Spectroscopic determination of the Boltzmann constant


The Bolztmann constant kB is one of the fundamental constant in physics. Its current CODATA value is 1.380 648 8(13)×10-23 J/K, with a relative uncertainty of 0.91 ppm. The international organization CIPM plans to define kB with a constant value of no errors, and use this constant to redefine the temperature unit Kelvin, thus replacing the present definition based on the triple point of water. In order to avoid unaccounted systematic errors of a singular method, CIPM requires that the planned new definition of kB must be based on by two independent methods of determination. However, the current level of uncertainty is overwhelmingly due to the acoustic method AGT. Meanwhile, spectroscopic methods based on Doppler broadening of atomic or molecular absorption lines are being developed. Under thermal equilibrium, the absorption lines of low-pressure gas molecules have a Doppler-broadened width (FWHM):

$$\omega_{D}=\nu_{0}\sqrt{\frac{8k_{B}Tln2}{mc^{2}}}$$
Here ν0 is the center frequency, T is the temperature of the gas, m is the mass of the molecule, and c is the speed of light. A precision measurement of the width at the triple point of water promises to determine kB with an uncertainty of ppm or even better.

Conventional laser spectroscopy detects an absorption line based on Beer’s absorption law: I = I0exp[-α(ν) L] , and the line width is determined by fitting the line shape α(ν). As for ro-vibrational transitions of molecules, in order to achieve the required absorption sensitivity, the gas pressure needs to be considerably high (~102 Pa), resulting in relatively complex line shape due to collisions. So far, no models of collision-induced lineshape have achieved the ppm-level of accuracy. Electronic transitions are strong and could be detected at very low pressures. However due to the interference from stray electronic and/or magnetic fields, hyperfine splitting, and power broadening effects, there are tremendous challenges in determining the accurate Doppler width.

Cavity Ring-Down Spectroscopy (CRDS) uses the ring-down time to measure light absorption:

$$\alpha(\nu)=\frac{1}{c\tau(\nu)}-\frac{1}{c\tau}$$

Here τ and τ0 are ring-down times with and without samples, respectively. They are of the order of micro seconds, and can be determined accurately. This method allows the use of near infrared laser sources. Even on weak infrared transitions of low-pressure gas molecules, sufficient detection sensitivity and dynamic range (Fig.1 vertical axis) can be achieved. We proposes to use a frequency locked CRDS technique, LL-CRDS, which allows controlled scan of laser frequency over 1 GHz (Fig.1 horizontal axis). We believe that LL-CRDS can be advantageous in determining kB.

We are developing a high precision setup of CRDS, and have already realized a detection sensitivity of 10-11/cm (equivalent to direct optical absorption path-length of several thousand kilometers) and frequency precision of 1 kHz. The temperature control is at the level of 1 mK. Preliminary results indicate a statistical uncertainty of kB at 10 ppm. At present, we are building a CRDS setup using a narrow-line diode laser and a cavity temperature at the triple point of water (273.16 K). The transition of CO molecules at 1.6 um is probed. This high-precision method can also find applications in studying molecular collisions, detecting forbidden transitions, and trace analysis for environmental applications.

Fig.1 CRDS and the Doppler width of an absorption line

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