Model of a Clean Superconductor Containing a Random Array of Vortices

Abstract

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 ($\mathrm{ql}\ensuremath{\ll}1$) and low frequencies, at low vortex densities ($\frac{B}{{H}_{c2}}\ensuremath{\ll}1$) near the critical temperature of the superconductor ($T\ensuremath{\simeq}{T}_{c}$). We find that ${{a}_{L}}^{\mathrm{II}}({T}_{c})$, the vortex cross section for ultrasonic attenuation of longitudinal waves propagating parallel to the vortex axis, equals 210 \AA{}, which agrees well with the experimental result of 240 \AA{} in vanadium.