Characteristics of potential well for nondegenerate electronic gas in a equilibrium dusty plasma

Calculated of the potential well for electrons of the dust-electron plasma depending on the temperature, size of the dust particles, the distance between them and the kind of material of which they consist. The shape of the potential well is significantly different from the rectangular approximation.


Introduction
Dusty plasmas in which the concentration of ionized gas atoms is negligible compared to the concentration of electrons, called dust-electron plasma [1][2][3][4][5][6]. Such a plasma is formed during combustion of various fuels at atmospheric pressure, plasma chemical reactors, MHD generator channels and several processes for obtaining the functional coating with the gas discharge [7,8]. At the temperature of the working gas from 1000 K to 2000 K, solid or liquid dust particles are positively charged as a result of electron emission from the surface and therefore the gas phase contains free electrons . The presence of charged dust particles leads to an uneven distribution of electric potential and the formation of potential wells for electrons.
Distribution of potential and concentration of electrons inside and around the dust particles depend on the parameters of the potential well. Usually in the approximate calculations believe that the potential well is as shown in Fig. 1, whereL -characteristic linear dimension of the solid particle, W n -the depth of the potential well. It is assumed that in the areasx<0 andx>Lfree electron concentration is equal to zero.
We show that this approach has some major drawbacks.Static equilibrium condition of the electron gas can be written as [9] const where µ-Fermi energy, q -the absolute value of the electron charge, φ -the electric potential.
We write the Gauss theorem to the electrostatic field  where ε -относительная диэлектрическая проницаемость, ε 0 -relative dielectric permittivity,  0electric constant, -the projection of the electric field on the external normal, Qcharge inside the surface S. If the S take the surface of a dust particle in the form of a sphere of radius R, then equation (2) takes the form If the electrons are only inside the particles in From (1) we find The Fermi energy [8] of a non-degenerate electron gas is Substituting this expression in (4) with the communication potential of the field intensity gives Taking into account the equation of state for non-degenerate electron gas The value dr dp is the power of pressure, acting on unit volume of the electron gas, and qE n F e  the force per unit volume of the electron gas with the electric field. If in the area R r  concentration of electrons is constant and equal n e0 , and when R r  it is zero, when R r  we have   dr dp and force   R qE n e equal to zero. Thus, equation (7) does not hold. Therefore, the assumption of a constant electron density in the area R r  and the absence of electrons in the area R r  is wrong. To resolve this issue, a closer look at the parameters of the potential well of the potential distribution and the concentration of electrons inside a dust particle in its vicinity.

Theoretical background
Consider this issue in the case of spherical particles. Let the inner radius of the solid particle is equal to R 1 , outer radius R, in the region of R 1 ≤ r ≤ R is equal to the hole concentration ni and the electron gas in the conduction band is not degenerate. We assume that the distance between dust particles is 2l. We approximately assume that the distribution of the parameters of a spherically symmetric.
We find the distribution of the potential φ (r) and the electron density n e (r) in the region 0 ≤ r ≤ l under the following conditions: φ'(0)=0, φ'(l)=0.
(9) Gauss's theorem (2)   The potential distribution   Constant C 5 and C 6 are determined from the conditions         . The condition φ'(l)=0 is used to determine the electron density n e0 at x = 0, which is included in all previous formulas.
To establish the shape of the potential well we use the following expression that determines the distribution of the potential energy of the electrons from the W coordinate r:

Results
The results of calculation using formula (19) in the case of spherical particles at T=1000 К,  Fig. 2. As can be seen, in r>R, W value increases due to escape of electrons from dust particles in the surrounding area.
In the particular case of R 1 =0, the resulting formulas are simplified. In particular, for potential distribution, formulas  10   l m presented in Fig. 3. As shown, the parameters of the potential wells are temperature dependent. With increasing temperature, the well depth increases. This effect is associated with an increase in electron gas pressure and as a result, high yield of electrons beyond the dust particles with increasing temperature. Thus, it was found that in general, the potential well is not rectangular but has a more complex geometry.