Ultrasonic Attenuation of Shear Waves in Superconducting Lead

Ultrasonic attenuation measurements have been made for shear waves propagating along the [001] and [110] directions in high-purity single crystals of lead at frequencies 10-50 Mc/sec and temperatures 2-15°K. It was found that the temperature-dependent dislocation attenuation is only a small fraction of the total electronic attenuation of shear waves at the transducer voltages used, and therefore reliable data on the attenuation in the superconducting state can be obtained. The rapid-fall electromagnetic attenuation just below the transition temperature was analyzed on the basis of Fossheim's model, and bulk values of the London penetration depth were obtained for the two directions of propagation. The results are in reasonable agreement with thin-film data. The residual attenuation remaining at temperatures below the rapid-fall region was analyzed on the basis of the BCS equation, using the zero-temperature energy gap $2\Delta \mathrm{}\left(0\right)\mathrm{}$ as an adjustable parameter. The apparent energy gap $2\Delta \mathrm{}\left(0\right)\mathrm{}$ is frequency-dependent for q∥[110] and $\mathcal{E}\parallel \left[1\mathrm{}\overline{1}0\right]$, varying from $4.4k{T}_c$ at 10 Mc/sec to $3.4k{T}_c$ at 50 Mc/sec. For q∥[110] and $\mathcal{E}\parallel \mathrm{}\left[001\right]\mathrm{}$, for q∥[001] and $\mathcal{E}\parallel \mathrm{}\left[110\right]\mathrm{}$, and for q∥[001] and $\mathcal{E}\parallel \mathrm{}\left[010\right]\mathrm{}$, the data indicate essentially frequency-independent energy gaps of $3.1k{T}_c$, $3.4k{T}_c$, and $3.4k{T}_c$, respectively. In addition to the above calculations of $2\Delta \mathrm{}\left(0\right)\mathrm{}$ using the ordinary technique of fitting the low-temperature portion of the data with the exponential in the BCS equation, the data were also fitted over the entire temperature range with BCS curves, using $2\Delta \mathrm{}\left(0\right)\mathrm{}$ to adjust the fit. The resulting values of $2\Delta \mathrm{}\left(0\right)\mathrm{}$ were somewhat larger than those given above, with the relative values remaining about the same. The range of gap values observed is in good agreement with the recent tunneling data of Rochlin.