Thermal expansion and compressibility of single-crystal silicon

Five years ago, the thermal expansion of single-crystal silicon was derived from length measurements in the temperature range from 7 K to 293 K and related to prior published results as well as the reference data of CODATA. The temperature interval covered so far has now been extended by the range up to 320 K and thereby supplemented by the simultaneous determination of the compressibility [1]. For this purpose, the absolute length of a silicon gauge block was investigated in a series of measurements over several weeks at different temperatures and air pressures. The parameter space covers temperatures from 285 K to 320 K and air pressures from atmospheric pressure to 10^-5 hPa. The measurement method used is imaging interferometry with refractive index compensation, taking into account a pressure-related correction of the refractometer used. From the data set obtained, the thermal expansion coefficient as well as the compressibility - or the compression modulus as its reciprocal - of monocrystalline silicon were determined. Since the choice of the model underlying the evaluation is not unique, a Bayesian model averaging approach was applied to account for possible model errors in the uncertainty evaluation.

For the thermal expansion, standard measurement uncertainties of less than 1 ⋅ 10^-9 K^-1 were achieved in the temperature range covered, thus falling below the uncertainty of the reference data. In view of the present results of both the current measurements and those in the cryogenic temperature range, an update of the previously recommended CODATA reference data can be considered. The 2019 updated mise en pratique for the meter also describes the secondary representation of the meter at the nanometer scale based on the silicon lattice spacing. Because of the dependence of the lattice spacing on temperature and pressure, accurate knowledge of thermal expansion and compressibility is important here.

The compressibility result is in agreement with the data obtained by different measurement techniques described in the literature. Moreover, taking into account the measurement uncertainties obtained, no significant dependence of the thermal expansion on the existing atmospheric pressure in the parameter space covered can be observed. This exemplarily justifies the general assumption that it is permissible to apply the values measured under vacuum conditions for thermal expansion corrections of length measurements performed at atmospheric pressure.

[1] Guido Bartl et al 2020 Meas. Sci. Technol. 31 065013