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Redetermination of the mass of two 28Si spheres for the Avogadro project


Taking the corrections into account which were determined by the BIPM for the national prototypes within the scope of the latest comparison measurements against the international prototype of the kilogram, the mass of two 28Si spheres was determined at PTB with a standard uncertainty (k = 1) of 7 µg.

In 2011, the Avogadro constant was determined by means of two 28Si spheres with a relative standard uncertainty (k = 1) of
3 × 10-8 [1, 2]. Numerous measurements (e.g. to determine the impurities and defects in the crystal, the molar mass, the lattice spacing, the surface layers as well as the mass and volume of the spheres) contributed to this result. A fundamental problem with regard to these measurements was that the surface of the Si spheres used had been contaminated by a thin metallic layer of atoms of nickel, copper and zinc during their manufacturing. In order to remove these contaminants, the surfaces of the spheres had to be etched and further processed. To calibrate the re-processed spheres, among other things, the mass of the spheres was redetermined with the smallest possible uncertainty by the national metrology institutes in Japan (NMIJ) and Germany (PTB) as well as the Bureau International des Poids et Mesures (BIPM) in Paris [3].

At PTB, the mass was determined in air as well as in vacuum using the kilogram prototype No. 70 as a reference mass in air. In addition, air buoyancy artefacts were used to determine the air density during the comparison measurements in air. Platinum-iridium sorption artefacts served as transfer standards for the transfer between air and vacuum. To determine the mass, standard uncertainties (k = 1) of 12 µg in air and
7 µg in vacuum were achieved. The difference obtained between the mass of the sphere in air and the mass of the sphere in vacuum amounted, for both spheres, to approx. 8.5 µg. Related to the surface of the 1 kg spheres, this corresponds to a difference in the sorption coefficient of approx. 30 ng/cm² when changing between air and vacuum. This relatively low value emphasizes the high quality of the Si spheres' surfaces and is in good agreement with the values published in [4] and [5].

Figure 1 shows a comparison between the results obtained by the BIPM, NMIJ and PTB for the determinations of the mass in vacuum. For both spheres, the agreement between the results lies within the indicated standard uncertainties (k = 1). The reference values of the spheres' masses, which are important to calculate the Avogadro constant, were determined as a weighted mean value with a standard uncertainty of 3.5 µg, taking the correlations into account [3]. The analysis of the correlation between the results published in 2011 [2] and in 2015 [3] for the mass of the 28Si spheres yielded a correlation coefficient of approx. 0.3 [6]. The results take the corrections into account which were determined by the BIPM for the national prototypes within the scope of the latest comparison measurements against the international prototype of the kilogram [7].



Figure 1: Comparison between the results of the mass determinations in vacuum for the 28Si spheres AVO28-S5c (a) and AVO28-S8c (b). The uncertainty ranges are shown as standard uncertainties (k = 1). BIPM: Bureau International des Poids et Mesures; NMJ: National Metrology Institute of Japan; PTB: Physikalisch-Technische Bundesanstalt, WM: Weighted mean



[1]  Andreas, B.; et al.: Counting the atoms in a 28Si crystal for a new kilogram definition. Metrologia 48  (2011) S1–S13
[2]  Picard, A.; Barat, P.; Borys, M.; Firlus, M.; Mizushima, S.: State-of-the-art mass determination of 28Si spheres for the Avogadro project. Metrologia 48 (2011) S112–S11
[3]  Azuma, Y.; et al.: Improved measurement results for the Avogadro constant using a 28Si-enriched crystal. Metrologia 52 (2015), 360–375
[4]  Picard, A.; Fang, H.: Methods to determine water vapour sorption on mass standards Metrologia 41 (2004) 333–339
[5]  Picard, A.: Mass determination of a 1 kg silicon sphere for the Avogadro project Metrologia 43 (2006) 46–52
[6]  Mana, G.; Massa, E.; Sasso, C. P.; Stock, M.; Fujii, K.; Kuramoto, N.; Mizushima, S.; Narukawa, T.; Borys, M.; Busch, I.; Nicolaus, A.; Pramann, A.: The correlation of the NA measurements by counting 28Si atoms. J. Phys. Chem. Ref. Data 44 (2015), No. 3, 031209, 1-8
[7]  Stock, M.; Barat, P.; Davis, R.; Picard, A.; Milton, M. J. T.: Calibration campaign against the international prototype of the kilogram in anticipation of the redefinition of the kilogram part I: comparison of the international prototype with its official copies. Metrologia 52 (2015), pp. 310 - 316 

Contact person:

Michael Borys, Dept. 1.1, WG 1.11, e-mail: Opens window for sending emailmichael.borys(at)ptb.de