Download the multidimensional plasma-sheath equation for low pressure

XXXII conference on plasma physics and CF, February 14 – 18, 2005, Zvenigorod
THE MULTIDIMENSIONAL PLASMA-SHEATH EQUATION FOR LOW PRESSURE
DISCHARGE POSITIVE COLUMN
Dvinin S.A.
M.V. Lomonosov Moscow State University, Physical faculty, Moscow, Russia,
e-mail: [email protected]
The multidimensional equation for plasma with multidimensional nonuniformity and a spatial
charge layer, generalizing Lengmuir and Tonks equation, is received and the analysis of its
decisions is carried out. The basic assumptions and a method of a conclusion the equations, used in
the work, are similar to suggested by Lengmuir and Tonks [1]. In contrary to one dimensional case
electric field strength lines and ions trajectories do not coincide with each other. The ions density in
a viewing point u0 can be represented as
n u0    du S ne u S  i F u S ,u0  ,
L
where integration goes on curve L. The trajectories of ions, borne in points of this curve pass
through a viewing point. To determine F uS ,u0  , except for quantities, calculated by Lengmuir and
Tonks, it is necessary found divergence of ions trajectories at their movement. A final kind for one
of the simplest form of plasma and sheath equation in cylindrical coordinates r,  is (  e kTe –
dimensionless potential, VS – ionic - sound speed, i – frequency of ionization).
r

r
drS
2
exp   r rDe
  1  i  e  rS ,  r ,  S

r  rS ,    r, 
2VS 0
r
 r
  
dr
dr
 2  


 .
 1 

 r 2  rS ,    r,  r  rS ,    r,   rS  r rS r 2  
S
S


If azimuthal derivatives are equal to zero, this equation coincides with result of Lengmuir and
Tonks [1]. Additional composed describe growth of ions density due to azimuthal drift (because
ions come from area with higher electron density) and azimuthal smoothing of potential and both
processes appear nonlocal as depend on potential properties in all points which the ion has visited.
Ionization frequency of is own value of the equation and depends on discharge chamber geometry.
The given equation can be a basis for calculation of spatial positive column plasma
characteristics in a condition of free ions run to the wall at the any form discharge boundary. As
against a one-dimensional problem the received equation appears approached, and has small
parameter – the ration of curvature radius of ion trajectory to the characteristic size of a plasma
column. The outcomes are interesting to investigations of processes in ionic sources and plasma
chemical reactors of low pressure.
References.
[1]. Langmuir I., Tonks L. Phys. Rev. 1929, 34, 876.
[2]. Granovsky V.L. Electric current in gas. M.: Nauka, 1971, p. 235 (in Russian).
[3]. Dvinin S.A., Dovzhenko V.A., Kuzovnikov A.A. Plasma Physics Report, 1999, v. 25, No.11,
p. 957.