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

Abstract

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.

Influence of the variation of the association energy on the equilibrium concentration of dipole aggregates in ionic crystals

“…A detailed analysis shows that this inequality is a t the best fulfilled for i = 1. With ( 5 ) follows 1 4 co> exp (2) and, because c, 5 C,, all the more I n Fig. 1 the niininial C, is drawn in for z1 = 12 and different dipole association energies El assuruing that inequality (9) is fulfilled sufficiently, when C,, is equal to five tinies the right-hand side of (9).…”

“…The dipoles are indexed by i = 1 and the isolated defects by i = 0, both the impurity ions and the cation vacancies, because the concentration of t h e latter ones is equal due t o elect,rical neutrality. The configuration entropy is[ 2 ] with thc abbreviation where TL, is the number of the i-th aggregate per volunie, Li the number of lattice sites occupied by this aggregate, zi the number of the orientation possibilities averaged on all configurations of this type of aggregate, and N the number of cation sites per volume. of the i-th aggregate follows by minimization of (1) and assuming E, = 0, i.e.…”

The role of Eu++ aggregation on optical absorption, emission, luminescence time decay, and ITC plots in NaCl:Eu++

“…Two possible explanations can be tentatively supported by the following arguments : a ) The fornier assumes that the luminescenee time decay signal is the superposition of a series of decay signals, a set of them arising froni I V dipoles and the other sets coining froni dimers arranged in different geometrical configurations. I n fact, different dinier configurations are possible : nine, if only those in which each defect in the aggregate has at least one neighbouring defect [25], many more, if loosely bound diniers are considered as well [26]. Hence the luminescence time decay related to diniers couples the complexity arising from the fact that E u + + exhibits a set of closely spaced levelsas previously reported for the monomerto the fact that many different dinier configurations can coexist, each of them exhibiting its own set of emitting levels and radiative lifetimes.…”

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

“…where E , is the association energy of the i-th aggregate with the energy of the isolated defects as zero level [2] and Xccnf the configuration entropy of the crystal. This entropy is given by [2] (2)…”