Electron and metastable spatial distributions for positive columns with ionization from a metastable state

The spatial distributions of electrons and metastables, n /sub e/ and n /sub M/, for a positive column with direct and two-step ionization are governed by two coupled continuity equations, -D /sub e/ 2n /sub e/ = n /sub e/ (ν /sub i/ + C /sub Mi/ n /sub M/ = n /sub e/ (ν /sub M/ - C /sub M/ n /sub M/ ), whose solutions depend only on two parameters: P is identical to C /sub M/ n /sub M/ (O)/ν /sub M/, Q is identical to ν /sub M/ C /sub Mi/ /ν /sub i/ C /sub M/. At low currents P→O while at high currents the concentration of metastables saturates and P→1. Solutions were obtained for a wide range of values of Q and Oless than or equal toP < 1, for cylindrical and planar geometries and for vanishing densities at the boundary. For P = O n /sub e/, n /sub M/ take the usual Bessel (cosine) shape. As P increases n /sub e/ is slightly modified but returns to the Bessel (cosine) shape as P→1; meanwhile n /sub M/ becomes progressively flatter. The average densities n /sub e/, n /sub M/ can be charaterized by appropriately defined diffusion lengths Λ /sub e/, Λ /sub M/. Λ /sub e/ → R/2.4(L/π) as P→O and P→1; Λ /sub M/ → R/2.4 (L/π) as P→O and diverges as P→1. These results together with Boltzmann calculations provide similarity laws of E/N vs. NR for I/R = const and of n /sub M/ /N vs. I/R for NR = const; an application to argon is discussed