Dispersion relations lattice vibrations

The dispersion relationships of lattice waves may be simply described within the first Brillouin zone of the crystal. When all unit cells are in phase, the wavelength of the lattice vibration tends to infinity and k approaches zero. Such zero-phonon modes are present at the center of the Brillouin zone. The variation in phonon frequency as reciprocal k) space is traversed is what is meant by dispersion, and each set of vibrational modes related by dispersion is a branch. For each unit cell, three modes correspond to translation of all the atoms in the same direction. A lattice wave resulting from such displacements is similar to propagation of a sound wave hence these are acoustic branches (Fig. 2.28). The remaining 3N-3 branches involve relative displacements of atoms within each cell and are known as optical branches, since only vibrations of this type may interact with light. 

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... 

The solution arrived at in our linear elastic model may be contrasted with those determined earlier in the lattice treatment of the same problem. In fig. 5.13 the dispersion relation along an arbitrary direction in g-space is shown for our elastic model of vibrations. Note that as a result of the presumed isotropy of the medium, no g-directions are singled out and the dispersion relation is the same in every direction in g-space. Though our elastic model of the vibrations of solids is of more far reaching significance, at present our main interest in it is as the basis for a deeper analysis of the specific heats of solids. From the standpoint of the contribution of the thermal vibrations to the specific heat, we now need to determine the density of states associated with this dispersion relation. 

This classic result exemplifies many aspects of the dynamics of lattices. The eigenvalue, 0)v, is a periodie function of k (conventionally written (ajljc) we shall find it convenient to use the k co ) form, 10.1.2). There is no longer a single valued vibrational frequency, as found for the isolated molecule. This dependence of vibrational frequency on k, in Eq. (4.45), is called the vibration dispersion relation or more simply its dispersion that of Eq. (4.45) is shown in Fig. 4.8. 

The transition arises dynamically through the interaction between the electrons and the quantized lattice vibrations of the solid, phonons (437). The phonons, in a manner similar to electrons, are assigned a wavevector q = 2jt/A where k is the wavelength of the lattice vibration. There is an energy associated with each phonon of wavevector q, as indicated schematically in Fig. 12. The actual dispersion relation is a function of the mass of the atoms in a... 

The dispersion relation contains the most important information concerning vibration normal modes in a crystal. Lattice vibrations can be measured experimentally by means of classical vibration spectroscopic techniques (infrared and Raman) or neutron inelastic scattering. However, only the latter technique allows one to measure the full spectrum in a range of k vectors, whereas with infrared and Raman spectroscopy, only lattice vibrations at T can be detected. This limitation for measuring phonon dispersions is serious, becuase neutron scattering experiments are demanding. 

The crystal lattice vibration and the force coefficients are the subject of Chapter 12. We describe the experimental dispersion curves and conclusions that follow from their examination. The interplanar force constants are introduced. Group velocity of lattice waves is computed and discussed. It allows one to make conclusions about the interatomic bonding strength. Energy of atomic displacements during lattice vibration (that is propagation of phonons) is related to electron structure of metals. 

At this point it becomes necessary to choose a model for the crystal so that the derivatives, (p, of the potential energy can be calculated and so that the density of states available to the phonons produced by the decay can be determined. For this purpose one needs either the lattice dynamics or the dispersion relations for the crystal, plus either the enharmonic components of the vibrational potential or certain of its derivatives. 

Equation (2.128) is the dispersion relation for the collective vibrations of adsorbed molecules arranged in a 2D lattice, i.e., for 2D phonons of the overlayer. 

The cosine curve duplicates the results and in its middle part represents the upper LO curve in Fig.2.7. We note that the optical vibration at q = 0 of the diatomic chain becomes an acoustic vibration at q = n/d of the mono-atomic chain. The upper part AB of the sine curve in Fig.2.7 can be obtained by folding out the LO branch AC, or equivalently, by translation of the LO branch CD through 2-n/a, that is, by a reciprocal lattice vector t = M. This is called an Umptapp process. The dispersion relation (2.43) can of course be obtained if we start directly with the Hamiltonian of the monoatomic chain which follows easily from (2.3) and then solving the resulting equations of motion by assuming a solution of the form... 

In a non-exhaustive literature search, a brief account is given here on the study of phonon and its vibrations. Corso et al. did an extensive study of density functional perturbation theory for lattice dynamics calculations in a variety of materials including ferroelectrics [93]. They employed a nonlinear approach to mainly evaluate the exchange and correlation energy, which were related to the non-linear optical susceptibility of a material at low frequency [94], The phonon dispersion relation of ferroelectrics was also studied extensively by Ghosez et al. [95, 96] these data were, however, related more with the structure and metal-oxygen bonds rather than domain vibrations or soliton motion. In a very interesting work, a second peak in the Raman spectra was interpreted by Cohen and Ruvalds [97] as evidence for the existence of bound state of the two phonon system and the repulsive anharmonic phonon-phonon interaction which splits the bound state off the phonon continuum was estimated for diamond. 

Lattice vibrations can be described as phonon quasiparticles, which carry momentum and heat energy in the lattice. The phonon dispersion relation provides a linear relation between energy and momentum at moderate phonon momenta. 

---