Helen C Mcevoy

Dave H Lowe

Robin Underwood

Michael De Podesta

Graham Machin

Maria-Jose J Martin

Jose M Mantilla

Joaquin Campos Acosta

Mohamed Sadli

Frederic Bourson

J Campos

S Briaudeau

S G R Salem

K Anhalt

M Waehmer

D R Taubert

X J Feng

J T Zhang

F Lu

H Yoon

Methodologies and uncertainty estimates for T -T 90 measurements over the temperature range from 430 K to 1358 K under the auspices of the EMPIR InK2 project

Keywords: temperature, thermodynamic temperature, ITS-90, blackbody, radiation thermometer, radiometer, thermodynamic, primary thermometry

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The current International Temperature Scale of 1990 (ITS-90) is the underpinning scale for temperature measurement worldwide. It defines instruments and methodologies (such as interpolation equations) for establishing a scale which is close to thermodynamic temperature, for temperatures above 0.65 K [1]. The instrumentation includes reference artefacts (fixed points which are based on phase transitions of pure materials) to calibrate specified thermometers: standard platinum resistance thermometers (SPRTs) for temperatures from ~ 14 K up to 1235 K and infrared radiation thermometers above 1235 K. The temperatures assigned to the fixed points were based on measurements made using the best determinations of thermodynamic temperature prior to 1990. Subsequent measurements have highlighted small departures of the ITS-90 from thermodynamic temperature across the range from 20 K to 1235 K [START_REF] Fischer | Present Estimates of the Differences Between Thermodynamic Temperatures and the ITS-90[END_REF], with the uncertainty in the difference being especially large above about 693 K (the Zn point) and corrections are needed to the ITS-90 values to obtain thermodynamic temperature.

The European Metrology Programme for Innovation and Research (EMPIR) project 'Implementing the new kelvin 2' (InK2) was established to investigate the differences between thermodynamic temperature, T, and the ITS-90 temperature, T90, using different thermodynamic techniques. Its predecessor the European Metrology Research Programme (EMRP) project 'Implementing the new kelvin 1' (InK1) [START_REF] Machin | Summary of achievements of the EMRP project implementing the new kelvin[END_REF] determined thermodynamic temperatures for a selected set of metal carbon eutectic and the copper fixed points (≥ 1358 K) [START_REF] Woolliams | Thermodynamic temperature assignment to the point of inflection of the melting curve high-temperature fixed points Phil[END_REF] as well as a range of temperatures below the indium point (430 K). InK2 is concerned with completing the lower temperature evaluation of T -T90, as well as determining T -T90 values from 430 K to 1358 K. The thermodynamic temperature data underpinning the specialist low temperature scale, the Provisional Low Temperature Scale of 2000 (PLTS-2000), was also a focus of study in the InK1 and InK2 projects. This paper describes the methodology and progress towards achieving the target uncertainties for measurements of the differences T -T90 over the temperature range from 430 K up to 1358 K using two different thermodynamic techniques: acoustic thermometry and primary radiation thermometry (or radiometry). Measurements using these techniques are being carried out at: the National Physical Laboratory (NPL), UK; Centro Español de Metrologia (CEM) in conjunction with the Agencia Estatal Consejo Superior de Investigaciones Cientificas (IO-CSIC), Spain; the Conservatoire national des arts et metiers (Cnam) and the Laboratoire national de métrologie et d'essais (LNE), France; Physikalisch-Technische Bundesanstalt (PTB), Germany; and the National Institute of Metrology (NIM), China. The final results will be presented in a subsequent paper.

INSTRUMENTATION AND METHODOLOGY

The instrumentation and methodology being employed at each of the partner institutes, namely acoustic (NPL and NIM) and radiometric (NPL, CEM/ IO-CSIC, LNE-Cnam, PTB and NIM), are described in the following sections along with anticipated target uncertainties.

ACOUSTIC GAS THERMOMETRY

Introduction

Acoustic Gas Thermometry (AGT) [START_REF] Moldover | Review article -Acoustic gas thermometry[END_REF] makes use of the simple relationship between thermodynamic temperature T and the limiting zero-pressure speed of sound, u0, in a monatomic gas: u0 2 =γ0kT/m, where k is the Boltzmann constant, m is the average molecular mass of the gas, and γ0 is the limiting low-pressure value of the adiabatic index, which is equal to exactly 5/3 in a monatomic gas.

In high accuracy AGT, the speed of sound u(T, p) is determined from the acoustic resonance frequencies of a gas-filled cavity. The shape of this cavity is either cylindrical, spherical or quasi-spherical. The resonance frequencies of a subset of acoustic modes are measured at several gas pressures and a single temperature (an isotherm), and a physical model describing the pressure dependence of u(T, p) is fitted to the data to find u0,T.

Measurements at NPL

Equipment and Methodology

In the previous InK project (InK1) NPL carried out extensive measurements of T -T90 over the temperature range from 118 K to 303 K using the NPL-Cranfield combined microwave and acoustic quasi-spherical copper resonator filled with argon. The measurements were made using relative AGT [START_REF] Moldover | Review article -Acoustic gas thermometry[END_REF]. In relative AGT measurements are made of the ratios of 𝑢 0 2 at the triple point of water, TTPW, (273.16 K) to 𝑢 0 2 at an unknown temperature T. T is then calculated from:

𝑇 273.16 K = 𝑚 T 𝑚 TPW ( 𝑢 0,T 𝑢 0,TPW ) 2 (1)
where mT and mTPW are the average molecular masses of the gases used at T and TTPW, respectively. The mass terms will cancel if the same gas is used for both temperatures, and if the composition of this gas remains stable with temperature and time.

The T -T90 values obtained with this method and apparatus agreed with previous estimates of T -T90 but had significantly lower uncertainties. In addition, previously undetected detail was observed in the T -T90 curve due to the use of small temperature intervals between measurements [START_REF] Underwood | Further estimates of (T -T90) close to the triple point of water[END_REF][START_REF] Underwood | Estimates of the difference between thermodynamic temperature and the ITS-90 in the range 118 K to 303 K Phil[END_REF].

The NPL AGT work in the InK2 project builds on the capability developed in the previous project and extends the measurement range to higher temperatures. In order to do this, it has been necessary to rethink every aspect of the system design, but the most fundamental change has been in moving from a quasispherical resonator to a cylindrical resonator [START_REF] Underwood | Microwave-dimensional measurements of cylindrical resonators for primary acoustic thermometry[END_REF]. This change is favoured because of the combination of several effects.

The acoustic transducers typically used in acoustic resonators have a maximum operating temperature of 423 K, and so at high temperatures they can no longer be placed directly on the resonator surface. One option for overcoming this limitation is to couple sound into and out of the resonator using acoustic waveguides. In order to minimally perturb the resonances, the acoustic waveguides must be relatively small in diameter, and hence will have limited highfrequency transmission. However, as the speed of sound increases with temperature (𝑢 ∝ √𝑇), the frequency of acoustic resonances in a resonator of a fixed size also increases. Thus, to have as many resonances to measure as possible, it is necessary to increase the size of the resonator.

Making increasingly larger spheres is one option, but the overall system volume and mass scale as the resonator radius squared and cubed respectively. When coupled with issues of material creep and operation at high temperatures and pressures, there comes a point where switching to a long cylindrical resonator becomes advantageous. This is because, in a long cylinder, the extra axial length results in a number of non-degenerate modes with relatively low frequencies (Figure 1). Additionally, manufacturing is simpler and all entry and exit ports (for acoustics, microwaves, and gas) can be positioned on a single endplate, simplifying access (Figure 2). One final advantage of the cylindrical geometry is the length-to-radius ratio can be chosen to minimise the number of accidental degeneracies. The reason for wanting to measure as many modes as possible is that the key uncertainties are likely to arise from inadequate understanding of the physical processes occurring within the resonator. Whereas measurements of even a single acoustic mode would be sufficient to estimate the temperature in principle, the extent of agreement between multiple acoustic modes with different characteristics provides a strong test of the completeness of the theoretical understanding of the physics underlying the temperature estimates. As with quasispherical resonators, simultaneous microwave measurements facilitate compensation of the acoustic measurements for temperature-and pressure-induced dimensional changes. At higher temperatures, dimensional changes in the cylindrical resonator from material creep are also likely to be significant.

In cylinders, mode-degeneracy is also a complicating factor for certain electromagnetic mode classes. For example, the TE0pq modes (which have the advantages of high quality-factors and zero current density at the lid-cylinder interface [START_REF] Underwood | Microwave-dimensional measurements of cylindrical resonators for primary acoustic thermometry[END_REF]) are frequency-degenerate with the low quality-factor TM1pq modes. This degeneracy can be lifted by making the cylinder crosssection slightly non-circular (Figure 3). Alternatively, if a multi-port vector network analyser is available, two antennas may be used to detect resonances in different locations in the cylinder. Because of the orthogonal nature of the electromagnetic field patterns on resonance, the two signals may then be combined to isolate each of the degenerate resonances in turn.

Early results and target uncertainties

Figure 4 shows the spontaneous fractional changes in length and radius observed in an aluminium resonator held at 429 K (the In point). This so-called 'creep' arises from growth of precipitated impurities introduced to harden the alloy for use at lower temperatures. Although the resonator changed size significantly during the measurement period, the rate of change was slow enough to allow acoustic resonance data to be compensated, thus allowing the speed of sound to be estimated.

Figure 5 shows mode-by-mode estimates of thermodynamic temperature derived from acoustic measurements at 323 K and 429 K using absolute primary acoustic gas thermometry i.e. the results are based directly on u0 2 =γ0kT/m rather than ratio of speed of sound measurements described in Equation 1. The results are shown as differences from the expected results based on estimates of the difference between T and the ITS-90 published by BIPM CCT in 2011 [START_REF] Fischer | Present Estimates of the Differences Between Thermodynamic Temperatures and the ITS-90[END_REF]. The magnitude of the differences is typically ± 20 parts in 10 6 in the speed of sound which amounts to approximately 13 mK at 323 K and 17 mK at 429 K. Agreement at this level is indicative of being able to fully model the physics within the resonator. Notice that since the deviations are systematic, simply taking an average over all the modes will not guarantee convergence to the correct answer.

Most significant is the similarity of the deviations from one temperature to another. Because of this we expect that relative primary acoustic gas thermometry (Equation 1), in which modeby-mode ratios of changes in the speed-of-sound squared are taken, will produce thermodynamic temperature estimates with uncertainties of just a few millikelvin.

Next steps

Going forward, NPL will use a cylindrical resonator constructed out of a copper alloy -CuCrZr. This alloy is known to retain its hardness up to 873 K while still retaining 80 % -90% of the thermal and electrical conductivity of copper. This new resonator will be used to perform T -T90 measurements up to higher temperatures than can be reached using the aluminium resonator.

Measurements at NIM

Since 2013 NIM has been working to develop acoustic gas thermometry for use up to the copper point (1358 K) using cylindrical resonators [START_REF] Feng | Microwave-cavity measurements for gas thermometry up to the copper point[END_REF][START_REF] Zhang | Acoustic and microwave tests in a cylindrical cavity for acoustic gas thermometry at high temperature Phil[END_REF][START_REF] Zhang | Microwave measurements of the length and thermal expansion of a cylindrical resonator for primary acoustic gas thermometry[END_REF]. The design of the acoustic transducers was optimised to obtain a good signal-to-noise ratio at high temperature [START_REF] Chen | Optimization of acoustic waveguides for acoustic gas thermometry[END_REF]. Because acoustic waveguides with larger inner diameter (ID) and shorter length have smaller acoustic attenuation but larger perturbation to the acoustic resonant frequencies, it is necessary to design the waveguides to get a good balance between the sound attenuation and perturbation.

The basic idea of the waveguide design is to use a smaller ID duct near the resonator cavity and then use a larger ID duct for most of the sound transfer. We have calculated the effects for different sizes of acoustic waveguides based on the impedance theory [START_REF] Gillis | Perturbations from ducts on the modes of acoustic thermometers[END_REF]. By using the variable inner diameter acoustic waveguides, the perturbation due to the resonant frequency from the acoustic waveguides can be controlled to less than 1 x 10 -5 , and the random error for the acoustic resonant frequencies, which refers to the typical fitting uncertainty of one resonance, can be reduced to less than 2 x 10 -6 at 900 K for the acoustic resonant measurements with a welded cylindrical cavity. The largest uncertainty component for the thermal expansion determination is from the inconsistency between different microwave modes, which is about 1.3 x 10 -5 near 900 K [START_REF] Zhang | Acoustic and microwave tests in a cylindrical cavity for acoustic gas thermometry at high temperature Phil[END_REF].

Heat pipes are used to provide a uniform and stable temperature environment for the cylindrical cavity [START_REF] Zheng | Investigation on precision measurement of thermal expansion coefficient for cavity based on microwave resonance[END_REF]. At temperatures from 323 K to the 505 K, a homemade water heat pipe is used. By the additional controlling of the temperature of the pressure vessel, the temperature stability of the cylindrical resonator is within ±1 mK over 30 minutes, and the uniformity of the resonator is 2 mK near 505 K.

Continuous improvements and measurements with high temperature acoustic gas thermometry at NIM are ongoing with the objective of achieving uncertainties for the (T -T90) measurement near 500 K, 1000 K and 1358 K of 10 mK, 50 mK and 120 mK, respectively, with k = 2.

PRIMARY RADIATION THERMOMETRY AND RADIOMETRY

Introduction

Primary radiometry techniques determine thermodynamic temperatures by measuring the absolute spectral radiance (W m -2 sr -1 nm -1 ) of a blackbody source [START_REF] Anhalt | Thermodynamic temperature by primary radiometry Phil[END_REF][START_REF] Hartmann | High-temperature measurement techniques for the application in photometry, radiometry and thermometry[END_REF]. The method requires the determination of the radiance (radiant flux emitted per unit area and unit solid angle) in a finite wavelength band (nm) with optical power measurements traceable to a cryogenic electrical substitution radiometer. Several different methods based on this principle exist [START_REF] Machin | Practical implementation of the mise en pratique for the definition of the kelvin above the silver point[END_REF], differing in where the defining geometry (two parallel, circular, collinear apertures of known physical dimensions and separation) is located in the calibration chain and whether an imaging system is used.

For InK2 a number of different primary radiation thermometers and radiometer systems, operating at different wavelengths, are being used to determine T, either directly or indirectly, and hence infer T -T90. Measurements are being made of:

• fixed-point blackbody (FPB) sources of tin (505 K), zinc (693 K), aluminium (933 K), silver (1235 K), gold (1337 K) and copper (1358 K) • variable temperature heat pipe blackbody (VTBB) sources which incorporate contact thermometers which have been calibrated according to the ITS-90.

The instrumentation, calibration process and measurement regime used by each of the participants (NPL, LNE-Cnam, CEM/ IO-CSIC, PTB and NIM) will now be described.

Measurements at NPL

Equipment and methodology

The primary radiation thermometer used for the NPL measurements is the Near Infrared Radiation Thermometer 3 (NIRT3) developed by the National Institute of Standards and Technology (NIST), USA [START_REF] Yoon | SSE-and noiseoptimized InGaAs radiation thermometer[END_REF]. It is an InGaAs detector-based radiation thermometer operating at a wavelength of ~1600 nm, with good optical and linearity characteristics. It is shown schematically in Figure 6. In order to determine T from the thermometer output the instrument needs to be calibrated in terms of absolute spectral responsivity. The spectral responsivity of the NIRT3 was measured at NIST using the NIST spectral irradiance responsivity and spectral radiance responsivity (SIRCUS) facility [START_REF] Brown | Facility for spectral irradiance and radiance responsivity calibrations using uniform sources[END_REF] traceable to the NIST Primary Optical Watt Radiometer. However, due to some technical difficulties (and therefore larger uncertainties) associated with absolutely calibrating an InGaAs-type detector (rather than a silicon photodiode detector) using this facility, it was decided instead to anchor the absolute calibration of the NIRT3 using, as a reference, the NIST gold FPB. The thermodynamic temperature of the NIST gold point was determined using the NIST 650 nm Absolute Pyrometric Transfer Radiometer (APXR650), which had been calibrated using SIRCUS. The thermodynamic temperature of the NIST gold point was determined to be 1337.36 K using the APXR650 (i.e. 30 mK above the current ITS-90 value) with an estimated uncertainty of 81 mK (k = 2). To determine thermodynamic temperatures with the NIRT3 the expected signals at each required temperature are calculated from the spectral responsivity (from the NIRT3 SIRCUS calibration) and Planck's law. The expected signals are then adjusted using the results of the measurements at the gold fixed point (with assigned T = 1337.36 K). The adjusted expected signals are then compared to the actual signals obtained at each temperature to determine T -T90. For measurements with the NIRT3 at NPL the reference is the NPL gold FPB (estimated emissivity 0.99998). This is assumed to have the same thermodynamic temperature as the NIST gold point within the measurement uncertainties.

At NPL T -T90 values are being determined using both FPBs and heat pipe VTBBs. The former are high emissivity (at least 0.99997) graphite blackbody cells, containing high purity metals of, respectively, tin, zinc, aluminium, silver and copper. The copper cell [START_REF] Mcevoy | A comparison of the NPL and LNE-Cnam silver and copper fixed-point blackbody sources, and measurement of the silver/copper temperature interval[END_REF] is installed in an electrical furnace with three heated zones. The other cells are installed within a heat pipe liner within a furnace: a potassium heat pipe in a furnace with three heated zones for the tin and zinc cells and a sodium heat pipe in a single-heated-zone furnace for the aluminium and silver cells.

The heat pipes improve the temperature uniformity of the heated zone in which the cell sits, and hence improve the fixed-point freezing plateau duration and shape. (Note that the tin point temperature is below the minimum operating temperature of the potassium heat pipe, but using a furnace with three heated zones enables the required level of temperature uniformity to be achieved at the tin point temperature.)

One of the NPL VTBBs is a bespoke pressure-controlled heat pipe (PCHP) system from Advanced Cooling Technologies (ACT), USA to NPL's design and technical specification.

The facility consists of two heatpipes: a potassium PCHP for the 673 K to 1173 K temperature range and a sodium PCHP for 973 K to 1373 K range, with a single control unit. The potassium heat pipe is constructed from Inconel whereas the sodium heat pipe is constructed from Haynes alloy -a nickel-chromium tungsten-molybdenum alloy materialwhich gives it more structural strength and enables it to reach the Cu fixed-point temperature (1358 K). Each PCHP incorporates a high emissivity (~0.99996) blackbody cavity nominally 400 mm long, 12.5 mm diameter with a 120° end cone. Each PCHP also contains five thermometer wells to insert calibrated thermometers (standard platinum resistance thermometers and Au/Pt thermocouples). The contact thermometers have been calibrated in terms of ITS-90 using fixedpoint cells. A schematic of the system is shown in Figure 7. Additionally, measurements are being performed using a 'passive' caesium heat pipe source which can cover the temperature range from 573 K to 873 K. This incorporates a blackbody cavity of nominally 15 mm diameter and 200 mm length, and which has an estimated emissivity of 0.9997. Four thermometer wells run from the rear of the heat pipe partially along the length of the blackbody cavity.

In every case T -T90 is determined by comparing the thermodynamic temperature as measured by the NIRT3 with the corresponding ITS-90 temperature, either ITS-90 temperature of the fixed-point metal or the temperature given by the calibrated contact sensors in the rear of the PCHP.

Target uncertainties

Example target uncertainties for the measurements of the FPBs and VTBBs at NPL are given in Tables 1 to 3 below. The uncertainties are comprised of both the uncertainty associated with the calibration of the NIRT3 and the uncertainty associated with the FPB or VTBB. The former includes components for the measurement of the spectral responsivity of the NIRT3, the measurement of the gain ratios and the uncertainty in the assignment of T to the NIST Au point since this was used to initially calibrate the NIRT3.

The uncertainties associated with the FPB and VTBB are estimated using the Consultative Committee of Thermometry non-contact thermometry working group (CCT-WG5) document 'Uncertainty Budgets for Calibration of Radiation Thermometers below the Silver Point' [START_REF] Saunders | Uncertainty budgets for calibration of radiation thermometers below the silver point[END_REF] and also include the uncertainty associated with the NPL Au point (U = 22 mK (k = 2)), since this is used as the reference to calibrate the NIRT3 in use. T -T90 measurements are being performed at CEM using two absolutely calibrated standard radiation thermometers (RTs) to measure both FPBs (Cu, Ag and Zn) and VTBBs. The standard thermometers are a silicon-detector-based LP4 (650 nm and 900 nm filters) and an InGaAs-detector based LP5 (1550 nm filter) [START_REF] Fischer | Metrological characterisation of a new transfer-standard radiation thermometer[END_REF][START_REF] Schreiber | Reduction of the size-of-source effect and distance dependency of a high precision infrared radiation thermometer[END_REF]. Both are calibrated by comparison with a Si trap radiometer and an InGaAs radiometer, respectively, using the radiance mode [START_REF] Machin | Practical implementation of the mise en pratique for the definition of the kelvin above the silver point[END_REF][START_REF] Martin | Performance of different light sources for the absolute calibration of radiation thermometers[END_REF]. The Si trap radiometer and the InGaAs radiometer are calibrated absolutely at IO-CSIC with an electrical substitution cryogenic radiometer.

The LP4 at 900 nm and LP5 at 1550 nm can also be used to estimate T by extrapolation (relative primary radiation thermometry) to lower temperatures from a Ag fixed point whose thermodynamic temperature, TAgFP, has been assigned a value of 1234.90 K (U = 170 mK (k = 2)) using the absolutely calibrated thermometer at 650 nm. The Ag cell is from ISOTECH and it is installed in a Na heat pipe ISOTECH OBERON R 426 furnace.

The method used to measure T (either absolute primary radiation thermometry or relative primary radiation thermometry) follows the scheme in Table 4. The indicated temperature ranges correspond to the range at which the radiation thermometers are most linear. The VTBBs used for measuring T -T90 are a Na heat pipe and a Cs heat pipe from KE Technologie (see Figure 8). The temperatures of the two VTBBs are measured using two ISOTECH standard platinum resistance thermometers (SPRTs) one of 25 Ω for T90  873 K and another of 0.25 Ω for T90 ≥ 873 K. The SPRTs have been calibrated at fixed points with an uncertainty of 10 mK and 20 mK, respectively. The SPRTs are placed in a thermowell within the heat pipe. For temperatures > 873 K, the SPRT is protected with a platinum sheath and air circulation (ISOTECH 416 system) is used to prevent contamination.

A number of corrections are applied to the measurement results: the SPRT reading is corrected for conduction within the Inconel tube; the radiation thermometer reading is corrected for the dark current, the size-of-source effect (to take into account the difference between the effective radii of the Ag FP and VTBB, which was assessed by measuring the temperature profiles of the sources [START_REF] Bloembergen | The characterization of radiation thermometers subject to the size-of-source effect Proceedings of the 6 th International[END_REF]) and the emissivity of the VTBB. The estimation of uncertainties has been carried out using the method described in Appendix B of [START_REF] Saunders | Uncertainty estimation in primary radiometric temperature measurement CC-WG-NCTh document[END_REF] and [START_REF] Machin | [END_REF]. The uncertainty calculation is based on the key spectral parameters of the thermometers and follows the approach described in [START_REF] Machin | [END_REF].

Table 5 shows the uncertainty budget for the temperature measurements using the LP4 at 650 nm. The components include the uncertainties associated with: the monochromator calibration and its drift; the standard deviation of the measurements of the spectral response; the uncertainty associated with H (the absolute calibration of the standard radiometer, its linearity and drift and the components related to the calibration by comparison), and the uncertainty associated with the radiation thermometer when in use (size-of-source effect, linearity and drift). The sources of uncertainty for the absolute calibration of the LP4 at 900 nm and the LP5 at 1550 nm are determined in a similar way with appropriate values for the components included in the budget. The expanded uncertainties for the LP4 at 900 nm are 260 mK, 220 mK and 170 mK at 1358 K, 1235 K and 1073 K, respectively. Those for the LP5 at 1550 nm are 110 mK, 140 mK, 180 mK and 220 mK at 673 K, 773 K, 873 K and 973 K, respectively. The drift of the LP5 is considered negligible because it is measured every day at the reference Ag FP.

The uncertainties associated with the radiation thermometers (RTs) are combined with the uncertainties associated with the T90 measurements (SPRT calibration, uncertainty due to the conduction at the Inconel block, conduction and rotation of the SPRT in the VTBB insert, the resistance bridge and standard resistor, self-heating of the SPRTs, stability of the VTBB and drift of the SPRT) to give the overall uncertainties shown in Table 6.

Uncertainties of the T -T90 measurements using relative primary radiation thermometry

For the relative primary radiation thermometry measurements, the LP4 at 900 nm is measured at the CEM Ag fixed point, with assigned T = 1234.90 K, as described above. Using the relative spectral responsivity of the LP4 T is then calculated in a relative way. The uncertainty calculation is again based on [START_REF] Machin | [END_REF] and includes the uncertainty in the LP4 calibration (monochromator calibration and drift, standard deviation of the measurements of the relative spectral response, uncertainty in assigning Tref etc.) along with the uncertainty associated with the use of the thermometer (size-of-source effect, non-linearity and drift). Table 7 shows the overall uncertainty budget. For the LP5 at 1550 nm the calculation is as above using appropriate values for the uncertainty components. The overall expanded uncertainties are 90 mK, 70 mK, 90 mK and 110 mK at 673 K, 773 K, 873 K and 973 K, respectively.

Table 8 shows the total measurement uncertainty including components associated with the SPRT and VTBB, and Table 9 summarises all the values from both radiometric methods highlighting the best values (indicated by an asterisk) when both methods are available. LNE-Cnam is using a radiance method to measure the thermodynamic temperature, T, of the Zn, Al and Ag fixed points, as well as a heatpipe VTBB. The measurements are carried out by comparing the radiance of the fixed points to that of a monochromatic source, which is absolutely calibrated against the LNE-Cnam cryogenic radiometer. The comparison is performed using LNE-Cnam's monochromator-based radiance comparator. This is generally used in the visible range but, for this work, it has been adapted to work in the infrared spectral range (around 1500 nm) using a suitable grating and an InGaAs photodiode detector.

Figure 9 shows an overview of the radiance comparator which, for the fixed point/ monochromatic radiance source comparison, is translated in front of the bench containing the sources. The instrument is comprised of four main parts: A, B, C and D defining respectively the geometrical extent, the spectral range, the shape of the beam and the detection system. The monochromatic source consists of an integrating sphere of 100 mm in diameter into which a laser is injected by means of an optical fibre. The power of the laser is 20 mW and it is set to a stable reference by the means of a feedback loop. The system is mounted on different translation and rotation stages to allow its full characterisation.

The radiance comparator slit scattering function is first measured by scanning the wavelength of the radiance source over the full spectral bandwidth of the instrument. The comparator is then translated in front of the fixed-point blackbody at the freezing plateau to measure its absolute radiance. The thermodynamic temperature of the fixed point is calculated from the following equation, deduced from the ratio of Planck's Law for the two measurements:

where: 𝑐 1 is the first radiation constant for spectral radiance, 𝑐 2 is the second radiation constant, 𝑛 air is the refractive index of air,  eff is the effective wavelength of the slit scattering function, 𝑆 laser is the monochromatic source signal as measured by the radiance comparator at the effective wavelength, 𝐺 bb is the radiance comparator amplifier transimpedance, 𝑆 bb ( eff , 𝑇) is the fixed point signal as measured by the radiance comparator at the effective wavelength, 𝐺 laser is the trap detector amplifier transimpedance and 𝐿 laser () is the radiance of the monochromatic source.

The comparison between the absolute radiance source and the blackbody is performed by fixing the effective wavelength of the radiance comparator at the wavelength of the laser. The ratio of the two signals is corrected using the optical responsivity of the radiance comparator. This correction is estimated by scanning the spectral bandwidth of the instrument with the monochromator, the comparator being in front of a blackbody.

The absolute radiance of the monochromatic source 𝐿 laser (), defined in the equation above, is measured by a so-called "radiancemeter". The instrument is composed of a trap detector directly calibrated against the cryogenic radiometer, and two precise apertures of 6 mm and 8 mm nominal diameter separated by a known distance. The aperture diameters are measured with a non-contact method, while the distance between them, typically 500 mm, is measured by a precise gauge. The 6-mm aperture is positioned in front of the trap detector. This aperture size was chosen to fit with the photodiode of 10 mm in diameter in order to collect all the radiant flux going through the geometrical extent. A baffle is set up between the sphere source and the trap detector to reduce the stray light detected by the trap detector. The horizontal rotation of the sphere by 90° allows it to be moved so that it can be viewed by both the radiance comparator and the trap detector alternately over a short time period.

Figure 10 shows a 3D drawing of the radiancemeter and the monochromatic source. Both the radiancemeter and the monochromatic source are placed inside a black-painted box to protect them from stray light. This radiance method requires an extensive characterisation of the possible biases and associated uncertainties induced by the radiancemeter such as the radiance stability of the sphere, its spatial homogeneity, and the stray light detected by the trap; as well as those associated with the radiance comparator such as the linearity, size-of-source effect and the outof-band transmittance.

Target uncertainty budgets for the LNE-Cnam measurements

Table 10 shows the uncertainty components u(TZn), u(TAl) and u(TAg) at the fixed points of zinc, aluminium and silver respectively and the combined uncertainty. The intermediate quadratic sums of the main components that relate to each instrument (radiancemeter, radiance comparator and fixed points) are shown. The uncertainty components relating to the radiancemeter are defined as follows:

λlaser refers to the uncertainty associated with the wavelength laser stability -Trap response refers to the stability of the trap within a daily measurement -Amplifier refers to the amplifier transimpedance uncertainty for the trap detector -Trap responsivity refers to the uncertainty of the calibration of the trap detector, including the spatial homogeneity, the stability during the calibration process and the repeatability of the measurements -Distance between apertures is the uncertainty associated to the distance that separates the two apertures -r1 and r2 are the uncertainty components associated to the radius of the sphere and the trap apertures respectively -K diffusion is the uncertainty associated to the correction of the stray light in the radiancemeter -K sphere inhomogeneity is the uncertainty associated to the correction of the radiance inhomogeneity in the plane of the sphere aperture

The uncertainty components relating to the radiance comparator are defined as follow:

-Effective wavelength refers to the uncertainty associated to the wavelength stability of the laser -Slit scattering function is the uncertainty that refers to the repeatability of the slit scattering function integral, defined by ∫ 𝐿 𝑙𝑎𝑠𝑒𝑟 (𝜆 𝑙aser 𝜆 𝑝2 𝜆 𝑝1 -𝜆 p ) 𝑑𝜆 p where 𝐿 laser (𝜆 𝑙aser -𝜆 p ) is the spectral radiance of the sphere measured by the radiance comparator at the effective wavelength 𝜆 p . [𝜆 p2 -𝜆 p1 ] is the spectral range of integration defined by the interference filter used to cut the out-of-band transmittance of the monochromator -Amplifier (sphere) refers to the uncertainty of the amplifier of the radiance comparator when it measures the sphere radiance -Response comparator is the uncertainty component related to the stability of the radiance comparator response, when a fixed point is measured at the effective wavelength 𝜆 p = 𝜆 𝑙aser -Amplifier (FP) refers to uncertainty of the amplifier of the radiance comparator when it measures the fixed point -KOBB is the uncertainty associated to the correction of the out of band transmittance of the monochromator -KSSE is the uncertainty associated to the correction of the size-of-source-effect of the radiance comparator

The uncertainty components relating to the fixed points are defined as follow:

-Emissivity is the uncertainty in determining the total emissivity of the cavity -Impurities is the uncertainty due to the ingot impurities in the 5N pure metal as claimed by the supplier -Set point / plateau repeatability is the uncertainty due to the reproducibility of the freezing plateaux including the change of the furnace set points to initiate the phase change -Day to day repeatability is the standard deviation of repeated measurements 3.2.5 Measurements at PTB

Equipment and methodology

Measurements at PTB are being performed using a four-wavelength ratio filter radiometer (FRFR), which operates at four distinct wavelengths between 440 nm and 1550 nm (Table 11) and is comprised of three silicon photodiodes and one InGaAs photodiode. The arrangement consists of four separate filter radiometers mounted on a linear stage with a common 3 mm diameter aperture, as shown in Figure 11 [START_REF] Wähmer | Thermodynamic temperature of high-temperature fixed points traceable to blackbody radiation and synchrotron radiation[END_REF]. For a simplified calibration an additional reference detector can be set up on the radiometers' linear stage. The calibration of the filter radiometer itself is a two-step process. First, a broadband transfer detector (Si trap detector or InGaAs single photodiode) is calibrated against a cryogenic radiometer at several discrete laser emission lines. Then the transfer detector is used to continuously calibrate the filter radiometers at a spectral comparator. The main principles of the calibration process are described in detail in [START_REF] Taubert | Improved calibration of the spectral responsivity of interference filter radiometers in the visible and near infrared spectral range at PTB[END_REF]. The resulting spectral responsivity of each of the four filter radiometers is presented in Figure 12. For thermodynamic temperature measurements, the filter radiometers will be set up with PTB's large area blackbody (LABB). This blackbody furnace has been developed for accurate realisation of the spectral irradiance in the red and near-infrared spectral ranges. Its temperature is measured by standard platinum resistance thermometers according to the International Temperature Scale of 1990 (ITS-90). The blackbody consists of two concentric sodium heat pipes, the inner one with temperature stability of a few millikelvin forming the radiating cavity.

The filter radiometer will also be used to measure T -T90 using the double heat pipe blackbody source (Figure 13 and [START_REF] Fischer | The double-heatpipe black body: a radiance and irradiance standard for accurate infrared calibrations in remote sensing[END_REF][START_REF] Stock | The double-heatpipe black body: a high-accuracy standard source of spectral irradiance for measurements of T -T90[END_REF]).

Target uncertainties

An overview of the measurement uncertainties with respect to ITS-90 temperature determination of the LABB using the SPRTs is given in the Table below. The expected combined measurement uncertainty for T90 varies between 24 mK and 62 mK between 692 K and 1235 K (k = 2). For the spectral irradiance measurement the expected relative uncertainty components are given in Table 13. The uncertainties originate from the measurement geometry (distance measurement, aperture area, diffraction correction), the filter radiometer calibration (which varies spectrally as the uncertainty of the reference detectors increases considerably towards the near infrared), and the measurement in front of the LABB (random uncertainty, emissivity, repeatability, amplifier gain).

For temperatures of 692 K, 933 K and 1235 K the corresponding temperature uncertainty is calculated for two filter radiometers with a centre wavelength at 900 nm and 1550 nm respectively. For the filter radiometer at 1550 nm the uncertainties range between 440 mK at 1235 K and 138 mK at 692 K (k = 2). For the 900 nm filter radiometer the expected measurement uncertainties are 120 mK at 1235 K and 68 mK at 933 K (k = 2). At 692 K a relatively high signal-to-noise ratio is expected for the 900 nm filter radiometer and this temperature point will be omitted for this filter radiometer.

The differences in achievable measurement uncertainty illustrate well-known principles in radiation thermometry (which can be derived from Planck's law of thermal radiation): the centre wavelength of the filter radiometer should be chosen to be as short as possible in order to achieve a low temperature uncertainty from a given uncertainty in spectral irradiance. Yet, the spectral irradiance of a blackbody decreases with decreasing wavelength on the short wavelength side of its maximum and consequently, the uncertainty of the radiance below a certain radiance level is dominated by the limited detectivity of the detector in the filter radiometer. Another measurement principle in radiation thermometry allows the determination of the temperature of a blackbody at unknown temperature Tx using a blackbody of known temperature Tref by a ratio measurement of spectral radiance using Planck's law of thermal radiation. The measurement uncertainty then propagates according to:

𝑈(𝑇 X ) = ( 𝑇 X 𝑇 ref ) 2 𝑈(𝑇 ref ) (3)
Thus, the uncertainty U(Tx) is larger than U(Tref) for Tx > Tref, and smaller than U(Tref) for Tx < Tref. Consequently, the low measurement uncertainty at 1235 K with the FR900 can be utilised towards lower temperatures by using the FR1550 for a spectral irradiance ratio measurement of the LABB operated at 692 K and 933 K with respect to 1235 K (the reference temperature measured with FR900).

The resulting expected measurement uncertainties for T at 692 K and 933 K are given in Table 14.

Measurements at NIM

For measuring the spectral radiance of the Ag fixed point directly, a new design of absolute radiation thermometer (ART) was established at NIM. This system consists of a 50 mm diameter chromatic doublet lens with 400 mm focal length, a 20 mm diameter knife-edge aperture behind the lens and a filter radiometer at a distance of 1400 mm from the lens. In the ART, the 900 nm filter radiometer (called FR900 in the text), which has a peak wavelength of 900 nm and a bandwidth of 40 nm, was used to determine the temperature of the Ag fixed point blackbody. The arrangement is shown schematically in Figure 14. The irradiance responsivity of the FR900 was calibrated at PTB and NIM. The results were compared and are shown in Figure 15.

400 500 600 700 800 900 1000 1100 1200 1300 10 -14 The uncertainty components associated with the FR900 are given in Table 15, and the corresponding uncertainties associated with the measurements of the Ag fixed point, expressed in terms of kelvin, are given in Table 16. The silver fixed point was measured by the ART following its calibration and characterisation. A dual direction laser was used to help the alignment of the ART with the furnace. The image of the cell aperture was observed directly on the cover of the filter radiometer aperture. The measurement results of the fixed point and uncertainties are shown in Table 17 below.

CONCLUSIONS

Under the auspices of the EMPIR InK2 project measurements of T -T90 are being carried by a number of different NMIs over the temperature range from 430 K to 1358 K using two thermodynamic techniques: acoustic gas thermometry and primary radiation thermometry or radiometry, as summarised below (note that all uncertainties are expressed as k = 2).

Acoustic gas thermometry measurements are being carried out at NPL using an aluminium cylindrical resonator to higher temperatures than the measurements carried out in the InK1 project. Measurements of T -T90 using relative primary acoustic gas thermometry are anticipated to produce thermodynamic temperature estimates with uncertainties of just a few mK. An additional resonator has been constructed out of CuCrZr has greater hardness than copper and will enable the measurement range to be extended further, up to 873 K.

At NIM a cylindrical acoustic resonator is being used with the aim of extending acoustic gas thermometry up to 1358 K. Heat pipes are being used to provide a uniform and stable environment for the cylindrical cavity. Target measurement uncertainties at 500 K, 1000 K and 1358 K are 10 mK, 50 mK and 120 mK, respectively (k = 2).

In addition, T -T90 measurements are being performed at NPL using primary radiation thermometry, using an InGaAs-based radiation thermometer operating at around 1600 nm, a set of fixed-point cells, a bespoke pressure-controlled heat pipe system and a passive caesium heat pipe. Together these can cover the temperature range from 505 K to 1358 K. The thermodynamic temperature, T, of the sources is determined from the spectral response of the radiation thermometer anchored to a gold fixed-point blackbody with an assigned T = 1337.36 K (as measured at NIST). The target expanded uncertainties were estimated and range from around 30 mK for measurements with the lower temperature fixed points (Sn, Zn) up to around 200 mK for measurements with the Na PCHP at the Ag point.

T -T90 measurements are being performed at CEM using two absolutely calibrated standard radiation thermometers, namely an LP4 operating at 650 nm and 900 nm and an LP5 operating at 1550 nm, along with Cu, Ag and Zn fixed points and VTBBs. The calibration of the standard thermometers is via a Si trap radiometer and an InGaAs radiometer which are calibrated absolutely at IO-CSIC with an electrical substitution cryogenic radiometer. T may be determined either by absolute primary radiation thermometry or relative primary radiation thermometry using a silver fixed-point blackbody with assigned T = 1234.90 K. All the uncertainties have been calculated for both methods and thermometers. The best expanded uncertainties range from 80 mK to 190 mK, from 673 K to 1358 K. The relative primary thermometry gives better uncertainties than absolute primary thermometry at T < 1223 K, due to the large calibration uncertainty at 900 nm and 1550 nmwhere U (absolute responsivity of the standard radiometers) is 0.15%, twice the uncertainty at 650 nm (0.075%).

T -T90 measurements are being performed at LNE-Cnam using a monochromator-based radiance comparator which has been adapted to the infrared spectral range (1600 nm). Measurements of T of the blackbody source (FPB or VTBB) is performed by comparing the radiance of the source with that of an integrating sphere, whose radiance is measured, in turn, with a transfer trap detector calibrated using the LNE-Cnam cryogenic radiometer. Target expanded uncertainties for the T -T90 measurements range from 84 mK with the Zn fixed point (692 K) to 272 mK with the Ag fixed point (1235 K).

T -T90 measurements are being carried out at PTB using a four-wavelength radiometer which operates at four distinct wavelengths between 440 nm to 1550 nm. The radiometers are calibrated with reference to a transfer detector which has been calibrated against a cryogenic radiometer. T -T90 measurements are being carried out using a large area double heat pipe VTBB. For absolute primary radiation thermometry, target uncertainties are from around 440 mK at 1235 K to around 140 mK at 692 K for the 1550 nm radiometer, and from around 70 mK at 933 K to 120 mK at 1235 K for the 900 nm radiometer. Relative primary radiation thermometry towards lower temperatures can be carried out with the 1550 filter radiometer, by using reference measurements at around 1235 K with the 900 nm filter radiometer. The resultant target expanded uncertainties range between 38 mK at 692 K to 120 mK at 1235 K.

Measurements at NIM are being carried out using an absolute radiation thermometer operating at 900 nm. This is being used to measure the thermodynamic temperature of a silver fixed-point blackbody (T90 = 1234.93 K). The target uncertainty for the measurements of the silver cell is 260 mK (k = 2).

Once all the measurements have been completed the results will be pooled in order to provide provisional estimated values of T -T90 over the whole range from 430 K to 1358 K for submission to the Consultative Committee on Thermometry (CCT). By pooling all the data it is hoped that it will be possible achieve lower uncertainties than are achievable by one laboratory operating in isolation.

Figure 1 :Figure 2 :

 12 Figure 1: Graph showing the quality factor of acoustic modes available for measurement in cylinders (blue) and spheres (red) as a function of frequency. The calculation is made for a cylinder of radius 50 mm and length 220 mm and a sphere of radius 62 mm at the melting temperature of aluminium (T = 933 K) in one atmosphere (p = 100 kPa) of argon. Notice that for frequencies below 15 kHz, many more modes are available for study in the cylinder compared with the sphere, however their quality factor is substantially lower.

Figure 3 :

 3 Figure 3: The lifting of degeneracy of microwave modes in a cylinder with an ellipsoidal cross section. The TM111 doublets are raised and lowered in frequency, allowing access to the unperturbed TE011.

Figure 4 :Figure 5 :

 45 Figure 4: Dimensional creep in an aluminium cylinder held at 429 K. The changes in length and radius are very significant, but slow enough that that microwave measurements can be used to compensate.

Figure 6 :

 6 Figure 6: Schematic of the NIST NIRT3 (courtesy of NIST)

Figure 7 :

 7 Figure 7: Schematic of one of the PCHPs (courtesy of Advanced Cooling Technologies Inc)

Figure 8 :

 8 Figure 8: Measurements of t-t90 in heat pipes using a SPRT and a radiation thermometer (RT). (Note: the field-of-view of the RTs is not blocked by the head of the SPRT)

Figure 9 :

 9 Figure 9: Schematic view of the radiance comparator in front of the sources showing the: field stop and entrance slit of the monochromator (1), exit slit (2), grating (3), Lyot pupil (4), entrance pupil (5), signal output (6, 7 and 8), gold coated mirrors (10 to 20)

Figure 10 :

 10 Figure 10: Radiancemeter and monochromatic source of radiance. Trap detector (1), monochromatic source (2), precise gauge (3), baffle (4)

Figure 11 :

 11 Figure 11: A schematic of the four-wavelength filter radiometer system

Figure 12 :

 12 Figure 12: Schematic of the operational wavelength bands of the filter radiometer system

Figure 13 :

 13 Figure 13: Schematic drawing (not to scale) of the double-heat pipe blackbody [from 30, 31]

2 2

 2 Sum in quadrature, relative standard uncertainty 6 14 Temperature equivalent at Zn point/ mK (k = 1) -69 Temperature equivalent at Al point/ mK (k = 1) 34 126 Temperature equivalent at Ag point/ mK (k = 1

Figure 14 :

 14 Figure 14: A schematic of the absolute radiation thermometer and the Ag fixed point furnace

Figure 15 :

 15 Figure 15: Irradiance responsivity of the filter radiometer FR900

Table 1 : Example target uncertainties associated with the NIRT3 at the fixed-point temperatures of Zn (693 K) and Ag (1235 K) with reference to the NPL Au point Source of Uncertainty At Zn point u/ mK At Ag point u/ mK

 1

	NIRT spectral responsivity calibration	16	50
	Amplifier gain drift/calibration	2	0
	Au point uncertainty	3	10
	Voltmeter accuracy	1	1
	NIRT3 SSE/ thermal profile	1	1
	Total

uncertainty (k = 1)/ mK 16 51 Total expanded uncertainty (k = 2)/ mK 33 102Table 2 : Example target uncertainties for the measurements of the FPBs using the NIRT3 Source of uncertainty At Zn point u/ mK At Ag point u/ mK

 2

	Impurities	0.3	0.7
	Plateau Identification	2.0	2.0
	Isothermal blackbody emissivity	0.4	2.9
	Emissivity calculation	0.9	1.2
	Reflected ambient radiation	0.0	0.0
	Radiant heat exchange	0.3	3.2
	NIRT3 component	16.0	51.0
	Total uncertainty (k = 1)/ mK	16	51
	Total expanded uncertainty (k = 2)/ mK	32	102

Table 3 : Example target uncertainties for the measurements of the PCHPs with the NIRT3 Source of uncertainty Using the K PCHP at the Zn point, u (k = 1)/ mK Using the Na PCHP at the Ag point

 3

	, u (k = 1)/ mK

Table 4 : Different ways of measuring T at CEM

 4

				Absolute	Relative
	Standard RT	Wavelength/ nm	Temperature range/ K	primary radiation	primary radiation
				thermometry	thermometry
	LP4	650	> 1223	x
	LP4	900	> 1073	x	x
	LP5	1550	673 -1235	x	x

Table 5 : Uncertainty budget for absolute measurements with the LP4 at 650 nm, at T(Cu), T(Ag) and 1223 K showing u (total standard uncertainty, k = 1) and U (total expanded uncertainty, k = 2)

 5

					Contribution to the
	Uncertainty component	Value	Unit	Standard uncertainty	uncertainty/ mK 1357.77 1234.93 1223.15
					K	K	K
	λ0, monochr. calib.	0.03	nm	0.0300	43	41	41
	λ 0, drift	0.05	nm	0.0289	42	40	39
	σ, bandwidth	0.07	nm	0.0700	14	16	16
	H, trap detector (TD) calib.	0.075	%	0.0375	31	26	25
	H, aperture area	0.1	%	0.0500	42	35	34
	H, distance	0.02	%	0.0100	8	7	7
	H, spatial homog. source	0.12	%	0.0346	29	24	23
	H, source stability	0.01	%	0.0100	8	7	7
	H, diffraction	0.01	%	0.0100	8	7	7
	H, stray light	0.01	%	0.0029	2	2	2
	H, out of band	0.01	%	0.0029	2	2	2
	H, signal/noise	0.01	%	0.0029	2	2	2
	H, TD drift	0.05	%	0.0300	24	20	20
	H, TD linearity	0.1	%	0.0289	24	20	20
	In use S, SSE	0.004	%	0.0012	1	1	1
	In use S, linearity	0.06	%	0.0173	14	12	12
	In use S, drift	0.03	K	0.0087	9	9	9
	Total

uncertainty (k =1)/ mK 95 84 83 Total expanded uncertainty (k = 2)/ mK 190 170 170

Table 6 : Uncertainty budget for T -T90 measurements (absolute primary radiation thermometry)

 6

	Uncertainty component	T/ K	U/ mK u/ mK	Total combined uncertainties T/ K u (T-T90)/ K U (T-T90)/ K
		673	39	19.5	673	58	120
		773	39	19.5	773	73	150
		873	39	19.5	873	92	180
	T90, SPRT and VTBB	973 1073	41 41	20.5 20.5	973 1073	112 87	220 170
		1223	41	20.5	1223	87	170
		Ag	-	-	Ag	85	170
		Cu	-	-	Cu	95	190
		673	110	55
		773	140	70
		873	180	90
	T, RT	973 1073	220 170	110 85
		1223	170	85
		Ag	170	85
		Cu	190	95

Table 7 : Uncertainty budget for the T measurements with the LP4 at 900 nm, at 1223 K and 1073 K.

 7

					Contribution to the
	Uncertainty component	Value	Unit	u	uncertainty/ mK
					At 1223 K	At 1073 K
	λ0, monochr.	0.03	nm	0.0300	0.4	4.7
	λ0, drift	0.05	nm	0.0289	0.4	4.5
	σ, bandwidth	0.05	nm	0.0500	0	0.7
	Tref	170	mK	85	83.4	64.2
	Sref	4.00 x 10 -14	A	4.00 x 10 -14	0.1	0.1
	In use S, SSE	0.006	%	0.0017	1.6	1.2
	In use S, linearity	0.062	%	0.0179	16.7	12.9
	In use S, drift	30	mK	9	8.7	8.7
	Total uncertainty (k =1)/ K				86	66
	Total expanded uncertainty (k = 2)/ K				170	130

Table 8 : Uncertainty budget of T -T90 measurements (relative primary thermometry)

 8

	Uncertainty component	T/ K	U/ mK u/ mK	Total combined uncertainties T/ K u(T-T90)/ mK U(T-T90)/ mK
		673	39	19.5	673	49	100
		773	39	19.5	773	40	80
	T90, SPRT and VTBB	873 973	39 41	19.5 20.5	873 973	49 59	100 120
		1073	41	20.5	1073	68	140
		1223	41	20.5	1223	87	170
		673	90	45
		773	70	35
	T, RT	873 973	90 110	45 55
		1073	130	65
		1223	170	85

Table 9 : Summary of T -T90 measurement uncertainties at CEM for different thermometers and primary radiation thermometry methods.

 9

	Temperature/ K	Radiation thermometer	U(T-T90)/ mK (k = 2) Absolute primary Relative primary thermometry thermometry
	1357.77	LP4, 650 nm	190	-
	1234.93	LP4, 650 nm	170	-
	1234.93	LP4, 900 nm	220	-
	1223	LP4, 900 nm	220	170*
	1073	LP4, 900 nm	170	140*
	973	LP5, 1550 nm	220	120*
	873	LP5, 1550 nm	180	100*
	773	LP5, 1550 nm	140	80*
	673	LP5, 1550 nm	110	100*

Table 10 : Uncertainty budget u(TZn), u(TAl) and u(TAg) for the thermodynamic temperatures TZn, TAl and TAg measured at LNE-Cnam at 1550 nm using an InGaAs trap detector as the radiometric reference

 10

	Component	Uncertainty Unit	u (TZn) /mK	u (TAl) /mK	u (TAg) /mK
	λlaser (air)	5.0 x10 -12	m	0	0	1
	Trap response	3.3 x 10 -5	V	10	19	33
	Amplifier (Glaser)	1.0 x 10 2	Ω	1	9	16
	Trap responsivity	6.0 x 10 -4	A/W/m	31	56	99
	Distance between apertures	3.5 x 10 -6	m	0	1	2
	r1 (radius aperture sphere)	2.5 x 10 -7	m	5	8	15
	r2 (radius aperture trap)	2.5 x 10 -7	m	6	11	19
	K diffusion (stray light)	6.0 x 10 -4	1	20	38	66
	K sphere inhomogeneity	1.0 x 10 -4	1	5	9	16
	Total (radiancemeter) (k = 1)			39	73	128
	Effective wavelength λeff	5.0 x 10 -12	m	2	3	4
	Slit scattering function (λeff)	4.6 x 10 -13	V.m	10	19	33
	Amplifier (sphere) (GBB)	1.0 x 10 3	Ω	5	9	16
	Response comparator Sbb	1.4 x 10 -5	V	4	4	11
	Amplifier (FP) (GBB)	9.9 x 10 3	Ω	5	9	16
	KOOB	6.0 x 10 -5	1	3	6	10
	KSSE	2.0 x 10 -5	1	1	2	3
	Total radiance comparator (k = 1)			14	24	43
	Emissivity	1 x 10 -4	1	2	3	5
	Impurities	5 x 10 -6	1	5	5	5
	Setpoint / plateau repeatability		1	5	8	10
	Total fixed point (k = 1)		1	7	10	12
	Day to day repeatability		1	5	22	17
	Combined uncertainty (k=1)/ mK			42	77	136
	Expanded uncertainty (k = 2)/ mK			84	154	272

Table 11 : Characteristics of the four FRs used within the FRFR

 11

	Centre wavelength 440 nm 650 nm 900 nm 1550 nm
	Photodiode	Si	Si	Si	InGaAs

Table 12 : Expected uncertainty budget for the determination of the ITS-90 temperature of the large area blackbody (LABB)

 12

	Uncertainty contribution	At Zn point u/	At Al point u/	At Ag point u /
	(k = 1)	mK	mK	mK
	Realisation of t90	3	5	10
	Gradient at cavity bottom	2	3	5
	Stability of LABB	10	4	25
	Stability of SPRT	5	10	15
	Self heating of SPRT	2	2	2
	Combined (k = 1)/ mK	12	12	31
	Expanded (k = 2)/ mK	24	24	62

Table 13 : Uncertainty budget for the spectral irradiance measurements

 13

Table 14 : Expanded uncertainty estimates of the LABB at lower temperatures by extrapolating down in temperature from the silver point (1235 K) at 900 nm

 14

	U (k = 2)/ mK

Table 15 : Relative uncertainty associated with the absolute radiation thermometer (ART) when measuring fixed-point cells

 15

	Source of uncertainty	FR900
	Integral of FR	8.4 x 10 -4
	Correction of non-linearity	5.0 x 10 -4
	Lens transmittance	2.4 x 10 -4
	Effective area of the blackbody	3.0 x 10 -4
	Lens aperture area	1.4 x 10 -4
	Distance of aperture	1.0 x 10 -4
	Digital voltmeter and amplifier	1.0 x 10 -4
	Air index	0.2 x 10 -4
	Combined standard uncertainty (k = 1)	1.1 x 10 -3

Table 16 : Uncertainties of the freezing point of Ag

 16

	Source of uncertainty	At Ag/ mK
	Calibration of ART	120
	Wavelength/ 0.02nm	10
	FP realisation	10
	Repeatability	40
	Combined standard uncertainty (k = 1)/ mK	130
	Total expanded uncertainty (k = 2)/ mK	260

Table 17 : Results and uncertainty for T measurements at the Ag fixed-point Fixed point T/ K U (k = 2)/ mK

 17

	Ag	1234.84	260

ACKNOWLEDGEMENTS

This work has received funding from the EMPIR programme co-financed by the Participating States and from the European Union's Horizon 2020 R&I programme through the InK2 project.

NIM's research was supported by the National Key R&D Program of China (number 2016YFF0200101)


A D W Todd

K Anhalt

P Bloembergen

B B Khlevnoy

D H Lowe

G Machin

M Sadli

N Sasajima

P Saunders

On the uncertainties in the realization of the kelvin based on thermodynamic temperatures of high-temperature fixed-point cells

Keywords: radiation thermometry, high-temperature fixed-point cells, thermodynamic temperature

In

Introduction

The redefinition of the kelvin in terms of the Boltzmann constant allows for new methods to realize thermomodynamic temperature traceably to the SI. For non-contact thermometry §4.2 of the mise-en-pratique for the definition of the kelvin (MeP-K-19), [START_REF]Consultative Committee for Thermometry 2017 Mise en pratique of the realization of the kelvin[END_REF] describes absolute and relative methods of primary radiometric thermometry, with relative methods: "extrapolation from one fixed point, which requires only knowledge of the relative spectral responsivity of the detector and filter; interpolation or extrapolation on the basis of two fixed points, which requires only the bandwidth of the responsivity; interpolation or extrapolation from three or more fixed points, for which detailed measurements of responsivity are not required". The relative method requires at least one fixed-point cell ‡ with a known thermodynamic phase transition temperature. At present, the MeP-K-19 appendix lists thermodynamic temperatures for cobalt-carbon (∼1597 K), palladium-carbon (∼1765 K), platinum-carbon (∼2011 K) and rheniumcarbon (∼2748 K). In contrast to the ITS-90, the uncertainty of each fixed-point temperature is given. It is expected that in the future this list will be expanded. These fixed points, possibly in conjunction with one or more of the silver, gold or copper fixed-points, can be used to establish a thermodynamic temperature above the silver fixed point (∼1235 K). If combined with ITS-90 specified fixed points and documented differences with uncertainties of T -T 90 there is considerable flexibility in setting up a thermodynamic scale. This paper considers the uncertainties that need to be taken into account when using the high-temperature fixed-points. Uncertainties and descriptions of the methods for absolute primary radiometric thermometry are given in [START_REF]Consultative Committee for Thermometry 2017 Mise en pratique of the realization of the kelvin[END_REF] and [START_REF] Saunders | Uncertainty Estimation in Primary Radiometric Temperature Measurement[END_REF].

Research into the fabrication, use, and determination of the thermodynamic melting temperatures of high-temperature metal-carbon or metal carbide-carbon eutectic and peritectic alloys has been intense since they were first suggested by Yamada et al in 1999 [START_REF] Yamada | [END_REF]. These are known generically as high-temperature fixed-points (HTFPs) and a summary of the history, construction, performance and applications of HTFP cells is given by Machin [START_REF] Machin | AIP Conf. Proc[END_REF].

This paper is a summary of the report of the Consultative Committee for Thermometry's Non-Contact Thermometry Working Group (CCT WG-NCTh) Task Group on high-temperature fixed-point uncertainties (HTFPU.) We describe the methodology of gathering and reporting the uncertainty components and values (Section 2), discuss the uncertainty components and their values intrinsically related to the HTFPs (Section 3), suggest uncertainties to be considered when HTFPs are used in practice (Section 4), present other considerations (Section 5), and recommend areas that require further research (Section 6). This work follows and complements earlier work on the uncertainties in realising ITS-90 using radiation thermometry where a CCT report [START_REF] Fischer | Uncertainty budgets for realisation of scales by radiation thermometry[END_REF] was subsequently published [START_REF] Fischer | Uncertainty budgets for the realisation of ITS-90 by radiation thermometry[END_REF] so as to make the information for users as widely available as feasible. This work can be used as a guide to future reviews of calibration and measurement capabilities (CMCs).

Methodology

Since the goal of this work was to determine a definitive list of uncertainties for HTFP cells and classify them as being either well known or in need of further research we confined ourselves to the types of HTFP cells that are in common use now: smallaperture fixed-point blackbody cells with diameters ∼24 mm and lengths from 43 mm to 50 mm. This work is also limited to those furnaces used for realizing HTFP cells for non-contact thermometry at the time the CCT report was being prepared.

In the tradition of [START_REF] Fischer | Uncertainty budgets for realisation of scales by radiation thermometry[END_REF] and [START_REF] Fischer | Uncertainty budgets for the realisation of ITS-90 by radiation thermometry[END_REF] we have attempted to include best and normal values for each uncertainty component. Best and normal are defined as those that "can be obtained with considerable effort by the small number of leading workers in the field" and those that "can easily be obtained at present in National Metrology Institutes", respectively. Some uncertainty components -those requiring further research -are not sufficiently well understood to be categorized in this way and thus simple "worst case" values are given.

Additionally, the components and uncertainties herein are based upon the most studied HTFP cells, namely Co-C, Pt-C and Re-C. To compile this list of uncertainties and their values we relied heavily on work done in establishing definitive melting temperatures as part of the CCT WG-NCTh/InK project, which is summarised in [START_REF] Machin | AIP Conf. Proc[END_REF], [START_REF] Woolliams | [END_REF], [8], [9].

When using HTFP cells to realize a temperature scale, there are two slightly different schemes which depend on how the temperature is assigned to the local cell or cells.

• Scheme 1. A given high-temperature fixed-point cell is taken to have the temperature value, within the stated uncertainty, given by the MeP-K-19. The stated uncertainty of each high-temperature fixed-point is included as a component in the full uncertainty budget, as summarized in Tables 1 and2.

• Scheme 2. A high-temperature fixed-point cell is taken to have the temperature value determined by a primary thermometric method such as described in [START_REF] Saunders | Uncertainty Estimation in Primary Radiometric Temperature Measurement[END_REF]. It is the uncertainty of that determination that is included in the full uncertainty budget.

It is possible to combine these two schemes to produce a hybrid scale where one HTFP -or more -uses values with uncertainty(ies) as Scheme 1 and one or more as Scheme 2. Most of the uncertainty components herein apply to both methods but some may only apply to one scheme, for example if the temperature of a local cell is assigned by primary radiometry at the same wavelength it will be used at, the uncertainty in the emissivity of the cell does not need to be considered.

Figure 2 shows a graphical representation of the hierarchy of relative and absolute primary radiometric thermometry and of the two relative HTFP schemes.

MeP-K

Absolute primary radiometric thermometry

Relative primary radiometric thermometry

MeP-K Annex

T by primary radiometry Cell T from primary radiometry

(Scheme 2)
Cell T from literature

(Scheme 1)
In-use uncertainty components useable T realization Figure 1. A roadmap from the MeP-K-19 to a realized high temperature scale

Uncertainties intrinsic to HTFP cells

In contrast to the fixed points used in ITS-90, which have zero uncertainty by definition, the high-temperature fixed-points in the MeP-K-19 each have an associated uncertainty in the assigned temperature which has to be combined with the uncertainty of the physical realization of the actual fixed-point cell being assessed. The uncertainty components discussed in this section are considered to be intrinsic to the HTFP cell itself. Some components, for example the size-of-source effect, are considered in Section 4 as they are a function of not only of the fixed point, but also the furnace and radiometer performance.

The uncertainty components to be considered are: temperature drop, structure effect (for Co-C and Fe-C), identification of the point-of-inflection (POI) or identification of the liquidus point, HTFP stability, impurities, furnace effect, emissivity and uncertainties of unknown origin ("unknown" uncertainties).

Temperature drop

Because of heat exchange between parts of the fixed-point cell and a furnace, there is a heat flux and thus a temperature gradient, across the wall of the cell cavity from the solid/liquid interface in the melting ingot to the cavity surface. The temperature drop uncertainty refers to our lack of knowledge of this temperature gradient. While the gradient can be calculated given knowledge of the dimensions and materials, the actual thickness of the cavity is not so well known since some of the graphite wall is consumed during the formation of the alloy ingot. The present best-estimates come from modelling the cells used in the CCT-WG5-High Temperature Fixed Point Research Plan/Implementing the new kelvin project (InK) [10] and are: for Co-C, Pt-C, and Re-C the uncertainties after correction are 0.004 K, 0.011 K and 0.048 K, respectively. Corrections from [10] were divided by 2 (as in [8]) to account for the thinning of the wall and these values were used as both the correction and uncertainty for the assigned values in the MeP-K-19, and these values correspond to the best uncertainty. The temperature drop uncertainty does not need to be taken into account when the HTFP cell's temperature has been determined directly for that HTFP cell under Scheme 2. But note that the temperature drop might depend on the furnace wherein the HTFP cell is subsequently used, which might be different from the one in which the HTFP cell was calibrated. However, this effect is small compared to the overall temperature drop uncertainty [10].

To estimate the normal uncertainty, Eq. 8 of [START_REF] Fischer | Uncertainty budgets for realisation of scales by radiation thermometry[END_REF] was used and it was assumed that the wall thickness was between 2 mm and 1 mm. Using typical graphite properties (thermal conductivity 81 W m -1 K -1 ; emissivity 0.85) and assuming a rectangular distribution, we obtain 0.009 K, 0.023 K and 0.080 K for Co-C, Pt-C and Re-C, respectively.

Structure effect

The microstructure of a binary eutectic alloy is formed during solidification, and the scale and degree of undercooling -the solid-liquid interface being at lower temperature than would be found in equilibrium -depends on the velocity of the interface. The effect that the previous freeze rate has on the subsequent melting temperature has been termed the structure effect [11]. This effect is small and has been observed only in Fe-C and Co-C. The smaller dispersion of the melting temperature with varying rate of the preceding freeze observed for Co-C is because the diffusion of carbon in the liquid state during the building up of the eutectic structure during solidification is significantly faster for Co-C than for Fe-C, given the higher eutectic temperature of Co-C. Note that the diffusion coefficient in question increases rapidly with temperature. As a consequence, at a given freezing rate we expect the eutectic structure of Co-C to be coarser than that of Fe-C.

On the other hand, at a given growth rate the melting range of the Co-C sample appeared to be larger than that of the Fe-C sample, since the impurity content of Co-C in [11] was significantly larger than that of Fe-C. This led to the structure effect uncertainty appearing to be larger for Co-C than for Fe-C: uncertainty values of 0.010 K and 0.014 K for Fe-C and Co-C, respectively, were arrived at in [11]. The value for Co-C is used here as both the best and normal uncertainty. Note that the uncertainties here were arrived at considering the POI and not the liquidus temperature. The uncertainty for the structure effect should be taken into account for both Scheme 1 and Scheme 2.

Identification of the point-of-inflection and the liquidus point

It is well known that HTFP cells do not have the well defined, minimal temperature range melting and freezing that can be achieved with the fixed-point cells used for ITS-90. Whether this is due to the microstructure of a binary alloy compared to pure metal, the difficulty of getting the very best purity (6N) material, the difficulty of getting good thermal uniformity at higher temperatures or some other reason is not well established. As defined in [11], the liquidus point should be determined as being halfway between the upper-limit point (the intersection of a line fitted to the slope of the melting curve at the inflection point, and a line fitted to the run-off after the melt) and the point-ofinflection (POI). This procedure, known as the specific limits approach, was described in [8] §

The uncertainty in the liquidus temperature should be taken as the difference between the upper-limit and point-of-inflection temperatures divided by 2 √ 3. In [8], this amounted to, on average, 0.024 K, 0.044 K and 0.069 K for Co-C, Pt-C and Re-C, respectively, although values can be calculated for each individual plateau. These values are taken to be both the best and normal uncertainties.

The uncertainty in determining the point-of-inflection was estimated to be 0.003 K, 0.005 K and 0.001 K for Co-C, Pt-C and Re-C, respectively, from [8]. These values were the average differences between the original determination of the points-of-inflection from [START_REF] Woolliams | [END_REF] and the reanalysis of the same data in [8]. But in practice other methods can be used to determine the POI, so a value (for all HTFPs) of 0.006 K (0.010 K/ √ 3) should be considered achievable. In the case where the POI is used, and temperatures of the cells are not directly measured, one must also consider how different the cells are from the set for which the temperature determination was made. Including an uncertainty component based upon the measured melting range could be used to estimate this. For example, the difference between the POI and the upper limit temperature measured on the local cell could be used to estimate the uncertainty in the difference between the local set and the set for which the temperature was determined. This uncertainty component should be accounted for when using both Scheme 1 and Scheme 2.

Stability

There is the possibility of changes in a fixed-point cell as, for example, the ingot dissolves some of the graphite crucible or impurities segregate given repeated melting and freezing. A re-assessment of cells used in [START_REF] Woolliams | [END_REF] was performed in [START_REF] Yamada | Stability of High-Temperature Fixed-Point Cells during Thermodynamic Temperature Determination Poster presented at Tempmeko[END_REF] and these values constitute our current best estimates for drift. Here, the worst-case drift from [START_REF] Yamada | Stability of High-Temperature Fixed-Point Cells during Thermodynamic Temperature Determination Poster presented at Tempmeko[END_REF], divided by 2 √ 3, was used and amounts to: 0.038 K, 0.070 K and 0.031 K for Co-C, Pt-C and Re-C, respectively. These values were taken as the best uncertainties. These values are § "At each iteration a cubic function is fitted to the new data range, keeping the fitted range centred on the point of inflection from the previous iteration: The point of inflection is identified; The analytical slope at the point of inflection is found; A linear function is fitted to a fixed fraction of the cubic fitted data set (the effect of choice of fraction having been evaluated and found not to affect the outcome)" consistent with, albeit slightly larger than, the results from [START_REF] Sadli | [END_REF]. An increase in the melting range of a fixed-point cell should be taken as an indication that a cell has drifted and requires verification. There has been a case [14] where a drift of 0.5 K was observed with a Re-C cell. As reported in [START_REF] Woolliams | [END_REF], one Re-C cell was similarly observed to have drifted by 0.5 K. For the normal uncertainty for Re-C a value of 0.5/2 √ 3 K = 0.144 K was used.

The uncertainty component related to the stability of the HTFP should be considered for both Scheme 1 and Scheme 2.

Impurities

Materials with impurities at the level of 10 µg g -1 ("5N") or better are recommended [START_REF] Fischer | Uncertainty budgets for the realisation of ITS-90 by radiation thermometry[END_REF]. In the determinations of the liquidus temperatures for Co-C, Pt-C, and Re-C [8] uncertainty values for the effect of impurities of 0.038 K, 0.037 K and 0.106 K, respectively, were used. These values were estimated from the glow discharge mass spectromometry (GDMS) assays of some of the metals used to fill the cells, with additional components added to account for the lack of knowledge of the purity of the other cells based on the comparison carried out as part of the pre-selection [15] of the cells for the CCT-WG5-High Temperature Fixed Point Research Plan/Implementing the new kelvin project [8]. These values were taken to be the normal uncertainties. To estimate the best uncertainties, an Overall Maximum Estimate (OME) [START_REF] Fellmuth | Methodologies for the estimation of uncertainties and the correction of fixed-point temperatures attributable to the influence of chemical impurities[END_REF] was assuming that 5N pure materials are used. For Co-C, Pt-C and Re-C the OMEs obtained were 0.008 K, 0.010 K and 0.011 K, respectively.

For Co-C and Fe-C, impurities have been covered in detail in [11]. In [17] impurity parameters m (liquidus slope) and k (distribution or partition coefficient) were calculated for a range of impurities by means of the software package Thermo-Calc; estimated uncertainties are included. In calculating the correction T E -T liq , and the associated uncertainty u(T E -T liq ) (where T E is the equilibrium transition temperature), three schemes underlying the uncertainty analysis were compared and the hybrid SIE-IE-IRE (which combines the sum of individual estimates (SIE), individual estimates (IE) and individual random estimates (IRE)) approach was shown to be a reasonable compromise [17].

Since in Scheme 2 each HTFP has had its temperature determined directly, there is no need to account for the effect of impurities. For Scheme 1 the uncertainty associated with the impurities needs to be accounted for since the temperature associated with the HTFP used this way is the published liquidus temperature.

Furnace effect

The "furnace effect" is where differences are seen with the same fixed-point cell in different furnaces which are larger than can be explained by the uncertainty assessment [18]. The furnace effect is an area that requires more work to understand its origins and magnitude. The most recent estimate of the uncertainties due to the furnace effect comes from [19], which gives 0.08 K, 0.125 K and 0.240 K for Co-C, Pt-C and Re-C, respectively. These values were taken as both the best and normal uncertainties. Thermal inertia [20] and the temperature gradient [START_REF] Bourson | Determination of the furnace effect of two high-temperature furnaces on metal-carbon eutectic points[END_REF] over the crucible length probably contribute to the furnace effect. Results from [18] suggest that the furnace effect is still not fully understood. Note that in addition the furnace co-determines the effective emissivity, temperature drop and the impact of the SSE. The furnace effect should be accounted for in both Scheme 1 and Scheme 2 (although the use of the same furnace for all measurements could reduce or eliminate this component).

Emissivity

Present best estimates, based on steady state and taken from [START_REF] Woolliams | [END_REF] for Co-C, Pt-C and Re-C, give equivalent uncertainties of 0.014 K, 0.018 K and 0.025 K, respectively. These values were taken as both the best and normal uncertainties. Emissivity should be accounted for when using both Scheme 1 and Scheme 2. However, if, using Scheme 2, the temperatures for each HTFP were determined at the same wavelength as they will be used at, the emissivity does not need to be accounted for.

There is a need to investigate how the effective emissivity changes dynamically as the temperature in the cavity changes.

Unknown uncertainties

Ideally, the uncertainty in the temperature of a HTFP can be characterized by the components listed above. However, in practice the differences between cells can be larger. The analysis in [START_REF] Woolliams | [END_REF] identified that the uncertainties given by participants for the determination of thermodynamic temperature were inconsistent with the model used. This could be because one or more of the components of uncertainty is actually larger than current estimates, or there could be additional sources of uncertainty as yet unidentified. An additional uncertainty component is required. This uncertainty component was estimated from the cell selection work [15] done prior to the thermodynamic measurements of Co-C, Pt-C and Re-C [START_REF] Woolliams | [END_REF]. To estimate the best value the full range of the deviation of qualified and spare cells (i.e. excluding the disqualified and broken cells) divided by 2 √ 3 was used, resulting in 0.016 K, 0.012 K and 0.044 K for Co-C, Pt-C and Re-C, respectively. To estimate the normal uncertainty values, all the cells (including the disqualified and broken cells) were included, resulting in values of 0.061 K, 0.069 K and 0.070 K for Co-C, Pt-C and Re-C, respectively. These "unknown" uncertainties only need to be considered for Scheme 1.

Temperature

The uncertainty in the temperature of the HTFP cells used must also be taken into account. If the values for the liquidus or POI temperature from MeP-K-19 [START_REF] Machin | MeP-K Relative Primary Radiometric Thermometry Edition[END_REF] are used, the uncertainties given there can be used for Co-C, Pt-C and Re-C as 0.07 K, 0.11 K and 0.22 K, respectively and for the point-of-inflection, the uncertainties are somewhat lower at 0.065 K, 0.090 K and 0.175 K.

If the local HTFP cells have their temperatures determined directly, then this uncertainty component is derived from the primary radiometric uncertainty. In [START_REF] Woolliams | [END_REF], this ranged from 0.10 K to 0.22 K for Co-C, 0.15 K to 0.32 K for Pt-C and 0.28 K to 0.57 K for Re-C.

Summary of the uncertainty components pertaining to the HTFP cells

Tables 1 and2 summarize the uncertainty values associated with the HTFP cells when using the values and uncertainties for liquidus temperature, T liq , given in the MeP-K-19.

Uncertainties related to the use of HTFP cells

In addition to the uncertainties intrinsic to the fixed points described in Section 3, there are additional uncertainty components that need to be considered when implementing HTFP cells. The components may depend on the specific way the HTFP cells are used and often are determined by the HTFP cells and other factors that arise when HTFP cells are used. The size-of-source effect (SSE) is important to the measurement of HTFP cells since the blackbody typically has a small ∼3 mm aperture and the fixed-point cell is housed inside a furnace which is 5 K to 30 K hotter than the fixed-point transition temperature. If used to calibrate an instrument, the SSE needs to be accounted for when measuring radiance sources of different sizes. For guidance and best and normal uncertainties, see [START_REF] Fischer | Uncertainty budgets for realisation of scales by radiation thermometry[END_REF]. For an approach tailored to calibration of radiation thermometers by fixed point blackbodies see [20,[START_REF] Yamada | Size-of-source-effect correction for radiation thermometers calibrated by fixed-point blackbodies[END_REF][START_REF] Bloembergen | [END_REF]. In [START_REF] Bloembergen | [END_REF], an uncertainty between 0.012 K and 0.015 K was arrived at for the SSE at the Re-C point.

Related to the radiation thermometer, the HTFP cell and the furnace used, the SSE should be considered for both scheme 1 and scheme 2. In principle, the cell-furnaceradiometer combination determines the SSE uncertainty.

To use HTFP cells to calibrate an instrument, interpolation between or extrapolation beyond the available fixed points is required. The various schemes have been described in [START_REF] Machin | MeP-K Relative Primary Radiometric Thermometry Edition[END_REF], or its extensions [START_REF] Machin | Realisation and dissemination of thermodynamic temperature above the silver point (1234.93 K)[END_REF] and [START_REF] Saunders | [END_REF], and these documents should be used as guides for the laboratory to evaluate the uncertainty in interpolation/extrapolation appropriate to the scheme it has chosen. The use of the Planck version of the Sakuma-Hattori equation is recommended where one, two or three fixed points are used (n = 1, n = 2 or n = 3), or a least squares approach where more than three are used (fixed points can be a mixture of HTFP and pure metal fixed points).

The case where two HTFP cells are used (the n = 2 scheme) has been studied by Bloembergen and Yamada [27] and Saunders et al. [START_REF] Saunders | Uncertainty in temperatures realised by radiation thermometry using two fixed points[END_REF], although the temperatures for the Co-C, Pt-C and Re-C fixed points need to be updated using the most recent temperatures from [8] and [START_REF] Woolliams | [END_REF]. As an example, at present using the n = 2 scheme with Cu and WC-C, the uncertainty ranges from approximately ∼0.05 K to ∼0.5 K over the range from 1200 K to 4000 K [27]. While related to both the choice of HTFP cells used and the radiation thermometer used, the uncertainty due to interpolation and/or extrapolation should be considered in both schemes.

The sensitivity coefficients required to evaluate the propagation of uncertainties from each of the fixed points can be be found in [START_REF] Machin | Realisation and dissemination of thermodynamic temperature above the silver point (1234.93 K)[END_REF] and [START_REF] Saunders | [END_REF].

Other considerations

Cell evaluation

The way HTFP cells have been filled and the extent to which voids and/or graphite inclusions are present in the ingot can significantly affect the plateau shape and the melting range, and consequently the point-of-inflection position on the plateau as well as the form and position of the run-off at the end of the plateau. In the case of incomplete filling, the most obvious effects would be: a lack of reproducibility of the melting temperature of the cell; a pronounced run-off before full melting was attained; and the sensitivity of the POI to changing the thermal environment (such as furnace set-point temperature, temperature gradients, etc.). The tests below can be used to evaluate the quality of a HTFP cell. Of course, it is up to the user to decide what is considered sufficiently high quality for their purpose.

Cell melting range

In the cell selection process [15] conducted prior to the measurements in [START_REF] Woolliams | [END_REF], selection criteria were established to ensure that only the best HTFP cells were chosen for the definitive thermodynamic temperature assignment. Three main criteria had to be respected for the cell selection: small melting range, high purity of the metals and high melting temperatures, i.e., close to the melting temperature of the pure alloy system (which is most probably correlated with the first two criteria given that for most impurities the distribution coefficient, k, is less than one.) Therefore, in order to ensure high-quality HTFPs, it is important to ensure that the cells to be used as temperature references present the following characteristics: completeness of filling (though the density of the M-C eutectics is not very well known, experience in filling cells can help determine the filling procedure and ensure that the cell is filled to over ∼95 % of its volume); high purity of the metals (99.999 %); and small melting range. The melting range is a reliable indication of the: purity of the materials; adequacy of the positioning of the cell inside the furnace (effect of the temperature distribution); and filling of the cell (an imperfectly filled cell with voids will show poor plateaus with large melting ranges in the case of large melt initiation temperature steps). For HTFP cells to be of the highest quality, it is recommended to ensure that the melting range obtained with a reference cell is compliant with the selection criteria in [15], namely about 0.12 K at Co-C, 0.30 K at Pt-C and 0.35 K Re-C. 5.1.2. Effect of melt initiation temperature step on the melting range During the assessment of the effect on the plateau melting range with different melt initiation steps, as above, HTFP cells that show large differences in their melting temperature between plateaus realized with different offset temperatures, for example 10 K, 20 K and 30 K, should be considered inadequate for use as high quality reference cells. In the authors' experience with Co-C, Pt-C and Re-C, differences in the melting ranges between the different offset temperatures of 0.07 K, 0.10 K and 0.15 K, respectively, should be considered as limits for the highest quality cells.

For less demanding applications, these criteria can be relaxed with correspondingly larger uncertainties.

Comparison of methods

Using the point-of-inflection as the reference point rather than the liquidus is likely to have smaller uncertainties, but does require knowing that the realization has been made under the same or sufficiently similar conditions as during the assignment of temperature. For this reason the point-of-inflection is well suited to scheme 2. If the values for point-of-inflection listed in the MeP-K-19 are to be used, this means the same construction requirements need to have been met and also comparable realisation conditions, such as what type of furnace is used, and how well it has been optimized. Alternatively, a direct chain of comparison measurements to one of the HTFP cells used in [START_REF] Woolliams | [END_REF] would be needed.

The use of the liquidus overcomes these limitations. The liquidus is bounded by limits, which gives larger uncertainties, but the limits are valid regardless of the conditions of use: poorer realisations will increase the spread of possible values but will still include the reference value albeit with a larger uncertainty. This method is thus best suited to new cells where there is no or limited suitable radiometric temperature assignment capability. There is a trade-off between the uncertainty in the published value compared to the local thermodynamic capability, and the need to assess the effect of impurities in a new cell.

Limitation of scope

Finally, it should be noted that the information here refers to eutectic alloys and the use of melting to identify the liquidus or a proxy for it. A number of peritectic alloys have also been investigated and at least one is under investigation for addition to the MeP-K-19. It may be possible that in evaluating the peritectic based HTFP the freeze is preferred to the melt. In that case additional uncertainties may need to be added to those considered in this paper.

Recommendations for further study

The materials used for HTFPs and the way references are realized have not been studied in as much depth as pure metal fixed points. There are aspects that relate to assessing uncertainties that are not fully understood. The two most important areas that need research are to better understand the furnace effect and to determine the temperatures of other HTFP.

• Further work to understand the furnace effect is the most important area of study.

The observed effect should be investigated to fully clarify its physical origin. Some recent work [START_REF] Yamada | [END_REF] has already been published. There are additional areas that would benefit from further investigation and might reduce or better characterize uncertainties.

• Results of the stability tests in Work Package 1 of the CCT-WG5-High Temperature Fixed Point Research Plan/Implementing the new kelvin project only detected drift for one cell. There is work presently underway to better understand HTFP stability. An update to the stability uncertainties should be made once this work is complete.

• If possible, experimental confirmation of the temperature drop calculations in [10] and further simulations should be attempted.

• The correlations between the temperatures of the fixed points used in any interpolation/extrapolation scheme should be considered. Uncertainties due to interpolation/extrapolation should be reconsidered given the present knowledge of fixed-point melting temperatures and their uncertainties.

Conclusion

The uncertainty components and their magnitudes discussed in this work represent the state of knowledge at the time of writing. The authors acknowledge that this is an incomplete picture and expect that these values will be become refined as the high-temperature metrology community gains more experience implementing, testing through comparisons, and reviewing calibration and measurement capabilities [30] based on relative primary thermometry.

•

  As previously mentioned, Co-C, Pt-C and Re-C are the most studied HTFPs. Future work should include determination of the thermodynamic temperatures of other HTFP types such as the WC-C and Pd-C points. This could be carried out using scales derived from the published values for Co-C, Pt-C and Re-C or via a direct radiometric determination.

Table 1 .

 1 For scheme 1, a summary of the components for the use of T liq of HTFP cells and present best and normal estimates of their uncertainties

	Item	Source Present best estimates of uncertainty (k =1) / mK
			Co-C		Pt-C		Re-C
		Best Normal Best Normal Best	Normal
	Temperature drop	4	9	11	23	48	80
	Structure effect	14	14	N/A	N/A	N/A	N/A
	Identification of T	24	24	44	44	69	69
	Stability	38	38	70	70	31	144
	Impurities	8	38	10	37	11	106
	Furnace effect	80	80	125	125	240	240
	Emissivity	14	14	18	18	25	25
	Unknown	16	61	12	69	44	70
	Liquidus T	70	70	110	110	220	220
	Total	119	138	188	204	342	393

Table 2 .

 2 For scheme 2, a summary of the components for the use of T liq of HTFP cells and present best and normal estimates of their uncertainties

	Item	Source Present best estimates of uncertainty (k =1) / mK
			Co-C		Pt-C		Re-C
		Best Normal Best Normal Best	Normal
	Structure effect	14	14	N/A	N/A	N/A	N/A
	Identification of T	3	6	5	6	1	6
	Stability	38	38	70	70	31	144
	Furnace effect	80	80	125	125	240	240
	T (Radiometer)	96	221	150	316	280	567
	Total	131	239	207	347	370	632