The calculation of the population density of excited levels of ions or neutrals in a plasma is usually complicated. However, it can be simply done in some cases. The first case is called (partial) local thermodynamic equilibrium (LTE/pLTE) that is valid when the electron density is high and the electron temperature is relatively low. In this case, the population densities of excited levels are determined by the Boltzmann relation. The second case is called corona model that is valid when the electron density is low and the electron temperature is high. In the corona model, all the excited levels are populated by the electron impact excitation from the ground level and the depopulation of excited levels is achieved by radiative transitions.

However, these two cases rarely occur in semiconductor plasma processing and laboratory plasmas. In these plasmas, a collisional-radiative (CR) model that considers all the collisional and radiative processes occurring in a plasma must be used. The excited levels of neutrals or ions can be populated not only by the direct excitation from the ground level but also by the excitation from other excited levels. The depopulation of the excited level occurs not only by radiative transitions but also by excitation/deexcitations. In addition, under certain conditions, the metastable diffusion should be considered and the emitted photons from radiative transitions are reabsorbed before they escape the plasma which is known as radiation trapping.

We developed an argon neutral CR model [Chai and Kwon, JQSRT 227, 136 (2019).]. In our CR model, the ground level and 14 excited levels are considered including two metastable levels. The population density of the 14 excited states (N_i) is obtained by solving the rate balance equation below.

Here, α^ex, α^I, and ν^d are the rate coefficients of excitation, ionization (direct, stepwise, heavy particle collision), and metastable diffusion loss, respectively. And A and η are the transition probability and the escape factor related to the radiation trapping. The terms on the left hand side represent the populating term and the terms on the right hand side represent the depopulating term. On the other hand, in the rate balance equation, the rate coefficient is usually obtained by integrating the cross section of the target reaction with respect to the electron energy distribution function. If the cross section is not available, the rate coefficient can be obtained by using the empirical formula.

In addition to the argon CR model, our group has developed CR models for helium and hydrogen plasmas.