научная статья по теме EXCITED ATOMS IN ARGON GAS DISCHARGE PLASMA Физика

Текст научной статьи на тему «EXCITED ATOMS IN ARGON GAS DISCHARGE PLASMA»

EXCITED ATOMS IN ARGON GAS DISCHARGE PLASMA

V. P. Afanas'ev", B. M. SmirnovK D. A. Zhilyaevh**

" National Research University "Moscow Power Engineering Institute'

b Joint Institute for High Temperatures. Russian Academy of Sciences 125412, Moscow, Russia

Received April 23, 2013

General principles are discussed for a gas discharge plasma involving excited atoms where electron-atom collision processes dominate. It is shown that an optimal kinetic model of this plasma at not large electric field strengths may be based on the rate constants of quenching excited atom states by electron impact. The self-consistent character of atom excitation in gas discharge plasma is important and results in the tail of the energy distribution function of electrons being affected by the excitation process, which in turn influences the excitation rate. These principles are applied to an argon gas discharge plasma where excitation and ionization processes have a stepwise character and proceed via formation of argon atom states with the electron shell 3f/'4.s.

DOI: 10.7868/S0044451014070177

1. INTRODUCTION

A gas discharge plasma is in principio a nonoquilib-rium system fl 3] because an electric field energy is first injected into the gas through plasma electrons and then electrons transfer this energy to atoms. Hence, this system requires a kinetic description [4] based 011 the cross sections and rate constants of elementary processes. A general approach to this problem [5, 6] is based 011 the simultaneous analysis of the kinetic equation for the energy distribution function of electrons and the balance equations for excited atoms based 011 the parameters of elementary processes in the gas discharge plasma. Usually, elastic and inelastic collisions are important for the kinetics of a gas discharge plasma, and the peculiarity of this description is such that the theory does not allow evaluating the cross sections of electron atom processes reliably, and therefore experimental data or certain scaling models based 011 experimental results are required.

Currently, there are numerous computer simulations based 011 this approach (see, e.g., [7 11]), but all these raise questions. First, processes of formation of fast electrons and excited atoms have a self-consistent character, that is, the process of atom excitation leads

E-mail: bmsmirnov'fflgmail.com E-mail: zhiliay'ögmail.com

to a sharp decrease in the electron distribution function with the increasing electron energy, and this in turn causes a decrease in the excitation rate. In this paper, the coupling of these processes is taken into account for an argon gas discharge plasma with a moderate electron number density Arc < 1013 cm-3. Second, the dependence of the atom excitation cross section on the electron energy is accounted for in this paper based on the quenching rate constants that are independent of the electron energy at low energies.

2. ENERGY DISTRIBUTION FUNCTION OF ELECTRONS

The gas discharge plasma under consideration is an ionized gas that is supported by an external stationary electric field. We consider the regime of high electron number densities where the electron equilibrium results from electron electron collisions, which leads to the Maxwell distribution function

where v is the electron velocity, 111, is the electron

mass,

Nc is the electron number density, and Tc is the electron temperature. The normalization condition for the electron distribution function has the form

J fo(i') ■ 4irv2dv = A>. (2.2)

Fig. 1. The electron temperature T,-: in an argon gas discharge plasma as a function of the reduced electric field strength x = E/Na given in townsends (1 Td = 10-1T V- cm2)

In the simplest method to account for the influence of inelastic processes, we take the distribution function to be zero at the atom excitation energy Ae, which leads to the following form of the distribution function [12]:

tfo(v) = Nc

n>, 2rTc

3/2

exp

IT,

exp

As

T,

A - » T,. (2.3)

where e = mcv2/2.

The difference of the electron T, and atom T temperatures at not large electric field strengths E is established in elastic electron atom collisions and is given by [131

T, —T =

Ma2 (v2/vc

(v2vca)

(2.4)

Here, M is the atom mass, a = eE/mc, where e is the electron charge, E is an electric field strength, vca = Mav(T*a, and <T*Q is the diffusion cross section of electron atom scattering. Being guided by the argon gas discharge plasma, we use formula (2.4) to determine the relation between the electron temperature T, and the reduced electric filed strength x = E/Na, assuming the electron distribution function (2.3) and the cross sections for elastic electron collisions with an argon atom in [14]. The results are given in Fig. 1.

It follows that the energy distribution function of electrons in the regime of high electron number densities has the Maxwell form in the main part and becomes distorted at the tail of the distribution function.

This distortion is schematically taken into account in formula (2.3), and below we consider the tail of the electron distribution function under certain conditions. We use a general approach to this problem [5, C] based on the kinetic equation for the distribution function of electrons and the balance equation for excited atoms. With the self-consistent character of processes of formation of fast electrons and excited atoms, we have the kinetic equation for the distribution function /0 of electrons in the form

- J hx(s),fo(e)d£ ■

Áe

Nm I kq(e-&£)fo(e-&e)(le

Me

(2.5)

where Ica(fo) and Icc(/o) are the electron atom and electron electron collisions, kcx is the rate constant of excitation of atoms in the ground state by electron impact, kq is the rate constant of quenching of an excited atom in collision with electrons, and we restrict ourself to one exited state for simplicity. In the equation, we include the processes

e + A <r+ e + A*

(2.6)

and Eq. (2.5) is coupled to the balance equation for excited atoms

dN„ dt

=-K / kcx(e)fo(e)d£

OO

Nm J kq(e - A ")/„(•• - Ae) de, (2.7)

wliere ATm is tlie nunibcr density of excited atonis. The set of equations (2.5) and (2.7) allows constructing tlie electrón distribution function in tlie rango tliat is responsible for atoni excitation, and Fig. 2 gives tliis distribution function for tlie metastable state ?'P-2 of tlie argón atoni. In tliis figure, rango 1 accounts for tlie character of formation of fast electrons in a gas dis-cliarge plasma; tlie distribution function drops sharply in rango 2 dúo to excitation of tliis state, and tlie equilibrium between excitation and qucnching processes is established in tlie región 3. We note tliat tlie energy electrón distribution function in rangos 2 and 3 aro determined by differont processes of electrón kinetics. In-doed, niost part of tlio electrons penotrate in rango 2

11 >K9T<£>, Bbiii. 1 (7)

161

Fig. 2. Reduced energy distribution function of electrons in an argon gas plasma as a function of their energy s, where f\> is the electron distribution function and y?0 is Maxwell distribution function (2.1). Arrows indicate the atom excitation threshold As and the atom ionization potential J. Curve a corresponds to the electron temperature T,-: = 2 eV, curve b relates to T,-: = 3 eV, and curve c corresponds to T,-: = 4 eV

as a result of diffusion in the energy space from the range of lower energies and account for the loss of fast electrons as a result of atom excitation. On the contrary, range 3 results from quenching of excited atoms by slow electrons.

3. PROCESSES IN ARGON GAS DISCHARGE PLASMA

We start the analysis of processes in an argon gas discharge plasma involving electrons from inelastic processes:

e + Ar(3/ )*+<> + A r (:'>/>"' l.s-). (3.1)

The principle of detailed balance establishes a relation between the atom excitation cross section crcr(e) and the cross section aq(e — As) of quenching an excited atom by electron impact as [15]

,9oeocx{e) = g*(e - Ae)aq(e - Ae), (3.2)

where g0 and g„ are the statistical weights for the ground and excited states. The excitation cross section near the excitation threshold Ae depends on the energy e of the incident electron as (Jex \j£ - A£ flC 18]. Therefore, the quenching rate constant kq is independent of the electron energy at moderate energies. Hence, including the quenching rate constants in

the kinetic scheme, we can take the energy dependence of other cross sections and rate constants into account, and such a kinetic model would be optimal at not large electric field strengths.

In considering atom excitation by an electron impact, we divide this process into two parts such that the first corresponds to the formation of fast electrons that are able to excite the atom and the second self-consistent process is atom excitation due to electrons located in the tail of the energy distribution function of electrons. The rate constant k> of formation of fast electrons is determined by diffusion of electrons in the space of electron energies due to collisions between electrons and owing to the action of the electric field in accordance with Eq. (2.5). Using the Landau collision integral [19] at large electron energies, we obtain the rate constant in the form [20]

^\Z277 . 'A 111 A ( A \

= ~^pi^Ta °xp \J\.) *

where cc = Nc/Na is the concentration of electrons, In A is the Coulomb logarithm, and we set In A = 7 here and hereafter.

The rate constant k> of atom excitation by fast electrons at energies above the atom excitation energy follows from the kinetic equation for fast electrons that in the stationary case has the form

^LjLflL'Il] ' v> lllA ±

'■'>r- ilr \ /.', „ ilr J 'irnfv2 dv

where vcx = Nakcx is the excitation rate. We use the somiclassical solution of this equation in the form [21, 22]

/ — \ v5/4

fo(£) = .f(A£)e-*. S = J . (3.5)

with the parameters

K = VrII = (3.6)

Here, vq = Nav(T*a(v0) is the rate constant of elastic electron atom collisions at the excitation threshold,

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