## Abstract

The thermodynamic temperature of the point of inflection of the melting transition of Re-C, Pt-C and Co-C eutectics has been determined to be 2747.84 ± 0.35 K, 2011.43 ± 0.18 K and 1597.39 ± 0.13 K, respectively, and the thermodynamic temperature of the freezing transition of Cu has been determined to be 1357.80 ± 0.08 K, where the ± symbol represents 95% coverage. These results are the best consensus estimates obtained from measurements made using various spectroradiometric primary thermometry techniques by nine different national metrology institutes. The good agreement between the institutes suggests that spectroradiometric thermometry techniques are sufficiently mature (at least in those institutes) to allow the direct realization of thermodynamic temperature above 1234 K (rather than the use of a temperature scale) and that metal-carbon eutectics can be used as high-temperature fixed points for thermodynamic temperature dissemination. The results directly support the developing *mise en pratique* for the definition of the kelvin to include direct measurement of thermodynamic temperature.

## 1. Introduction

This paper describes the assignment of, and provides values for, low-uncertainty thermodynamic temperatures to the points of inflection of the melting transition curves of the metal-carbon eutectics Re-C, Pt-C and Co-C, thus realizing the hope expressed in Yamada's original 1999 papers [1,2] that these should become high-temperature fixed points (HTFPs) with assigned thermodynamic temperatures [3,4]. It also provides thermodynamic temperature measurement results of the freezing temperature of the Cu fixed point.

In this work, the thermodynamic temperatures of these fixed points have been determined through direct measurement of the radiance of a blackbody cavity surrounded by the fixed-point material. This measurement [5] does not rely on other reference temperatures, but is determined from Planck's law and hence directly linked to the Boltzmann constant.

The evolving *mise en pratique* for the definition of the kelvin (*MeP*-K) [6,7] will, in its second edition, encourage the realization and dissemination of thermodynamic temperature either directly (primary thermometry) or indirectly (relative primary thermometry) via fixed points with assigned reference thermodynamic temperatures and associated uncertainties. In this paper, we show that primary thermometry realized by radiometric methods is sufficiently mature at high temperatures to reconcile results from nine National Metrology Institutes (NMIs). We also provide reference thermodynamic temperatures, and uncertainties, to enable relative primary thermometry.

In addition, these fixed points could provide the basis for new temperature references for the calibration of radiation thermometers at temperatures above the freezing point of silver (1234.93 K), and will potentially be able to reduce the uncertainties associated with high-temperature scale realization [8] compared with the International Temperature Scale of 1990 (ITS-90) [9].

The collaborative research that has led to this paper was initiated in 2007 by Working Group 5 of the Consultative Committee for Thermometry (CCT)^{1} [10,11]. The aim of the research was to

— develop stable, robust cells for Re-C, Pt-C and Co-C;

— to perform long-term stability tests on these cells [12,13];

— to provide a ‘definitive reference set’ of fixed points [14,15];

— to understand uncertainties associated with furnace effects [16–18] and cell impurities [19]; and

— to perform an initial comparison of filter radiometric techniques [20].

The work of this paper is the culmination of this programme and was performed as part of the EU-funded project ‘Implementing the new Kelvin’ (InK) [21].

This paper initially covers the design of the measurement campaign, a brief description of the design, manufacture and selection of the cells used, the methods for measuring thermodynamic temperature radiometrically and the corrections and cell and furnace effect uncertainties. It then describes the method used to analyse and combine the measured values and finally provides the reference point-of-inflection temperatures (freezing temperature of Cu). We have produced a document provided as part of the electronic supplementary material that gives more details about most of these aspects, including the individual temperature values measured by the participants.

## 2. Design of the measurement campaign

### (a) Measurements made

The purpose of the measurement campaign was to obtain a consensus estimate of the transition temperature of the three metal-carbon eutectics Co-C, Pt-C and Re-C and of the freezing point of Cu. Four HTFP cells of each type were selected from a larger group [14,15] to be the cells used in the measurement campaign (§3). These cells were circulated, in two groups (designated cells A and B and designated cells C and D) to the participating laboratories between February 2013 and April 2015.

This main measurement campaign started and ended with relative temperature measurements by NPL to provide information on the stability of the cells (§7). Thermodynamic temperature measurements were made using radiometric techniques (§4) by the other participating NMIs. Cells were measured in the following sequence: cells A and B by NPL, NRC, NIST, PTB, NIM, NMIA, NPL, and cells C and D by NPL, VNIIOFI, PTB, LNE-Cnam, CEM, NPL, NMIJ.

### (b) Different melt and freeze steps

Measurements were made at each institute by heating the cells within a furnace and monitoring the radiance of the emitting cavity using a spectroradiometric instrument. The measurement campaign protocol specified for each cell between 8 and 10 ‘cycles’, spread over 2 days, where a cycle is defined as taking a cell from a temperature below the melt, melting it, holding it above the melting temperature and then cooling it through the freeze. To test the eutectics for sensitivity to melt and freeze, the protocol specified for each cycle one of three furnace freeze-initiation temperature steps and one of two furnace melt-initiation steps. Some participants performed all the defined cycles and others were able to perform only a subset of these. For copper, participants were encouraged to ‘vary the operational conditions’ with no furnace conditions specified.

### (c) Pre-analysis by participants

All the raw measurement data by the participants (the temperature of the cavity during individual melt and freeze cycles as measured from the radiance) were pre-analysed in a common way using a supplied spreadsheet. For the eutectic HTFPs the spreadsheet calculated, for each cycle, the point of inflection of the melt transition, as well as the so-called ‘upper-limit temperature’ needed to estimate the more fundamental liquidus temperature [18,19]. The point of inflection (time and temperature) was calculated using a method described in [22–24]. The method defines the ‘start of melt’ and the ‘end of melt’ as the points where the magnitude of the first derivative of the melt curve reaches a maximum and determines the point of inflection analytically from a cubic function fitted to the central half of the melting plateau (figure 1). In some cases, it was necessary to shorten the section of the melt curve over which the cubic function was fitted, because the end-of-melt was not always a single well-defined point. For Cu, the measured temperature was determined by taking the mean temperature in the central third of the freeze plateau. The temperature values obtained from these calculations were taken to be the ‘measured temperatures before corrections’.

## 3. The high temperature fixed point and copper cells

### (a) Cell design, manufacture and selection

The cells consist of a graphite crucible incorporating a blackbody cavity and containing the fixed-point material and were built as part of the CCT international research programme [11]. The long-term stability tests had shown that cells of a ‘hybrid design’ (with a sacrificial inner sleeve and C/C sheet to take the stress of different thermal expansion coefficients of the ingot and graphite) were more robust and of better performance than those of a simpler design [12], and so hybrid design cells were produced. Nine NMIs developed [15] altogether five to seven cells of each type (i.e. Re-C, Pt-C, Co-C, Cu). All cells had an outer diameter of 24 mm, a length between 40 mm and 49 mm, with a blackbody cavity of 3 mm aperture and a length between 27 and 36 mm. These different cells were investigated and tested [14] to select the most suitable cells for measurement. The details of the construction, selection and the sources of the metals used in the selected cells are given in [23].

### (b) Cell impurities

The cells are all made from materials with some level of impurities (in all cases impurity levels were nominally less than 20 ppm by mass). The electronic supplementary material includes information about impurity assays performed on the cell material and the calculations performed [19] to estimate the effect of these impurities on the measured transition temperature. In practice, different impurity assays, even when performed on nominally identical metal powders (obtained by different NMIs from the same supplier, although with different batch numbers), show very different results. Because of these differences, the impurity information was not used to provide a correction to the determined temperature, but instead was used as a way to evaluate the cell effect uncertainty due to impurities (table 3).

## 4. Radiometric temperature measurements

### (a) Measurement of radiance

Primary radiometric thermometry requires the measurement of the absolute spectral radiance (units W m^{−2} sr^{−1} nm^{−1}) of a blackbody cavity. Planck's law is then used to convert radiance to temperature. The measurement of radiance requires determining the optical power (W) in a particular geometry (m^{2} sr^{1}) in a finite wavelength band (nm), and is usually performed using a filter radiometer [25,26]. There are four methods (power, irradiance, hybrid and radiance) proposed in [5] for the future edition of the *MeP*-K for calibrating a filter radiometer. These methods all rely on an optical power measurement with a cryogenic electrical substitution radiometer, a monochromatic optical radiation source and a defining geometry of two parallel, circular, collinear apertures of known physical dimensions and separation. The methods differ only in the location of that geometry in the calibration chain. In this work, all four methods were used and, in addition, a fifth method, the ‘illuminance method’ [27] has been used.

The radiance method and the hybrid method calibrate instruments that include an imaging lens (or mirror), which can be used directly to measure the small cavities of the HTFPs in a *single-step process*. The irradiance, illuminance and power methods are used to calibrate systems without a lens; these can only determine the temperature of a large area blackbody. Therefore, NMIs using these methods have introduced a *two-step process*. Here the filter radiometer is used to measure a reference large area blackbody and this is then used to calibrate an imaging radiation thermometer, which measures the smaller HTFP cavity. Two laboratories using radiance and hybrid methods also carried out a two-step process. In this case, the radiation thermometer was used to compare the circulating cells with their own HTFP cells and at a later, more convenient, date their own HTFP cells were calibrated with their primary radiometric method.

All methods are spectral methods where the radiance of the blackbody is measured over a narrow spectral band. Three methods were used for spectral filtering. Some instruments, including all the radiation thermometers used in a two-step process and some of the filter radiometers, relied on a narrowband interference filter. Other filter radiometers relied on a broadband glass filter. One participant had a radiance-mode instrument where the spectral selection was through the grating and slit of a monochromator.

### (b) Approaches used by the participants

The calibration approaches for the participants measuring absolute radiometric temperatures are described in table 1. NPL took additional measurements using a narrow band radiation thermometer (650 nm) at the start and end of the measurement campaign to determine cell stability. These measurements were made by reference to a copper point blackbody.

The uncertainties in participants’ measured values (calculated according to [40,41]) were categorized into those associated with effects that were (i) common to all measured values made by the participant, e.g. calibration of the instrument, size-of-source effect, etc., and (ii) random (varied from measured value to measured value, e.g. noise, fitting quality). The standard uncertainties are given in table 2.

## 5. Emissivity and the temperature drop effect

### (a) Emissivity

Emissivity calculations [42] have been made for the blackbody cavities in the cells used in this measurement campaign. The emissivity of a cell depends on its geometry (which is sufficiently similar for all cells here) and on the emissivity of the wall material (graphite). The largest source of uncertainty is from the unknown emissivity of graphite. There is a negligible sensitivity of the cavity emissivity to the temperature gradient of the furnace in front of the cell and hence to the specific furnace used. For the purposes of this analysis, all cells were assumed to have an (isothermal) emissivity of 0.999 72 with a standard uncertainty (from the modelling assumptions) of 0.000 11.

For all measurements a correction to the measured transition temperature was made based on this emissivity and the centroid wavelength of the instrument used to measure the circulating HTFPs. The correction that has been added to the measured temperature is based on Wien's approximation and the assumption that the detector is narrowband and is
5.1where *c*_{2} is the second radiation constant, *T*_{meas} is the measured temperature, *λ*_{0} is the centroid wavelength and *ε* is the cavity emissivity. The correction was calculated from the emissivity value separately for each participating NMI; the associated uncertainty is given in table 3.

Where participants had performed a two-step process and the filter radiometer and pyrometer are at different wavelengths, the emissivity of the reference blackbody affects the transfer. This effect is included in those participants’ uncertainty budgets.

### (b) Temperature drop

The temperature drop effect refers to the difference in temperature between the solid–liquid interface and the back wall of the cavity where the temperature is measured. In effect, it is the temperature drop across the thickness of the cavity (back) wall. As we do not know with precision what that thickness is after cell construction and as this effect is similar for all furnaces and for all cells [16,42,43], no correction has been applied for temperature drop. The estimated uncertainty, taken to be the full value of the calculated temperature drop, is given in table 3.

## 6. Furnace effect

It has been observed that use of different furnaces can affect the realized (point of inflection) melting temperature of a HTFP, e.g. [16,18]. Different furnaces were used by the participants. NRC and VNIIOFI used a high-temperature blackbody furnace (the HTBB3500 made by VNIIOFI) for melting the cells [44,45]. PTB, NIM, CEM, NMIJ (and NPL for the relative measurements) used a Chino IR-R80 furnace [46], and NIST and NMIA used a Thermogauge furnace [47]. LNE-Cnam used both a Chino IR-R80 and a VNIIOFI HTBB3200 furnace (with Cell C and Cell D in different furnaces), and used a homebuilt 3-zone furnace for the Cu cells.

There are at least two ways in which the type of furnace could affect the measured temperature. First, there is a small effect from the furnace temperature profile on the temperature drop and emissivity of the cavity (§5). Studies [16,42] considered extreme (theoretical) furnace temperature profiles for different back-wall thicknesses and the maximum difference between furnaces is approximately 40 mK for Re-C, 10 mK for Pt-C and 3 mK for Co-C. Secondly, both the furnace temperature profile *and* the furnace inertia (which can be characterised by the time the furnace takes to achieve the set point above the melt) can affect the shape of the melt curve, and hence the calculated point-of-inflection temperature [18].

In [48], measurements were made of the effect of the furnace temperature profile and the position of the cell in both a Chino IR-R80 and a VNIIOFI HTBB3200 furnace by displacing within the furnace to positions of lower thermal uniformity. In both furnaces, it was found that when there was a temperature gradient of 10 K along the cell the measured temperature for the melting curve point of inflection drops by 25 mK, 75 mK or 100 mK for Co-C, Pt-C and Re-C, respectively. It is possible to optimize the furnace to improve the uniformity [45,49] and all NMIs performed some pre-assessment to help ensure that the cells were placed in the most uniform part of the furnace. Nevertheless, it is probable that cell placement within the furnace explains some of the variability in cell-to-cell differences described in §7.

Recent studies [50] give a difference of approximately 140 mK for two different Co-C cells in two different furnaces (one low thermal inertia and the other high thermal inertia). Taking this observed difference as the extreme of a rectangular distribution (and hence dividing by to obtain a standard uncertainty), and adding, in quadrature, a contribution for the effect of furnace temperature gradients, yields an estimate of approximately 50 mK furnace effect uncertainty for Co-C. Then, by extrapolating the former effect with the ratio of temperature for Cu, Pt-C and Re-C and adding in quadrature the effects of furnace temperature gradients gives a 30 mK furnace effect uncertainty for Cu, 90 mK for Pt-C and 120 mK for Re-C. These uncertainties are included in the analysis; see table 3.

## 7. Observed cell stability and cell-to-cell differences

The electronic supplementary material includes information on the cell-to-cell transition temperature differences as measured during the pre-assessment of the cells [14], at the beginning and the end of the measurement campaign by NPL through relative measurements, and by PTB during the measurement campaign. The solution to the least-squares analysis (LSA, §8) also provides information about differences between cells. The results show that cell-to-cell differences at any individual NMI can be of the order of 150 mK for the eutectics and 50 mK for Cu.

The only significant observable trend during this measurement campaign is for Re-C cell A. The difference between cell A and the other cells increases from the pre-assessment to NPL's round 1, to PTB's measurement and to NPL's round 2 measurement. These differences suggest a real drift of this cell and this is accounted for in the analysis model (§8b). For the other cells there is no clear trend—for example, the NPL round 1 measurements of Pt-C suggest that cell C has the lowest temperature, whereas the PTB measurements suggest it has the highest temperature. The observed small cell-to-cell differences are likely to be due to cell positioning within the furnace on the day of measurements (§6) rather than real biases between the cells. This effect is likely to explain the ‘missing random component’ indicated by the *χ*^{2}-test (§9a).

## 8. A model for data analysis

### (a) Basic model

During the measurement campaign a total of 108, 138, 145 and 73 measured values were taken for Re-C, Pt-C, Co-C and Cu, respectively. For determining the point-of-inflection of the melting curve of the eutectics and the mean freezing temperature for Cu, these measured values are considered equivalent estimates. They are not, however, independent estimates as they have an associated correlation.

Measured values by a single NMI have correlations because they rely on a single calibration of a single radiometer. Measured values of the same cell by different participants are also correlated because impurities within that cell affect all measurements of that cell. There may be a small correlation between measured values of cells in the same type of furnace.

The model for a measured value of temperature by a participant takes into account those correlations:
8.1where *T*_{ijkl} is the result of the *l*th measurement by laboratory *i* of cell *j* in furnace *k*; *δT*_{ε,i} is the correction to this value from the emissivity calculation for the wavelength of NMI *i*'s instrument (§5a); *T*_{0} is the joint best estimate of the transition temperature of that HTFP (which we want to determine); *S*_{i} is the systematic offset of laboratory *i*, arising from scale effects: radiometer calibration and operational conditions at that laboratory; *I*_{j} is the systematic offset of cell *j*, arising from cell impurities, emissivity and temperature drop effects; *F*_{k} is a systematic offset for measurements made in the type of furnace *k* and *R*_{ijkl} is the random offset for that specific measurement.

Our initial best estimates for all parameters on the right-hand side of expression (1.2), except the desired transition temperature *T*_{0}, is that they are all zero (the only term for which we make a correction is the emissivity effect), but they have uncertainties. We denote the standard uncertainty associated with a quantity *X* by *u*(*X*),

*u*(*S*_{i}) is the standard uncertainty described in laboratory *i*'s uncertainty budget for systematic effects,

*u*(*R*_{ijkl}) is the standard uncertainty described in laboratory *i*'s uncertainty budget for random effects for the measurement of cell *j* in furnace *k* for the specific measurement *l*,

*u*(*I*_{j}) is the standard uncertainty for cell effects, and

*u*(*F*_{k}) is the standard uncertainty for furnace effects.

We obtain *u*(*S*_{i}) and *u*(*R*_{ijkl}) from the provided uncertainty budgets of the NMIs, as discussed in §4b, whereas *u*(*I*_{j}) depends on all effects that vary from cell to cell. These effects include emissivity (for post-correction uncertainty; see §5a), the temperature drop effect (§5b) and impurities (§3b). This cell effect uncertainty should explain the differences observed from cell to cell. The combined cell uncertainty due to these three effects is given in table 3. There is an additional line labelled ‘cell effect uncertainty to use’. This uncertainty was used in the analysis and was either the combined uncertainty from the three known effects or the largest measured difference between cells during the pre-assessment, whichever was the greater (under the assumption that there were additional effects, not understood in the uncertainty analysis that explained this difference). Table 3 also gives the uncertainty associated with furnace effects, *u*(*F*_{k}), as discussed in §6. Note Re-C cell A's uncertainty is increased to account for the drift of this cell (§8b(ii)), and that for Co-C cell B is increased as this was made with less pure material.

### (b) Variations in approach for Re-C

For Re-C, the basic model needed two modifications. One was to account for a change in VNIIOFI's measurement process and the other was to account for a drift in cell A.

#### (i) Re-C VNIIOFI measurements

VNIIOFI made measurements for Re-C cell D with the radiation thermometer at a different distance to that used for all the other cells (and from the calibration distance). This means that there was an uncertainty component that was common for all measurements made of cell D, but not for the measurements made of cell C. Therefore, we define for Re-C only:
8.2where *S*_{2VNIIOFI,D} is an additional parameter for the cell D measurements also having an expectation value of zero and an associated uncertainty given by VNIIOFI's uncertainty budget.

#### (ii) Re-C cell A's instability

Re-C cell A drifted during the measurement campaign (§7). To account for this, for Re-C cell A only, we add an additional term to the model, which becomes
8.3There are (separate) *δI*_{A,i} terms for each of the five NMIs that measured cell A, which allows the cell to have a different nominal value at each NMI. All of these have an expected value of zero and an associated uncertainty of 0.2 K. The cell effect uncertainty, i.e. *u*(*I*_{A}), is increased to 0.3 K (cf. 0.16 K for the other cells, as in table 3). Note that this drift is treated as an uncertainty term and is not corrected for.

### (c) Approach to solving the model

In an earlier paper [23], we have suggested that the model would be solved via a generalized weighted mean. However, during the analysis we made the decision to use an LSA instead as this would provide additional diagnostic information. The generalized weighted mean, and the simple mean, were also calculated to understand the sensitivity of the result to the method of analysis. For LSA, the first requirement is to obtain a covariance matrix for the measured values and the second requirement is to obtain the least-squares solution.

#### (i) Developing a covariance matrix

Covariance comes from common effects in the model, i.e. a common value of *S*_{i},*I*_{j} or *F*_{k} for two measured values (by definition *R*_{ijkl} takes a different value for each measurement). We can, therefore, calculate the covariance between any two measured values as the sum of the squares of all uncertainty components that are common to the two measured values. This calculation is described in more detail in the electronic supplementary material.

We simplify calculation of the covariance matrix by introducing a design matrix ** A**, which has rows representing every individual measurement and columns representing all the parameters, i.e. the cell effect for all four cells, the laboratory offset for all the participants and the furnace effect for all the different types of furnace. An element of the design matrix takes the value 1 when the parameter is relevant for that measured value and 0 when it is not. We also create a covariance matrix for the parameters (cell effects, laboratory offsets and furnace effects),

*U*_{par}, which is a square matrix, with the diagonal terms being the variance associated with each of these parameters and all off-diagonal elements being zero.

The covariance matrix associated with the set of measured values is obtained through simple matrix algebra:
8.4where diag{*u*^{2}(*R*_{ijkl})} is a square matrix with the squared uncertainties in random effects on the diagonal and zeros elsewhere.

#### (ii) Least-squares solution

Our model contains 17 unknown parameters, namely *T*_{0}, the four quantities *I*_{j} (one for each cell), nine quantities *S*_{i} (one for each NMI), and three quantities *F*_{k} (one for each furnace type). The rank of the design matrix ** A** is 12, which implies there are only 12 independent parameters. Therefore, to solve the model we reduce the number of parameters to 14 by not solving for the three furnace values

*F*

_{k}(nor the additional terms introduced for Re-C in §8b). We also introduce two resolving constraints, increasing the rank of the matrix by two to 14. These are 8.5The first constraint says that the sum of the cell offsets is zero. The second constraint states that the weighted mean of the laboratory systematic biases is zero. The weights used are proportional to the reciprocals of the variances in the NMIs’ measured values. Note that this second constraint is making the assumption that there is no systematic error to the radiometric techniques the participants used and that there was sufficient variation in technique (§4b) that ‘on average the participants obtained an unbiased estimate of the true measured vale’. The first constraint will provide a bias to the determined measured value, because the cells will show a bias from the true melting temperature. This is discussed in §9b(ii).

To obtain a least-squares solution, we create a new design matrix ** B**. This is similar to design matrix

**but with an extra column for**

*A**T*

_{0}taking the value 1 for all the temperature measurement rows and two extra rows representing the two constraints. It also excludes the furnace offsets and the extra columns to account for variations in the measurement model (§8b). The solution gives a vector,

**, containing estimates of**

*X**T*

_{0}and the cell offsets and NMI offsets. Formally, 8.6here

**is the design matrix as above;**

*B*

*U*_{T2}is the covariance matrix for the measured values with the two extra rows and columns of mostly zeros and an arbitrary diagonal term for the constraints and

*T*_{2}is the column vector listing all the measured temperature values followed by two zeros for the constraints.

### (d) Testing the model

The consistency of the model and the measured values was assessed using the *χ*^{2} test. The observed *χ*^{2} value was obtained with an equation that considers the full covariance matrix [51]:
8.7here ** E** is a column vector of the model residuals, namely the differences between the measured temperatures and the calculated

*T*

_{0}(i.e. it contains the

*T*

_{ijkl}−

*T*

_{0}) and

*U*_{T}is the covariance matrix for the measured values. It is also useful to consider the Birge ratio [52] when testing the model, which normalizes the observed

*χ*

^{2}, through 8.8where

*ν*is the degrees of freedom, the number of measured values minus the number of model parameters. The Birge ratio is expected to have a value of 1. Values higher than 1 indicate that the model does not explain the data or that some of the uncertainties are underestimated. Values lower than 1 indicate that uncertainties have been overestimated. The

*χ*

^{2}test can be considered to pass if the Birge ratio is less than 1.

If the model is inconsistent with the data from which its parameters are estimated, an increase in some of the data uncertainties can achieve model-data consistency. Any increase should of course be justified on scientific grounds. Such consistency should be obtained whenever, as here, inferences are to be made from the modelling. In the analysis below, we add a small value to the uncertainty associated with random effects for the participants to ensure a Birge ratio less than 1. This suggests that the random effects may have been slightly underestimated. The physical explanation for this may be the sensitivity of the cell to positioning within the furnace. It should be noted that the added uncertainty values were extremely small compared with all other uncertainties and had almost no effect on the determined temperature or its uncertainty.

## 9. Results

### (a) Individual measured results and model consistency

#### (i) Re-C results

The Re-C results are shown graphically in figure 2. The individual measured values from the participants are given, along with the temperature *T*_{0} and its associated uncertainty (§9b(i)), both calculated from the LSA. All individual measured results are statistically consistent with the determined temperature.

The calculated *χ*^{2}_{cov} value using the full covariance matrix (equation (8.7)) is 325, compared with 94 degrees of freedom (d.f.). That *χ*^{2}_{cov} value is statistically too large; however, adding 27 mK to the uncertainty associated with random effects for all measured values, reduces the *χ*^{2}_{cov} value to 94, providing an ‘ideal’ Birge ratio of 1 (§8d). The consequence is to alter the calculated temperature *T*_{0} by 4 mK and its uncertainty by less than 1 mK.

To understand the sensitivity to the analysis technique, the simple mean (which does not use the covariance matrix) is 48 mK lower than the value *T*_{0} and the generalized weighted mean (which uses the covariance matrix) is 38 mK higher than *T*_{0}.

#### (ii) Pt-C results

The Pt-C measurement results are given in figure 3. Here there is very good consistency between the individual measured values and the calculated temperature *T*_{0}.

The calculated *χ*^{2}_{cov} using the full covariance matrix (equation (8.7)) is 178, compared with 124 d.f. This *χ*^{2}_{cov} value is statistically too large; however, adding 5 mK to the uncertainty associated with random effects for all measured values, reduces the *χ*^{2}_{cov} value to 124 and provides a Birge ratio of 1 (§8d). This change alters both the calculated temperature *T*_{0} and its uncertainty by less than 1 mK.

Note, to understand the sensitivity to the analysis technique, that the simple mean is 9 mK lower than the value *T*_{0} and the generalized weighted mean (which uses the covariance matrix) is 15 mK higher than *T*_{0}.

#### (iii) Co-C results

The Co-C measurement results are given in figure 4. Here there is a good statistical consistency between the individual measured values and the calculated temperature *T*_{0}. There do appear to be offsets between different cells measured by the same participant and, in some cases measurements of the same cell on different days (e.g. in the NIM data, cycles 67–71 were on one day and cycles 72–76 on a second day). It is likely that these steps are to do with reproducibility of positioning within the furnace.

The calculated *χ*^{2}_{cov} using the full covariance matrix (equation (8.7)) is 167, compared with 131 d.f. This is statistically too large; however, adding 4 mK to the uncertainty associated with random effects for all measured values, reduces the *χ*^{2}_{cov} value to 131 and provides a Birge ratio of 1 (§8d). This alters both the calculated temperature *T*_{0} and its uncertainty by less than 1 mK.

Note, to understand the sensitivity to the analysis technique that the simple mean is 13 mK lower than the value *T*_{0} and the generalized weighted mean (which uses the covariance matrix) is 5 mK higher than the value *T*_{0}.

#### (iv) Cu results

The Cu measurement results are given in figure 5 (NMIA did not provide data for Cu). Here there is a good consistency between the individual measured values and the calculated temperature *T*_{0} for most participants, but a discrepancy in the NIST results. The NIST values have been given a weight of zero in the constraint equation (equation (8.5)) so that they are not included in the calculation of the reference value.

The differences of the NIST measurements from the mean at Cu-point temperatures can be eliminated if the spectral out-of-band radiance responsivities beyond 700 nm are increased by a factor of 15. The NIST assignments of the metal-carbon eutectic melting temperatures are also in better agreement with the global mean values in table 4 if the out-of-band responsivities are increased. Further measurements are being performed at NIST to resolve these discrepancies.

The calculated *χ*^{2}_{cov} using the full covariance matrix (equation (8.7)) is 42, compared with 60 d.f. This suggests that here uncertainties may be slightly overestimated. No correction was made, however.

Note, to understand the sensitivity to the analysis technique that the simple mean is 18 mK lower than the value *T*_{0} and the generalized weighted mean (which uses the covariance matrix) is 4 mK higher than the value *T*_{0}.

### (b) Thermodynamic temperatures for the fixed points

#### (i) Calculated values

The determined thermodynamic temperatures for the fixed points are those of the points of inflection of the melting curves for the eutectics and the mean of the central third of the freezing plateau for Cu. These values are given in table 4.

#### (ii) Transition temperatures

The values and uncertainties given in table 4 are the values obtained from the circulating cells measured during this measurement campaign. They are not the ‘true’ transition temperature of these fixed points for the following reasons.

First, the temperatures of the eutectic melting transitions correspond to the point of inflection. It has been discussed previously [22,53] that this point is not the fundamental quantity. The fundamental quantity is the liquidus temperature obtained under equilibrium conditions. The raw data has information that could be used to estimate this temperature (i.e. measured values were made for different previous freezing rates and upper-limit temperatures were also calculated), but this paper has concentrated on the more ‘pragmatic’ point-of-inflection temperature.

Second, the cells are all made from the purest material available, and were preselected for high melting temperature and flat plateaus, which suggests that the purest cells were used. However, there will be some residual impurity in the cells. Similarly, there will also be a residual temperature drop at the cavity bottom for which account was not made. Although cell impurity and cell temperature drop uncertainties were included in the analysis, this was to understand cell-to-cell differences, and the constraint equation (equation (8.5)) forced the average cell effect to be zero. An additional uncertainty (and possibly correction) should be added to the obtained uncertainty values to account for these cell effects if a more fundamental temperature is required, and our recommendation is to include an additional uncertainty component that is as large as the cell effect uncertainty given in table 3 to be added in quadrature with the uncertainties of table 4.

Third, §7 shows that at least one Re-C cell was observed to drift during this measurement campaign. To confirm that the other cells have not drifted, we have begun measurements to compare the cells again against each other, against the selected cell that did not travel and against brand new cells, using the methods of the pre-selection [14].

It is hoped that results of the post-campaign drift tests and also a further analysis of the difference between the measured values and the eutectic liquidus temperature will be presented at the Tempmeko 2016 conference by Yamada and Lowe, respectively.

#### (iii) New fixed points

It is important to realize that the uncertainties given in §9b cannot be applied directly to other (newly constructed) HTFP cells of the same type. In the initial phase of this project, a large number of HTFP cells of each type were constructed from nominally identical raw materials. These showed temperature differences in the melt transitions of 100 mK to 200 mK [14]. This variation is probably due to variation in the impurity levels in the base raw materials, despite their having nominally the same purity and possibly due to variations in the temperature drop effect, due to variations in the thermal conductivity of graphite, differences in the cell design, etc. Any new cell built would have different impurities and conditions, and, without a similarly robust selection [15], is likely to be less pure than these cells. An additional uncertainty should be added to account for the differences between new cells and the ones reported here.

Furthermore, there remains an unresolved effect of the furnace, which means individual measured temperatures may also vary by approximately 100 mK to 200 mK depending upon which type of furnace is used (higher or lower thermal inertia), and to a lesser extent the temperature gradients within the furnace over which the fixed point is realized (assuming that the HTFP is realized in the lowest temperature gradient part of the furnace). A contribution would need to be added to the uncertainty of any individual cell.

Apart from Re-C cell A, all the HTFP cells showed good stability. However, if relative primary thermometry were used to realize thermodynamic high temperatures it would always be good practice to have a small cohort of cells as your reference rather than rely on one HTFP of a particular type, this is particularly important if Re-C is your reference.

As a result, the uncertainty associated with the transition temperature of a new HTFP will inevitably be somewhat larger than the uncertainties given in §9b and would include contributions from these effects.

## 10. Conclusion

This paper has presented the results of an 8-year highly collaborative research project. It has provided, in table 4, thermodynamic temperatures for the point of inflection of the melting transition of the metal-carbon eutectics Re-C, Pt-C and Co-C and for the freezing temperature of the Cu fixed point, along with their uncertainties.

Apart from the results of one participant for the Cu point measurements, all results are individually consistent with the obtained consensus temperatures and only a very small additional uncertainty was required to augment the uncertainties associated with random effects for a model-data consistency test to be passed. This consistency suggests that radiometric techniques are sufficiently mature to provide reliable direct realization of thermodynamic temperature, a concept that will be included in the *MeP-*K in due time, with uncertainties of some 300–500 mK at Re-C (§4b).

Only one cell (Re-C cell A) showed significant drift during the measurement campaign. It appears Re-C can and does change unpredictably and significantly with respect to the achievable radiometric uncertainties. We would not recommend using a single HTFP cell as a reference for radiation thermometer calibration, especially if using Re-C. The other cells showed the required long-term stability. More studies are needed to understand the source of the instability of the Re-C point and to quantify its limits. In parallel, studies of the alternative HTFP from the WC-C peritectic should be pursued [54], which provides a higher temperature than Re-C, to see if it is more stable.

This work has obtained measurements of the transition temperatures of HTFPs with the lowest uncertainty ever achieved and thus provides reference temperatures for their use to disseminate thermodynamic temperature at high temperatures. It is important to realize that to make the best use of these temperature values, any new cell should be made [15] and selected [14] with the rigour of the ones studied here. In addition uncertainties would need to be included alongside those reported here, to account for differences between cells and between furnaces.

Nevertheless, even with the additional uncertainty components included with the uncertainties reported here, the overall uncertainty in realizing thermodynamic temperatures using HTFPs [40] would be similar to or smaller than the uncertainties associated with the routine realization of temperature using ITS-90 and allow thermodynamic temperature, *T*, rather than the defined scale, *T*_{90}, to be disseminated.

## Data accessibility

The datasets supporting this article have been uploaded as part of the electronic supplementary material.

## Authors' contributions

E.R.W. was lead author of the paper. Led the preparation of the protocol and analysis templates, developed the analysis methodology and brought the results together. K.A. reviewed and provided corrections to the paper, wrote analysis technique for emissivity correction. Supported the development of the initial protocol. Led the experimental work and analysis at PTB. Experimental design, data interpretation, uncertainty assessment, produced PTB measurement report. M.B. reviewed and provided corrections to the paper. Experimental design and results analysis at NMIA. P.B. reviewed and provided corrections to the paper. Supported the development of the initial protocol. Strongly involved in the development of the analysis methodology and, particularly, the impurity and emissivity corrections. Also supported both NMIJ and NIM in their experimental design. F.B. reviewed and provided corrections to the paper. Experimental design, measurements, results interpretation, uncertainty assessment and reporting of the LNE-Cnam results. S.B. reviewed and provided corrections to the paper. Experimental design, measurements, results interpretation, uncertainty assessment and reporting of the LNE-Cnam results. J.C. reviewed and provided corrections to the paper. Design and calibration of the radiometer used by CEM. M.G.C. reviewed and provided corrections to the paper. Development of the analysis methodology and methods for consistency checking. D.d.C. reviewed and provided corrections to the paper and critically reviewed the analysis methodology. Analysis of the CEM results. M.R.D. reviewed and provided corrections to the paper. Supported the development of the initial protocol. Experimental design, measurements, data analysis and interpretation of the NPL measurements, produced the NPL measurement report. V.G. reviewed and provided corrections to the paper. Data analysis and interpretation of the VNIIOFI data. I.G. reviewed and provided corrections to the paper. Performed VNIIOFI measurements, initial data analysis of VNIIOFI data. M.L.H. reviewed and provided corrections to the paper. Design and calibration of the radiometer used by CEM. F.J. reviewed and provided corrections to the paper. Performed NMIA measurements, initial data analysis of NMIA data. B.K. reviewed and provided corrections to the paper. Supported the development of the initial protocol. Experimental design of VNIIOFI measurements, measurements, data analysis and interpretation, uncertainty analysis. Produced VNIIOFI measurement report. V.K. reviewed and provided corrections to the paper. Experimental design, measurements, data analysis and interpretation of the NIST measurements. D.H.L. reviewed and provided corrections to the paper. Supported the development of the initial protocol. Experimental design, measurements, data analysis and interpretation of the NPL measurements. Developed analysis spreadsheets for interpreting the impurity assays and also technique for fitting a cubic to the melt curve. Uncertainty analysis of NPL measurements. X.L. reviewed and provided corrections to the paper. Experimental design, measurements, data analysis and interpretation of the NIM measurements, produced the NIM measurement report. G.M. considerable editing of early draft of paper. Overall technical lead of project: initiated project and provided technical leadership to project direction. Supported the development of the initial protocol. Supported the development of the analysis techniques and the interpretation of the measured results. J.M.M. reviewed and provided corrections to the paper. Developed design of CEM radiometer and performed its calibration. Experimental set-up, measurements, data analysis and interpretation. Supported the preparation of the CEM measurement report. M.J.M. considerable editing of early draft of paper. Supported the development of the initial protocol. Experimental set-up, measurements, data analysis and interpretation of the CEM measurements. Prepared the CEM measurement report. H.C.M. reviewed and provided corrections to the paper. Experimental design, measurements, data analysis and interpretation of the NPL measurements, produced the NPL measurement report. B.R. reviewed and provided corrections to the paper. Experimental design, data analysis and interpretation of the LNE-Cnam measurements. M.S. reviewed and provided corrections to the paper. Supported the development of the initial protocol. Led the experimental work and analysis at LNE-Cnam. Experimental design, data interpretation, uncertainty assessment, produced LNE-Cnam measurement report. S.G.R.S. reviewed and provided corrections to the paper. Experimental design, measurement, data analysis and interpretation of the LNE-Cnam measurements. N.S. reviewed and provided corrections to the paper. Experimental design, measurement, data analysis and interpretation of the NMIJ measurements. D.R.T. reviewed and provided corrections to the paper. Calibration of the PTB radiometer and experimental design at PTB. A.D.W.T. considerable editing of early draft of paper. Supported the development of the initial protocol. Led the experimental work and analysis at NRC. Experimental design, data interpretation, uncertainty assessment, produced NRC measurement report. Also produced test software to check the main (overall) analysis software and reviewed the implementation of the analysis technique. R.V.d.B. reviewed and provided corrections to the paper; produced the results graphs. Developed and tested the main (overall) analysis software that calculated the transition temperatures. Improved the analysis model based on measured values. E.v.d.H. reviewed and provided corrections to the paper. Led the experimental work and analysis at NMIA. Experimental design, data interpretation, uncertainty assessment, produced NMIA measurement report. T.W. reviewed and provided corrections to the paper. Experimental design, data analysis and interpretation of the NIM measurements. D.W. reviewed and provided corrections to the paper. Produced the initial analysis spreadsheet for individual measurement cycles that was the basis of the one used by participants. Produced the initial analysis spreadsheet for cell impurities. Experimental design, data analysis and interpretation of the NIM measurements. A.W. reviewed and provided corrections to the paper. Experimental design and measurements at NPL. B.W. reviewed and provided corrections to the paper. Experimental design, measurements and data analysis at PTB. D.J.W. reviewed and provided corrections to the paper. Experimental design, equipment set-up, measurements and data analysis and interpretation at NRC. J.T.W. reviewed and provided corrections to the paper. Experimental design, measurements and data analysis at NIST. Yo.Y. reviewed and provided corrections to the paper. Technical leadership and review to the whole project. Provided information and interpretation on cell and furnace effects. Experimental design, data analysis and interpretation of the NMIJ measurements. Y.Y. reviewed and provided corrections to the paper. Calibration of NMIJ thermometer, experimental design, data analysis and interpretation of the NMIJ measurements. Prepared the NMIJ measurement report. H.W.Y. reviewed and provided corrections to the paper. Experimental design, measurements and data analysis at NIST. Z.Y. reviewed and provided corrections to the paper. Experimental design, measurements and data analysis at NIM.

## Competing interests

We have no competing interests.

## Funding

National Physical Laboratory, Physikalisch-Technische Bundesanstalt, LCM LNE-Cnam, Spanish National Research Council, Centro Español de Metrologia acknowledge funding from the European Metrology Research Programme (EMRP) through which the Implementing the new kelvin (INK) project was funded. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. National Institute of Metrology acknowledges the support from the National Key Technology Support Program (no 2006BAF06B03) and the National Natural Science Foundation of China (no 51206151). A.D.W.T. and D.J.W. acknowledge funding by the National Research Council Canada. The All-Russian Research Institute for Optical and Physical Measurements acknowledges funding from the Ministry of Education and Science of the Russian Federation (identification number RFMEFI59214X0001).

## Acknowledgements

The National Physical Laboratory authors acknowledge Subrena Harris who provided a radiometric calibration of the LP3 instrument used and Peter Machin with the initial analysis of the NPL measurement data. The VNIIOFI authors acknowledge Denis Otryaskin and Maxim Solodilov for their support with the VNIIOFI measurements, and Rainer Winkler of NPL for providing calibrated trap detectors to use as references. All authors acknowledge Victoria Montag, who provided excellent project management for ‘Implementing the New Kelvin (InK)’ EMRP project under which most of this work was performed. She also reviewed the first draft of this paper and suggested significant improvements.

## Footnotes

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

↵1 In 2014 renamed as the CCT Working Group for Non-Contact Thermometry.

- Accepted August 25, 2015.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.