Defect of the well-known (classical) expression for the ionization rate in gas-discharge plasma and its modification

Abstract

A critical analysis is given of the well-known expression for the electron-impact ionization rate constant αi of neutral atoms and ions, derived by linearization of the ionization cross section σi(e) as a function of the electron energy near the threshold I and containing the characteristic factor (I + 2kT). Using the classical Thomson expression for the ionization cross section, it is shown that in addition to the linear slope of σi(e), it is also necessary to take into account the large negative curvature of this function near the threshold. In this case, the second term in parentheses changes its sign, which means that the commonly used expression for αi (∼4kT/I) already at moderate values of the temperature (kT/I ∼ 0.1). The source of this error lies in a mathematical mistake in the original approach and is related to the incorrect choice of the sequential orders of terms small in the parameter kT/I. On the basis of a large amount of experimental data and considerations similar to the Gryzinski theory, a universal two-parameter modification of the Thomson formula (as well as the Bethe—Born formula) is proposed and a new simple expression for the ionization rate constant for arbitrary values of kT/I is derived.