Calculation of the formation volume of vacancies in solids

By considering the creation of a vacancy as an isothermal and isobaric process the Gibbs formation energy $g^f$ can be expressed as $g^f=CB\Omega$, where $B$ is the isothermal bulk modulus and $\Omega$ the mean volume per atom. In former publications a method was proposed in which $C$ was assumed constant. It led to explicit expressions for the formation entropy $s^f$ and enthalpy $h^f$ as functions of $T$. Extending this method to the formation volume $v^f$ and to the parameters ${\beta }^f=\left(\frac{1}{v^f}\right){\left(\frac{\partial v^f}{\partial T}\right)}_P$ and ${\kappa }^f=-\left(\frac{1}{v^f}\right){\left(\frac{\partial v^f}{\partial P}\right)}_T$ explicit relations with the pressure and temperature derivatives of the bulk properties $B$ and $\beta$ (the volume thermal-expansion coefficient) are derived. The main results can be summarized as follows. (i) The parameters $v^f$, ${\beta }^f$, and ${\kappa }^f$ are found to depend on $P$ and $T$ but in a different way than the bulk properties $\Omega$, $\beta$, and $\kappa$. (ii) The volume $v^f$ consists of two terms $v_h^f={\left(\frac{\partial h^f}{\partial P}\right)}_T$ and $v_s^f=-T{\left(\frac{\partial s^f}{\partial P}\right)}_T$. At high temperatures the term $v_s^f$ is never negligible in comparison to $v_h^f$. (iii) The thermal-expansion coefficient of $v_h^f$ is approximately equal to $-\beta$ whereas that of $v_s^f$ is positive and one order of magnitude larger. In order to check the reliability of the proposed method numerical values of $v^f$ were calculated by using elastic data. In all types of solids investigated close agreement with experiment was found. The calculated values of ${\kappa }^f$ and ${\beta }^f$, for alkali halides, agree with the corresponding values that can be extracted from the curvature of the plots of conductivity and diffusion versus pressure at various temperature. The same holds true for tracer experiments on Na and Cd.