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tokens: He atom, which includes 4 levels for He II and 1 level for He III, as well as a 5-level-plus continuum H atom. By calculating the full non-LTE radiative transfer for each model we find the energy level populations and thus ionization densities for both atomic species. The cylinder is orientated such that its axis is parallel to the solar surface. We then calculate the brightness temperature for a set of horizontal lines-of-sight through the prominence cylinder. We follow the same methods for brightness temperature calculation as outlined in Rodger Labrosse (2017) with one notable change in the calculation of the thermal gaunt factor. In this study we no longer estimate the thermal gaunt factor through the assumption given in Dulk (1985) for plasma of temperature 2 × 10 5 K, but instead take our values by interpolating the table of calculated thermal gaunt factors of van Hoof et al. (2014) , as described in Gayet (1970) and Simões et al. (2017) . Whilst the optical thickness diagnostic discussed in this study assumes purely thermal bremsstrahlung absorption, the results from these numerical models are calculated using both thermal bremsstrahlung and neutral hydrogen absorption. For this analysis we first consider a set of isothermal prominence models to see how well they agree with Eq. ( 4 ) and see how this changes when considering a multi-thermal plasma instead. Whilst nominally the models used here describe solar prominences, the results are applicable to any off-limb solar atmospheric structure. For on-disc structures, the spectral gradient will follow a similar relation to Eq. ( 4 ) with an additional term corresponding to the contribution of the background solar continuum spectrum. On-disc structures would therefore require knowledge of the brightness temperature of the structure and the background emission illuminating it from the solar disc. This will likely be problematic unless the structure in observation is formed above the chromosphere, the forming region for most millimetre radiation, or where the structure is transient in nature such that measurements of both the background and enhanced brightness temperature phases are obtainable. 3.1. Isothermal models The set of isothermal models we use here are the same as the “ t ” models as described in Gouttebroze Labrosse (2009) . They each consider a prominence cylindrical cross-section of radius 0.5 Mm at an altitude of 10 Mm. The isobaric pressure is 0.1 dyn cm −2 , the helium abundance is 0.1 and the micro-turbulent velocity is 5 km s −1 in all models. The set of temperatures considered is {6000, 8000, 10 000, 15 000, 20 000, 30 000, 40 000, 50 000, 65 000, 80 000, 100 000} K. From Fig. 1 it can be seen that whilst the relationship between the modelled spectral gradient and the optical thickness follows a similar trend to that expected by Eq. ( 4 ) the values are slightly lower. This discrepancy is caused by the assumption that the gaunt factor is approximately constant over the