Phonons lattice dynamics

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. 

This potential was subsequently used in self-consistent phonon lattice dynamics calculations [115] for a and y nitrogen crystals. And although the potential—and its fit— were crude by present day standards, lattice constants, cohesion energy and frequencies of translational phonon modes agreed well with experimental values. The frequencies of the librational modes were less well reproduced, but this turned out to be a shortcoming of the self-consistent phonon method. When, later [ 116,117], a method was developed to deal properly with the large amplitude librational motions, also the librational frequencies agreed well with experiment. 

The density of states (DOS) of lattice phonons has been calculated by lattice dynamical methods [111]. The vibrational DOS of orthorhombic Ss up to about 500 cm has been determined by neutron scattering [121] and calculated by MD simulations of a flexible molecule model [118,122]. 

With development of ultrashort pulsed lasers, coherently generated lattice dynamics was found, first as the periodic modulation in the transient grating signal from perylene in 1985 by De Silvestri and coworkers [1], Shortly later, similar modulation was observed in the reflectivity of Bi and Sb [2] and of GaAs [3], as well as in the transmissivity of YBCO [4] by different groups. Since then, the coherent optical phonon spectroscopy has been a simple and powerful tool to probe femtosecond lattice dynamics in a wide range of solid... 

Material response in THz frequency region, which corresponds to far- and mid-infrared electromagnetic spectrum, carries important information for the understanding of both electronic and phononic properties of condensed matter. Time-resolved THz spectroscopy has been applied extensively to investigate the sub-picosecond electron-hole dynamics and the coherent lattice dynamics simultaneously. In a typical experimental setup shown in Fig. 3.5, an... 

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],... 

Contents Lattice Dynamics. - Symmetry. - Inter-molecular Potentials. - Anharmonic Interactions. - Two-Phonon Spectra of Molecular Crystals. -Infrared and Raman Intensities in Molecular Crystals. 

The dispersion curves of surface phonons of short wavelength are calculated by lattice dynamical methods. First, the equations of motion of the lattice atoms are set up in terms of the potential energy of the lattice. We assume that thejxitential energy (p can be expressed as a function of the atomic positions 5( I y in the semi-infinite crystal. The location of the nth atom can be... 

Recently, we hav measured the surface phonon dispersion of Cu(l 10) along the rx, rF, and F5 azimuth of the surface Brillouin zone (Fig. 13) and analyzed the data with a lattice dynamical slab calculation. As an example we will discuss here the results along the TX-direction, i.e. the direction along the close-packed Cu atom rows. 

The frequency of the MS7 mode is well suited to give some fix points for the lattice dynamical calculation. This is obvious by inspection of Fig. 19 which shows the displacement pattern of this mode at the T point (at T the MS7 phonon corresponds to the A, symmetry group). The motions of the atoms being shear vertical, the lattice layers remain rigid planes i.e. the frequency of... 

Vibrations in a real crystal are described by the lattice dynamical theory, discussed in section 2.1, rather than by the atomic oscillator model. Each harmonic phonon mode with branch index k and wavevector q then has, analogous to Eq. 

A very much simplified lattice-dynamical model is that of Debye. In the Debye approximation, discussed in the following section, a single phonon branch is assumed, with frequencies proportional to the magnitude of the wavevector q. 

Here the argument of the arctan does not diverge at any to. Therefore the integral (17) is real (i.e., y, = 0). Note that equation (17) is well known in the lattice dynamics it describes the contribution of an impurity to the phonon part of the free energy of the crystal (in H = 1 units) at 7 = 0, supposing that the impurity does not change the point symmetry of the crystal lattice [30]. 

Two of the more direct techniques used in the study of lattice dynamics of crystals have been the scattering of neutrons and of x-rays from crystals. In addition, the phonon vibrational spectrum can be inferred from careful analysis of measurements of specific heat and elastic constants. In studies of Bragg reflection of x-rays (which involves no loss of energy to the lattice), it was found that temperature has a strong influence on the intensity of the reflected lines. The intensity of the scattered x-rays as a function of temperature can be expressed by I (T) = IQ e"2Tr(r) where 2W(T) is called the Debye-Waller factor. Similarly in the Mossbauer effect, gamma rays are emitted or absorbed without loss of energy and without change in the quantum state of the lattice by... 

The shell model has its origin in the Born theory of lattice dynamics, used in studies of the phonon dispersion curves in crystals/ Although the Born theory includes the effects of polarization at each lattice site, it does not account for the short-range interactions between sites and, most importantly, neglects the effects of this interaction potential on the polarization behavior. The shell model, however, incorporates these short-range interactions. 

The earliest applications of the shell model, as with the Born model, were to analytical studies of phonon dispersion relations in solids.These early applications have been well reviewed elsewhere.In general, lattice dynamics applications of the shell model do not attempt to account for the dynamics of the nuclei and typically use analytical techniques to describe the statistical mechanics of the shells. Although the shell model continues to be used in this fashion, lattice dynamics applications are beyond the scope of this chapter. In recent decades, the shell model has come into widespread use as a model Hamiltonian for use in molecular dynamics simulations it is these applications of the shell model that are of interest to us here. 

The formation of the nickel excitons results in the lattice distortion near them and the induced lattice vibrations. Conditions of their occurrence are defined by the charged impurity because the removed hydrogen-like type carrier practically does not influence the deformation of the lattice near the charged impurity. Thus, our analysis of the vibrational background of the zero phonon line of the EA spectrum of the nickel exciton is based on results of a simulation of the lattice dynamics of the ZnO crystal with NE or Ni ions. 

This technique, besides allowing determination of the Lamb-Mossbauer factor, provides direct access to the density of phonon states for the probe isotope in a solid. It thus provides information about lattice dynamics that is excluded by the limitations of Mossbauer spectroscopy. This technique could be valuable in investigations of adsorption with the adsorbing element as the probe and showing the modifications brought about by the adsorbate on the dynamic properties of the probe. 

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. 

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