Rigid Ion Research Articles

Lattice dynamics and debye-waller factor near the 195 K phase transition in RbCaF 3

The experimental temperature dependence of the mean square amplitudes of atoms in RbCaF 3 is compared with the theoretical values calculated from both an harmonic and a quasi rigid ion model. In the light of these results the anhamonicity of the cubic phase is discussed.

Infrared and Raman Spectra and Lattice Vibrations of Some Oxide Spinels

The infrared transmission and reflection spectra of Co II Co III 2 O 4 , CoAl 2 O 4 , ZnFe 2 O 4 and MnFe 2 O 4 and the single crystal Raman spectra of Co II Co III 2 O 4 and CoAl 2 O 4 have been measured. The frequencies of the transverse (TO) and longitudinal (LO) modes of the infrared active species have been obtained from the observed reflectivities. The TO-LO splitting supplies a criterion for the ionic character. The lattice dynamical analysis of the optically active vibrations has been made for Co II Co III 2 O 4 , CoAl 2 O 4 and ZnFe 2 O 4 , in which the rigid ion model for the long-range Coulomb potential and the central force field for the short-range potential have been used. The obtained short-range potential constants have been discussed, in relation to the bond character.

Third and fourth order elastic constants of fluoperovskites CsCdF 3, TlCdF 3, RbCdF 3, RbCaF 3

The hydrostatic dependence of the second order elastic constants of several fluoperovskites has been measured and studied from a macroscopic point of view and through a force field analysis. The first part of this paper is devoted to the experimental determination of some combinations of the third and fourth order elastic constants. The second part analyses the pressure dependence of the second order elastic constants from a macroscopic point of view. The large values of the second order anharmonicity coefficients b ij = 1 2 ( ∂ 2C ij ∂P 2 ) 0C (0) ij are connected to the occurrence of a phase transition in some compounds, whereas the first order coefficients ( ∂C ij ∂P ) 0 have an order of magnitude which agrees with a quasiharmonic hypothesis. The third part is devoted to the microscopic interpretation of the third order elastic constants. A rigid ion model is used with either classical Born Mayer potentials or a modification of these potentials with an axial term (Cowley and Rousseau's models).

Dynamical study of phonons in PbTiO3

Abstract Lattice dynamical formalism, using rigid ion model, was applied to the crystal of PbTiO3 in the tetragonal phase. The model includes long range Coulomb forces and short range axially symectric forces. The radial part of the short range force constants was assumed to be of Born-Meyer type. The parameters of the model were determined through a least squares analysis of zone center and a few zone boundary phonons. Dispersion curves for [100] and [001] directions are presented.

Elastic constants of calcium tungstate

The seven independent adiabatic elastic constants of CaWO4 are calculated using Born's longwave method and the rigid ion model of Steinman et al. (1972). Good agreement exists between the calculated elastic constants and those measured by Gluyas et al. (1973). The sign of C16 is consistent with the observation of Farley et al. (1972). Further the dynamical equilibrium conditions are also found to be satisfied for this model.

Lattice dynamics of BaFBr

A normal coordinate and force constant calculation for BaFBr was carried out on the basis of the rigid ion model. There is a strong anisotropy of the effective dynamical charges of the barium, fluoride, and bromide ions, which are 1.57e, −0.78e, and −0.79e for the degenerate vibrations and 1.04e, −0.71e, and −0.33e for the vibrations in the z-direction. In accordance with the ionic nature of BaFBr the short range force constants have only small values.

Low energy phonon dispersion curves of KZnF 3 and CsCaF 3

Ultrasonic wave propagation and neutron inelastic scattering experiments have been performed upon the cubic perovskites KZnF 3 and CsCaF 3. The low frequencies phonon dispersion curves calculated with a rigid ion model are given.

Low-frequency lattice vibrations in β-alumina

The elastic constants of M β-alumina (M = Na, K, Rb, Ag and TL) are obtained from polarized Brillouin scattering spectra. The frequency versus wave-vector dispersion curves near the Γ-point are calculated from the elastic constants obtained in the harmonic approximation by using a rigid ion model in which a spinel block composed of Al 3+ and O 2- is considered as one rigid ion. The vibrational frequency or the attempt frequency of the mobile ion in β-alumina is related not only to the mass of the mobile ion but also to the force constant between the spinel block and the mobile ion, which participates in the elastic constant C 44.

Structural disorder and phase transitions in ZrO 2-Y 2O 3 system

The phase transitions in the ZrO 2-YzO 3 system have been investigated as a function of temperature using Raman scattering. The cubic phase was found to be structurally disordered and the spectrum has been compared with a one phonon density of states derived from a rigid ion model. The frequencies and symmetries of the modes were obtained for the cubic, tetragonal and monoclinic phases and comparisons have been made between the various temperatures and compositions.

Lattice vibrational modes in potassium thiocyanate

Calculations on long-wavelength lattice vibrations have been made for cystallline KNCS within the framework of the rigid ion model. The short range force constants and effective charges of the ions have been determined from the observed frequencies. Unobserved frequencies are predicted, and normal coordinates have been obtained and schematized.

Lattice Dynamics of 1T-TiSe2

Phonon dispersion curves for the high temperature phase of a layer compound 1 T -TiSe 2 , which reveals a second order structural phase transition at T c =200 K, are calculated using the rigid ion model and the shell model, The calculated phonon dispersion curves, especially with the shell model, are in good agreement with the observation with regard to their gross features. In particular, the calculated dispersion curve for acoustic phonons propagating in the layer and polarized perpendicular to the layer shows an upward bend as this is observed. However,the observed dip at the L point cannot be reproduced. with our models in which only central two-body forces are assumed; the L point corresponds to the wave vector of the observed lattice distortion. The results are discussed in connection with other possible effects.

Sharp-line absorption, luminescence, Raman studies for the 5d3hexafluororhenate(IV) ion in pure and host crystal environments

Five electronic transitions have been observed by absorption spectroscopy at liquid helium temperature for the ReF6 -2 ion in a single Cs2ReF6 crystal and in a Cs2GeF6 cubic host lattice. Also, luminescence has been observed from the Γ7(2 T 2g ) electronic state to the Γ8(4 A 2g ) ground state in the pure Cs2ReF6 system. Temperature dependent absorption hot bands, Raman and luminescence results provide information about the symmetry of the ReF6 -2 ion in a pure Cs2ReF6 crystal. The observed electronic transitions can be assigned with a crystal field model to give B, C, Dq and λso values for ReF6 -2. The detailed phonon structure observed for the t 2g 3 →t 2g 3 transitions in both the pure and mixed crystal cases, can be assigned to lattice modes, internal modes of the ReF6 -2 entity, and combination modes of the proper symmetry. We have assigned the low energy lattice modes by use of recent rigid ion lattice dynamical model results for Cs2SiF6.

Lattice dynamics of KNO3, LiNbO3 and KNbO3

We report results of quasi-harmonic lattice dynamical studies of KNO3, LiNbO3 and KNbO3. A rigid ion model (with charges and radii as parameters) is employed for calculation of phonon dispersion relation (PDR) in their non-cubic phases. External mode formalism is adopted for KNO3. Results are in fair agreement with neutron data; data for α-KNO3 were obtained by us at Trombay. We have interesting comments to make on PDRs in β and γ-KNO3, temperature dependence of soft modes in niobates, soft mode dependence in K NbO3 on a single model parameter, transferability of radii parameters etc.

An evaluation of finite strain equations of state using a lattice model

Analytic expressions have been derived for ρ ρ 0 , V p V b0 and V s V b0 as functions of p K 0 for rigid ion models of the NaCl, CsCl and ZnS structures. In this dimensionless form, the only independent parameter in the lattice model is the repulsive parameter n (or equivalently K′ 0). Analytic first and second pressure derivatives of the bulk and shear moduli calculated from the lattice model were used as parameters in Murnaghan and Birch equations of state (EOS) of first and second order. Comparison between the various finite strain equations and the lattice model for the three crystal structures leads to the following general conclusions. (1) For each finite strain EOS, V p is in better agreement with the lattice model than V s; (2) the Birch EOS is in better agreement with the lattice model than is the Murnaghan EOS of comparable order; (3) no first-order EOS is capable of extrapolating V s to p K 0 > 0.2 without deviating by more than 5% from the lattice model, and (4) the second-order Murnaghan EOS gives very little improvement over the first-order equations, while the second-order Birch EOS allows V s to be extrapolated to p K 0 ≈ 0.8 without deviating from the lattice model by more than 5%. For close-packed oxides and silicates this covers the pressure range throughout the lower mantle.

Raman and Infrared Spectra and Lattice Vibrations of KNO3Crystal

Polarized Raman and infrared reflection spectra of KNO 3 crystals have been measured at room temperature. Raman and infrared transmission spectra of the high temperature phase of KNO 3 have been also measured. The frequencies of the transverse (TO) and longitudinal (LO) modes of the infrared active vibrations of the crystal have been obtained from the observed reflectivities. The lattice dynamical analysis of the optical active vibrations of the room temperature phase of KNO 3 has been made based on a rigid ion model which includes effective ionic charges and short-range repulsive force constants. Short-range force constants for the non-bonded metal-oxygen and oxygen-oxygen interactions have been determined by the least squares method using the observed Raman frequencies.

On the pressure dependence of the effective ionic charge of cesium halides

The importance of temperature effects in the use of the second Szigeti relation to calculate the volume dependence of the effective ionic charge of cesium halide crystals is studied. The effect obtained is larger than the one for alkali halide crystals of the NaCl structure. Positive values of (∂ ln s/∂ In v) are obtained using low temperature experimental data with the Grüineisen parameter γ t ( T → 0K) calculated using the generalized first Szigeti relation. Values of γ t are also obtained with a rigid ion model and they differ little from the previous ones.

Structure of molten alkaline-earth halides; SrCl2and CaF2

Recently de Leeuw has reported Monte-Carlo simulations for solid and liquid SrCl2 and CaF2 based on the rigid ion model. In the absence of adequate diffraction experiments for these liquids, his results form effective experimental information in this respect. We therefore thought it of interest to test the predictions of the ‘general primitive model’ of electrolytes—mixtures of oppositely charged hard spheres of unequal diameters, for these molten salts. For the solution of this model we have employed the mean spherical approximation using the procedure developed for arbitrary diameter ratio by Abramo et al. The overall agreement of the calculated radial distribution functions and the coordination numbers with those obtained by de Leeuw is fair. This correspondence improves when the ratio of ionic diameters is taken to be the same as in the pair potential used in the rigid ion model.

Lattice relaxation around tetragonal and rhombic defects in alkali halides

The relaxation of the crystal lattice around impurity pair defects in alkali halides is calculated based on a rigid ion model. Pairs with tetragonal symmetry—fourth nearest neighbors along the [001] axis—and those with rhombic symmetry—second nearest neighbors along the [110] axis— are both considered, for all cases where experimental evidence exists for resonant or gap modes. These are: NaCl with F−, Li+, and Ag+; KI with Cl−, Br−, Na+, and Rb+; KCl with Na+; and NaBr with Li+. Relaxation spaces with 72 ions (tetragonal) and 50 ions (rhombic) are used and results reported for all ions within the spaces. The nearest neighbor central force constants coupling the impurity ions and their nearest neighbors are calculated in the rigid ion approximation and the results compared with earlier dynamical model calculations. Because the dynamical models assumed that only one perturbed force constant was needed to characterize all the impurity–host interactions, and significant directional inhomogeneity of the force constants was a result of the present calculations, only qualitative comparisions could be made.

Lattice dynamics of zincblende structure compounds using deformation-dipole model and rigid ion model

We present calculations of the Raman scattering spectra by the long-wavelength vibrations of Ga1−xAlxSb mixed crystals for three different cation concentrations. Each mixed crystal is approached using a primitive cell 64 times larger than the primitive cell of the bulk constituents GaSb and AlSb. The phonon modes are calculated on the basis of an 11 parameter Rigid Ion Model and the Raman spectra are calculated using the Bond Polarizability Model (BPM), away from resonance conditions. The parameters of this model (BPM) are not arbitrarily approximated but we have obtained them on the basis of certain relations, involving directly measurable quantities, such as dielectric and elastooptic constants of the bulk crystal. It is shown that for small concentrations the Al ions are not randomly distributed over the whole crystal but almost all tend to concentrate in neighboring lattice planes. Further, we have reproduced the Raman spectra close to resonance conditions, assuming that the value of the first order polarizability of AlSb is increased by an amount of 50% close to resonance conditions. Finally it is shown that disorder produces asymmetric Raman lines spectra with the intensities of the two strongest peaks in the optic frequency ranges of the bulk constituents being concentration dependent.

Structure of molten KCl

The radial distribution function of molten KCl has been determined by X-ray diffraction and compared with those derived from computer simulations with the shell model and the rigid ion model.