Pressure Variation of the Elastic Constants of Sodium

Abstract

The pressure variation of the single crystal elastic constants of sodium has been measured using a modified ultrasonic pulse echo method. The values found for the pressure derivatives of the elastic constants are: $\frac{d{C}_{44}}{\mathrm{dP}}=1.63, \frac{d{C}^{\prime }}{\mathrm{dP}}=0.226, \frac{d{B}_s}{\mathrm{dP}}=\mathrm{3.60.}\mathrm{}$ The notation $C^{\prime }=\frac{(C_{11}-C_{12})}{2}$ and $B_s=\frac{(C_{11}+2\mathrm{}{C}_{12})}{3}$ has been used. The experimental observation that the elastic anisotropy ratio $\frac{C}{C^{\prime }\left(C\equiv C_{44}\right)}$ does not depend on pressure indicates that one can positively neglect interaction of ion cores as a contribution to the elastic stiffness of sodium. The results are interpreted in terms of Fuchs' theoretical calculation of the Coulomb contribution to the shear stiffnesses of the alkali metals. The interpretation indicates that as sodium is compressed, the value of the electronic wave function at the boundaries of the atomic polyhedra increases more rapidly than ${\Omega }^{-\frac{1}{2}}$, where $\Omega$ is the volume of the atomic polyhedron. The volume variation of the value of the wave function of the lowest electronic state at the boundaries of the atomic polyhedra is found to be: ${\left[\frac{d\ln {\mu }_0\left(r_s\right)}{d\ln \Omega }\right]}_{\Omega ={\Omega }_0}=-\mathrm{0.27.}\mathrm{}$