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

Abstract

Within the scope of the efforts concerning the redefinition of the SI base unit "kg" the final results of the international research project aiming at the redetermination of the Avogadro constant are ready to be announced. Among other quantities the volume of two spheres which are made of a highly enriched 28 Si single crystal had to be determined. For this purpose a special Fizeau interferometer for the measurement of the spheres' diameters has been developed at PTB. This paper reports the final results of the volumes, the uncertainties and also the latest findings regarding systematic corrections including the effects of surface layers on the pure silicon core. The results are confirmed by density comparison measurements.

Realization of the kilogram by the XRCD method

“…The first example is a spherical interferometer developed at PTB [15]. Figure 18 shows the principle of diameter measurement using the spherical interferometer, where the sphere is placed in an etalon formed by two spherical reference surfaces.…”

“…Later, Zerodur cubes were similarly used as primary density standards to determine the density of silicon crystals by hydrostatic weighing and to determine the Avogadro constant with a relative standard uncertainty of 1.1 parts in 10 6 [11]. After the development of a polishing technique for manufacturing 1 kg Si spheres [12], it became possible to determine the density of a Si crystal by direct measurements of its diameter and mass thereby eliminating the need for hydrostatic weighing, leading to substantial reduction of uncertainty in the density measurement [13][14][15]. In order to further reduce the abovementioned uncertainty, an accurate evaluation of surface layers of the Si sphere became necessary.…”

Determination of the Avogadro constant by the XRCD method using a28Si-enriched sphere

“…NMIJ-2015 . In the sphere volume measurement by PTB in 2015, an optical interferometer with a spherical etalon was used [8,35]. The optical configuration and phase-shifting algorithm of the PTB interferometer were completely different from those of the NMIJ interferometer used in 2015 [8,15] and 2017 [11].…”

Volume determination of two spheres of the new28Si crystal of PTB

“…The measurement of a sphere consists of pairs of alternating measurements, the basic principle of which is explained in the following (details can be found in [2]). …”

“…This is considered in the evaluation by means of a fit of real spherical harmonics, in which the uncertainties of the single values are included as weighting factors. Then the volume of the sphere is the integral over its radius representation ϕ ϑ R , ( ) via real spherical harmonics as described in [2]:…”