Einstein-Kanzaki model of static and dynamic lattice relaxation: Application to vacancies in metallic hydrogen

A method is proposed for calculating the formation energy of localized defects in crystalline solids with pair forces of arbitrary range. The theory is most useful in the cases of small mass or high temperature for which, in addition to the usual static relaxation, changes in the lattice vibrations make a significant contribution. Defect migration is not described however. A self-consistent Einstein approach is used, each particle in the crystal oscillating with its own frequency about an average position. The total free energy is minimized with respect to all of these frequencies and positions. This minimization is made tractable by the assumption that large changes in frequency and position occur only for a finite number of particles near the defect; the changes for all the other particles are treated linearly. The result is very similar to Kanzaki's $\overrightarrow{\mathrm{k}}-$ space "lattice statics" formalism. However, instead of being 3 × 3 the lattice Green's function becomes a 4 × 4 matrix, thereby encompassing changes in Einstein frequencies as well as particle positions. The method is applied to calculate the free energy of vacancy formation in metallic hydrogen.