Non-local thermodynamic equilibrium stellar spectroscopy with 1D and <3D> models - I. Methods and application to magnesium abundances in standard stars

We determine Mg abundances in 6 Gaia benchmark stars using theoretical one-dimensional (1D) hydrostatic model atmospheres, as well as temporally- and spatially-averaged 3D model atmospheres (<3D>). The stars cover a range of Teff from 4700 to 6500 K, log g from 1.6 to 4.4 dex, and [Fe/H] from -3.0 dex to solar. Spectrum synthesis calculations are performed in local thermodynamic equilibrium (LTE) and in non-LTE (NLTE) using the oscillator strengths recently published by Pehlivan Rhodin et al. We find that: a) Mg abundances determined from the infrared spectra are as accurate as the optical diagnostics, b) the NLTE effects on Mg I line strengths and abundances in this sample of stars are minor (although for a few Mg I lines the NLTE effects on abundance exceed 0.6 dex in <3D> and 0.1 dex in 1D, c) the solar Mg abundance is 7.56 +/- 0.05 dex (total error), in the excellent agreement with the Mg abundance measured in CI chondritic meteorites, d) the 1D NLTE and <3D> NLTE approach can be used with confidence to analyse optical Mg I lines in spectra of dwarfs and sub-giants, but for red giants the Mg I 5711 A line should be preferred, e) low-excitation Mg I lines are sensitive to the atmospheric structure; for these lines, LTE calculations with <3D> models lead to significant systematic abundance errors. The methods developed in this work will be used to study Mg abundances of a large sample of stars in the next paper in the series.