Electron Contribution to the Temperature Dependence of the Elastic Constants of Cubic Metals. I. Normal Metals

By direct calculation of the electron energy of a metal crystal as a function of elastic strain and temperature, it is shown that the elastic constants can exhibit a $T^2$ dependence at low temperatures. This temperature dependence arises from the displacement of the Fermi surface during strain and the simultaneous transfer of electrons across Brillouin zone boundaries. The results for face-centered cubic and body-centered cubic metals are obtained in terms of the energies of symmetry points in the Brillouin zone and the electron density of states and its first and second derivatives with respect to energy. The magnitude and algebraic sign of the temperature dependence are shown to depend critically on the shape of the Fermi surface and the electron density distribution. It is also shown that the form of the electron contribution to the temperature dependence of the elastic constants is directly analogous to that derived for the thermal variation of the paramagnetic susceptibility of metals at low temperatures.