Ultrasonic Determination of the Superconducting Energy Gap in Vanadium

Ultrasonic attenuation measurements on single-crystal cylinders axially oriented along the $〈110〉$ crystallographic direction were used to determine the temperature-dependent superconducting energy gap $2{\epsilon }_0\mathrm{}\left(T\right)\mathrm{}$ of three samples of vanadium of different purities. An echo-pulse technique using longitudinal waves was used. The propagation direction was along the $〈110〉$ for all the experiments. The zero-temperature energy gap for the three samples was found to be $(3.4\mathrm{}\pm 0.2)k{T}_c$, $(3.5\mathrm{}\pm 0.2)k{T}_c$, and $(3.5\mathrm{}\pm 0.1)k{T}_c$. These values were computed by averaging the results obtained from 200 or 300 experimental points. They are in agreement with the Bardeen, Cooper, and Schrieffer result that $2{\epsilon }_0\mathrm{}\left(0\right)\mathrm{}$ should be equal to $3.5k{T}_c$ for all superconductors. The dependence of the normalized energy gap on the reduced temperature of these samples is closely predicted by the BCS theory. These experiments were performed within the frequency range 100 to 450 Mc/sec, and in all of them the product of the ultrasonic wave vector times the electron mean free path was smaller than one. The temperature dependence of the attenuation in the superconducting state showed small oscillations. It is found that the amplitude of these is a small effect for very pure samples. Moreover, evidence has been collected to show that these oscillations are not an inherent property of the superconducting state in vanadium but are probably associated with acoustic modes of wave propagation.