The redefinition of several physical base units planned for 2018 requires precise knowledge of the values of certain fundamental physical constants. Scientists are working hard to meet the deadlines for realizing the ultimate International System of Units.

The creation of the decimal metric system during the French Revolution and the subsequent deposition of two platinum standards representing the metre and the kilogram, on 22 June 1799 in the Archives de la République in Paris, was the first step in the development of the present International System of Units. Following the signing of the Convention of the Metre on 20 May 1875, new international prototypes for the metre and the kilogram were established in 1889. These units, together with the astronomical second (based on the mean solar day) as the unit of time, constituted a mechanical unit system. Following the introduction of the ampere, the kelvin and the candela as the units for electric current, thermodynamic temperature and luminous intensity, respectively, the name Système International d'Unités, with the abbreviation SI, was given to the system in 1960. In 1971, the mole, the unit for amount of substance, completed the present SI.

Together with the derived units, the SI constitutes a coherent set of units (that is, without conversion factors) by which any measurable quantity of interest in research, industry, trade or society can be quantified. The signatory states of the Metre Convention represent about 98% of the world's economy, so the SI is the very basis of international trade, constituting a global measurement quality infrastructure through the national metrology institutes.

## Defining constants

Extraordinary advances have been made in relating SI units to truly invariant quantities such as the fundamental constants of physics and the properties of atoms. Recognizing the importance of linking SI units to such invariant quantities, the 26th General Conference on Weights and Measures in 2018 will ratify a new definition of the SI based on the use of a set of seven such constants as references for the definitions. The value of any one of these seven constants is written as the product of a numerical coefficient and a unit, *Q* = {*Q*} [*Q*], where *Q* denotes the value of the constant and {*Q*} its numerical value when expressed in the unit [*Q*]. By fixing the exact numerical value — that is, not assigning any uncertainty to it — the unit becomes defined, as the product of the numerical value and the unit must equal the value of the constant, which is invariant.

The so-called defining constants are the following: the frequency of the ground-state hyperfine splitting of the caesium-133 atom Δ*ν*(^{133}Cs)_{hfs}, the speed of light in vacuum *c*, the Planck constant *h*, the elementary charge *e*, the Boltzmann constant *k*, the Avogadro constant *N*_{A} and the luminous efficacy *K*_{cd} (see Table 1 and Fig. 1). Combinations of these constants define the units second (s), metre (m), kilogram (kg), ampere (A), kelvin (K), mole (mol) and candela (cd) — and thus the entire SI.

One of the consequences of the redefinitions (Table 1) is that a distinction between (the former) base units and derived units is no longer necessary. As a second important feature, the definition and practical realization of the units will be decoupled in the new SI. Indeed, although the definitions of units may remain unchanged over a long period of time, their practical realizations can be established by many different experiments with ever-increasing accuracy — described in so-called *mise-en-pratiques* — thus allowing for experiments yet to be devised.

## Implications of the new definitions

The second and the metre have already been successfully defined through fundamental constants. Fixing the numerical value of the hyperfine splitting of caesium defines the second as the duration of 9,192,631,770 periods of the radiation that corresponds to the transition between the two hyperfine levels of the caesium atom. Having defined the second, fixing the numerical value of the speed of light means that the metre is the path length travelled by light in vacuum during a time interval of 1/299,792,458 of a second.

In 2018, the units kilogram, ampere, kelvin and mole will be defined in a similar way. The effect of fixing the numerical value of the Planck constant is a definition of the unit kg m^{2} s^{−1} (the unit of the physical quantity called action). Together with the definitions of the second and the metre, this leads to a definition of the kilogram; macroscopic masses can be measured in terms of *h*, *Δν*(^{133}Cs)_{hfs} and *c*. One way of establishing a mass scale is by counting the number of atoms in a silicon single-crystal sphere using the X-ray crystal density (XRCD) approach — probing the regular arrangement of atoms in a perfect lattice — and multiplying it by the known mass of a silicon atom (the ^{28}Si isotope)^{1}. Another route to the kilogram is based on balancing electric and gravitational forces in a so-called watt balance^{2}: in this scheme, the weight of a test mass is compared with the force generated by a coil, the electric power of which is measured very accurately by making use of the Josephson and quantum-Hall effects. The number chosen for the numerical value of the Planck constant will be such that at the time of adopting the definition, the kilogram is equal to the mass of the international prototype currently used for the definition of mass, within the uncertainty of the combined best estimates of the value of the Planck constant at that moment. These estimates are calculated regularly by the CODATA task group on fundamental constants^{3}. (The mission of CODATA, the Committee on Data for Science and Technology, is to improve the quality, reliability, management and accessibility of data of importance to all fields of science and technology.) Subsequently, the mass of the international prototype will become a quantity to be determined experimentally.

The impact of fixing the numerical value of the elementary charge is that the ampere will become the electric current that corresponds to the flow of 1/(1.6021766208 × 10^{−19}) elementary charges per second. Electrical quantities (such as voltage, current and resistance) will be defined by fixing the value of *e* (and Δ*ν*(^{133}Cs)_{hfs}) instead of the permeability of vacuum *μ*_{0} (which will have an uncertainty equal to that of the fine structure constant *α* = *μ*_{0}*e*^{2}*c*/2*h*). The conventional (defined) values of the Josephson and von Klitzing constants *K*_{J-90} and *R*_{K-90} will no longer be needed; these were introduced because they allowed a more precise realization of the electrical units than via today's definition of the ampere. In the future, the numerical values of the Josephson and von Klitzing constants *K*_{J} and *R*_{K} will be fixed in terms of the constants *e* and *h* (via the relationships *K*_{J} = 2*e/h* and *R*_{K} = *h/e*^{2}). In particular, the volt and the ohm will be directly realizable, thus making the quantum realizations of the electrical units consistently embedded in the new SI. In fact, there will be a step change in the electrical units realized from quantum standards when the numerical values *K*_{J-90} and *R*_{K-90}, which have been in use for more than 20 years, are abrogated in favour of the new fixed numerical values.

Fixing the numerical value of the Boltzmann constant means that the kelvin is equal to the change of thermodynamic temperature that results in a change of thermal energy *kT* by 1.38064852 × 10^{−23} J. Today's definition of the kelvin is based on an exact temperature value assigned to the triple point of water^{4}. After redefinition, this value will exhibit an uncertainty equal to that of the (currently experimentally determined) Boltzmann constant. With the new definition, it is evident that thermodynamic temperature can be realized directly at any point in the scale without referring to the singular temperature of the triple point of water. At present, thermometry relies on international temperature scales (ITSs), which were developed to give results that are in close agreement with the thermodynamic temperature, and are derived from a series of temperature fixed points (and interpolations between them) that are given conventional values approximating the corresponding thermodynamic temperatures^{5}. Deviations between an ITS — the current ITS was agreed upon in 1990 — and the corresponding thermodynamic temperatures are made available to the user community by the Consultative Committee for Thermometry (CCT). It is expected that the new route with direct traceability to the SI will initially only be used in temperature ranges where primary thermometric methods offer lower uncertainties than ITS-90^{6} (for example, below 20 K and above 1,300 K).

Fixing the numerical value of the Avogadro constant means that the mole is the amount of substance of a system that contains 6.022140857 × 10^{23} specified elementary entities. The value for *N*_{A} will be based on the result of the XRCD experiment. The current definition of the mole defines the value of the molar mass of carbon-12, *M*(^{12}C), as exactly 0.012 kg mol^{−1}. In the future, *M*(^{12}C) will no longer be exactly defined but will have an associated uncertainty equal to that of the molar Planck constant *N*_{A}*h*, which is far smaller than that achieved in any practical chemical measurement^{7}.

## Requirements for the defining constants

To guarantee minimal changes when adopting the new definitions, the Consultative Committee for Mass and Related Quantities (CCM) passed a recommendation^{8} based on a review published in 2010^{9} to impose quantitative requirements on the results of determinations of *h*. Among others, the following essential conditions must be met: (1) at least three independent experiments, including watt balance and XRCD experiments, should yield consistent values of the Planck constant with relative standard uncertainties not larger than 5 parts in 10^{8}; and (2) at least one of these results should have a relative standard uncertainty not larger than 2 parts in 10^{8}. Here and in the following all uncertainty values given are relative standard uncertainties.

At the time of publication of the 2010 CODATA evaluation of the fundamental constants, there was a significant discrepancy between the available experimental values for the Planck constant (Fig. 2)^{3}. This concerned values (NIST-98 and NIST-07) obtained at NIST (National Institute of Standards and Technology, USA) with different watt balance experiments^{10,11} and a value (IAC-11) of the International Avogadro Collaboration based on the XRCD method^{12}.

An extensive review of the NIST watt balance data led to a revised value of the Planck constant published in 2014^{13} with an increased uncertainty (compared with NIST-07) of 4.5 parts in 10^{8}. The result from NRC (National Research Council, Canada), also published in 2014^{14}, has an uncertainty of only 1.8 parts in 10^{8}, which meets the second CCM condition. Most recently, NIST researchers have reported a single value (NIST-15) for *h* with an uncertainty of 5.6 parts in 10^{8} based on all the data obtained using their current watt balance^{15}, thereby replacing their values reported in 2007 and 2014 (marked by grey dots in Fig. 2). The value obtained with an earlier apparatus (NIST-98) stands as an independent result.

In 2015 the IAC collaboration reported a new value (IAC-15) with an uncertainty of 2.0 parts in 10^{8}, which also fulfils the second CCM condition^{16}. Moreover, IAC researchers successfully addressed issues concerning the correlation between the 2011 and 2015 IAC values, thereby allowing both sets of data to be used for the 2014 adjustment. All this led to an uncertainty of only 1.2 parts in 10^{8} for the 2014 adjusted value of *h*.

The CCT similarly imposed requirements on the value of the Boltzmann constant. They recommended two conditions^{17} that are to be met before the new definition can be adopted: (1) the uncertainty of the adjusted value of *k* should be less than 1 part in 10^{6}; and (2) the determination of *k* should be based on at least two fundamentally different methods, of which at least one result for each must have an uncertainty of less than 3 parts in 10^{6}.

Measured values of *k* taken into account in the 2014 CODATA adjustment^{18} are shown in Fig. 3. The 2010 CODATA adjustment^{3} had an uncertainty of 9.1 parts in 10^{7}, which fulfils the first condition of the CCT. The second condition was not met at all, as only experiments at NIST, LNE (Laboratoire National de Métrologie et d'Essais, France) and NPL (National Physical Laboratory, UK) — all based on acoustic gas thermometry — achieved uncertainties below 3 parts in 10^{6} (ref. 19). Based on these results, and thanks to the resolving of a discrepancy between the values of LNE-11 and NPL-13^{20}, the uncertainty of the 2014 adjusted value of *k* has been significantly reduced to 5.7 parts in 10^{7}. Experiments using dielectric-constant gas thermometry developed at PTB (Physikalisch-Technische Bundesanstalt, Germany)^{21} and noise thermometry at NIM (National Institute of Metrology, China)^{22} achieved uncertainties close to 4 parts in 10^{6}. These two experiments are certainly candidates for meeting the second CCT requirement^{23}.

## Summary

Encouraging progress has been made in lowering the uncertainties of *N*_{A}, *h* and *k* to the extent that the conditions for redefinition are either met or within reach. Moreover, new results for *h* and *N*_{A} are expected to be published by NIST and the IAC collaboration, and for *k* by PTB and NIM, before the deadline (1 July 2017) set by the Consultative Committee for Units (CCU) for data that will be used to fix the numerical values of the constants for the SI redefinitions. A number of coordination and communication initiatives have been launched, such as a CCM–CCU roadmap for the kilogram^{26} and a new-SI awareness campaign initiated by the CIPM (Comité International des Poids et Mesures, France).

Redefining the SI units by fixing the numerical values of natural constants — as suggested by Max Planck in 1900^{27} — will have far-reaching benefits for innovations in industry, serving society and fostering science and research. In Planck's visionary words, these units will “necessarily retain their validity for all times and cultures, even extra-terrestrial and nonhuman,” meaning that they are stable and realizable everywhere. New primary methods are emerging for measuring, for example, masses at the milligram level or temperature over extended scales, and most importantly, uncertainties in electrical metrology will be significantly reduced. There is no doubt that these redefinitions will open the door to continuous technical developments towards ever-decreasing uncertainties in the realization of the units, without the need for further redefinition.

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### Joachim Fischer and Joachim Ullrich are at the Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, 10587 Berlin, Germany, and at the Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany

- Joachim Fischer
- & Joachim Ullrich

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