Off-centre instability induced by exchange forces

Theory of the off-centre instability is developed for the impurity ions (Li', CuC, and Ag') in alkali halides on the basis of the charge transfer excitation model which is different in the involved electronic processes from the earlier theory based on the polarizable-point charge ion model. The lowest potential minima for the impurity ion along < 11 1 ), ( 110 ), and < 100 ) crystal axes are calculated. As the lattice constant increases, the normal lattice site becomes an unstable position for the impurity ion and the most stable off-centre position appears at first along ( 11 1 ), then along ( 110 ), and finally along ( 1W > axis. This result is explained with the exchange interaction arising from the electron exchange. The off-centre instability for impurity ions (Lif , Cu+, and AgC) in alkali halides has been studied theoretically by many authors [I-41. They calculated the minimum-energy configurations for the systems using the following model : constituent ions have a point charge, Born-Mayer repulsion, and polarizability. In their treatments, the polarization effect of ions, especially in the neighbouring halogen around the impurity ion [1], plays an essential role in the off-centre instability. We shall call the model polarizable-point charge ion (PPCI) model. Though the PPCI model has explained the off-centre instability very well in the case of Lif ion in KC1 and KBr [2,3], it has not always succeeded in the other systems; calculations showed that the minimum-energy configurations had always < 11 1 ) symmetry for impurity ions Li+ [2, 31, and CuC and Agf [4] in alkali halides when the instability occurred. But it has been confirmed experimentally that some impurity ions lie in < 110 ) direction away from the on-centre position as for Fin NaBr [5] and Ag+ in RbCl and RbBr [6,7]. However, when we probe into the electronic processes involved in the off-centre instability, there is an ambiguous point in the PPCI model. Because the polarizability of ion comes from intra-ionic excitations, the most important electronic processes in the PPCI model would be the intra-ionic excitations of the halogen around the impurity ion. But it seems more plausible that the lowest excited state is a charge transfer excitation (CTE) state around the impurity ions (LiC, CuC, and Ag+) [8] i.e. an electron is removed from a neighbouring halogen and is added into the impurity ion. Such state may be a localized state around the impurity ion [9] because ionization energy of the impurity atoms are larger than that of host alkali atoms. See table I. Table I. Ionization energies of the impurity atoms and alkali atoms, in eV. Cu 7.724 Li 5.390 K 4.339 Ag 7.574 Na 5.138 Rb 4.176 From American Institute of Physics Handbook, 2nd ed. (McGraw HilI, New York) (1963). In the present note, we wish to develop the theory of the off-centre instability on the basis of the CTE model for impurity ions (Lif, Cuf , and Ag+) in alkali halides. In order to study the problem, we shall treat a cluster of the form Mf Xg in a lattice, where M + signifies the impurity ion with closed shell structure, and Xa halide ion. Total lattice configuration will be expressed by (R,, R . .., R,, . ..), where index 0 represents the impurity ion, indices 1 6 the neighbouring halogen, and indices 7 the ions in the surroundings. Hereafter we express the total ionic configuration by R. The lowest energy electronic configuration of the cluster [M+ X; . . . X;] will be expressed as ( 0 : (W ), where ionic configuration is fixed at R. Suppose one Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19806105 OFF-CENTRE INSTABILITY INDUCED BY EXCHANGE FORCES C6-413 of the valence p-electron of the i-th (1 < i < 6) halogen is removed and is added into s-orbital of the impurity ion i.e. [MO X; ... Xp ... X;]. It corresponds to the CTE state. For such CTE state, two types of spin states are possible; a singlet and a triplet state. We will consider only the singlet state, which will be expressed as I i%: R ), where K specifies the three-fold degeneracy of the removed p-orbital. Now we assume that low lying states of the cluster are given in terms of linear-combinations of these states. The diagonal elements of the total energy H of the system will be given by Then the energy matrix (7 x 7) of the cluster is obtained as Eigenvalues of the matrix are obtained as