Dynamical Matrices Research Articles

Thermal Diffuse Scattering in X-Ray Diffraction by Aluminium

The purpose of this work is to investigate the diffuse streak patterns in X-ray diffraction quantitatively by numerical calculation of the three dimensional intensity distribution of the thermal diffuse scattering in the reciprocal space of aluminium at 293 K. The contributions of the harmonic first- and second-order thermal diffuse scatterings and the lowest-order anharmonic scattering were evaluated individually. The calculation was carried out by the use of the inverse dynamical matrices for up to 400 samples of wave vectors within the first Brillouin zone. The data of the interatomic force constants were taken from Gilat and Nicklow's article (Phys. Rev. 143 (1966) 487), in which the axially symmetric Born-von Karman force constant model including nearest neighbours from the first to the eighth was used. The results of our calculation agree with photographic observation of the diffuse streaks well.

Anharmonic Thermal Scattering of X-Rays from Cubic Metals

Thermal diffuse scatterings of X-rays by the lowest-order anharmonic vibration of cubic metals are investigated. The scattering which can be regarded as an interference between the harmonic one-phonon and two-phonon scatterings gives rise to asymmetry in the scattering around the reciprocal lattice point. The intensity formula is expressed as products of the elements of the inverse dynamical matrices in the high-temperature approximation. In terms of the inverse dynamical matrix elements for wave vectors up to 4000 in the first Brillouin zone, the intensity distributions of the anharmonic scatterings are evaluated numerically on the symmetry axes of aluminium and lead at room temperature. The observable asymmetries in the intensity distribution of the diffuse scatterings due to the anharmonic scattering are found near the reciprocal lattice points of the both crystals.

A New Model for the Lattice Dynamics of Alkali Halides

Qualitative properties of the electronic band structure of a KCl crystal are used to derive a simple two-parameter model for the correlation function c(q, q') which arises in the theory of lattice dynamics. This function relates the adiabatic deformation of the electronic band structure to the vibration of the ion cores. Expressions for the dynamical matrices are derived. The model is ready for a direct computer evaluation of the phonon frequencies to test its validity.

On phonon dispersion in hcp metals

A new expression for the electron-ion interaction has been obtained by averaging Bhatia's expression for the force at the ionic centres over the whole Wigner-Seitz sphere. Central interaction up to fourth neighbours has been calculated with the Born-Von Karman method. The resulting dynamical matrices have been solved along the [0001] and [01 1 0] symmetry directions for six hcp metals, namely, Mg, Zr, Y, Ti, Tl and Hf. Theoretical results are compared with the experimental neutron-scattering data for Mg, Zr and Y. Results for Hf, Tl, Ti have been compared with our earlier theoretical results.

A lattice-dynamical treatment of the thermal-motion bond-length correction

The effect of the thermal motions of the atoms on interatomic distances is treated in terms of lattice dynamics. It is not only shown how the anisotropic vibration tensors of the atoms can be derived from the dynamical matrices of the crystal but also how mean binary-product coupling tensors are obtained. Each of these tensors expresses the coupling of the motions of two atoms in the unit cell as an average over time and lattice and thus is suitable for formulating the bond-length correction. Hence the discussion centres around the coupling tensors. The coupling tensors cannot be determined by experiment; but in order to calculate them and the bond-length correction, one is forced to conceive dynamical models of motion for the atoms in the unit cell. The corrections for the known models of uncorrelated motion, rigid-body motion and riding motion are rederived by using the coupling tensors.

Contributions of Three-body Overlap Effects to the Force Fields in CaF2-type Shell-model Lattices

Abstract Due to the exclusion principle the distribution of electronic charge in an ionic crystal differs from a superposition of free-ion charge densities even in the simple Heitler-London picture. This charge density deformation engenders three-body long-range forces the influence of which on lattice vibrations is not accounted for by the usual Kellermann matrix. To obtain a better separation of long-range from short-range forces in CaF2 , SrF2, and BaF2 , i. e. to avoid absorbing long-range interactions in an adjustable short-range force field, explicit formulae are derived for three-body contributions to the shell-model equations of motion. The additional dynamical matrices pertain to arbitrary wavelengths. In adding to the force field terms which are not purely volume dependent the present descirption of three-body forces is somewhat at variance with recent work on alkali halide dynamics. The deviation from pure volume dependence originates in overlap charges residing in internuclear regions.

Lattice dynamics and magnetic ordering in the chalcogenide spinels CdCr2S4 and CdCr2Se4

AbstractA detailed analysis is presented of the k = 0 phonons of CdCr2S4 and CdCr2Se4 which is based on a simple force model with four adjustable short range parameters. With the help of the derived dynamical matrices for the different symmetry species, the analysis can be performed easily for any compound with the normal spinel structure. From the measured frequencies of the Raman and infrared active phonons of CdCr2S4 and CdCr2Se4 the force constants and eigenvectors of the different normal modes are calculated. We firstly discuss the derived force fields of these two compounds and secondly the coupling between the internal and external lattice vibrations. Finally, with the help of the calculated normal coordinates, the observed influence of the magnetic ordering on the Raman‐ and infrared active phonons of CdCr2S4 and CdCr2Se4 is studied in detail.

Hermitian character of the dynamical matrix - a comment on ‘non-hermitian dynamical matrices in the dynamics of low symmetry crystals’

The suggestion by Powell that dynamical matrices in low symmetry crystals may be non-Hermitian is shown to be based upon an incorrect use of the translation invariance condition. Correct application of the condition may lead to new requirements that must be satisfied by any physically acceptable force constant model. Die These nach Powell, wonach dynamische Matrizen der niedrigen symmetrischen Kristalle einen nicht Hermiteschen Charakter annehmen konnen, weist auf eine inkorrekte Anwendung der Translations-invarianz hin. Eine Korrekte Anwendung konnte zu neuen Bedingungen fuhren, wenn die Kraftkonstante bei jedem physikalisch annehmbaren Model vorhanden ist.

A new method for calculating the influence of force constant changes on the mean square displacements of surface atoms

The temperature dependence of the mean square displacements and the mean square momenta of surface atoms is studied using a series expansion. The elements of the series are calculated—even in the high temperature limit—by inverting dynamical matrices. Two parallel crystal surfaces are considered as a ‘force constant defect’ of a crystal with cyclic boundary conditions. Only elements of the dynamical matrices corresponding to atoms in the first two layers parallel to the surfaces are affected by the surfaces. As a consequence matrix partitioning permits a calculation of the inverse dynamical matrices by inverting matrices of much smaller dimension. Calculations are performed for the (100) surface of f.c.c. lattices using two Born-von Karman models with interactions up to third nearest neighbours. In the high temperature limit the influence of a change in the coupling constants for central forces between nearest neighbours at the surface is investigated. The coupling constant for central forces between an atom in the first layer and its nearest neighbours in the same layer and the coupling constant for central forces between an atom in the first layer and its nearest neighbours in the second layer are varied independently. A comparison between theoretical results and experimental data from low-energy electron diffraction is given.

Non-Hermitian dynamical matrices in the dynamics of low symmetry crystals

It is shown that in the Born-von Kármán theory of lattice dynamics the dynamical matrix may, under particular conditions, be non-hermitian for low symmetry crystals. Normal modes can then be introduced by using both right and left eigenvectors of the matrix. The hermitian character of the matrix can be maintained by imposing constraints on the interatomic force constants.

Symmetry Properties of the Kinetic Energy Matrix and Their Applications to Problems of Molecular Vibrations

In the present paper it is shown that G, the inverse of the kinetic energy matrix in the internal coordinates space, provides all the possible information on the symmetry of molecular vibrations. Namely, once the G matrix of a molecule is known, the point group G of the molecule, the coordinates which provide a basis for a completely reduced representation of G (symmetry coordinates), and the minimal set of independent force constants which must be chosen can be derived. Corresponding computer programs have been written which provide an automatic way for the factorization of the dynamical matrices of large and/or highly symmetrical molecules.

Frequency Distribution, Specific Heat, and Young’s Moduli of Orthorhombic Polyethylene with Skeletal Approximation

Abstract In preliminary studies on the frequency distribution, specific heat and Young’s moduli of the orthorhombic polyethylene crystal, methylene groups were treated as single dynamic units. Intermethylene force constants were estimated from the intermolecular potential function of methane and from deBoer’s potential function. Dynamical matrices of acoustic vibrations were constructed. A practical method was worked out for deriving the frequency distribution of acoustic vibrations from the volume enclosed in constant-frequency surfaces in a three-dimensional phase-difference space. With reference to specific heat in the lowest temperature region, inter-methylene force constants were adjusted. An efficient method was used for calculating the frequency distribution of crystal vibrations of chain-polymers. The frequency distribution of orthorhombic polyethylene crystal and specific heats in low temperature region were calculated in agreement with experimental results. A simple approximate method was used for treating Young’s moduli of orthorhombic chain-polymer crystals. Applicability of the skeletal approximation is discussed.

Stability of Crystal Lattices

The condition that a given lattice model be stable against all small deformations is that all the eigenvalues of all the dynamical matrices be positive. The stability of several lattices is studied by calculating the dynamical matrices for a large number of wave vectors in the Brillouin zone. Central potential interactions represented by Lennard-Jones and Rydberg forms are used, and the nearest-neighbor distance $\ensuremath{\epsilon}$ is allowed to vary throughout a range of \ifmmode\pm\else\textpm\fi{}10% of the value ${\ensuremath{\epsilon}}_{0}$ which minimizes the static lattice potential. The fcc and hcp lattices are stable for all central potentials and all values of $\ensuremath{\epsilon}$ studied; the bcc is stable for all values of $\ensuremath{\epsilon}$ for long-range central potentials; and the diamond is stable for a range of $\ensuremath{\epsilon}>{\ensuremath{\epsilon}}_{0}$ for all central potentials studied. The sum of the static lattice-potential plus the harmonic zero-point energy is minimized as a function of $\ensuremath{\epsilon}$, and it is found for all stable models except diamond that this procedure increases $\ensuremath{\epsilon}$ from ${\ensuremath{\epsilon}}_{0}$ by about ${\ensuremath{\epsilon}}_{0}\ensuremath{\kappa}$, where $\ensuremath{\kappa}=\frac{\ensuremath{\hbar}}{{M}^{\frac{1}{2}}{D}^{\frac{1}{2}}{\ensuremath{\epsilon}}_{0}}$, with $M=\mathrm{mass}\mathrm{of}\mathrm{ions}$ and $D=\mathrm{static}\mathrm{lattice}\mathrm{binding}\mathrm{energy}$. For diamond, however, there is no minimum in the range of $\ensuremath{\epsilon}$ for which the lattice is stable, for physically reasonable values of $\ensuremath{\kappa}$. Born has suggested that if a lattice is stable for long waves, it is stable for short waves; a counterexample has been found in the present study for the diamond lattice. A comprehensive table of accurate lattice sums is given in an Appendix.

Lattice Dynamics and Elasticity of Stressed Crystals

A conservative system composed of a collection of interacting ions plus externally applied forces is considered. The mechanical problem of motion is defined for such a system, and the equilibrium and the translational and rotational invariance conditions discussed. It is shown that the second equilibrium condition of Born and Huang, namely that the stresses must vanish in the equilibrium configuration for the infinite lattice model, is not a requirement of the theory. The dynamical matrices, whose eigenvalues are simply related to the phonon frequencies, are shown to be of the same form for a homogeneously strained crystal as for an ideal unstrained crystal. The elastic constants of a homogeneously strained crystal are calculated by the method of homogeneous deformation, and also by the method of long waves. The quantities which are observed in the measurement of sound velocities in strained crystals are the effective elastic constants, which differ from the elastic constants by terms involving the stress components. These effective elastic constants are also calculated in the lattice theory, and their properties are discussed.

Calculation of the Moments of the Vibrational Frequency Distributions of Simple Crystals

A method of obtaining moments of frequency distributions of vibrations of simple crystal lattices is demonstrated. The moments are obtained directly from corresponding powers of the dynamical matrices in terms of simple displacement coordinates.