Comment on Gor'kov and Eliashberg's theory for far-infrared absorption by small metallic particles

Numerous errors exist in Gor'kov and Eliashberg's expressions for the electric susceptibility $\chi \left(\omega \right)$ of a small metallic particle. We have reevaluated the integral for $\chi \left(\omega \right)$ for Dyson's circular ensembles and use the results to examine the dependence of the far-infrared absorption coefficient $\alpha \left(\tilde{\nu }\right)$ and its derivative $\frac{d\alpha }{d\tilde{\nu }}$ on particle size and the geometric standard deviation $\sigma$ of a log-normal distribution of sizes. We conclude that experimental observation of the oscillations characteristic of quantum-size effects by far-infrared absorption requires use of derivative spectroscopy and a distribution of particles with $\sigma \lesssim 1.05$ for the symplectic ensemble, which is the most favorable case. Thus, experimental verification of the applicability of the Gor'kov-Eliashberg theory is unlikely with currently available methods of sample preparation.