Simplified coefficients in the Pekarek-Lee theory of moving striations

Volume 49A, number 6

PHYSICS LETTERS

21 October 1974

SIMPLIFIED COEFFICIENTS IN THE PEKAREK—LEE THEORY OF MOVING STRIATIONS M. SHIMADA Department of Physics, Osaka Kyoiku University, Osaka, Japan Received 2 July 1974 The coefficients in the Pekarek—Lee theory of moving striations are given in more simplified forms by making inwstigation into the equation of energy balance for electrons.

Moving striations are the low frequency waves which are universally observed in the weakly ionized plasma, and many investigations about them have been made experimentally and theoretically [1—3]. The theory originally presented by Pekarek [4] and furthermore developed by Lee et al. [5] is, in spite of its approximate treatment, one of the most excellent theories that describe the phenomena of moving striations under various discharge conditions. The theory comprises not only the backward positive striations usually observed in the rare gas discharge of low pressure, but also the for~vardpositive waves found in the constricted channel of medium pressure [6]. It is the object of the present letter to show that the coefficients a, b, c1 and c2 in their theory can be written in more simplified forms with discharge parameters by investigating the equation of energy balance forThe electrons by Granowski [7]. energygiven balance for electrons in a positive column plasma is expressed by the following equation dUe dx

—

2

8

3E

~

tron temperature in the stationary state. When there are moving striations, the field and the electron temperature are described with their small deflections fromt he stationary state, —

e

(1)

where Ue = kT~/qis the electron temperature in mean volt, 1~eis the E is the intensity of axial electric field, free path of electrons, K is the mean energy loss at impact with a neutral gas atom, and the direction of x is chosen opposite to the electric field. When no striations exist, electrons attain to the energy balance after traveling sufficiently in the electric field: dUe/dX = 0. From eq. (1), therefore,we have ~ (2) eO

whereE~and U~are the axial electric field and the elec-

e

0

where UeO ~ 0 I and E0 ~ I e I. Substituting the above quantities into eq. (1), we get the balance equation derived by Pekarek et al. [8] dO/dx—aO=—be.

(4)

In this equation, the direction of x is chosen to the electric field and coefficients a and b are given as follows 16 2 a= U~K/E0X~ (5) 2 8 —

b ,,~2 ‘

e +

E=E

=

+—

U~/E~X~.

With substitution of eq. (2), we have simplified forms a = ~EOIUeo b 4 (6) ,

~.

The growth rate 0 and the dispersion relation of moving striations are derived by Lee et al. [5] ac 2 2+C =

~Daa

1—

2

2+o2), a +a

2

(7) (8)

wc2a/(a where Da is the ambipolar diffusion coefficient, a is the wave number, w is the angular frequency, and the 455

Volume 49A, number 6

PHYSICS LETFERS

coefficients c1 and c2 have been defined c1 =Z~jbUe0,

c2

=

Z1b(aUeo +E0).

(9)

By using our simplified a and b, the coefficients c1 and c2 also can be rewritten in simplified forms ci4Z~L4eü, c 2=~Z~E0, (10) where Z’ is the slope of the dependence of the ionization frequency Z on the electron temperature. The ionization frequency Z(sec~)is given by von Engel [9] in the following equation exp 2 ( ViIUe) (11) Z 6.69 X 10~aPVjUe~ 1cm~)is the initial slope of the where ~ (V~Torr curve of ionization efficiency, P(Torr) is the gas pressure, and V~(V)is the ionization potential. The wave number a~,and wave length X~,of the dominant moving striations that have the maximum growth rate are derived from eq. (7) 2 2 2ir~ 4E0 4E0 Up = = ~ — ~ ~). (12) The phase velocity (Vph)P of the dominant waves is also given by eq. (8) (Vph)P = U~0\T7Z~pj/3 ,

(13)

where ,u~is the ion mobility. The frequency of the dominant waves is obtained by the relationf~= (VPh)PfAp. The square of wave number being positive, it is concluded from eq. (12) that the dominant waves exist under the following condition ‘2 16 (14) /.ti

21 \UeO /

has a positive sign for a > a, and a negative sign for a < a. The dominant moving striations are, therefore, classified into the following two types: Forward positive striations 2for 64 (Eo ~2 z~ 16 (Eo)

(16)

,

21 U~i i~ 21 backward positive striations for 64 —> ~

( u—~ ~

(17)

the characteristics of moving can be Thus, described with the axial electric fieldstriations E 0, the electron Ueo, thethe gasionization pressure Ppotential and the kind of gastemperature (which determines V 1, the initial slope of ionization efficiency the ion mobility for 1 Torr jt11). ~,

References [1] G. Francis,

Handbuch der Physik ed. S~FlUgge (Springer,

Berlin), Vol. 22 (1956) 140. [21 R.S. Palmer and A. Garscadden, United States Air Force Report, ARL65-120 (1965). [3] N.L. Oleson and A.W~Cooper, Advances in Electronics and Electron Physics, ed. L Marton (Academic Press, New York and London), Vol. 24 (1968) 155. [4] L. Pekarek, Proc. VI littern. Conf. on Ionization phenomena in Gases ed. P. Hubert (SERMA, Paris), Vol. 2 (1963) 133. [5] D.A. Lee, P. Bletzmger and A. Garscadden, J. Apprl. Phys. 37 (1966) 377. [6] A. Garscadden and D.A. Lee, Intern. J. Electron 20 (1966) [7] 567. W.L Granowski, Der elektrische Strom im Gas (AkademieVerlag, Berlin, 1955) 289.

And the dominant waves are spontaneous in case of >0. The group velocity derived from eq. (8) 2—a2)/(a2+a2)2 (15) V~=c2(a ,

456

21 October 1974

[8] andIonized V. Kejci, Czech. J. Phys. B12 (1962) [9] L. A. Pekarek von Engel, gases (Oxford, Clarendon Press,296.

1965).