Ultrasonic Attenuation in Normal Metals at Low Temperatures

Expressions for the attenuation αd and αₜ of plane dilatational and shear sound waves are obtained by solving the Boltzmann transport equation for the electron distribution function f without assuming the existence of a relaxation time τ for the collision term in this equation. Instead the collision integral is considered to arise explicitly from the interaction of electrons with thermal phonons and impurities. Making the usual "ideal metal" assumptions, it is found that the attenuation in general depends on a set of effective relaxation times τLM which are associated with the various terms in the expansion of f in a series of spherical harmonics YLM(θ, φ); the τLM are independent of the subscript M, and hence the same set {τL} determines both αd and αₜ. Explicit expressions for τL are derived.For the case in which all the τL equal to one another and equal to τ say, the analytical expressions for αd and αₜ obtained here are the same as those of Pippard. However, usually τL are not equal to one another. It is then found that when λl (λ is the wavelength of the sound wave and l a mean free path of the electrons), τ in Pippard's expressions must be replaced by τ₂ and, contrary to what is usually assumed, α would not be in general proportional to the electrical conductivity σ (σ∝τ₁). When λl, the attenuation, with one exception, is independent of {τL} and is the same as that given by Pippard. For λ~l, and τL not equal to one another, α may be calculated numerically if the ratios τLτ₁ are known; the results of one such calculation show that the deviations from Pippard's analytical expressions are at most about 20%, provided τ in the latter is identified as τ₂.Lastly, the possible influence of electron-electron collisions on attenuation is briefly discussed.