Measurement, characterization and volume determination of approximately spherical objects

The roundness and sphericity of manufactured single crystal silicon spheres of nearly perfect geometry have been measured using a high-accuracy rotating probe instrument. Automated data collection, spindle error separation and fitting algorithms have been applied to generate 3D sphericity data. A two-dimensional roundness error profile of any great circle is determined with a standard uncertainty of 2.3 nm, and three dimensional sphericity error surface is determined with a standard uncertainty of 2.8 nm. When scaled with absolute diameter data, the sphericity error surface is transformed to a 3D data set of absolute radial dimensions. Two methods of computing the volume of an approximate sphere are presented. They are direct numerical integration and spherical harmonics. The algorithmic precision of the volume calculation is twelve significant digits for the direct numerical integration method and ten significant digits for the spherical harmonics method. These accuracies are several orders of magnitude better than the target uncertainty of 0.1 ppm or seven significant digits. Assuming a standard uncertainty in absolute diameter measurement of 2.8 nm, the volume is shown to be calculable with a standard uncertainty of 0.1 ppm.