Model of a Clean Superconductor Containing a Random Array of Vortices

Equations for the Green's function and correlation functions of a superconductor containing a random array of vortices are derived. These differ from the equations for a superconductor containing paramagnetic impurities inasmuch as here, the scattering potentials are macroscopic and cannot be treated in Born approximation. The scattering self-energy is approximated by the T matrix of scattering theory, which has been calculated elsewhere for real energies. In particular, the ultrasonic attenuation is derived for phonons of long wavelength (ql1) and low frequencies, at low vortex densities (BHc21) near the critical temperature of the superconductor (TTc). We find that aLII(Tc), the vortex cross section for ultrasonic attenuation of longitudinal waves propagating parallel to the vortex axis, equals 210 {}, which agrees well with the experimental result of 240 {} in vanadium.