Amplitude-Independent Longitudinal Ultrasonic Attenuation in Superconducting Lead

The amplitude-independent attenuation of longitudinal ultrasound propagating in the [001] crystal-lographic direction in single-crystal superconducting lead was investigated, using the pulse-echo method, at frequencies between 10 and 210 MHz. 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. The electron mean free path was evaluated for the specimens used in the experiment, and numerical data on the superconducting-to-normal electronic-attenuation ratio for well-characterized specimens are given for reduced temperatures greater than 0.6 where the data retain sufficient accuracy to be useful. The attenuation ratio derived from the measurements generally depends both on the ultrasonic frequency and impurity concentration, in disagreement with the prediction of the BCS theory. No unique value of the superconducting energy gap can be derived from the data using that theory. The decrease in the attenuation ratio near the transition is too rapid to be consistent with the BCS prediction. For reduced temperatures less than about 0.5 (depending on frequency and impurity concentration), the experimentally determined attenuation ratio decreases less rapidly than the BCS prediction. In pure deformed lead to temperatures where the mean freepath is phonon-limited, the attenuation ratio is frequency-dependent and departs from BCS most strongly for the lowest ultrasonic frequencies. This result is to be compared with recent results reported by Deaton for pure, well-annealed lead, where departures from BCS were found to be most pronounced at the highest ultrasonic frequencies. When the electron mean free path was impurity-dominated, the attenuation ratio was found to be nearly frequency-independent in the frequency range studied, but is dependent on impurity concentration. The data agree most closely with the BCS prediction for high-impurity concentrations. The frequency dependence of the electronic attenuation in the normal and superconducting states is in qualitative agreement with calculations of the electronic thermal conductivity, which show that the phononlimited electron mean free path in the superconducting state is less than the corresponding free path in the normal state. An attempt is made to interpret the data semiquantitatively using this idea, and numerical estimates of the superconducting-to-normal ratio of an effective (energy-independent) phonon-limited free path are given in the reduced temperature interval $0.6\le t\le 1.00$. The mean-free-path analysis reproduces the main qualitative freatures exhibited by the data just below the transition, but fails to quantitatively explain some of the less prominent features of the data in this temperature range. For reduced temperatures less than about 0.5, the mean-free-path mechanism is incapable of explaining the behavior of the observed attenuation ratio. The results of the present experiment are briefly compared with results recently reported by others.