In situ evaluation of the linear thermal expansion coefficient of Cu-ETP from 4.3 K to 299 K

In this study, the linear thermal expansion coefficient of electrolytic through pitch copper (Cu-ETP) was used as a resonator material in the single-pressure refractive-index gas thermometer and was evaluated in situ at high precision via the multi-mode microwave resonance method in the temperature range of 4.3 to 299 K. Two experimental measurement schemes, cooling method (5–299 K) and temperature control method (4.3–26 K), are employed for different temperature ranges. These methods adopt the same calculation method, wherein the relation between the length and temperature is obtained first, and then the polynomial fitting is used to obtain the linear thermal expansion coefficient of the resonator. The resonator installed in the cryostat has a quasi–spherical shape, with similar radii in the x, y, and z axes; for example, if the radius in one direction is R, then the radii in the other two directions are 1.001R and 1.0005R. The accurate radius of the quasi–sphere in low temperature can be measured by the multi-mode microwave resonance method, which is a mature method with a significant non-ideal correction to reduce the difference between the actual and ideal environments. For the cooling method, to reduce the impact of random errors, we collect five microwave modes (TM11, TE11, TM12, TE12, and TE13) and repeat four experiment runs (Run9, Run10, Run12, and Run17), assuming the average value as the final result. The max radius deviation during the different modes is 0.37 μm, indicating that the result has a good mode consistency. Then, the measurement uncertainty of the radius is analyzed, with all values within 0.27 μm and the mode consistency being the main influencing item. The linear thermal expansion coefficient can be calculated by the polynomial fitting method with the standard uncertainty of 2.2×10⁻⁷ K⁻¹, with repeatability being the main source of uncertainty. As for the controlling method, the same analyzing procedure is implemented, the max deviation of the radius during the four modes (TM11, TE11, TM12, and TE13) is 0.12 μm, and the deviation of different runs from the average value is within 0.0056 μm, smaller than the radius uncertainty, which has good repeatability. The standard uncertainty of radius is within 0.12 μm in the entire range and the non-ideal correction and frequency stability are the two main influencing factors. The standard uncertainty of the linear thermal expansion is 2.9×10⁻⁹ K⁻¹, and the two main sources are the microwave mode consistency and repeatability. Due to the higher stability of temperature control and lower microwave measurement noise, the results determined by the temperature control method are more accurate. Finally, equations for the linear thermal expansion coefficient of Cu-ETP are further developed to realize a high-precision correlation between the experimental data and temperature.