Phonon collisions

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. 

Photoelectron spectroscopy of valence and core electrons in solids has been useful in the study of the surface properties of transition metals and other solid-phase materials. When photoelectron spectroscopy is performed on a solid sample, an additional step that must be considered is the escape of the resultant photoelectron from the bulk. The analysis can only be performed as deep as the electrons can escape from the bulk and then be detected. The escape depth is dependent upon the inelastic mean free path of the electrons, determined by electron-electron and electron-phonon collisions, which varies with photoelectron kinetic energy. The depth that can be probed is on the order of about 5-50 A, which makes this spectroscopy actually a surface-sensitive technique rather than a probe of the bulk properties of a material. Because photoelectron spectroscopy only probes such a thin layer, analysis of bulk materials, absorbed molecules, or thin films must be performed in ultrahigh vacuum (<10 torr) to prevent interference from contaminants that may adhere to the surface. 

In electrically insulating solids, heat is transferred in the form of elastic waves or phonons [1], Anything that affects the propagation of the phonons through the solid affects the thermal conductivity of the solid. In a pure crystalline ceramic, the intrinsic thermal conductivity is limited by the energy dissipated during phonon-phonon collisions or so-called Umklapp processes [15], Commonly, the intrinsic thermal conductivity of solids is described by (5). 

Electron-electron. At room temperature the mean distance between electron-electron collisions is about 10 times that of electron-phonon collisions so electron-phonon scattering is dominant. 

The classical theory of absorption in dielectric materials is due to H. A. Lorentz and in metals it is the result of the work of P. K. L. Drude. Both models treat the optically active electrons in a material as classical oscillators. In the Lorentz model the electron is considered to be bound to the nucleus by a harmonic restoring force. In this manner, Lorentz s picture is that of the nonconductive dielectric. Drude considered the electrons to be free and set the restoring force in the Lorentz model equal to zero. Both models include a damping term in the electron s equation of motion which in more modem terms is recognized as a result of electron-phonon collisions. 

In recent decades the thinking of physicists has largely been dominated by attempts to describe systems in terms of linear differential equations and their solutions. Deviations fi om their harmonic behaviour, which lead to non-linear terms in the differential equations, have been treated as perturbations by introducing interactions between the quasi-particles, correspond to the harmonic solutions (electron-electron and electron-phonon collisions, etc.). The idea of the soliton concept is to solve the non-linear differential equations, not by numerical approximations but analytically and to associate new quasi-particles wifli exact solutions, the solitons. 

Hot Carriers. Another assumption of the conventional model (10, 19) is that the energy of injected photogenerated carriers is given by the position of the minority carrier band edge at the semiconductor-electrolyte interface. That is, carriers are accelerated to the surface by the electric field in the semiconductor space charge layer, but they lose energy (as heat) in the process via carrier - phonon collisions. As a result, the carriers are in thermal equilibrium with the lattice before injection. 

Fig. 4.7 Dynamics of a neutral kink K along a ring of 139 sites with an initial velocity uk = I0 during 1060 time steps At = 1.2 X 10 s. Time evolution of the kink velocity nk (top left) and the kink half-width (top right). Bottom Time evolution of bond length alternation 6, = (-l) (

The development of ultrashort laser pulses down to pulse durations of 5 x 10 s has opened access to studies of extremely fast transient phenomena. Examples are the relaxation of electrons in semiconductors after their excitation by a short light pulse. The electrons excited with a definite energy tuo into the conduction band thermalize within 10 s by electron-phonon collisions (Figure 10). With a much longer decay constant, they recombine with holes in the valence band before they can be excited again by the next pulse. Such time-resolved studies give important information on the limiting processes for the maximum speed of computers. 

Illustration of an N-process in which the collision is elastic and a U-process in which the collision is inelastic. If the resulting vector from a phonon-phonon collision falls outside the first Brillouin zone (the square box), the momentum is Bragg reflected back into the first Brillouin zone with a transfer oi G1r momentum to the lattice in the form of crystal momentum. 

Thermal conductivities of (a) diamond and (b) cubic SiC. The conductivity of diamond reaches a peak of 12,000 W/m -K at 100 K, 30 times greater than Cu. The thermal conductivities are limited by heat capacity below 100 K and by phonon collisions above 100 K. (Data taken from Adachi, Handbook of Physical Properties of Semiconductors.)... 

At low temperatures, phonon-phonon collisions disappear, and X is determined by the distance between imperfections in the crystal. The phonons collide with imperfections such as impurity atoms, dislocations, intercrystalline boundaries, or finally, the specimen boundaries. The distance between these does not depend on temperature, so X becomes constant at low temperatures. 

The type of thermal conductivity curve suggested from all of these considerations is again exemplified by that of corundum in Fig. 3.17. Above temperatures corresponding to about 0 >/lO, phonon-phonon collisions govern. These collisions decrease at lower temperatures to cause a maximum near 0/>/2O. Below this temperature the size of the crystal limits the mean free path, and the dependence causes a decrease in fc, to zero at 0 K. 

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