Lattice Dynamics of Alkali Halide Crystals

Abstract

The paper comprises theoretical and experimental studies of the lattice dynamics of alkali halides. A theory of the lattice dynamics of ionic crystals is given based on replacement of a polarizable ion by a model in which a rigid shell of electrons (taken to have zero mass) can move with respect to the massive ionic core. The dipolar approximation then makes the model exactly equivalent to a Born-von K\'arm\'an crystal in which there are two "atoms" of differing charge at each lattice point, one of the "atoms" having zero mass. The model has been specialized to the case of an alkali halide in which only one atom is polarizable, and computations of dispersion curves have been carried out for sodium iodide. We have determined the dispersion $\ensuremath{\nu}(\mathrm{q})$ relation of the lattice vibrations in the symmetric [001], [110], and [111] directions of sodium iodide at 110\ifmmode^\circ\else\textdegree\fi{}K by the methods of neutron spectrometry. The transverse acoustic, longitudinal acoustic, and transverse optic branches were determined completely with a probable error of about 3%. The dispersion relation for the longitudinal optic (LO) branch was determined for the [001] directions with less accuracy. Frequencies of some important phonons with their errors (units ${10}^{12}$ cps) are: TA[0,0,1]1.22\ifmmode\pm\else\textpm\fi{}0.04, LA[0,0,1] 1.82\ifmmode\pm\else\textpm\fi{}0.06, TA[\textonehalf{},\textonehalf{},\textonehalf{}]1.52\ifmmode\pm\else\textpm\fi{}0.05, LA[\textonehalf{},\textonehalf{},\textonehalf{}]2.32\ifmmode\pm\else\textpm\fi{}0.06, TO[0,0,0] ${3.6}_{0}$\ifmmode\pm\else\textpm\fi{}0.1, TO[0,0,1]${3.8}_{0}$\ifmmode\pm\else\textpm\fi{}0.1, TO[\textonehalf{},\textonehalf{},\textonehalf{}]${3.5}_{0}$\ifmmode\pm\else\textpm\fi{}0.1. The agreement between the experimental results and the calculations based on the shell model, while not complete, is quite satisfactory. The neutron groups corresponding to phonons of the LO branch were anomalously energy broadened, especially for phonons of long wavelength, suggesting a remarkably short lifetime for the phonons of this branch.