Casimir problem of spherical dielectrics: numerical evaluation for general permittivities

• Published in: Physical review. E, Statistical, nonlinear, and soft matter physics Statistical, nonlinear, and soft matter physics, 2002-08, Vol.66 (2 Pt 2), p.026119-026119
• Main Authors: Brevik, I, Aarseth, J B, Høye, J S
• Format: Article
• Language: English
• ISSN: 1539-3755
• DOI: 10.1103/PhysRevE.66.026119

The Casimir mutual free energy F for a system of two dielectric concentric nonmagnetic spherical bodies is calculated, at arbitrary temperatures. The present paper is a continuation of an earlier investigation [Phys. Rev. E 63, 051101 (2001)], in which F was evaluated in full only for the case of ideal metals (refractive index n= infinity ). Here, analogous results are presented for dielectrics, for some chosen values of n. Our basic calculational method stems from quantum statistical mechanics. The Debye expansions for the Riccati-Bessel functions when carried out to a high order are found to be very useful in practice (thereby overflow/underflow problems are easily avoided), and also to give accurate results even for the lowest values of l down to l=1. Another virtue of the Debye expansions is that the limiting case of metals becomes quite amenable to an analytical treatment in spherical geometry. We first discuss the zero-frequency TE mode problem from a mathematical viewpoint and then, as a physical input, invoke the actual dispersion relations. The result of our analysis, based upon the adoption of the Drude dispersion relation at low frequencies, is that the zero-frequency TE mode does not contribute for a real metal. Accordingly, F turns out in this case to be only one-half of the conventional value at high temperatures. The applicability of the Drude model in this context has, however, been questioned recently, and we do not aim at a complete discussion of this issue here. Existing experiments are low-temperature experiments, and are so far not accurate enough to distinguish between the different predictions. We also calculate explicitly the contribution from the zero-frequency mode for a dielectric. For a dielectric, this zero-frequency problem is absent.