Influence of the distribution of measuring points on the mean diameter determination of the Avogadro project's silicon spheres

Abstract

The members of the International Avogadro Project are aiming at the redetermination of Avogadro's constant with a relative uncertainty of less than 2×10−8 in order to be qualified for the redefinition of the kilogram. Therefore, among other quantities, the volume of a sphere made of a silicon single crystal has to be determined very precisely with a diameter uncertainty of 0.3 nm. A special Fizeau interferometer with spherical reference faces has been developed at PTB providing the required precision in absolute interferometry and a generous coverage of the surface of the sphere at the same time. Due to the intrinsic imbalanced distribution of measurement positions given by the setup, an equalization has to be performed without introducing a numerical error. In this paper an appropriate procedure using point distributions on a sphere is described and characterized with regard to the number of points involved.

Improvements to the Volume Measurement of ²⁸Si Spheres to Determine the Avogadro Constant

“…The total number of measurement points was 770 and the experimental standard deviation of the mean volume for the 11 sets of measurements was 3.6 × 10 −8 . The contribution from this uncertainty was relatively large contrary to the results of investigation on the effect of the number of measurement points on the uncertainty of the mean diameter determined for 1-kg Si spheres [10].…”

Volume determination of the Avogadro spheres of highly enriched28Si with a spherical Fizeau interferometer

“…The whole set of measurements gave 320000 and 440000 single diameter values, respectively, distributed all over the sphere and covering the topographies thoroughly. According to [9] the surfaces of the spheres were subdivided into small cells of almost equal area and all measurement values inside a cell were assigned to the center position of this cell. Doing so, a map of the topography and of the distribution of the measuring point densities can be determined, which are shown…”

Contributions of precision engineering to the revision of the SI

“…This form of the Kibble balance relation is easily solved for the Planck constant, and can be solved in a reciprocal fashion for a unit of mass using standards of time (frequency), length, voltage, and resistance all derived from fundamental constants once the kilogram is redefined. Of course for this to happen, Kibble balance experiments must achieve a combined relative standard uncertainty < 5 x 10 -8 , a tremendous challenge given that the practical realizations of length, voltage, and resistance quantities have relative uncertainties themselves typically in the order of a few parts in 10 9 .…”

“…A possible goal is to move the coil along the gravitational axis, and to simultaneously record both the velocity of the coil and the induced voltage. Ultimately, the ratio of voltage to velocity during the dynamic mode must provide a determination of the magnetic product B x L that is accurate, with a standard relative uncertainty below 5 parts in 10 9 .…”

“…A full topography mapping of the sphere is realised by subsequent measurements in different orientations. Data evaluations were performed to take into account the distribution of measurement points on the sphere surface [9] and by using redundant information in the overlapping surface areas, also a radius (or absolute form) topography map can be reconstructed from the interferometric data [11].…”