Thermodynamically Consistent Vibrational-Electron Heating: Generalized Model for Multi-Quantum Transitions

Abstract

Accurate prediction of electron temperature ($T_{\mathrm{e}}$) is critical for non-equilibrium plasma applications ranging from hypersonic flight to plasma-assisted combustion. We recently proposed a thermodynamically consistent model for vibrational-electron heating [Phys. Fluids 37, 096141 (2025)] that enforces the convergence of $T_{\mathrm{e}}$ to the vibrational temperature ($T_{\mathrm{v}}$) at equilibrium. However, the original derivation was restricted to single-quantum transitions, limiting its validity to low-temperature regimes ($T_{\mathrm{e}}\lesssim 1.5$ eV). In this Letter, we generalize the model to include multi-quantum overtone transitions, extending its applicability to high-energy regimes. We demonstrate that previous models neglecting hot-band transitions incur a systematic heating error of $\exp (-\hat{I}{¸}_{\mathrm{v}}/T_{\mathrm{v}})$, where $\hat{I}{¸}_{\mathrm{v}}$ is the characteristic vibrational temperature. This error exceeds 40% when $T_{\mathrm{v}}$ is greater than $\hat{I}{¸}_{\mathrm{v}}$, effectively preventing thermal relaxation. To correct this, we derive a formulation where the total heating rate is a summation of channel-specific cooling rates $Q_{\mathrm{e}-\mathrm{v}}^{(m)}$, each associated with a quantum jump $m$, scaled by a thermodynamic factor $\exp (m\hat{I}{¸}_{\mathrm{v}}/T_{\mathrm{e}}-m\hat{I}{¸}_{\mathrm{v}}/T_{\mathrm{v}})$. This generalized model preserves thermodynamic consistency by ensuring zero net energy transfer at equilibrium.

4 pages, 1 figure