Sublattice Displacement Research Articles

Dynamical parameters and phonon spectrum of crystal In4Te3

Calculations of the phonon spectrum of an In4Te3 crystal in the central pair-interaction model neglecting long-range forces are reported. The model developed contains 19 unknown parameters, which were determined from experimental values of the displacements of individual sublattices in the unit cell and from the change in the basis vectors as a result of hydrostatic pressure. The phonon spectrum obtained contains a large number of low-frequency optical branches, deforming the acoustic branches.

Higher order elastic constants and generalised gruneisen parameters of elastic waves and low temperature thermal expansion of dysprosium

The expressions for the 19 fourth order elastic constants, 10 third order elastic constants and six second order elastic constants are derived using the method of homogenous deformation with interactions extending up to second neighbours using the sublattice displacements to the second degree in strains. These expressions are used to obtain the higher order elastic constants and their pressure derivatives in dysprosium. Dysprosium belongs to the hexagonal class of crystals. The higher order elastic constants are used to find out the generalised Gruneisen parameters of the elastic waves propagating in different directions in dysprosium. The Brugger gammas are evaluated and the low temperature limit of the Gruneisen gamma is obtained. The results are compared with the available reported values.

Role of covalence and of correlation in the dielectric polarization of oxides

The dynamical charges measure the macroscopic polarization induced by a relative sublattice displacement in a polar insulator. When the material has a mixed ionic/covalent character such charges present nontrivial features. Apparently similar oxides have strikingly different behaviors: some (like the ferroelectric perovskites) have giant dynamical charges, while others (like ZnO) are quite normal and display dynamical charges close to the nominal static ones. The rationale for such differences is found, upon closely scrutinizing the covalence mechanism. Furthermore, the effects of electron-electron interaction on the polarization of a model highly correlated dielectric are discussed.

Higher Order Elastic Constants and Generalized Gruneisen Parameters of Elastic Waves and Low Temperature Thermal Expansion of Titanium

The expressions for the 19 fourth order elastic constants, 10 third order elastic constants, and 6 second order elastic constants of a solid are derived using the method of homogenous deformation with interactions extending up to second neighbors using the sublattice displacements to the second degree in strains. These expressions are used to obtain the higher order elastic constants and their pressure derivatives in titanium. Titanium belongs to the hexagonal class of crystals. The higher order elastic constants are used to find the generalized Gruneisen parameters of the elastic waves propagating in different directions in titanium. The Brugger gammas are evaluated and the low temperature limit of the Gruneisen gamma is obtained. The results are compared with the available reported values.

Higher order elastic constants and generalized Gruneisen parameters of elastic waves and low temperature thermal expansion of gadolinium

Expressions for the higher order elastic constants are derived using the sublattice displacements to the second degree in strains. These expressions are used to obtain the higher order elastic constants and their pressure derivatives in gadolinium. The higher order elastic constants are used to find out the generalized Gruneisen parameters of the elastic waves propagating in different directions in gadolinium. The Brugger gammas are evaluated and the low temperature limit of the Gruneisen gamma is obtained. The results are compared with the available reported values.

Fourth order nonlinear elastic coefficients of hexagonal close packed lattice

The expressions for the fourth-order elastic constants of a hexagonal close packed lattice is derived using the sublattice displacements to the second order in strains. These expressions together with the expressions for the second- and third-order elastic constants have been used to obtain the higher order nonlinear elastic coefficients in some hexagonal metals. These expressions are used to find the second-order, third-order and fourth-order elastic constants of cadmium. The pressure derivatives of their elastic constants are also obtained. The results obtained are compared with the available experimental results.

Ab Initio and Model Calculations on Different Phases of Zirconia

AbstractIn order to get a better understanding of the energetics of ZrO2 (zirconia) ab initio calculations with the full potential linear muffin tin orbital method ( fp LMTO) have been performed on the tetragonal structure over a range of c/a and sublattice displacement. A new semi-empirical shell model is developed which makes use of Hartree-Fock calculations and includes compressible anions and quadrupolar distortions. The empirical model predicts energies for tetragonal distortion in agreement with the fp LMTO calculations. Furthermore, it enables us to understand why the seven-fold coordinated monoclinic phase is the low temperature equilibrium structure.

Ionicity dependence of surface bond lengths on the (110) cleavage faces of isoelectronic zincblende structure compound semiconductors: GaP, ZnS, and CuCl

Previously measured low-energy electron diffraction (LEED) intensity data for the (110) surfaces of the isoelectronic series GaP, ZnS, and CuCl, are reanalyzed using dynamical LEED calculations, a new selection of independent structural variables, and an automated optimization procedure. The bond lengths of the atoms in the top layer and perpendicular displacements of the cation-anion sublattices in the top two layers are selected as the independent structural variables. All other bond lengths are taken to be equal to their bulk values. An automated gradient method search procedure is utilized to obtain these structural variables by optimizing the fit of the measured intensities by the calculated ones as determined by the x-ray R factor. This procedure permits a precise comparison of the bond lengths of species in the top layer from one material to another. Our major finding is that the bond lengths, d, between the anion and cation in the top layer depend on the spectroscopic ionicity, f, via [(d−dB)/dB]=0.025−0.14f, in which dB is the bulk value of the bond length. The bonds between surface and subsurface species do not exhibit a systematic dependence on f.

Fourth-order elastic constants of nonideal hcp crystal Mg and Er

The expressions for the 19 fourth-order elastic constants of an hcp crystal with nonideal axial ratio have been derived with interactions extending up to second neighbors and using sublattice displacements to the second degree in strains. These formulas have been used to evaluate the fourth-order elastic constants of magnesium and erbium. The results reveal higher-order elastic anisotropy in erbium. The second pressure derivatives of the effective second-order elastic constants of magnesium and erbium have been obtained using their fourth-order elastic constants and are positive. The magnitudes of the second pressure derivatives of C112, C133, and C111 in magnesium and that of C112 in erbium are extremely large, lending support to the phase changes that occur in these metals when subjected to hydrostatic pressure.

Vibronic theory of a structural phase transition and a tricritical point in IV-VI compounds.

The vibronic theory of a structural phase transition in IV-VI compounds is reformulated in terms of the mean-field approximation, taking account of the spin-orbit coupling between conduction and valence bands and anharmonic couplings between TA and TO modes. The effect of the electron-hole pair polarization is incorporated into the self-energies of the TA phonon as well as the TO phonon, so as to give the carrier concentration dependence of shear modulus ${C}_{44}$. This yields the proper dependence of the ratio \ensuremath{\zeta} of temperature derivatives of the squared TO-phonon frequencies in the cubic and rhombohedral phases on the carrier concentration, and also that of the jump of ${C}_{44}$ in the acoustic anomaly at the transition point. It is shown that the ratio \ensuremath{\zeta} becomes divergent at the tricritical point, and the TA mode softens at the superheating point (in case of first-order transition) or at the tricritical point. The spin-orbit coupling leads to peculiar structures of Fermi surfaces and to an increase in the electron chemical potential with growth of sublattice displacement. It is also shown in the fourth-order anharmonicity of the TO mode that the lattice anharmonicity is suppressed by the vibronic anharmonicity due to the spin-orbit coupling.

Interpenetrating incommensurately modulated lattices with icosahedral symmetry.

This paper derives an infinite class of quasiperiodic lattices, composed of incommensurately modulated sublattices with displacive modulations, whose diffraction patterns have icosahedral symmetry. The sublattice reference lattices, as well as the sublattice displacement fields, belong to the space group $R\overline{3}m$ and are equivalent under the icosahedral point group. These quasiperiodic lattices are fundamentally different from those constructed by the projection and grid methods. Nevertheless, a special displacement field reproduces a three-dimensional generalization of the Penrose tiling.

Effect of metal substitution in ZrSe3-type compounds: Vibrational states of Zr1-tHftS3.

The pseudobinary ${\mathrm{ZrS}}_{3}$--${\mathrm{HfS}}_{3}$ system exhibits a complete range of solid solubility, the structure being of the ${\mathrm{ZrSe}}_{3}$ type with Zr and Hf distributed at random over the metal sublattice in the ternary region. The unit-cell dimensions, as well as the virtually-temperature-independent diamagnetic susceptibility of ${\mathrm{Zr}}_{1\mathrm{\ensuremath{-}}\mathrm{t}}$${\mathrm{Hf}}_{\mathrm{t}}$${\mathrm{S}}_{3}$, evolve approximately linearly with the compositional parameter t. Examination of the vibrational states of ${\mathrm{Zr}}_{1\mathrm{\ensuremath{-}}\mathrm{t}}$${\mathrm{Hf}}_{\mathrm{t}}$${\mathrm{S}}_{3}$ by Raman spectroscopy has proven to be an efficient tool for probing the lattice properties of the parent members ${\mathrm{ZrS}}_{3}$ and ${\mathrm{HfS}}_{3}$. One-mode behavior of most of the long-wavelength optical modes in ${\mathrm{Zr}}_{1\mathrm{\ensuremath{-}}\mathrm{t}}$${\mathrm{Hf}}_{\mathrm{t}}$${\mathrm{S}}_{3}$ confirms the identity of the eigenvectors, which remain unperturbed under the Zr-to-Hf substitution as they involve essentially only the motions of the sulfur atoms. Continuous and slight variation of the eigenfrequencies with the composition reflects minor changes in force constants when one metal atom is replaced by another. The two-mode behavior of the rigid sublattice displacements of the metal against the sulfur atoms perpendicular to the b axis gives rise to a broad structure in the Raman spectrum of ${\mathrm{HfS}}_{3}$ resulting from an impurity mode of the Zr originally present in the Hf starting material. The accidental degeneracy of the ${A}_{g}$ and ${B}_{g}$ modes at 151 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ in ${\mathrm{ZrS}}_{3}$ is lifted by the metal substitution, thus confirming previous findings obtained under hydrostatic pressure.

The electronic band structure of Pb1-xSnxTe alloys. II. Temperature dependence through the structural and band inversion transitions

For pt.I see ibid., vol.15, p.317 (1985). The Shubnikov-de Haas effect has been studied in samples of Pb1-xSnxTe which have normal or inverted band configurations at low temperatures. The measurements have been made over a range of temperatures which take the samples through the structural phase transition from the rhombohedral to the cubic rock-salt phase. The changes in Fermi-surface cross-sectional areas with temperature are interpreted in terms of a temperature-dependent band-structure calculation which includes the interaction between electrons and soft transverse optic phonons associated with the phase transition and the temperature-dependent sublattice displacement and shear strain in the rhombohedral phase. The basic band-structure parameters are those deduced from the orientation-dependent study of Pb1-xSnxTe at 4.2K in the distorted phase, and the variations with temperature of the energy gap in the alloys is used to yield further parameters. The resulting fit to the variation of the cross-sectional area with temperature is excellent. Measurements of the temperature dependence of the transverse effective mass have also been made in this temperature range and it is found that a temperature-dependent scattering parameter has to be included in the theory to obtain agreement with the observations. This parameter is related to the soft-TO phonon scattering and to the formation of domain walls near the transition.

Effects of sublattice positions and of asphericity of 4f and 5d charge densities on X-ray diffraction intensities in Tb and Dy

Aspherical 5d charge densities in a model for 5d electrons in the heavy rare-earth metals developed by the present author produce small differences in X-ray diffraction intensities from those predicted by more traditional theories, in which anisotropy is significant only for 4f shells. Diffraction intensities in Tb and Dy depend both on asphericities of charge densities and on the value of a parameter measuring the relative positions of sublattices in the low-temperature orthorhombically distorted structure of these metals. Predictions of ratios of intensities are made for several pairs of lines in each metal. Relative intensity measurements to an accuracy of the order of 1% should be sufficient to determine the sublattice displacements, but to distinguish between the model with large 5d electric quadrupole and octupole moments and traditional theories would require accuracies of the order of 0.3%.

Stresses and elastic constants of metals with classical ion motion

For a metallic system of ions with interaction potentials of the general form given by pseudopotential theory, and with the motion of the ions treated by classical statistics, strain derivatives of the canonical partition function are constructed. Expressions are obtained for the stresses and isothermal elastic constants, for any crystal in which there are no sublattice displacements accompanying a homogeneous deformation. These expressions are in the form of canonical-ensemble averages, and can be evaluated by standard computer simulation techniques. The nature and magnitude of quantum corrections are discussed.

Preparation and structural analysis of SnTe/Sb composition modulated structures

A new composition modulated structure composed of semiconducting SnTe and semimetallic Sb has been grown epitaxially on mica substrates with differing SnTe to Sb ratios. The samples were characterized by x-ray diffraction using the transmission Laue patterns and θ-2θ diffractometer scans with scattering vectors both normal and in the plane of the film. We observed that the modulation satellites were 180° out of phase for films with short modulation wavelengths, but were in phase for long wavelengths. This change in the phase of the modulation can be understood by assuming that Sb adopts a quasicubic structure at short modulation wavelength, due to the coherency strain, but relaxes back to a distorted rhombohedral structure, with the loss of coherency, at long wavelengths. A simple 1-D model was used to estimate the modulation wavelength dependence of the sublattice displacement of Sb.

The relation of the stress dependence ofF(H)2to inner compliance and internal strain tensors in 20 simple crystal structures

Internal strain tensors describe the relative displacements of sublattices in stressed crystals in which some atoms occupy sites lacking inversion symmetry. The variation of |F(H)|2 in such crystals is shown to depend on inner compliance tensors which are essentially products of internal strain tensors with the macroscopic elastic compliance tensor. The forms of the inner compliance tensors and the uniaxial stress derivatives of |F(H)|2 are tabulated for 20 simple crystal structures in which the number of atoms per lattice point is 2, 3 or 4.

Band Edge Structure Transformation Due to Ferroelectric Transition in Pb1-xGexTe Alloy Semiconductors

The band edge structure modification of Pb 1- x Ge x Te due to the structural phase transition has been investigated by means of the interband absorption and the Shubnikov-de Haas (SdH) effect. Below the phase transition temperautre ( T c ) the band gap determined by the optical absorption increases with lowering temperatures or at least its temperature coefficient changes. From the SdH effect below T c , we obtained the information of the carrier redistribution among the conduction (or valence) bands caused by the phase transition. These results can be explained only when we take into account the effects of relative sublattice displacement and the homogeneous strain at the same time.

Study of the tetragonal-to-tetragonal phase transition of theA15pseudobinary alloyNb3Sn1−xSbxby neutron and x-ray scattering

High-momentum-resolution neutron and x-ray scatterings have been carried out on recently grown ${\mathrm{Nb}}_{3}{\mathrm{Sn}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$ single crystals. It was shown that, in addition to the well-known cubic-to-tetragonal martensitic phase transition at ${T}_{M}$, another phase transition takes place at ${T}_{t}$ between two tetragonal phases with the identical space group ${D}_{4h}^{9}$ but with opposite tetragonal distortions $agc$ and $alc$. The transition at ${T}_{t}$ in the Sb concentration range $0.05\ensuremath{\lesssim}x\ensuremath{\lesssim}0.15$ was found to be of the first order. In addition, the magnitude and direction of the sublattice displacements of the Nb atom were determined in both tetragonal phases. The Nb atom shifts so as to avoid the closer contact with its four neighboring Sn or Sb atoms. The tetragonal-to-tetragonal transition observed in the present system offers a rare example in which no orthorhombic phase exists between two tetragonal phases with opposite distortions. Conditions for the absence of an intermediate phase are discussed by Axe and Yamada.

Second-order inner displacement in the hexagonal close-packed structure

Expressions given recently by Ramanand et al. for second-order sublattice displacements in the hexagonal close-packed structure are shown to be in error. The use of inner elasticity theory leads to expressions that conform fully to the requirements of thermodynamics and crystal symmetry.