Dispersion force constant

In order to determine the phonon dispersion of CuZn and FeaNi we made use of an expanded tight binding theory from Varma and Weber . In the framework of a second order perturbation theory the dynamical matrix splits in two parts. The short range part can be treated by a force constant model, while the T>2 arising from second order perturbation theory is given by... 

Figure 7. Phonon dispersion including the electron-phonon interaction for bcc CuZn. Force constants have been obtained from ah initio calculations. Dashed line is the phonon dispersion without the V-i contribution. Diamonds mark experimental data. ...

In Fig. 20 we show a theoretical dispersion plot using these parameters and a tensile stress = 2.7 x 10 dyn/cm. Due to the symmetry of the modes at X the stress tensor tpy does not affect the surface eigenmodes at this symmetry point. In addition, we have softened the intralayer force constant 4>ii in the first layer by about 10%. With these parameters, we find good agreement between experimental data and theoretical dispersion curves. 

I have not described the calculation of the eigenvalues, which requires the solution of the equations of motion and therefore a knowledge of the force constants. The shell model for ionic crystals, introduced by Dick and Overhauser (1958), has proved to be extremely useful in the development of empirical crystal potentials for the calculation of phonon dispersion and other physical properties of perfect and imperfect ionic crystals. There is now a considerable literature in this field, and the following references will provide an introduction Catlow etal. (1977), Gale (1997), Grimes etal. (1996), Jackson et al. (1995), Sangster and Attwood (1978). The shell model can also be used for polar and covalent crystals and has been applied to silicon and germanium (Cochran (1965)). 

Ph. Ghosez, X. Gonze and J.-P. Michenaud, "Ab Initio phonon dispersion curves and interatomic force constants of barium titanate," accepted for publication in Ferroelectrics. 

A dispersion formula results, based on frequency dependency on masses, force constant and distance between the two masses, such as... 

London forces (or "dispersion forces") — are forces attracting apolar molecules due to their mutual polarizability. London forces are also components of the forces between polar molecules. The London equation approximately describes respective energy of interactions, Vi = -C/r6, where C is constant dependent on energy of ionization and polarizabilities of both molecules and r is the distance between the molecules. See also - van der Waals forces, and - Casimirforce. 

The dispersion curve for the Weber model, showing the effect of adding the interbond force constant/. ... 

This is undertaken by two procedures first, empirical methods, in which variable parameters are adjusted, generally via a least squares fitting procedure to observed crystal properties. The latter must include the crystal structure (and the procedure of fitting to the structure has normally been achieved by minimizing the calculated forces acting on the atoms at their observed positions in the unit cell). Elastic constants should, where available, be included and dielectric properties are required to parameterize the shell model constants. Phonon dispersion curves provide valuable information on interatomic forces and force constant models (in which the variable parameters are first and second derivatives of the potential) are commonly fitted to lattice dynamical data. This has been less common in the fitting of parameters in potential models, which are onr present concern as they are required for subsequent use in simulations. However, empirically derived potential models should always be tested against phonon dispersion curves when the latter are available. 

For the calculation of the dispersion effect we shall consider the interaction of two dipolar linear oscillators. Let each of these consist of a positive and negative charge. The frequency of oscillation Vq of each oscillator is related to the force constant k by the equation... 

Fig. 3. The upper two diagrams are the calculated dispersion curves for ice Ih and ice Ic based on a simple LD model containing O atoms only. An 0-0-0 bending and an 0-0 stretching force constants, G = 0.33 eV/Rad and K = 1.1 eV/A were used. The diagram second below shows three curves of g(co) for the three particular reciprocal directions in ice Ic. The lowest diagram is the completely BZ integrated g(o)) for both ice Ic and Ih. It differ considerably from the curves above, indicating the incomplete BZ integration can be misleading.

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