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

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) = hc/k(B)T satisfies the condition L << lambda(T) << 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.