A relaxation-accelerated propagator method for calculations of electron energy distribution function and electron transport parameters in gas under dc electric fields

Abstract

A propagator method (PM), a numerical technique to solve the Boltzmann equation (BE) for the electron velocity or energy distribution function (EVDF/EEDF) of electron swarms in gases, was customized to obtain the equilibrium solution quickly. The PM calculates the number of electrons in cells defined in velocity space using an operator called the propagator or Green's function. The propagator represents the intercellular transfer of electrons corresponding to the electron velocity change due to the acceleration by the electric field and the collisional events with gas molecules. The relaxation of the EVDF to its drift equilibrium solution proceeds with iterative propagator operations for the EVDF. Merits of the PM are that the series expansion of the EVDF as done in the BE analyses is not required and that time evolution of the electron swarm can be observed if necessary. On the other hand, in case only the equilibrium solution of the EVDF is wanted, the relaxation can be accelerated numerically. A demonstration achieved a shortening of the computational time by about three orders of magnitude. Furthermore, this scheme was applied to calculations of a set of electron transport parameters required in fluid-model simulations, i.e. the effective ionization frequency, the centroid drift velocity and the longitudinal diffusion coefficient, using the zeroth-, first-and second-order moment equations derived from the BE. A detailed description on the PM calculation was presented.

Configuration of propagator method for calculation of electron velocity distribution function in gas under crossed electric and magnetic fields

“…The present PM calculation follows the principle summarized in Ref. [29,30]. In this section, the cell configuration and the PM scheme are detailed from a viewpoint of practical coding.…”

A Computational Scheme of Propagator Method for Moment Equations to Derive Real-Space Electron Transport Coefficients in Gas Under Crossed Electric and Magnetic Fields

“…3) Renewal sequence in the ϕ direction (outer loop): The following forward and reverse cycles alternate every other relaxation cycles. In the forward cycle, the renewal is operated from k = k max (ϕ = 2π) to k = 1 (ϕ = 0) by decrement of k. In the reverse cycle, the renewal for the cells in the region of v x < 0 is unchanged; from k = 3 4 k max (ϕ = 3 2 π) to k = 1 4 k max + 1 (ϕ = 1 2 π) by decrement of k. On the other hand, that for the cells in the region of v x > 0 is operated from k = 3 4 k max + 1 to k = k max by increment of k and from k = 1 to k = 1 4 k max by increment of k. This sequence was adopted in the EVDF relaxation [18], and is applied here also to the relaxations of m C 1,r (v) and m C 2,r (v) in the same manner.…”

“…One way to obtain m C n,r (v, t) in equilibrium is to follow their temporal relaxation processes as performed for electron swarms in dc E fields [14], [15], [16], [17]. On the other hand, the equilibrium solutions of m C n,r (v, t) can be obtained faster by a numerically accelerated relaxation scheme [3] under the following assumptions.…”

“…Under assumptions (i), (ii), and (iii) in section II-D, which are described as (27)- (29), µ C n,r,i,j,k in (49) satisfies the following equation on the balance between the outflow and inflow in equilibrium [3]:…”

“…Therefore, availability of reliable numerical tools to obtain the EVDF and associated electron transport coefficients from the electronmolecule interaction cross sections is of an important interest for investigators in the field of plasma analyses. A brief history of the development of the PM in the research field of gas discharges and plasmas was presented in a recent paper [3]. The PM analyses of the EVDF started from those under static electric fields [4], [5].…”