Collisions between phonons

The specific heat is the lattice specific heat of the solid. Its variation with temperature is plotted on the same temperature scale in Fig. 3.16b. The mean velocity v of the phonons is the mean of the velocity of sound and varies only slightly with temperature as shown in Fig. 3.16c. The phonon gas differs from a real gas in that the number of particles varies with the temperature, increasing in number as the temperature is increased. At high temperatures, the large number of phonons leads to more collisions between phonons. Thus, as the temperature increases, X decreases, as shown in Fig. 3.16d. 

Physically, the Brillouin spectrum arises from the inelastic interaction between a photon and the hydrodynamics modes of the fluid. The doublets can be regarded as the Stokes and anti-Stokes translational Raman spectrum of the liquid. These lines arise due to the inelastic collision between the photon and the fluid, in which the photon gains or loses energy to the phonons (the propagating sound modes in the fluid) and thus suffer a frequency shift. The width of the band gives the lifetime ( 2r)-1 of a classical phonon of wavenumber q. The Rayleigh band, on the other hand, represents the... 

There is a very simple model for estimating the trapping probability in atomic adsorption due to a phonon-excitation mechanism. In the hard-cube model (HCM) [6, 7], the impact of the atom on the surface is treated as a binary elastic collision between a gas phase atom (mass m) and a substrate atom (mass Mc) which is moving freely with a velocity distribution Pc(uc). This model is schematically illustrated in Fig. 1. If the depth of the adsorption well is denoted by Ead, the adsorbate will impinge... 

When a positron with a well-defined energy is injected from a vacuum into a polymer, it is either reflected back to the surface or it penetrates into the polymer. The fraction of positrons that enter the polymers increases rapidly as a function of the positron energy. As the positrons enter the polymer, inelastic collisions between the positron and molecules slow down the positrons by ionization, excitation and phonon processes. The implantation—stopping profile P(z,E) of the positrons varies as a function of depth as [1, 2] ... 

Like in any porous insulation material, the total heat losses are the sum of the skeletal conduction (phonons), gas conduction (collisions between gas molecules), and radiation contributions. The particular mesoscopic structure of aerogels leads to a considerable reduction of the gas-phase conduction contribution due to a trapping effect of the pore gas. For most terrestrial applications, the thermal conductivity at ambient temperature and pressure is relevant, which although well studied, is still a challenge to determine accurately [203], primarily due to the lack of large-area homogeneous aerogel specimens. 

The terms and in (8.28) are those responsible for collision-induced phonon excitation. These terms depend upon time through the time-dependence of the trajectory. If only the linear terms are included, the oscillators are called linearly forced. From extensive studies on the linearly forced (and also more general) atom-molecule interactions, we know that good agreement between the semiclassical description and the full quantum description of the excitation process is obtained through the so-called symmetrized Ehrenfest approach. This approach is based upon the following two conjectures ... 

The dynamics of ion surface scattering at energies exceeding several hundred electronvolts can be described by a series of binary collision approximations (BCAs) in which only the interaction of one energetic particle with a solid atom is considered at a time [25]. This model is reasonable because the interaction time for the collision is short compared witii the period of phonon frequencies in solids, and the interaction distance is shorter tlian the interatomic distances in solids. The BCA simplifies the many-body interactions between a projectile and solid atoms to a series of two-body collisions of the projectile and individual solid atoms. This can be described with results from the well known two-body central force problem [26]. 

Static defects scatter elastically the charge carriers. Electrons do not loose memory of the phase contained in their wave function and thus propagate through the sample in a coherent way. By contrast, electron-phonon or electron-electron collisions are inelastic and generally destroy the phase coherence. The resulting inelastic mean free path, Li , which is the distance that an electron travels between two inelastic collisions, is generally equal to the phase coherence length, the distance that an electron travels before its initial phase is destroyed ... 

An elementary treatment of the free-electron motion (see, e.g., Kittel, 1962, pp. 107-109) shows that the damping constant is related to the average time t between collisions by y = 1 /t. Collision times may be determined by impurities and imperfections at low temperatures but at ordinary temperatures are usually dominated by interaction of the electrons with lattice vibrations electron-phonon scattering. For most metals at room temperature y is much less than oip. Plasma frequencies of metals are in the visible and ultraviolet hu>p ranges from about 3 to 20 eV. Therefore, a good approximation to the Drude dielectric functions at visible and ultraviolet frequencies is... 

The velocity relevant for transport is the Fermi velocity of electrons. This is typically on the order of 106 m/s for most metals and is independent of temperature [2], The mean free path can be calculated from i = iyx where x is the mean free time between collisions. At low temperature, the electron mean free path is determined mainly by scattering due to crystal imperfections such as defects, dislocations, grain boundaries, and surfaces. Electron-phonon scattering is frozen out at low temperatures. Since the defect concentration is largely temperature independent, the mean free path is a constant in this range. Therefore, the only temperature dependence in the thermal conductivity at low temperature arises from the heat capacity which varies as C T. Under these conditions, the thermal conductivity varies linearly with temperature as shown in Fig. 8.2. The value of k, though, is sample-specific since the mean free path depends on the defect density. Figure 8.2 plots the thermal conductivities of two metals. The data are the best recommended values based on a combination of experimental and theoretical studies [3],... 

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