Phonon description

Following the solid-state approach, equations have been derived [8,9] also for the electron spin relaxation of 5 = V2 ions in solution determined by the aforementioned processes. Instead of phonons, collisions with solvent should be taken into consideration, whose correlation time is usually in the range 10"11 to 10 12 s. However, there is no satisfactory theory that unifies relaxation in the solid state and in solution. The reason for this is that the solid state theory was developed for low temperatures, while solution theories were developed for room temperature. The phonon description is a powerful one when phonons are few. By increasing temperature, the treatment becomes cumbersome, and it is more convenient to use stochastic theory (see Section 3.2) instead of analyzing the countless vibrational transitions that become active. 

The diametrically opposite treatment to the phonon description in the previous sections is to regard the impurity and its immediate neighbours in the solid as an isolated unit, or a molecule. The vibrational motion of the atoms in such a molecule is represented by the molecular normal modes, with amplitudes Qr7 (having dimensions as in (2.5) of length x square root of the reduced mass M) and their-mo-menta-conjugates Pr7 (T is an irreducible representation of the molecular point group and 7 its component). The electron-vibration coupling enters the Hamiltonian as... 

In the case of liquid lattices, the difficulty in adequately characterizing their structures renders them unsuitable for treatment by quantum mechanics. Accordingly, liquid lattices are often treated classically by considering the effects of molecular rotations and translations, with characteristic correlation times, r, on time-dependent magnetic and electric fields that may influence relaxation processes. Accordingly, it becomes convenient to depart from the phonon description of sound and adopt a classical view of this as the sinusoidal propagation of a pressure wave through a medium. 

Fig.2.9. Oscillator and phonon description of the vibrations of atoms in a crystal. The index s stands for (qj), where q is the wave number and j specifies the branch...

Wlrile tire Bms fonnula can be used to locate tire spectral position of tire excitonic state, tliere is no equivalent a priori description of the spectral widtli of tliis state. These bandwidtlis have been attributed to a combination of effects, including inlromogeneous broadening arising from size dispersion, optical dephasing from exciton-surface and exciton-phonon scattering, and fast lifetimes resulting from surface localization 1167, 168, 170, 1711. Due to tire complex nature of tliese line shapes, tliere have been few quantitative calculations of absorjDtion spectra. This situation is in contrast witli tliat of metal nanoparticles, where a more quantitative level of prediction is possible. 

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 the analysis of crystal growth, one is mainly interested in macroscopic features like crystal morphology and growth rate. Therefore, the time scale in question is rather slower than the time scale of phonon frequencies, and the deviation of atomic positions from the average crystalline lattice position can be neglected. A lattice model gives a sufiicient description for the crystal shapes and growth [3,34,35]. 

The generally accepted theory of electric superconductivity of metals is based upon an assumed interaction between the conduction electrons and phonons in the crystal.1-3 The resonating-valence-bond theory, which is a theoiy of the electronic structure of metals developed about 20 years ago,4-6 provides the basis for a detailed description of the electron-phonon interaction, in relation to the atomic numbers of elements and the composition of alloys, and leads, as described below, to the conclusion that there are two classes of superconductors, crest superconductors and trough superconductors. 

The above dynamical description of the polymerisation strongly parallels that of nonradiative transitions and this is not accidental althouth the monomer crystal from which the polymeric one is issued, do fluoresce, the polymeric one does not, despite its strong absorption at 2 eV. This strongly indicates efficient nonradiative relaxation of the excitation and strong electron-phonon coupling. 

Quantization (the idea of quantums, photons, phonons, gravitons) is postulated in Quantum Mechanics, while the Theory of Relativity does not derive quantization from geometric considerations. In the case of the established phenomenon the quantized nature of portioned energy transfer stems directly from the mechanisms of the process and has a precise mathematical description. The quasi-harmonic oscillator obeys the classical laws to a greater extent than any other system. A number of problems, related to quasi-harmonic oscillators, have the same solution in classical and quantum mechanics. 

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],... 

Every example of a vibration we have introduced so far has dealt with a localized set of atoms, either as a gas-phase molecule or a molecule adsorbed on a surface. Hopefully, you have come to appreciate from the earlier chapters that one of the strengths of plane-wave DFT calculations is that they apply in a natural way to spatially extended materials such as bulk solids. The vibrational states that characterize bulk materials are called phonons. Like the normal modes of localized systems, phonons can be thought of as special solutions to the classical description of a vibrating set of atoms that can be used in linear combinations with other phonons to describe the vibrations resulting from any possible initial state of the atoms. Unlike normal modes in molecules, phonons are spatially delocalized and involve simultaneous vibrations in an infinite collection of atoms with well-defined spatial periodicity. While a molecule s normal modes are defined by a discrete set of vibrations, the phonons of a material are defined by a continuous spectrum of phonons with a continuous range of frequencies. A central quantity of interest when describing phonons is the number of phonons with a specified vibrational frequency, that is, the vibrational density of states. Just as molecular vibrations play a central role in describing molecular structure and properties, the phonon density of states is central to many physical properties of solids. This topic is covered in essentially all textbooks on solid-state physics—some of which are listed at the end of the chapter. 

For a molecular crystal, the description can be simplified considerably by differentiating between internal and external modes. If there are M molecules in the cell, each with nM atoms, the number of external translational phonon branches will be 3M, as will the number of external rotational branches. When the molecules are linear, only 2M external rotational modes exist. For each molecule, there are 3nM — 6 (3nM — 5 for a linear molecule) internal modes, the wavelength of which is independent of q. Summing all modes gives a total number of N M(3nM — 6) + 6M = 3nN, as required, because each of the modes that have been constructed is a combination of the displacements of the individual atoms. 

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). 

The concept of a mobility edge has proved useful in the description of the nondegenerate gas of electrons in the conduction band of non-crystalline semiconductors. Here recent theoretical work (see Dersch and Thomas 1985, Dersch et al. 1987, Mott 1988, Overhof and Thomas 1989) has emphasized that, since even at zero temperature an electron can jump downwards with the emission of a phonon, the localized states always have a finite lifetime x and so are broadened with width AE fi/x. This allows non-activated hopping from one such state to another, the states are delocalized by phonons. In this book we discuss only degenerate electron gases here neither the Fermi energy at T=0 nor the mobility edge is broadened by interaction with phonons or by electron-electron interaction this will be shown in Chapter 2. 

The aim of this chapter is to clarify the conditions for which chemical kinetics can be correctly applied to the description of solid state processes. Kinetics describes the evolution in time of a non-equilibrium many-particle system towards equilibrium (or steady state) in terms of macroscopic parameters. Dynamics, on the other hand, describes the local motion of the individual particles of this ensemble. This motion can be uncorrelated (single particle vibration, jump) or it can be correlated (e.g., through non-localized phonons). Local motions, as described by dynamics, are necessary prerequisites for the thermally activated jumps responsible for the movements over macroscopic distances which we ultimately categorize as transport and solid state reaction.. 

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