A 2007 Consultative Committee for Mass discussion paper (Becker et al, Metrologia 44, 1) catalogued the open scientific issues that had to be settled before the kilogram and mole could be redefined via fixed fundamental constants: mutual consistency of watt-balance (Kibble) determinations of the Planck constant hh and silicon-sphere (XRCD) determinations of the Avogadro constant NAN_{\mathrm{A}} at about 2×10−82\times10^{-8} relative uncertainty, the exactness in hh and ee of the Josephson and quantum Hall relations underlying electrical standards, joint verification of E=hνE=h\nu and E=mc2E=mc^2 at definitional accuracy, validated realization procedures (mises en pratique), and management of the molar-mass factor (1+κ)(1+\kappa). The experimental and institutional preconditions were met well enough for the 26th General Conference on Weights and Measures to adopt the revised SI in 2018, effective 2019, fixing hh, ee, kk and NAN_{\mathrm{A}}. Two core physical issues from that list remain unresolved: (1) the exactness of the Josephson relation KJ=2e/hK_J=2e/h and the quantum Hall relation RK=h/e2R_K=h/e^2 — now exact by definition, yet still neither proven for realistic devices nor jointly tested at the 10−810^{-8} level via the quantum metrological triangle (Josephson effect, quantum Hall effect, single-electron transport); and (2) mutual agreement of independent primary realizations of the kilogram, which still disagree beyond uncertainties, so that worldwide mass dissemination continues through an interim consensus value rather than individual realizations.

Background

From 1889 until 2019 the SI unit of mass was defined by a single platinum–iridium artifact, the international prototype of the kilogram (IPK) kept at the International Bureau of Weights and Measures (BIPM) in Sèvres. An artifact definition has no guaranteed long-term stability: comparisons of national copies with the IPK indicated relative drifts of order 10−810^{-8} per decade, and any such drift changes the kilogram itself, on which the old definitions of the ampere and mole also hung. By the mid-2000s metrology institutes therefore planned to redefine the kilogram — together with the ampere, kelvin and mole — by fixing exact numerical values of fundamental constants, chiefly the Planck constant hh for the kilogram and the Avogadro constant NAN_{\mathrm{A}} for the mole.

Two experimental routes can link a macroscopic mass to hh. A Kibble balance (originally called a watt balance) compares the weight of a test mass with an electromagnetic force on a coil moving in a magnetic field and, through a second measurement phase, relates that mechanical power to electrical power measured via the Josephson and quantum Hall effects, so the mass is expressed in terms of hh. The x-ray crystal density (XRCD) method instead effectively counts the atoms in a 1 kg single-crystal sphere of highly enriched silicon-28: from the measured lattice parameter, sphere volume, molar mass and mass one obtains NAN_{\mathrm{A}}, and since the product NAhN_{\mathrm{A}}h is known independently, an NAN_{\mathrm{A}} determination equally fixes the kilogram. In the mid-2000s the two routes disagreed at the relative level of about 10−610^{-6}.

The CCM (Consultative Committee for Mass and Related Quantities) discussion paper by Becker et al (2007) reviewed the candidate definitions and their preconditions. It favored linking the kilogram to the mass of a particle or to hh; it discussed, critically, a definition fixing hh through the Compton frequency of a material particle, a route that presumes the simultaneous exactness of the quantum relation E=hνE=h\nu and the relativistic relation E=mc2E=mc^2; and it proposed redefining the mole by fixing NAN_{\mathrm{A}}, noting that a simultaneous change of kilogram and mole makes the molar-mass constant depart from exactly 10−310^{-3} kg/mol by a small factor (1+κ)(1+\kappa) with non-zero uncertainty. The paper's open question is the list of scientific issues requiring resolution before adoption: mutual consistency of the hh and NAN_{\mathrm{A}} routes at about 2×10−82\times10^{-8}; the theoretical status of the Josephson and quantum Hall relations; empirical verification of the quantum-relativistic mass–frequency link; practical-realization procedures (mises en pratique, the official documents describing how a definition is physically realized); and management of the molar-mass factor.

On the electrical side, the ac Josephson effect relates the voltage across an irradiated superconducting junction to frequency through V=(h/2e)νV=(h/2e)\nu, defining the Josephson constant KJ=2e/hK_J=2e/h; the integer quantum Hall effect (QHE) quantizes the Hall resistance of a two-dimensional conductor in integer multiples of the von Klitzing constant RK=h/e2R_K=h/e^2. That these relations are exact in terms of hh and ee, with no corrections, is an assumption underlying all primary voltage and resistance standards. It can be tested jointly with single-electron transport (a pump driven at frequency ff delivers current I=efI=ef) by closing the quantum metrological triangle: a measurement chain combining all three quantum effects so that V=RIV=RI links KJK_J, RKR_K and ee with no assumed exactness. No such closure at the 10−810^{-8} level exists.

The institutional process codified numerical preconditions: the CCM's 2013 recommendation required at least three independent experiments (Kibble and XRCD) yielding consistent values of hh with relative standard uncertainties not larger than 5×10−85\times10^{-8}, at least one with uncertainty not larger than 2×10−82\times10^{-8}, traceability of the relevant mass standards to the IPK, and validated mises en pratique (Milton, Davis and Fletcher 2014). The XRCD target was met by the International Avogadro Coordination (IAC), which reported NA=6.022 140 76(12)×1023N_{\mathrm{A}}=6.022\,140\,76(12)\times10^{23} mol−1^{-1} with relative uncertainty 2.0×10−82.0\times10^{-8} (Azuma et al 2015); Kibble-balance results with uncertainties of 99–13×10−913\times10^{-9} followed. The CODATA 2017 special adjustment (CODATA is the Committee on Data for Science and Technology, which issues internationally recommended values of the fundamental constants) gave h=6.626 070 150(69)×10−34h=6.626\,070\,150(69)\times10^{-34} J s with relative uncertainty 1.0×10−81.0\times10^{-8} (Newell et al 2018; Mohr et al 2018), and the 26th CGPM (General Conference on Weights and Measures, the body that formally decides SI changes) declared the conditions met and adopted the revised SI, effective 20 May 2019, fixing hh, ee, kk and NAN_{\mathrm{A}} exactly (CGPM 2018 Resolution 1). In the revised SI, KJK_J and RKR_K are exact by definition, the conventional values KJ-90K_{J\text{-}90} and RK-90R_{K\text{-}90} were abrogated, and the molar mass of carbon-12 is no longer exactly 12 g/mol.

Problem Statement

The revised SI defines the kilogram by fixing the Planck constant hh and the ampere by fixing the elementary charge ee; consequently the Josephson constant KJ=2e/hK_J=2e/h and the von Klitzing constant RK=h/e2R_K=h/e^2 are exact as definitions, and the kilogram may be realized by any method that links mass to hh. The open problems, inherited from the unresolved core of Becker et al (2007), concern the physical faithfulness of these fixed-constant definitions at relative accuracies of a few parts in 10−810^{-8}. They are two separate issues from the 2007 list of pre-adoption scientific issues, and each is an independent target whose resolution does not require resolving the other:

(1) Exactness of the Josephson and quantum Hall relations as physics. Determine — by theoretical derivation for realistic macroscopic devices (with dissipation, finite size, contacts, edge structure and interactions) or by experiment — whether the ac Josephson relation V=(h/2e)νV=(h/2e)\nu and the integer quantum Hall relation R=h/(ie2)R=h/(ie^2) on plateau ii hold with no physical corrections at a relative accuracy of a few parts in 10−810^{-8}, the level at which they are assumed exact in primary electrical standards. Either outcome resolves this target: a rigorous demonstration that the relations hold with no corrections at that level, or a rigorous demonstration that a physical correction exists (an inconsistent quantum-metrological-triangle closure result at that level being one such demonstration). One experimental route is closure of the quantum metrological triangle: a measurement chain combining a Josephson voltage standard, a quantum Hall resistance standard, and a single-electron current source delivering I=efI=ef, with the single-electron source's transfer error and current level adequate for a 10−810^{-8}-level consistency test. No such closure exists to date, and only a partial theoretical result (vanishing higher-order corrections for the interacting integer-quantum-Hall linear response) is available.

(2) Convergence of independent primary realizations of the kilogram. Bring the Kibble-balance and XRCD (silicon-sphere) realizations of the kilogram into mutual agreement within their combined uncertainties at the required relative accuracy of about 2×10−82\times10^{-8}, by identifying and suppressing the systematic effects responsible for the present excess scatter; agreement at the tighter 1.0×10−81.0\times10^{-8} level (CGPM 2018 Resolution 1, Appendix 2), the post-2018 realization accuracy of the kilogram, is an optional stronger target. Ending worldwide dissemination through the interim consensus value in favor of individual primary realizations, a decision reserved to the Consultative Committee for Mass, is an institutional motivation for this target, not part of its physical completion criterion.

The two issues are related in motivation — each asks whether the independent physical routes through the fixed constants hh and ee — electrical and mechanical, single-particle and macroscopic — agree as physics and not merely as convention — but they are separate targets, and progress or completion on one does not presuppose the other; both targets are required for a complete resolution.

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