Determination of the Boltzmann Constant by Means of Precision Measurements of ${\mathrm{H}}_2{}^{18}\mathrm{O}$ Line Shapes at $1.39\mu \mathbf{m}$

We report on a new implementation of Doppler broadening thermometry based on precision absorption spectroscopy by means of a pair of offset-frequency locked extended-cavity diode lasers at $1.39\mu \mathrm{m}$. The method consists in the highly accurate observation of the shape of the $4_{4,1}\to 4_{4,0}$ line of the ${\mathrm{H}}_2{}^{18}\mathrm{O}$ ${\nu }_1+{\nu }_3$ band, in a water vapor sample at thermodynamic equilibrium. A sophisticated and extremely refined spectral analysis procedure is adopted for the retrieval of the Doppler width as a function of the gas pressure, taking into account the Dicke narrowing effect, the speed dependence of relaxation rates, and the physical correlation between velocity-changing and dephasing collisions. A spectroscopic determination of the Boltzmann constant with a combined (type $A$ and type $B$) uncertainty of 24 parts over ${10}^6$ is reported. This is the best result obtained so far by means of an optical method. Our determination is in agreement with the recommended CODATA value.

Figure 1

Schematic diagram of the dual-laser absorption spectrometer. OI, optical isolator; BS, beam splitter; AOM, acousto-optic modulator; EOM, electro-optic modulator; $f_{\mathrm{beat}}$, beat frequency; $\lambda /4$, quarter wave plate; M, mirror; PBS, polarizing beam splitter; Pd, photodiode; FPd, fast photodiode; PS, radio frequency power-splitter; OFL, offset-frequency locking; HFC, high-finesse cavity; SA, spectrum analyzer; PC, personal computer.

Figure 2

Experimental ${\mathrm{H}}_2{}^{18}\mathrm{O}$ spectrum near $\lambda =1.39\mu \mathrm{m}$, at a pressure of about 2 Torr and a temperature of 273.1550 K. The signal-to-noise ratio was measured to be $\sim 10000$. Absolute residuals are also reported, resulting from the nonlinear least-squares fit to the PCSDHC profile. The root-mean-square value of the residuals amounts to about $200\mu \mathrm{V}$, while the reduced ${\chi }^2$ value is 0.22. The same level of agreement between theory and experiment has been obtained in the whole pressure range.

Figure 3

Spectroscopic determination of the Boltzmann constant. The red line represents the current CODATA value, namely $1.3806488(13)\times {10}^{-23}\mathrm{J}/\mathrm{K}$. Error bars, corresponding to one standard deviation, are the internal errors resulting from the fitting procedure.