A general framework for photoionization calculations applied to nonthermal gas discharges in air

A general framework for photoionization rate calculations in a constant pressure gaseous medium is introduced. The formulation includes the number of photons emitted per unit volume per unit time per unit wavelength due to a radiating source, photobasorption cross sections and density of species comprising the medium, and the photoionization probability (i.e. photoionization yield) of the species being photoionized. We derive a standard integral representation of the photoionization problem that may be readily converted to a set of Helmholtz differential equations for efficient calculation of the photoionization rate. The model is applied to the photoionization problem in air in which b¹Π_u, b′¹Σ_u ⁺, c₃ ¹Π_u, o₃ ¹Π_u, c₄′¹Σ_u ⁺ singlet states of N₂ are excited due to collisions with electrons generated as a result of nonthermal discharges. Radiative decay from these states gives rise to respective band systems Birge-Hopfield I, Birge-Hopfield II, Worley-Jenkins, Worley, and Carroll-Yoshino, and to photons which are generally energetic enough to ionize O₂. The excitation rates and contribution of each band system to photoionization of O₂ in air are calculated for the first time. Using recently measured electron impact excitation cross sections of these states, and the recently measured extreme ultraviolet (XUV) spectra of N₂, we quantify the emission from each singlet state. Absorption of emission is modeled using measured photoabsorption cross sections of N₂ and O₂. The photoionization rate of O₂ upon absorption of a photon with a certain energy is calculated using experimental values for the photoionization yield of O₂. Finally, we introduce a set of coefficients which define the differential representation of the problem of photoionization in air. The developed modeling framework allows accurate solution of photoionization problems in air for the broad range 10^-2 < p_O2 R < 10⁴ Torr cm, where p_O2 is the partial pressure of O₂ in air in units of Torr (p_O2 = 152 Torr at atmospheric pressure) and R in cm is a characteristic spatial dimension of the system of interest. The model performance is demonstrated using a set of artificial sources leading to photoionization over a representative range of p_O2 R values and a realistic problem of dynamics of a double-headed streamer in air that was used in previous photoionization literature. The validity of the modeling framework is demonstrated by comparisons with the photo-ion yield function in air, Ψ (p_O2 R), derived from the classic photoionization model of Zheleznyak et al (1982) and more recent experimental data on photoionization in air.