## Abstract

We describe a method for determining the density of helium via measurements of optical refractivity. In combination with the equation of state, this allows realization of the pascal. Our apparatus is based on the integration of a gas triple-cell into a quasi-monolithic heterodyne interferometer: the stability of the interferometer is $\pm 50\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{pm}$ over 10 h. We claim the contribution of cell window thinning to pathlength uncertainty can be canceled within an uncertainty of 0.37 fm/Pa per window pass, of which for our 25 cm cell length corresponds to a fractional error of $9.3\times {10}^{-6}$ in the measure of helium refractivity. We report the ratio ${(n-1)}_{{\mathrm{N}}_{2}}/{(n-1)}_{\mathrm{He}}=8.570354(13)$ at $p=367.420(4)\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kPa}$, $T=293.1529(13)\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{K}$ and $\lambda =632.9908(6)\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, which can be used to calibrate less-accurate refractometers. By measuring helium refractivity at known temperature and pressure, we determined the Boltzmann constant with standard uncertainty ${k}_{\mathrm{B}}=1.380652(17)\times {10}^{-23}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{JK}}^{-1}$.

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