Harmonic phonon frequencies

The Kieffer approach uses a harmonic description of the lattice dynamics in which the phonon frequencies are independent of temperature and pressure. A further improvement of the accuracy of the model is achieved by taking the effect of temperature and pressure on the vibrational frequencies explicitly into account. This gives better agreement with experimental heat capacity data that usually are collected at constant pressure [9],... 

The phonon coupling broadens the absorption and emission spectra. The vibrational ground state has the wavefunction of a simple harmonic oscillator with phonon frequency co,... 

The atom-atom potential fitted to the ab initio data gives fairly re stic results for the equilibrium structure (unit cell parameters and molecular oriratations in the cell), the cohesion energy and the phonon frequencies of the molecular crystal. The latter have been obtained via both a harmonic and a self-consistent phonon lattice dynamics calculation and they were compared with and Raman spectra. About some of the aninncal hydrocarbon atom-atom potentials which are fitted to the crystal data, we can say that they correspond reasonably well with the ab initio results (see figs. 6, 7, 8), their main defect being an underestimate of the electrostatic multipole-multipolc interactions. 

If the anharmonicity is small, it can be accounted for in the framework of the body of mathematics used in harmonic approximation. Such analysis was made elsewhere [88], and it was shown that the formulas for the rate constant remain unchanged, but the parameters involved—equilibrium coordinates and phonon frequencies—turn out to be temperature-dependent effective values. The applicability criteria for harmonic approximation were also obtained. 

The scaling with the inverse of the square of the phonon frequency follows from harmonic phonon theory, such that low frequency modes have the larger amplitudes. The rotations of the octahedra will give a net reduction in volume ... 

For the interpretation of the results of the IR and Raman spectroscopy one can use phonon frequencies calculated by using the so-called harmonic approximation widely used in solid-state physics [26]. The selection rules for... 

So Sap(q, co) describes the spectrum of density fluctuations at wave vector q. At low temperatures the crystal dynamics consist of phonon vibrations and Fap(q, t) is a superposition of harmonic oscillations so 5a (q, phonon frequencies corresponding to wave vector q. At higher temperatures translational motion occurs and the associated correlations should simply decay in time, giving rise to a peak in Sa/ (q, co) that is centred on co = 0, and therefore called the quasi-elastic peak, with subsidiary phonon peaks at the appropriate values of co. A nice example is shown in a paper by Gillan (1986). 

Recoil-free fraction measurements for soUd krypton between 5 K and 85 K are also available [19,20], but in this case the / values are lower than expected. Some of the first calculations [20] omitted the effects of anharmonicity upon the phonon frequency spectrum. Inclusion of an harmonic effect [21] gives better agreement between experiment and theory, but the zero-temperature limit of / is anomalously low and has still not been explained. 

In a perfectly harmonic crystal the elastic constants would be strictly independent of temperature. However, due to the existence of third- and fourth-order anharmonic terms in the crystal potential there is a coupling between the homogeneous strains and the phonon coordinates. This will lead to a background temperature dependence of the elastic constants. It can be described within a quasiharmonic approximation (Ludwig 1967), in which the anharmonic contributions to the crystal potential are implicitly included by assuming a strain dependence of the phonon frequencies which can be characterized by the... 

At LT, the crystal behaves like a harmonic solid and the dependence of the phonon frequencies on temperature is negligibly small allowing one to ignore the... 

I tested the GAP models on a range of simple materials, based on data obtained from Density Functional Theory. I built interatomic potentials for the diamond lattices of the group IV semiconductors and I performed rigorous tests to evaluate the accuracy of the potential energy surface. These tests showed that the GAP models reproduce the quantum mechanical results in the harmonic regime, i.e. phonon spectra, elastic properties very well. In the case of diamond, I calculated properties which are determined by the anharmonic nature of the PES, such as the temperature dependence of the optical phonon frequency at the F point and the temperature dependence of the thermal expansion coefficient. Our GAP potential reproduced the values given by Density Functional Theory and experiments. 

The main idea of SCF phonons is to calculate variationally the vibrational energy levels (and consequently phonon frequencies), the trial wavefunctions being harmonic oscillator wavefunctions or modifications of these. Those wavefunctions are exact wavefunctions of some harmonic potential, and the parameters characterizing the wavefunctions and the potential are its curvature, or effective force constant, and its minima, or... 

In the dielectric screening method the electron density response due to the motion of the ions around their equilibrium positions is calculated in first order perturbation theory. The potential energy of the crystal for an arbitrary configuration of the ions is expanded to second order in the ionic displacements from equilibrium. The expansion coefficients of the second order term form a matrix. The Fourier transform of this matrix is the dynamical matrix whose eigenvalues yield the phonon frequencies. The dynamical matrix has an ionic and electronic part. The electronic part can be expressed in terms of the electron density response matrix and of the ionic potential. This method has the advantage over the total energy difference m ethod that the phonon frequencies for any arbitrary wave vector can be calculated without additional difficulties. Furthermore in this method the acoustic sum rule is automatically satisfied as a consequence of the way the dynamical matrix is derived. However the dielectric screening method is limited to harmonic phonons. 

We are encountering a feature that is inherent in all "direct" approaches to lattice dynamics as the total energy can only be calculated with finite displacements u, the harmonic terms appear intertwined with the enharmonic contributions, and we have to treat all the expansion terms simultaneously from the very beginning. In addition to the frozen phonon frequencies, we then obtain detailed Information on the anharmonicity of the mode in question, data which are difficult to find by other means, both theoretical and experimental. In some cases, it can be verified that the displacement u is small enough and does not give rise to any noticeable enharmonic effects. With most displacement patterns, however, the total energy has to be evaluated for several magnitudes of the displacement (typically 5 to 25 values of ) ... 

Table 4.1 Calculated phonon frequencies and the mode Grunelsen parameters y = -(dtu/io) / (dV/V) compared with experiment. All eigenfrequencies correspond to the harmonic part of the expansion E (u), i.e. to vibrations with u- 0. All... 

The calculation of the vibrational spectrum from an (AI)MD trajectory involves Fourier-transforming the time-dependent velocity autocorrelation function [60] an alternative approach involves calculating the phonon frequencies by diagonalizing the Hessian matrix of a model obtained by structural optimization of the classical MD structure [53]. The AIMD-VACF approach naturally include finite-temperature anharmonic effects missing in the Hessian-harmonic approximation, but it does not produce accurate IR intensities (for which an autocorrelation function based on the exact dipole moments would be needed [61-63]). Despite these issues, it turns out that, in the case of 45S5 Bioglass , the two methods give similar frequencies of the individual modes [53]. 

Here it is our intention to show that for a system constituted by substrate phonons and laterally interacting low-frequency adsorbate vibrations which are harmonically coupled with the substrate, the states can be subclassified into independent groups by die wave vector K referring to the first Brillouin zone of the adsorbate lattice.138 As the phonon state density of a substrate many-fold exceeds the vibrational mode density of an adsorbate, for each adsorption mode there is a quasicontinuous phonon spectrum in every group of states determined by K (see Fig. 4.1). Consequently, we can regard the low-frequency collectivized mode of the adsorbate, t /(K), as a resonance vibration with the renormalized frequency and the reciprocal lifetime 7k-... 

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