A method is derived for the calculation of the free energy of a crystal with aggregates of divalent cation impurities and cation vacancies. The free energy is explicitly given for the case of isolated impurities and vacancies, dipoles, dimers, and trimers. The equilibrium condition for the free energy yields a simple formula for the equilibrium concentration of all aggregates. If the association energy for all aggregates is known an algebraic equation of sixth order has to be solved. As an example the concentrations in the case of the validity of the point charge approximation of the association energies are computed.
The formation and association energies for the different aggregates of divalent metal impurities and cation vacancies are derived by considerations of the sums over all lattice vectors. This gives a simple formula in the point charge approximation if an infinite lattice is assumed. Because of the neglection of the polarization of the ions this method yields a higher value for these energies, as a comparison with the literature for the case of dipoles shows. But nevertheless it is a first possibility for a correct estimation of these energies also for higher aggregates.Bildungs-und Assoziationsenergien fur die verschiedenen Aggregate aus 2-wertigen Verunreinigungen und Kationenleerstellen werden durch Betrachtungen der Summen iiber alle Gittervektoren abgeleitet. Dies fuhrt zu einer einfachen Formel in Punktladungsniiherung fur ein unendlich ausgedehntes Gitter. Das Verfahren liefert wegen der Vernachliissigung der Polarisation der Ionen zwar systematisch zu hohe Werte fur diese Energien, wie ein Vergleich mit den aus der Literatur bekannten Werten fur Dipole zeigt, ergibt jedoch erstmals die Moglichkeit fur eine korrekte Abschiitzung dieser Energien auch fur hohwe Aggregate.
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