Investigation of the temperature dependence of the Casimir force between real metals

We investigate the Casimir force acting between real metals at nonzero temperature. It is shown that the zero-frequency term of the Lifshitz formula is difficult to interpret in the case of a real metal described by the Drude model. This is because the scattering theory underlying the Lifshitz formula is not well formulated when the dielectric permittivity takes into account dissipation. To give the zeroth term of the Lifshitz formula a definite meaning, different prescriptions have been used recently by different authors with diverse results. These results are shown to be improper and in disagreement with experiment and the general physical requirements. We propose a prescription that is a generalization of the Schwinger, DeRaad, and Milton recipe formulated earlier for ideal metals. On this basis, detailed numerical and analytical computations of the temperature Casimir force are performed in the configurations of two plane plates and of a spherical lens (sphere) above a plate. The corrections due to nonzero temperature and finite conductivity found here are in agreement with the limiting case of a perfect metal and fit all experimental and theoretical requirements. Among other facts, previous results obtained in the framework of the plasma model are confirmed. It appears that they are the limiting case of Drude model computations when the relaxation parameter goes to zero. A comparison with the Casimir force acting between dielectric test bodies is made.