Low-Temperature Amplitude-Independent Longitudinal Ultrasonic Attenuation in Normal Lead

The amplitude-independent normal-state attenuation of longitudinal ultrasound propagating in the [001] crystallographic direction in single-crystal lead was investigated using the pulse-echo method at frequencies between 10 and 210 MHz in the temperature range 1 to 8°K. A method for the detection of amplitude-dependent effects is presented; the work was limited to lightly cold-worked high-purity lead and well-annealed tin- and thallium-doped lead, where the amplitude independence of the results was experimentally demonstrated. A method for the evaluation of the electron mean free path from normal-state attenuation data is presented; the electron mean free path was evaluated for the specimens used in the experiment. The phonon-limited mean free path was found to be proportional to $T^{-a}$ with $a=4.02\mathrm{}\pm 0.05$. Below 4.2°K, relative phonon-limited mean-free-path values derived from the acoustic data agree closely with the resistivity data of Van den Berg, but above 7.2°K the acoustic and resistivity results differ. The $\mathrm{ql}$ and frequency dependence of the electronic attenuation is in good agreement with the free-electron calculations of Pippard, and the magnitude of the Sommerfeld-Wilson deformation-potential constant, as derived from the present data, is $\mathrm{}\left|C\right|=(1.63\mathrm{}\pm 0.25)$ eV. In the low-$\mathrm{ql}$ limit, the magnetic field dependence of the electronic attenuation is not in agreement with results based on the free-electron model, nor is it consistent with the empirical behavior of the zero-field attenuation. The data do not support the electron-damped dislocation mechanism proposed by Mason to account for the anomalous superconducting-to-normal electronic-attenuation ratio that has been reported for lead. The implications of the present results as applied to Mason's theory are briefly discussed.