Eur. Phys. J. D (2026) 80: 51
https://doi.org/10.1140/epjd/s10053-026-01161-8

Research - Plasmas

The semi-forbidden Ti I$3\to 2$ line: collisional–radiative modelling and conditions for astrophysical maser amplification

¹ Department of Computer Science, Ben-Gurion University of the Negev, P.O.B. 653, Beer Sheva, Israel
² Department of Physics, Ariel University, 40700, Ariel, Israel

^a This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 17 December 2025
Accepted: 2 April 2026
Published online: 4 May 2026

Abstract

Semi-forbidden metal lines can act both as sensitive diagnostics of dense partially ionised plasmas and, under favourable conditions, as astrophysical maser transitions. In this work, we investigate a semi-forbidden transition in neutral titanium Ti I as a candidate maser line and develop a unified framework that links ab initio atomic-structure calculations, collisional–radiative modelling, and a practical single-line diagnostic for observed spectra. Using DFT/TDDFT supplemented by configuration-interaction calculations with explicit spin–orbit coupling, we construct a compact reduced four-level Ti i subsystem with an intercombination transition $3\to 2$. For this scheme, we determine level energies, statistical weights, oscillator strengths, and Einstein coefficients and use them as input to a stationary collisional–radiative model that includes electron-impact collisions, radiative transitions, and external radiation over a wide range of $(T_{\text{e}},N_{\text{e}},N_{\text{Ti,tot}},L_{\text{eff}},P_{14})$. From the collisional–radiative solution, we obtain maps of population differences and optical depth and analyse how the central optical depth ${\tau }_0$ scales with total neutral-titanium abundance, effective path length, and radiative pumping rate. We show that, for astrophysically realistic parameters representative of a cool stellar atmosphere, the Ti I$3\to 2$ line operates in the regime of ordinary absorption, yet remains sensitive to the onset of population inversion and potential maser amplification in neighbouring regions of parameter space. On this basis, we implement a simple single-line diagnostic in which the observed central depth of the semi-forbidden line is used to infer ${\tau }_0$ and the effective neutral-titanium concentration $N_{\text{Ti,tot}}^{(\mathrm{best})}$ that reproduces the observed line depth (and the profile shape within the adopted broadening model) in the collisional–radiative framework. We demonstrate that the Ti I$3\to 2$ transition can serve as a quantitative probe of dense partially ionised plasmas and as a tool for assessing proximity to maser conditions in objects that exhibit anomalously narrow or unusually strong metal lines. Robustness tests with respect to the uncertain oscillator strength, direct collisional coupling, effective omitted-channel losses, and external pumping show that the qualitative interpretation of the reference observed profile remains stable within the present reduced-model framework.