Einstein frequency diffusion

For an aqueous suspension of crystals to grow, the solute must (a) make its way to the surface by diffusion, (b) undergo desolvation, and (c) insert itself into the lattice structure. The first step involves establishment of a stationary diffusional concentration field around each particle. The elementary step for diffusion has an activation energy (AG ), and a molecule or ion changes its position with a frequency of (kBT/h)exp[-AGl,/kBT]. Einstein s treatment of Brownian motion indicates that a displacement of A will occur within a time t if A equals the square root of 2Dt. Thus, the rate constant for change of position equal to one ionic diameter d will be... 

In Fig. 3 the pressure and concentration (n) variations in the wave are shown. The diffuse structure in this case may be explained qualitatively by the fact that the high-frequency waves necessary to generate a steep front propagate with a speed greater than D and, moving forward away from the wave, are damped (absorbed). As Einstein showed, waves with an oscillation period less than t decrease in amplitude by e times at a distance of order cr. 

Once the base resistance problems are solved, the base transit time will be mi obstacle for high frequency operation due to the small diffusion constant, as estimated from the small mobility and the Einstein relation. A theoretical study on base transit time on GaN/InGaN HBTs [7] showed that a base transit time of less than 0.1 ps can be achieved with an appropriate base design by employing composition grading and/or doping grading. 

For Brownian motion, the collision frequency function is based on Fick s first law with the particle s diffusion coefficient given by the Stokes-Einstein equation. The Stokes-Einstein relation states that... 

The ability to correctly reproduce the viscosity dependence of the dephasing is a major accomplishment for the viscoelastic theory. Its significance can be judged by comparison to the viscosity predictions of other theories. As already pointed out (Section II.C 22), existing theories invoking repulsive interactions severely misrepresent the viscosity dependence at high viscosity. In Schweizer-Chandler theory, there is an implicit viscosity dependence that is not unreasonable on first impression. The frequency correlation time is determined by the diffusion constant D, which can be estimated from the viscosity and molecular diameter a by the Stokes-Einstein relation ... 

In deviations from the Nernst-Einstein equation in a molten salt, one hypothesis involved paired-vacancy diffusion. Such a model implies that holes of about twice the average size are available at about one-fifth the frequency of averagesized holes. Use the equation in the text for the distribution of hole size to test this model. 

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

For ionic solids, measurement of the ionic conductivity, <7 , has long provided a method for studying their atomic diffusion [25, 209, 225, 226] (see also Chapter 3). The measurements are usually made with an alternating current (AC) bridge operating at a fixed frequency, f (typically >1 kHz), to avoid polarization effects. The early studies were restricted to measurements on single crystals, and in this case (7i and the tracer diffusion coefficient were seen to be related by the Nernst-Einstein equation [25] ... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation Dx lkT)a = C holds, which forms the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the t5q)e of boundary conditions used in deriving Stokes law. It follows that the diffusion coefficient ratio is given by ... 

By equating Pick s second law and the Stokes-Einstein equation for diffusivity, Smoluchowski (1916,1917) showed that the collision frequency factor takes the form... 

Brownian motion is a random thermal motion of a particle inside a fluid medium. The collision between the fluid molecules and suspended microparticles are responsible for the Brownian motion. The Brownian motion consists of high frequencies and is not possible to be resolved easily. Average particle displacement after many velocity fluctuations is used as a measure of Brownian motion. The mean square diffusion distance, is proportional to DAt, where D is diffusion coefficient of the particle given by Einstein relation as... 

---