Escape factors for Paschen 2p-1s emission lines in low-temperature Ar, Kr, and Xe plasmas

Radiation trapping phenomenon is often observed when investigating low-temperature plasmas. Photons emitted from the upper excited states may be reabsorbed by the lower states before they leave the plasmas. In order to account for this effect in the modelling and optical diagnostics of plasmas, either an 'escape factor' of a function of the optical depth or a strict solution of the radiation transfer equation can be employed. However, the former is more convenient in comparison and thus is widely adopted. Previous literatures have provided several simple expressions of the escape factor for the uniform plasmas. The emission line profiles are assumed to be dominated by the Doppler broadening, and the line splitting due to the hyperfine structure is not considered. This kind of expression is only valid for small atoms, e.g. Ar in low-pressure uniform discharges. Actually, the excited state density in many of the low-temperature plasmas is non-uniform and the emission line profile can be significantly influenced by the collisional broadening at medium and high pressures. In these cases, a new escape factor equation should be calculated. In this work, we study the escape factor equations for the often used 2p-1s transitions (Paschen's notation) of the Ar, Kr, and Xe atoms. Possible non-uniform density profiles are considered. In addition, we include the line splitting due to the hyperfine structure for Kr and Xe. For the low-pressure plasmas, an escape factor expression mainly based on the Gaussian line profile is given and particularly verified by an experiment in a low-pressure capacitive discharge. For the high-pressure plasmas, an equation based on the Voigt line profile is also calculated. In this way, the new escape factor expression is ready for use in the modelling of the Ar, Kr, and Xe plasmas from low to atmospheric pressure.