Material dependence of Casimir forces: gradient expansion beyond proximity

TL;DR

This paper develops a derivative expansion method to accurately quantify curvature corrections to the proximity force approximation for Casimir forces between curved surfaces, including temperature effects and material dependence.

Contribution

It introduces an explicit expression for the leading curvature correction to PFA for metals and insulators at room temperature, extending the accuracy of Casimir force estimates beyond the PFA.

Findings

Derived explicit formula for PFA correction amplitude $ heta_1$

Found logarithmic scaling of corrections in the non-retarded limit

Discovered strong temperature dependence of $ heta_1$ for gold

Abstract

A widely used method for estimating Casimir interactions [H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948)] between gently curved material surfaces at short distances is the proximity force approximation (PFA). While this approximation is asymptotically exact at vanishing separations, quantifying corrections to PFA has been notoriously difficult. Here we use a derivative expansion to compute the leading curvature correction to PFA for metals (gold) and insulators (SiO$_2$) at room temperature. We derive an explicit expression for the amplitude $\hat\theta_1$ of the PFA correction to the force gradient for axially symmetric surfaces. In the non-retarded limit, the corrections to the Casimir free energy are found to scale logarithmically with distance. For gold, $\hat\theta_1$ has an unusually large temperature dependence.