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Refractive index gas thermometry (RIGT) uses the dependence of the relative dielectric permittivity on the density of a noble gas (He), typically measured with a temperature-controlled resonator. On an isotherm, to a good approximation, the slope of the dielectric permittivity versus pressure gives the thermodynamic temperature T. To measure T with a low uncertainty, one must measure the absolute value of the pressure with a low uncertainty and know the compressibility of the resonator wall. This article shows how to use RIGT in a novel way between 5 K and 25 K. Instead of changing the pressure on an isotherm (J.W. Schmidt et al., Phys. Rev. Lett., 98, 254504 (2007)), a constant pressure of pure helium gas is maintained at multiple temperatures. After calibration of the resonator under vacuum at different temperatures, all thermometry is performed at a single pressure and referred to a fixed point of the International Temperature Scale of 1990 (ITS-90) (here the neon triple point at 24.5561 K). The quantity that yields the temperature is the ratio of the resonance frequencies of a microwave mode in the resonator measured at the fixed point and at the unknown temperature. This paper describes the theoretical model of Single-Pressure Refractive Index Gas Thermometry (SPRIGT) and analyses the shifts and broadenings due to non-ideal behaviour. The technique should enable a helium-based measurement of thermodynamic temperature with a resolution better than 25 μK and an uncertainty of around 250 µK.

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