The [C ii] / [N ii] ratio in 3 < z < 6 sub-millimetre galaxies from the South Pole Telescope survey

We present Atacama Compact Array and Atacama Pathfinder Experiment observations of the [N II] 205 $\mu$m fine-structure line in 40 sub-millimetre galaxies lying at redshifts z = 3 to 6, drawn from the 2500 deg$^2$ South Pole Telescope survey. This represents the largest uniformly selected sample of high-redshift [N II] 205 $\mu$m measurements to date. 29 sources also have [C II] 158 $\mu$m line observations allowing a characterization of the distribution of the [C II] to [N II] luminosity ratio for the first time at high-redshift. The sample exhibits a median L$_{[C II]}$ /L$_{[N II]}$ $\approx$ 11 and interquartile range of 5.0 to 24.7. These ratios are similar to those observed in local (U)LIRGs, possibly indicating similarities in their interstellar medium. At the extremes, we find individual sub-millimetre galaxies with L$_{[C II]}$ /L$_{[N II]}$ low enough to suggest a smaller contribution from neutral gas than ionized gas to the [C II] flux and high enough to suggest strongly photon or X-ray region dominated flux. These results highlight a large range in this line luminosity ratio for sub-millimetre galaxies, which may be caused by variations in gas density, the relative abundances of carbon and nitrogen, ionization parameter, metallicity, and a variation in the fractional abundance of ionized and neutral interstellar medium.