Sphericity analysis of solid density standards

An analysis of dimensional sphericity measurement data is presented, applied to near-perfect single-crystal silicon spheres used as solid density standards and for the precise determination of the Avogadro constant. Using the formal correspondence of the expansion of the three-dimensional measurement data set in terms of spherical harmonics with the theory of angular momentum in quantum mechanics, invariant quantities are determined, allowing objective assessment of fabrication and measurement process variables. Two methods for sphericity analysis are presented and applications to solid density standards are given. Within the limitations of the quantity and quality of the measurement data tested, our results are consistent with qualitative assessments of sphere topography and have revealed a structure in the spectral distribution of spherical harmonic amplitudes.