Maintaining consensus for the redefined kilogram: reconciling discrepant independent realizations for consistent mass dissemination

Independent primary realizations of the kilogram, redefined in 2019 via the fixed numerical value of the Planck constant, are produced by Kibble/joule balances and the x-ray crystal density (XRCD) method, but they disagree by tens of micrograms at 1 kg — beyond their quoted standard uncertainties of roughly 9–60 µg. Wübbeler, Bettin, Härtig and Elster (Metrologia 55, 722, 2018) analyzed two meta-analysis-based correction procedures designed to keep international mass dissemination consistent despite this disagreement: a fixed-effects procedure that subtracts time-constant laboratory offsets identified through a weighted-sum constraint, and a random-effects procedure that inflates uncertainties by the estimated between-laboratory heterogeneity. The two procedures make different assumptions about the temporal behavior of potential inconsistencies; only the fixed-effects variant met the CCM uncertainty thresholds, at the cost of an unverifiable identifiability constraint, an unknown temporal structure of laboratory effects, and consistency tests of limited power, and the authors critically assess the attempt to re-establish consistency through a statistical correction procedure. Since February 2021 the International Bureau of Weights and Measures (BIPM) has instead disseminated the kilogram through an interim 'consensus value' with an assigned 20 µg standard uncertainty, to be maintained until the dispersion in values from independent realizations becomes compatible with their individual uncertainties. The open problem is to replace this interim arrangement with a scientifically validated one: a principled statistical framework for using key-comparison data on discrepant realizations, and the identification and elimination of the physical systematic effects causing the disagreement. The decision to retire the consensus value and move to dissemination from individual realizations (phase 3) is an institutional determination of the Consultative Committee for Mass and Related Quantities (CCM), made against transition criteria already described in the CCM's dissemination document. As of the third consensus value (implemented 1 March 2026) dissemination still rests on this arrangement — the two smallest-uncertainty realizations now agree within their expanded uncertainties, but the consensus value, whose 20 µg uncertainty dominates the BIPM's 21 µg total calibration uncertainty, remains the basis of worldwide mass traceability — so both scientific components remain open.

Background

Since 20 May 2019 the kilogram has been defined by fixing the numerical value of the Planck constant $h$. The definition can be realized independently by any laboratory: a Kibble balance (also called a watt balance) relates the weight of a 1 kg mass to an electromagnetic force measured with quantum electrical standards, linking mass to $h$; a joule balance is a closely related variant; and the x-ray crystal density (XRCD) method determines the mass of a silicon sphere highly enriched in the isotope $^{28}$Si by counting its atoms from the lattice spacing and the isotopic composition. Under the CIPM Mutual Recognition Arrangement (MRA) — the framework under which national metrology institutes (NMIs) recognize each other's calibration certificates — NMIs check the equivalence of their standards in key comparisons: interlaboratory exercises whose statistical analysis yields a key comparison reference value (KCRV) and pairwise 'degrees of equivalence' between laboratories.

Before approving the redefinition, the Consultative Committee for Mass and Related Quantities (CCM) required in 2013 that at least three independent experiments, including Kibble-balance and XRCD work, yield consistent values of $h$ with relative standard uncertainties not larger than $5\times10^{-8}$, and at least one with $2\times10^{-8}$. The 2017 determinations failed the consistency part: converted to deviations of a 1 kg mass, the results ranged over tens of micrograms — for example NRC $+2.6$ µg with standard uncertainty $u=9.1$ µg versus IAC $-38.5$ µg with $u=11.6$ µg — so independent realizations would disagree well beyond their quoted uncertainties (a few parts in $10^{8}$). To proceed nonetheless, the CCM's 2017 recommendation proposed using the results of an ongoing key comparison to correct future measurement results so that dissemination remains consistent.

Wübbeler, Bettin, Härtig and Elster (2018) formalized this idea with two correction procedures taken from meta-analysis. Writing laboratory $i$'s result as $x_i=\mu+\lambda_i+\epsilon_i$, with $\mu$ the common measurand, $\lambda_i$ a laboratory-specific effect and $\epsilon_i\sim N(0,u_i^{2})$ the quoted measurement uncertainty, the fixed-effects procedure assumes the $\lambda_i$ are constant in time: they are identified through the weighted-sum constraint $\sum_i \lambda_i/u_i^{2}=0$ and subtracted from that laboratory's future results, preserving small uncertainties. The random-effects procedure treats $\lambda_i$ as fresh draws from $N(0,\sigma_\lambda^{2})$: the between-laboratory heterogeneity $\sigma_\lambda$ is estimated and future uncertainties are inflated by it. The two procedures make different assumptions about the temporal behavior of potential inconsistencies, and only the fixed-effects correction produces sufficiently small uncertainties for corrected results in the sense of the CCM 2013 recommendation; the authors critically assess the attempt to re-establish consistency through such a statistical correction procedure. The fixed-effects correction rests on a constraint the comparison data cannot verify, since any bias common to all laboratories can be shifted into the $\lambda_i$ without changing the likelihood, and corrected uncertainties then reflect the adopted convention as much as the measurement itself. Obstacles of this kind remain for any statistical correction of discrepant comparison data: the identifiability of common versus laboratory-specific bias; the unknown temporal structure (constant, random, or mixed) of laboratory effects, not estimable from a single comparison; the low power of chi-squared consistency tests at the small participant numbers typical of primary-realization groups; the need for metrological investigation of the physical causes of disagreement; and the question whether corrections that may mask common biases are appropriate for official dissemination under the MRA.

In practice the CCM chose a different interim device: since 1 February 2021 the International Bureau of Weights and Measures (BIPM) has disseminated the kilogram through a consensus value (CV) — the arithmetic mean of the 2014 value traceable to the International Prototype of the Kilogram (IPK, the artefact that defined the unit until 2019), the 2016 pilot-study reference value, and the latest key-comparison KCRV — with an assigned standard uncertainty of 20 µg, updated after each new key comparison of realizations. Dissemination is to remain based on the CV until the dispersion in values from realization experiments becomes compatible with their individual uncertainties; the agreed end state (phase 3 of the CCM's dissemination plan) is dissemination from individual realizations, whose calibration and measurement capabilities would then be assessed through the ordinary key-comparison machinery. The criteria for that transition are already described in the CCM's dissemination document, and the transition itself is an institutional CCM determination.

Problem Statement

Independent primary realizations of the redefined kilogram — Kibble and joule balances (electromechanical experiments that link a 1 kg mass to the fixed Planck constant through quantum electrical standards) and x-ray crystal density (XRCD) experiments (which fix the mass of an enriched $^{28}$Si sphere from its lattice spacing and isotopic composition) — disagree by tens of micrograms at 1 kg, beyond their quoted standard uncertainties of roughly 9–60 µg. International dissemination of the unit therefore rests on the interim consensus value disseminated by the International Bureau of Weights and Measures (BIPM), with its assigned 20 µg standard uncertainty. The open problem — posed by the critical assessment of Wübbeler, Bettin, Härtig and Elster (2018) and by the interim character of the consensus-value arrangement of the Consultative Committee for Mass and Related Quantities (CCM, the body that coordinates world mass metrology) documented by Davidson and Stock (2021) — is to replace this interim arrangement with a scientifically validated one, comprising two linked parts:

(1) A statistically principled and empirically validated procedure for using key-comparison results to keep calibrations issued by different laboratories mutually equivalent — by correcting future realization results, their uncertainties, or both. This must confront three obstacles inherent in the comparison data: a single key comparison cannot separate a systematic bias common to all laboratories from laboratory-specific offsets; the temporal structure of laboratory effects (time-constant, randomly varying, or a mixture of both) is unknown and cannot be decomposed from one comparison; and chi-squared-type consistency tests have limited power at the participant numbers involved (about 5–10 national metrology institutes, NMIs).

(2) The identification and elimination of the physical systematic effects responsible for the disagreement between the realizations, such that the dispersion in values from independent realization experiments becomes compatible with their individual quoted uncertainties — the end condition stated by the CCM for maintaining the interim consensus value. This is a group-level consistency criterion across the participating realizations, not merely pairwise agreement of the smallest-uncertainty results at an expanded-uncertainty level.

The two parts serve one overarching objective — a mutually consistent, scientifically defensible dissemination of the SI unit of mass at the smallest achievable uncertainty: a validated statistical treatment of discrepant comparison data together with realization experiments whose dispersion is compatible with their individual quoted uncertainties. Both parts must be addressed for a complete answer, and each is satisfied by an analysis rather than by the execution of a new experimental programme: part (1) by a statistical procedure, with the assumptions it rests on, its validation against key-comparison data, and the corrections or uncertainty inflations it yields; part (2), whose full resolution depends on new realization measurements beyond any single study, by an analysis that identifies the candidate physical systematic effects and specifies the evidence needed to attribute and eliminate them so that the dispersion of independent realizations can be brought into compatibility with their quoted uncertainties. A submission that addresses only one of the two parts does not complete the task. The decision to retire the consensus value and move to dissemination from individual realizations (phase 3 of the CCM's dissemination plan) is an institutional CCM determination made against transition criteria already described in the CCM's dissemination document; it is context of the problem rather than a scientific deliverable.

References

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