Deviation of Temperature Determined by ITS-90 Temperature Scale from Thermodynamic Temperature Measured by Acoustic Gas Thermometry at 79.0000 K and at 83.8058 K

Abstract

A difference was determined between temperature defined by International Temperature Scale (ITS-90) and thermodynamic temperature measured by relative acoustic gas thermometry at 79.0000 K and at 83.8058 K. Measurement were carried out using misaligned spherical acoustic resonator filled with helium at different pressures. The isotherms were fitted for four different acoustic modes simultaneously assuming the same thermal accommodation coefficient and third acoustic virial coefficient for all modes. Our measurements yield the following differences between the thermodynamic temperature T and ITS-90 temperature $T_{90}$: $T-T_{90}=-4.81 \pm 1.02$ mK at 83.9058 K and $T-T_{90}=-4.47 \pm 0.97$ mK at 79.0000 K.

Appendix: Calculation of Type A Uncertainties of T Measurements

Contributions to type A uncertainty of T measurement due to noise of microwave and acoustic signal were calculated based on the T determination procedure. The following expressions were used:

$$\begin{aligned} u_{Afm}(T)= & {} \sqrt{\frac{T^2}{f_{m0}(T)^4} u_{Af}(f_{m0}(T)^2)^2 + \frac{T^2}{f_{m0}(T_W)^4} u_{Af}(f_{m0}(T_W)^2)^2} \end{aligned}$$

(53)

$$\begin{aligned} u_{Af}(f_{m0})= & {} \sqrt{\frac{1}{3} \frac{\sum \limits _{j=1}^3 (f_{mi} + g_{mj})^2 (u_A (f_{mj})^2 + u_A (g_{mj})^2)}{\sum \limits _{j=1}^3 (f_{mj} + g_{mj})^2}} \end{aligned}$$

(54)

$$\begin{aligned} u_{Af}(f_{mj})= & {} \sqrt{ u_A (Re S_{21})^2 \sum \limits _{j=1}^{N_m} \left( \frac{\partial f_{mj}}{\partial Re S_{21j}} \right) ^2 + u_A (Im S_{21})^2 \sum \limits _{j=1}^{N_m} \left( \frac{\partial f_{mj}}{\partial Im S_{21j}} \right) ^2} \end{aligned}$$

(55)

$$\begin{aligned} u_{Af}(g_{mj})= & {} \sqrt{ u_A (Re S_{21})^2 \sum \limits _{j=1}^{N_m} \left( \frac{\partial g_{mj}}{\partial Re S_{21j}} \right) ^2 + u_A (Im S_{21})^2 \sum \limits _{j=1}^{N_m} \left( \frac{\partial g_{mj}}{\partial Im S_{21j}} \right) ^2} \end{aligned}$$

(56)

where $N_m =400$ is the amount of points in the frequency dependence of microwave transmission coefficient.

$$\begin{aligned} u_{Afa0l}(T)= & {} \sqrt{\frac{T^2}{f_{a0l}(T)^4} u_{Af}(f_{a0l}(T)^2)^2 + \frac{T^2}{f_{a0l}(T_W)^4} u_{Af}(f_{a0l}(T_W)^2)^2} \end{aligned}$$

(57)

$$\begin{aligned} u_{Af}(f_{a0l}^2)= & {} \sqrt{\sum \limits _{k=1}^4 \sum \limits _{j=1}^{N_{ap}} \left( \frac{\partial f_{a0l}^2}{\partial a_{0k}} \frac{\partial a_{0k}}{\partial \varphi _{kj}} \right) ^2 u_{Af} (\varphi _{kj})^2 } \end{aligned}$$

(58)

$$\begin{aligned} u_{Af}(\varphi _{kj}^2)= & {} \left| \frac{\partial \varphi _{kj}}{\partial f_{akj}} \right| u_A (f_{akj}^2) \end{aligned}$$

(59)

$$\begin{aligned} u_{Af}(f_{akj})= & {} \sqrt{ u_A (Re U)^2 \sum \limits _{j=1}^{N_a} \left( \frac{\partial f_{akj}}{\partial Re U_j} \right) ^2 + u_A (Im U)^2 \sum \limits _{j=1}^{N_a} \left( \frac{\partial f_{akj}}{\partial Im U_j} \right) ^2} \end{aligned}$$

(60)

where $N_a$ = 30 is the amount of point in the frequency dependence of acoustic signal. Derivatives of resonance frequencies were calculated using following expressions:

$$\begin{aligned}&\frac{\partial f_{m(ak)j}}{\partial Re (Im) S_{21} (U)_j} = Re (- Im) \frac{\partial F_{m(ak)j}}{\partial S_{21} (U)_j} \end{aligned},$$

(61)

$$\begin{aligned}&\frac{\partial g_{m(ak)j}}{\partial Re (Im) S_{21} (U)_j} = Im (Re) \frac{\partial F_{m(ak)j}}{\partial S_{21} (U)_j} \end{aligned}.$$

(62)

Fitting procedure minimizes sum of squared deviations Q of experimental points from fitting function $G(P, x_j)$. Thus, derivatives of Q over fitting parameters $p_k$ are equal to zero for best values of fitting parameters. Therefore,

$$\begin{aligned}&\frac{\partial p_k}{\partial y_i} = \sum \limits _{l=1}^{N_p} || W ||_{kl}^{-1} \frac{\partial ^2 Q}{\partial p_l \partial y_i} \end{aligned}$$

(63)

$$\begin{aligned}&\frac{\partial p_k}{\partial x_i} = \sum \limits _{l=1}^{N_p} || W ||_{kl}^{-1} \frac{\partial ^2 Q}{\partial p_l \partial x_i} \end{aligned}$$

(64)

$$\begin{aligned}&W_{kl} = \frac{\partial ^2 Q}{\partial p_k \partial p_l} \end{aligned}$$

(65)

$$\begin{aligned}&Q = \sum \limits _{j=1}^N |y_j - G(P,x_j)|^2 d_j \end{aligned}$$

(66)

where $P=(p_1, p_2,\ldots , p_{N_p})$ is the array of fitting parameters, $d_j$ the weight, and N the amount of point in fitting dependence.