Experimental determination of the volume of a crystalline silicon sphere using spherical harmonics

For determination of the Avogadro constant from the X-ray crystal density method, calculation of the volume of a nearly perfect single-crystalline silicon sphere was performed. A scanning-type optical interferometer was used to measure the diameters of the sphere, and the diameter data were fitted to a series of spherical harmonics. A method of least squares, which minimizes chi 2 , was applied in order to determine the coefficients of the harmonics and to estimate their uncertainties. Results of the fitting have shown that the volume uncertainty arising from geometrical imperfections of the sphere is less than 0.07 ppm. The total volume uncertainty, including systematic error in the diameter measurements, is estimated to be 0.21 ppm.