The 2019 SI System of Units

Eric Weisstein, WRI/Wolfram|Alpha

ABSTRACT

On World Metrology Day May 20, 2019 (the 144th anniversary of the Metre Convention), four SI base units (the kilogram, ampere, kelvin, and mole) were redefined by setting the Planck constant, elementary electric charge, Boltzmann constant, and Avogadro constant to certain exact values when expressed in SI units.

This means that in addition to the second, meter, and candela, all seven base SI units now have exact definitions in terms of physical constants.

While the new definitions do not change the magnitude of any existing unit to within the precision measurable by existing techniques, the change from numeric values to exact numbers nonetheless required a significant update of unit and physical constant values and uncertainties in the Wolfram|Alpha unit and the Wolfram Language quantity systems.

These changes and their implications will be discussed and illustrated in this talk.

"PhysicalConstant" entity domain

In order to define units, it is convenient to curate and have access to detailed knowledge of physical constants on which many (and now even more) unit are based

As of Version 12, "PhysicalConstant" entities encode values, uncertainties, and other information about nearly 300 constants in directly computable form:

EntityList["PhysicalConstant"]//Length

Some "PhysicalConstant" entities:

RandomEntity["PhysicalConstant",20]

"PhysicalConstant" properties:

EntityProperties["PhysicalConstant"]

Curating values and uncertainties of physical constants

Consider the gravitational constant (one of the least precisely known fundamental constants).

Entity["PhysicalConstant", "GravitationalConstant"]["Value"]

InputForm[%]

Precision[QuantityMagnitude[%]]

EntityValue[Entity["PhysicalConstant","GravitationalConstant"],EntityProperty["PhysicalConstant","Value",{"Uncertainty"->"Around"}]]

Easy to represent uncertainties in different convenient forms:

TextGrid[Table[{unc,EntityValue[(Entity["gravitational constant", Entity["PhysicalConstant","GravitationalConstant"], "Entity[\"PhysicalConstant\", \"GravitationalConstant\"]", "physical constant"]),EntityProperty["PhysicalConstant","Value",{"Uncertainty"->unc}]]},{unc,{"Around","Interval","Precision"}}],]

Revisions of physical constants

While the physical constants are constant over time, the precision of their measurement generally increases over time.

Historical values of physical constants are curated in "PhysicalConstant" entities.

"ValueAssociation" gives all tabulated historical and measured values.

Take[vals=(Entity["gravitational constant", Entity["PhysicalConstant","GravitationalConstant"], "Entity[\"PhysicalConstant\", \"GravitationalConstant\"]", "physical constant"])["ValueAssociation"],3]//Column

TextGrid[G=Cases[Flatten[{#1,{"StandardYear","Value","StandardUncertainty"}/.#2}]&@@@(Entity["gravitational constant", Entity["PhysicalConstant","GravitationalConstant"], "Entity[\"PhysicalConstant\", \"GravitationalConstant\"]", "physical constant"])["ValueAssociation"],{source_,y_Integer,rest__}:>{source,DateObject[{y}],rest}],]

Gvals=Callout[{#2,Around[#3-Mean[G[[All,3]]],#4]},#1]&@@@G;

DateListPlot[Sort@Gvals,]

Sources of physical constant values

IAU (International Astronomical Union): General Assembly resolutions

Astronomical Almanac, GRS80, WGS84, ...

Planck collaboration values based on analysis of the European Space Agency Planck observatory; analyses released

2013 [https://arxiv.org/abs/1303.5076]

2015 [https://arxiv.org/abs/1502.01589]

2018 [https://arxiv.org/abs/1502.01589]

PDG (Particle Data Group): the Particle Physics Booklet  which is printed every two years and updated annually on the web

...others...

recently published literature not yet incorporated into standard value tabulations

CODATA (Committee on Data for Science and Technology): established in 1966 as an interdisciplinary committee of the International Council for Science which seeks to improve the compilation, critical evaluation, storage, and retrieval of data of importance to science and technology.  Its Task Group on Fundamental Constants publishes recommended values of many constants every four years [https://physics.nist.gov/cuu/Constants/]

Entity["PhysicalConstant", "GravitationalConstant"]["PrimarySource"]

Definitions of SI units

BIPM (International Bureau of Weights and Measures): intergovernmental organization established by the 1875 Metre Convention through which member states act together on measurement science and measurement standards, in particular the SI system of units

CGPM (General Conference on Weights and Measures): delegates of Member States of the BIPM meet in Paris once every four years to make recommendations on measurement and unit standards

CGPM recommendations are released through Resolutions

The entire SI standard, incorporating Resolutions, is released through the "SI Brochure" The International System of Units (SI), historically published at intervals from 4-8 years. The last one was in 2006

9th edition (2019) [https://www.bipm.org/en/publications/si-brochure/]

8th edition supplement (2014)  [https://www.bipm.org/utils/common/pdf/si_supplement_2014.pdf]

8th edition (2006) [https://www.bipm.org/utils/common/pdf/si_brochure_8.pdf]

The 2019 redefinition of SI

On May 20, 2019, the 9th edition of the "SI brochure" was released in which the SI base units were redefined

First full revision of SI since 2006

This completes the process begun in 1960 when SI was first formally published. At that time, the definition of there meter was changed from the physical prototype meter to be based on a spectral line of a krypton-86, making it derivable from a universal natural phenomenon

In the interim, a total of three SI base units were defined in terms of fundamental constants:

the 13th CGPM (1967/68) replaced the definition of the second with the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom

The 16th CGPM (1979) defined the candela in terms of the luminous efficacy of monochromatic radiation of frequency 540 THz

The 17th CGPM (1983) replaced the 1960 definition of the meter with one based on the second by giving an exact definition of the speed of light in units of meters per second

The 2019 redefinition of SI, continued

Until this year, the kilogram remained defined by a physical prototype, leaving it the only artefact upon which the SI unit definitions depended

As part of the 2019 redefinition, the international prototype kilogram was retired, definitions of the kilogram, ampere, and kelvin were replaced, and the definition of the mole was revised

With the 2019 redefinition, the SI became wholly derivable from universal natural phenomena with unit values based on fundamental physical constants

The values of the constants for the 2019 redefinition were chosen to produce no practically relevant discontinuities in the sizes of the kilogram, ampere, kelvin, or mole

Exact definition of SI units

Prior to 2019:

The speed of light c is exactly 299792458 metres per second

The unperturbed ground state hyperfine structure transition frequency of the cesium-133 atom (Subscript[Δν, Cs]) is exactly 9192631770 hertz

The luminous efficacy (Subscript[K, cd]) of monochromatic radiation of frequency 540 THz  is exactly 683 lumens per watt

Defined in 2019:

The Planck constant h is exactly 6.62607015×(10^(-34)) joule-second

The elementary charge e is exactly 1.602176634×(10^(-19)) coulomb

The Boltzmann constant k is exactly 1.380649×(10^(-23)) joule per kelvin

The Avogadro constant (Subscript[N, A]) is exactly 6.02214076×(10^23) reciprocal mole

Consequences

The SI is now constructed and defined based on seven defining physical constants

Four of the seven SI "base" units (second, meter, kilogram, ampere, kelvin, mole, candela) are redefined, though definitions of the SI "derived" units in terms of the "base" units remain the same

The designation of SI "base units" is retained but is no longer essential to define SI measures since all units are now expressed in terms of seven defining constants

While the precision of measurements of physical constants will continue to improve, their values are now set in stone when expressed in SI units.  Instead of physical constants changing values, the magnitudes of SI units will change (very slightly)

The seven base physical constants (and any expressible through combinations of them) are now exact (with identically 0 uncertainty) when expressed in SI units

While many physical constants now have exact values when expressed in SI units, some previously exact one became approximate ((Subscript[μ, 0]), (Subscript[ϵ, 0]), (Subscript[Z, 0]))

Uncertainties of other derivable constants now depend on constants that can be measured most precisely, in particular on the fine structure constant α and Rydberg constant (Subscript[R, ∞])

The new SI was incorporated into the 2018 CODATA physical constant values

Because the relationships between the SI units remained the same and the new definitions were chosen to avoid discontinuities, most conversions between, to, and from SI units did not change

Values of some non-SI units defined in terms of physical constants did change (electronvolts, faradays)

Any computational framework that distinguishes between exact and approximate physical constants requires adjustment for the 2019 changes

Consequences for the WL quantity system and "PhysicalConstant" entities

PhysicalConstant entities

"ValueAssociation" had to be expanded to include new exact constant values

"ValueAssociation" values and uncertainties had to be updated for (many) constants derivable from the fundamental ones

CODATA2018 (incorporating the 2019 SI changes) values and uncertainties needed to be added and made primary

Quantities

Quantity definitions had to be similarly modified with new values and uncertainties

The previously essentially independent physical constants and quantity definitions were streamlined and largely unified

Wolfram|Alpha

Interface and formatting heuristics needed to be modified to prevent "unexpected" appearance of exact rational numbers

Planck constant

The meaning of "exactly" in the WL

"The elementary charge e is exactly 1.602176634×(10^(-19)) coulombs"

ExactNumberQ[1.602176634×(10^(-19))]

1.602176634×(10^(-19))//Precision

1602176634×(10^(-19-9))

N[%,20]

Entity["PhysicalConstant","ElementaryCharge"]["Value"]

"The Planck constant h is exactly 6.62607015×(10^(-34))joule-second"

662607015×(10^(-34-8))

N[%,20]

Entity["PhysicalConstant","PlanckConstant"]["Value"]

Derived constants

Josephson constant

Column[Entity["PhysicalConstant","JosephsonConstant"]["EquivalentForms"],Dividers->All]

2Entity["PhysicalConstant","ElementaryCharge"]["Value"]/Entity["PhysicalConstant","PlanckConstant"]["Value"]

Entity["PhysicalConstant","JosephsonConstant"]["Value"]

UnitSimplify[%-%%]

New exact values

SIConstants=EntityList[(EntityClass["SI exact constants", EntityClass["PhysicalConstant","SIExact"], "EntityClass[\"PhysicalConstant\", \"SIExact\"]", "physical constants"])]

The new values can be summarized by retrieving the relevant properties from "PhysicalConstant" entities:

TextGrid[Join[{Thread[Style[{"constant","symbol","new exact value"},Italic]]},EntityValue[SIConstants,{"Entity","Quantity","Value"}]/.],]

Physical constants in terms of units

quantityToFramedUnits[expr_]:=expr/.Quantity[x_,u_]:>(x/.None->1)(u/.s_String:>Framed[ToLowerCase[StringDrop[s,-1]],])

definingEquation[constant_]:=constant==quantityToFramedUnits[UnitConvert[constant["Value"],"SIBase"]/."Steradians"->1]

{eq["s"]=definingEquation[(Entity["Cs\[Hyphen]133 hyperfine splitting frequency", Entity["PhysicalConstant","Cesium133HyperfineSplittingFrequency"], "Entity[\"PhysicalConstant\", \"Cesium133HyperfineSplittingFrequency\"]", "physical constant"])],eq["m"]=definingEquation[(Entity["speed of light", Entity["PhysicalConstant","SpeedOfLight"], "Entity[\"PhysicalConstant\", \"SpeedOfLight\"]", "physical constant"])],eq["kg"]=definingEquation[(Entity["Planck constant", Entity["PhysicalConstant","PlanckConstant"], "Entity[\"PhysicalConstant\", \"PlanckConstant\"]", "physical constant"])],eq["A"]=definingEquation[(Entity["elementary charge", Entity["PhysicalConstant","ElementaryCharge"], "Entity[\"PhysicalConstant\", \"ElementaryCharge\"]", "physical constant"])],eq["K"]=definingEquation[(Entity["Boltzmann constant", Entity["PhysicalConstant","BoltzmannConstant"], "Entity[\"PhysicalConstant\", \"BoltzmannConstant\"]", "physical constant"])],eq["mol"]=definingEquation[(Entity["Avogadro constant", Entity["PhysicalConstant","AvogadroConstant"], "Entity[\"PhysicalConstant\", \"AvogadroConstant\"]", "physical constant"])],eq["mol"]=definingEquation[(Entity["Avogadro constant", Entity["PhysicalConstant","AvogadroConstant"], "Entity[\"PhysicalConstant\", \"AvogadroConstant\"]", "physical constant"])],eq["cd"]=definingEquation[(Entity["monochromatic radiation at 540 THz luminous efficacy", Entity["PhysicalConstant","MonochromaticRadiation540THzLuminousEfficacy"], "Entity[\"PhysicalConstant\", \"MonochromaticRadiation540THzLuminousEfficacy\"]", "physical constant"])]}

Units in terms of physical constants

Define the set of equations and solve for the seven "base" units as expressed through the constants.

newSIEquations=Thread[eq[{"s","m","kg","A","K","mol","cd"}]];

unitsThroughConstants=Sort@Solve[newSIEquations,DeleteDuplicates@Cases[newSIEquations,_Framed,∞]][[1]]

TextGrid[List@@@unitsThroughConstants,]

Units in terms of physical constants, factored form

factoredIntegerForm[i_Integer]:=Times@@@(HoldForm@@{FactorInteger[i]}/.{b_Integer,e_}:>b^e/.HoldPattern[b_^1]:>b)
factoredIntegerForm[Rational[num_,den_]]:=factoredIntegerForm[num]/factoredIntegerForm[den]

TextGrid[Prepend[{#1,#2/.e_Entity:>FromEntity[e]/.Quantity[x_,u_]:>Quantity[factoredIntegerForm[x],u]}&@@@unitsThroughConstants,Thread[Style[{"SI base unit","value"},Gray,Italic]]],
]

Units in terms of physical constants, derived units

Joules

UnitConvert[Quantity["Joules"]]

%/.q_Quantity:>quantityToFramedUnits[q]

%/.unitsThroughConstants

Newtons

UnitConvert[Quantity["Newtons"]]

%/.q_Quantity:>quantityToFramedUnits[q]

%/.unitsThroughConstants

Teslas

UnitConvert[Quantity["Tesla"]]

%/.q_Quantity:>quantityToFramedUnits[q]

%/.unitsThroughConstants

Uncertainties now have different contributors

Uncertainties of physical constants now depend on the uncertainties of the combination of most precisely measurable physical constants through which they can be obtained.  These frequently turn out to be the fine structure constant α and the Rydberg constant (Subscript[R, ∞]).

Column[ids=Entity["PhysicalConstant","ElectronMass"]["EquivalentForms"],Dividers->All]

me=ids[[1]]/.Quantity[v_,((u_)^(p_.))]:>v ((Entity["PhysicalConstant",u])^p)

ents=Cases[me,_Entity,∞]

EntityValue[ents,"RelativeUncertainty"]

{2,1,1,1}.%

Entity["PhysicalConstant","ElectronMass"]["RelativeUncertainty"]

Comparing uncertainties in constants after the SI redefinition

Summary

The 2019 SI redefinition provides a reformulation of the unit system used by (almost) every nation on Earth

The changes are structurally important in eliminating the use of an arbitrary reference standard in favor of physical constants whose values are fundamental and can be measured reproducibly (independent of time or location) by scientific experiment

The vast majority of unit users (both technical and non-technical) will notice no effect from the changes

Because the Wolfram Language treats expressions symbolically, WL users are among a select few of technical people who might notice the changes

In particular, the Quantity system in the Wolfram Language and unit framework in Wolfram|Alpha were retooled to support the 2019 SI changes

These updates were made available in existing Mathematica 12 installations via an automatic paclet update in June 2019

While physical constant values and unit conversions involving rational numbers having potentially large numerators and denominators at first may seem bizarre, they are a consequence of the SI 2019 redefinition and follow from the (good) decision to keep the magnitudes of SI units experimentally indistinguishable from their earlier definitions

Thanks

Michael Trott
Nick Lariviere
José Martín-García
Björn Zimmermann
Rachelle Bergmann

Epilog: Uncertainties for "all" constants after the SI redefinition

To examine the relative uncertainties of physicals constants, first retrieve the values for all constants, then remove constants with exact values, unknown uncertainties and the constants corresponding to atomic and natural units.

TextGrid[Take[uncertaintydata=DeleteCases[SortBy[EntityValue["PhysicalConstant",{"Entity","Symbol","RelativeUncertainty"}],Last],],20],]

Construct a new list of data in which relative uncertainties are converted to precisions. Also single out a few important constants to ensure they are labeled in the resulting plot using the LabelVisibility option.

labels={(Entity["fine structure constant", Entity["PhysicalConstant","FineStructureConstant"], "Entity[\"PhysicalConstant\", \"FineStructureConstant\"]", "physical constant"]),(Entity["gravitational constant", Entity["PhysicalConstant","GravitationalConstant"], "Entity[\"PhysicalConstant\", \"GravitationalConstant\"]", "physical constant"])};

precisiondata=Callout[-Log10[2#3],#1,LabelVisibility->If[MemberQ[labels,#1],All,Automatic]]&@@@uncertaintydata;

Now plot the sorted precisions of tabulated constants.

ListPlot[precisiondata,]

The precisions to which physical constants are known can also be nicely viewed as a word cloud, where symbols are shown with size weighted by the precision to which they are known (and where the precision has been scaled by cubing it to better emphasize the variations).

WordCloud[(-Log10[#3]^3->Tooltip[Style[#2/.,
],#1])&@@@uncertaintydata,MaxItems->∞]