Physikalisch-Technische Bundesanstalt

Working Group 3.24 “Avogadro Constant” coordinates the research at PTB and the collaboration with institutes worldwide for a more accurate determination of the Avogadro constant, in order to lay the foundations for a redefinition of the mass unit kilogram on the basis of a natural constant.

The Avogadro constant is the number of atoms or molecules in one mole of a substance. Since the mole is defined as the number of entities (atoms) in 12 g of the isotope ¹²C, the kilogram can be defined by a specified number of atoms, e.g., the mass m(¹²C) of a ¹²C atom or of the atomic mass unit  1 u = m(¹²C)/12. This is managed by “counting” the atoms in 1 kg spheres of silicon by exploiting the periodicity and perfection of the silicon single crystal. The Avogadro constant is calculated using the formula

            N_A = 8 M V / (m a³) .

Here, 8V/a³ is the number of atoms in a sphere of volume V, since a³ is the volume of the unit cell of the crystal, which includes 8 atoms. For the calculation of the Avogadro constant, the number of atoms in the sphere has to be divided by the amount of substance, i.e. the number of moles n = m/M , where m denotes the mass of the sphere and M the molar mass of the silicon.

For purchasing the isotopically enriched silicon (²⁸Si), which is necessary for the accurate determination of the molar mass, seven metrological institutes founded the International Avogadro Coordination (IAC) in 2004. Today, the IAC consists of the International Bureau of Weights and Measures (Bureau International des Poids et Mesures, BIPM), the Istituto Nazionale di Ricerca Metrologica (INRIM, Italy), the National Measurement Institute of Australia (NMIA), the National Metrology Institute of Japan (NMIJ) and PTB. In 2011, the IAC published in a special issue of the journal Metrologia its best result so far for the Avogadro constant:

            N_A = 6.022 140 82 ∙ 10²³ mol^-1

with a relative standard uncertainty of only 3 ∙ 10^-8 and thus the most accurate determination of the Avogadro constant (cf. Fig. 1). A short uncertainty budget is shown in Table 1.

The isotope enrichment was carried out by the Central Design Bureau of Machine Building in St. Petersburg by centrifugation of the silicontetrafluoride (SiF₄) gas. Then the SiF₄ gas was converted into silane (SiH₄) and the silicon was deposited as a polycrystal by the Institute of Chemistry of High-Purity Substances of the Russian Academy of Sciences in Nizhny Novgorod. At the Leibniz-Institut für Kristallzüchtung (IKZ, Leibniz Institute for Crystal Growth) in Berlin, the dislocation-free single crystal was grown (Fig. 2), the necessary samples of which – in particular two 1 kg spheres – were manufactured. Eventually, the Australian Centre for Precision Optics polished the spheres to an unroundness below 0.1 µm.

With the exception of the lattice parameter a, all quantities necessary for the determination of N_A are (also) determined at PTB:

The isotopic composition and the molar mass are measured in Working Group (WG) 3.11 “Inorganic Analysis” using isotope-dilution mass spectrometry (IDMS) (the relative atomic masses of the naturally existing silicon isotopes ²⁸Si, ²⁹Si and ³⁰Si are known accurately enough from different measurements).

The volume of the silicon spheres is determined at 20 °C and in vacuum in WG 5.41 “Interferometry on Spheres” using a spherical Fizeau interferometer. This WG also performs – in collaboration with WG 4.33 “X-ray Optics” – simulation calculations for the interferometer. The temperature measurements of the volume and lattice parameter determinations are compared with each other and with the temperature realisation of WG 7.42 “Applied Thermometry” using an electronic 20 °C temperature reference point that was developed in WG 3.23 “Thermal State Behaviour and Density”.

The mass of the spheres in air and in vacuum is determined in WG 1.11 “Realization of Mass”.

The necessary knowledge about thickness and structure of the surface layers on the spheres is investigated in the following WGs using the noted methods:

• WG 4.33 “X-ray Optics” using X-ray fluorescence spectroscopy (XRF),
• WG 5.13 “Surface Metrology on Nanostructures” using spectral ellipsometry (SE),
• WG 5.14 “Roughness Measuring Methods” using interference microscopy,
• WG 7.11 “X-ray Radiometry” using X-ray reflectometry (XRR) and X-ray fluorescence spectroscopy (XRF) at BESSY II.

WG 3.23 “Thermal State Behaviour and Density” performs density comparisons of the spheres and of other samples. Additionally, the density of the thermal oxide on silicon spheres was determined by density comparison measurements.

WG 4.33 “X-ray Optics” sets up X-ray interferometers for relative and absolute measurements of the lattice parameters. The optical part of the absolute interferometer is built by WG 5.21 “Length Graduation”.

The WG 3.11 “Inorganic Analysis” determines the concentration of the main impurities boron, carbon, oxygen and nitrogen using low-temperature infrared spectroscopy. Due to the impurities, the mass of the real silicon sphere differs slightly from the mass of a sphere that consists only of silicon atoms.

WG 4.33 “X-ray Optics” prepares samples from the ²⁸Si crystal and WG 5.56 “Manufacturing Technology” polishes spheres with the lowest possible shape deviations.

More detailed information can be found on the Internet sites of the Working Groups.

WG 3.24 “Avogadro Constant” also organises measurements at external institutes for some partial problems, e.g.

• at the University of Dresden, Germany, the measurement of the hydrogen concentration in the crystal using Deep Level Transient Spectroscopy (DLTS),
• at the University of Halle-Wittenberg, Germany, the measurement of the vacancy concentration in the crystal using positron annihilation experiments,
• at the Bundesamt für Metrologie (METAS, Federal Office of Metrology, Switzerland) the measurement of surface layers of silicon spheres using X-ray photoelectron spectroscopy (XPS), and
• at the National Institute of Metrology (NIM, China), the National Institute of Standards and Technology (NIST, USA) and the National Research Council of Canada (NRC) molar mass measurement of the isotopically enriched silicon using IDMS.

The PTB chairs the Working Group on the Avogadro Constant of the Consultative Committee for Mass and Related Quantities (CCM WGAC). Members of this Working Group are now the institutes of the International Avogadro Coordination (IAC) and the NIM.

The BIPM conducts mass determinations of the silicon spheres in air and in vacuum and ensures the traceability to the International Prototype of the Kilogram.

The INRIM performs lattice parameter measurements at 20 °C and in vacuum using a combined X-ray and optical interferometer.

The NMIA measures the sphere diameter at 20 °C and in vacuum using a Saunders interferometer.

The NMIJ measures the mass of the spheres in air and in vacuum, their diameter at 20 °C and in vacuum using an interferometer and the thickness of the oxide layer using spectral ellipsometry. Additionally, the NMIJ investigates the homogeneity of the lattice plane distances in the ²⁸Si crystal.

The NIM performs measurements in the fields of molar mass, volume, surface layers and mass.

The next occasion for a new definition of the kilogram will be at the General Conference on Weights and Measures (Conférence Générale des Poids et Mesures, CGPM) in 2014. It is intended to improve the accuracy of the Avogadro constant by then. Additionally, some quantities should be confirmed by independent measurements. Eventually, the attempt will be made – in collaboration with the watt balance experiments worldwide – to find the cause(s) for the diverging results of the Avogadro constant, thus ensuring a secure and accurate new definition of the mass unit.

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