The hopping process of a vacancy defect in a crystal

The average time associated with the movement of a vacancy defect between two adjacent sites is estimated using a classical density functional model. Our calculation shows that this time is much faster than that of the typical vacancy diffusion process over longer distances in the crystal lattice. The process of movement of a vacancy defect from one lattice site to its neighbor is depicted as a series of readjustments in the local crystalline structure. The free energy barrier to this process is obtained here using the classical density functional theory. The theory requires the structure factor of the corresponding homogeneous liquid and can therefore be studied with the basic interaction potential as the only necessary input. In this work we consider mono-atomic systems interacting respectively with hard sphere and Lennard-Jones potentials. We show that as the location of the point vacancy shifts from a lattice site to its nearest neighbor on the face-centered cubic lattice, the corresponding free energy curve constitutes two symmetric minima separated by a barrier. The height of this barrier is obtained in terms of the interaction potential from the present density functional theory model. Assuming a low concentration of vacancies, the escape rate of the vacancy defect from one lattice site to the adjacent one is then estimated by using Kramer's theory for crossing the energy barrier