Second- and Third-Order Elastic Constants of Alkali Metals

Abstract

The second- and third-order elastic constants of lithium, sodium, potassium, and rubidium in the bodycentered cubic structure are calculated. The relationship between Brugger elastic constants and Fuchs elastic constants is worked out. The Brugger elastic constants, which are defined as the derivatives of the energy with respect to the Lagrangian strain, are widely used to express experimental results. The Fuchs elastic constants, which are defined as the derivatives of the energy with respect to homogeneous expansion and volume-conserving homogeneous shear, are often more convenient for calculations in terms of atomistic considerations, and are particularly convenient for calculations with pseudopotentials. They are used here to calculate the contribution of the band-structure energy to the elastic constants using the local pseudopotential proposed by Ashcroft. This pseudopotential contains the core radius as the only adjustable parameter. The contribution of the band-structure energy to the elastic constants is represented as a summation of two kinds of derivatives of the wave-number characteristics over the reciprocal lattice points—those with respect to homogeneous expansion and those with respect to volume-conserving homogeneous shear. The core radius which gives the best fit to the experimental second-order elastic constants agrees with that determined by Ashcroft from data on the Fermi surface or on the resistivity of liquid metals. The band-structure energy term is found to make a small contribution to the second-order elastic constants but an indispensable contribution to the third-order elastic constants.