Many new techniques for ensuring traceable temperature measurements at the point of use are being developed and some are approaching maturity. The aim of this study is to examine the formalism associated with traceability to the SI kelvin for these practical techniques, as well as to identify areas of research which should be a priority. First, the status quo of thermodynamic temperature realization and dissemination is summarized. Then the state of the art of two main types of thermometry which can potentially provide in situ traceability is discussed. These are self-validating thermometers which make use of the phase change of materials, and practical primary thermometers, examples of which are given in order of decreasing commercial readiness: relative primary radiometry, acoustic gas thermometry (AGT), Johnson noise thermometry (JNT) and Doppler broadening thermometry (DBT). It is shown that relative primary thermometry is, in general, much more likely to become a day-to-day practical reality than absolute primary thermometry, and that this has a significant bearing on what the formalism might look like regarding metrological traceability and demonstrations of equivalence. This article is part of the Theo Murphy meeting issue 'The redefined kelvin: progress and prospects'.

Historically, the realization of the base unit kelvin was exclusively possible via International Temperature Scales (ITS). The new definition of the kelvin and the accompanying Mise en Pratique of the kelvin (MeP-K) enables the alternative to use primary thermometry for the realization of the kelvin. During the last years, considerable improvements in primary thermometry in the low-temperature range have been achieved. Thereby, above 300 K, still a lot of work is necessary to be competitive with the existing ITS realization. In this contribution, the interplay between primary thermometry, ITS and scale carriers (platinum resistance thermometers, thermocouples) in a historical perspective is sketched. After a short introduction to primary thermometry in general and the established techniques within the MeP-K, the focus will lie on the challenges connected with thermodynamic temperature measurement above 300 K. The advantages and disadvantages of the different techniques will be discussed briefly, and a summary will be given showing that thermodynamic temperature T, TITS and scale carriers are highly connected and improvement in one field relies on improvement in the others. The background of the discussion will be the deficiencies of the existing ITS and thus the requirements for the temperature-measurement community. This article is part of the Theo Murphy meeting issue 'The redefined kelvin: progress and prospects'.

In 2019, the basic unit of thermodynamic temperature, the kelvin, was redefined by fixing the value of the Boltzmann constant, opening new avenues for implementing and disseminating the kelvin with lower uncertainty, especially at temperatures below 25 K. In response, the Mise en Pratique for the definition of the kelvin (MeP-K) (2019) has recommended several primary thermometry methods, including acoustic gas thermometry (AGT), dielectric-constant gas thermometry (DCGT), refractive index gas thermometry (RIGT) and Johnson noise thermometry (JNT), as viable alternatives for realizing and disseminating the kelvin. Since the International System of Units (SI) revolution, significant progress on implementing the new kelvin has been made below 25 K. This progress indicates that primary thermometry, particularly its relative variants, can offer promising practical options for realizing and disseminating thermodynamic temperature directly linking to the new kelvin below 25 K with lower uncertainty. This is very important for metrological applications of science and industry, which require precise and accurate temperature calibrations. This article is part of the Theo Murphy meeting issue 'The redefined kelvin: progress and prospects'.

Here we give an overview of the European Partnership in Metrology (EPM) project; Dissemination of the redefined kelvin (DireK-T). The background context will be introduced, including the redefinition and progress with realising the redefined kelvin thus far. An overview of the aims and objectives of the DireK-T project will be given and the paper will end with a description of the project outcomes and impact.

Recommended values are given for the difference between the thermodynamic temperature T and T90, the temperature on the International Temperature Scale of 1990, for temperatures below 4.2 K. The recommendations are based on a combination of primary thermometry and thermodynamic calculations. Functions for both the difference T − T90 and its uncertainty are provided.

In 2011, a working group of the Consultative Committee for Thermometry published their best estimates of the differences between the thermodynamic temperature T and its approximation (T-90), the temperature according to the International Temperature Scale of 1990, ITS-90. Since 2011, there has been a change in the definition of the kelvin and significant improvements in primary thermometry. A recent paper [J. Phys. Chem. Ref. Data 51, 043105 (2022)] updates the (T - T-90) estimates by combining and analyzing the old and new data. The new data has been obtained by four types of gas thermometry. Their uncertainty estimates are now comparable with the uncertainties in the best measurements of T and the uncertainties in ITS-90 realizations. The new estimates are the basis for updating the annex Estimates of the differences T - T-90 of the Mise en pratique for the definition of the kelvin in the SI. For users without primary thermometry capability, it is now possible to access thermodynamic temperature values T below 335 K with comparably small uncertainties via an ITS-90 calibration and the transfer applying (T - T-90). This is a way to bridge the existing gap between enormous effort for T measurements and comparably good access to T-90. The applications in this way are divers and increase with demands for decreasing uncertainties. Three examples are treated in this paper in different fields, such as (1) alternative pressure standard applying invers dielectric-constant gas thermometry, (2) thermophysical properties, where ab initio calculations of gas properties have made enormous progress, and (3) optical clocks.

Johnson noise thermometry (JNT) is a purely electronic method of thermodynamic thermometry. In primary JNT, the temperature is inferred from a comparison of the Johnson noise voltage of a resistor at the unknown temperature with a pseudo-random noise synthesized by a quantum-based voltage-noise source (QVNS). The advantages of the method are that it relies entirely on electronic measurements, and it can be used over a wide range of temperatures due to the ability of the QVNS to generate programmable, scalable, and accurate reference signals. The disadvantages are the requirement of cryogenic operation of the QVNS, the need to match the frequency responses of the leads of the sense resistor and the QVNS, and long measurement times. This review collates advice on current best practice for a primary JNT based on the switched correlator and QVNS. The method achieves an uncertainty of about 1 mK near 300 K and is suited to operation between 4 K and 1000 K.

The harmonization of international temperature measurements requires the high-accuracy realization of many different temperature reference points. This results from the feature of the intensive measurand temperature that temperatures cannot simply be divided or multiplied. Thus, the points must cover the whole range of interest, at present from 1 mK to a few 1000 K. Furthermore, instruments are necessary for the interpolation between the non-continuous guide values. This led to the establishment of International Temperature Scales (ITS). The ITS prescribe interpolation instruments and assign fixed temperature values to suitable phase transitions without uncertainty. The large temperature range can only be covered by applying very different phase transitions. This includes the classical transitions, namely triple, melting, and freezing points, but also second-order transitions, as superfluid and superconducting ones, and the very new eutectic or peritectic points of metal-carbon compositions. A high-accuracy realization requires a reliable uncertainty estimation. This is, therefore, the central topic of this review. Since a given non-ideal condition of a sample, especially the impurity content, cannot be reproduced as accurate as necessary, the fixed- and reference-point temperatures are defined for ideal substances under ideal conditions. Thus, the estimation of the uncertainty of the realizations must be based on estimating the magnitude of all physical effects influencing the observed phase-transition temperature. The application of this methodology is discussed in the paper as unifying topic independent of the individual problems to be solved. Furthermore, recommendations of the Consultative Committee for Thermometry are summarized, and own experiences are supplemented.

Recent advances regarding the interplay between ab initio calculations and metrology are reviewed, with particular emphasis on gas-based techniques used for temperature and pressure measurements. Since roughly 2010, several thermophysical quantities - in particular, virial and transport coefficients - can be computed from first principles without uncontrolled approximations and with rigorously propagated uncertainties. In the case of helium, computational results have accuracies that exceed the best experimental data by at least one order of magnitude and are suitable to be used in primary metrology. The availability of ab initio virial and transport coefficients contributed to the recent SI definition of temperature by facilitating measurements of the Boltzmann constant with unprecedented accuracy. Presently, they enable the development of primary standards of temperature in the range 2.5-552 K and pressure up to 7 MPa using acoustic gas thermometry, dielectric constant gas thermometry, and refractive index gas thermometry. These approaches will be reviewed, highlighting the effect of first-principles data on their accuracy. The recent advances in electronic structure calculations that enabled highly accurate solutions for the many-body interaction potentials and polarizabilities of atoms - particularly helium - will be described, together with the subsequent computational methods, most often based on quantum statistical mechanics and its path-integral formulation, that provide thermophysical properties and their uncertainties. Similar approaches for molecular systems, and their applications, are briefly discussed. Current limitations and expected future lines of research are assessed.

In 2011, a working group of the Consultative Committee for Thermometry published their best estimates of the differences between the thermodynamic temperature T and its approximation (T90), the temperature according to the International Temperature Scale of 1990, ITS-90. These consensus estimates, in combination with measurements made in accordance with ITS-90, are an important alternative to primary thermometry for those requiring accurate measurements of thermodynamic temperature. Since 2011, there has been a change in the definition of the kelvin and significant improvements in primary thermometry. This paper updates the (T − T90) estimates by combining and analyzing the data used for the 2011 estimates and data from more recent primary thermometry. The results of the analysis are presented as a 12th-order polynomial representing the updated consensus values for the differences and a sixth-order polynomial for their uncertainty estimates.

In 2020, Nature Physics 16, 177 (2020) presented a primary gas‐pressure standard, based on temperature and electrical measurements and ab initio calculations of the thermophysical properties of helium. It is compared against the world's most accurate primary mechanical pressure standard and the test is successful with a relative uncertainty of 5 parts per million (ppm) at about 7 MPa. This is feasible for two reasons. First, the experimental setup developed for the determination of the Boltzmann constant is capable of achieving a combined relative uncertainty on the level of 2 ppm and the primary mechanical pressure standard 1 ppm. The latter consists of two pressure balances with five piston‐cylinder assemblies. Second, the uncertainties of the theoretical calculations in 2020 are on the level of about 4 ppm. Within the last two years, significant improvement has been achieved for all theoretical quantities involved, and the uncertainty contribution of the theory is below the level of 1 ppm. With doubled sensitivity, the thermodynamic gas‐pressure standard is compared to the most accurate mechanical pressure standard. This stress test for theory and experiment is once again successful.

The component caused by the influence of impurities often dominates the uncertainty budgets of realisations of temperature fixed points. The Consultative Committee for Thermometry recommended, therefore, in section 2.1 Fixed Points : Influence of Impurities of the Guide to the Realization of the ITS-90 ( www.bipm.org ) two methods for estimating this component. One is based on summing up the individual effects of all relevant impurities. This method causes huge efforts. The second one requires a reliable information on the overall impurity concentration, which is usually not supplied by the producers of the fixed-point materials. In the paper, it is analysed if the determination of the residual resistance ratio r 0 can aid in getting the required information [ r 0 means the ratio of the electrical resistivities ρ at a sufficiently low temperature and at 293 K, usually r 0 = ρ (4.2 K)/ ρ (293 K)]. The crucial criterion is the sensitivity of r 0 for the influence of an impurity on the fixed-point temperature, i.e. the magnitude of the derivative d T l /d r 0 of the liquidus-line temperature T l with respect to r 0 . As examples, the developed methodology is applied to three fixed points of the ITS-90, namely the freezing points of indium, zinc, and aluminium. It can be shown that the use of r 0 can reduce the number of impurities, for which the effortful determination of the individual concentrations is necessary, by an order of magnitude, namely to about 10 or below.

We have measured the refractive index of helium using a quasi-spherical copper microwave resonator at five different temperatures in the interval between the triple point of hydrogen at 13.8 K and the triple point of xenon at 161.4 K for pressures up to 380 kPa. From these results and additional measurements of the refractive index of neon near 54.4 K, 83.8 K and 161.4 K we determine the differences ( T − T 90 ) between the thermodynamic temperature T and its approximation T 90 by the International Temperature Scale of 1990 (ITS-90). We have estimated the isothermal compressibility of copper and the effective compressibility of our microwave resonator by different methods including resonant ultrasound spectroscopy (RUS) and microwave measurements with helium at 273.16 K. We compare the results of these compressibility estimates and discuss the associated uncertainty. From the refractive index measurements, we estimate the second density virial coefficient of helium and neon which are found in good agreement with the ab initio calculations of the same properties.

This short note is a supplement to the paper ‘Primary thermometry from 2.5 K to 140 K applying dielectric-constant gas thermometry’ (2017 Metrologia 54 141–7). It deals with thermodynamic temperature measurements at the boiling point of the heavy isotope of helium (4He, 4 K) and the triple points of hydrogen (14 K) and neon (25 K). This is of special interest because recent data published in this temperature range have an unexpectedly large spread. The results presented in this short note show that the International Temperature Scale of 1990, ITS-90, is thermodynamically correct at 4 K and 14 K, but too high by about half a millikelvin at 25 K. Furthermore, care is given to two aspects of dielectric-constant gas thermometry in the low-temperature range. First, dielectric-constant gas thermometry can be applied for practical primary thermometry. The measurement of only one isotherm in one day yields a temperature value with an uncertainty order of a few tenths of a millikelvin. Second, the use of recent ab initio values for the virial coefficients of helium as a measuring gas can reduce the efforts significantly. Even one data pair of pressure and dielectric constant can yield a thermodynamic temperature value of uncertainty less than a factor of two larger compared to the uncertainty of the usually used fit evaluation. For 4 K and 25 K, the achieved final uncertainty for T is still larger than the realization uncertainty of T 90. Thereby in the case of 14 K, both uncertainties are comparable.

We have measured the refractive index of helium using a quasi-spherical copper microwave resonator at five different temperatures in the interval between the triple point of hydrogen at 13.8 K and the triple point of xenon at 161.4 K for pressures up to 380 kPa. From these results and additional measurements of the refractive index of neon near 54.4 K, 83.8 K and 161.4 K we determine the differences (T − T90) between the thermodynamic temperature T and its approximation T90 by the International Temperature Scale of 1990 (ITS-90). We have estimated the isothermal compressibility of copper and the effective compressibility of our microwave resonator by different methods including resonant ultrasound spectroscopy (RUS) and microwave measurements with helium at 273.16 K. We compare the results of these compressibility estimates and discuss the associated uncertainty. From the refractive index measurements, we estimate the second density virial coefficient of helium and neon which are found in good agreement with the ab initio calculations of the same properties.

New interatomic potential energy and interaction-induced polarizability curves for two ground-state neon atoms were developed and used to predict the second density, acoustic, and dielectric virial coefficients and the dilute gas shear viscosity and thermal conductivity of neon at temperatures up to 5000 K. The potential energy curve is based on supermolecular coupled-cluster (CC) calculations at very high levels up to CC with single, double, triple, quadruple, and perturbative pentuple excitations [CCSDTQ(P)]. Scalar and spin-orbit relativistic effects, the diagonal Born-Oppenheimer correction, and retardation of the dispersion interactions were taken into account. The interaction-induced polarizability curve, which in this work is only needed for the calculation of the second dielectric virial coefficient, is based on supermolecular calculations at levels up to CCSDT and includes a correction for scalar relativistic effects. In addition to these first-principles calculations, highly accurate dielectric-constant gas thermometry (DCGT) datasets measured at temperatures from 24.5 to 200 K were analyzed to obtain the difference between the second density and dielectric virial coefficients with previously unattained accuracy. The agreement of the DCGT values with the ones resulting from the first-principles calculations is, despite some small systematic deviations, very satisfactory. Apart from this combination of two virial coefficients, the calculated thermophysical property values of this work are significantly more accurate than any available experimental data.

Since 20 May 2019, the temperature unit kelvin is no longer defined by fixing the value of the temperature of the triple point of water, but by a fixed value of the Boltzmann constant (k = 1.380649 x 10(-23) J/K). Through this, mechanical and thermal energy are directly connected, which is physically a principal progress. However, the new definition is very abstract because there is no reference to rules for the realization of the unit. Practical recommendations ("Mise en Pratique") have been, therefore, prepared by international committees. Primary thermometry, which is directly based on the laws of thermodynamics, is of special importance for this. But the new definition does not change the role of the International Temperature Scales. Thus, the efficient system of traceable temperature measurements is maintained.

New measurements of thermodynamic temperature T with Dielectric-Constant Gas Thermometry (DCGT) were performed at PTB from 50 K to 200 K. Particular care was taken to check for possible systematic sources of errors by performing experiments applying three working gases, namely helium, neon, and argon, the polarizability of which differs by a factor of up to eight. Together with former DCGT values of thermodynamic temperature the new results yield a consistent dataset in the range from 30 K to 200 K. This dataset is in good agreement with the newest results of Acoustic Gas Thermometry (AGT) and Refractive-Index Gas Thermometry (RIGT), which have quite different sources of uncertainty compared with DCGT. The combination of these DCGT, AGT, and RIGT data with the ‘Estimates of the differences between thermodynamic temperature and the ITS-90’, being as an appendix of the ‘Mise en pratique for the definition of the kelvin in the SI’ the present-day recommendation of the Consultative Committee for Thermometry, yields a new function T − T90 versus ITS-90 temperature T90 for the range from 35 K to 195 K, the uncertainty of which is reduced by a factor up to about four.

Since the redefinition of the base unit kelvin via fixing the Boltzmann constant in 2019, it has been possible to realise the unit applying different gas-thermometry methods in accordance with the 'Mise en pratique for the definition of the kelvin in the SI'. For this application, the use of data for the gas properties resulting from ab initio calculations is of special interest because it makes primary thermometry much easier. But since a rigorous estimation of the uncertainty of theoretical calculations is at least very complex, if not impossible, a check of the results by a comparison with highly-accurate experimental values is mandatory. Such a check is performed for the second virial coefficient of helium, which is a widely used measuring gas, in the temperature range from 3.7 K to 273 K. For obtaining highly-accurate second-virial-coefficient values (relative uncertainty at a few tenth of a percent level or even better), isotherms were measured with the PTB dielectric-constant gas thermometer. The highly-accurate isotherm data were evaluated by fitting, applying an extended working equation for the dependence of the gas pressure on the dielectric constant. The comparison with the results of the latest ab initio calculations shows coincidence within the combined uncertainty estimates.