The revision of the SI that took effect on 20 May 2019 fixed the Planck constant hh (defining the kilogram) and the Avogadro constant NAN_A (defining the mole) exactly and independently, ending the definitional tie between the atomic and macroscopic mass scales that the pre-2019 SI maintained through the exactly stipulated molar mass of carbon-12. Since then, the relation between the dalton (the unified atomic mass unit) and the kilogram — equivalently the molar mass constant MuM_u — rests entirely on experiment, and macroscopic mass must be realized from hh by primary experiments. Two independent realization methods are recognized: the Kibble balance (electromagnetic) and the X-ray crystal density (XRCD) method, which counts the atoms in a kilogram-scale sphere of isotopically enriched 28^{28}Si. The unresolved problem is the atom-counting half of this program: the XRCD realization stands at a relative standard uncertainty of about 2×10−82\times10^{-8}, just above the target level, and Kibble-balance and XRCD results were mutually inconsistent within their stated uncertainties at the CODATA 2017 special adjustment and during the early key comparisons, forcing the Consultative Committee for Mass (CCM) to disseminate the kilogram through an interim 'consensus value' since February 2021. Completing the link — atom-based realization of the kilogram and mole at or below 2×10−82\times10^{-8} with demonstrated cross-method consistency and reproducibility, including crystal isotopic homogeneity, surface-layer, volume and lattice-parameter corrections, and resolution of any inter-method discrepancies that remain — remains open; after the third key comparison CCM.M-K8.2024 (final report published February 2026) the CCM's published criteria for retiring the consensus value remain unmet, and dissemination still runs through a newly calculated consensus value.

Background

Until 2019 the International System of Units (SI) tied the atomic and macroscopic mass scales together definitionally. The kilogram was the mass of the International Prototype of the Kilogram (IPK), a platinum–iridium artefact kept at the International Bureau of Weights and Measures (BIPM), and the mole was defined so that the molar mass of carbon-12, M(12C)M(^{12}\mathrm{C}) — the mass of one mole of carbon-12 atoms — was exactly 12 g/mol. Equivalently, the molar mass constant Mu=M(12C)/12M_u = M(^{12}\mathrm{C})/12 was exactly 1 g/mol, so the dalton (the unified atomic mass unit, mu=m(12C)/12m_u = m(^{12}\mathrm{C})/12, the atomic mass scale used throughout chemistry) and the kilogram were connected through the Avogadro constant NA=Mu/muN_A = M_u/m_u, which links the number of entities in a mole to the mass of a single entity and had to be measured experimentally. Pre-redefinition analyses of the proposed 'new SI' — for example the stepwise-redefinition proposals of Isaev, Kononogov and Khruschov, which debated fixing NAN_A and the constraints on choosing its stipulated numerical value, and the proposal of Si to define the kilogram via the carbon-12 atom realized by a countable 'carbon onion' (a concentric-shell fullerene of 12^{12}C atoms) — recognized that fixing the Planck constant hh and NAN_A independently would break this definitional bridge: the mole would decouple from the kilogram, and MuM_u would become a quantity determined by experiment.

This is now the situation. The 26th General Conference on Weights and Measures (CGPM, 2018) fixed h=6.62607015×10−34h = 6.62607015\times10^{-34} J·s and NA=6.02214076×1023N_A = 6.02214076\times10^{23} mol−1^{-1} exactly, effective 20 May 2019. The molar mass constant is no longer exact: in the 2022 adjustment of the fundamental constants by CODATA (the Committee on Data for Science and Technology, which issues the internationally recommended values of the fundamental constants) it is Mu=1.000 000 001 05(31)×10−3M_u = 1.000\,000\,001\,05(31)\times10^{-3} kg/mol with relative standard uncertainty 3.1×10−103.1\times10^{-10}, determined not by weighing atoms but through the fine-structure constant α\alpha (the dimensionless coupling constant of electromagnetism, known from measurements of the electron gg-factor (the dimensionless magnetic moment of the electron) and of atom recoil) via mu=2hR∞/(α2c Ar(e))m_u = 2hR_\infty/(\alpha^2 c\,A_r(e)), where R∞R_\infty is the Rydberg constant and Ar(e)A_r(e) the relative atomic mass of the electron. The molar mass of carbon-12 now deviates from its pre-2019 exact 12 g/mol by 105(31)×10−11105(31)\times10^{-11} — a discontinuity exceeding three standard deviations that CODATA itself flags as an inconsistency to monitor.

Macroscopic mass, meanwhile, must be realized from the fixed hh by primary experiments named in the official 'mise en pratique' for the kilogram (the CIPM document listing the accepted practical ways to realize a unit definition). The first is the Kibble balance (formerly 'watt balance'), which balances the weight of a test mass against an electromagnetic force calibrated through the Josephson and quantum Hall effects, tying mass to hh via electrical quantum standards. The second is the X-ray crystal density (XRCD) method — literal atom counting: a sphere of ~1 kg made from a crystal of silicon highly enriched in the isotope 28^{28}Si is measured; the number of atoms follows from the sphere volume (measured by optical interferometry), the crystal lattice parameter (measured by X-ray interferometry) and the crystal structure (8 atoms per cubic unit cell); the molar mass and isotopic composition of the enriched crystal are measured by isotope-dilution mass spectrometry using multicollector inductively coupled plasma mass spectrometry (MC-ICP-MS IDMS); and the thin surface oxide layer is characterized by X-ray photoelectron spectroscopy (XPS) and ellipsometry so its mass can be subtracted. The sphere's mass then follows from the atom count and the mass of a single silicon atom computed from hh. The XRCD route descends from the International Avogadro Coordination determinations of NAN_A (relative standard uncertainty 2.0×10−82.0\times10^{-8} with that collaboration's first-generation 'AVO28' crystals) and from the new 'Kilogram-2' enriched crystals grown for exactly this purpose.

The two routes, however, did not agree. At the special CODATA 2017 adjustment performed to fix hh, the Kibble-balance and XRCD data sets were mutually inconsistent within their stated uncertainties (implied differences up to about 70 µg on 1 kg, i.e. 7×10−87\times10^{-8}), and the redefinition proceeded on their adjusted mean with relative uncertainty 10−810^{-8}. To guarantee a uniform world mass scale, the Consultative Committee for Mass (CCM) therefore organized comparisons of kilogram realizations — a 2016 pilot study, then the key comparisons CCM.M-K8.2019, CCM.M-K8.2021 and CCM.M-K8.2024 (Kibble balances, a joule balance — an energy-based Kibble-balance variant operated at the NIM in China — and XRCD experiments at NMIJ and PTB, joined by CMS/ITRI in the 2024 comparison) — and placed dissemination in phases: since 1 February 2021 the kilogram has been disseminated from a 'consensus value' (a mean of comparison results, standard uncertainty 20 µg on 1 kg) rather than from any single realization. The final phase, in which individual experiments may realize the kilogram independently, is gated on published CCM criteria: at least five consistent realization experiments with relative standard uncertainty at or below 4×10−84\times10^{-8} (at least three of them at or below 2×10−82\times10^{-8}), including both independent methods, each consistent with the key-comparison reference value (KCRV) and stable over two consecutive comparisons, with the final consensus value and KCRV differing by less than 5×10−95\times10^{-9}. In CCM.M-K8.2019 the overall chi-squared consistency test passed, but the two results with the smallest uncertainties were not in mutual agreement; in CCM.M-K8.2024 (final report published February 2026) the chi-squared test passed again and the difference between the NRC and PTB results was significantly reduced compared with CCM.M-K8.2021, yet the CCM is calculating a third consensus value rather than entering the final phase. The atom-counting realization itself — at NMIJ, with about 21–24 µg (≈2×10−82\times10^{-8}) standard uncertainty on 1 kg and five-year reproducibility of 2.4 µg — sits just above the target, and the isotopic homogeneity of the newest enriched crystals (urel(M)=1.4×10−9u_\mathrm{rel}(M) = 1.4\times10^{-9}, no significant inhomogeneity) has largely removed what was, at the time of the sources, an open materials question.

Problem Statement

In the present SI, where hh and NAN_A are fixed exactly and independently, the atomic and macroscopic mass scales are connected only through experiment. The problem is to complete the atom-counting half of this connection: realize macroscopic mass from a counted number of atoms — by the X-ray crystal density method with isotopically enriched 28^{28}Si spheres, or by any other atom-count construction — at a relative standard uncertainty competitive with Kibble-balance realizations of the kilogram, that is, at or below 2×10−82\times10^{-8}, and demonstrate the mutual consistency and long-term reproducibility of independent realization methods at that level, so that atom-counting realizations of the kilogram and the mole are sanctioned as primary independent realizations and the interim consensus-value dissemination can be retired (the completion boundary being the published CCM criteria for the final dissemination phase). This single objective jointly requires: (i) if the XRCD route is used, reducing every component of the XRCD uncertainty budget — molar mass and isotopic homogeneity of the enriched crystal, lattice parameter, sphere volume, and surface-layer (oxide) corrections — to a level consistent with a total at or below 2×10−82\times10^{-8}, and if another atom-count construction is used, reducing the corresponding uncertainty contributions of that construction to a level consistent with such a total, every such component being established for the crystal, sphere and construction actually used rather than adopted from earlier determinations; (ii) demonstrating reproducibility of the realization across successive CCM key comparisons; and (iii) demonstrating the mutual consistency within uncertainties of independent realization methods and, where residual inconsistencies remain, identifying the systematic effects responsible and showing corrected mutual consistency within uncertainties. The CCM's published criteria for the final dissemination phase describe ensemble-level agreement across the key comparisons and are the external boundary that ultimately retires the consensus value; they frame the problem but are not themselves the contribution sought, which is the atom-counting realization and the mutual-consistency and reproducibility evidence of (i)-(iii) at or below 2×10−82\times10^{-8}.

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