Volume of Quasi-spherical Solid Density Standards

In Bragg's approach to the determination of the Avogadro constant, the measurement of silicon density as the ratio of mass to volume is an essential step. With a view to achieving a relative uncertainty of 10 -7 in the determination of N A , the computation of the volumes of solid density standards manufactured in the form of optically-polished silicon balls was investigated. Apart from a negligible difference, the volume considered is that of a sphere having the same average diameter. The estimation of the average diameter is studied in the angular momentum domain by using spherical harmonics as a basis and results are assessed by Monte Carlo simulation. The N -point approximation is investigated in detail and a formula for the uncertainty and sampling which minimize aliasing is also given. If the spectrum is bandwidth limited, it is demonstrated that N -point means exist, which give the average diameter exactly. The work involves least-squares estimation and refers to discrete Fourier transforms, Nyquist sampling criterion and analogies with quantum mechanical formulae.