


The Avogadro constant and the revision of the SI
The Avogadro constant expresses the number of elementary entities per mole of substance. In the present SI, by definition, {N_{A}} atoms of Carbon12 weigh exactly 12 grams.^{[1]}
At present, a new approach to define and realize the SI unit of mass (the kilogram) is being investigated. Essentially, the kilogram could be defined as the mass of {N_{A}}*1000/12 carbon12 atoms by fixing the numerical value of N_{A}. For this redefinition to take place, N_{A} would need to be accurately known in the present SI. This is the purpose of the International Avogadro Coordination (IAC) project. In this context, a relative uncertainty of 2 × 10^{−8} on N_{A} has been recommended; see Recommendation G1 (2010), by the Consultive Committee for Mass and Related Quantities (CCM).
The redefinition of the kilogram could also be based on a fixed numerical value of the Planck constant, h. In this case, an accurate determination of N_{A} would still be crucial because it would provide an alternative method to determine h via the molar Planck constant, N_{A}h which is known with very small relative uncertainty (2014 CODATA recommended values of the fundamental constants):
u(N_{A} h) / N_{A} h = 4.5 × 10^{−10}
The Avogadro project
The International Avogadro Coordination (IAC) project formally began as an international effort whose scope was to determine the Avogadro constant N_{A} with a relative uncertainty equal to or less than 2 × 10^{−8} using an isotopically enriched silicon crystal.
The IAC initally ran from 2004 to 2011 as a collaboration between the BIPM, INRIM (Italy), the IRMM (Belgium), NIST (United States), the NMIA (Australia), the NMIJ/AIST (Japan), the NPL (United Kingdom), and the PTB (Germany). The collaboration was renewed in 2012 by a Memorandum of Understanding between the BIPM, INRIM, the NMIA, the NMIJ and the PTB.
Within the framework of the original IAC, in 2011 the Avogadro constant was determined as
N_{A} = 6.022 140 82(18) × 10^{23} mol^{−1} with a relative uncertainty of 3.0 × 10^{−8}
Improvements of the experiments during the continued collaboration resulted in the publication of the most recent determination of the Avogadro constant in 2015:
N_{A} = 6.022 140 76(12) × 10^{23} mol^{−1}
with a relative uncertainty of 2.0 × 10^{−8}
1. The Avogadro constant N_{A} = {N_{A}} [mol^{−1}].
Principle of the measurement of N_{A}
In order to relate the kilogram to an atomic mass, the number of silicon atoms in two 1 kg singlecrystal silicon spheres is determined by exploiting their ordered arrangement in the crystal. The spheres are highly enriched (99.995 %) with the ^{28}Si isotope. The expression of the Avogadro constant is based on the following equation:
where n, ρ, M and a are respectively the number of atoms per unit cell (n = 8), the density, the molar mass and the lattice parameter of the crystal.
This new approach by using an isotopically enriched silicon crystal overcomes the previously limiting problem of accurately determining the isotopic composition of a natural silicon crystal. The experiment uses isotope dilution mass spectrometry (IDMS) combined with multicollector inductively coupled plasma mass spectrometry to determine the molar mass of the enriched ^{28}Si material with unprecedented accuracy. The isotopic composition, molar mass, mass, volume, density and lattice parameter of two 1 kg ^{28}Si spheres were accurately determined and their surfaces were chemically and physically characterized at the atomic scale (see details in Metrologia, 2011, 48(2), S1S13). Impurity concentration and gradients in the enriched crystal were measured by infrared spectroscopy and taken into account.
Production of the two spheres (Fig. 2) began in 2004 with the isotopic enrichment of SiF_{4} gas by centrifugation at the Central Design Bureau of Machine Building in St Petersburg (Russian Federation). The enriched gas was subsequently converted into SiH_{4} and chemical vapour deposition was used to grow a polycrystal at the Institute of Chemistry of HighPurity Substances of the Russian Academy of Sciences. In 2007, a 5 kg ^{28}Si boule was grown by the LeibnizInstitut für Kristallzüchtung (Germany). Two ^{28}Si spheres, AVO28S5 and AVO28S8, were manufactured from the boule and were shaped into nearly perfect spheres by the Australian Centre for Precision Optics. Silicon was chosen because it can be grown into large, highpurity and almost perfect single crystals



Fig. 1. Silicon unit cell. The unit cell of silicon has a cubic packing arrangement of 8 atoms. The unit cell volume is measured by determining the lattice parameter a, which is the length of one of the sides of the cube. 

Fig 2. A silicon sphere (PTB image). 
A special issue of Metrologia on International determination of the Avogadro constant is dedicated to explaining the various measurements involved.
In the context of the IAC project, the mass m of two silicon spheres AVO28S5 and AVO28S8 needs to be determined. This mass m represents the mass of the core of the silicon single crystal m_{core} excluding the mass of the oxide layer m_{oxide}, of water sorption m_{wat.sorpt}, of point defects m_{def} and of any other possible contaminants (hydrocarbons, etc.), m_{cont}.
The mass of the core of the sphere can therefore be written as:

m_{core} = m_{tot} − m_{oxide} − m_{wat.sorpt} − m_{def} − m_{cont}  (1) 
where m_{tot} represents the total mass of the sphere, which is measured in vacuum with respect to a 1 kg Pt/Ir mass standard traceable to the International Prototype of the Kilogram, which is always maintained in air.
The mass m_{wat.sorpt} is composed of two different terms:

m_{wat.sorpt} = m_{water(rev)} + m_{water(irrev)}  (2) 
m_{water(rev)} corresponds to the reversible, physisorbed, part of the water vapour adsorbed on the surface of the sphere which is removed by placing the sphere in vacuum at room temperature. m_{water(rev)} was evaluated by using sorption artefacts following a technique developed by the BIPM (see Water vapour adsorption effects and [2]).
The irreversibly, chemisorbed, adsorbed water m_{water(irrev)} on the sphere can best be removed by baking under vacuum or in a neutral dry gas.
Any possible carbonaceous contamination layer can be removed sufficiently by an adequate cleaning procedure just before the mass measurement.
To determine the mass of the silicon core, the mass of the surface layers was subtracted from the mass of the sphere. In this project, the task of the mass laboratories was to determine the total mass of the sphere (m_{tot}) with highest accuracy in vacuum (10^{−3} Pa to 0.1 Pa) by taking into account the mass of physisorbed water (m_{water(rev)}). The other terms of equation (1) and (2) are estimated from separate experiments carried out by members of the IAC. The thickness and chemical composition of the surface layer were characterized independently of the mass determination by measurements using methods such as xray fluorescence analysis (XRF), nearedge xray absorption finestructure spectroscopy (NEXAFS), xray reflectometry (XRR), xray photoelectron spectroscopy (XPS) and optical spectral ellipsometry (SE). The results of these studies were considered for the determination of the mass of the silicon core and the Avogadro constant.
The determination of the Avogadro constant within the framework of the International Avogadro Coordination (IAC) project requires the masses of the silicon spheres to be determined. Therefore mass comparisons in air and under vacuum of the two ^{28}Si spheres (AVO28S5 and AVO28S8) against PtIr kilogram standards traceable to the International Prototype of the Kilogram (IPK) were carried out at the BIPM (pilot laboratory), the NMIJ and the PTB. The initial target uncertainty fixed by the IAC for the mass of the spheres was 5 µg (5 parts in 10^{9}). Suitable vacuum mass comparators have been available for a few years, which make it possible to determine the mass of 1 kg silicon spheres under vacuum conditions (10^{–3} Pa to 0.1 Pa). Owing to the airtovacuum transfers (water vapour adsorption effects^{[1]}) of the reference standards, sorption corrections had to be considered, which were determined by means of sorption artefacts.
The measurements carried out in this context at the BIPM in 2008 and in 2009 show that the mass difference between air and vacuum conditions for both spheres were within 7 µg. The agreement among the three laboratories was within 10 µg and 20 µg respectively for the spheres AVO28S5 and AVO28S8. The results of the three laboratories have been averaged by taking into account the correlations between the results. The results obtained for the spheres AVO28S5 and AVO28S8 demonstrate that by using air buoyancy artefacts^{[2]} and sorption artefacts, mass values can be achieved with a relative uncertainty of about 4 parts in 10^{9} ^{[3]}. The fact that transportation of the spheres between laboratories in ambient conditions did not represent a problem was due to the very efficient cleaning method developed by the NMIA.
Within the continued Avogadro collaboration started in 2012 the masses of the spheres were determined again by the same three laboratories. During this work, the surfaces of the spheres were decontaminated and repolished, in order to produce a surface without metallic contamination and with improved sphericity. The BIPM measurements were carried out in early 2014, during the time of the Extraordinary Calibration with respect to the IPK. This allowed the masses of the spheres to be established with respect to that of the IPK with a very small uncertainty – of about 4.4 µg in vacuum and 13 µg in air. The results of the BIPM, the NMIJ and the PTB agreed within the uncertainties. This allowed the calculation of the weighted mean of the three results with an uncertainty of 3.5 µg, i.e. 3.5 parts in 10^{9}. This uncertainty is negligible with respect to other contributions to the combined uncertainty of the determination of the Avogadro constant (20 parts in 10^{9}), published in 2015^{[4]}.
[1] Methods to determine water vapour sorption on mass standards 
Metrologia, 2004, 41, 333339 
[2] Mass comparisons using air buoyancy artefacts 
Metrologia, 2004, 41, 330332 
[3] Stateoftheart mass determination of ^{28}Si spheres for the Avogadro project 
Metrologia, 2011, 48, S112S119 
[4] Improved measurement results for the Avogadro constant using a ^{28}Sienriched crystal 
Metrologia, 2015, 52, 360375 
Published results
In 2011 the IAC published a determination of the Avogadro constant,
N_{A} = 6.022 140 82(18) × 10^{23} mol^{–1}
(relative uncertainty of 3.0 x 10^{–8})
which was at that time the most accurate determination of a fundamental constant which can be used for a new definition of the kilogram.
During the calibration campaign with the International Prototype of the Kilogram in 2014, an offset of the BIPM asmaintained mass scale was observed. Correction of the IAC (2011) result for this offset leads to a value of N_{A} = 6.022 140 99(18) × 10^{23} mol^{–1}.
The collaboration then continued with the objective of reducing the uncertainty. The two enriched ^{28}Si spheres were decontaminated from a metal contamination of their surface and repolished to improve sphericity. New measurements were made on the reworked spheres using improved methods and apparatus. A new result was published in 2015:
N_{A} = 6.022 140 76(12) × 10^{23} mol^{–1}
(relative uncertainty of 2.0 x 10^{–8})
Bibliography
A special issue of Metrologia on International determination of the Avogadro constant is dedicated to explaining the various measurements involved.
The results of the International Avogadro Coordination project were published in 2011 and 2015:
 Andreas B., Azuma Y., Bartl G., Becker P., Bettin H., Borys M., Busch B., Gray M., Fuchs P., Fujii K., Fujimoto H., Kessler E., Krumrey M., Kuetgens U., Kuramoto N., Mana G., Manson P., Massa E., Mizushima S., Nicolaus A., Picard A., Pramann A., Rienitz O., Schiel D., Valkiers S., Waseda A., Determination of the Avogadro constant by counting the atoms in a ^{28}Si crystal, Phys. Rev. Lett., 2011, 106, 030801
 Andreas B., Azuma Y., Bartl G., Becker P., Bettin H., Borys M., Busch I., Fuchs P., Fujii K., Fujimoto H., Kessler E., Krumrey M., Kuetgens U., Kuramoto N., Mana G., Massa E., Mizushima S., Nicolaus A., Picard A., Pramann A., Rienitz O., Schiel D., Valkiers S., Waseda A., Zakel S., Metrologia, 2011, 48, S1S13
 Azuma Y., Barat P., Bartl G., Bettin H., Borys M., Busch I., Cibik L., D'Agostino G., Fujii K., Fujimoto H., Hioki A., Krumrey M., Kuetgens U., Kuramoto N., Mana G., Massa E., Mee? R., Mizushima S., Narukawa T., Nicolaus A., Pramann A., Rabb S.A., Rienitz O., Sasso C., Stock M., Vocke R.D., Waseda A., Wundrack S., Zakel S., Improved measurement results for the Avogadro constant using a ^{28}Sienriched crystal, Metrologia, 2015, 52 360375
 Mana G., Massa E., Sasso C.P., Stock M., Fujii K., Kuramoto N., Mizushima S., Narukawa T., Borys M., Busch I., Nicolaus A., Pramaan A., et al., The correlation of the N_{A} measurements by counting ^{28}Si atoms, J. Phys. Chem. Ref. Data, 2015, 44, 031209
Other related publications concerning the BIPM's mass measurements of the spheres can be found at:
 Picard A., Bignell N., Borys M., Downes S., Mizushima S., Mass comparison of the 1 kg silicon sphere AVO#3 traceable to the International Prototype K, Metrologia, 2009, 46(1), 110
 Picard A., Primary mass calibration of silicon spheres, Meas. Sci. Technol., 2006, 17, 25402544
 Picard A., Mass determination of a 1 kg silicon sphere for the Avogadro project, Metrologia, 2006, 43(1), 4652




