MEASUREMENT, CHARACTERIZATION AND VOLUME DETERMINATION OF APPROXIMATELY SPHERICAL OBJECTS

The roundness and sphericity of manufactured single crystal silicon sp heres of nearly perfect geometry have been measured using a high-accur acy rotating probe instrument. Automated data collection, spindle erro r separation and fitting algorithms have been applied to generate 3D s phericity 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 un certainty of 2.8 nm. When scaled with absolute diameter data, the sphe ricity error surface is transformed to a 3D data set of absolute radia l dimensions. Two methods of computing the volume of an approximate sp here are presented. They are direct numerical integration and spherica l harmonics. The algorithmic precision of the volume calculation is tw elve significant digits for the direct numerical integration method an d ten significant digits for the spherical harmonics method. These acc uracies are several orders of magnitude better than the target uncerta inty of 0.1 ppm or seven significant digits. Assuming a standard uncer tainty in absolute diameter measurement of 2.8 nm, the volume is shown to be calculable with a standard uncertainty of 0.1 ppm.