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 N _ { 2 } H ^ { + } and N _ { 2 } D ^ { + } 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 N _ { 2 } H ^ { + } and N _ { 2 } D ^ { + } , 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 N _ { 2 } H ^ { + } does occur , starting at densities 5–7 \times 10 ^ { 5 } cm ^ { -3 } and reaching a factor of 6 ^ { +13 } _ { -3 } in abundance , iii ) deuterium fractionation reaches \sim 70 % at the core center , and iv ) the density profile is proportional to r ^ { -1 } out to \sim 4000 AU , and to r ^ { -2 } beyond .

Conclusions : Our NLTE code could be used to ( re- ) interpret recent and upcoming observations of N _ { 2 } H ^ { + } and N _ { 2 } D ^ { + } in many pre-stellar cores of interest , to obtain better temperature and abundance profiles .