Poisson equation plasma

In order to be able to explain the observed results plasma modeling was applied. A one-dimensional fluid model was used, which solves the particle balances for both the charged and neutral species, using the drift-diffusion approximation for the particle fluxes, the Poisson equation for the electric field, and the energy balance for the electrons [191] (see also Section 1.4.1). 

Earlier calculation on many electron atomic systems under plasma was performed by Stewart and Pyatt [58], who estimated the variation of IP of several atoms using a finite temperature TF model. Applications of the density functional theory on these systems were reviewed by Gupta and Rajagopal [57], The calculations on many electron systems are mostly concerned with the hot and dense plasmas with the application of the IS model, or from general solutions of the Poisson equation for the potential function. The discussions using the average atom model in Section 3.3, Inferno model of Liberman in 3.4, STA model in 3.5, hydrodynamic model in... 

This is an excellent assumption for the plasmas of interest since the Debye length is exceedingly small (10s of pm) compared to the reactor dimensions. Of course, the electroneutrality constraint can t be applied in the sheath, where the Poisson equation... 

Another important point, closely connected with electron kinetics, concerns the self-consistent treatment of electron kinetics, of the particle and/or power balance equations for heavy particles (such as different ions and excited atoms or molecules), and of the Maxwell equations (or a reduced version such as the Poisson equation or appropriate electric circuit equations) to obtain a more complete description of all important plasma components as well as of the internal electric field. This self-consistent treatment is usually tricky and is based on an iterative approach to the solution of the various types of equations involved (Loffhagen and Winkler, 1994 Uhrlandt and Winkler, 1996 Yang and Wu, 1996). To integrate the electron kinetic equation in such an approach adequately, a very effective solution procedure for this equation is of particular importance, although remarkable progress with respect to the speed of computation has been achieved in recent years. 

A low aerosol density, n rl <plasma chemistry. In this case, considered by Fridman (1976), the combination of the Poisson equation with the Boltzmatm distribution leads to equations for the potential cp r) and electron density n r) around a spherical thermally ionized particle ... 

The typical space-size characterizing a plasma is the Debye radius, which is a linear measure of electroneutrality and shielding of external electric fields. The typical plasma time scale and typical time of plasma response to the external fields is determined by the plasma frequency illustrated in Fig. 3-19. Assume in a one-dimensional approach that all electrons at X > 0 are initially shifted to the right on the distance xq, whereas heavy ions are not perturbed and remain at rest. This results in an electric field, which pushes the electrons back. If = 0 at X < 0, this electric field acting to restore the plasma quasi-neutrality can be found at x > xq from the one-dimensional Poisson equation as... 

Another technique, widely used in a different context (kinetic theory, plasma simulation) is the so-called Vortex-in-Cell Method (Hockney and Eastwood 1981, Birdsall and Langdon 1985). In the Vortex-in-Cell Method a two-dimensional computational domain is divided into cells and the vorticity is counted in each cell. The Poisson equation relating the vorticity and the Laplacian of the stream function is subsequently solved on the grid by an FTT algorithm, the velocity... 

In the case of thermodynamical equilibrium, an accurate description of nonlinear effects in grain screening can be obtained within the Poisson-Boltzmann (PB) approach describing the plasma as a two-component gas with Boltzmann distribution. The relevant equation for the case of a single spherical high-Z grain of a radius a in a plasma background reads,... 

We start from the conventional Poisson s equation for a single charged spherical grain of a radius a immersed in a plasma background,... 

The models for a plasma etcher are usually onedimensional (direction perpendicular to the electrodes in parallel plate geometry), use mobility limited description of the velocity field, use one energy equation that only involves the electron temperature, and write the electric potential in terms of Poisson s... 

External electric fields induce plasma polarization, which prevents penetration of the field inside the plasma. Poisson s equation for the space evolution of potential cp induced by the external field E is... 

D yachkov, L. G. 2005a. Screening of macroions in colloidal plasmas Accurate analytical solution of the Poisson-Boltzmann equation. Physics Letters A 340, no. 5-6 440-448. 

The polyelectrolyte chains exist in the effective medium created by the counterions and ions from the simple salt. The charges on the polymer chains interact among themselves mediated by the neutralizing plasma that makes up the system except the polymer molecules. We further assume that the mediation by the background is adequately described by the Debye-Hiickel theory. This assumption allows us to deduce the key features, without resorting to heavy numerical work that will be needed to solve the very complex Poisson-Boltzmann equations for such topologically correlated objects as flexible polymers. 

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