  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&#8722;3 and reaching a factor of 6 (+13/&#8722;3) in abundance, iii)
deuterium fractionation reaches &#8764;70% at the core center, and iv) the
density profile is proportional to r^-1 out to &#8764;4000 AU, and to
r^&#8722;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.
