The lattice dynamics of an anharmonic crystal

The theory of the physical properties of an anharmonic crystal is discussed by using the thermodynamic Green's functions for the phonons. A perturbation procedure is developed to obtain the Green's functions and it is shown that for some purposes a quasi-harmonic approximation is useful, in which the frequencies of the normal modes are those determined by infra-red or neutron spectrometry. The thermodynamic, elastic, dielectric and scattering properties of an anharmonic crystal are discussed in terms of the Green's functions, and detailed expressions are given for the more important contributions. Detailed numerical calculations are presented of the thermal expansion, dielectric properties and shapes of some of the inelastically scattered neutron groups, for sodium iodide and potassium bromide. The calculations, which give reasonable agreement with experiment, show that even at quite low temperatures, the lifetimes of some of the normal modes can be quite short. By using the quasi-harmonic approximation it is shown that the large temperature dependence of the normal modes in a ferroelectric crystal can be treated adequately.

Lattice Dynamics of Alkali Halide Crystals

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 Kármán 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 ν(q) relation of the lattice vibrations in the symmetric [001], [110], and [111] directions of sodium iodide at 110°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 10¹² cps) are: TA[0,0,1]1.22+/-0.04, LA[0,0,1] 1.82+/-0.06, TA[ 1/2 , 1/2 , 1/2 ]1.52+/-0.05, LA[ 1/2 , 1/2 , 1/2 ]2.32+/-0.06, TO[0,0,0] 3.6₀+/-0.1, TO[0,0,1]3.8₀+/-0.1, TO[ 1/2 , 1/2 , 1/2 ]3.5₀+/-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.

Lattice Dynamics of Alkali Halide Crystals. II. Experimental Studies of KBr and NaI

Dispersion curves for the lattice vibrations propagating in the [00ζ], [ζζ0], and [ζζζ] directions in NaI at 100°K and in KBr at 90°K have been measured using neutron spectrometry, and the results compared with calculations based on a simple shell model. Both substances obey the Lyddane-Sachs-Teller relation. In addition, some measurements were made on KBr at 400°K most frequencies showed a decrease of a few percent. At this temperature, the acoustical modes show no significant energy broadening, the transverse optical modes show slight broadening, and the longitudinal optical modes show very considerable broadening. The anomalous broadening of the LO modes is not yet understood and requires further study. It appears to be specimen-dependent as well as temperature-dependent

Lattice Dynamics of Alkali Halide Crystals. III. Theoretical

The "shell model" of an alkali halide is extended to take into account short-range forces between both first- and second-nearest neighbor atoms in the crystal, the polarizability of both ions, and the possibility that the ionic charge may be less than one electronic charge. The arbitrary parameters of the model have been obtained by means of a least-squares fit to the measured dispersion relations for the lattice vibrations, the dielectric constants, and the elastic constants. For NaI and KBr, models have been found which give excellent agreement both with these measurements and with measurements of the specific heat. This good agreement is, however, obtained only when the simple shell-model concept of the ions is to some extent abandoned. The reasons for this, and the connection with work of other authors, are discussed.

Cesar A Hernandez E
19.502.806
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