may also treat the complementary geometry, namely, the atom-disk system. The interaction energy when the atom is in the symmetry axis of the disk and with dominant pollarizability in this direction is [17] …”