Direct mode summation for the Casimir energy of a solid ball

TL;DR

This paper calculates the Casimir energy of a solid sphere in an infinite medium using direct frequency summation and contour integration, considering specific permittivity and permeability conditions, with implications for sonoluminescence.

Contribution

It introduces a direct frequency summation method for Casimir energy of a solid sphere under specific electromagnetic conditions, including a dilute approximation and Debye expansion considerations.

Findings

Derived a general expression for Casimir energy of a solid ball.

Performed calculations for dilute spheres considering first-order effects.

Discussed implications for explaining sonoluminescence via Casimir effect.

Abstract

The Casimir energy of a solid ball placed in an infinite medium is calculated by a direct frequency summation using the contour integration. It is assumed that the permittivity and permeability of the ball and medium satisfy the condition $\epsilon_1 \mu_1=\epsilon_2\mu_2$. Upon deriving the general expression for the Casimir energy, a dilute compact ball is considered $(\epsilon_1 -\epsilon_2)^2/(\epsilon_1+\epsilon_2)^2\ll 1$. In this case the calculations are carried out which are of the first order in $\xi ^2$ and take account of the five terms in the Debye expansion of the Bessel functions involved. The implication of the obtained results to the attempts of explaining the sonoluminescence via the Casimir effect is shortly discussed.