Deformation Dipole Research Articles

Two phonon density of states of MnO

Temperature weighted two-phonon density of states for MnO is reported. Lattice dynamics of MnO used in computation is based on the deformation dipole model.

Effect of hydrostatic pressure on the electronic dielectric constant of ionic crystals

Abstract The strain derivatives of the electronic dielectric constant in LiF, NaCl, KCl, KBr, KI, RbCl and MgO crystals have been calculated with the help of the Clausius-Mossotti model. The variation of electronic polarizabilities with compressive stress has been derived by taking into account the distortion of ions, within the framework of the deformation dipole model. It has been observed that consideration of the variation of polarizabilities with compressive stress improves the agreement between theory and experiment. The photoelastic behaviour of MgO is predicted to be of the opposite nature to that of alkali halides in conformity with the experimental observations.

Deformation dipole model and lattice dynamics of zinc-blende-structure compounds

The deformation-dipole-model (DDM) formalism is derived for a crystal of arbitrary symmetry and applied to the zinc-blende structure. In its present form, the model is more general than the one proposed by Karo and Hardy, the principal difference consisting in introducing interactions called nonlocal electric polarizabilities, analogous to the shell-shell interactions of the shell model. Both approaches are compared and it is shown that in ionic crystals the present version of the DDM provides the same description of core motion as the shell model. Parameters of one model are then expressed in terms of the other model, for ionic structures; the case of covalent crystals is also briefly discussed. A simplified form of the DDM was used in numerical calculations for 15 crystals with ${T}_{d}^{4}$ symmetry. Model parameters fitted to elastic, piezolectric, optical and, for eight compounds, also to neutron-diffraction measurements gave a good description of phonon frequencies. Some of the parameters appeared to be smoothly varying functions of the interatomic distance or ionicity.

The Debye–Waller factors of the rubidium and cesium halides

Computed mean-square displacements and Debye-Waller BK factors, derived using the deformation dipole model, are reported for the rubidium and cesium halides at 4.2, 80, and 300 °K. Experimental X-ray diffraction Debye-Waller factors, obtained at 300°K, are reported for CsCI, CsBr, and CsI. The calculated BK values are compared with these and other experimental values wherever such data exist. The differences between calculated and measured values at 4.2°K are not large but in three out of four cases lie outside the small experimental uncertainties quoted. At 300°K, on the other hand, the differences tend to be larger, but in about half of the cases are less than the experimental uncertainty.

Lattice dynamics of MnO

A deformation dipole model for Lattice dynamics of MnO has been formulated. Phonon dispersion curves calculated are in reasonably good agreement with experimental data. One phonon density of states of MnO is also reported.

The lattice dynamics of CsCl, CsBr and CsI

Calculated dispersion curves and frequency spectra are presented for CsCl, CsBr and CsI. These calculations have been made using the deformation dipole model including both first and second neighbor non-coulomb forces. The best fit to the experimental results is provided by a simple five parameter first neighbor model.

Two-phonon Raman spectra of LiH and LiD crystals

The experimental two-phonon Raman spectra of LiD and LiH are reported here. A deformation dipole model formulated elsewhere for LiD is used to compute the lattice dynamics of LiD and LiH required in the Raman intensity calculations. The use of the same model for LiH is justified by the good agreement between the peak positions of the temperature-weighted two-phonon density of states and the various experimental spectra. The Raman intensity calculations are carried out by treating the second-order expansion coefficients in the polarisability tensor as adjustable parameters. The need for the parameters associated with the next-nearest-neighbor ions is clearly demonstrated for all the spectra. A single set of six parameters for the T 2 8 spectra and thirteen parameters for the E 8 (or A 1 8 ) spectra is found to explain all the experimental spectra of LiD and LiH quite well. Since the polarisability of Li + is very small, the need for the second-neighbor positive-positive parameters reflects on the extended and highly polarisable nature of the H − or D − ion.

Anharmonic Models and Thermal-Expansion Calculations for KBr

The mode-Gr\uneisen parameters, the macroscopic Gr\uneisen function $\ensuremath{\gamma}(T)=3\ensuremath{\alpha}{B}^{\mathrm{is}}\frac{V}{{C}_{V}}$, and the coefficient of thermal expansion $\ensuremath{\alpha}$ are calculated for KBr. Three anharmonic models are used: the rigid-ion (RI) model, a deformation-dipole (DD) model in which the deformability of the ions does not depend on the volume $V$, and a DD model in which the deformability does depend on $V$. Allowing the deformability to depend on $V$ significantly improves the agreement with experiment. A definite minimum in the low-temperature values of $\ensuremath{\gamma}(T)$ is predicted by the DD models, but not by the RI model. The distance between nearest neighbors $r$ at which the potential energy of the lattice has its minimum value is found self-consistently. This value of $r$ is used in the determination of the derivatives of the short-range overlap potential $\ensuremath{\nu}(r)$. The force constants obtained differ significantly from those obtained by substituting room-temperature experimental data into formulas derived using the equilibrium condition. Significantly larger values for $\ensuremath{\gamma}(T)$ and $\ensuremath{\alpha}$, which are in better agreement with the experimental results, are obtained by allowing for the temperature dependence of the zero-pressure values of ${B}^{\mathrm{is}}$ and $r$.

Second-order Raman spectra and phonon spectrum of LiD

The second-order Raman spectra of LiD are studied at room and liquid nitrogen temperatures. These spectra are such that they can be compared with the temperature weighted two-phonon density of states to get information on phonons throughout the first Brillouin zone. This comparison is used to supplement the neutron scattering results to formulate a very good deformation dipole model for the lattice dynamics of LiD.

Lattice dynamics of silver bromide

Lattice dynamical calculations for AgBr using the deformation dipole model are presented. Significantly better agreement with experiment is obtained for the q ≈ 0 optical phonons and with specific heat data when next nearest neighbor and three body forces are included.

Frenkel Defects in Alkali Halides

The method of lattice statics and the deformation dipole model have been applied to Frenkel defects in ionic crystals. For isolated interstitials, all short-ranged repulsive and Coulombdefect host-lattice ion forces were evaluated at the relaxed configuration. The zero-order Coulomb force was allowed to act on the remainder of the host-lattice ions past first neighbors. We have calculated Frenkel-pair formation energies, ionic displacements, and ionic polarizations in the twelve Na, K, Rb halides. Our results show over all that, in ionic crystals of the rocksalt structure, Schottky defect pairs are more likely to be present than Frenkel defect pairs.

A thermal diffuse X-ray scattering investigation of the lattice dynamics of caesium iodide

Absolute measurements have been made of the total thermal diffuse scattering of CuK alpha X-rays from single oriented crystals of caesium iodide. Data have been obtained in the major symmetry directions and compared with the theoretical values of total thermal diffuse scattering calculated using the 'deformation dipole' (d.d.) lattice dynamical data of Karo and Hardy (1968). In particular the results are used to discriminate between two variations of the d.d. model, demonstrating the unsuitability of the d.d. (+) variation and the good overall agreement with the d.d. (-) variation. Where feasible, the experimental thermal diffuse scattering has been analysed to yield acoustic and optic phonon dispersion curves. Finally, the discrepancies between the total experimental thermal diffuse scattering and the d.d. (-) theory predictions have been investigated for any asymmetry.

Migration Energy of an Interstitial He Atom in LiH

The method of lattice statics is used to study the distortion of the LiH crystal with an interstitial He atom in the body-centered or face-centered position. In these calculations, one needs to know the defect-host-ion interaction potentials and the dynamical matrix of the undistorted host crystal. The Heitler-London method is used to estimate the defect-host-ion interaction potentials. The dynamical matrix of the perfect crystal is calculated on the basis of the deformation dipole model. The formation energies are computed for the two defect positions and their difference, 0.650 eV, gives the migration energy of an interstitial He atom in LiH. The calculated migration energy is discussed in light of the high-temperature experimental result for the activation energy of diffusion of He gas in LiH crystal (1.22 eV).

Pressure dependence of the optical properties of alkali halide crystals

Abstract An equation of Szigeti relating the compressibility of an ionic lattice to its optical properties is generalized so that it is valid at all pressures. This equation is then used to re-analyse the data of Jones (1967) on (∂ In e0/∂p)T, and leads to less anomalous estimates of the volume dependence of the effective ionic charge. However, detailed investigation of the most recent data still indicates that the effective charge varies less with volume than has been predicted theoretically. Calculations of (de0/dp) and the Gruneisen parameters yt and yl ; are carried out for two static deformation dipole models of KC1 with the volume dependences suggested respectively by Hardy (1962) and Mitskevich (1964). Disagreement of (de0/dp) with a low temperature value of (∂e0/∂p)T extrapolated from Jones's measurements confirms the existence of an apparent anomaly. The values of yt and yl differ little from those of a rigid ion model with the same interionic potential.

Lattice Dynamics and Second-Order Raman Spectrum of NaF

The second-order Raman spectrum of NaF has been measured using an argon laser source. For a light beam incident along the [010] direction, polarized along the [001] direction, and viewed along the [100] direction, the scattered radiation is almost completely polarized. It is shown theoretically that it is possible to account for the polarization and the detailed shape of the observed spectrum within the experimental resolution using a Raman polarizability tensor whose second derivatives with respect to nuclear displacements depend only on the positions of first neighbors in the crystal, provided one uses the phonon frequencies derived from a deformation dipole model that includes second-neighbor ${F}^{\ensuremath{-}}\ensuremath{-}{F}^{\ensuremath{-}}$ interactions. By assuming that the polarizability is affected only by central displacements of the nearest neighbors, the number of independent polarizability derivatives is reduced from eight to three. This assumption also gives a vanishing depolarized intensity in agreement with the experimental results. The second-order difference spectra have also been evaluated, and are found to be much more sensitive to the relative magnitudes of the various polarizability derivatives.

Lattice Dynamics of Imperfect Alkali Halides

In the framework of the T-matrix formalism the generalized phonon dispersion relations for a crystal containing a finite concentration of randomly distributed defects are introduced. The Green functions and the resonance conditions are given for the nearest-neighbor perturbation model of defects in NaCl-type lattices. Systematic calculations concerning monovalent positive and negative defects in Na, K, and Rb chlorides, bromides, and iodides are then performed in the framework of Hardy's deformation dipole (dd) model for the host lattice dynamics, and applications to the U center are given. In the same framework the nearest-neighbor effective force constant is defined as a function of the interionic distance and calculated for all the family of alkali halides. The results for several defects in NaCl and KCl are compared and found to be in good agreement with thermal conductivity data. Predictions concerning the offcenter systems are presented. Finally, theoretical acoustic dispersion curves, phonon relaxation rates, and infrared absorption spectra for copper- and silver-doped NaCl crystals are also reported.

Debye-Waller factors of alkali halides

The Debye-Waller factors of six alkali halides are tabulated as a function of temperature from absolute zero to 900°K. The calculations are based on the deformation dipole model. The results are in agreement with experiment at temperatures where comparison is possible, although in many cases the experimental errors are greater than the expected uncertainty in the computed results.

Lattice Dynamics and Specific-Heat Data of CsCl, CsBr, and CsI

Detailed calculations are presented for the frequency distributions and dispersion curves of the three cesium chloride-structure alkali halides: cesium chloride, bromide, and iodide. Three distinct models have been used: the rigid ion model, the polarization dipole model, and two variations of the deformation dipole model. In each case combined frequency spectra and dispersion curves are presented and detailed critical-point assignments made. Plots of the effective Debye temperature θD as a function of temperature are shown and compared with experiment. In addition, the moment functions of the frequency distributions ωD(n) = [13 (n + 3) μn] / n, where μn is the nth moment of a given distribution, have been computed, and plots of these results show that they provide a better means of discriminating between the various models than the θD curves, but unfortunately, no experimental data exist.

Theory of I.R. lattice vibration absorption by gap modes and resonance modes in KI

Conditions for the occurrence of resonance and gap modes associated with Cl − and Br − impurities in KI are investigated on the basis of a model for the impurity ion which takes into account the difference between the mass of the impurity and the mass of the ion it replaces and the changes in the nearest neighbor central and non-central force constants for the impurity-host crystal interaction. The dynamical properties of the host crystal are calculated on the basis of Hardy's deformation dipole model and Cochran's shell model. The absorption coefficients of KI containing Cr − and Br − impurities are calculated and compared with the corresponding experimental results. The results for the absorption coefficients calculated on the basis of the deformation dipole model are found to be in much better agreement with experimental results than are the theoretical results based on the shell model. However, both models of the host crystal fail to predict the magnitude of the experimental isotope shift in the frequencies of the gap modes associated with 35Cl − and 37Cl − impurities. In addition, in order to fit the experimental value of the Br − gap mode frequency both models require values of the change in the non-central, nearest neighbor, force constant which seem to be too large. Possible reasons for these discrepancies between the theoretical predictions and the experimental results are discussed.

Theory of infrared lattice vibration absorption by gap modes and resonance modes in KI

Abstract Conditions for the occurrence of resonance and gap modes associated with Cl − and Br − impurities in KI are investigated on the basis of a model for the impurity ion which takes into account the difference between the mass of the impurity and the mass of the ion it replaces and the changes in the nearest neighbor central and non-central force constants for the impurity-host crystal interaction. The dynamical properties of the host crystal are calculated on the basis of Hardy's deformation dipole model and Cochran's shell model. The absorption coefficients of KI containing Cr − and Br − impurities are calculated and compared with the corresponding experimental results. The results for the absorption coefficients calculated on the basis of the deformation dipole model are found to be in much better agreement with experimental results than are the theoretical results based on the shell model. However, both models of the host crystal fail to predict the magnitude of the experimental isotope shift in the frequencies of the gap modes associated with 35 Cl − and 37 Cl − impurities. In addition, in order to fit the experimental value of the Br − gap mode frequency both models require values of the change in the non-central, nearest neighbor, force constant which seem to be too large. Possible reasons for these discrepancies between the theoretical predictions and the experimental results are discussed.