Uniqueness theorem for the non-local ionization source in glow discharge and hollow cathode

The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to wide class of electric field configurations, and to the walls of discharge volume, which have a property of incomplete absorption of the electrons. Cathode is regarded as interior singular source, which is placed arbitrarily close to the wall. The existence of solution is considered also. During the proof of the theorem many of useful structure formulae are obtained. Elements of the proof structure, which have arisen, are found to have physical sense. It makes clear physical construction of non-local electron avalanche, which builds a source of ionization in glow discharge at low pressures. Last has decisive significance to understand the hollow cathode discharge configuration and the hollow cathode effect.


Introduction
The problem of creation of the hollow cathode theory, -the theory for a glow discharge device in gases, which was invented by Paschen almost yet hundred years ago [1], producing anomalously high currents at the same voltages of discharge compared the glow discharge devices, which have no geometry of hollow cathode, -considerably stipulated for non-local ionization. The classical theory of the Engel and Shteenbeck cathode sheath [2] for glow discharge in simple geometry used the Townsend formula for a source of ionization [3], which had in mind local dependence of ionization on the electric field. The local (or twofluid) models of a glow discharge [4] gave qualitatively true description of the electric field and the electron and ion current density distributions in the cathode sheath of the plane capacitor at low current densities and not too low pressures. However they couldn't catch really the specific of negative glow. Last has low electric field, and the ionization here is produced with electrons gained the energy in another place, namely in the high field of the cathode sheath. Therefore the region of negative glow in the local model could not be obtained in principle, and the cathode sheath here is always contacting with a positive column. For a hollow cathode the model with local ionization turned out unacceptable at all, because both cathode sheath and plasma of discharge have significantly non-local properties.
On the route to the non-local ionization theory there were developed hybrid modelsinstead of two-fluid models, -in which electrons produced ionization were considered apart from slow electrons, latter providing balance of current and electric charge [5]. In a hybrid model ions and slow electrons are described in terms of drift and diffusion, but fast electrons are described with the aid of the boltzmann equations or simulated with Monte-Carlo methods [6 -8]. It was shown [9] that hybrid models describe a density of plasma in a glow discharge much better, though they are more complicated.
Hybrid models set a problem of description of non-local ionization source, for which the Townsend formula is not available. The boltzmann equation for fast electrons is manydimensional, which makes difficulties in its use in calculations. The Monte-Carlo method is in essence a computing experiment, which supplies empirical data about numerical simulation of ionization source without understand reasons of results obtained.
The author of present paper had managed earlier to simplify this problem by use of original (not the hilbert [10]) averaging of the boltzmann equation for fast electrons, in result it was obtained the non-local equation for a source of ionization in glow discharge and hollow cathode [11,12]: second kind Volterra equation [13,14]. In a case, when the non-local effects can be neglected, Necessary conditions for existence of fundamental solution for auxiliary differential operator of the problem come to the Fredholm alternatives [15]. Investigation of sufficient conditions for existence of solution is not easy problem, and in the paper we make the assumption that necessary fundamental solution for auxiliary operator exists.
The proof is divided on five lemmas and final proving of the theorem. In lemma 1 it is proved uniqueness of zero solution for homogeneous differential equation, which is generated by auxiliary differential operator. This operator defines the distribution function for "fast" electrons, which are fortunate not suffer any inelastic scattering as long as they appear from a source of ionization (or from the cathode). Lemma 2 formulates analogues statement for conjugate operator. From these results it follows the uniqueness of fundamental solution for auxiliary differential operator (if it exists), so the lemma 3 is devoted to prove this. In lemma Thus, a set of useful formulae is obtained during the proof of the theorem. The constructions arisen have clear physical sense.

Definitions and properties of physical values
Consider the domain ( ) here e is an elementary charge, ion ε is the electron impact ionization energy of the atom, eV, e m is a mass of electron.
The operator of elastic scattering The operator of inelastic scattering at electron impact excitation and ionization conserves a number of electrons involved into inelastic scattering with atoms (here we mind primary electrons, -we do not include secondary electrons, which appear in ionization in addition): in ε is the smallest of the energy thresholds for inelastic processes, eV. On physical reasons in ion ε ε > because atomic ionization energy always exceeding excitation energy of any level from discrete energy spectrum.
It follows from restriction (5) that integration in (4) could be narrowed to the ball from which follows that the average rate ( )  lists mutually insolated areas where the inequalities for small kinetic energy are true (slow electron cannot penetrate from one area to another through potential barrier).
Define a zone of "fast" electrons as The subset of 3D velocity space 1) a normal to the hypersurface of energy 2) a direction of phase flow (phase trajectories of collision-free electron motion) 3) a spatial-like direction having spatial component, which is orthogonal to the velocity and electric field 4) a velocity-like direction, having velocity component, which is orthogonal to the velocity and electric field ( )

5) a velocity-like direction, which is orthogonal to
6) a direction, which is orthogonal to five ones above So defined directions do not consist mutually orthogonal set of directions in 6D space (because a scalar product: is not equal to zero in general case), but they consist a complete basis of directions in (regular) point ( ) in Indeed, first four 6D-vectors: . The functions are simultaneously 1) finite functions, support of which is belonging to closing in Ξ of the domain in Ξ ; 2) continuously differentiable along direction t Ν of the phase flow; 3) twice continuously differentiable on v in local section 4) having its continuous extension and continuous extension of all listed derivatives into the piece of boundary 5) as the problem has no differential operators along the rest of three directions ε Ν , ⊥ Ν r and 5 Ν , we guess necessary the continuity of function f in in Ξ itself, nothing about its derivatives in these directions.
In analogy we define the function classes ( , and so on, having in mind other domains of definitions in the item 1).
Let us designate as ( ) ( classes of finite functions, which are continuous on closings of domains of their definition.
The operator of electron scattering on atoms of gas also operator of phase flow for collision-free motion of electron in the electric field can be considered as acting from as a fundamental solution of stationary boltzmann equation with a point source in the right hand side, and satisfying the here Ω ∂ n is an external unit 3D normal in the boundary Ω ∂ of the domain Ω , 1 0 : ≤ ≤ β β is the reflection factor for electrons in the boundary of discharge.
Physical sense of the condition of incomplete absorption is: the boundary Ω ∂ of discharge volume Ω absorbs the part β − 1 of all electrons in average, which get it, the rest of electrons suffer elastic ("mirror") reflection by the boundary. In this way, the cathode -a boundary emitter of electrons -is convenient to simulate as a given singular source, which is located inside Ω , but arbitrarily close to its real boundary position. This -has no answer in this paper. It is tied with formulation of sufficient conditions for existence of stationary solution, discussed above. So, we guess here: it is not empty.
, then for velocity of electron before reflection The proof of the theorem below is main goal of this paper:

The theorem
A solution of integral equation is an arbitrary function, which is integrable under the Lebesgue sense in Ω , g is defined with (7) and (8), at the condition of existence of generalized solution 1 g of auxiliary which also satisfying boundary conditions (8), exists and is unique.

Lemma 1
The solution of homogenious equation

The proof of lemma 1
An existence of solution is obvious.
Make transformations: First two integrals are reduced to integration along the boundary Ε Ξ ∂ in . It consists of three kinds of sections: 1) the section, which is conditioned with the boundary of spatial domain of the section, which is conditioned with upper boundary of energy ; 3) the section, which is conditioned with the lower energy thresholds for inelastic processes , which consists of First integral vanishes in sections Ε Γ and in Γ on a reason of orthogonality of the phase flow to the normal of the hypersurface of constant energy. In section Ω ∂ Γ it gives the This expression is nonnegative due to the condition of incomplete absorption (8).
Second integral vanishes in all three sections of the boundary due to orthogonality of the boundary normal to the vector of flow density (the intergrand here is a divergence of 3Dvector, which is orthogonal to v ).
In order that sum of three integrals with continuous nonnegative integrands be equal to zero it is necessary that every of integrands be equal to zero in all points of the domain of integration. Taking The equality (12) The equality (14) . But the subset of low is not empty and it needs additional investigation (it includes "fast" electrons having low kinetic energy, but total mechanical energy sufficient for inelastic processes). Substitution the expression (15) into equation (11) for the subset mentioned with ( ) 0 can be presented as a sum of its non-crossing connected components: Here the curve C is meant as longest. If two of such curves connect the same pair of local maximum and local minimum of potential, they define identical set c Ω .
Indeed, suppose contrary: for such curves 1 C and 2 C exists a value max min : u u u u < ′ < ′ of parameter, for which corresponding points of curves belong two different connected components of potential:  (17) can be rewritten as: here χ is any arbitrary-differentiable function. With the use of (15), in the connectedness domain of the potential at 0 ≠ v we obtain: 1.
2) The set of singular points is a closed set. Consider any its connected component. Obviously, the potential has constant value along it. The equation (17) gives Thus, all solutions of equation (11) where Q is a number of connected components of hypersurface of constant energy. This number can differ the number P of connected components of the potential: , because connected components of equal potential , such one, that its crossing with Ε Ξ in is not empty (it means, that the electron, which is arranged over given "potential pit", has the kinetic energy in some place, which exceeds the threshold of inelastic processes in ε ). In the point ( ) 2) The set one can find such Ε , that this point would belong to the set Ε Ξ in also, therefore the lemma condition is extended onto The lemma is proved.

The remark
The domain of uniqueness of zero solution of homogeneous equation (11)

The conjugate equation
. Multiply non-homogeneous equation and integrate over the domain Ε Ξ in : Making integration by parts in the left hand side of the equation, we obtain: (23) From the obtained equality (21) it is apparently, that if function h satisfies conjugate homogeneous equation also the "conjugate" condition of incomplete absorption: (in which a sign of scalar product of the velocity on the normal is opposite to that was in (8) vanishes, and consequently, the condition of the orthogonality of the right hand side of (20) to all solutions of the homogeneous problem (24), (25) becomes the condition of solvability of the equation (20) (the Fredholm condition). But for this problem the next lemma is true:

Lemma 2
The unique solution of conjugate homogeneous equation (24) at "conjugate" condition of incomplete absorption (25) on class of functions ( ) ( )

The proof
is completely analogues to the proof of the lemma 1.

Lemma 3
Generalized function ( ) ( ) , which satisfies the equations: if it exists, is unique. The support of the function belongs to the set

The proof
Existence and uniqueness. The equation (26)  The last was proved by lemma 1.
The support of 1 g . The proof of the lemma 1 (also the lemma 2) remains true in that case, if from the set in Ξ one excludes any layer . At that, it is not necessary to put any boundary condition in the boundary of the layer, because the vector field of phase flow is tangential to the boundary in any point, and the flux through the boundary from appropriate integral terms vanishes. In result, the equalities (13) and (14)  dr r r C π η be the normalized "cap" [15]. Build a sequence of "caps", which are convergent to the deltafunction of right hand side of (26) or (27): , .
(Here the convergence treats in a sense of weak convergence of linear functionals, see [16].) The support of any "cap" is concentrated in the right product of balls If now one chooses , than we obtain that the equation The lemma is proved.

Lemma 4
The solution ( ) Physical sense of inequality (30): the electron, which has arisen in glow discharge in the phase point v r ′ ′, at the energy ( ) v r ′ ′, ε , can not gain more energy in the process of motion under elastic and inelastic scattering on "motionless" atoms of gas, because inelastic processes make its mechanics of motion dissipative.

The proof
Rewrite the equation (7) in the form (otherwise the factor µ vanishes). Using the superposition principle [15] p. 195, let us represent the solution of equation (7) as a sum of solutions with elementary sources from (26). It enables to represent the equation (7) in integral form Because of property (5) one can change the order of integration by the velocities v v ′ ′ ′ ′ ′ , : Due to (5) the integration here is made over set From structure of this set we have inequality: also. This make possible to change the order of integration mentioned.
We obtained the integral equation where Using (5), we obtain: In the operator form (where the names of integral operators correspond to its kernels) the equation (31) takes form: . Its formal solution is the Neumann series: However, because the energy is restricted from the bottom, but every action of the operator Thus the series has finite number of summands: Existence and uniqueness of ( ) The lemma is proved.

Normalized spaces
In manifolds . Such definition of the norm is convenient in the sense, that the norms of the operator 1 D and its reverse operator  The lemma is proved.

The proof of the theorem
The equation (9) The formula (37) obtained defines identically the solution of the equation (9), and thus it proves an existence of the solution (at the condition of existence of solution the equation (26)) and its uniqueness.
The theorem is proved.

Discussion of results and conclusions
In the proof of the uniqueness theorem for solution of equation ( The inelastic losses of energy generate a source of electrons having lower energy than primary ones had, this process is described with operation 1 1 g s µ = . The distribution 2 g of lower energy electrons gives operation 1 2 g K g = . If distribution 2 g describes "fast" electrons, they are able to loose its energy again: Here general inequality for norms is substituted with equality, because all operator summands are non-negative and act on non-negative source functions. When "more than unit" (> 1) can be substituted with "much more than unit" (>> 1), the primary ionization source (first summand of series in (37) at 0 = m ) could be very small in comparison with all sum, this is named "avalanche". As distinct from the Townsend local avalanche, where increase of number of electrons occur along one of spatial coordinates in the direction opposite to electric field, the increase in the non-local avalanche is not spatial, but rather energetic one. It is a reason why the light of hollow cathode looks like it is very uniform. Yakovenko, which were aimed to make paper more clear.

Acknowledgments
The work is made under support of the grant number 5499 of The Ministry of Science of Russian Federation.