Computer solution of a kinetic equation for electrons
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Physical and mathematical approaches are presented for the behavior of a weakly ionized plasma in a thermoelectronic converter. Numerical solutions are obtained by computer methods. The distribution function for the electrons is examined in series form for a Boltzmann kinetic equation subject to boundary conditions; the coefficients of the series are deduced via moment equations. The electric field is incorporated in the quasineutrality approximation. An exampIe envisaging only electron-atom collisions is presented. Consider two unbounded planar electrodes (cathode and anode) heated to different temperatures, between which lies a weakly ionized plasma subject to a potential difference. From the electrodes flow ion and electron fluxes into the plasma, where ionization and recombination can occur. The quantities to be determined are the current, the potential distribution, the temperature, and the charge density. This problem occurs for a cesium converter in the arc mode. If the volume ionization can be neglected, the processes are ctosely described by the diffusion theory , but it is desirable to have more detailed information about the distribution function for the electrons when ionization, excitation, and recombination become important. The diffusion theory is then replaced by a Boltzmann kinetic equation, but this greatly increases the computational difficulties. The present approach envisages the use of computers.
The method of solution is basically as follows. The electron-distribution function in the kinetic equation is replaced by a series in some complete set of functions of the velocity coordinates. There is a second system of independent functions; these are mukiplied by the two parts of the kinetic equation, whereupon integration over velocity space gives differential equations of first order in the spatial coordinates for the parameters of the series for the distribution function. These are balance equations or equations for the moments with respect to the above system of independent functions (usuaIly these are polynomials in the velocity coordinates).
We select from this system a subsystem of functions, which we multiply by the boundary conditions for the kinetic equation and integrate over the region where they are given (i.e., with respect to the velocity of the electrons leaving the electrode). This gives the boundary conditions for the differential equations for the moments. Varieties of this method are to be seen in Grad's  and Weitzsch's  methods in gas dynamics, or the method of spherical harmonics [4, 8] in neutron physics; see  for review. The method of expansion used here differs from Grad's method in that I use functions of the energy and spherical angles in velocity space, whereas Grad used functions of the cartesian coordinates of the velocity. Moreover, the zero-th-approximation function is taken as the isotropic exp(--mu a/2kT) instead of Grad's anisotropic exp[--m(v-v0)/2kT] (m is electron mass, T is temperature, k is Boltzmann's constant, v is particle velocity, and v0 is the mean particle velocity). These differences are introduced for the following reasons. The electrons in a weakly ionized plasma collide frequently with neutral atoms, so there is more rapid relaxation in momentum than in energy , and the distribution function differs little from isotropic. On the other hand, a principal purpose here is to examine the inelastic processes of ionization and excitation, and the major feature is the energy distribution of the electrons without reference to the orientation of the momentum vector. Hence we need take only the first two terms in the expansion with respect to the spherical coordinate ~=vx/U (the Pt approximation in the method of spherical harmonics).
We also take account of the electric field set up by the space charge.
Let d be the distance between cathode and anode, V be the potential differences, r the Oebye radius, n+ and n. the concentrations of ions and electrons, and q the charge on an electron. As in , we consider the case where the main change in the electrical potential U occurs near the electrodes in regions of scale r, while the rest of the region obeys the quasineutrality condition ∣n+ - n_∣ ≪ n+ (Fig. 1). The size of the space-charge regions is less than the mean free path for any of the bulk processes, so no scattering occurs in these regions, while their presence is allowed for by the additional potential barriers U t (cathode) and U 2-V (anode), Both physical conditions are obeyed for r sufficiently small.
We also assume that the potential changes monotonically in the space-charge regions, as in Fig. 1, where 1 is the cathode, 2 is the anode, a are the space-charge regions, and 4 is the quasineutral plasma.
KeywordsKinetic Equation Spherical Harmonic Velocity Space Moment Equation Independent Function
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