il riassetto del si t i t i l di itàsistema internazionale...

Il riassetto del Il riassetto del Si t I t i l di itàSi t I t i l di itàSistema Internazionale di unitàSistema Internazionale di unità

W. Bich, INRIM, TorinoW. Bich, INRIM, Torino

Università di Bologna, 3 maggio 2011Università di Bologna, 3 maggio 2011

A unique system of unitsA unique system of units1788 France: about 2000 units among which 200 different1788, France: about 2000 units, among which 200 different

livres!

1789: “Throughout the whole kingdom there should be but1789: Throughout the whole kingdom there should be but one code of laws, one system of weights and measures”.(From the cahier de doléances of the Nobility of Blois)

1795, 7th April: French law on “republicans” weights and measures

1840, 1st January: metric system in France

1875: The Metre Convention1875: The Metre Convention

1960: International System of Units (SI)

2Università di Bologna, 3 maggio 2011

The present SIThe present SI

Seven base units for as many quantities

Quantity Unit Symbol

length metre m

kil kmass kilogram kg

time, duration second s

l t i t Aelectric current ampere A

thermodynamic temperature kelvin K

luminous intensity candela cdluminous intensity candela cd

amount of substance * mole mol

3

* From 1971

Università di Bologna, 3 maggio 2011

Kinds of definitionsKinds of definitions

B d t f t t d dBased on an artefact standard:

The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram.

kg = m(K)kg m(K)



B d t l t d dBased on a natural standard:

The second is the duration of 9 192 631 770 periods of the di ti di t th t iti b t th tradiation corresponding to the transition between the two

hyperfine levels of the ground state of the caesium 133 atom.atom.

It follows that the hyperfine splitting in the ground state f th i 133 t i tlof the caesium 133 atom is exactly

9 192 631 770 hertz, ν (hfs Cs) = 9 192 631 770 Hz.



B d f d t l t t IBased on a fundamental constant I:

The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a

dsecond.

It follows that the speed of light in vacuum is exactlyIt follows that the speed of light in vacuum is exactly

299 792 458 metres per second, c0 = 299 792 458 m/s.


Kinds of definitionsKinds of definitionsBased on a fundamental constant II:The ampere is that constant current which, if maintained in

two straight parallel conductors of infinite length, of li ibl i l ti d l d 1 tnegligible circular cross-section, and placed 1 metre

apart in vacuum, would produce between these conductors a force equal to 2 x 10–7 newton per metre ofconductors a force equal to 2 x 10 newton per metre of length.

Wh i th t it f fWhere is the newton, unit of force.

It follows that the magnetic constant, μ0, is exactly

74p x 10–7 henries per metre, μ0 = 4p x 10–7 H/m.


Base units in the present SIBase units in the present SIBase quantities are q

conventionally regarded as independent base

m

kgindependent, base units are not:

kgmol

scd

AK

Università di Bologna, 3 maggio 2011 8

Derived unitsDerived units

D i d it d fi d d t f f thDerived units are defined as products of powers of the base units. When the product of powers includes no numerical factor other than one, the derived units arenumerical factor other than one, the derived units are called coherent derived units.

Some of the coherent derived units in the SI are given special namesspecial names


Some derived quantitiesSome derived quantitiesQuantity Unit Symbol In terms of

other SI unitsIn terms of SI base unitsother SI units base units

frequency hertz Hz s-1

f 2force newton N m kg s-2

energy, work, amount of heat

joule J N m m2 kg s–2

power, radiant flux watt W J/s m2 kg s–3

electric charge, t f l t i it

coulomb C s Aamount of electricityelectric potential difference,

l t ti f

volt V W/A m2 kg s–3 A–1

electromotive forceelectric resistance ohm Ω V/A m2 kg s–3 A–2


Areas of improvementAreas of improvement

Structure of the systemStability of the mass unitStability of the mass unitUncertainty of fundamental constantsSituation of electrical units


Structure of the systemStructure of the systemMany of the present definitions are circular (base units areMany of the present definitions are circular (base units are

defined in terms of derived units), and/or incomplete (they are defined in terms of another base unit)


Stability of the mass unit

Th kil i th itThe kilogram is the unitof mass; it is equal to the mass of the internationalmass of the internationalprototype of the kilogram

(Third CGPM, 1901)

(By courtesy of the BIPM)

13

(By courtesy of the BIPM)


G. Girard, Metrologia, 1994, 31


A provisional solutionA provisional solution

• The CIPM declared that, pending further research, thereference mass of the international prototype is thatp ypimmediately after cleaning and washing by a specifiedmethod (PV, 1989, 57, 104-105 and PV, 1990, 58, 95-97).

(Brochure SI, 8th ed., 2006)


Uncertainty of fundamental constants

M t f d t l t t i t llMost fundamental constants are experimentally determined in terms of SI units.

Therefore, their recommended SI values have an i t d t t i t dassociated measurement uncertainty and are

periodically updated.


Present situationQuantity Symbol Unit Rel. std.

unc. ur

speed of light in vacuum c,c0 m s−1 exactp g , 0

mass of the Prototype kg m(K) kg exactfrequency of hyperfine splitting of 133Cs

ν (hfs Cs)

s−1 exact133Cs Cs)magnetic constant μ0 N A−2 exactmolar mass of 12C M(12C) g/mol exacttemperature of water triple point Ttpw K exactspectral luminous efficacy K cd sr/W exactPlanck constant h J s 5 0 x 10−8Planck constant h J s 5.0 x 10 8

elementary charge e C 2.5 x 10−8

electron mass me kg 5.0 x 10−8

Avogadro constant NA, L mol-1 5.0 x 10−8

Boltzmann constant k J K-1 1.7 x 10−6


Situation of electrical unitsSituation of electrical unitsThe volt and the ohm are nowadays represented in a highly

d ibl b f J h dreproducible way by means of Josephson and quantum Hall standards, depending on the Josephson constant

and the von Klitzing constant

The volt and the ohm (as the other electrical units) realized in terms of the ampere (involving electromechanicalin terms of the ampere (involving electromechanical experiments) are affected by much higher uncertainties.


Early proposalEarly proposal

1963


2005

2006


The 23rd General Conference 2007

On the possible redefinition of certain base units of the International System of Units (SI)y ( ) The 23rd General Conference, (omissis)

d th t N ti l M t l I tit t d th BIPMrecommends that National Metrology Institutes and the BIPMpursue the relevant experiments so that the International Committee can come to a view on whether it may be possible to redefine the kilogram, the ampere, the kelvin, and the mole using fixed values of the fundamentalampere, the kelvin, and the mole using fixed values of the fundamental constants at the time of the 24th General Conference (2011), (omissis) and requests the International Committee to report on these issues to the 24th General Conference in 2011 and to undertake whatever preparations are considered necessary so that, if the results of experiments are found to be satisfactory and the needs of users met, formal proposals for changes in the definitions of the kilogram ampere the kelvin and mole can be put tothe definitions of the kilogram, ampere, the kelvin and mole can be put to the 24th General Conference.


The essence of the proposalThe essence of the proposal

T i ti ll t SI l t it blTo assign conventionally exact SI values to seven suitable invariants of nature.

Thi i li d fi iti f th b d fThis implies re-definition of the seven base and of any conceivable derived unit in terms of the seven invariants.

Whi h l ?Which values?

The best available (CODATA) at the moment of the re-d fi itidefinition.

Which invariants?

Debate is still open, but broad consensus exists on a (sub)set of invariants.


The proposed invariantsThe proposed invariants


The future scenarioQuantity Symbol Unit Rel. std. unc. ur

speed of light in vacuum c c m s−1 exactspeed of light in vacuum c,c0 m s exactmass of the Prototype kg m(K) kg £ 2.0 x 10−8

frequency of hyperfine splitting of 133C

ν (hfs C )

s−1 exact133Cs Cs)magnetic constant μ0 N A−2 2.5 x 10−8

molar mass of 12C M(12C) g/mol £ 5.0 x 10−8( ) gtemperature of water triple point Ttpw K £ 1.7 x 10−6

spectral luminous efficacy K cd sr/W exactPlanck constant h J s exactelementary charge e C exactelectron mass me kg £ 1.4 x 10−9electron mass me kg £ 1.4 x 10Avogadro constant NA, L mol-1 exactBoltzmann constant k J K-1 exact


Resistenze alle ridefinizioni proposte Difficoltà di ordine sperimentale

Difficoltà di ordine concettuale


Difficoltà di ordine sperimentalepDeterminazione della costante di Planck h (anche attraverso la costante di Avogadro NA): decisiva per ridefinire il kilogrammo


Difficoltà di ordine sperimentalepDeterminazione della costante di Boltzmann k : decisiva per ridefinire il kelvin

m atomic mass (40Ar)γ = 5/3 for gas of single atomsu0

2 = γ kT / mtransducer

receiver

γ g gu0 speed of sound

u0 γ kT / m

Obiettivo:Confermare (o contraddire) l'attuale unico dato di riferimento con

dunico dato di riferimento con accuratezza confrontabile (1,7 × 10-6)

Non ancora raggiuntoNon ancora raggiunto

risonanza elettromagneticag

risonanza acustica32


Difficoltà di ordine concettualeDifficoltà di ordine concettuale

Resistenza ad abbandonare la logica tradizionale del campione:

l'unità è basata sul campione riproducibile con la minima incertezzala definizione dell'unità “insegue” l'evoluzione tecnologica dei campionila realizzazione dell'unità non garantisce la coerenza tra le unità

a favore di una logica di sistema:a a o e d u a og ca d s ste al'unità è basata su un invariante fondamentale delle teorie scientifichela definizione è resa obsoleta solo dal decadimento di quelle teorie la realizzazione dell'unità garantisce intrinsecamente la coerenza del sistemala realizzazione dell'unità garantisce intrinsecamente la coerenza del sistemala ridefinizione può comportare un'aumento dell'incertezza di realizzazione


Difficoltà di ordine concettuale

La difficoltà può essere superata:definendo l' unità con riferimento a una costante

fondamentale incorporando la realizzazione dell'unità nel miglior campioneincorporando la realizzazione dell unità nel miglior campionemantenendo distinte le componenti di incertezza che intervengono

separatamente nella pratica metrologica (misurazioni dirette o indirette)

nuova realizzazionedell'unitàu incertezza del campione

indirette)

uS

uS I incertezza di realizzazione

uS incertezza del campione(tarature, misurazioni dirette)

uSIS I

(misurazioni indirette)


Difficoltà di ordine concettualeDifficoltà di ordine concettuale

Acquisire un consenso sul tipo di definizione da adottare per vecchie e nuove definizioni di unità:vecchie e nuove definizioni di unità:

Definizione del tipo “explicit-unit” (tradizionale)

Definizione del tipo “explicit constant” (CCU)Definizione del tipo explicit-constant (CCU)

Definizione del tipo “explicit-reference” (nuova proposta)


Explicit-unit definitions

Used so far in the SI documents.

Example:The metre is the length of path travelled by light in vacuum during a time interval of 1/299 792 458 of a secondinterval of 1/299 792 458 of a second

Advantages: • Familiarity

• Easy to understand and visualize

Disadvantage: It must indicate an ideal experimental context to identify in words the relation between constant and unit, which can ,become of quite difficult comprehension when involving quantum physics

Comment: The present SI survived for a long time with this disadvantage

36

Comment: The present SI survived for a long time with this disadvantage.


Explicit-constant definitionsA novelty of the draft Ch 2 SI Brochure (CCU/10-3.1, 2010)

Example:The metre, unit of length, is such that the speed of light in vacuum is equal to exactly 299 792 458 metres per second.

Claimed advantages:

• Simplicity• Simplicity

• “... draws attention to the implications of the definition for fundamental physics”

• No reference to, or suggestion of a specific experiment for the realization

p y


Explicit-constant definitions.Diffi ltiDifficulties

A good step towards rationalization (all definitions have the same format).

However, there are some difficulties:

• Incompleteness

• Circularity

• Bi-univocal association of a unit with a constant• Bi-univocal association of a unit with a constant

• Mysterious wording

All these difficulties stem from a common weakness, intrinsic

38

in the choosen kind of definition (more on this later).


Explicit-constant definitions.I l tIncompleteness

Incompleteness is the main concern. Excluding the second, which is thep g ,first definition in the list, and the mole, which is now independent of thekilogram, each definition implies knowledge of at least one of thepreceding definitions.

Examples: The metre, unit of length, is such that the speed of light in vacuum is equal to exactly 299 792 458 metres per second

This definition is not self-consistent, as requires the second to be

vacuum is equal to exactly 299 792 458 metres per second.

separately defined.

Taken alone, the definition establishes a necessary and not sufficient condition.

There are infinitely many metres satisfying the definition, depending on how the second is defined

39

depending on how the second is defined.


Explicit-constant definitions.I l t f th l

The kilogram, unit of mass, is such that the Planck constant is equal

Incompleteness - further exampleg , , q

to exactly 6.626 068 96 ×10−34 joule second.

Al thi d fi iti i t lf i t t it i li k l d fAlso this definition is not self-consistent, as it implies knowledge of the dimension of the joule and the definition of the second.

Again taken alone the definition establishes aAgain, taken alone, the definition establishes a necessary and not sufficient condition.

Th 2 kil ti f i th d fi itiThere are ∞2 kilograms satisfying the definition, depending on how the second and the metre are defined.

The change in the order of definitions, intended to avoid defining a unit interms of another which is defined later, is a necessary but not sufficientmeasure The result is still unsatisfactory

40

measure. The result is still unsatisfactory.


Explicit-constant definitions.pCircularity

A base unit/quantity is defined in terms of a unit/quantity derived from itself.

Examples:The metre unit of length is defined in terms of metre perThe metre, unit of length, is defined in terms of metre per second, unit of speed, the time derivative of length.

Th kil it f i d fi d i t f j l dThe kilogram, unit of mass, is defined in terms of joule second, unit of action, thus involving the joule, a unit derived from the kilogram.

This is a serious concern, especially in a system in which the distinction is maintained between base and derived units.


Explicit-constant definitions.Explicit constant definitions.Bi-univocal association of a unit

ith t twith a constant With the present form of explicit-constant definitions, an unnecessary and potentially misleading bi-univocal relationship between a given unit and a constant is established.

The metre and the speed of light, the kilogram and the Planck constant, the l d th A d t t th d th h f th l tmole and the Avogadro constant, the ampere and the charge of the electron

etc.

In almost every case, it must be intended that also some other constants have been fixed by other definitions.


Explicit constant definitionsExplicit-constant definitions.Mysterious wording

(a comparatively minor concern, half aesthetic, half semantic)

The international system of units, the SI, is the system of units scaled so that ..... (from CCU/10-3.1)

Th it f i h th t i l t tl

And, again from CCU/10-3.1, a format common to all definitions:

The …, unit of …, is such that …., is equal to exactly ….

In the lack of any indication about an experimental context the relationIn the lack of any indication about an experimental context, the relation between the unit and the relevant constants is completely left to the intuition of the reader, which finds the definition unsatisfactory.


Explicit constant definitionsExplicit-constant definitions.Mysterious wording

(a comparatively minor concern, half aesthetic, half semantic)

I personally have problems in understanding how a system can be “scaled”, and how a unit alone can be “such that”… something else has a given valuehas a given value.

These sentences can perhaps be grasped intuitively, but would not stand a rational or semantic analysisnot stand a rational or semantic analysis.

Not casually, they cannot be written as explicit mathematical relations.

I anticipate considerable problems in teaching.


Why these difficulties?Why these difficulties?

All the difficulties arise because in the definitions the base units [QB]i andthe associated constants Ci refer to different quantities. For example, thekilogram is defined in terms of an action.

It might be said that the international system of units SI (or perhaps theunderlying system of quantities ISQ) and the international system ofunderlying system of quantities ISQ) and the international system ofconstants are based on two different sets of base quantities.

This causes incompleteness and circularity.

Th b di fl t th li ti i l d i d fi i th itThe obscure wording reflects the complication involved in defining the unit of a quantity in terms of units of different quantities.


Is there a possible way out?p y

A further type of unit definition is proposed, which could becalled “explicit-reference”.

It derives from a systemic approach, with no necessaryassociation of the unit with a single constant.

Also the distinction between base and derived quantitiesbecomes unnecessary and could be dropped.


ProposalThe proposed structure for any unit definition:

Proposal

"The unit of [quantity], [special name], is equal to [numericalcoefficient] times [monomial expression of the reference constants]"

The monomial expression of the reference constants is a constantquantity homogeneous with the unit being defined.q y g g

With the present definitions the construction of the unit system is sequential (each successive definition fixes the value of an additional constant)

With the proposed definitions, the construction is systemic (from a set of

(each successive definition fixes the value of an additional constant).

t t e p oposed de t o s, t e co st uct o s syste c ( o a set ostatements on the reference constants and their fixed values to a system of unit definitions).


ProposalProposalSet of initial statements (for constants proposed by CCU):

the ground state hyperfine splitting frequency of the caesium 133 atom Δν(133Cs)hfs is exactly 9 192 631 770 hertz;

the speed of light in vacuum c is exactly 299 792 458 metres per second;

th Pl k t t h i tl 6 626 068 96 10 34 j l dthe Planck constant h is exactly 6.626 068 96 ×10–34 joule second;

the elementary charge e is exactly 1.602 176 487 ×10–19 coulomb;

the Boltzmann constant k is exactly 1.380 650 4 ×10–23 joule per kelvin;

the Avogadro constant N is exactly 6 022 141 79 ×1023 per mole;the Avogadro constant NA is exactly 6.022 141 79 ×10 per mole;

the spectral luminous efficacy Kcd of monochromatic radiation of frequency 540 ×1012 hertz is exactly 683 lumen per watt

48

540 ×10 hertz is exactly 683 lumen per watt.


Proposal

Dn(133Cs)hfs = 9 192 631 770 s–1

c = 299 792 458 m s–1

The same statements in form of equations are:

h = 6.626 068 96 ×10−34 m2 kg s−1

e = 1.602 176 487 ×10−19A s

k 1 380 650 4 10−23 2 k −2 K 1k = 1.380 650 4 ×10 23 m2 kg s 2 K–1

NA= 6.022 141 79 ×1023 mol–1

K = 683 cd m−2 kg−1 s3

Whose solution yields the explicit-reference definitions:

the unit of time, second, is equal to9 192 631 770 times 1 / Dn(133Cs)hfs;the unit of time, second, is equal to 9 192 631 770 times 1 / Dn( Cs)hfs;the unit of length, metre, is equal to 30.663 318 99 times c / Dn(133Cs)hfs;

the unit of mass, kilogram, is equal to 1.475 521 66 â 1040 times h Dn(133Cs)hfs / c2; 8 133the unit of electric current, ampere, is equal to 6.789 687 44 â 108 times e Dn(133Cs)hfs;

the unit of temperature, kelvin, is equal to 2.266 668 times h Dn(133Cs)hfs / k; the unit of amount of substance, mole, is equal to 6.022 141 79 â 1023 times 1 / NA;, , q A;

the unit of luminous intensity, candela, is equal to 2.615 â 1010 times h K Dn(133Cs)hfs2."


ProposalProposal

Whichever set of base constants is selected, a monomial combination of them can easily be found which has the dimension of an elementary length an elementary mass and so on for all the SIQ's either base orlength, an elementary mass and so on for all the SIQ s, either base or derived quantities.

The monomial expressions of constants and the numerical coefficients linking them to the corresponding units can be found solving a g p g ggeneralised system, published together with partial solutions (See Cabiati and Bich, Metrologia 46, 2009).


CommentsComments

Th di d f ili di t th t i i l d fi iti• The wording sounds familiar, corresponding to the most original definitions

• Each definition is self-consistent (no need for other definitions)

• Circularity is avoided

• The relation between the unit and the constants involved in eachdefinition is clearly shown in elementary algebraic terms, independent ofany possible physical interpretationy p p y p

• The definitions are rational and intuitive at the same time. This greatlyhelps understanding and teaching themhelps understanding and teaching them


Straying on forbidden grassStraying on forbidden grass…

This way of defining cruelly brings into light how all units except the moledepend on Δν(133Cs)hfs, definitely not a universal constant, rather aninvariant of nature which is ultimately nothing else than a standard of the

f ( f f )same kind of the Prototype kilogram (the former natural, the latter artificial).

Should a better transition be found, all these definitions would need to bechanged accordingly. This is not a proof of the superiority of the explicitconstant definitions On the contrary the latter simply hides the problemconstant definitions. On the contrary, the latter simply hides the problem,does not solve it.


St i f biddStraying on forbidden grass…

The natural invariant Δν(133Cs)hfs is inadequate as base constant, andwould be better placed as a recommended transition frequency in amise en pratique.

An indirect proof of the inadequacy of Δν(133Cs)hfs as a baseconstant is the large value of the numerical coefficient for thekilogram. This value is questioned in some quarters as unphysical.

Clearly, it is not appropriate to attach a special physical meaning toy pp p p p y gcombinations of universal constants and a specific natural invariant.


Straying on forbidden grassStraying on forbidden grass…The adequate base constant to complete the transformation of the SIq pwould be the electron mass me.

Including this important constant within the reference set of the SI,

Several other constants would benefit from an exact valueof the electron

the definition of the kilogram would very naturally beThe kilogram, unit of mass, is equal to exactly 1,097 769 29 × 1030 me .

Several other constants would benefit from an exact valueof the electronmass and other unit definitions from the introduction of that base constant.In particular, the Compton wavelength h/(mec) and Compton frequencym c2/h would become the reference quantities for the length and time unitsmec /h would become the reference quantities for the length and time unitsrespectively.

Contrary to their belief, the time&frequency community would not eventi th h i th d fi iti f th d B t thi i diff tnotice the change in the definition of the second. But this is a different

story…


Sviluppi recentiSviluppi recenti

(http://www bipm org/en/si/new si/)(http://www.bipm.org/en/si/new_si/)


Ritocchi… (http://www.bipm.org/en/si/new_si/)

The metre, m, is the unit of length; its magnitude is set by fixingg g y gthe numerical value of the speed of light in vacuum to be equalto exactly 299 792 458 when it is expressed in the unit m s-1.

The kilogram, kg, is the unit of mass; its magnitude is set byfixing the numerical value of the Planck constant to be equal toexactly 6.626 06X x10−34 when it is expressed in the unit s−1 m2 kg,y p g,which is equal to J s.


Nuovi dati…

As-stated uncertainties – May 2011

7

5

06

METAS watt bal.

3

2606

7 Js

) 10

NPL watt bal. 28Si Avogadro

1(h-6

,6 Weighted mean

NIST watt bal.

Si Avogadro

-1

57

-3


… e nuove resistenze (http://www.bipm.org/en/si/new_si/)

RECOMMENDATION G 1 (2010) Considerations on a new definition of the kilogram The Consultative Committee for Mass and Related Quantities (CCM) …omissis

recommends

that the following conditions be met before the kilogram is redefined in terms ofthat the following conditions be met before the kilogram is redefined in terms of fundamental constants:

1. at least three independent experiments, including work both from watt balance and from International Avogadro Coordination projects, yield values of the relevant constants with relative standard uncertainties not larger than 5 parts in 108. At least one of these results h ld h l ti t d d t i t t l th 2 t i 108should have a relative standard uncertainty not larger than 2 parts in 108,

2. for each of the relevant constants values provided by the different experiments befor each of the relevant constants, values provided by the different experiments be consistent at the 95 % level of confidence,

3.

58

traceability of BIPM prototypes to the international prototype of the kilogram be confirmed,


th t th CODATA d d l b d t d f th l tthat the CODATA recommended values be adopted for the relevant fundamental constants,

that the associated CODATA relative standard uncertainties be suitablythat the associated CODATA relative standard uncertainties be suitably considered when the initial uncertainty is assigned to the mass of the international prototype of the kilogram,

that a pool of reference standards be established at the BIPM to facilitate the dissemination of the new definition of the kilogram,

that the BIPM and a sufficient number of National Metrology Institutesthat the BIPM and a sufficient number of National Metrology Institutes continue to develop, operate or improve facilities or experiments that allow the realization of the kilogram to be maintained with a relative standard uncertainty not larger than 2 parts in 108.uncertainty not larger than 2 parts in 10 . that the uncertainty component arising from the practical realization of the unit be suitably taken into account.


Conclusione

Tutto è rinviato (almeno) al 2015

Grazie per l’attenzione!


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