Phonons dynamical matrix

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... 

To calculate the phonon frequencies, one should diagonalize the dynamic matrix of the crystal for the center of the Brillouin zone (BZ)... 

The expressions used in calculating the properties referred to above from these derivatives are discussed in greater detail in Reference 9. For more detailed discussions of the calculation of phonon dispersion curves from the second derivative or dynamical matrix W, the reader should consult References 41 and 42. Finally, we note that by the term lattice stability we refer to the equilibrium conditions both for the atoms within the unit cell, and for the unit cell as a whole. The former are available from the gradient vector g, while the latter are described in terms of the six components ei- ee, which define the strain matrix e, where... 

We now consider a more quantitative model of the vibrational density of states which makes a remarkable linkage between continuum and discrete lattice descriptions. In particular, we undertake the Debye model in which the vibrational density of states is built in terms of an isotropic linear elastic reckoning of the phonon dispersions. Recall from above that in order to effect an accurate calculation of the true phonon dispersion relation, one must consider the dynamical matrix. Our approach here, on the other hand, is to produce a model representation of the phonon dispersions which is valid for long wavelengths and breaks down at... 

Although the preferred method of calculating a spectrum is to perform an ab initio calculation on an extended solid, extracting frequencies and displacements across the Brillouin zone, on a fine A-grid, this approach can be computationally very expensive. In plane wave codes like CASTEP [18], CPMD [19], TWSCF [20], VASP [21], ABINIT [22], and some others, the number of plane waves that are taken into consideration, the selected correlation fimctional and the choice of pseudopotential will all have an impact on the quality of the calculations. Some codes (e.g. ABINIT) alleviate the problem by permitting frozen phonon calculations at the symmetry zone boundary, i.e. (0,0,0), (l/2,0,0), (l/2,l/2,0) and (l/2,l/2,l/2) and so determine the dynamical matrix at these points. The code then interpolates values of the d3mamical matrix for all the points within the Brillouin zone and uses these to calculate the solution to the vibrational problem inside the zone. 

II) The diagonalization of the resulting larger dynamical matrix leads to a coupling of translational ("phonon") and rotational ("libron") degrees of freedom within the first Brillouin zone. 

Behavior remarkably similar to that revealed by the one-dimensional model crystals is generally observed for lattice vibrations in three dimensions. Here the dynamical matrix is constructed fundamentally in the same way, based on the model used for the interatomic forces, or derivatives of the crystal s potential energy function, and the equivalent of Eq. (7) is solved for the eigenvalues and eigenvectors [2-4, 29]. Naturally, the phonon wavevector in three dimensions is a vector with three components, q = (qx, qy, qz)> and both the fiequency of the wave, co(q), and its polarization, e q), are functions... 

In the case of a periodic solid the vibrational modes become phonons and the dynamical matrix becomes a function of a reciprocal lattice vector k chosen from the Brillouin zone. This means that in constructing D(k) all interactions are multiplied by the phase factor exp(ikrjj), where rp is the interatomic vector. A more detailed discussion of the theory of phonons can be found elsewhere (Dove 1993 Chapter 13 by Kubicki). 

The advantage of this method is that it can be used for more complicated systems, where explicit calculation of the full dynamical matrix would be extremely expensive. Furthermore, we can calculate the temperature dependence of the phonon spectrum by simply performing molecular dynamics simulations at different temperatures. The temperature dependence of the phonon spectrum is due to anharmonic effects, i.e., at larger displacements when terms higher than second order contribute to the potential energy in Eq. 5.4. 

We evaluated the phonon frequencies, resulting from the dynamical matrix (Eq. 69) including the first order polarizability (Eq. 75, 77, 78). However, for Na and Al, the effect turned out to be of the order of 1 to 2... 

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. 

Diagonalization of the dynamical matrix yields the phonon frequencies for the wave vector q under consideration. It should be noted that phonon frequencies of an arbitrary wave vector can be calculated. This is contrary to the total energy method described in the lecture notes of S. Louie, K. Kune and R.M. Martin in this volume. 

Thus, from Eqs. 12 and 13, the phonon frequency can be evaluated from the curvature of the calculated energy vs. displacement curve for small displacements. These results can be extended to the case of compounds and to general wave vectors where the lack of symmetry requires the calculation and dlagonalization of the dynamical matrix to obtain the phonon frequencies and polarization vectors. Moreover, this approach allows a detailed investigation of the role of core-core, electron-core, electron kinetic, and electron-electron energies to determine the vibrational frequencies of the solids examined. This kind of information has been valuable in analyzing and understanding phonon anomalies in semiconductors and transition metals. 

This difficulty could be avoided by applying linear response theory, which is widely used in solid-state physics to determine directly the dynamic matrix, polarization, and frequency-dependent dielectric functions, as well as phonon dispersion curves in the harmonic approximation. This method has the great advantage that it requires only a band structure at the equilibrium geometry of the solid (chain), i.e., one does not have to determine a potential hypersurface. However, since this theory has not yet been applied to polymers and involves a rather complicated formalism, we cannot enter into details here but refer the reader to standard solid-state physical works and an application to a simple solid (Si). ... 

This text of the two-volume treatment contains most of the theoretical background necessary to understand experiments in the field of phonons. This background is presented in four basic chapters. Chapter 2 starts with the diatomic linear chain. In the classical theory we discuss the periodic boundary conditions, equation of motion, dynamical matrix, eigenvalues and eigenvectors, acoustic and optic branches and normal coordinates. The transition to quantum mechanics is achieved by introducing the Sohpddingev equation of the vibrating chain. This is followed by the occupation number representation and a detailed discussion of the concept of phonons. The chapter ends with a discussion of the specific heat and the density of states. 

Having established the form of the potential energy and its parameters a and e, we now evaluate the dynamical matrix D(q) and the phonon frequencies Wj(q). According to (3.140), the dynamical matrix for a monoatomic crystal with atomic mass m is given by... 

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