Table of contents

This issue of Metrologia collects papers about the results of an international research project aimed at the determination of the Avogadro constant, N_A, by counting the atoms in a silicon crystal highly enriched with the isotope ²⁸Si. Fifty years ago, Egidi [1] thought about realizing an atomic mass standard. In 1965, Bonse and Hart [2] operated the first x-ray interferometer, thus paving the way to the achievement of Egidi's dream, and soon Deslattes et al [3] completed the first counting of the atoms in a natural silicon crystal.

The present project, outlined by Zosi [4] in 1983, began in 2004 by combining the experiences and capabilities of the BIPM, INRIM, IRMM, NIST, NPL, NMIA, NMIJ and PTB. The start signal, ratified by a memorandum of understanding, was a contract for the production of a silicon crystal highly enriched with ²⁸Si. The enrichment process was undertaken by the Central Design Bureau of Machine Building in St Petersburg. Subsequently, a polycrystal was grown in the Institute of Chemistry of High-Purity Substances of the Russian Academy of Sciences in Nizhny Novgorod and a ²⁸Si boule was grown and purified by the Leibniz-Institut für Kristallzüchtung in Berlin. Isotope enrichment made it possible to apply isotope dilution mass spectroscopy, to determine the Avogadro constant with unprecedented accuracy, and to fulfil Egidi's dream.

To convey Egidi's 'fantasy' into practice, two ²⁸Si kilogram prototypes shaped as quasi-perfect spheres were manufactured by the Australian Centre for Precision Optics; their isotopic composition, molar mass, mass, volume, density and lattice parameter were accurately determined and their surfaces were chemically and physically characterized at the atomic scale. The paper by Andreas et al reviews the work carried out; it collates all the findings and illustrates how Avogadro's constant was obtained. Impurity concentration and gradients in the enriched crystal were measured by infrared spectroscopy and taken into account; Zakel et al relate these measurements in detail. Next, Pramann et al illustrate how the molar mass of the enriched crystal was measured by exploiting isotopic enrichment and isotope dilution mass spectrometry. Valkiers et al report about remeasurement of the molar mass of a natural Si crystal, a measurement prompted by the exigency of clarifying the origin of the discrepancy between the N_A value given in the present issue and the value obtained using natural Si crystals. A consistency analysis of the different isotopic-composition determinations is illustrated in the paper by Bulska et al. As reported in two papers by Massa et al, to determine the lattice parameter an x-ray interferometer was manufactured from the material between the already mentioned spheres. The measurement result was combined with lattice comparisons between different crystal samples and with the impurity gradient to extrapolate the sphere's lattice-parameter. Ferroglio et al's contribution analyzes the self-weight deformation of the x-ray interferometer. Fujimoto et al report about the lattice-perfection investigations carried out by a novel self-referencing diffractometer at the National Laboratory for High-Energy Physics (KEK) in Japan. A really great effort was made to characterize the sphere surfaces and to correct for the oxide layer and the contaminating atoms. The results of these investigations are given by Busch et al. The sphere diameter and topography were measured by optical interferometry to nanometer accuracy; the papers of Bartl et al and Kuramoto et al describe how the sphere volumes were determined. Andreas et al's paper describes the calculation of phase corrections for the diameter measurements. The results of mass comparisons against the Pt–Ir standards of the BIPM, NMIJ and PTB are given by Picard et al.

The results reported in the present issue need to be completed. One of the necessary activities is to relate the mass of the ²⁸Si atom to its Compton wavelength to test the mass–energy–frequency equivalence. Another effort is to monitor the stability of the Pt–Ir prototype: the technologies described in the present issue can be refined and finalized to calculate the mass variation of 1 kg ²⁸Si spheres by monitoring the surface evolution without weighing them on a balance. The last activity is the determination of the mass of a ²⁸Si sphere by electrical measurements using a watt balance and without any reference to the Pt–Ir prototype. In this framework, it will be necessary to demonstrate the mutual consistency and the stability of both the electrical and crystal mise en pratique of a kilogram definition based on a conventional value of the Planck constant. A related issue is to develop suitable procedures and protocols to disseminate the unit of mass from the new realizations.

Since the molar Planck constant is well known via the measurement of the Rydberg constant, the accurate measurement of N_A also provides an accurate and independent determination of the Planck constant, h. A comparison of the values of the Planck constant obtained via the watt-balance experiment and the N_A determination tests quantum mechanics. In fact, the watt-balance value of h depends on solid state physics through the theories of Josephson and quantum Hall effects, whereas the value of h derived from N_A depends on atomic physics through the energy level differences in hydrogen and deuterium, whose associated transition frequencies yield information on the Rydberg constant.

Grateful thanks are addressed to H-J Pohl for his outstanding project management in Russia, to A K Kaliteevski and his colleagues of the Central Design Bureau of Machine Building and the Institute of Chemistry of High-Purity Substances for their dedication and the punctual delivery of the enriched material, to H Riemann and his staff of the Institut für Kristallzüchtung for the crystal growth, to our directors for their advice and financial support, and to our colleagues for their daily work.

Special thanks are addressed to Peter Becker, to whom this issue is dedicated on the occasion of his retirement from work at the Physikalisch-Technische Bundesanstalt. In 1974, young Peter joined the PTB's Avogadro group which, under the direction of Peter Seyfried, followed Bonse's work and improved the measurements of the lattice parameter and the Avogadro constant [5, 6]. In 2004, Peter proposed and backed this project by taking on his shoulders the risks, the management burden and the coordination of the many relevant activities.

References

[1] Egidi C 1963 Phantasies on a natural unity of mass Nature200 61–2

[2] Bonse U and Hart M 1965 An x-ray interferometer Appl. Phys. Lett.6 155–6

[3] Deslattes R D et al 1974 Determination of the Avogadro constant Phys. Rev. Lett.33 463–6

[4] Zosi G 1983 A neo-Pythagorean approach towards an atomic mass standard Lett. Nuovo Cimento38 577–80

[5] Becker P et al 1981 Absolute measurement of the (220) lattice plane spacing in a silicon crystal Phys. Rev. Lett.46 1540–3

[6] Seyfried P et al 1992 A determination of the Avogadro constant Z. Phys. B 87 289–98

This paper concerns an international research project aimed at determining the Avogadro constant by counting the atoms in an isotopically enriched silicon crystal. The counting procedure was based on the measurement of the molar volume and the volume of an atom in two 1 kg crystal spheres. The novelty was the use of isotope dilution mass spectrometry as a new and very accurate method for the determination of the molar mass of enriched silicon. Because of an unexpected metallic contamination of the sphere surfaces, the relative measurement uncertainty, 3 × 10⁻⁸N_A, is larger by a factor 1.5 than that targeted. The measured value of the Avogadro constant, N_A = 6.022 140 82(18) × 10²³ mol⁻¹, is the most accurate input datum for the kilogram redefinition and differs by 16 × 10⁻⁸N_A from the CODATA 2006 adjusted value. This value is midway between the NIST and NPL watt-balance values.

Within the scope of re-determining the Avogadro constant, the content of impurities in ultra-pure, mono-isotopically enriched silicon 'Si28' was measured by infrared spectrometry. The calibration factors required for the calculation of the concentrations from the maximum absorption coefficients had to be determined in this context for the isotopically enriched material. For this purpose a model was prepared, which enables conversion of the factors for Si with natural isotopic abundance. On the basis of these results, the concentrations of the three most decisive impurities in the enriched material were determined. These are oxygen atoms on interstitial sites and substituting carbon and boron atoms which occupy silicon sites. Whereas carbon and oxygen were measured in accordance with standardized procedures modified for the particular metrological use, the absorption measurement of the boron was carried out differently to the conventional standard. The concentrations determined for the three types of crystal impurities are high enough to make a correction of the lattice parameter for the 'Si28' material necessary.

The molar mass of a new silicon crystal material highly enriched in ²⁸Si ('Si28', x(²⁸Si) >99.99%) has been measured for the first time using a combination of a modified isotope dilution mass spectrometry (IDMS) technique and a high resolution multicollector-ICP-mass spectrometer. This work is related to the redetermination of the Avogadro constant N_A with an intended relative measurement uncertainty u_rel(N_A) ⩽ 2 × 10⁻⁸. The corresponding experimental investigations of the International Avogadro Coordination (IAC) were performed using this novel 'Si28' material. One prerequisite of the redetermination of N_A is the determination of the isotopic composition and thus molar mass of 'Si28' with u_rel(M('Si28')) ⩽ 1 × 10⁻⁸. At PTB, a molar mass M('Si28') = 27.976 970 27(23) g mol⁻¹ has been determined with an associated relative uncertainty u_rel(M('Si28')) = 8.2 × 10⁻⁹, opening the opportunity to reach the target uncertainty of N_A.

A procedure is described to obtain SI-traceable silicon isotope amount ratios in the BaSiF₆ form. On the basis of a gravimetric approach, three synthetic isotope mixtures were made, very similar to natural Si, by blending silicon isotopically enriched with ²⁸Si, ²⁹Si and ³⁰Si. These mixtures served as references to calibrate measurements of (SiF₃)⁺ current ratios. In this way, it was possible to make SI-traceable measurements of silicon isotope amount ratios, without any assumptive correction. The isotopic composition of the natural Si crystal WASO17.2 was remeasured and its molar mass was redetermined.

To determine the Avogadro constant by counting the atoms in quasi-perfect spheres made of a silicon crystal highly enriched with ²⁸Si, the isotopic composition of the crystal was measured in different laboratories by different measurement methods. This paper examines the consistency of the measurement results.

The spacing of the {2 2 0} lattice planes of a ²⁸Si crystal, used to determine the Avogadro constant by counting silicon atoms, was measured by combined x-ray and optical interferometry to a relative accuracy of 3.5 × 10⁻⁹. The result is d_2 2 0 = (192 014 712.67 ± 0.67) am, at 20.0 °C and 0 Pa. This value is greater by (1.9464 ± 0.0067) × 10⁻⁶d_2 2 0 than the spacing in natural Si, a difference which confirms quantum-mechanics calculations. This result is a key step towards a realization of the mass unit based on a conventional value of the Planck or the Avogadro constant.

The Avogadro constant has been determined by atom counting in two enriched ²⁸Si spheres. Atoms were counted by exploiting their ordered arrangement in the spheres and calculating the ratio between sphere and the unit-cell volumes. This paper describes how the values of the sphere lattice parameters and, consequently, their unit-cell volumes were obtained.

To achieve the smallest uncertainty in the measurement of the ²⁸Si lattice parameter, an x-ray interferometer with an unusually long analyser crystal has been considered. The simulation of the gravitational bending by the finite-element method allowed the interferometer to be optimally designed, the residual lattice strain to be accurately predicted and the self-weight contribution to the uncertainty budget to be kept below 1 nm m⁻¹.

The homogeneity of the lattice spacings of silicon single crystals from different origins was characterized. Strain measurements were performed on single crystals from NRLM3, NRLM4 and 28Si-10Pr11 ingots, which are all used to determine the Avogadro constant. NRLM3 and NRLM4 both exhibited clear striations, whereas almost no pattern was obtained for 28Si-10Pr11. The standard deviation of the lattice spacing of the single crystal obtained from 28Si-10Pr11 was 4.7 × 10⁻⁹, which enabled the lattice spacing to be determined with a standard uncertainty of 3 × 10⁻⁹.

For the accurate determination of the Avogadro constant, two ²⁸Si spheres were produced, whose macroscopic density, in addition to other values, must be determined. To make a contribution to the new definition of the kilogram, a relative standard uncertainty of less than 2 × 10⁻⁸ has to be achieved. Each silicon surface is covered by a surface layer (SL). Consequently, correction parameters for the SL are determined to be applied to the mass and volume determination of the enriched spheres. With the use of a large set of surface analysing techniques, the structure of the SL is investigated. An unexpected metallic contamination existing on the sphere surface enlarges the uncertainty contribution of the correction parameters above the originally targeted value of 1 × 10⁻⁸. In the framework of this investigation this new obstacle is resolved in two ways. A new combination of analytical methods is applied to measure the SL mass m_SL and the thickness d_SL, including this new contamination, with an uncertainty of u(m_SL) = 14.5 µg and 14.4 µg, respectively, and u(d_SL) = 0.33 nm and 0.32 nm for the ²⁸Si spheres AVO28-S5 and AVO28-S8, respectively.

In the second part of the work, the chemical composition of these metallic contaminations is found to be Cu, Ni and Zn silicide compounds. For the removal of this contamination, a special procedure is developed, tested and applied to the spheres to produce the originally expected surface structure on the spheres. After the application of this new procedure the use of x-ray reflectometry directly at the spheres will be possible. It is expected to reduce the uncertainty contribution due to the SL down to 1 × 10⁻⁸.

The volumes of two 1 kg silicon spheres, AVO28-S5 and AVO28-S8, fabricated from a ²⁸Si-enriched crystal were measured to determine the Avogadro constant by the x-ray crystal density method in the International Avogadro Coordination Project. The volumes were determined from diameter measurements of the two spheres using a laser interferometer with a flat etalon. In the diameter measurement, the sphere was placed between the two flat etalon plates. The gaps between the sphere and the etalon plates and the distance between the etalon plates were measured by phase-shifting interferometry with optical frequency tuning. The apparent volumes of the ²⁸Si spheres were determined from the diameter measurement in many directions with relative combined standard uncertainties of 5.0 × 10⁻⁸ and 4.4 × 10⁻⁸ for AVO28-S5 and AVO28-S8, respectively. The effect of the surface layer on the diameter measurement was evaluated on the basis of the results of characterizing the sphere surface. By taking into account the effect of the surface layer, the silicon core volume excluding the surface layer and the actual volume including the surface layer were also determined. Details of the interferometer, data analysis and the uncertainty in the measurement are described.

Within the scope of the efforts concerning the redefinition of the SI base unit 'kilogram' 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 ²⁸Si 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.

To determine the volume of solid density standards, manufactured as Si-crystal spheres, an optical interferometer is used to measure their diameter at the NMIJ. To support and complement these measurements, the effect of the Gouy phase has been studied analytically and numerically. In measurement, the sphere is placed between the end-mirrors of a Fizeau cavity and the distances between the cavity mirrors and the sphere are measured, as well as the cavity length. The present analysis outlines a model of the interferometer operation and quantifies the Gouy-phase correction in the diameter measurement.

For a new determination of the Avogadro constant to play a role in the future redefinition and realization of the mass unit, the mass measurements involved need to be carried out at a very high level of accuracy. From a mass comparison among two 1 kg ²⁸Si spheres and the 1 kg platinum–iridium (Pt/Ir) mass standards, the mass of the spheres must be determined with a combined standard uncertainty of less than 5 µg. The BIPM, the PTB and the NMIJ have carried out such a state-of-the-art mass comparison in air and under vacuum in order to reach this target set by the International Avogadro Coordination (IAC). The results obtained for the spheres AVO28-S5 and AVO28-S8 involved in the comparison have demonstrated that by using air buoyancy artefacts and sorption artefacts it is possible to achieve a relative uncertainty of 4.1 × 10⁻⁹. The reference value for each sphere has been determined, taking into account the traceability of the masses to the International Prototype of the kilogram, , in due consideration of the correlations among 17 standards used directly or indirectly in this comparison.