A CURRENT-VOLTAGE CHARACTERISTIC OF NON-SELF-MAINTAINED DISCHARGE

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A CURRENT-VOLTAGE CHARACTERISTIC OF NON-SELF-MAINTAINED DISCHARGE

J. Londer

To cite this version:

J. Londer. A CURRENT-VOLTAGE CHARACTERISTIC OF NON-SELF-MAINTAINED DISCHARGE. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-203-C7-204.

�10.1051/jphyscol:19797100�. �jpa-00219070�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ&ment au n07, Tome 40, JuiZZet 1979, page C7- 203

A CURRENT-VOLTAGE CHARACTERISTIC OF NON-SELF-IMADJTAINED DISCHARGE.

J.I. Londer.

AZZ-Union EZectrotecknicaZ I n s t i t u t e , Moscow U.S.S.R.

'I. The analytical expression of the non- -self- maintained current-voltage charac- teristic is obtained. The ion current flo- wing into cathode sheath from plasma is impotant factor in the considered model, This ion current was not taken into consi- deration in the classical Thomson's paper /I/, when the analogous problem was solved In paper /2/ the ion current fpom plasma was taken into consideration for construc- tion of probe theory in non-self-maintain- ed discharge, but the rate of ionization in the cathode sheath was not taken into account.

Let us consider the plane system of el- ectrodes. We'll solve equations for plas- ma and cathode sheath separately and then make join of solutions on the plasma- -sheath boundary.

2, The equations for plasma region are given by:

- - -

Here Q_ is the rate of ionization created by extgrnal source, d, is the effective recombination coefficient. Index k de- pends on the type of the recombination.

We'll consider the cases when k-2 and 1.

Take the origin ,of coordinates at the plasma-sheath boundary and introduce non-

dimensional concentration B and nondimen- sional length y:

N = LiL

.

Here is ambipolar diffusion- coeff ici-

ent, Zd is the recombination length, is unperturbed plasma concentration.F~om (1) and (2), taking into account (5), we get:

4'

&a&'

,A

z - dlv ⁼ f -fi, ^-N, = i,' / V $ / r - N ) @ $ i ~ = 2 ( 8 1 Eere d%=J2e&dl/d+4) is the elect6ic field

- " "

in far from boundary. Integrating (7), we can obtain the potential drop on the plasma. _-

-f 2@

-2

V,Ew&-xo)+63_z,h~

Here L is the anode-cathode distance, xo is the cathode sheath thickness, No is the concentration on the plasma-sheath b o m d ~ y .

3. If the heat energy of electrons is far less than the cathode drop, then el- ectrons will be found in the sheath in the strong repulsive field and ni>> ne on the whole sheath except narrow region near plasma-sheath boundary. Therefore we can neglect n, in comparison with ni

-

in Poisson's equation for sheath. The distribution of the field and ni must be determined from joint solution of equa- tions:

with boundary cindition ji(xo)=(l+~"jb

b/

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797100

under bombardment of the cathode by ions.

The solutions must be joined at the plasma-sheath boundary. So, joining deri- vative of the field, we obtain algebraic equation for ion concentration at the bo- undary (ma):

Substituting the value No, found from (13), in (6) and (7) k e obtain values of ion current jo and electric field Eo on.

the plasma-sheath boundary, These values jo and Eo will be boundary conditions for equations (10)-(12). Introducing nondi- mensional field z=Eh0, nondimensional coordinate t=l+eqx/ and nondimensional

10 -2

parameter ~ ~ = % t $ e ~ ~ d jo , we integrate

Now we can find the cathode c G p V : d i f

Kg2dx

,$&-&

~ubgtitutin~ (14) in (16) we obtain equa- tion of the current-voltage characteris- tic of the cathode sheath.

The behaviour of the expression (17) de- pends greatly upon the value 8

.

of (6) and (7) it can be obtained that:

(18) The value of parameter S is less or about 1 in electropositive gases at the pressures up to several atmospheres.Under this condition it can be neglected loga- rith&Lc term in' expression (17). In ele- ctronegative gases, beaause of strong at- tachment of electrons, the parameter 6

can become mueh lmger than I,In this case it cann9t be neglcsoted logarithmic term in (17)- However, there are two li- mits, when (17) can be aimplkfied inde-

pendently of k. The first limit takes pla- ce if:

tmax >> mar

p; s2]

The second limit takes place if:

(tmax -I ) -<< min-[f; 8

'1

From (17) and (14) we obtain for cathode drop Vc and electric field Ec near catho- de: if the criterion (19) is carried out then known expressions take place /I/:

If the csiterion (20) is carried. out, then we have:

4. Using (6), (15) and (16) we can ob- t ain:

~i=h$-

5'7

One can see from (23); that there are two characteristic lengths xo and of forming current in considered model.

Under xo>> %$'the discharge current is formed mainly in the cathode sheath regi- on, and under x o 4 ~ % % h e characteristic length of forming current is 5% ^{.In the}

first case tma,>> 1 and criterion (19) can be carried out; in the second case

-1 ) << I and criterion (20) can be ( tmax

carried out, If criterion (19) takes place, then transition region with length

~,r/,/can be neglected and the sheath can be joined with plasma without considera- tion jo and Eo, that was made in /I/, If criterion (20) is carried out, ionization in the cathode sheath can be neglected in comparison with ioniaation on the length _-

%No/, that was made in /2/. And general expression (17) must be used for deter- mining PC in the intermediate case when the values xo and zdZ,/V,' have the same or- der of magnitude.

'Phe author thanks K.N.Ulyano~ for useful discussions.

References

1. J.J,Thornson, G.P.Thomson. Conduction of Electricity through 5ases.v.l,ehap.IV

2. K.l'?.Ulyanovr XI11 Intern,Conf.on Phenom-in Ioniz,Gases. p.113 Berlin1977