Temperature dependence of the thermal-expansion coefficient of vacancies

Self-diffusion experiments on Zn and Cd under pressure show that the thermal-expansion coefficient ${\beta }^{\mathrm{act}}$ of the activation volume decreases with temperature according to the law $\frac{1}{T}$. On the other hand in the case of Na recent calculations show that ${\beta }^{\mathrm{act}}$ increases with temperature. By combining thermodynamics with a recent model [Varotsos, Ludwig, and Alexopoulos, Phys. Rev. B 18, 2683 (1978)] it is shown that ${\beta }^{\mathrm{act}}$ decreases with temperature when the diffusion experiments are carried out at temperatures well above the Debye temperature. This is the case of Zn and Cd. On the other hand measurements at lower temperatures give a ${\beta }^{\mathrm{act}}$ that increases with temperature as found in the case of sodium. Furthermore, a criterion is given according to which when one knows the elastic and expansivity data of a solid he can predict whether the $\frac{1}{T}$ behavior is obeyed or not.