Ultrasonic Attenuation of Longitudinal Waves in Pure Superconducting Niobium in High Magnetic Fields

By adapting the approach of Kadanoff and Falko to the two-band model, the ultrasonic attenuation coefficient for longitudinal waves in two-band superconductors is obtained. The attenuation coefficient is expressed in terms of various correlation functions. Using the techniques developed by Maki for evaluating the various correlation functions for pure superconductors in high fields, the longitudinal attenuation for arbitrary $\mathrm{ql}$ is obtained for the case in which the propagation vector of the sound wave is parallel to the magnetic field. The ${(H_{c2\mathrm{}}-H)}^{\frac{1}{2}}$ field dependence is still a feature of the two-band expression. This expression is used to explain the purity dependence observed in the ultrasonic attenuation in superconducting niobium placed in high magnetic fields.