Showing papers on "Anomalous diffusion published in 1971"

Plasma Diffusion in Two Dimensions

Abstract: Diffusion of plasma in two dimensions is studied in the guiding center model. It is shown that in this model diffusion always exhibits the anomalous 1/B variation with magnetic field. The velocity correlation function and the diffusion coefficient are calculated in detail using functional probabilities. In addition to the 1/B field dependence, the diffusion coefficient is unusual in that it depends weakly on the size of the system. These theoretical results are compared with those from computer experiments and their significance for real plasma is discussed.

Anomalous Plasma Resistivity at Low Electric Fields

Abstract: Experiments have shown the appearance of “nonclassical” resistivity, higher than that predicted by the known transport theory based on electron‐ion collisions, in magnetically confined plasmas where the electron gyrofrequency is larger than the Langmuir frequency and the electric field is well below the runaway threshold. A theoretical interpretation of the anomalous resistivity onset is proposed observing that (at long but finite mean free paths) nonresonant modes, associated with the effects of electron‐electron collisions, are excited when the electron flow velocity along the magnetic field is roughly larger than the ion sound‐wave velocity. The magnitude of anomalous resistivity is then discussed in relation with the nonlinear effects of resulting fluctuations and of related modes with wavelengths shorter than the mean free path. A comparison between several experiments on plasma confined in different stellarator and tokamak configurations is given. The differences in physical regimes that lead to various mechanisms for the appearance of “nonclassical” resistivity and the relationship with anomalous diffusion are pointed out.

The effect of oxidation on anomalous diffusion in silicon

Abstract: The diffusion of impurities into silicon is usually carried out in the presence of an oxidizing environment and an oxide film is therefore formed on the surface of a silicon wafer during the diffusion process. The growth of this oxide takes place at the oxide-silicon interface and in order to accommodate the inward growing oxide, some of the material near the interface must be transported into the silicon matrix. This material flow will contain some impurity atoms and thus gives rise to an additional component in the impurity flux. The effect of this will be most apparent near the interface where the impurity concentration is greatest and gives rise to enhanced impurity diffusion. A model based on this effect has been analysed in detail and shown to account quantitatively for the anomalous profiles reported by other workers. For an interstitialcy diffusion mechanism an equation describing the profiles can be readily derived but it is necessary to treat the problem by irreversible thermodynamics a...

Brownian diffusion in a nonuniform gas

Abstract: An analysis is made of the effects on the diffusion of Brownian particles whose Knudsen number is large compared to unity, of nonuniformities in the host gas. As examples, in one type of nonuniformity of the host gas, the Chapman-Enskog velocity distribution function for the gas molecules is used; in the other, the host gas is a free-molecule Couette flow. In both cases, a new force on the Brownian particles appears. Two techniques are used (extending Kramers' method and utilizing the Chapman-Enskog method) to transform the new Fokker-Planck equation into generalized Smoluchowski and convective diffusion equations. In these equations, the diffusion coefficient appears as a second-order tensor. Thus, it is demonstrated that Brownian diffusion in a nonuniform gas is anisotropic.

Diffusion Equation for a Pair of Brownian Particles

Abstract: A diffusion equation for a pair of Brownian particles is derived by a Chapman‐Enskog‐type method from a two‐particle kinetic equation. The diffusion tensors are given explicitly in terms of the friction tensors, and are evaluated hydrodynamically in the long‐range limit.

Anomalous diffusion of Al into SiC

Abstract: It has been stated by other authors that Al diffusion into SiC is a normal diffusion process. The results of our own investigations confirm this statement only in the case that the Al diffusion takes place from an Al-containing vapour phase as the diffusion source. We found that quite a different, anomalous diffusion process occurs when an Al-containing epitaxial SiC layer is growing on an Al-free SiC substrate. The differences between the Al diffusion from the layer into the substrate, or ‘solid source diffusion,’ and the normal ‘vapour source diffusion’ are very striking. In the former case the penetration depths are shallower by roughly a factor 100 and the concentration gradients are roughly a factor 100 steeper than in the latter case. Moreover the profile practically does not change any more for growing/diffusion periods increasing beyond a certain limit. We can provide no satisfactory explanation for this highly anomalous behaviour.

Non-Gaussian Implantation Profiles

Abstract: The distribution profile of ions implanted into single crystals frequently departs from the Gaussian form expected in amorphous solids. The reasons for this include ion channelling and various types of anomalous diffusion. The experimental evidence for such processes is reviewed, mostly for the case of silicon, and it is shown that while under some circumstances one type of mechanism may be dominant, at other times a different one is important. It is proposed that, besides the diffusion of interstitial or substitutional phosphorus after implantation into silicon, another possible transport mechanism is by the migration of phosphorus-vacancy pairs through the lattice. Some implications of these ideas from the point of view of device applications will be briefly discussed.

Effects of Particle Shape and Size Distribution on the Transient Solution of the Diffusion Equation

Abstract: THE phenomenon of zeolitic sorption has attracted much attention recently because of the increasing practical importance of synthetic zeolites, both as specific adsorbents and as catalysts. It is customary to correlate kinetic data for the sorption of gases in zeolites in terms of diffusion coefficients, calculated by matching experimental sorption curves to the appropriate transient solution of the diffusion equation. This of course assumes that the diffusivity is independent of concentration but, provided the change in concentration is small, this is a reasonable approximation. For a system of uniform spherical particles of radius r subjected at t=0 to a step change in sorbate concentration, the solution of the diffusion equation is1illustration where mt= mass of sorbate adsorbed or desorbed during time t, m∞=mass of sorbate adsorbed or desorbed as t←∞ and D=diffusivity. It is generally assumed that this expression can be used to represent the sorption curves for zeolites, using a mean equivalent spherical radius to account for the effect of crystal shape2. For short times equation (1) fits the experimental data well, but significant deviations are commonly observed at intermediate and large times. This led Kehat and Heineman3 to conclude that the sorption process is non-Fickian. In this communication we shall show that the deviations from equation (1) observed at intermediate and large times probably arise because the effects of crystal shape and size distribution are neglected in the analysis of the sorption curves. Open image in new window

Motion of a Particle in a Random Field

Abstract: The motion of a particle in a random force field is investigated using the diffusion approximation.

The effect of the observation time on turbulent diffusion in the atmosphere

Abstract: Turbulent diffusion from a fixed point-source is classified according to the length of the observation time with the aid of modern diffusion theory. The diffusion zone in the downwind is shown to consist of the linear, cuspidal, and parabolic zones. The effect of the observation time on the diffusivity coefficients and plume spreading is also discussed.