Abstract

At the CCT’s request, Working Group 4 (WG4) critically reviewed all available measurements of $T - T_{90}$ including constant-volume gas thermometry, acoustic gas thermometry, spectral radiation thermometry, total radiation thermometry, noise thermometry, and dielectric-constant gas thermometry. Consensus estimates are provided for $T - T_{90}$ , for selected measurements from $4.2 K$ to $1 358 K$ , as well as a recommendation for analytic approximations to $T - T_{90}$ for the range $0.65 K$ to $1 358 K$ .

Estimates of the Differences between Thermodynamic Temperature and the ITS-90

1. Table of Differences

Table 1 summarizes the best estimates of $T - T_{90}$ above $4.2 K$ as of 2010. In general, a weighted average was formed using the uncertainties identified by WG4. For details see [1]. The data are shown in Figure 1 and Figure 2.

Table 1. Estimates of $T - T_{90}$ between $4.2 K$ and $1 358 K$ . The transitions of the defining fixed pointsand secondary reference points of the ITS-90 are marked in the 2^nd and 6^th columns. All uncertainties are standard uncertainties ( $k = 1$ ). The differences for temperatures above $1 358 K$ are under investigation by Working Group 5. The results presented here may be extrapolated above $1 358 K$ using Planck’s law.

$T_{90}$ ( $K$ )		$T - T_{90}$ ( $mK$ )	$u$ ( $mK$ )	$T_{90}$ ( $K$ )		$T - T_{90}$ ( $mK$ )	$u$ ( $mK$ )
$4.2$		$- 0.02$	$0.12$	$161.405$	Xe	$- 8.43$	$1.8$
$5$		$0.10$	$0.12$	$195$		$- 6.97$	$1.8$
$6$		$0.04$	$0.13$	$234.315 6$	Hg	$- 3.25$	$1.0$
$7$		$- 0.08$	$0.09$	$255$		$- 1.64$	$0.9$
$8$		$0.01$	$0.10$	$273.16$	TPW	$0$	$0$
$9.288$	Nb	$0.13$	$0.11$	$290$		$2.19$	$0.4$
$11$		$0.27$	$0.12$	$302.914 6$	Ga	$4.38$	$0.4$
$13.803 3$	e-H₂	$0.44$	$0.14$	$335$		$7.62$	$0.5$
$17.035$	e-H₂	$0.51$	$0.16$	$373.124$	H₂O	$9.74$	$0.6$
$20.27$	e-H₂	$0.32$	$0.17$	$429.748 5$	In	$10.1$	$0.8$
$22.5$		$0.10$	$0.18$	$505.078$	Sn	$11.5$	$1.3$
$24.556 1$	Ne	$- 0.23$	$0.20$	$600.612$	Pb	$9.21$	$6.1$
$35$		$- 0.53$	$1.0$	$692.677$	Zn	$13.8$	$6.9$
$45$		$- 0.75$	$1.4$	$800$		$22.4$	$6.4$
$54.358 4$	O₂	$- 1.06$	$1.6$	$903.778$	Sb	$27.6$	$7.6$
$70$		$- 1.57$	$1.9$	$933.473$	Al	$28.7$	$6.6$
$77.657$		$- 3.80$	$1.2$	$1 052.78$	Cu/Ag	$40.9$	$26$
$83.805 8$	Ar	$- 4.38$	$1.3$	$1 150$		$46.3$	$20$
$90$		$- 5.30$	$1.1$	$1 234.93$	Ag	$46.2$	$14$
$100$		$- 6.19$	$1.2$	$1 337.33$	Au	$39.9$	$20$
$130$		$- 8.07$	$1.6$	$1 357.77$	Cu	$52.1$	$20$

2. Interpolation Functions

If it is not convenient to use Table 1, the differences $T - T_{90}$ may be approximated by the following expressions. Above $70 K$ , the relative differences of the interpolation functions (with respect to the values of Table 1) are less than 15 %, except at $600 K$ and the gold point.

From $0.65 K$ to $2 K$ , use the polynomial for the temperature scale PTB-2006 (based on the ³Helium vapor-pressure) [2] with

$T - T_{90} \equiv T_{2 006} - T_{90} .$

Below $1 K$ , $T_{2 006}$ is identical to $T_{PLTS- 2 000}$ .

From $2 K$ to $8 K$ ,

$T - T_{90} \equiv 0 .$

From $8 K$ to $273.16 K$ ,

$(T - T_{90}) / mK = \sum_{i = 0 . . 7} b_{i} \times {(\log_{10} (T_{90} / 273.16 K))}^{i + 1}$ (1)

with the coefficients:

$b_{0} = 4.424 57 \times 10^{1} b_{1} = - 1.763 11 \times 10^{2} b_{2} = - 1.539 85 \times 10^{3} b_{3} = - 3.636 85 \times 10^{3}$

$b_{4} = - 4.198 98 \times 10^{3} b_{5} = - 2.613 19 \times 10^{3} b_{6} = - 8.419 22 \times 10^{2} b_{7} = - 1.103 22 \times 10^{2}$

The derivative $d (T - T_{90}) / d T_{90}$ at the triple point of water is $7.0 \times 10^{- 5}$ .

From $273.16 K$ to $1 357.77 K$ (copper point):

$(T - T_{90}) / mK = (T_{90} / K) \sum_{i = 0 . . 4} c_{i} \times {(273.16 K / T_{90})}^{2 i}$ (2)

with the coefficients:

$c_{0} = 0.049 7 c_{1} = - 0.303 2 c_{2} = 1.025 4 c_{3} = - 1.289 5 c_{4} = 0.517 6$

The derivative at the triple point of water is $10.1 \times 10^{- 5}$ , resulting in a discontinuity of $3.1 \times 10^{- 5}$ between Equation (1) and Equation (2), see Figure 1. This is consistent with the values from recent thermodynamic measurements and measurements of platinum resistance thermometers that conform to ITS-90.

Figure 1 — Overview of consensus estimates for $T - T_{90}$ with emphasis on the range above the triple point of water. The smooth functions (Equation (1) and Equation (2), black line) are interpolating the mean values (black dots). Error bars represent uncertainties with $k = 1$ .

Figure 2 — Enlargement of the range between $4.2 K$ and $80 K$ of consensus estimates for $T - T_{90}$ . The smooth function (Equation (1), black line) interpolates the mean values (black dots) above $8 K$ . Error bars represent uncertainties with $k = 1$ .

References

[1] ] J. Fischer, M. de Podesta, K. D. Hill, M. Moldover, L. Pitre, R. Rusby, P. Steur, O. Tamura, R. White, L.Wolber, Int. J. Thermophys. 32, 12-25 (2011).

[2] ] J. Engert, B. Fellmuth, K. Jousten, Metrologia 44, 40-52 (2007).