Phonons force-constant matrix

Because of the term 5K, when we evaluate dco(q,a)/dV, as demanded by eqn (5.72) in our deduction of the thermal expansion coefficient, it will no longer vanish as it would if we were only to keep K. The phonon frequencies have acquired a volume dependence by virtue of the quasiharmonic approximation which amounts to a volume-dependent renormalization of the force constant matrix. 

Obtain an analytic expression for the zone edge phonons in fee Cu using the Morse potential derived in the previous chapter. To do so, begin by deriving eqn (5.37) and then carry out the appropriate lattice sums explicitly for the Morse potential to obtain the force constant matrix. In addition, obtain a numerical solution for the phonon dispersion relation along the (100) direction. 

Phonons using the Pair Functional Formalism Generalize the discussion given in this chapter based on pair potentials in order to deduce the force constant matrix for a total energy based on pair functionals. The result of this analysis can be found in Finnis and Sinclair (1984). Having obtained this expression for the force constant matrix, use the Johnson potential presented in the previous chapter and compute the phonon... 

DFT calculations of the static lattice were performed within the local density approximation (LDA) using planewave basis sets and ultrasoft pseudopotentials, and the results were used to construct a force constant matrix within a large superceU model. The phonon spectrum was then evaluated as a function of temperature using quasi-harmonic models that allowed us to constmct mode Gruneisen relationships [66] (Fig. 4.4). The results indicated a dip in the V(T) relation at 80 K, that was slightly smaller than that observed for diamond-structured Si (Fig. 4.5). 

The phonon dispersion curves and corresponding density of states, calculated from the force constant matrix pRik) with the help of methods described in Section 9.1, are shown in Figure 9.7 and atomic displacements of normal modes for = 0 are sketched in Figure 9.8. The Raman spectrum of all-/ran5-PA, measured by Shirakawa et shows two... 

To obtain the last expression we have also used Eqs. (6.10) and (6.11) to relate the displacements to the force-constant matrix and the frequency eigenvalues Combining the expressions for the kinetic and potential energies, we obtain the total energy of a system of phonons ... 

Lattice vibrations are calculated by applying the second order perturbation theory approach of Varma and Weber , thereby combining first principles short range force constants with the electron-phonon coupling matrix arising from a tight-binding theory. 

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... 

It is expected that deuteration of matrix molecules leads to a low-energy shift (red shift) of lattice vibrations or local phonon modes (cage modes) as compared to those of the protonated material, because the vibrating masses become larger, while the force constants are almost unchanged. Indeed, this behavior is observed in the emission spectra, when the low-energy satellites relative to the electronic origins are compared (Table 5). 

Complete dispersion curves along symmetry directions in the Brillouin zone are obtained from calculated force constants. Calculations of enharmonic terms and phonon-phonon interaction matrix elements are also presented. In Sec. IIIC, results for solid-solid phase transitions are presented. The stability of group IV covalent materials under pressure is discussed. Also presented is a calculation on the temperature- and pressure-induced crystal phase transitions in Be. In Sec. IV, we discuss the application of pseudopotential calculations to surface studies. Silicon and diamond surfaces will be used as the prototypes for the covalent semiconductor and insulator cases while surfaces of niobium and palladium will serve as representatives of the transition metal cases. In Sec. V, the validity of the local density approximation is examined. The results of a nonlocal density functional calculation for Si and... 

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