Lattice dynamics of fcc Ca

Abstract

A large single crystal of fcc Ca was grown and was used to study the lattice dynamics of this divalent metal by coherent inelastic neutron scattering. The phonon dispersion curves were measured, at room temperature, along the [$\ensuremath{\xi}00$], [$\ensuremath{\xi}\ensuremath{\xi}0$], [$\ensuremath{\xi}\ensuremath{\xi}\ensuremath{\xi}$], and [$0\ensuremath{\xi}1$] symmetry directions. The dispersion curves bear a striking resemblance to those of fcc Yb, which is also a divalent metal with an electronic band structure similar to that of Ca. In particular, the shear moduli ${c}_{44}$ and $\frac{({c}_{11}\ensuremath{-}{c}_{12})}{2}$ differ by a factor of 3.4, which implies that fcc Ca (like fcc Yb) is very anisotropic with regard to the propagation of elastic waves. The frequencies of the ${T}_{1}$[$\ensuremath{\xi}\ensuremath{\xi}0$] branch for $\ensuremath{\xi}$ between approximately 0.5 and 0.8 are slightly above the velocity-of-sound line determined from the low-frequency measurements. Since a similar effect has been observed in fcc Yb, it is natural to assume that the anomalous dispersion exhibited by the ${T}_{1}$[$\ensuremath{\xi}\ensuremath{\xi}0$] branches of these metals is due to an electronic effect. To provide further support for this assumption we have performed a band theoretical calculation of the generalized susceptibility $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ of fcc Ca. The results suggest that, for $\ensuremath{\xi}$ between approximately 0.6 and 0.8, there is a relative decrease in the electronic screening of the vibrational motion of the nuclei, which may account for the positive dispersion exhibited by the ${T}_{1}$[$\ensuremath{\xi}\ensuremath{\xi}0$] branch in this range of $\ensuremath{\xi}$ values. The data were used to evaluate the elastic constants, the phonon density of states, and the lattice specific heat of fcc Ca.