Relation between Third-Order Elastic Moduli and the Thermal Attenuation of Ultrasonic Waves in Nonconducting and Metallic Crystals

Abstract

The background loss of all materials, and sometimes the principal loss in single crystals, is connected with the conversion of acoustic waves into thermal phonons. One source of conversion is the thermoelastic loss resulting from the flow of heat from the compressed (hotter) part of the wave to the expanded (cooler) part. This source accounts for about half the thermal attenuation in a metal, but only about 4% of that in an insulator. Another source of thermal attenuation—first suggested by Akhieser—is connected with the separation of phonon-mode temperatures by a suddenly applied stress, followed by an equilibration of these temperatures with a relaxation time τ. This source has been called “phonon viscosity.” A method for evaluating this source of attenuation was proposed in previous papers [W. P. Mason and T. B. Bateman, J. Acoust. Soc. Am. 36, 644–652 (1964); W. P. Mason, Physical Acoustics (Academic Press Inc., New York, 1965), Vol. 3B, Chap. VI, pp. 235–286]. It involves the use of the third-order elastic moduli to determine the energy stored by the phonon-mode temperature separations, together with a relaxation time τ to equilibrate this energy. Recently, new determinations of these third-order moduli have been made for NaCl, KCl, MgO, and Y3Fe5O12 (YIG). The present paper shows that the formulas predict the measured attenuations (longitudinal and shear waves) within about 25%. Silicon, without oxygen, has a higher thermal conductivity than silicon with dissolved oxygen. This is reflected in the higher attenuation of oxygen-free material, in agreement with theory. A theoretical evaluation of the attenuation is made for a longitudinal wave propagated along a (111) axis. For metals, it is shown that the “phonon-viscosity” attenuation is in the same order as thermoelastic attenuation. A method for evaluating this loss in single-crystal lead is devised. The attenuation measured over a temperature range is consistent with the sum of the thermoelastic attenuation and the “phonon-viscosity” attenuation.