577

Thermophysics and Aeromechanics, 2007, Vol. 14, No. 4

SPATIOTEMPORAL EVOLUTION OF THEDISTRIBUTION FUNCTION OF ELECTRONSIN SIGN-CHANGING ELECTRIC FIELD*

A.V. FEDOSEEV1 and G.I. SUKHININ1,2

1Kutateladze Institute of Thermophysics SB RAS, Novosibirsk, Russia

2Novosibirsk State University, Novosibirsk, Russia

(Received April 16, 2007)

A numerical model for solving the Boltzmann unsteady non-local kinetic equation for the distributionfunction of electrons over energy is constructed . The Boltzmann equation for isotropic part of the distributionfunction written in natural variables the kinetic energy ⎯ the coordinate was solved by the pseudo-unsteadymethod. The model was applied for describing the spatiotemporal evolution of the distribution function ofelectrons in a uniform electric field. For a model distribution of the electric field with the “negative” value inthe Faraday dark space and the “positive” value in the positive column of the glow discharge, the main macro-scopic parameters of electrons are obtained, the diffusion mechanism of the electron current transfer in thenegative electric field region is confirmed.

INTRODUCTION

Glow discharges in low-pressure gases are widely used in various technologies andscientific applications. There are numerous applications of the glow discharge for theplasma-chemical deposition of thin films and coatings in microelectronics, in plasmadisplay panels, for gas activation in plasma-chemical reactors, for cleaning the surfacesof materials, for creating active media of gas-discharge lasers and different light sources,in gas-discharge commuting devices, etc. To describe the plasma of low-pressure dis-charges with a low current density it is necessary to consider the non-local electron ki-netics. This is especially true for non-uniform plasma regions such as the near-electrodelayers, the Faraday dark space (FDS), the plasma of a stratified positive column (PC) ofthe discharge. The distribution function of electrons (DFE) is not in an equilibrium withthe local electric field but depends on the prehistory of electrons motion. The high-energy part of the DFE depends substantially on specific conditions in the discharge,namely: the DFE high-energy part determines the constants of ionization processes, thevelocities of molecules excitation, etc. The foundations of the non-local theory at a con-sideration of discharges are contained in the works [1, 2].

The mechanism of the incipience and propagation of ionization waves in positivecolumn of the discharge may vary under different discharge pressures and currents. Ininert gases under low currents and not very high pressures, when the electron-electroncollisions do not play a substantial role, the DFE formation and the energy balance aredetermined by elastic collisions of electrons. The stratification mechanism at lower den- * The work was financially supported by the Russian Foundation for Basic Research (Grant No. 07-02-00781-a)

and by State Contract No. 02.513.11.3242.

© A.V. Fedoseev and G.I. Sukhinin, 2007

578

sities is related to resonance effects, when the energy balance is determined by inelasticcollisions of electrons (the energy relaxation length in inelastic processes is much less

than the relaxation length in elastic processes, �in << �

el). In this case, the energy losses

in elastic collisions are small, and the electrons reach the excitation energy at length

L ≥ U

ex/(eE0) with a subsequent energy loss in inelastic collisions (U

ex is the excitation

threshold for the atom electron state, E0 is the mean strength of the electric field). Thelength L determines the field non-uniformity scale that is the strata length. This mecha-

nism assumes that the energy relaxation length in elastic collisions �el

~ (M/m)1/2

� (� isthe electron free path, m and M are the masses of the electron and atom, respectively) is

much higher than the strata length, �el

>> L. At low gas densities, the non-local nature ofthe electron kinetics and DFE has been established exactly, and the distribution functionof electrons is formed on the entire potential profile in the stratum. The opposite case isrealized under elevated pressures when the elastic collisions dominate in the energy bal-

ance, and the inequality �el

< L holds. The less the ratio �el

/L, the closer the DFE to thelocal distribution function.

There are many attempts to describe the strata in PC with the aid of the non-localkinetic theory based on solving the Boltzmann kinetic equation. With the aid of the ex-pansion of the distribution function of electrons in the series in Legendre polynomials(the two-term approximation is usually employed) the spatially non-uniform Boltzmannequation is transformed into a partial differential equation for the DFE isotropic part.The terms describing the elastic and inelastic collisions of electrons with atoms enter theequation. They contain the arguments shifted in energies. A numerical scheme for solv-ing such an equation written in the variables the total energy of electrons ⎯ the coordi-nate (instead of the kinetic energy ⎯ the coordinate) under the corresponding boundaryconditions was presented in the work [3]. A passage from the kinetic energy of electronsto the total energy, which was proposed in [4, 5], simplifies greatly the solution of theBoltzmann equation, the parabolic equation for the DFE isotropic part was solved in [3]for the given total energy value by the tridiagonal inversion method using already knownDFE values at higher total energy of electrons. The method enabled the consideration ofthe DFE and corresponding macroscopic parameters of electrons in a given spatial dis-tribution of the electric field and also the consideration of the influence of electric fieldnon-uniformities on the DFE formation. Such numerical schemes were used with somechanges in other works [6−8] to describe the DFE relaxation in inert gases in a constantelectric field upon introducing some field perturbation at the PC beginning or whilestudying the electron relaxation in sinusoidal (“strata-similar”) electric fields. A methodwas presented in the work [9] for solving the unsteady non-local Boltzmann equationwritten in the variables “the total energy of electrons ⎯ the coordinates”, which wassolved by a method based on the method from the work [3]. Using the model the spatio-temporal DFE evolution in a given uniform electric field was obtained ⎯ the process ofthe formation of spatial structures in time in the interval the positive column ⎯ the anode.

There are the regions in gas discharges, where the electric field may change itssign. In a region between the cathode layer and the positive column (at the FDS begin-ning) the electric field becomes negative [10−13]. There are some indications of the factthat a reverse field may arise in the region of strong strata and in a low-voltage arc,which is directed from the anode to the cathode [13]. A potential well for electrons arisesin such regions, that is the potential in discharge becomes nonmonotonous, and themodel of solving the Boltzmann equation in the variables “the total energy ⎯ the coor-dinate”, which was developed in [3], becomes unacceptable. It is necessary to considerthe Boltzmann equation in natural variables “the kinetic energy of electrons ⎯ the coor-dinate”. The unsteady Boltzmann equation for the DFE isotropic part, which was written

579

in natural variables “the kinetic energy of electrons ⎯ the coordinate”, was solved in[14] by the pseudo-unsteady method starting from some initial DFE distribution. Theprocess of DFE formation in space and time was described in the model for differentgases, electric field values, and boundary conditions, and the different transport coeffi-cients of electrons were also found.

In the present work, a method for solving the Boltzmann non-local unsteady equa-tion for the DFE in the variables “the kinetic energy of electrons ⎯ the coordinate”(which is similar to [14]) is presented. The constructed model enables a description ofthe DFE spatiotemporal evolution, an explanation of the kinetic mechanism of the originof a stratification of the discharge positive column in inert gases in the given distributionof the electric field similarly to model [3]. Besides, the constructed scheme for solvingthe Boltzmann equation by the pseudo-unsteady method enables the obtaining of theDFE spatial distribution in the case of a sign-changing distribution of the electric field.A computation has been carried out in the work for a model steady distribution of theelectric field with a constant field value in the discharge positive column and a depres-sion ahead of the PC. The electric field value in the depression has a weak negativevalue and corresponds to the FDS. The results of the work confirm a diffusion mecha-nism of the electron current transfer in the gas discharge region with the negative electricfield described in [12].

1. MODEL

To describe the spatial and temporal evolution of electrons of the glow dischargeplasma the unsteady non-local Boltzmann equation for the distribution function

( , , )F x tυ� �

of electrons over the velocities was used:

el in0 ( ) ( ),ke k

eF F FE S F S F

t x mυ

υ∂ ∂ ∂+ − = +∂ ∂ ∂ ∑

�

�

� � (1)

where Sel

is the integral of elastic collisions, inkS is the integral of inelastic collisions

(includes several processes of inelastic collisions of electrons with the argon atoms), −e0

is the charge, and me is the electron mass, ( , )E x t�

�

is the applied local electric field. For a

discharge in a tube, the boundary effects on walls were neglected, and it was assumed

that the electric field distribution is only axial; the field is directed along the z axis,

( , ) ( , ).zE x t i E z i=� �

�

The distribution function of electrons ( , , )F x iυ� �

is symmetric with

respect to the z axis and reduces to the dependence F(υ, υz /υ, z, t) of the quantity

,υ υ= �

the directing cosine cos ϑ = υz /υ, the spatial coordinate z, and time t. Under the

assumption of a weak anisotropy, only the first two terms were retained in the DFE ex-

pansion in Legendre polynomials (the so-called two-term approximation):

0 1( , / , , ) ( , , ) ( , , ) .zzF z t f z t f z t

υυ υ υ υ υυ

= + (2)

Here f0 is the DFE isotropic part, f1 is the DFE anisotropic part. If one substitutes expan-

sion (2) in equation (1) and integrates over 2πd cosϑ and 2π cosϑdcosϑ, then one ob-

tains after passing to the kinetic energy 2

2

mU

υ= a system of two equations (3) and (4)

for the isotropic and anisotropic parts of the distribution function of electrons over theenergies (DFEE):

580

1/ 21/ 2 2 el0 01 1

0

in in in in0 0

( , ) ( )12 ( )

2 3 3

( ) ( ) ( ) ( , , ) 0,

e eg

g k k g k k kk k

m f e E z t mf UfU U U N Q U f

t z U U M

UN Q U f U U N Q U U f U U r t

∂ ∂ ∂ ∂⎛ ⎞ ⎡ ⎤+ − − +⎜ ⎟ ⎢ ⎥∂ ∂ ∂ ∂⎝ ⎠ ⎣ ⎦

+ − + + + =∑ ∑ (3)

1/ 21/ 2 0 01

0 1( , ) ( ) 0,2

em f ffU e E z t H U f

t z U

∂ ∂∂⎛ ⎞ + − + =⎜ ⎟ ∂ ∂ ∂⎝ ⎠ (4)

where Ng is the density of neutral particles of mass M, Qel

(U) is the transport cross sec-

tion of scatter in elastic collisions, Qkin

(U) is the cross section of the kth inelastic colli-

sion with a loss of energy Uk by the electron, the coefficient H(U) = NgQel

(U) +

ΣkNgQkin

(U). Gas atoms were assumed to be at rest. The collisions of electrons with ionsand metastable particles were neglected. An arbitrary scatter of electrons in elastic colli-sions and isotropic scatter in inelastic collisions were assumed. The last term in equation(3) with argument U + Uk is responsible for the appearance of an electron with kineticenergy U due to the energy Uk lost by the electron in the kth inelastic process. The re-laxation rate of the DFEE isotropic and anisotropic parts in time in equations (3)−(4) wasdetermined by a ratio of the frequencies of the impulse and energy losses in elastic andinelastic processes as well as by the rate of the electric field E(z, t) change. In mostgases, the frequency of the impulse loss usually exceeds the energy loss frequencies bothin elastic and inelastic processes. Since the relaxation rate of the DFEE anisotropic partis much higher than the relaxation rate of the DFEE isotropic part, the time derivative inequation (4) was neglected. With regard for this, let us express the DFEE anisotropicpart using (4)

0 01 0

( , , ) ( , , )1( , , ) ( , ) .

( )

f U z t f U z tf U z t e E z t

H U z U

∂ ∂⎛ ⎞= − −⎜ ⎟∂ ∂⎝ ⎠ (5)

Substituting (5) in (3), we obtain a partial differential equation for the DFEE isotropic part

1/ 21/ 2 0 0 0

0

00 0

in in0 0

, )

( ) ( , )2

( ) ( ( )

( ) ( ) ( , , ) 0,

e

k k k kk k

t

m f f fU B U e E z t

t z z U

fB U e E z C U f

U U U

G U f G U U f U U z t

⎡ ⎤∂ ∂ ∂∂⎛ ⎞ ⎛ ⎞− − +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤∂∂ ∂⎛ ⎞+ − +⎡ ⎤⎢ ⎥⎜ ⎟ ⎣ ⎦∂ ∂ ∂⎝ ⎠⎣ ⎦

+ − + + =∑ ∑

(6)

where the coefficients

B(U) = 1

3 ( )

U

H U, C(U) = 2 el2 ( ),e

gm

U N Q UM

Gk(U) = ( ).ing kUN Q U

Equation (6) describes the spatiotemporal evolution of the DFEE isotropic part, which isdetermined by an accumulation of electrons energy in the electric field and by its loss invarious elastic and inelastic electron-atom collisions. The electrons scattering cross sec-tions in argon were taken from the data base [15]. All ionization collisions were inter-preted only as the processes of the energy loss that is they did not alter the number ofparticles. The finite-difference form of equation (6) has the first order of accuracy intime t and the second order of accuracy in the z coordinate and energy U. Starting from

581

some initial distribution of the

DFEE isotropic part 00 ( , )tf U z=

in the coordinate plane (U − z),equation (6) was solved by thepseudo-unsteady method. Sinceequation (6) for the DFEE iso-tropic part represents a nonlinearimplicit dependence 0 0( , , , )f f U z t , it was solved by an implicit method. At the moment

of time tk = 1, the first approximation of the distribution function was found. The initial

function 00 ( , )tf U z= was used as the zeroth iteration. The obtained solution correspond-

ing to the first iteration was used again instead of the zeroth iteration, etc., until the dif-ference in solutions for neighboring iterations satisfied some criterion (in the presentwork, the relative difference in the DFEE is less than 10−5). After that, the solution wasfound for the next moment of time tk+1. In the given model, the number of iterations did

not exceed 2−3. The time step at such an approach has the order �t ~ 10−11 s for certainvalues of the step in energy and coordinate. The computations were carried out until themoment of time of the order tmax ~ 10−6−10−4 s, at which the solution for the DFEE con-verged to a final value independent of the form of the chosen initial distribution.

The region of solving equation (6) is shown in Fig. 1. The U−z coordinate planewas partitioned into the subdomains with nodes Du = {U0 = 0, U1, …, Ui, …, UI = Umax},

Dz = {z0 = 0, z1, …, zj, …, zJ = zmax}. The number of grid nodes in the plane was varied

within the limits I = 100−500, J = 100−500. The electric field was directed to the oppo-site direction of the z axis in such a way that the electrons accelerated in the positivedirection of the axis. At the i, j point the first- and second-order derivatives and themixed derivatives were found from the known DFEE values from the preceding iterationat points (i ± 1, j ± 1), (i ± 1, j), (i, j ± 1), (i, j).

At z = 0, the stationary anisotropic part of the distribution function of electrons wasspecified by the Gauss function

2

1( , 0) exp ,mU Uf U z cU

U

⎡ ⎤−⎛ ⎞⎢ ⎥= = −⎜ ⎟Δ⎢ ⎥⎝ ⎠⎣ ⎦ (7)

which models the electron beam. Different values of the mean energy of electrons andthe beam distribution widths were used. The normalizing constant c was determinedfrom the condition of the equality of the electron flow density at the inlet of 1 mA/cm2 ofthe computational region. Knowing the distribution of the DFEE anisotropic part at thecathode boundary of the region under consideration and assuming the initial DFEE iso-tropic part distribution to be uniform (∂f0 /∂z = 0, t = 0) in the uniform distribution ofthe electric field, we find with the aid of expression (5) the initial distribution

0 max( , 0 , 0)f U z z t≤ ≤ = :

0 10

1( , 0) ( ) ( , 0) .

( 0)U

f U t H U f U z dUe E t

∞

= = == ∫ (8)

The maximum kinetic energy Umax under consideration is chosen in such a way that forthe given value of the reduced electric field E/Ng, the distribution function of electrons

Fig. 1. Solution domain of equation(6).

582

with energy Umax equaled zero everywhere. The anisotropic part of the distribution func-tion equals zero at the kinetic energy of electrons equal to zero (this follows from (6)).According to this condition and condition (5), the value of the DFEE isotropic part was

found at the lower boundary (U = 0). The condition ∂2f0/∂z

2 = 0 was posed on the right

boundary (z = zmax), which did not restrict neither the value of f0 nor its derivative valueat the boundary.

2. RESULTS

As a result of the numerical solution of equation (6), spatiotemporal evolution ofthe nonmonotonous distribution function formation was obtained. Computations werecarried out for the argon pressure p = 1 Torr in the constant electric field E = 8 V/cm.Figure 2 shows the logarithm of the DFEE isotropic part measured along the verticalversus the kinetic energy U and the z coordinate at different moments of time: the initialuniform distribution, at times t������������������ ��������������������������������i-cially specified DFEE isotropic part (see Fig. 2, �) corresponding to some energy distri-bution of the entering beam of electrons (see (7) and (8)) rapidly adjusts at the initialstage to the scatter cross sections specified for argon. The DFEE transforms in the waythat the energy accumulation by electrons in a constant electric field and the energy lossby electrons in elastic and inelastic collisions are balanced. The DFEE initial distributionis seen to be slightly narrower than the distribution at time t����������������������� �b),the distribution function tail has shifted into the region with a higher kinetic energy.Since the energy dependence of the DFEE anisotropic part at the PC cathode boundary isspecified and is time-invariable, the narrowed beam of electrons shifts with increasing zcoordinate into the region with a higher kinetic energy U. Upon reaching the first energythreshold related to the loss of electrons energy in the inelastic collision, a hump in theDFEE arises in the region with the zero kinetic energy (see Fig. 2, b). Such a periodicprocess of the electrons energy accumulation in the electric field and the losses in elec-tron-atom collisions develops from the PC cathode side in space and in time. The results

Fig. 2. Three-dimensional representation of the isotropic part of the distribution function at differ-ent moments of time.

Initial distribution (�), 0.168 (b) 1.68 (c), 8.43 (d) �s.

583

computed for t = 1.68 �s are shown in Fig. 2, �. One can see 5−6 formed strata. The finalDFEE distribution is shown in Fig. 2, d and corresponds to the time t = 8.43 �s.

A complete potential drop in the interval under consideration equals 80 V and cor-responds to about 6.5�L, from where the potential drop at the stratum length L is obtainedto be equal to L(eE0) ≈ 12.3 eV and exceeds slightly the first excitation thresholdU1 ≈ 11.3 eV. Interestingly, in the absence of elastic losses and in the presence of onlyone process of the energy loss with threshold U1 with a high value of scatter section (theblack wall approximation) the final distributions of macroscopic parameters would benon-decaying, and the distance between the strata would correspond strictly to the lengthL = U1/eE0 of the energy accumulation U1 in electric field [16]. Such a mechanism of thestratification in PC, when the electrons periodically accumulate energy in electric fieldand loose it in inelastic collisions, was indicated already in 1952 in the work [17]. A de-tailed analysis, which was indeed the first one, of the kinetic mechanism of the stratifi-cation phenomenon was carried out by L.D. Tsendin [16].

The obtained solutions depend strongly on the boundary condition on the PC cath-ode side. In particular, if the mean energy Um is given a lower value in the energy distri-bution of the electrons beam (see (7)), then the electrons will accelerate up to the valueL(eE0) ≈ 12.3 eV during a longer time. The DFEE hump will reach this value furtheralong the coordinate axis z, the remaining strata will also shift along the z axis. One candraw here an analogy with the stratum phase. The obtained solution also depends on thewidth of the specified electrons beam. At the specification of a narrower beam of elec-trons the solution obtained in the entire region will be more pronounced. If one substi-tutes the solution of the homogeneous Boltzmann equation for the case of a uniformelectric field as the initial uniform distribution of the DFEE isotropic part then no strataare formed since the DFEE is adjusted at each z point to the scatter sections, and thesame accumulation of electrons energy and the energy loss in collisions occur at eachpoint in the field. For the formation of strata at such an initial (and steady cathodeboundary) condition, the presence of a depression in the electric field distribution, inwhich the electron beam would “thermolize”, is necessary [3]. When the thermolizedelectrons beam gets into a region with a higher electric field, the narrowed distribution ofelectrons over energies will lead to the interval stratification.

The spatial distribution of the DFEE isotropic part is found from equation (6), theDFEE anisotropic part is expressed with the aid of relation (5). Knowing the DFE wecan determine several macroscopic parameters of electrons by integrating the distribu-tion function over the energy. In particular, the distribution of electron density and of themean energy of electrons are found by integrating the DFEE isotropic part:

1 20

0

( , ) ( , , ) ,en z t U f U z t dU∞

= ∫ (9)

3 20

0

( , ) ( , , ) .eT z t U f U z t dU∞

= ∫ (10)

The density of electrons flux and the density of the flux of electrons energy are de-termined by the integrals of the DFEE anisotropic part:

10

1 2( , ) ( , , ) ,

3ze

j z t Uf U z t dUm

∞

= ∫ (11)

21

0

1 2( , ) ( , , ) .

3ue

j z t U f U z t dUm

∞

= ∫ (12)

584

Preliminary computations show that in the given region of parameters under con-sideration (the reduced electric field E/p ~ 8 V/cm/Torr), variation of the particles num-ber at the expense of impact ionization is insignificant. The ionization of argon atomsbegins to play its role for E/p > 10 V/cm/Torr. In the given model, the ionization can,however, be easily handled. Accounting for that only the conservative terms enter theintegral of inelastic collisions, we obtain the equations for the balance of particles andenergy at the corresponding integration of equations (3) and (4) over the energies:

( , ) ( , ),e zn z t j z t

t z

∂ ∂= −

∂ ∂ (13)

0( , ) ( , )

( , ) ( , ) ( , ) ( , ),el ine uz k

k

u z t j z te E z t j z t P z t P z t

t z

∂ ∂= − − − −

∂ ∂ ∑ (14)

where the term of energy loss in elastic collisions

20

0

( , ) 2 2 / ( ) ( , , )el dee

mP z t m U NQ U f U z t dU

M

∞

= ∫

and the term of energy loss at the kth inelastic collision

00

( , ) 2 / ( ) ( , , ) .in in ink k e kP z t U m UNQ U f U z t dU

∞

= ∫

Equations for the particles (13) and energy (14) balance must be satisfied at any spatialpoint z under consideration at any time t and serve a criterion for the correctness of ob-tained solutions. At the numerical computations, the initial and boundary conditions, thesteps in time �t, in the energies �U and in the coordinate �z, the values of Umax, zmax aswell as the external parameters (the values of electric field E and of the gas density Ng)were chosen in such a way that the relative deviation from the satisfaction of balances(13) and (14) did not exceed one percent.

Figure 3 shows the distributions of the electrons density and the rate of change ofthe electrons density at different moments of time 0.168 (�), 1.68 (b), 3.37 (c), 8.43 (d) �s.Figure 3, � corresponds to Fig. 2, b. At the initial moment of time, the electrons startaccelerating in the electric field, a gradient of the density of electrons flux arises. Ac-cording to the equation of the particles balance (13) the electrons density starts varying.The rate of change of the electrons density is presented in the lower graphs of Fig. 3. It isseen from the figures that the maximum rate of change of the electrons density shiftswith time into the region with a higher value of z. At t���������� �������������������the electrons density near the PC cathode boundary becomes nearly equal to zero. Att > 8.43 �s, the local density of electrons stops vary everywhere. The gradient of the den-sity of electrons flux becomes equal to zero, and the electron current density is1 mA/cm2 in accordance with the normalization on the PC cathode side. The final distri-bution of electrons density is shown in Fig. 3, d. The formed equidistant strata are visi-ble, whose amplitude weakly decays with the increasing coordinate. As was alreadynoted, the spatial decay occurs because of energy losses in elastic collisions. With thegas pressure increase the energy losses in elastic collisions increase, a stronger decay ofstrata occurs towards the anode. In argon at the pressure p > 2 Torr, the energy losses inelastic collisions prevail, and the strata do not develop in the given model.

As can be seen from the results of the present work, at a low gas density the distri-bution of electrons density in the PC is strongly non-uniform, and it is to be supposedthat the distribution of density of ions, which are much slower than the electrons, mustdiffer from the distribution of electrons density. Due to this, a non-uniform resonance

585

distribution of the electric field must arise, in which the hump of the distribution functionwould transform to narrow peaks moving along the resonance trajectories (the so-calledbunching effect [18]). It is necessary to consider the electric field at a self-consistentlevel for a more detailed description of the given effect.

To investigate the properties of the electrons distribution function in a sign-changing electric field the computations were carried out in a model field with the meanvalue −E0 = 3 V/cm and a depression at z = 3 cm, −E(z = 3 cm) = −0.5 V/cm (Fig. 4, �).For a spatially non-uniform electric field, it is difficult to choose the initial distributionof the DFEE isotropic part. At a non-uniform distribution of the electric field, it is neces-sary to know the initial spatial and energy distribution of the DFEE, which has, generallyspeaking, no local dependence of the field. At a specification of an inadequate initialDFEE function (for example, the uniform one) for this field distribution (see Fig. 4, �),the solution of equation (6) by the given method diverges even at a specification of

Fig. 3. Distribution of electrons density and the rate of its change at different moments of time.0.168 (�), 1.68 (b), 3.37 (�), 8.43 (d) �s.

586

a very small time step. In order toavoid this problem the DFEE initialdistribution and the electric field dis-tribution were set to be spatially uni-form. The initial constant electricfield –E(z) = 3 V/cm was slowly trans-formed to the distribution shown inFig. 4, �:

( )2( , ) 3 3.5exp ( 3)E z t z− = − − − ×

( )1 exp( ) .rt t× − − (15)

The quantity tr was chosen to bemuch higher than the time step,tr >> �t, and exceeded the time of theDFEE formation at a given point

��������������������� ��E/Ng). In the course of numerical computation of equation (6)in time one can assume the field distribution quasi-stationary at each moment of time.The results presented below are the converged solution of the steady equation (6) for astationary electric field at the moment of time t���!�����

Figure 5 shows the logarithm of the DFEE isotropic part in the coordinate planeU − z for the given electric field distribution. It is seen that the FDEE is narrower in en-ergy in the field depression region than in other intervals of the solution region. Thestrata formed after the field depression are also visible in the zone of a high electric field.The presented electric field distribution imitates the field distribution in the FDS regionand the following positive column of a glow gaseous discharge. The electron beamentering from the cathode side is thermolized in the FDS region with a low and even

Fig. 4. Model distribution of the electricfield E(z) with a depression at z = 3 cm(�), the final distribution of the electronsdensity ne(z) (b), the final distributionof the mean temperature of electrons Te(z) (c).

Fig. 5. Logarithm of the distribution of the DFEE isotropic part for the model field case (see Fig. 4, �).

587

negative value of the electric field, and thenit gets into the region with some field valuein the discharge positive column. Figure 4, �shows the distribution of the mean tem-perature of electrons Te(z) = 3/2ue(z).

Of special interest is the electron cur-rent behavior in the case of a sign-changingelectric current, taking into account the factthat the drift component of the electron cur-rent is proportional to the electric fieldvalue. Let us present the expression for the density of electrons flux jz(z, t) in the form(following from expressions (5) and (11))

0

0

( , , )1 2( , )

3 ( )ze

f U z tUj z t dU

m H U z

∞ ∂= − +

∂∫

00

0

( , , )1 2( , ) .

3 ( )e

f U z tUe E z t dU

m H U U

∞ ∂+

∂∫ (6)

Here the first term represents the diffusion component of electron current jd(z, t), thesecond one represents the drift component jf (z, t), jz(z, t) = jd(z, t) + jf (z, t). Figure 6shows the final distributions of the electron flux density, of the drift jf (z, t) and diffusionjd(z, t) components. It is seen that the drift component of current becomes negative andhas its minimum at z = 3 cm. The final stationary distribution of the electron flux densityjz(z) is constant along z (cf. (13)). It was pointed out in the paper [12] that the longitudi-nal diffusion of electrons in the discharge in the FDS region may take the function ofelectric current transfer in the region of an abrupt drop of electrons density behind thedensity maximum. Figure 4, b shows the distribution of electrons density. The densitymaximum indeed lies at z < 3 cm, and the maximum gradient of the electrons densityand, respectively, the maximum of electrons diffusion lies at z = 3 cm.

CONCLUSIONS

The model for solving the unsteady non-local kinetic Boltzmann equation pre-sented in the work is applicable for description of the spatiotemporal evolution of thedistribution function of electrons in a weakly ionized plasma under the electric field ef-fect. The unsteady Boltzmann equation for the DFEE isotropic part written in naturalcoordinates the kinetic energy ⎯ the coordinate was solved by the pseudo-unsteadymethod under the corresponding initial and boundary conditions.

The spatiotemporal evolution of the DFEE and the main macroscopic parameters ofelectrons in a given constant electric field have been obtained with the aid of the model.The development of a stratified DFEE distribution is caused by the given energy distri-bution of the electron beam entering the positive column, by the periodic accumulationof electrons energy in the electric field, and by the energy loss in different electron-atomcollisions. The results of the present work were compared with the solution of theunsteady Boltzmann equation by the method of [9]. The final steady solution and the

Fig. 6. Distribution of the density of electronflux jz(z) (1), of the diffusion jd (z) (2) and driftjf (z) (3) components of the electron current.

588

states intermediate in time correspond completely to one another in both models. It isworth mentioning that the CPU time in the presented model is less than the time of com-puting by the Monte-Carlo method, but exceeds the time for solving the Boltzmannequation in the variables “total energy ⎯ coordinate”. However, at a nonmonotonousdistribution of the potential the model of the solution of the Boltzmann equation devel-oped in [3] in the variables “the total energy ⎯ the coordinate” becomes inapplicablebecause the computational region is partitioned into the subregions with potential wells.

For the given sign-changing distribution of the electric field, the correspondingDFEE distribution, the distribution of the density and mean temperature of electronsheve been found with the aid of the model. It is shown that in the region with oppositefield directed from the anode to the cathode the longitudinal transfer of electric current isperformed by the diffusion of electrons. The author of paper [12] indicated such amechanism of the electric current transfer in the region between the cathode layer andthe positive column with negative electric field.

The models and approaches mentioned in the work enables the DFEE determina-tion in experimentally determined or a priori given electric fields. Along with the kinet-ics of electrons it is necessary to consider also the ions. The electric field in plasma insuch an approach will be determined by the distribution of charges and will be found in aself-consistent way from the Poisson equation. The authors of [19] made an attempt todescribe the stratification effect at a self-consistent level: the model was based on a cou-pled solution of the Boltzmann kinetic equation for the DFE by the method of [3], of theunsteady continuity equation for ions, and the Poisson equation for the self-consistentelectric field. Under certain conditions, a resonance sign-changing electric field arose: inthe stratum region, the field transformed into narrow peaks, and in the region betweenthe strata the field became negative, which led to the divergence of the solution of theBoltzmann equation by the method of [3]. At an attempt to describe the phenomenon ofthe gas discharge stratification with the aid of a hybrid model based on the solution ofthe Boltzmann equation and the solution of unsteady drift and diffusion equations forions and electrons jointly with the Poisson equation [20−21], the negative electric fieldalso arose in the FDS region and in the PC between strong strata. The model described inthe present work was just developed for describing the behavior of electrons in sign-changing electric fields.

REFERENCES

1. L.D. �sendin, Electron kinetics in non-uniform glow discharge plasmas, Plasma Sources Sci. and Technol.,�""! �#����$ �%���� �������−211.

2. V.I. Kolobov and V.A. Godyak, Nonlocal electron kinetics in collisional gas discharge plasma, IEEETrans. Plasma &��� ��""! �#�����' �%���$ ����!�'−531.

3. F. Sigeneger and R. Winkle, Response of the electron kinetics on spatial disturbances of the electric fieldin nonisothermal plasmas, Contrib. Plasma Phys., 1996, Vol. 36, No. 5, �. 551−571.

4. I.B. Bernstein and T. Holstein, Electron energy distributions in stationary discharges, Phys. Rev., 1954,Vol. 94. �. 1475–1482.

5. L.D. Tsendin, Distribution of electrons over energy in weakly ionized plasma with current and transversenon-uniformity, Sov. Phys.-JETP, 1974, Vol. 66, Iss. 5, P. 1638−1650.

6. Yu. B. Golubovskii, V.A. Maiorov, V.O. Nekuchaev, J. Behnke, and J.F. Behnke, Kinetic model ofionization waves in a positive column at intermediate pressures in inert gases, Phys. Rev. E, 2001, Vol. 63,P. 036409-1–10.

7. Yu. B. Golubovskii, V.A. Maiorov, I.A. Porokhova, and J. Behnke, On the non-local electron kinetics inspatially periodic striation-like fields, J. Phys. D: Appl. Phys., 1999, Vol. 32, �. 1391−1400.

8. F. Sigeneger, G.I. Sukhinin, and R. Winkler, Kinetics of the electrons in striations of spherical glow dis-charges, Plasma Chem. Plasma Proc., 2000, Vol. 20, No. 1, �. 87−110.

9. D. Loffhagen and R. Winkler, Spatiotemporal relaxation of electrons in non-isothermal plasmas, J .Phys.D: Appl. Phys., 2001, Vol. 34, �. 1355−1366.

10. Yu.P. Raizer and M.N. Shneider, The non monotonicity of transition from Faraday dark space to positivecolumn and emergence of standing strata behind the cathode region of glow discharge, High Temp., 1997,Vol. 35, No. 1, P. 19−24.

589

11. J.P. Boeuf and L.C. Pitchford, Field reversal in the negative glow of a DC glow discharge, J. Phys. D:Appl. Phys., 1995, Vol. 28, �. 2083−2088.

12. Yu.P. Raiser, Current level of the understanding of phenomena in cathode parts of glow discharge, HighTemp., 1986, Vol. 24, P. 984−993.

13. Yu.P. Raizer, Gas Discharge Physics, Springer-Verlag, Berlin et al., 1991.14. M.O.M. Mahmoud and M. Yousfi, Boltzmann equation analysis of spatiotemporal electron swarm devel-

opment, J. Appl. Phys., 1997, Vol. 81. P. 5935.15. W.L. Morgan, J.P. Boeuf, and L.C. Pitchford, Siglo Data Base, CPAT and Kinema Software, 1998, http,

www.csn.net/siglo.16. L.D. Tsendin, Kinetics of ionization and ionization waves in neon, Sov. Phys-JETP, 1982, Vol. 52, Iss. 4,17. B.N. Klarfeld, Formation of strata in gas discharge, Sov. Phys.–JETP, 1952, Vol. 22, Iss. 1, P. 66–77.18. Yu.B. Golubovsky, I.A. Porokhova, J. Behnke, and V.O. Nekutchaev, On the bunching effect of elec-

trons in spatially periodic resonance fields, J. Phys. D: Appl. Phys., 1998, #����'� �����$$(−2457.19. A.V. Fedoseev and G.I. Sukhinin, Self-consistent kinetic model of the stratification plane and spherical

low-pressure discharges in argon, Plasma Phys., 2004, Vol. 30, No. 12, P. 1061−1070.20. A.V. Fedoseev and G.I. Sukhinin, Spherical glow discharge. Drift-diffusion approximation, Thermo-

physics and Aeromechanics, 2003, Vol. 10, No. 1, P. 61−67.21. G.I. Sukhinin and A.V. Fedoseev, Abnormal and subnormal regimes of a spherical glow discharge in the

drift-diffusion approximation, Plasma Phys. Repts., 2003, Vol. 29, No. 12, P. 1062−1069.