Context. The study of pre-stellar cores (PSCs) suffers from a lack of undepleted species to trace the gas physical properties in their very dense inner parts. Aims. We want to carry out detailed modelling of N2H+ and N2D+ cuts across the L183 main core to evaluate the depletion of these species and their usefulness as a probe of physical conditions in PSCs. Methods. We have developed a non-LTE (NLTE) Monte-Carlo code treating the 1D radiative transfer of both N2H+ and N2D+, making use of recently published collisional coefficients with He between individual hyperfine levels. The code includes line overlap between hyperfine transitions. An extensive set of core models is calculated and compared with observations. Special attention is paid to the issue of source coupling to the antenna beam. Results. The best fitting models indicate that i) gas in the core center is very cold (7$pm$ 1 K) and thermalized with dust, ii) depletion of N2H+ does occur, starting at densities 5-7E5 cm−3 and reaching a factor of 6 (+13/−3) in abundance, iii) deuterium fractionation reaches ∼70% at the core center, and iv) the density profile is proportional to r^-1 out to ∼4000 AU, and to r^−2 beyond. Conclusions. Our NLTE code could be used to (re-)interpret recent and upcoming observations of N2H+ and N2D+ in many pre-stellar cores of interest, to obtain better temperature and abundance profiles.
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