The present definition of kelvin is that the thermodynamic temperature of the triple point of water (TPW) is exactly 273.16 K.^[1] Because it is very difficult to control the macroscopic quality of the water cells, including the isotopic abundances and contents of impurities, ^[2] the inconsistency among different national primary TPW cells can be as large as 0.1 mK. Mills et al. proposed^[3] to redefine the kelvin unit on an exact value of Boltzmann constant k_B, which directly relates the thermodynamic temperature to thermal energy. In a similar way, base units of kilogram, ampere and mole will be redefined by linking them to exactly known values of the Planck constant h, elementary charge e, and Avogadro constant N_A, respectively. The new definitions would be independent of any material substance, techniques of implementation and environments. The proposal has been accepted by the International Committee for Weights and Measures (Comité international des poids et mesures, CIPM). As a rule, new definitions should be based on the values of the fundamental constants agreeing with the best available measurements, to maintain the units invariant to current definitions.

The k_B value under present definition of kelvin is 1.3806488 (13) × 10^{– 23} J/K recommended by CODATA2010.^[4] It has been determined using different methods. Acoustic gas thermometry (AGT)^[5] determines the speed of sound in a gas at thermal equilibrium, by measuring acoustic resonant frequencies of a cavity, usually cylindrical or spherical, in which the acoustic resonator eigenvalues are known from theory. The experimental approach of AGT is to determine the speed of sound c₀(T, p) of a monatomic gas, usually helium or argon, at around the TPW temperature and at different pressures, and an extrapolation to the zero pressure limit gives c₀(T, p = 0). The molar gas constant R = k_BN_A can be derived according to the equation

where M is the molar mass of the gas, γ ₀ ≡ C_p/C_v is the ratio of the specific heat capacity at constant pressure to that at constant volume, which is 5/3 for ideal monatomic gases. Since the relative uncertainty in N_A is only 4.4 × 10^{– 8} in CODATA2010, k_B can be derived from the R value determined in AGT. Dielectric constant gas thermometry (DCGT)^[6] is similar to the refractive index gas thermometry (RIGT).^[7] They measured the polarizability of ⁴He, and k_B is derived from the comparison to theory.^[8] DCGT technique measures the change in capacitance of a capacitor with and without helium gas, and RIGT measures the index of refraction of helium gas in a microwave resonator. Johnson noise thermometry (JNT)^[9] is a purely electronic approach, which determines the quotient of the Boltzmann and Planck constants, k_B/h, in which h has a relative uncertainty of 4.4 × 10^{– 8} in CODATA2010. JNT measures the Johnson noise voltage in a bandwidth of frequency across a resistor in thermal equilibrium at TPW temperature.

The present CODATA2010 recommended value of k_B is inferred from a group of results obtained from AGT, ^{[10– 15]} RIGT, ^[7] and JNT.^[9] The combined relative uncertainty of k_B is 0.91 ppm. An uncertainty of 0.71 ppm based on AGT method has been recently reported.^[16] The CODATA2010 included k_B values and recent results^{[6, 16– 19]} are presented in Fig. 1.

Fig. 1.

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Fig. 1. The Boltzmann constant kB determined by different groups using different methods. INRIM: Istituto Nazionale di Ricerca Metrologica (Italy); [14] LNE: Laboratoire national de mé trologie et d’ essais (France); [12, 15] NPL: National Physical Laboratory (UK); [10, 13, 16] NIM: National Institute of Metrology of China; [18] NIST: National Institute of Standard and Technology (US); [7, 11] PTB: Physikalisch-Technische Bundesanstalt (Germany); [6, 17] U-Nap: Seconda Università di Napoli (Italy).[19]

Since the uncertainties in the RIGT and JNT values of k_B are 9.1 ppm and 12 ppm, respectively, there is an increasing concern that the new value of k_B may be solely determined from AGT measurements. As recommended by the Consultative Committee of Thermometry (CCT), the redefinition of the kelvin should be based on measurements applying different types of primary thermometry, to avoid the risk of unrevealed systematic deviation in one single method. Therefore, measurements using alternative methods other than AGT, with sufficiently low uncertainty (< 7 ppm), are required to fulfil the CCT conditions.

An optical determination of the Boltzmann constant was first demonstrated by Daussy et al.^[20] in 2007. Denoted as the Doppler broadening thermometer (DBT), it determines the Boltzmann constant k_B from the Doppler width of a transition of atoms or molecules in thermodynamic equilibrium at the TPW temperature. The Doppler width, γ _D (full width at half maximum, FWHM), relates with k_B and the temperature T following the equation:

where ν ₀ is the central frequency of the transition, m is the mass of the molecule, and c is the speed of light. Note that in Eq. (2), the speed of light, defined as 29979.2458 m/s, is a constant without uncertainty, masses of quite a few atoms and molecules have been determined with an accuracy at the 10^{– 8} level or better, ^[4] and frequencies of many atomic or molecular transitions can be straightforwardly measured with an accuracy better than 10^{– 9}, therefore, precise measurements of the sample temperature T and the Doppler width of the transition will result in a spectroscopic determination of k_B.

In 2007, the Université Paris 13 group (France) obtained a k_B with a relative uncertainty of 2 × 10^{– 4[20]} by measuring an absorption line of NH₃ near 10 μ m with a frequency-stabilized CO₂ laser. By using a multi-pass cell to improve the signal-to-noise ratio (SNR) in the absorption spectrum, they reduced the uncertainty to 5 × 10^{– 5}.^{[21, 22]} They recently proposed that the accuracy can possibly be improved to 1 ppm.^[23] Yamada et al. reported an optical determination of k_B with a relative uncertainty of 1200 ppm^[24] by measuring a ¹³C₂H₂ line near 1.5 μ m with a comb-stabilized diode laser. The Italian group at Seconda Università di Napoli obtained a k_B with an accuracy of 160 ppm^[24] by measuring an absorption line of CO₂ at 2 μ m with a distributed feed-back (DFB) diode laser. Recently they have improved the accuracy to 24 ppm^[19] by measuring an absorption lines of near 1.39 μ m. It is the best result to date given by DBT.

To the best of our knowledge, all the reported DBT measurements are based on direct absorption spectroscopy. We proposed that cavity ring-down spectroscopy (CRDS) is more advantageous to an optical determination of the Boltzmann constant.^[26] CRDS has been first implemented by O’ Keefe and Deacon^[27] in 1988. The main idea of CRDS is to measure the decay rate of the laser light emitted from a resonant cavity composed of two high-reflectivity mirrors. The absorption coefficient, α , of the sample gas can be obtained by measuring the ring-down time, following the equation:

where c is the speed of light, τ and τ ₀ is the ring-down time with and without absorption, respectively. The sensitivity of CRDS, often denoted as the noise equivalent absorption, can be as good as 1 × 10^{– 11} cm^{– 1}· Hz^{– 1/2}.^[28] Consequently, for a line with a central absorption coefficient of 10^{– 6} cm^{– 1}, it is possible to determine the α value with a relative uncertainty at the ppm level.

To record a precise line profile, it is also necessary to achieve sufficient frequency precision during the spectral scan. It can be accomplished by locking the laser to an external reference, either a frequency-stabilized laser^[29] or a thermo-stabilized etalon.^[30] We have proposed to apply a laser-locked cavity ring-down spectrometer as a Doppler broadening thermometer.^[26] The main advantage of using CRDS instead of conventional absorption techniques is that the superior sensitivity of CRDS allows precise measurements with low gas pressures and narrow-linewidth near-infrared lasers. Measurements with low gas pressures will reduce the uncertainty from the complicated collision broadening effect which has not yet been well investigated with a precision at the ppm level. Using mature narrow-linewidth near-infrared lasers secures the frequency accuracy in the measurement which also helps to reduce the statistical uncertainty.

In this section, we will investigate the sources of uncertainties which need to be considered for DBT measurements toward an accuracy at the 1 ppm level.

We are developing a CRDS instrument for Doppler broadening thermometry and the configuration is presented in Fig. 2. The ring-down cavity is composed of a pair of mirrors with a reflectivity of 0.99995. The 50-cm long cavity is installed in a vacuum chamber with a three-layer structure to maintain a temperature stability of better than 1 mK. From outside to inside, the first layer is a stainless steel vacuum chamber and it is also used as a heat sink. The second layer is made of aluminum and it is heated with a wire attached on its surface. A feed-back circuits is applied to stabilize the temperature of this aluminum layer by controlling the current in the heating wire. The third layer is also made of aluminum, and it is used as a heat shield to isolate the RD cavity from the residual temperature drift of the environment. Two platinum resistance thermisters are placed at each side of the RD cavity with a separation of about 40 cm. The thermisters and the readout (MKT 50, Anton Parr) have been calibrated with the Chinese national TPW reference in National Institute of Metrology of China. We have tested the temperature stability of our RDC at 300 K. The recorded typical temperature drift on the second layer is within 10 mK, while the temperature fluctuations of the ring-down cavity can be less than 2 mK within a few days.

Fig. 2.

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	Fig. 2. Configuration of the CRDS instrument for optical determination of kB. Abbreviations: AOM, acousto– optical modulator; EOM, electro– optical modulator; F– P filter, Fabry– Pé rot filter; PBS, polarized beam splitter; PD, photo– detector; PZT, lead zirconate titanate piezoelectric actuator; ULE– FPI, Fabry– Pé rot interferometer made of ultra-low-expansion glass.

The probe laser is locked to a longitudinal mode of a Fabry– Pé rot interferometer (FPI) using the Pound– Drever– Hall (PDH) method. The slow and fast feed-back control signals are delivered to the laser controller and the driver of an acousto– optical modulator (AOM 1), respectively. The 10-cm-long FPI is made of ultra-low-expansion glass (ULE) and installed in a high vacuum chamber which is thermo-stabilized at about 303 K with a temperature drift below 5 mK. The frequency drift of the longitudinal modes of the ULE-FPI is estimated to be less than 10 kHz within several months.

Figure 3 shows an example of the recorded spectrum of acetylene near 12696.4123 cm^{– 1}, which is around the R(9) line of the ν ₁ + 3ν ₃ band of ¹²C₂H₂. The sample pressure is 3 Pa, resulting with a pressure broadening width γ _P estimated to be about 1 × 10^{– 6} cm^{– 1}, which is over 4 orders of magnitude less than the Doppler width γ _D. In this case, if a Voigt function is applied for the line profile, and the resulted line width is approximately , the contribution from pressure broadening to the total line width should be less negligible (≪ 1 ppm). It agrees with the spectral fitting of the observed spectrum using different line profile models taking into account the collision effects, which are shown in Fig. 3.

Fig. 3.

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	Fig. 3. CRDS spectrum of the R(9) line of the ν 1 + 3ν 3 band of C2H2. The sample pressure was 3.3 Pa. (a) Experimental (open circles) and simulated (solid line) spectra, panels (b)– (e) are fitting residules using different profiles: Gaussian, Voigt, and quadratic speed dependent Voigt (qSDVP), respectively.

The spectrum shown in Fig. 3 was obtained from a spectral scan during about 100 s. The Gaussian width could be derived from fitting of the observed spectrum with a Gaussian function. Results from about 300 scans obtained in about 10 h are depicted in the upper panel of Fig. 4. A statistics of the results shown in the same figure indicates the relative uncertainty of the derived Doppler width, δ γ _D/γ _D, roughly follows , where N is the number of averaged scans. Therefore, the relative uncertainty decreases to about 10 ppm when over 100 scans are averaged. However, we have observed a systematic deviation of the line width: the observed one is about 100 ppm larger than the calculated Doppler width according to the temperature. It could be a result of weak C₂H₂ transitions located in the vicinity of the R(9) line. Although the selected R(9) line looks well isolated as shown in Fig. 3, a measurement using much larger sample pressures reveals many weak lines, which are weaker than the target R(9) line by over three orders of magnitude but are not reported in the literature. These “ hidden” weak lines could distort the derived line profile.

Fig. 4.

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	Fig. 4. (a) Doppler widths γ D derived from fitting of the observed spectra of the 787 nm line of C2H2. (b) The statistics of the uncertainty in γ D.

International Committee for Weights and Measures (CIPM) accepted a proposed new definition of the unit kelvin based on a fixed value of Boltzmann constant k_B, which relates the thermodynamic temperature to thermal energy. At the present stage, it is necessary to determine the value of k_B using different methods to check the consistency, avoiding any possible systematic deviation inherited from a single method. The Doppler broadening thermometry (DBT) determines k_BT by measuring the Doppler width of an absorption line of atoms or molecules. The method takes the advantage of rapid progress in precision spectroscopy and laser techniques, and could potentially achieve an accuracy comparable to the acoustic methods. There are miscellaneous sources of the uncertainties in DBT measurements, including the noise and nonlinearity of the detectors, laser line width, and frequency drift, collision-induced broadening, temperature fluctuation and gradients, and weak parasitic absorption lines. The uncertainty in complicated line-shape models due to collisions is the main obstacle for an optical determination of k_B with an accuracy at the ppm level.

Cavity ring-down spectroscopy for DBT measurements has some remarkable advantages over the conventional direct absorption method. Its superior sensitivity allows DBT measurements under much lower gas pressures without sacrificing the signal-to-noise ratio, and therefore possibly circumvents the difficulty due to our insufficient knowledge in collision-induced line profiles. We have built a laser-locked CRDS instrument combined with a temperature-stabilized gas cell, which will be applied for DBT determination of k_B. Preliminary experiments have been carried out using a ro-vibrational transition of C₂H₂ near 787 nm using acetylene gas samples with pressures of a few Pa. The results indicate that we can reduce the statistical uncertainty in derived line width to about 10 ppm per day, which is promising for a DBT measurement toward the 1 ppm precision. A cavity ring-down spectrometer working at the triple-point-of-water temperature is under construction in our laboratory.

The selection of proper transitions for DBT measurements is also very important. We are going to use the ro-vibrational transition of carbon monoxide in the 1.56-μ m region, which belongs to the second overtone (V = 3) of CO. These transitions have quite a few advantages that are very useful for DBT measurements: they have moderate line strengths suitable for CRDS measurement; mature narrow-linewidth lasers are available in this spectral region; the inter-molecular interaction of CO is relatively weak, which helps to reduce the influence from collisions; ¹²C¹⁶O has no hyperfine structure and its accurate mass is known; and the relatively simple vibrational band structure of a diatomic molecule eliminates the possible influence from weak lines near the target line. A test measurement shows that we can find some “ truly isolated” transitions of CO in this region: we cannot observe any evidence of weak nearby CO lines with strengths more than one part in million of the target line.