Theory Of Lattice Dynamics Research Articles

On b.c.c. 3He, a Highly Anharmonic Crystal

The self consistent (SC) theory of lattice dynamics for highly anharmonic crystals is derived via expansion and selected re-summation of the one phonon Green function. Since perturbation treatments of weakly anharmonic crystals use the same expansion, the two cases can then be viewed as variations in a single method. The SC theory to lowest (SCH) and second (SC2) order and a re-order potential power series in which each coefficient appears averaged over the vibrational motion is derived.The SCH theory with the leading correction in the new series, the cubic anharmonic term, is applied to b.c.c. 3He. Phonon frequency dispersion curves, lifetimes, sound velocities, and elastic constants are computed. The phonons are well defined and the elastic constants and isotropy agree quite well with experiment. Although the cubic correction is significant, it suggests that the re-ordered series converges.

Lattice dynamics of ionic and covalent crystals

Abstract Lattice dynamics has some claim to being the oldest branch of solid state physics. Planck's Theory of Radiation and the Theory of Specific Heat was published by Einstein in 1907, On Vibrations in Space Lattices by Born and von Karman in 1912, and On the Theory of Specific Heat by Debye in the same year. Other early papers on lattice dynamics included those by Debye and by Waller on the effect of temperature op the scattering of x-rays by a crystal. In the 1920's Peierls made fundamental contributions to the theory of thermal conductivity and of electrical conductivity involving lattice dynamics, and in the 1930's Blackman brought to an end the acceptance of the complete validity of the Debye theory of specific heat and also contributed the first papers on anharmonic effects in the absorption of infrared radiation by ionic crystals. The work we have mentioned used the formal theory of lattice dynamics to account for the thermal and other properties of crystals. It was not until 1940 that Kellerman...

Elastic shear moduli and crystal stability at highPandT

This paper presents new theoretical expressions for the elastic velocities (of a cubic or isotropic homogeneous solid) as functions of density and temperature (equations 7 and 8). These equations have been derived within the fourth-order anharmonic theory, i.e., the extension of the theory of lattice dynamics into the regime of finite strain, and they include implicitly the effects of noncentral forces, distant-neighbor interaction, and thermal vibration. Application of the equations to garnet, spinel, and olivine (considered as Voigt-Reuss-Hill isotropic bodies) confirms and lends rigor to the present consensus on the elasticity of the mantle. The equations indicate further that the observed instabilities of these materials under upper mantle conditions are not directly related to vanishing of an elastic modulus.

Phenomenological lattice-dynamical theory of ferroelasticity

A phenomenological lattice-dynamical investigation is made into ferroelasticity and some of its associated phenomena, such as optical birefringence and elastooptic effect, in ferroelastic crystals that are nonmagnetic and may or may not be ferroelectric at the same time. Two different types, provisionally called optical type and elastic type, are conceivable for ferroelastics. Ferroelastics of the optical type are optically soft and elastically hard. They have a special optical mode of lattice vibration that is Raman-active and lowers rapidly in eigenfrequency as temperature falls toward the Curie point. Ferroelastics of the elastic type are elastically soft and optically hard. In them, a certain elastic stiffness when the coordinates of all lattice vibration modes with zero wave number are fixed diminishes rapidly as temperature falls toward the Curie point. The investigation is carried out for both these different types of ferroelastics, which are found to behave similarly in respects proper to ferroelasticity and to behave differently in not a few other important respects.

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.

A classical theory of lattice dynamics

A new formulation of the classical theory of lattice dynamics is presented which correctly reproduces all the features of the quantum theoretical results. This new approach starts with a more detailed description of the various potential functions that contribute to the total potential energy and uses these to define a new set of potential functions that depend on the initial configuration only. The expansion of the total potential energy in powers of nuclear displacements is then found to have a particularly simple form in terms of these configuration dependent potentials.

Phase Transitions and Soft Librational Modes in Cubic Crystals

Landau's theory of second-order phase transitions and the theory of lattice dynamics are combined in a presentation of an approach to the understanding of the crystallographic phase transitions of complex fcc crystals. The approach is applied to the antifluorite-type crystal ${\mathrm{K}}_{2}$Re${\mathrm{Cl}}_{6}$, which exhibits four phase transitions. The results of a series of crystallographic and spectrographic experiments are described, and are interpreted in light of the developed theory. Finally, a model based on the theory is constructed and is shown to describe the spectral observations. The resultant model leads to the conclusion that two of the phase transitions are driven by a single branch of a "soft" phonon.

Lattice-Dynamics Approach to the Theory of Elastic Dielectrics with Polarization Gradient

A continuum theory of elastic dielectrics including polarization gradient is obtained as the long-wave approximation of a theory of lattice dynamics for the shell model of cubic ionic crystals. The additional energy associated with the formation and relaxation of a free surface is obtained by considering the unbalanced charges and dipole moments on the surface. The material coefficients of the continuum theory are related to the lattice properties, and their numerical values are calculated for NaI, NaCl, KI, and KCl. The surface energy density of a free half-space predicted by the continuum theory is compared with experimental results as well as other theoretical results based on discrete models.

Motional Averaging of the Electric Field Gradient at Chlorine Nuclear Sites inK2PtCl6andK2PdCl6by the Lattice Vibrations

Temperature variations of the $^{35}\mathrm{Cl}$ nuclear-quadrupole-resonance (NQR) frequencies in ${\mathrm{K}}_{2}$Pt${\mathrm{Cl}}_{6}$ and ${\mathrm{K}}_{2}$Pd${\mathrm{Cl}}_{6}$ are discussed. The dominant thermal motions of the lattice responsible for the averaging of the electric field gradient at a nuclear site are assumed to take place in the plane perpendicular to the symmetry axis of the field gradient. Contributions to the NQR frequency shifts due to the internal modes ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$, and ${Q}_{6}$ of the ${[M{\mathrm{Cl}}_{6}]}^{=}$ octahedra are calculated using the results of the harmonic theory of lattice dynamics and the Bayer-Kushida theory of NQR frequencies. Differences between the experimental and calculated frequency shifts are assumed to be due to the rotary lattice modes; the frequencies of these modes are deduced in ${\mathrm{K}}_{2}$Pt${\mathrm{Cl}}_{6}$ and ${\mathrm{K}}_{2}$Pd${\mathrm{Cl}}_{6}$ for temperatures in the range $100\ifmmode^\circ\else\textdegree\fi{}\mathrm{K}lTl340\ifmmode^\circ\else\textdegree\fi{}\mathrm{K}$. In each case the torsional oscillation frequency is found to be relatively insensitive to the temperature.

Self-Consistent Phonon Formulation of Anharmonic Lattice Dynamics

The self-consistent phonon theory of anharmonic lattice dynamics is derived via a stationary functional formulation. The crystal dynamics is approximated by a set of damped oscillators, and these are used to construct a trial action, analytically continued into the complex time-temperature plane. Using the action, a free-energy functional is required to be stationary with respect to the trial oscillators. The resulting phonon modes are undamped at the first order of approximation, whereas to second order the phonon spectral function is determined self-consistently. Expressions are obtained in first order for various thermodynamic derivatives, such as pressure, elastic constants, specific heats, and thermal expansion.

Effect of static uniaxial stress on the Raman spectrum of silicon

We have performed the first experimental investigation of the effects of static uniaxial stress on the frequency of the q≈0 optical phonons in a diamond-type crystal. Application of uniaxial stress along [001] or [111] is found to cause splittings and shifts of the Raman peaks of the q≈0 triply degenerate optical phonons. From the observed splittings we have obtained experimental values for the phenomenological coefficients which describe the changes in the ‘spring constant’ of these phonons with strain. Our results are in reasonable agreement with lattice dynamical theory.

The quantum theory of lattice dynamics. I

The dynamics of a crystal is examined on the basis of the adiabatic approximation. In part I we examine the form of the dynamical and anharmonic tensors on the assumption that the effective nuclear potential energy can be represented as a simple two-body interaction. In part II we derive expressions for the electronic contributions on the basis of the Hartree-Fock approximation and show that these electron-nuclear interactions are more complex than a simple two-body interaction. In part III we examine these interactions in more detail and find that the two-body approximation is equivalent to a rigid-ion model and that this approximation becomes exact in the limit q = 0.

The quantum theory of lattice dynamics. IV

The total electronic energy for condensed systems is examined as a function of nuclear displacements from the equilibrium configuration. The expansion is derived to all orders both for the exact solutions to the many-electron Schrödinger equation and for the approximate Hartree-Fock solutions. The results are shown to be translationally invariant to all orders. Expressions are derived for the electronic contributions to the force-constant tensors in real space and it is shown that the effective nuclear potential has a simple two-body form for a Bravais lattice.

Theory of Anharmonic Crystals. I. General Formulation

AbstractThe theory of lattice dynamics of an anharmonic crystal is developed without using the harmonic approximation as starting point. On the basis of the double‐time Green's function method the self‐consistent system of equations determining the equilibrium parameters of the lattice, the vibrational frequency and their damping is obtained. The free energy and the internal energy of an anharmonic crystal are calculated.

Session H: Phase transitions and critical scattering

The one-phonon cross-section from the Rayleigh mode calculated with a point-ion lattice dynamical theory accounts for all salient inelastic scattering intensities seen in He-LiF(001)〈100〉 measurements published by Brusdeylins, Doak and Toennies [J. Chem. Phys. 75 (1981) 1784]. Scattering cross-sections are found by the Rayleigh method, wherein the repulsive part of the scattering potential is approximated as a hard-wall. The solution is in the form of a perturbation expansion in the elastic corrugation and phonon motion of the surface. For each phonon order the solution is infinite in the elastic corrugation. Resonant intensity enhancement occurs either through selective desorption or selective adsorption, both of which have amplitudes which explicitly appear in the perturbation expansion and can be separately calculated. Intensity enhancement also occurs by kinematical focusing. We find a new type of kinematical focusing in addition to that discussed by Benedek [Phys. Rev. Letters 35 (1975) 234].

Elastic and Dielectric Anomalies and the Lattice-Dynamical Theory of Ferroelectric Phase Transitions

The lattice-dynamical theory of ferroelectric phase transitions relates the dielectric anomaly with an anomalous of a TO mode as $q\ensuremath{\rightarrow}0$. It is shown that the elastic anomalies, which are commonly observed at ferroelectric phase transitions, are predicted by the lattice-dynamical theory of ferroelectricity, within a diatomic approximation. The softening of the TO mode is associated with variations in $\frac{d({\ensuremath{\omega}}_{\mathrm{TA}})}{\mathrm{dq}}$ which are reflected in the behavior of the transverse and shear elastic constants. The discussions are extended to more complicated systems and are illustrated by an examination of the available neutron scattering data on the temperature dependence of the dispersion curves of the TO and TA modes of SrTi${\mathrm{O}}_{3}$ and KTa${\mathrm{O}}_{3}$.

The quantum theory of lattice dynamics

It can be shown that the adiabatic and Hartree-Fock approximations allow one to define a unique electronic charge distribution for each atomic nucleus in a crystal and that this distribution moves rigidly with the nucleus in the harmonic approximation. The dynamics of the crystal can be shown to depend solely on the electrostatic interactions of these rigid ions.

New Theory of Lattice Dynamics at 0°K

It has previously been shown that the optimum choice of harmonic Hamiltonian with which to approximate a crystal Hamiltonian is one is which the force constants are equal to the ground-state expectation value of the second derivative of the crystal potential, the expectation value being computed self-consistently with the ground-state eigenfunction of the harmonic Hamiltonian. It is shown here that the appearance of ground-state averages of various derivatives of the potential is due to an explicit or implicit expansion of the potential in a Hermite polynomial series, and that such an expansion is superior to the conventional Taylor-series expansion for anharmonic systems. In addition, it is shown that one can systematically treat very anharmonic systems by expanding the Hamiltonian in a set of polynomials orthogonalized with respect to weight function chosen to cut off the potential at short range, and that explicit incorporation of an easily optimized Gaussian factor in this weight function provides a computationally convenient way of introducing certain desirable features into the general expansion.

Properties of Crystalline Argon and Neon in the Self-Consistent Phonon Approximation

The self-consistent phonon theory of anharmonic lattice dynamics, devised independently by several authors using varying techniques and implemented computationally by Koehler, is here applied to the crystals of neon and argon. A Lennard-Jones 6-12 interatomic potential is assumed. The quantities calculated are the phonon spectrum and the bulk thermodynamic properties of thermal expansion, compressibility, and specific heat, all as a function of temperature at zero pressure. Although the computations are intended primarily to explore in detail the content of the self-consistent phonon approximation preparatory to incorporating the more elaborate expressions of the next higher approximation, comparison is made with the existing experimental data.

Lattice dynamics of quantum crystals

A theory of lattice dynamics for quantum crystals is developed. This is done by summing an infinite class of diagrams of the usual anharmonic expansion and by avoiding the harmonic approximation as starting point. The zero order of the expansion given in this paper corresponds to the harmonic approximation with an effective potential. Higher orders correspond to higher anharmonic corrections with the same potential. Since the new potential varies more slowly the expansion seems to converge more rapidly than the usual anharmonic expansion. Numerical calculations on bcc He3 show that the ground state energy is lowered by about 3–4 cal/mol by taking into account long range correlations due to phonons. The elastic constants and the Debye temperature are calculated in zero and second order. The lowering of the bulk modulus due to the second order is about 10%. Experiments agree quite well with the second order results.