Casimir Interaction Between a Plane and a Sphere: Correction to the Proximity-Force Approximation at Intermediate Temperatures

TL;DR

This paper derives an analytical correction to the proximity-force approximation for the Casimir interaction between a plane and a sphere at intermediate temperatures, accounting for geometry and temperature effects.

Contribution

It provides the first analytical correction formula for the plane-sphere Casimir interaction at intermediate temperatures, including zero-frequency contributions.

Findings

Correction involves temperature-dependent logarithmic terms.

Zero-frequency Matsubara contribution significantly affects the correction.

Analytical formula applicable under typical experimental conditions.

Abstract

We consider the Casimir interaction energy between a plane and a sphere of radius $R$ at finite temperature $T$ as a function of the distance of closest approach $L$. Typical experimental conditions are such that the thermal wavelength $\lambda_T=\hbar c/k_\mathrm{B}T$ satisfies the condition $L\ll \lambda_T\ll R$. We derive the leading correction to the proximity-force approximation valid for such intermediate temperatures by developing the scattering formula in the plane-wave basis. Our analytical result captures the joint effect of the spherical geometry and temperature and is written as a sum of temperature-dependent logarithmic terms. Surprisingly, two of the logarithmic terms arise from the Matsubara zero-frequency contribution.