## Abstract

At its 25th meeting, the General Conference on Weights and Measures (CGPM) approved Resolution 1 ‘On the future revision of the International System of Units, the SI’, which sets the path towards redefinition of four base units at the next CGPM in 2018. This constitutes a decisive advance towards the formal adoption of the new SI and its implementation. Kilogram, ampere, kelvin and mole will be defined in terms of fixed numerical values of the Planck constant, elementary charge, Boltzmann constant and Avogadro constant, respectively. The effect of the new definition of the kelvin referenced to the value of the Boltzmann constant *k* is that the kelvin is equal to the change of thermodynamic temperature *T* that results in a change of thermal energy *kT* by 1.380 65×10^{−23} J. A value of the Boltzmann constant suitable for defining the kelvin is determined by fundamentally different primary thermometers such as acoustic gas thermometers, dielectric constant gas thermometers, noise thermometers and the Doppler broadening technique. Progress to date of the measurements and further perspectives are reported. Necessary conditions to be met before proceeding with changing the definition are given. The consequences of the new definition of the kelvin on temperature measurement are briefly outlined.

## 1. Introduction

Thermometers whose basic relation between the measurand and the thermodynamic temperature *T* can be written down explicitly without having to introduce unknown, temperature-dependent constants are called primary thermometers. In the basic equations for these primary thermometers, the temperature always appears as thermal energy *kT*. Therefore, it is evident that primary thermometers can be used to determine either *T* or the Boltzmann constant *k*. When the temperature of the system is known, e.g. by operating the thermometer at the triple point of water (TPW), the Boltzmann constant can be determined by the same experiment. In future, the unit of thermodynamic temperature *T*, the kelvin, will be related to the Boltzmann constant with a fixed numerical value. In this report, the different primary thermometers such as acoustic gas thermometers, dielectric constant gas thermometers, noise thermometers and Doppler broadening thermometers currently contributing to determinations of the Boltzmann constant are reviewed in more detail.

At its 25th meeting, the General Conference on Weights and Measures (CGPM) approved Resolution 1 ‘On the future revision of the International System of Units, the SI’ [1], which sets the path towards redefinition of four base units at the next CGPM in 2018. This constitutes a decisive advance towards the formal adoption of the new SI and its implementation. The changes proposed for the International System of units (SI) will not actually be adopted until the experimental results on the new definitional constants that are proposed have reached a further stage of refinement. Accordingly, the Consultative Committee for Thermometry (CCT) recommended [2] that the International Committee for Weights and Measures (CIPM) requests the Committee on Data for Science and Technology (CODATA) to adjust the values of the fundamental physical constants only when the following two conditions are met:

(1) the relative standard uncertainty of the

*adjusted*value of*k*is less than 1×10^{−6};(2) the determination of

*k*is based on at least two fundamentally different methods, of which at least one result for each shall have a relative standard uncertainty less than 3×10^{−6}.

In its 2010 adjustment of fundamental constants [3], the CODATA Task Group on Fundamental Constants (TGFC) recommended for the Boltzmann constant a value with a relative uncertainty equal to 9.1×10^{−7}. (All uncertainties in this article are standard uncertainties with coverage factor *k*=1 corresponding to a 68% confidence interval.) However, this value was based on only one experimental method, namely acoustic gas thermometry (AGT, §2). Nevertheless, with the 2010 CODATA adjustment the first condition was fulfilled. The second condition was only met by one method: several AGT experiments obtained uncertainties well below 3×10^{−6}. The most promising other methods for meeting the second condition are dielectric-constant gas thermometry (DCGT, §3) and noise thermometry (§4). The Doppler broadening technique (DBT, §5), using optical measurements, has only recently been proposed for the purpose of determining the Boltzmann constant. Thus, an improved value of the Boltzmann constant for the new definition of the kelvin would ideally have been determined by at least three fundamentally different methods and be corroborated by optical measurements with larger uncertainty [4].

In §§2–5, we will briefly review the state of the art of the above-mentioned methods to determine the Boltzmann constant. The interested reader should consult the separate reviews of the methods [5–7] for more details. In §6, a discussion of the prospects of further research envisaged by the involved groups up to mid-2017 will show that a reduction in the uncertainty of the methods independent of AGT to meet the second CCT requirement is highly probable.

## 2. Acoustic gas thermometry

For an ideal gas, the relation between the speed of sound *u*_{0}, the thermodynamic temperature *T* and the gas constant *R* is given by
2.1with *γ*=*C*_{p}/*C*_{V} being the ratio of the constant-pressure to constant-volume specific heat and *M* the molar mass of the gas. Taking into account the precise knowledge of the Avogadro constant *N*_{A}, with *R*=*kN*_{A}, the Boltzmann constant can be derived, adding no significant uncertainty. Usually *u*_{0} is determined by extrapolation to zero gas density [6]. The main challenge of the method is the measurement of *u*_{0} with the necessary uncertainty. Moldover *et al.* [8] developed at the National Institute of Standards and Technology (NIST) a spherical resonator and used it during 1986 to re-determine the gas constant *R* with an estimated relative uncertainty of 1.7×10^{−6}, a factor of 5 smaller than the uncertainty of the best previous measurement [9]. From the known density of mercury, the dimension of the resonator was deduced. For spherical acoustic resonators, many of the major perturbations, which influence the determination of the resonance frequencies, are well understood. Cavities with a quasi-spherical geometry permit operation as combined acoustic and electromagnetic resonators and their quasi-spherical form lifts the degeneracy of the microwave normal modes. The microwave resonances are used to determine the dimensions of the sphere. State-of-the-art dimensional measurements and pycnometry of the spherical resonators validated the microwave approach. The problem of determining the speed of sound is shifted from an independent measurement of time and length to measuring the ratio of acoustic and microwave frequencies as a function of pressure corrected for the main known perturbative effects. The ultimate accuracy achievable depends on the capability of modelling appropriate corrections for the acoustic and microwave eigenfrequencies to account for the imperfections of a real resonator [6].

Between 2008 and early 2011, an iMERAPlus joint research project [10] was coordinating the European activities to determine the Boltzmann constant in Denmark (Danish Fundamental Metrology, DFM), France (Laboratoire National de Métrologie et d’Essais, LNE-CNAM, and Laboratoire de Physique des Lasers at Université Paris Nord, LPL), Italy (Istituto Nazionale di Ricerca Metrologica, INRiM, and the Universities of Naples and Milan), Spain (Universidad de Valladolid and Centro Español de Metrología, CEM), UK (National Physical Laboratory, NPL) and Germany (Physikalisch-Technische Bundesanstalt, PTB). This large research collaboration project resulted in major progress and essential developments for AGT. The AGT measurements of LNE-CNAM [11,12], NPL [13] and INRiM [14] all used resonators jointly developed within the iMERAPlus project and achieved the smallest uncertainties of all methods. All results are highly consistent (table 1) and agree very well with the CODATA value of 2006 [34], based mainly on [8]. These four new determinations were exploited by the CODATA TGFC in its 2010 adjustment of fundamental constants together with [8,9] (table 2, box). As a result, the CODATA TGFC recommended a value for *k* with a relative uncertainty of 9.1×10^{−7} [3], which is about a factor of two smaller than the previous *u*(*k*)/*k* of 1.7×10^{−6}.

In 2013, NPL published high-precision measurements using argon gas in a 62 mm radius sphere with a relative uncertainty of 0.71×10^{−6}, the lowest uncertainty in the determination of the Boltzmann constant achieved so far [25]. The NPL estimate of *k* was 2.7×10^{−6} higher than the LNE-CNAM estimate with argon in 2011 [12] and this difference was inconsistent with the combined relative uncertainty of both determinations of 1.4×10^{−6}. Examination of the contributions to the measurement uncertainty in *k* showed that the largest single component (24%) is the uncertainty of the molar mass of atmospheric argon determined by Lee *et al*. at the Korea Research Institute of Standards and Science (KRISS) [35]. If that estimate were to be revised then NPL's estimate for *k* would need to be adjusted accordingly. The possibility that the difference is due to an error in the estimate of the molar mass by either NPL or LNE-CNAM was under joint investigation in 2014 and is summarized below.

The LNE-CNAM Boltzmann value was based on the molar mass (*M*_{Ar}) determinations of the Institute for Reference Materials and Measurements, Belgium (IRMM). The NPL estimate was based on a comparison at the Scottish Universities Environmental Research Centre (SUERC) of the isotopic composition of the experimental gas with the isotopic composition of argon from atmospheric air, and referenced to the KRISS 2006 estimate for the isotopic composition of atmospheric argon [35]. Three important studies were performed during the period October to December 2014 [36]. In the first study, the isotopic composition of samples of argon gas previously measured at IRMM was examined at KRISS. The analysis showed disagreements of *M*_{Ar} by up to 3.5 ppm. In the second study, the isotopic composition of a sample of argon gas used in the NPL experiment [25] of 2013 was examined. The estimate of *M*_{Ar} was 2.73 ppm lower than the corresponding SUERC estimate. In the third very comprehensive study, extensive measurements were made of a series of samples from LNE-CNAM, INRiM, the National Metrology Institute of China (NIM), the National Metrology Institute of Japan (NMIJ/AIST) and NPL on which the corresponding speed of sound measurements had been made at LNE-CNAM. The results showed a clear correlation between the two analysis techniques. Based on the measurements at KRISS, only the value of the Boltzmann constant *k* determined in the NPL 2013 experiment was corrected by Δ*k*/*k*=−2.73×10^{−6}, resolving the discrepancy between both results. For the CODATA adjustment of 2014, both the LNE-CNAM 2011 and the NPL 2013 results are considered to rely on the KRISS molar mass determinations and no longer on the IRMM and SUERC results, introducing a strong correlation between the 2011 LNE-CNAM and the 2013 NPL results, generated as a consequence of the referencing to the KRISS molar mass determinations. Taking account of the uncertainty of the KRISS measurement increases the 2011 LNE-CNAM and the 2013 NPL total relative uncertainties of *k*, now estimated at 1.42×10^{−6} and 0.92×10^{−6}, respectively. For NPL, there is a further correction required based on the revised estimate of the thermal conductivity of argon, which lowers the NPL estimate by a further 0.192 ppm and decreases the relative uncertainty to 0.91×10^{−6} [37].

LNE-CNAM published in 2015 the series of measurements of May 2012 and January 2013 using helium gas with the 0.5 l copper quasi-sphere BCU3 (50 mm radius) [29]. The value of *k* is in good agreement with the earlier measurements in argon (table 1) and has a relative uncertainty of 1.02×10^{−6}. The values of the uncertainty contributions are nearly equally spread over the measurements of acoustic frequency, resonator volume, molar mass and temperature, the last being the lowest. For 2016, new results are expected with the 3.1 l quasi-sphere BCU4, which has a 90 mm radius and is operated with helium and argon.

INRiM has pursued an accurate determination of the speed of sound in helium at 273.16 K with a 3 l volume copper sphere assembled in 2013. Acoustic and microwave results indicate that the performance of the experiment has significantly improved with respect to previous INRiM achievements. Also, temperature measurement and thermal gradients across the resonator are now satisfactory. A cross-check of the current estimate of helium impurities was made by using the mass spectrometry facilities made available by PTB. Thus, for all of the four major uncertainty contributions significant progress has been made. INRiM determined a new value for the Boltzmann constant with a large reduction in the relative uncertainty to 1.06×10^{−6} [30].

NIM compensated for the disadvantages of cylindrical resonators by developing a special two-cylinder regime. As proof of concept, a single 130 mm long cylindrical cavity was used (c-AGT) [22]. In 2013, advanced results were obtained with the single resonator with a relative uncertainty of 3.7×10^{−6} [26], where the shape was closer to that of a perfect cylinder and the thermometry was also improved. The largest component of the uncertainty resulted from inconsistent values of *k* determined with the various acoustic modes and is 2.9×10^{−6}. Finally, NIM applied the two-cylinder regime by developing a special virtual resonator approach. The main advantage of measuring the acoustic resonances in the new regime with two cylindrical cavities of identical diameters and lengths *l* and 2*l* is the removal of some major perturbations causing otherwise large corrections. The predominant perturbation caused by the bending of the two endplates of the cylinders could be effectively removed. However, the perturbations caused by the thermal-viscous boundary layers formed on the cylindrical shells do not cancel out. Comprehensive experimental and theoretical studies showed that such perturbations can be well corrected by model calculations making use of the accurate knowledge of the viscosities of the gas sample. NIM used two cavities of lengths 80 and 160 mm for a test of the virtual regime for the determination of *k*. An undesired resonant coupling between the two cavities or the pressure vessel is presently limiting the performance of the experiment [38].

At the Universidad de Valladolid in collaboration with CEM a stainless steel misaligned spherical resonator with a radius of 40 mm was developed to determine the Boltzmann constant. Owing to the material, small size and surface quality of the cavity the relative uncertainty was limited to 20×10^{−6} [32].

## 3. Dielectric constant gas thermometry

DCGT is based upon the variation of the dielectric constant (permittivity) *ε* with temperature. The gas particles act in a capacitor as induced dipoles with the static electric dipole polarizability *α*_{0} according to the Clausius–Mossotti equation. The density in the state equation of an ideal gas is replaced by the dielectric constant [7]. Thus, it can be written in the form
3.1where *ε*_{0} is the exactly known electric constant and *p* is the pressure of the gas. A similar approach using optical or microwave resonators measures the refractive index *n* of the gas and is therefore called refractive index gas thermometry (RIGT). The basic equation is derived from (3.1) by substituting the dielectric constant *ε* by *n*^{2}*ε*_{0}. This method was used in 2007 to determine the Boltzmann constant [15]. Both methods require the polarizability *α*_{0} to be known with the necessary accuracy. Nowadays, this condition is fulfilled for helium, which became a model substance for evaluating the accuracy of *ab initio* calculations of thermophysical properties. Recent progress has decreased the uncertainty of the *ab initio* value of *α*_{0} well below one part in 10^{6} [39].

A proof of concept of DCGT was performed at PTB with a low-temperature thermometer [16]. A dedicated experimental set-up at the TPW has been constructed [23] consisting of a large-volume thermostat, a vacuum-isolated measuring system, stainless steel 10 pF cylindrical capacitors, an autotransformer ratio capacitance bridge and a high-purity gas-handling system including a mass spectrometer. The pressure was generated by special pressure balances with traceably calibrated piston–cylinder assemblies with effective areas of 2 cm^{2}. In the pressure range from about 1 to 7 MPa, 10 helium isotherms have been measured at the TPW and a value has been determined for *k* with a relative uncertainty of 9.2×10^{−6}. Earlier low-temperature DCGT experiments yielded a value with a relative uncertainty of 15.9×10^{−6} [40]. The weighted mean of the two values has a relative uncertainty of 7.9×10^{−6} [23].

A significant reduction in uncertainties was achieved in 2013 by the use of tungsten-carbide cylindrical capacitors featuring at least a factor of two lower effective compressibility. Further essential progress was the design and the assembly of the measuring capacitors, the determination of its effective compressibility, the sensitivity of the capacitance bridge, the influence of stray capacitances, the purity of the measuring gas, the pressure measurement and the scattering and evaluation of the data. The resulting new value has a relative uncertainty of 4.3×10^{−6} [27]. Activities to decrease the uncertainty of the pressure measurement to a level of 1×10^{−6} were successfully completed in 2014 [41]. This included extensive cross-float comparisons between six independent primary piston–cylinder assemblies to improve the consistency of their effective areas and pressure-distortion coefficients. Moreover, an uncertainty reduction came from careful comparisons of the pressure-distortion coefficients up to 7 MPa. The resulting new relative uncertainty for the Boltzmann constant amounts to 4.0×10^{−6} [7]. The largest uncertainty contributions are now the type A estimate and the determination of the effective compressibility.

Additional progress in decreasing the uncertainty is expected by using measuring capacitors of a quite different design, namely ring cross capacitors. Cross capacitors have several advantages such as insensitivity to dielectric films on the electrodes and an effective compressibility very close to that resulting from the volume compressibility of the electrode material. Considering the progress achieved during the DCGT experiments, it seems realistic that the relative uncertainty of the Boltzmann constant determination will be decreased to a level of only about 2×10^{−6} before 2017.

## 4. Noise thermometry

The noise thermometer is based on the temperature dependence of the mean-square noise voltage, 〈*U*^{2}〉, developed in a resistor. From thermodynamic calculations, one can derive
4.1valid for frequencies *f*≪*kT*/*h*, where *R*_{el} is a frequency-independent resistance, Δ*f* is the bandwidth of the detection system and *h* is Planck's constant. From the statistical nature of the measured quantity, long measuring times arise. One of the main problems is the accurate measurement of the very small voltages developed by avoiding extraneous sources of noise and maintaining constant bandwidth and gain of the amplifiers. The noise generated by the wires which connect the sensor to the amplifiers has to be eliminated from the measurement. This is conveniently done by making a four-wire connection to the sensor and using two amplifier chains. The most successful technique to date is the switched input digital correlator. The correlator is implemented by digitizing the signals from the two channels and carrying out the multiplication and averaging function by software. In use, the thermometer switches between a reference and the thermal noise source at the temperature *T*. The switching also removes the effects of drift in the gain and bandwidth of the amplifiers and filters [42].

NIST developed a new approach using the perfect quantization of voltages from the Josephson effect [43]. This approach retains the proven elements of the switched correlator, but separates the roles of the temperature reference and the voltage reference. The sensing resistor in the reference arm of the comparator is replaced by a quantum-voltage noise source. The quantum-voltage noise measurement of the Boltzmann constant is quite different from gas-based measurement techniques in that it is a purely electronic approach that links the kelvin to quantum-based electrical measurements. The quantum-voltage noise source, which is a low-voltage realization of the superconducting Josephson arbitrary waveform synthesizer, is programmed to produce multi-tone pseudo-noise voltage waveforms with small (less than 1 μV peak) amplitudes. The voltage pulses have time-integrated areas perfectly quantized in integer values of *h*/2*e*, where *e* is the elementary charge. The synthesized voltage is intrinsically accurate because it is exactly determined from the known sequence of pulses, the clock frequency and fundamental physical constants. The resistance *R*_{el} of the thermal resistor is determined traceably to the quantum Hall effect. With this new approach, NIST achieved a relative uncertainty of 12 parts in 10^{6} [21] for the Boltzmann constant. NIST is now developing an advanced system for a more efficient measurement: the two-channel system will be replaced by a four-channel system in a very compact setting with a four-channel analogue-to-digital converter readout. The bandwidth of the system will be increased by switching to amplifiers with increased bandwidth, with lower or comparable noise and higher linearity. In the most recent measurements, NIST used a 200 Ω sense resistor that reduced the statistical uncertainty by 25% in the same measurement period compared with that of a 100 Ω sense resistor [44].

A completely new noise thermometer was developed recently at NIM in cooperation with NIST [45]. It measures *k* by comparing the thermal noise across a 100 Ω resistor with the noise synthesized by a bipolar pulse-driven quantum-voltage noise source. The flat ratio between the thermal noise and the calculated quantum-voltage noise up to 800 kHz, and self-consistent fitting results with different bandwidths, indicate that the systematic uncertainties are greatly reduced. Finally, NIM compared the thermal noise power of a 200 Ω sensing resistor directly immersed in a TPW cell with the noise power of a quantum-voltage noise source of nominally equal noise power. Measurements integrated over a wide bandwidth of 575 kHz and a total integration time of 33 days gave a relative uncertainty of 3.9×10^{−6} [31]. In the course of this determination very accurate measurements of the nonlinearity of the detection system have been performed, contributing only 0.1×10^{−6} to the relative uncertainty. The dominating uncertainty of 3.8×10^{−6} arises from the ratio of thermal and quantum-voltage noise powers including a 3.2×10^{−6} statistical uncertainty.

NMIJ/AIST has been developing a quantum-voltage calibrated noise thermometer system to first measure thermodynamic temperatures near the TPW or the Boltzmann constant *k*, and then aims to extend the temperature range to higher temperatures. In this system, most of the key elements have been built from scratch independently from NIST and NIM; especially, the Josephson junction array was built in house at AIST, and its driving method differs from that of either NIST or NIM. The values for *k* should ideally be flat and independent of the bandwidth. However, the adjustment level of the system is still premature and consequently there is some bandwidth dependency in the measurements [46]. Also some electromagnetic interference in the measurements has to be eliminated. Currently, NMIJ/AIST is expecting further progress in the Boltzmann constant measurements through improvements in the hardware.

## 5. Doppler broadening technique

In the DBT, a laser beam propagates through an absorption cell containing a gas of atoms or molecules with uniform temperature *T*. The Gaussian Maxwell probability density for the velocity of the particles with atomic mass *m* translates into the corresponding Doppler-broadened absorption line profile with the Doppler width Δ*ν*_{D} given by
5.1where the absorption frequency is denoted by *ν*_{0} and *c*_{0} is the speed of light. This relation allows the determination of the Boltzmann constant by spectroscopic measurement of a Doppler-broadened absorption line profile and determination of its width. In principle, the measurement can be done using standard laser-spectroscopic techniques. As a main advantage compared with other optical methods such as absolute radiation thermometry, the Doppler profile can be determined by relative radiation measurements since it is only its width which is of interest here. Moreover, laser frequencies can be controlled with extremely small uncertainties. However, at the 10^{−6} uncertainty level, various other sources of uncertainty will have to be investigated in detail. Apart from the quadratic Doppler effect, these include, among many others, the effects brought about by interatomic interactions, notably the additional line broadening (collisional, transit time and saturation broadening) and the reduction in Doppler broadening caused by a finite mean free path length (Dicke narrowing). The exact modelling of the absorption line shape is the main drawback, but semiclassical models can be successfully employed, provided that the Doppler width is retrieved from an extrapolation to zero pressure with the required uncertainty. The gas pressure varies from 0.1 to 500 Pa, which is orders of magnitude smaller than the pressure of the AGT and DCGT experiments. Therefore, this method is very complementary to the other gas-based thermometers.

LPL in cooperation with LNE-CNAM undertook in 2007 a first proof of concept of the DBT and obtained a relative uncertainty of 1.9×10^{−4} [17] for the Boltzmann constant. They used an ammonia line (^{14}NH_{3}) probed by a CO_{2} laser spectrometer with a wavelength of 10.35 μm. In further refinements, LPL evaluated a series of 1420 ammonia spectra applying a Voigt line profile [19], and subsequently a Galatry line profile [20], and corrected it for the hyperfine structure of the absorption lines [24]. The French group deduced a relative uncertainty of 5×10^{−5} [24]. In 2013, LPL published a complete analysis of the line shape of ammonia considering a speed-dependent Voigt profile corrected for hyperfine structure. The absorption length of the gas cell is 37 cm in a single-pass configuration (SPC) or 3.5 m in a multipass configuration (MPC). The pressures range from 1 to 25 Pa in an SPC and from 0.1 to 2.5 Pa in an MPC. Simulated speed-dependent Voigt profile spectra corresponding to the optimal experimental conditions in an MPC (pressure below 2 Pa) were fitted and a type B uncertainty contribution to *k* of 2.3×10^{−6} is expected [47]. LPL is currently testing a compact spectrometer based on a widely tunable laser source—a quantum cascade laser. The tunability of the spectrometer is increased by more than three orders of magnitude. Together with the higher available intensity and the potentially lower amplitude noise of the quantum cascade laser, a large reduction in the time needed to record absorption spectra is expected. The group now believes that the speed-dependent Galatry profile is the most suitable line shape. They included three additional uncertainty contributions in the budget: laser linewidth, frequency sweeping range and absorption saturation. To avoid significant saturation and to finish with the same uncertainty estimate as in [47] the laser power has to be reduced considerably from 1 to 0.02 μW [48].

An Italian set-up at the Second University of Naples, in collaboration with the Polytechnic of Milan and INRiM, probed initially a CO_{2} line at a wavelength of 2 μm. The group obtained a relative uncertainty of 1.6×10^{−4} [18,49]. Experiments were performed at pressures between 70 and 130 Pa, a range for which Dicke-narrowing and speed-dependent effects could be neglected. Thus, the line shape is given by a Voigt profile that accounts for both Doppler and collision broadening mechanisms. The laser–gas interaction took place in a 12 cm long cell consisting of a cylindrical cavity inside an aluminium block. The group switched then to a water (H_{2}^{18}O) line at a wavelength of 1.39 μm and analysed in 2013 the spectra recorded at different gas pressures between 200 and 500 Pa [28]. This implementation of DBT was based on a pair of offset frequency-locked extended-cavity diode lasers. A refined spectral analysis procedure was 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. The determination of the Boltzmann constant resulted in a relative uncertainty of 24×10^{−6} [28]. This is the best result obtained so far by means of an optical method. The two single uncertainties with the largest contributions are type A and line shape modelling. A further reduction in uncertainties is expected by the use of a low-pressure (10 Pa) absorption cell with a maximum path length of 12 m, by increasing the number of spectra, by improving the signal-to-noise ratio and by removing the dither on the reference laser. As for this last upgrade, they have presently implemented a technique known as noise-immune cavity-enhanced optical heterodyne molecular spectroscopy for the highly sensitive detection of the sub-Doppler line. They then quantified the effects of absorption saturation and detector nonlinearity and considered three additional uncertainty components: finite detection bandwidth, the relativistic Doppler effect and the influence of spontaneous emission of the probe laser [50]. These contributions do not increase previous uncertainty estimates.

The group at the University of Adelaide, Australia, used an extended cavity diode laser and probed rubidium vapour at 780 nm and a caesium transition at 895 nm. To suppress the unresolvable shift of a Zeeman transition in caesium, a magnetic shield was built inside the vacuum chamber containing the absorption cell. By fitting a Voigt profile to the caesium absorption line they obtained a value for *k* with a relative uncertainty of 71×10^{−6} [33]. The dominating uncertainty comes from defining the Lorentzian component of the line shape. Other significant uncertainty contributions are optical pumping and etalon effects which arise from stray reflections.

The researchers at Hefei University, China, developed a cavity ring-down spectrometer for the determination of the Boltzmann constant. Compared with conventional direct absorption methods, the high sensitivity of cavity ring-down spectroscopy allows sufficient precision to be reached at lower sample pressures, which also reduces the influence due to collisions. By recording the spectrum of acetylene, C_{2}H_{2}, at 787 nm, they demonstrated a statistical uncertainty of 6 ppm in the determined linewidth values in a measurement of several hours at a sample pressure of only 1.5 Pa [51].

## 6. Summary and outlook

Table 1 gives a summary overview of the achievements since 1979 of the relevant primary thermometers in terms of the applied method, publication date, value of the Boltzmann constant and uncertainty. In addition, the relative differences from the 2014 CODATA adjusted value of *k* recently published on the Internet are given. The last column lists the preliminary, estimated weights of the contributions for the 2014 CODATA adjustment without taking into account the correlations between the determinations. Only those results with weights larger than 0.01 were usually considered by the CODATA TGFC. There are several determinations of *k* with AGT featuring a relative uncertainty of around 1×10^{−6}. Because of these and because the discrepancy between the LNE-CNAM 2011 and NPL 2013 experiments is resolved now, the relative uncertainty of the *adjusted* value of *k* is now 5.7×10^{−7}. A second group of experiments using DCGT and noise thermometry achieved relative uncertainties close to 4×10^{−6}. Among these experiments, there are certainly candidates suitable to meet the second CCT requirement for the new definition to have a relative standard uncertainty of less than 3×10^{−6}.

Table 2 has been deduced from the meeting of the CCT Task Group on the SI (TG-SI) in February 2015 and compares the findings described in the reports of 2012–2014. The uncertainties of the six determinations taken into account in the CODATA adjustment of 2010 in the fourth column are included in a box. The uncertainties of the determinations taken into account in the 2014 adjustment are marked in bold and are plotted in figure 1. In addition, the low-uncertainty determination of INRiM [30] published after the deadline for the 2014 adjustment and the CODATA adjusted values of 2006, 2010 and 2014 are shown in figure 1. By inspection of figure 1 the major progress achieved since 2006 is evident.

The unit of thermodynamic temperature *T*, the kelvin, is presently defined by the temperature of the TPW. The measurement uncertainty of the adjusted value of *k* would be transferred to the value of *T*_{TPW} if that *k*-value were taken to be the exact value of the Boltzmann constant and used to define the kelvin. Hence, if, for example, the 2014 CODATA-recommended value were fixed as the exact value of *k* tomorrow, the best estimate of *T*_{TPW} would still be 273.16 K. However, this value would no longer be exact as it is now the result of the current definition of the kelvin, but would become uncertain by 5.7×10^{−7}, which corresponds to only 0.16 mK.

In Resolution 1 of the CGPM ‘On the future revision of the International System of Units, the SI’ [1], the closing date for the publication of new data to be considered by the CODATA TGFC for the adjustment of the fundamental constants in preparation for the redefinition is set for 1 July 2017. We conclude here that the experiments currently underway to measure *k* will lead by mid-2017 to a relative standard uncertainty that will be small enough for the redefinition of the kelvin to be adopted by the 26th CGPM in 2018. Summarizing the progress described in this review, it is very likely that uncertainties of the Boltzmann constant based on different independent measurements with the required low uncertainties below 3×10^{−6} will be achieved.

To put the new definition of the kelvin into practice, a *mise en pratique* has already been recommended to the CIPM by the CCT [52]. The *mise en pratique* will allow direct determination of thermodynamic temperatures, particularly at temperatures far away from the TPW, in parallel with the realization described in the International Temperature Scale. Sections describing the operation of several primary thermometers have been prepared [53]. In the high-temperature range, this will considerably reduce the uncertainty of the realization of the kelvin for many purposes for which the need to refer back to the TPW is anomalous, such as radiation thermometry.

## Competing interests

I have no competing interests.

## Funding

I received no funding for this study.

## Acknowledgements

The comprehensive support of R. Gavioso (INRiM), G. Machin (NPL), M. Moldover (NIST), L. Pitre (LNE-CNAM), A. Pokhodun (VNIIM), P. Rourke (NRC), R. White (MSL), K. Yamazawa (NMIJ/AIST), I. Yang (KRISS) and J. Zhang (NIM), the members of the CCT Task Group on the SI and M. de Podesta, is greatly acknowledged.

## Footnotes

One contribution of 16 to a Theo Murphy meeting isssue ‘Towards implementing the new kelvin’.

- Accepted July 13, 2015.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.