Surface-impedance approach solves problems with the thermal Casimir force between real metals

Abstract

The surface impedance approach to the description of the thermal Casimir effect in the case of real metals is elaborated starting from the free energy of oscillators. The Lifshitz formula expressed in terms of the dielectric permittivity depending only on frequency is shown to be inapplicable in the frequency region where a real current may arise leading to Joule heating of the metal. The standard concept of a fluctuating electromagnetic field on such frequencies meets difficulties when used as a model for the zero-point oscillations or thermal photons in the thermal equilibrium inside metals. Instead, the surface impedance permits not to consider the electromagnetic oscillations inside the metal but taking the realistic material properties into account by means of the effective boundary condition. An independent derivation of the Lifshitz-type formulas for the Casimir free energy and force between two metal plates is presented within the impedance approach. It is shown that they are free of the contradictions with thermodynamics which are specific to the usual Lifshitz formula for dielectrics in combination with the Drude model. We demonstrate that in the impedance approach the zero-frequency contribution is uniquely fixed by the form of impedance function and does not need any of the ad hoc prescriptions intensively discussed in the recent literature. As an example, the computations of the Casimir free energy between two gold plates (or the Casimir force acting between a plate and a sphere) are performed at different separations and temperatures specific for the regions of the anomalous skin effect and infrared optics. The results are in good agreement with those obtained by the use of the tabulated optical data for the complex refraction index and plasma model. It is argued that the surface impedance approach lays a reliable theoretical framework for the future measurements of the thermal Casimir force.

Results from electrostatic calibrations for measuring the Casimir force in the cylinder-plane geometry

“…The finite temperature contribution added to the quantum fluctuations has originated a lengthy debate about the interplay of the thermal contribution with the finite conductivity properties of the surfaces (see for instance [48][49][50][51][52][53][54][55][56][57][58][59] for the initial steps of the debate). On the experimental side, attempts to evidence the thermal contribution discriminating various models have been reported for the sphere-plane geometry [9], while proposals using torsional balances in the parallel-plane configuration [60,61] are under development.…”

Some Mathematical Aspects of the Lifshitz Formula for the Thermal Casimir Force

“…The fallacy in the calculations of [12] concerning the type of the impedance is that it disregards the requirement that the reflection properties for virtual photons on a classical boundary should be the same as for real photons. Paper [6] demonstrates in detail that by enforcing this requirement the exact and Leontovich impedances coincide at zero frequency and lead to the conclusions of [13] which are in perfect agreement with the Nernst heat theorem.…”

“…(1.1) and (3.9) lay down the theoretical foundation for the calculation of the thermal Casimir effect. In fact the approaches of [2,3,4] and the impedance approach of [6,13] predict quite different magnitudes of the thermal corrections to the Casimir force. Up to separation distances of a few hundred nanometers, the thermal correction predicted by the impedance approach is negligibly small.…”

“…The contour C 1 is chosen so as to avoid all these branch points and enclose all the photon eigenfrequencies. After the integration by parts and some rearrangement [13], we find the equivalent but more simple expression for the Casimir free energy…”

Casimir energy for a spherical boundary within the surface impedance approach: Obtaining negative values

“…In [3] the authors showed that, under suitable conditions, the transition between attractive and repulsive regime only depends on the Surface Impedance (SI) of the material constituting the boundary. Since then a lot of effort in that direction has been done, see [9,10,11,12] and references there in. As far as we know the first ones to use the SI to compute Casimir energy were Mostepanenko and Trunov in [13].…”