The association energy of aggregates of divalent impurities and vacancies in NaCl‐type crystals (point charge approximation)

Abstract

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.

Comparison of Experimentally and Theoretically Determined Dipole Concentrations of NaCl:Ca++ Considering Aggregation Phenomena

“…An exact comparison of these values with experimentally determined association energies is only possible for dipoles, becaii se higher aggregates can exist in different configurations with different interaction energies, and so far it is impsosible to distinguish these configurations experimentally. I n the case of dipoles the values experimentally determined by ionic conductivity measurement for different divalent impurity ions in NaCl are 20 to 50% lower than the theoretical ones (see the coniparison in [ 5 ] ) . An analogous tendency has to be expected for the higher aggregates of dipoles.…”

The equilibrium concentration of aggregates of cation vacancies and divalent impurities in NaCl-type lattices (Point-Charge Approximation)

“…2 to 5 the concentrations of the aggregates are drawn versus the temperature for NaCl and KC1 crystals. Because of the point-charge approximation there is no influence of the special kind of impurity on the results [3]. Therefore this is a first approximation to the problem.…”

“…This gives 2nothe third term in the bracket of (10) can be developed, so the the equilibrium concentrations areThe second term in the exponential can be neglected as compared with the first one. With [El > 0.2 eV (see[3]) and T < lo3 K is valid. Because 2c, 5 2 x 1 0 -3 for most cases this neglect, is justified.Thus (14) can be written(15)…”

“…Stirling formula the configuration entropy is given by Sconf = k 2 { n tIn (Lizi) + N i In N ini In ni -(Nini) In (Nini)}The Equilibrium Concentration of the AggregatesThe free energy of the whole crystal i s here U i is the energy for the i-th aggregate. If one chooses the energy for isolated defects as the zero level of these energies, that means u,,, = us,, = 0 9 (8) G. BERG, F. FROHLICH, and S. SIEBENHUNER one gets the association energies E i for the aggregates[3]. The condition to preserve the inass of all impurities 317 no is the number of impurities per volume) gives the possibility to eliminat,e nS1, of (7).…”