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Ion Free Fall Gas Discharge Regime Modeling By Schrödinger Equation

N.A. Azarenkov, A.V. Gapon
Karazin Kharkiv National University, 31 Kurchatov Ave., Kharkov
61108, Ukraine
Keywords: gas discharge, free-fall regime, opticsU˝mechanics analogy , 2D Langmuir model
Modeling of low pressure gas discharges is helpful for plasma technology purposes. Whereas diffusion regime
modeling has not encounter serious obstacles, free-fall regime is more difficult for calculation. One of the difficulties
is ion component movement in fixed ambipolar potential calculation.
Because of ion movement is going under potential force it can be considered with the help of Hamilton-Jacoby
equation, which is identical with eikonal equation due to optics-mechanics analogy. But for the reason of complexity
of eikonal equation treatment the Schrödinger equation is used, because of as says H. Goldstein in [1], Hamilton-
Jacoby equation is short-wave Schrödinger equation assimptotic.
Numerical algorithm consists of subsequently doing two steps by turns: calculation of electron density and ambipolar
potential for given ion density and calculation of ion density for given ambipolar potential. Electron density and
ambipolar potential are found from Poisson equation solved with the assumption of Boltzmann electron distribution
in the potential. Ion density is found from 2D generalization of well known Langmuir model of particle transport in
collisionless discharge regime. To obtain ion density in certain point one should to summarize contributions of all
remote sources of ions, ion trajectories from which pass through this point. Instead trajectories finding, we consider
each surface of equal potential as one radiated De Broglie wave in the potential descending direction. Strength of ion
sources along the surface is determined by ionization intensity. Ion density should be obtained as summa of partial
waves amplitudes squares to avoid interferention effects.
In fact, we set Plank constant big enough to avoid calculation difficulties. This leads to great deviation from ray
optic limit in the vicinity of an ion point of birth, where particle moment is small and it’s De Broglie wavelength tends
to infinity. Nevertheless, ion density calculated in such way seems to be not differing substantially from one found
from Newton dynamic equations integration.
REFERENCES
1. Goldstein H. Classical Mechanics, Addison-Wesley, Reading, MA 1980, 672 p.

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