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On the path to a “nanometer standard”

Absolute measurement of the thermal expansion and compressibility of single-crystal silicon

PTBnews 3.2021
Especially interesting for

materials characterization

fundamentals of metrology

nanoscale length measurement

At PTB, absolute length measurements on a single-crystal silicon gauge block have been performed using imaging interferometry. These measurements have a smaller measurement uncertainty than all previous measurements. They provide more accurate values for the CODATA reference data and are valuable for a new secondary realization of the meter.

Gauge block made of single-crystal silicon. Dimensions: 197 mm × 35 mm × 9 mm

Since it was necessary to have a reference material for high-accuracy measurements of thermal expansion, a large number of measurements used to be performed on silicon over a wide temperature range. Due to its diamond-like crystalline structure, single-crystal silicon expands uniformly in all spatial directions, meaning that it is isotropic with regard to thermal expansion. In addition, high-grade silicon is readily available at an industrial scale.

As early as six years ago, PTB had already presented results of thermal expansion measurements between 7 K and 293 K obtained by means of imaging interferometry. A systematic deviation from the CODATA reference values was, however, noticed in this temperature range. In contrast to dilatometric measurements obtained by others, PTB’s results were derived from absolute length measurements. The present thermal expansion study is based on this work. In this study, the temperature range has been extended to 320 K and the measurement uncertainty reduced. In addition, the study includes the simultaneous determination of the compressibility of silicon.

The measured data were analyzed by means of a new method that provides for the fact that the thermal expansion coefficient (calculated by derivation) is a quantity that is sensitive to the data evaluation model chosen. The approach is based on Bayesian model averaging (BMA) and allows different models to be dealt with at the same time and also to be taken into account when calculating model probabilities.

The results have shown that in the temperature and pressure ranges covered, the thermal expansion coefficient hardly depends on the ambient pressure. The new measurements provide more accurate values than the previous reference values. Furthermore, the measurement uncertainty is smaller than that of previously obtained results by up to one order of magnitude.

Since the latest revision of the mise en pratique for the definition of the meter in the SI refers to the lattice spacing of silicon as a basis for nanoscale secondary realization methods for the meter, these findings can also be used in this context.


Guido Bartl
Department 5.4
Interferometry on Material Measures
Phone: +49 531 592-5430
Opens local program for sending emailguido.bartl(at)ptb.de

Scientific publications

G. Bartl, C. Elster, J. Martin, R. Schödel, M. Voigt, A. Walkov: Thermal expansion and compressibility of single-crystal silicon between 285 K and 320 K. Meas. Sci. Technol. 31, 065013 (2020)
Opens external link in new windowDOI: 10.1088/1361-6501/ab7359

J. Martin, G. Bartl, C. Elster: Application of Bayesian model averaging to the determination of thermal expansion of single-crystal silicon. Meas. Sci. Technol. 30, 045012 (2019)
Opens external link in new windowDOI: 10.1088/1361-6501/ab094b