Theory Of Lattice Dynamics Research Articles

A quantum field theory of lattice dynamics and melting in fcc crystals

Phonons in crystals are constructed by particle-hole excitations of ion cores. The Ward-Takahashi relations play an essential role in constructing the lattice dynamics and determining the melting temperature. The phonon dispersion and the melting temperature in fcc crystals are presented. In principle, our theory covers metallic, covalent and molecular crystals.

Nonlinear lattice-dynamical theory of mixed-valence compounds

by M. Cardona The unusual temperature dependence of anomalies of phonon dispersion curves in Sm 0.75Y 0.25S is interpreted by means of nonlinear electron-phonon coupling. The γ 1-symmetry coupling is attributed primarily to the fluctuations of Sm electrons between f and d states. In as much as the anomalies in the phonon dispersion curves are seen only in the longitudinal modes in the Λ direction, they can be treated exactly in a one-dimensional projection.

A quantum field theory of melting and lattice dynamics

The melting and the lattice dynamics are investigated from the view-point of a quantum field theory at finite temperature (thermo field theory). Our theory starts with ion cores and valence electrons interacting with each other as the first principle. The Ward-Takahashi relations play an essential role in determining the melting temperature and constructing the lattice dynamics. Phonons in crystals are constructed by particle-hole excitations of ion cores and valence electrons. Phonon spectra, phonon propagators and the dynamical map of the density of ion cores are calculated at finite temperature. In principle, our theory covers metallic, covalent and molecular crystals. The conventional secular equation for the lattice dynamics is obtained by neglecting some quantum effects in the equation of the density-density correlation of ion cores.

Theory of lattice-dynamical properties of compressed solids.

The Thomas-Fermi-Dirac statistical model is generalized to treat the mechanical and thermal properties of compressed solids where the effect of the periodic lattice is taken into account in the muffin-tin approximation. A general lattice-dynamical theory is presented in the same approximation, where the core effect on the dynamical matrix is treated consistently with the statistical model. Calculated results are given for the elastic constants, the Debye temperature, the Gr\"uneisen constant, and the melting temperature for Al and partly for Fe covering a wide range of density. The present theory has been examined particularly for phonon spectra at the normal density, and is in good agreement with the experiments for Al.

Effect of a magnetic field on the vibrations of an ionic lattice.

If an ionic solid is regarded as a vibrating lattice of point charges, a static magnetic field will exert a Lorentz force on each ion. In this paper the standard lattice dynamical theory is modified to allow for such forces, and a perturbation approach is developed. The frequencies of lattice vibrations and the velocity of propagation of acoustic waves are found to be perturbed, with the perturbation parameter being the ratio of the ion-cyclotron frequency in the magnetic field to the characteristic frequency of lattice vibrations. It is found that in non-centro-symmetric crystals the perturbation will be first order in this parameter (and hence in the magnetic field), while for centro-symmetric crystals, with certain exceptions, they will be second order. The exceptions are for degenerate modes, such as 〈100〉 and 〈110〉 propagation in cubic crystals, when a first-order effect which is quadratic in the wave number is predicted. The possibility of experimental measurements of the first-order effects is pointed out. A simple two-dimensional model is examined for more detailed illustration.

A quantum field theory of lattice dynamics at finite temperature

Abstract A quantum field theoretical treatment of three-dimensional cubic crystals at finite temperature is presented from the view-point of the spontaneous breakdown of the spatial translational invariance using thermo field theory. The effective interaction Hamiltonian is constructed by taking into account the dynamical map of the molecular density operator which is obtained from the Ward-Takahashi relations. The acoustic phonons are expected to be the excitation of particle-hole pairs. The conventional secular equation for the lattice vibrations is obtained by neglecting some quantum effects in the Bethe-Salpeter equation for the molecular density fluctuations. The phonon spectra, the phonon propagators and the dynamical map of the molecular density operator are calculated at finite temperature.

One-phonon scattering of He from the LiF(001)〈100〉 Rayleigh mode

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].

On the Nonlocal Theory of Wave Propagation in Elastic Plates

Propagation of longitudinal waves in isotropic homogeneous elastic plates is studied in the context of the linear theory of nonlocal continuum mechanics. To determine the nonlocal moduli, the dispersion equation obtained for the plane longitudinal waves in an infinite medium is matched with the parallel equation derived in the theory of atomic lattice dynamics. Using the integroalgebraic representation of the stress tensor and the Fourier transform, the system of two coupled differential field equations is solved in the standard manner giving the frequency equations for the symmetric and antisymmetric wave modes. It is found that the short wave speed in the Poisson medium differs by about 13 percent from the speed established in the classical theory. A numerical example is given.

Modulated piezoreflectivity and the photoelasticity of molecular crystals

Analysis of the optical response of a molecular crystal undergoing an external, periodic modulation is presented in terms of classical dipole theory. The case of modulated strain is treated in particular and identification of such strain with acoustic phonons with q=0 permits use of lattice dynamical theory to elucidate the acoustic phonon–exciton interaction. It is shown that extrema in the piezomodulated reflectivity (PMR) locate transverse and longitudinal exciton–polariton frequencies under specific experimental conditions. Model calculations demonstrate the great difference in the PMR for strong and weakly coupled systems. Relation of the PMR intensity to the elasto-optic coefficients and the storage of mechanical energy by the crystal permits extension of the technique to the study of such phenomena as internal strains and phase transitions where the partition of the mechanical energy among the various crystal excitations is of interest.

Plane waves in nonlocal micropolar elasticity

Dispersion relations are obtained for the transverse plane waves in linear, nonlocal micropolar elastic solids. By equating the frequency of the transverse acoustical (TA) branch at the end of the Brillouin zone, the nonlocal attenuation function is determined. The resulting (TA) curve for the frequency versus wave number ω(k) fits well with the result known from the Born-Kármán theory of lattice dynamics, in the entire Brillouin zone. By using experimental values of ω at the origin and the end of the Brillouin zone for Mg crystals, the micropolar elastic moduli are determined.

Improved version of group-theoretical analysis of lattice dynamics

Title of program: GROUP THEORY LATTICE DYNAMICS 2 Reference to other published version of this program: Catalogue number: ACMM Catalogue number: ACMI; Title: GROUP THEORY OF LATTICE DYNAMICS: Ref. in CPC: 3 (1972) 82 Program obtainable from: CPC Program Library, Queen’s University of Belfast N. Ireland (see application form in this

Single-kink dynamics in a one-dimensional atomic chain: A nonlinear atomistic theory and numerical simulation

A nonlinear lattice-dynamical theory of single kinks is presented which involves a simple equation of constraint. A set of coupled equations of motion is derived for the kink and discrete lattice fluctuations, which retains the full details of their mutual interaction. The theory is used to study ${\ensuremath{\varphi}}^{4}$-lattice kinks. For low kink velocities the kink equation of motion reduces to a generalized Langevin form. The static kink energy is found to vary periodically with the lattice spacing, implying that thermal kink motion is an activation process at low temperatures. Numerical studies reveal that kinks undergo damped motion, oscillatory motion, and trapping between lattice sites. Damping and oscillatory motion vanish as the continuum limit is approached. A phenomenological theory of damped kink propagation is developed and compared with numerical simulation.

Lattice-dynamical evaluation of temperature factors in non-rigid molecular crystals: a first application to aromatic hydrocarbons

Evaluation of temperature factors from a harmonic lattice-dynamical theory of molecular crystals has been applied to some aromatic hydrocarbons in the non-rigid case: i.e. mixing of lattice vibrations with the lowest-frequency internal modes. The calculations start from known atomic coordinates, unit-cell parameters and symmetry operations, using Cailfano-Neto potentials for in-plane modes, Williams IVa potentials for intermolecular interactions, and an overall value of 0.9 nN Å for any twisting around C-C bonds. Molecular motions inside the crystal are described on the basis of normal coordinates of the isolated molecule, by applying Gwinn's method. For some molecules (benzene and naphthalene) the rigid-body approximation is adequate for the carbon-atom framework, and practically no coupling occurs between the lattice vibrations and the internal motion. For anthracene and phenanthrene, there are differences between rigid-body and non-rigid estimates of temperature factors. Molecular vibration tensors T, L, and S can be calculated, and also a general molecular displacement matrix (tensor). including even the internal modes (W), which permits extension of the rigid-body analysis to non-rigid molecules.

Theory of anharmonicity in the lattice dynamics of crystals of rigid molecules

The use of the rigid-molecule approximation in the theory of lattice dynamics of molecular crystals leads to higher-order kinetic energy terms when anharmonic effects are allowed for. The conventional theory of lattice dynamics is extended to take account of these terms.

Local-field effects and phonon screening in polar semiconductors

We study here the redistribution of the electronic charge associated to zone-center phonon modes in polar semiconductors, within the linear-response microscopic theory of lattice dynamics. In a polar material zone-center phonons are only well defined in the longwavelength limit. Both transverse and longitudinal phonons correspond, in the ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}_{0}\ensuremath{\rightarrow}0$ limit, to the same ionic displacements in the unit cell; owing to the presence of long-range forces, they induce a different microscopic polarization. For a TO phonon the electronic response is exactly the same as induced by a frozen-in distortion having the lattice periodicity (${\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}_{0}=0$). We develop in detail the formalism needed to describe the screening difference between LO and TO phonons, which is basically due to nonanalytic terms in the dielectric matrix. We show that this longitudinal-transverse difference is closely related to other basic local-field effects, which have recently been studied. Detailed calculations are presented for ZnSe; in order to illustrate the trends with increasing ionicity the results for Ge, which is nonpolar and isoelectronic to ZnSe, are also shown. Both calculations are based on first-principles dielectric matrices recently published. We show that the effects of the local fields are very important and essential to account for the basic trends with increasing ionicity. The same feature has recently been proved for the calculated zone-center phonon frequencies.

Thermodynamics of strongly anharmonic crystals

AbstractA self‐consistent phonon theory of lattice dynamics based on the thermodynamic double‐time Green's function method is applied to the investigation of the thermal properties of the anharmonic face‐centred and body‐centred cubic crystals with the Morse potential as a model for nearest‐ neighbour central force interaction. The temperature and pressure dependence of the thermodynamic functions of cubic metals in the high and low temperature limits are given and compared with experimental data.

Internal size effect in condensed BaTiO3 ferroelectric films

Polycrystalline BaTiO3 films of equal thickness and with various size of the coherent scattering regions prepared by rf sputtering are studied. A distinct correlation between the dielectric permittivity and the size of coherent scattering region is established. It is shown that at a critical size of the coherent scattering region, D*≈ 300 A, the dielectric permittivity considerably decreases and the ferroelectric state becomes unstable. The data obtained are interpreted in terms of the lattice dynamical theory of size effects in ferroelectrics. [Russian Text Ignored].

Dynamics of the crystallized one-component plasma

The dynamics and energy of the crystallized one-component plasma (OCP) is evaluated using the self-consistent-phonon (SCP) theory of lattice dynamics. Melting of the crystal is also examined. The OCP crystal is harmonic for particle rms vibrational amplitudes as large as 25% of the interparticle spacing. This is due to the soft ( r −1) core of the Coulomb potential. Anharmonic effects are, however, entirely responsible for the eventual mechanical instability, identified here with melting, of the crystal at large enough rms amplitudes. This takes place at r s = 180 at T = 0 K for the most sophisticated SCP theory. In the classical limit, this SCP theory predicts the crystal to be more stable than does the “exact” Monte Carlo study of melting by Pollock and Hansen. This suggests that including further anharmonic terms in the SCP theory leads to melting at even larger r s at T = 0 K. However, comparison of crystal and fluid energies by Ceperley, Hansen, and Mazighi suggest melting in the range r s = 65 to 135. Near melting the anharmonic contributions shift the phonon frequencies by a factor of 2 and the phonon lifetimes become very short.

Xe monolayer adsorption on Ag(111): Statistical mechanics

Interactions contributing to the lateral energy of a xenon monolayer are evaluated. Thermodynamic properties of the monolayer are derived from a statistical mechanical treatment of a realistic model of the interactions. At low temperatures the quasiharmonic theory of lattice dynamics is used; at intermediate temperatures the cell model approximation is used. The thermal expansion of the unconstrained two-dimensional solid, the lateral compressibility, and lateral energy contributions to the isosteric heat are evaluated; the results agree well with recent experimental data. Two-dimensional second virial coefficients are also evaluated.

Phonon dispersion curves and phonon lifetimes in crystalline ammonia

The theory of lattice dynamics of molecular crystals using atom—atom and multiple—multipole potentials developed in previous papers is extended to take account of anharmonic effects. Calculations of the dispersion curves and the density of states for the harmonic phonons in solid ammonia are presented. The finite phonon lifetimes due to anharmonicity are calculated for the zero wave vector modes and compared with experiment.