Multiple solutions in the Te–Ne parameter space of argon CR model via Monte Carlo optimization

The accurate determination of the electron temperature (Te) and density (Ne) is critical for understanding argon plasma characteristics, yet inverse solutions to the collisional-radiative (CR) model often suffer from non-uniqueness, complicating data interpretation. The core contribution of this study is the visualization of band-like solution structures within the CR inverse problem and the development of a practical strategy for handling this non-uniqueness. The results reveal band-like residual regions in the Te-Ne space, representing continuous zones where fitting residuals are insensitive to coupled Te-Ne variations, thereby corresponding to multiple solutions. These regions exhibit a change in monotonicity at Te ≈ 1.77 eV under the Maxwellian assumption, reflecting a transition in the dominant electron-collision kinetics governing the excited states. The structures of these regions are highly sensitive to spectral line selection and EEPF assumptions: higher excited-level sets shift the solutions toward higher Ne to balance the relative population distribution, whereas EEPF assumptions with greater high-energy depletion shift them toward higher Te to maintain the necessary excitation rates. The proposed optimization approach efficiently identifies these solutions while maintaining physical consistency. This strategy resolves the non-uniqueness of the inverse problem, ensuring reliable tomographic determinations of Te and Ne in argon inductively coupled plasmas.