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

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 λT=ℏc/kBT satisfies the condition L≪λT≪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.

Casimir Physics beyond the Proximity Force Approximation: The Derivative Expansion

“…[57], in Ref. [60], the authors explored the Casimir interaction between a plane and a sphere of radius R at a finite temperature T, in terms of the distance of closest approach, a. Noting that, under the usual experimental conditions, the thermal wavelength λ T satisfies a ≪ λ T ≪ R, they evaluated the leading correction to the PFA, applicable to such intermediate temperatures.…”

“…Defining the variables x = a/R and τ = a/λ T , and the deviation δF(T) = F(T) − F(0), in the intermediate temperature regime x ≪ τ ≪ 1, it is found in Ref. [60] that…”