Some properties of dielectric constant and phonon frequency of a superconductor

Abstract

The dielectric constant and the phonon frequency in a super-conductor have been investigated by means of Green’s functions for the electronic and the total charge density fluctuation. Making use of the linearization approximation the exact expressions for the electronic dielectric constant are obtained in both the high-frequency limit and the low-frequency limit. The proof of the sum rule formula given by Rickayzen has been confirmed in the small wave number limit. The relationship between the effective interaction between electrons and the dielectric constant has been examined by considering matrix elements of the density fluctuation between pair states. In the Appendix, for a simplified effective interaction truncated in the neighborhood of the Fermi surface, the explicit expressions of the dielectric constant and the phonon frequency are obtained. The differences in the superconducting state from the values in the normal metal have been pointed out using these expressions. Some vertices discarded in Rickayzen’s theory give rise to a change in the location of a singularity to appear in the expression for the dielectric constant at a lower frequency than the characteristic frequency corresponding to the energy gap. There appears a change in the phonon frequency at very short wave lengths but no change appears in the sound velocity.