Anharmonic broadening of localized vibrational modes in alkali halide crystals containing H− ions

Abstract

Calculations are presented of the anharmonic broadening of the high-frequency localized mode associated with H− ion impurities (U centers) in KI and KBr, and also of the gap mode which occurs in the KI:KH crystal. Expressions are obtained which give the half widths of the infrared absorption peaks induced by these modes in terms of matrix elements of the impurity Green function of Lifshitz. It is shown that by taking specific forms for the localized modes and for the anharmonic terms in the Hamiltonian, closed expressions result which may be evaluated taking into account the perturbation of the lattice by the impurity. The gap mode is assumed to be completely localized on the nearest neighbors of the impurity and the high-frequency localized mode on the impurity ion alone. Only anharmonic terms derived from the nearest-neighbor, short-range forces are considered. These forces are assumed to be of the Born–Mayer form and are described by anharmonic coefficients pertinent to the pure crystal, with the exception of the force between the impurity and its nearest neighbors. In this case the cubic coefficient deduced from the observed relative intensity of the U center sideband is used and the quartic coefficient estimated from this.The numerical results show that the high-temperature behavior of the half width of the gap mode peak in the KI:KH sideband is essentially determined by that of the high-frequency localized mode peak. At low temperatures the theoretical result is smaller than the experimental value by a factor close to two. This is probably a consequence of using an oversimplified anharmonic Hamiltonian. The half width of the HFLM peak is found to be determined essentially by decay processes at low temperatures, whereas at high temperatures the contribution of scattering processes is dominant, producing the expected T2 dependence. Comparison of theory and experiment is restricted to high temperatures by the available data. In this region the theoretical half widths are too small by a factor of the order of 10. It is shown that this discrepancy may result from the neglect, in calculating the contribution to the broadening of indirect scattering processes, of three-phonon interactions in which none of the interacting phonons is the high-frequency localized mode phonon. Additional processes which are possible when such interactions are included are briefly discussed.