Application of the Method of Lattice Statics to Vacancies in Na, K, Rb, and Cs

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Abstract
We have calculated the lattice distortion produced by a single vacancy in Na, K, Rb, and Cs. The calculations have been carried out using the technique of lattice statics, which is based on the Fourier transformation of the direct-space equilibrium equations, making consistent use of discrete lattice theory. Three distinct types of potential have been used to describe the interactions between the host atoms. The first of these applies only to sodium, and contains an ion-electron-ion term derived from the measured phonon dispersion curves. The second applies only to potassium, and has been similarly obtained. The third is based on a model pseudopotential and applies to all four metals. Comparison has been made between our displacements due to a single vacancy in Na, using the first of these potentials, and analogous results obtained by a semidiscrete method in which only the atoms in the first five shells are allowed to relax. The agreement is reasonable for atoms in the first two neighbor shells about the vacancy, but poor for atoms farther away. The calculated displacements have been used to calculate the dilatations and relaxation energies associated with single vacancies in alkali metals. There is a large discrepancy between the magnitudes of these quantities calculated using the first Na potential and those obtained using the second Na potential, and a similar discrepancy exists between the two sets of K results. We have also used the method of lattice statics to determine the strain-field interaction energies between several types of vacancy pairs in these metals. In every case we find the next-nearest-neighbor configuration to be the most stable, whereas in the nearest-neighbor configuration, the two vacancies repel one another. The magnitudes of these binding energies depend strongly on which model potential is used.

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