Casimir Interaction between Plane and Spherical Metallic Surfaces

Abstract

We give an exact series expansion of the Casimir force between plane and spherical metallic surfaces in the non trivial situation where the sphere radius R, the plane-sphere distance L and the plasma wavelength λP have arbitrary relative values. We then present numerical evaluation of this expansion for not too small values of L/R. For metallic nanospheres where R, L and λP have comparable values, we interpret our results in terms of a correlation between the effects of geometry beyond the proximity force approximation (PFA) and of finite reflectivity due to material properties. We also discuss the interest of our results for the current Casimir experiments performed with spheres of large radius R ≫ L.The Casimir force is a striking macroscopic effect of quantum vacuum fluctuations which has been seen in a number of dedicated experiments in the last decade (see for example [1,2] and references therein). One aim of the Casimir force experiments is to investigate the presence of hypothetical weak forces predicted by unification models through a careful comparison of the measurements with quantum electrodynamics predictions. This aim can only be reached if theoretical computations are able to take into account a realistic and reliable modeling of the experimental conditions. Among the effects to be taken into account are the material properties and the surface geometry, these effects being also able to produce phenomena of interest in nanosystems [3,4].A number of Casimir measurements have been performed with gold-covered plane and spherical surfaces separated by distances L of the order of the plasma wavelength (λ P ≃ 136nm for gold), making material properties important in their analysis [5]. As those measurements use spheres with a radius R ≫ L, they are commonly analyzed through the Proximity Force Approximation (PFA) [6], which amounts to a trivial integration over the sphere-plate distances. An exception is the Purdue experiment dedicated to the investigation of the accuracy of PFA in the sphere-plate geometry [7], the result of which will be given as a precise statement below.In the present letter, we give for the first time an exact series expansion of the Casimir force between a plane and a sphere in electromagnetic vacuum, taking into account the material properties via the plasma model (see Fig. 1). We present numerical evaluation of this expansion which are limited to not too small values of L/R, because of the multipolar nature of the series. We show below that these new results lead to a striking correlation between the effects of geometry and imperfect reflection when evaluated for nanospheres, with R, L and λ P having comparable values. In the end of this letter, we also discuss the interest of these results for the Casimir experiments performed with large spheres R ≫ L [7].Our starting point is a general scattering formula for the Casimir energy [8]. Using suitable plane-wave and multipole bases, we deduce the Casimir energy E PS be- tween a plane and a spherical metallic surface in electromagnetic...

Going beyond PFA: A precise formula for the sphere-plate Casimir force

“…It turns out that to estimate the Casimir force and its gradient with an accurary of 10 −4 , as we seek, one needs l max 6R/a, m max 6 R/a, n max 10 λ T /a. Previous simulations [46,47] with l max = 45 were used to compute quite precisely the force for R/a < 20. As we mentioned earlier, a new large simulation [31] reached l max = 2 × 10 4 , making it possible to probe aspect ratios up to R/a ∼ 4 × 10 3 .…”

Casimir microsphere diclusters and three-body effects in fluids

“…To obtain a finite matrix A, the number of spherical waves (or spherical harmonics Y m ) is truncated to a finite order . Because this expansion converges exponentially fast for spheres [24,27], we find that ≤ 12 suffices for < 1% errors with the geometries in this paper. (Conversion from planewaves to spherical waves is performed by a semi-analytical formula [24] that involves integrals over all wavevectors, which was performed by a standard quadrature technique for semi-infinite integrals [28].)…”

The Casimir effect: medium and geometry

“…The equation in a medium has the same form (53) in the coordinate space, where the operation rot in (53) is a standard one in the coordinate space.…”

“…In my opinion, one of the principal questions in the Casimir effect is to understand the conditions under which it is possible to use the approximation (52) and the resulting equation (53) in the Casimir effect, i.e. when it is possible to neglect spatial dispersion.…”