After a short review of the history of the flotation method, the flotation apparatus of the Physikalisch-Technische Bundesanstalt (PTB) and the measurement procedures are described in detail. A comprehensive description of the method used to achieve flotation by adjusting the hydrostatic pressure in the liquid is given, covering the heating by compression, surface effects and a calculation of the theoretical detection limit. Data evaluation of the single measurement is described as well as the least-squares fit for a group of samples using Lagrange multipliers to take constraints into account. Results of comparisons of density standards and samples measured to determine the molar volume of silicon and the Avogadro constant including the uncertainty budget and correlation coefficients are reported.
After a short overview of flotation methods and experiments, basic equations for the pressure-of-flotation method are given including a temperature and composition dependence of the liquid's thermal expansion and compressibility. Then, the pressure-of-flotation experiment of PTB is described. A new calibration procedure using doped silicon samples is presented. Now, density comparisons of silicon samples can be performed with an uncertainty less than 0.1 ppm for density differences up to 10 ppm, i.e. for practically all undoped silicon samples. Current applications of the method and an outlook to future developments are discussed. Relative uncertainties of the order of 10−9 seem to be possible and would allow new applications.
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