Phonon spectra

The set of equations in (16) are greatly simplified if we make some assumptions which allow the exciton and phonon site nondiagonal matrix elements to be neglected. In the case of an Einstein phonon spectrum... 

It is possible to make elastic scattering corrections to the algorithm (24) in the case of an Einstein phonon spectrum and purely local exciton-phonon coupling. If we calculate the energy of the polaron state at the value E ss nuio only the matrix elements 5 " should be considered in Eqs.(16). In this case... 

While being very similar in the general description, the RLT and electron-transfer processes differ in the vibration types they involve. In the first case, those are the high-frequency intramolecular modes, while in the second case the major role is played by the continuous spectrum of polarization phonons in condensed 3D media [Dogonadze and Kuznetsov 1975]. The localization effects mentioned in the previous section, connected with the low-frequency part of the phonon spectrum, still do not show up in electron-transfer reactions because of the asymmetry of the potential. 

In order to discuss electron transport properties we need to know about the electronic distribution. This means that, for the case of metals and semimetals, we must have a model for the Fermi surface and for the phonon spectrum. The electronic structure is discussed in Chap. 5. We also need to estimate or determine some characteristic lengths. 

Independent of specific theoretical models for the phonon spectrum of a solid matrix, the recoil-free fraction can be given in terms of the y-energy Ej and the mean local displacement of the nucleus from its equilibrium position ([2] in Chap. 1) [5] ... 

Several types of spin-lattice relaxation processes have been described in the literature [31]. Here a brief overview of some of the most important ones is given. The simplest spin-lattice process is the direct process in which a spin transition is accompanied by the creation or annihilation of a single phonon such that the electronic spin transition energy, A, is exchanged by the phonon energy, hcoq. Using the Debye model for the phonon spectrum, one finds for k T A that... 

At higher temperatures, the two-phonon (Raman) processes may be predominant. In such a process, a phonon with energy hcOq is annihilated and a phonon with energy HcOr is created. The energy difference TicOq — ha>r is taken up in a transition of the electronic spin. In the Debye approximation for the phonon spectrum, this gives rise to a relaxation rate given by... 

If optical phonons are responsible for the Raman processes, the Einstein model for the phonon spectrum is more appropriate. In this case, one finds... 

In the molecular approximation used in (14) only the L = 3W — 6 (W is the number of atoms) discrete intramolecular vibrations of the molecular complex in vacuo are considered. In general these vibrations correspond to the L highest optical branches of the phonon spectrum. The intermolecular vibrations, which correspond to the three acoustical branches and to the three lowest optical branches are disregarded, i.e., the center of mass and - in case of small amplitudes - the inertial tensor of the complex are assumed to be fixed in space... 

Here it is our intention to show that for a system constituted by substrate phonons and laterally interacting low-frequency adsorbate vibrations which are harmonically coupled with the substrate, the states can be subclassified into independent groups by die wave vector K referring to the first Brillouin zone of the adsorbate lattice.138 As the phonon state density of a substrate many-fold exceeds the vibrational mode density of an adsorbate, for each adsorption mode there is a quasicontinuous phonon spectrum in every group of states determined by K (see Fig. 4.1). Consequently, we can regard the low-frequency collectivized mode of the adsorbate, t /(K), as a resonance vibration with the renormalized frequency and the reciprocal lifetime 7k-... 

Fig. 4.1. Coupling of the adsorbate low-frequency mode with substrate phonons K level of the adsorbate (a) initial quasicontinuous phonon spectrum of the substrate not perturbed by the adsorbate, bold lines designating the levels which correspond to the specified wave vector K (b) level shifts in the K subsystem caused by the coupling of the adsorbate K mode and substrate phonons (c).

Vibrational properties can also be computed using the total energy formalism. Here, the atomic mass is required as input and the crystal is distorted to mimic a frozen-in lattice vibration. The total energy and forces on the atoms can be computed for the distorted configuration, and, through a comparison between the distorted and undistorted crystal, the lattice vibrational (or phonon) spectrum can be computed. Again, the agreement between theory and experiment is excellent. [22]... 

The successful prediction of superconductivity in the high pressure Si phases added much credibility to the total energy approach generally. It can be argued that Si is the best understood superconductor since the existence of the phases, their structure and lattice parameters, electronic structure, phonon spectrum, electron-phonon couplings, and superconducting transition temperatures were all predicted from first principles with the atomic number and atomic mass as the main input parameters. 

The phonon spectrum of a crystal surface consists of two parts. The bulk... 

This paper summarizes the results of our study of PE and APE waveguides in LiNb03 and EiTa03. We foeused on the optical and structural characterization of PE layers formed on Z-eut substrates. The reffaetive index ehange was measured and the propagation losses were estimated. Raman speetroseopy was used as a method providing direct information about the phonon spectrum. The latter was related to the structure and ehemieal bonds of a given erystalline phase. Sueh information may be useful for eorreet identification of both phase eomposition and the microscopic mechanisms responsible for the observed variation of the properties from phase to phase. 

With further increase of the concentration (in p, phase range for H cLii cNb03) many new bands were observed. The fact that the low concentration boundary of the P phases is approximately x = 0.5 leads to the assumption for some kind of ordering of Li and as reported. On one hand, it can be assumed that the protons form a (nearly) ordered sub-lattice. Such a structure would have a phonon spectrum different from that of a pure LiNbOs, see . On the other hand, the PE probably leads to a reduction of the crystal symmetry, i.e. due to the incorporation of H, the two Li sites in the unit cell may become non-equivalent. In such case, the symmetry would be reduced from Csv to C3. As a result, the number of molecules per unit cell would remain the same, but new bands would appear in the vibration spectrum. 

The dynamical behaviour of the atoms in a crystal is described by the phonon (sound) spectrum which can be measured by inelastic neutron spectroscopy, though in practice this is only possible for relatively simple materials. Infrared and Raman spectra provide images of the phonon spectrum in the long wavelength limit but, because they contain relatively few lines, these spectra can only be used to fit a force model that is too simple to reproduce the full phonon spectrum of the crystal. Nevertheless a useful description of the bond dynamics can be obtained from such force constants using the methods described by Turrell (1972). 

Growing degree of substitutional disorder results in a reduction of the other above-mentioned quantities. However, the microscopic mechanism, which mediates disorder to Tc and to the other physical quantities, is not yet clarified. Typical scenarios for disorder effects could be the peak of the density of states at the Fermi level, N(Ef), may be broadened or the phonon spectrum may be modified by disorder (Manalo et al. 2001) or the scattering rate of the conduction electrons may increase. 

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