A steady-state theory of the discharge column is derived which is applicable in the pressure range where the ion mean free path is neither much greater than nor much less than the column radius, and which goes over in the low- and high-pressure limits to the free-fall and ambipolar diffusion theories, respectively. Solutions for planar and cylindrical geometry are given for the density and potential profiles. The plasma-sheath boundary is discussed and the sheath potential drop is estimated. The theory is shown to agree well with measurements in mercury discharges over the pressure range 1–20 μ Hg.

Simple procedures and formulas for tracing the characteristics of a spherical Gaussian beam through a train of lenses or mirrors are described which are analogous to those used in geometrical optics to trace repeated images through an optical train.

The plasma-sheath problem for the low-pressure discharge in plane geometry is treated exactly, that is, with no arbitrary division into plasma and sheath regions. Numerical solutions are presented for various values of the parameter α, which is of the order of the ratio of the Debye length to the discharge width for 10−3 ≤ α ≤ 10−1; and for three assumptions regarding the ion generation rate, namely generation uniform, proportional to electron density, and proportional to the square of electron density. For the higher values of α, corresponding to weak laboratory discharges, there is a smooth transition from a quasi-neutral plasma region to a thick sheath. At the smaller values of α, the conventional model of a quasi-neutral plasma region passing rather abruptly into a narrow sheath region is substantiated. In all cases, accurate values for the potential profile throughout the plasma and sheath regions are given and compared with the separate asymptotic plasma and sheath solutions for α = 0. The ion current density, wall potential, space-charge density, mean ion energy, and sheath thickness are discussed.

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