Thermodynamic-temperature data from 30 K to 200 K

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


Introduction
The appendix 'Estimates of the differences between thermodynamic temperature and the ITS-90' [1] of the Mise en Pratique ('practical realisation') of the definition of the kelvin [2] gives recommended estimates of the differences between thermodynamic temperature T and temperature T 90 on the International Temperature Scale of 1990 (ITS-90) [3] at a predefined set of temperatures ('base temperatures'), including the fixed points of the ITS-90 and additional points that are often secondary reference points. Additionally, for the temperature ranges from 8 K to 273 K and 273 K to 1358 K, Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. respectively, functions T − T 90 versus T 90 are given. (The function for the low-temperature range is called in the present paper (T − T 90 ) LT2011 .) These estimates and functions allow corrections to be made to T 90 values when accurate measurements of T are needed. The appendix is based on an evaluation of the existing thermodynamic data finished in 2010 by a working group of the Consultative Committee for Thermometry (CCT) [4]. In some temperature ranges below the triple point of water, the uncertainty of the estimates approaches 2 mK, while the absolute differences T − T 90 are smaller than 10 mK. To overcome this unfavourable situation of high relative uncertainty, thermodynamic temperature measurements were performed mainly at three Metrological Institutes [5][6][7].
In this paper, the results of most recent measurements with the Dielectric-Constant Gas Thermometer (DCGT) performed at PTB are described in section 2. The new data T DCGT − T 90 (T DCGT is the DCGT temperature value) are combined in section 3 with former thermodynamic results obtained with DCGT at PTB, which are summarised in [8], but not considered in [4], to obtain a consistent function T DCGT − T 90 versus T 90 for the temperature range from about 30 K to 200 K, called (T − T 90 ) PTB2020 . Finally, in section 4, this function is in turn combined with the 'Estimates of the differences between thermodynamic temperature and the ITS-90' [1] as well as with thermodynamic data obtained applying Acoustic Gas Thermometry (AGT) at the National Physical Laboratory (NPL), UK [5], and Refractive-Index Gas Thermometry (RIGT) at the National Research Council (NRC), Canada [6]. The final result is a function T − T 90 versus T 90 for the temperature range from 35 K to 195 K with reduced uncertainty that is called (T − T 90 ) ARD2020 , where ARD is an abbreviation for AGT-RIGT-DCGT.

DCGT method
DCGT is based on replacing the density in the equation of state of a gas by the dielectric constant ε. The dielectric constant is determined via the change of the capacitance C(p) of a suitable capacitor measured with (C(p)) and without the measuring gas (C(0)). For performing primary thermometry in the temperature range from 50 K to 200 K applying helium, neon, and argon as measuring gases at pressures up to 0.3 MPa, the following fourth-order working equation for the pressure p considers sufficiently the interaction of the gas particles: with and as well as where A ε is the molar polarizability, R is the molar gas constant, and ε r = ε/ε 0 (ε 0 is the electric constant). The linear term describes the ideal-gas behaviour. Its coefficient A 1 contains the target measuring quantity, the thermodynamic temperature T. κ eff denotes the effective compressibility of the capacitor, which describes the change of the capacitance only due to the mechanical deformation caused by the measuring gas.
(The term 'effective' indicates the fact that each capacitor is a composite because small pieces of insulator materials are necessary to isolate the electrodes electrically.) As shown in [9], the coefficients A 2 , A 3 , and A 4 of the second-, third-and fourth-order terms, respectively, can be expressed as functions of A 1 , A ε , and κ eff as well as of both the density and dielectric virial coefficients, which describe the interaction of the gas particles.
For determining A 1 , isotherms have to be measured. Then, a polynomial fit without a constant term according to the working equation (1) to the obtained p versus µ dependence is performed. The calculation of T from the fitting coefficient A 1 requires knowing R and A ε with sufficiently small uncertainty. (κ eff must be determined individually for the measuring capacitor, see the following section.) The newest values of the fundamental constants including R are published in [10]. For helium, a sufficiently accurate value for the static electric dipole polarizability of a gas particle has been obtained from ab initio calculations [11][12][13][14][15]. This value, having a relative uncertainty below one part per million (1 ppm), yields together with the newest values of the fundamental constants A 4 He ε = 5.1725409(5) × 10 -7 m 3 mol −1 for the heavy helium isotope 4 He (the number in brackets is the standard uncertainty given as multiple of the last digit). For neon and argon, the uncertainty of theoretical calculations of the polarizability is yet too large for primary thermometry. Experimental values having the smallest uncertainty have been obtained by DCGT at the triple point of water [9]: A Ne ε = 9.947114(24) × 10 -7 m 3 mol −1 and A Ar ε = 4.140686(10) × 10 -6 m 3 mol −1 .

Experimental setup
The DCGT setup of the second generation (DCGT2), leading to the present results, is already described extensively in [16] and in full detail in [17]. Therefore, in this section only a very short summary of the major changes and improvements achieved, in comparison with this original equipment, is given.
(1)For stabilising temperatures between 50 K and 200 K at a level of a few 10 µK, a tank cryostat with four cooling stages (nitrogen tank, helium tank and two pumped helium vessels) was used. The cryostat has five modes of operation: (i) liquid helium (lHe) and liquid nitrogen (lN 2 ) in the two tanks, respectively, and pumping of the two vessels filled with lHe from the tank via capillaries having well-dimensioned flow impedances; (ii) lHe and lN 2 in the two tanks and empty vessels; (iii) pumped solid nitrogen in the helium tank and lN 2 in the nitrogen tank; (iv) lN 2 in both tanks; (v) circulation of temperature-controlled alcohol through a cooling tube, which is normally used for fast cooling with liquid nitrogen. For covering the temperature range from 50 K to 200 K modes (iii) to (v) were necessary. The temperature control must be optimised individually for the different modes. This includes the temperature distribution along the pressure-sensing tubes, which go completely through vacuum. The temperature T 90 according to the ITS-90 was measured with three capsule-type standard platinum resistance thermometers (CSPRTs) situated at opposite sides of the measuring copper block housing in between two measuring capacitors. The calibration of the CSPRTs is described in [18,19]. The CSPRTs carrying an ITS-90 calibration were compared at all measuring temperatures, and the maximum difference between their readings was below 50 µK. At all isotherm temperatures, the CSPRTs showed an agreement between 30 µK and 50 µK. These comparison results were used as an estimate of thermal gradients in the system and included in the uncertainty component 'Fit coefficient A 1 ' of the budget in table 3.
(2)The relative uncertainty of the pressure measurement was decreased to 3.6 parts per million (3.6 ppm) by calibrating the piston-cylinder unit (PCU) of the pressure balance used, which has a nominal effective area of 3 cm 2 , against the special PCUs characterised for the measurement of the Boltzmann constant [20].
(3)The equipment used for measuring capacitance changes was developed as described in [7,21], which resulted in a relative uncertainty of one part per billion (1 ppb).
(4)The two newest-generation cylindrical measuring capacitors called C1 and C2 have the same general design as those used by Luther et al [22]. The electrodes and the housing were machined from the copper-beryllium-cobalt alloy C17500 (nominal content of (0.4-0.7)% beryllium and (2.4-2.7)% cobalt). The inner electrode has a length of 45 mm and an outer diameter of 17 mm. The inner length of the outer electrode is 70 mm and its inner diameter is 22 mm. The electrodes are fixed on a grounded bed-plate using mica disks of thickness 0.1 mm for electrical isolation. The determination of κ eff in dependence on temperature as described in [23] is based on the measurement of isotherms at the TPW as starting point, see [8]. Furthermore, it was necessary to measure the following properties of the electrode material as described in [23]: specific heat capacity (77 K to room temperature), thermal expansion (2 K to 300 K), and adiabatic bulk modulus (230 K to 330 K).
(5)DCGT measurements cause extreme requirements regarding the purity of the measuring gas especially for helium. Impurities should not cause a relative change of the result by more than 1 ppm. To prevent contamination of the measuring gas during handling, gas purifiers (getter and adsorber) have been incorporated in the ultra-high-purity gas-handling system. (The specified upper limit for water and the other relevant impurities beside noble gases is 10 parts per billion (ppb) for the getter, and for the adsorber, the specified upper limit is 0.1 ppb.) After each measurement of an isotherm, which lasted usually one day, the measuring gas was analysed with the aid of a mass spectrometer to check for a possible contamination especially due to outgassing from the different pieces inside the experimental chamber (the detection limit for noble gas contaminations is 10 ppb). The most problematic impurity is water because it has a polarizability 160 times larger than that of helium. Thus, the detection limit of 20 ppb would cause an uncertainty component of order 1 ppm applying a rectangular distribution. But additional analysis of helium gas remaining in the system for weeks led to an even lower uncertainty component. Therefore, considering the specification of the adsorber, 1 ppm is a reliable upper estimate for an overall uncertainty component including all relevant impurities.

Isotherm data
New triplets of temperature T 90 , pressure p and dielectric constant (represented by the DCGT measuring quantity µ ≈ (ε r − 1)/3, see equations 2 and 3) were measured with the working gases helium (purity 99.99999 %) at 50 K, 51 K, 60 K, 70 K, 79 K, 100 K, and 200 K, neon (purity 99.9993 %) at 51 K, 60 K, 70 K, 79 K, 100 K, 120 K, and 200 K, and argon (purity 99.99999%) at 200 K. The complete dataset is given in table 1, in which the temperature is given on the ITS-90 (T 90 ) together with its standard uncertainty (u(T 90 )), and the p and µ values are in each case the mean of the specified number n of single values. (The overall number of final triplets amounts to 276.) The uncertainty values u(T 90 ) have been estimated applying the procedures recommended in [24] for propagating the uncertainty and considering the non-uniqueness. Based on recent international intercomparisons, especially on a star intercomparison of sealed triple-point cells [25], the standard uncertainty of the fixed-point temperatures used for the calibration of CSPRTs amounts to 0.1 mK. The relative standard uncertainty of the pressure measurement is dominated by the uncertainty of the calibration of the pressure balance used (3.6 ppm, see section 2.2), whereby the resolution of the balance of 0.1 Pa is included in the Type A uncertainty component 'Fit coefficient A 1 ' (see table 3). Furthermore, the uncertainty of the correction of the pressure difference caused by the gas column (head correction) has been included in table 3. The relative uncertainty of the measurement of capacitance changes of order 1 ppb yields a relative uncertainty of the µ values of 10 -9 /µ. But in reality, the uncertainty of a single µ value is only partly relevant for the temperature result (see the Type B component "Susceptibility measurement (capacitance change)" in table 3). This is because several pairs of p and µ are fitted on an isotherm, which results in a Type A uncertainty component for the errors of the inductive voltage divider used for balancing the capacitance bridge. This component is included in the component 'Fit coefficient A 1 '. The agreement of the zero-pressure capacitance ratio before and after the isotherm measurement was in all cases below the detection limit of the capacitance bridge.

Data evaluation and uncertainty estimation
All isotherms were fitted to polynomials of second, third, and fourth order according to the working equation characterised briefly in section 2.1. (In view of the limited number of triplets and the spacing of the measuring temperatures, only singleisotherm fits were performed.) The A 1 values resulting from fits of second order cannot be used directly for determining the thermodynamic temperature value T DCGT because they are distorted by neglecting the third-order term, which is mainly connected with the third density virial coefficient C. But these values can yield a useful additional information if they are corrected for the influence of the third coefficient in equation (1) (A 1 A 3 ). A derivation of the complete formula is given in [9]. Neglecting higher order terms in κ eff , the third dielectric virial coefficient as well as the combination of the second density and dielectric virial coefficients, this term has the form A 1 3 (RT) −2 C. Theoretical values of the third virial coefficient as constraints were taken from [26] for helium, from [27] for neon, and from [28] for argon (for more details see also [9]). For the evaluation with the constraints, a relative uncertainty component of 5% was added for all three gases. This is far above the theoretical estimates and, therefore, sufficiently conservative. The resulting correction of the A 1 values from The new DCGT dataset between 50 K and 200 K is listed. For each isotherm, the specific capacitor C1 and C2, respectively, as well as the associated compressibility and the uncertainty is listed. Furthermore, the triplets T 90 , p and µ are given, were µ is the measuring quantity used in the DCGT measurements (see section 2.1, equation (3)). The p and µ values are in each case the mean of n single values. The nominal purity stated by the manufacturer, before further purifying procedures (see text), is listed, whereby 7N stands for 99.99999% of the measuring gas and 5.3N stands for 99.9993%. The reference temperature T 90 is the PTB realisation of the ITS-90 applying capsule-type standard platinum resistance thermometers. The temperature T DCGT is the weighted mean of the results of second, third and fourth order fits (for more details see text). u(T 90 ) and u(T DCGT ) as well as u(T − T 90 ) are the individual standard uncertainties of the temperatures T 90, T DCGT and the difference T − T 90 , respectively (for more details see table 3).

Gas
Capacitor second-order fits in temperature equivalent is largest for Neon at 51 K with 6 mK. It is continuously decreasing to a level of 0.7 mK with raising temperature. For argon, the correction at 200 K amounts to 5 mK, and for helium it ranges from 2.5 mK to 0.4 mK. The smallness of the correction together with the very conservative uncertainty attributed to it leads to the fact that for this approach, no high-level ab initio calculations are needed. Thus, this approach would also work with semi-classical calculations available since many years. Therefore, it is intentionally different from other approximations using high-level ab initio calculations as constraints for both the second and the third virial coefficient, where the corrections due to real-gas properties are on a level of 1 K or more [6]. Nevertheless, the use of highly-accurate ab initio values not only for the density but also for the dielectric virial coefficients of neon and argon is continuously improving [29,30], and it will allow similar evaluation techniques, which are at the moment mostly restricted to helium. For each isotherm, the A 1 value obtained with the constraint was used together with the A 1 values of the thirdand fourth-order fits to calculate a weighted mean, with the weights being the inverse uncertainty estimates squared. Since the three single values are largely correlated, the uncertainty of the result was not estimated by the weighted-mean uncertainty, but by the smallest uncertainty of the three A 1 values. This approach corresponds to the case of full correlation.
For calculating T DCGT from the weighted-mean A 1 value, the effective compressibility κ eff of the measuring capacitor is needed. Its determination in dependence on temperature is briefly described in section 2.2. The resulting consistent dataset T DCGT − T 90 in dependence on T 90 obtained in the range from 50 K to 200 K with DCGT is given in table 2 together with the standard uncertainty u(T DCGT − T 90 ) as well as the measuring gas, and the capacitor used. New data is listed for 50 K, 51 K, 60 K, 70 K, 79 K, 100 K, 120 K and 200 K. If different results exist, the weighted mean has been estimated in steps considering correlations. The isotherms obtained with neon at the triple point of argon at 84 K are already given in [8] but the evaluation is shifted due to the new determination of the polarizability of neon with reduced uncertainty [31]. Therefore, the weighted-mean value listed in table 2 for 84 K and, due to an additional neon isotherm (the helium isotherms at 120 K are already listed in [8]), the value at 120 K are revised compared with [8]. Table 2. Consistent new dataset T DCGT − T 90 in dependence on T 90 in the temperature range from 50 K to 200 K obtained with the second generation of DCGT equipment (see section 2.2). u(T DCGT − T 90 ) is the standard uncertainty of the weighted-mean value T DCGT − T 90 that has been estimated in steps considering correlations. The data of this work have been obtained using capacitor C1 and/or C2. Uncertainty budgets to estimate u(T DCGT − T 90 ) have been established individually. Examples are given in table 3 for the three temperatures 51 K, 100 K, and 200 K. In accordance with a recommendation of the Consultative Committee for Thermometry, the non-uniqueness is included in the estimation of the uncertainty of the realisation of the ITS-90 via the calibration of standard platinum resistance thermometers [24].

Consistent dataset and function T DCGT − T 90 versus T 90 from 30 K to 200 K
The data presented in table 2 of reference [8] for T DCGT − T 90 and u(T DCGT − T 90 ) in the temperature range from 28.5 K to 140 K were not considered in [4]. The combination of this data with the results listed in table 2 of the present paper yields, therefore, a consistent dataset for the temperature range from about 30 K to 200 K that is independent from the evaluation in [4].
For further comparison with literature data, the new consistent dataset was fitted with a fourth-order polynomial. Both unweighted and weighted fits have been compared against a spline interpolation. (The squared invers uncertainties were used as the weights.) The maximum difference between these three approximations was used to estimate an uncertainty component applying a rectangular distribution. Figure 1 shows the final interpolation function T DCGT − T 90 versus T 90 being the fourth-order polynomial obtained from the unweighted fit: with a 0 = 0.3260, a 1 = 0.013628, a 2 = −0.001506, a 3 = 1.0079•10 −05 , a 4 = −1.7443•10 −08 , which is valid for the temperature range from 30 K to 200 K. Figure 1 shows also data points (T DCGT − T 90 ; T 90 ) together with bars representing their standard uncertainty estimates u(T DCGT − T 90 ). The data points taken from literature are discussed in the following section.

Updated function T − T 90 versus T 90 from 35 K to 195 K
For updating the information on T − T 90 versus T 90 , it seems to be the easiest and most clearly arranged way to combine the recommendations of the Consultative Committee for Thermometry with the new data at the 'base temperatures' selected in [4]. Rourke [6] [4] and a spline interpolation between the weighted-mean values for combination (1) given in table 4. The corresponding shaded areas display the confidence interval corresponding to the standard uncertainty. Figure 2 gives the impression that the estimates of the thermodynamic temperature, being based on the critical review of previous T − T 90 determinations performed by a working group of the Consultative Committee for Thermometry [4], are slightly too low, but the order of magnitude of the uncertainty is realistic. The new data, combined in the weightedmean values for combination (1), have an uncertainty, which is    [8]; Half grey and half red dots: Values from [8] readjusted at 84 K due to a new polarizability value of neon and at 120 K due to an additional neon isotherm, respectively. The red line represents a fourth-order polynomial obtained from an unweighted fit to the DCGT data from about 30 K to 200 K (Polynomial 5). The black line displays the best fit of a critical review of previous T − T 90 determinations performed by a working group of the Consultative Committee for Thermometry [4] (This function is called in the present paper (T − T 90 ) LT2011 ). In addition, literature data are included for comparison: Blue stars: AGT by Underwood et al 2016 [5]; Black filled squares: RIGT by Rourke 2020 [6]. Table 4. Overview of results for the differences between thermodynamic temperature T and temperature T 90 on the International Temperature Scale of 1990 (ITS-90) at 'base temperatures' selected in [4]. The second to seventh column contain recent data obtained by AGT [5], DCGT (this paper), and RIGT [6], respectively, and the accompanying standard uncertainty estimates u(T − T 90 ). The AGT and DCGT data have been deduced applying fourth-order polynomials fitted to the experimental pairs (T 90 ; (T − T 90 )), see text. Columns eight to eleven show weighted-mean values together with their uncertainty estimates. Combination (1) considers the information given in the second to seventh column. Combination (2) includes also the values from [4]. The (T − T 90 ) values for combination (2) have been approximated by function (T − T 90 ) ARD2020 (Polynomial 6). All differences and uncertainty estimates are given in mK.

AGT
DCGT RIGT Combination (1) Combination (2)  much smaller (up to a factor of about four) than the uncertainty estimates presented in [4]. This leads in turn to a significant reduction of the uncertainty of the updated function T − T 90 versus T 90 from 35 K to 195 K, called (T − T 90 ) ARD2020 (Polynomial 6), which approximates the overall weighted-mean values (combination (2)). It should be emphasised that considering the combined uncertainties, the DCGT results are consistent with the AGT ones. This is of crucial importance because the sources of error of these two primary-thermometry methods are quite different [32].

Summary and conclusions
New data is presented for the difference between the thermodynamic temperature measured with DCGT, T DCGT , and the temperature on the ITS-90, T 90 , at 50 K, 51 K, 60 K, 70 K, 79 K, 84 K, 100 K, 120 K, and 200 K. Particular care was taken to check for possible systematic sources of uncertainty by performing experiments applying three working gases, namely helium, neon, and argon, the polarizability of which differs by a factor of up to eight. The new data T DCGT − T 90 are combined with former thermodynamic results obtained with DCGT at PTB to obtain a consistent function T DCGT − T 90 versus T 90 for the temperature range from about 30 K to 200 K, called (T − T 90 ) PTB2020 . In turn, this function is combined with the 'Estimates of the differences between thermodynamic temperature and the ITS-90' [1,4] as well as with thermodynamic data obtained applying AGT [5] and RIGT [6]. The final result is an updated function T − T 90 versus T 90 for the temperature range from 35 K to 195 K with significantly reduced uncertainty (up to a factor of about four), called (T − T 90 ) ARD2020 .
Since the sources of error of the two primary-thermometry methods DCGT and AGT are quite different, the observed consistency is a good basis for a high quality of updating the T − T 90 estimates. The presented data as well as unpublished and therefore not included results from other institutes were driven by different joint European projects during the last years (for more details see [33][34][35]). The function (T − T 90 ) ARD2020 will be submitted to the Working Group for Contact Thermometry of the Consultative Committee for Thermometry (CCT-WG-CTh) for preparing new recommendations on T − T 90 versus T 90 in the low-temperature part of the ITS-90 (0.65 K to 273 K).