Phonon velocity

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Phonon transport is the main conduction mechanism below 300°C. Compositional effects are significant because the mean free phonon path is limited by the random glass stmcture. Estimates of the mean free phonon path in vitreous siUca, made using elastic wave velocity, heat capacity, and thermal conductivity data, generate a value of 520 pm, which is on the order of the dimensions of the SiO tetrahedron (151). Radiative conduction mechanisms can be significant at higher temperatures. 

A pre-factor 1/r contains a time scale r or a frequency which for instance corresponds to the hard phonon or to an atomic frequency. The growth rate of the crystal is proportional to this rate (23). As will be shown later, the nucleus once formed expands in a time scale shorter than the one necessary for nucleation. If the process consists of a series of sequential subprocesses, the global velocity is governed by the slowest one. Therefore, this nucleation process determines the growth rate of a faceted surface. 

In order to study the vibrational properties of a single Au adatom on Cu faces, one adatom was placed on each face of the slab. Simulations were performed in the range of 300-1000"K to deduce the temperature dependence of the various quantities. The value of the lattice constant was adjusted, at each temperature, so as to result in zero pressure for the bulk system, while the atomic MSB s were determined on a layer by layer basis from equilibrium averages of the atomic density profiles. Furthermore, the phonon DOS of Au adatom was obtained from the Fourier transform of the velocity autocorrelation function. ... 

Koehler attributed the cross-gliding to thermal activation, but it was found experimentally that it increases with dislocation velocity, which is inconsistent with thermal activation, so Gilman (1997) proposed that it is associated with flutter of screw dislocations caused by phonon buffeting. 

Debye phonon velocity) and lower in the case of very dissimilar materials. For example, the estimated Kapitza resistance is smaller by about an order of magnitude due to the great difference in the characteristics of helium and any solid. On the other hand, for a solid-solid interface, the estimated resistance is quite close (30%) to the value given by the mismatch model. The agreement with experimental data is not the best in many cases. This is probably due to many phenomena such as surface irregularities, presence of oxides and bulk disorder close to the surfaces. Since the physical condition of a contact is hardly reproducible, measurements give, in the best case, the temperature dependence of Rc. 

The application of an electric field E to a conducting material results in an average velocity v of free charge carriers parallel to the field superimposed on their random thermal motion. The motion of charge carriers is retarded by scattering events, for example with acoustic phonons or ionized impurities. From the mean time t between such events, the effective mass m of the relevant charge carrier and the elementary charge e, the velocity v can be calculated ... 

Brillouin scattering of laser light in liquids has been studied by several authors. Shapiro etal. 233) measured hypersonic velocities in various liquids and obtained a Brillouin linewidth of 0.011 cm" in methylene chloride but of less than 0.002 cm in benzene, carbon disulfide and chloroform. The broadening of the Brillouin components arises from damping of thermal phonons and is closely connected with the viscosity coefficient of the medium. From the measured linewidths, the lifetimes of the phonons responsible for Brillouin scattering at 89 45 were calculated to be 4.8 x 10 sec for methylene chloride and 7.6 x 10 sec for toluene. 

An expression for the electrical conductivity of a metal can be derived in terms of the free-electron theory. When an electric field E is applied, the free carriers in a solid are accelerated but the acceleration is interrupted because of scattering by lattice vibrations (phonons) and other imperfections. The net result is that the charge carriers acquire a drift velocity given by... 

These equations must be solved numerically. In our first calculations [46] the value of to was taken as 0.3 eV, the value obtained from an ab initio calculation of the energy of a pair of Gs as a function of the distance between them [47] for a separation of 3.4 A. The electron-phonon coupling constant a, the derivative of t with respect to displacement, was obtained as 0.6 eV/A from the results of [47]. Subsequently, when a value of to other than 0.3 eV was used a was scaled accordingly. Although to and a values depend on the particular pair of neighboring bases, the calculations were simplified by using the same value for all pairs of bases. The value of the elastic constant K was taken as 0.85 eVIA, derived [46] from the measured value of the sound velocity in DNA. 

Dislocations move when they are exposed to a stress field. At stresses lower than the critical shear stress, the conservative motion is quasi-viscous and is based on thermal activation that overcomes the obstacles which tend to pin the individual dislocations. At very high stresses, > t7crit, the dislocation velocity is limited by the (transverse) sound velocity. Damping processes are collisions with lattice phonons. 

This is a very unique situation for superconductivity, since in the previous experience inhomogeneity is almost always harmful to superconductivity. Why is the superconductivity in the cuprates so different While further research is clearly required to answer this puzzle, one possibility is that the spatial confinement produces the vibronic resonant state of phonon and charge that enhances HTSC [15,23], The benefit of spatial confinement on HTSC has been strongly advocated for some time by Phillips with the idea of filamental superconductivity [24] and more recently by Bianconi [25] as the shape resonance effect. In both cases the effect arises due to the enhancement of the local density of states (DOS). An additional, and possibly more central, effect of confinement is to reduce the group velocity of electrons and bring it comparable to the phonon velocity, thus... 

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