The energy of formation of Schottky defects in ionic crystals

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Abstract
This paper calculates the energy of formation Ws of Schottky defects in ionic crystals by methods derived from the original method of Mott and Littleton. The main aim of these calculations is to examine the variations in the predicted energies resulting from reasonable variations in the assumed models. The types of variation considered are: (1) changes in the description of the long-range displacement and polarization field around the vacancies, corresponding to the inclusion of (a) purely elastic distortions and (b) electronic deformations of the ions resulting from the polarization of the lattice; (2) changes in the magnitude of the second neighbour Born-Mayer interactions corresponding to different available choices for ionic radii and to the inclusion or otherwise of Van der Waals forces; and, more briefly, (3) changes from the Born-Mayer form to the Born—Mayer-Yerwey form of closed-shell repulsion potential. The results are given in tables but are broadly as follows. For a given closed-shell repulsive potential, the inclusion of elastic terms in the displacement field (in addition to polarization displacements resulting from the effective charges on the defects) raises Ws, while inclusion of electronic deformations lowers it again. The net result of including both is to increase Ws by ∼½ev. For given assumptions about the displacement and polarization field changes in the Born-Mayer constants can be important; thus use of the conventional Goldschmidt radii in place of the more recent radii of Tosi and Fumi leads to a lowering of the Schottky energies by amounts which are larger for the compounds of small cations and large anions—for NaCl by ∼1/2ev. These results are consistent with those of Tosi and result from the importance of the first-neighbour interactions in determining Ws. Use of a Born-Mayer—Yerwey potential increases Ws compared to the values for the corresponding Bom-Mayer form. Experimental values of W s are known for some alkali halides. Reasonable agreement with those for Nad, NaBr and KC1 results from using the Tosi-Fumi constants in a Born-Mayer potential, but for the Li halides and KBr the calculated values are too low. For these substances a somewhat stiffer potential seems necessary. Our results, especially those on the effect of the changes in the closod-shell repulsions, are important for the extension of this model to simple oxides.