In order to determine the Avogadro constant by the x-ray crystal density method, the molar volume of silicon in a single crystal grown by the float-zone method was determined by absolute measurements of the density and the molar mass of the crystal. The density was directly determined by mass and volume measurements of 1 kg spheres fabricated from the crystal. For the volume determination, a scanning-type optical interferometer was developed to measure the diameters of the spheres. The molar mass measurements were calibrated using synthetic isotope mixtures prepared from enriched silicon isotopes. Two spheres for density measurements, and fourteen samples for molar mass measurements, were systematically prepared from the crystal to evaluate distributions of the density and the isotopic composition of the crystal. The results give a molar volume of 12.058 8646 cm 3 /mol with a standard uncertainty of 0.000 0026 cm 3 /mol, at 22.500 C and 0 Pa after correction for impurities. When this value is combined with the lattice constant of the crystal determined using a combined x-ray/optical interferometer, a value of 6.022 1550 10 23 mol -1 with a standard uncertainty of 0.000 0016 10 23 mol -1 is obtained for the Avogadro constant. Recent data on the absolute measurement of the lattice constant by combined x-ray/optical interferometers have shown a consistent result within the measurement uncertainties [5] when the observed values are corrected for impurity effects. Relative differences

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A new interferometer has been developed for the accurate determination of the density of a silicon crystal, in which a single-crystal silicon sphere of nearly perfect geometry is placed in a Fabry–Perot etalon of accurately known plate distance, and the diameters are obtained by measuring the two gaps between the etalon and the adjacent surface of the sphere. A new method is used to measure the sum of the length of the two gaps by scanning the etalon against the sphere. Two wavelengths, 633 nm from a frequency-stabilized He–Ne laser and 441 nm from a free-running He–Cd laser, are used to determine the order of interference by applying the method of exact fractions. The diameter of about 94 mm has been measured with a resolution of 0.5 nm. Diameter measurements from uniformly distributed directions have shown that the mean diameter has been determined with a standard deviation of 8.6 nm, corresponding to 0.28 ppm in the volume determination. The total uncertainty of the volume is estimated to be 0.34 ppm. Effects of a thin oxide layer and impurities on the bulk density are discussed.

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