Debye velocity

In case of harmonic solids with Debye-like low-frequency dynamics and vibrational isotropy, that is, polycrystalline sample, or single-crystalline one with a cubic Bravais lattice, the Debye velocity of sound vd can be precisely determined from the expression [107]... 

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. 

For very dilute solutions, the motion of the ionic atmosphere in the direction of the coordinates can be represented by the movement of a sphere with a radius equal to the Debye length Lu = k 1 (see Eq. 1.3.15) through a medium of viscosity t] under the influence of an electric force ZieExy where Ex is the electric field strength and zf is the charge of the ion that the ionic atmosphere surrounds. Under these conditions, the velocity of the ionic atmosphere can be expressed in terms of the Stokes law (2.6.2) by the equation... 

Dislocation motion in covalent crystals is thermally activated at temperatures above the Einstein (Debye) temperature. The activation energies are well-defined, and the velocities are approximately proportional to the applied stresses (Sumino, 1989). These facts indicate that the rate determining process is localized to atomic dimensions. Dislocation lines do not move concertedly. Instead, sharp kinks form along their lengths, and as these kinks move so do the lines. The kinks are localized at individual chemical bonds that cross the glide plane (Figure 5.8). 

FIGURE 11.32 Flow profiles in microchannels, (a) A pressure gradient, - AP, along a channel generates a parabolic or Poiseuille flow profile in the channel. The velocity of the flow varies across the entire cross-sectional area of the channel. On the right is an experimental measurement of the distortion of a volume of fluid in a Poiseuille flow. The frames show the state of the volume of fluid 0, 66, and 165 ms after the creation of a fluorescent molecule, (b) In electroosmotic flow in a channel, motion is induced by an applied electric field E. The flow speed only varies within the so-called Debye screening layer, of thickness D. On the right is an experimental measurement of the distortion of a volume of fluid in an electroosmotic flow. The frames show the state of the fluorescent volume of fluid 0, 66, and 165 ms after the creation of a fluorescent molecule [165], Source http //www.niherst.gov.tt/scipop/sci-bits/microfluidics.htm (see Plate 12 for color version). 

We wish to note that 0D can also be calculated from the measured ultrasonic velocities, and data should be equal to those obtained from specific heat measurements. The Debye s temperature evaluated from data of ultrasonic velocity is (see e.g. [14,15]) ... 

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. 

Even better agreement is observed between calorimetric and elastic Debye temperatures. The Debye temperature is based on a continuum model for long wavelengths, and hence the discrete nature of the atoms is neglected. The wave velocity is constant and the Debye temperature can be expressed through the average speed of sound in longitudinal and transverse directions (parallel and normal to the wave vector). Calorimetric and elastic Debye temperatures are compared in Table 8.3 for some selected elements and compounds. 

It may appear surprising at first sight that we should keep explicitly y% terms in Eq. (372) because in Eq. (371) the average velocity very small, of the order of the inverse Debye length k. In the limit of infinite dilution, we may reach a regime where... 

We have seen in Section V that the classical theory of electrophoresis is intimately connected with the existence of a velocity field of the fluid around a given moving ion, which in turn contributes to a supplementary friction acting on the Debye atmosphere of this ion. 

Historically, one of the central research areas in physical chemistry has been the study of transport phenomena in electrolyte solutions. A triumph of nonequilibrium statistical mechanics has been the Debye—Hiickel—Onsager—Falkenhagen theory, where ions are treated as Brownian particles in a continuum dielectric solvent interacting through Cou-lombic forces. Because the ions are under continuous motion, the frictional force on a given ion is proportional to its velocity. The proportionality constant is the friction coefficient and has been intensely studied, both experimentally and theoretically, for almost 100... 

The Debye temperature of the solid defines the form of the vibrational spectrum in the acoustic zone (low frequency) and is related to the molar volume of the solid V and to the mean velocity of acoustic waves through... 

The Debye model assumes that there is a single acoustic branch, the frequency of which increases with constant slope (proportional to the average velocity of sound in the crystal) as q increases, up to the boundary of the Brillouin zone. The boundary is assumed to be of spherical shape, with a radius qD determined by the total number of normal modes of the crystal. Thus,... 

The thickness of the diffuse-charge region is given by K 1,i.e., the Debye-Hiickel parameter (Section 6.6.4). Thus, if the fluid velocity is v at the distance k" then the velocity gradient is given by... 

Equations (6.4.43a-c) yield the central result of this section—the following expression for the electro-osmotic slip velocity ua under an applied potential and concentration gradient, in the Debye-Hiickel approximation for a thin double layer... 

The Debye theory [220] in which a sphere of volume V and radius a rotates in a liquid of coefficient of viscosity t has already been mentioned. There is angular momentum transfer across the sphere—liquid interface that is, the liquid sticks to the sphere so that the velocity of the sphere and liquid are identical at the sphere s surface. Solution of the rotational diffusion equation... 

Let us summarize by modeling the velocity autocorrelation function using Debye-Huckel type interactions between charged point defects in ionic crystals, one can evaluate the frequency-dependent conductivity and give an interpretation of the universal dielectric response. 

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