Theory of cylindrical Langmuir probes in weakly ionized, non-thermal, stationary and moderately collisional plasmas

Abstract

A finite length cylindrical Langmuir probe is modelled as an ellipsoid of revolution with spheroidal equipotential surfaces and confocal orthogonal hyperboloidal electric field lines. The theory is applicable in the transition regime of probe operation between the collisionless and fully collisional limits. The plasma is assumed to be weakly ionized, non-thermal and stationary, being characterized by frozen reactions and constant temperatures. It is further assumed that in an isotropic plasma the cold ions follow the field lines, as a result of ion–neutral collisions, in the presheath and sheath regions with collisionless Maxwellian electrons. The governing system of equations is derived and solved numerically with the results presented of the presheath and sheath solutions in collisionless and collisional regimes. These show convergence to the respective collisionless and collisional radial motion limits for spherical and cylindrical probes. Analytical approximations are also obtained for the sheath width (defined as the point where the ions reach the Bohm speed) and the Bohm potential over a wide range of collisionality. The collisional presheath drop according to the perturbation theory of Shih and Levi, as applied to cylindrical probes, is shown to significantly underestimate the numerical results. These are in better agreement with the collisional presheath drop for spheres even for long probes. Application of the theory to experimentally derived probe characteristics is also discussed.