livres! 1960: International System of Units (SI)

Transcription

1 Il riassetto del Sistema Internazionale di unità W. Bich, INRIM, Torino

2 A unique system of units 1788, France: about 2000 units, among which 200 different livres! 1789: 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, 7 th April: French law on republicans weights and measures 1840, 1 st January: metric system in France 1875: The Metre Convention 1960: International System of Units (SI) 2

3 The present SI Seven base units for as many quantities Quantity Unit Symbol length metre m mass kilogram kg time, duration second s electric current ampere A thermodynamic temperature kelvin K luminous intensity candela cd amount of substance * mole mol * From

4 Kinds of definitions Based on an artefact t standard: d The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. kg = m(k) 4

5 Kinds of definitions Based on a natural standard: d The second is the duration of periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. It follows that the hyperfine splitting in the ground state of the caesium 133 atom is exactly hertz, ν (hfs Cs) = Hz. 5

6 Kinds of definitions Based on a fundamental constant t I: The metre is the length of the path travelled by light in vacuum during a time interval of 1/ of a second. It follows that the speed of light in vacuum is exactly metres per second, c 0 = m/s. 6

7 Kinds of definitions Based on a fundamental constant II: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible ibl 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 of length. Where is the newton, unit of force. It follows that the magnetic constant, μ 0, is exactly 4p x 10 7 henries per metre, μ 0 = 4p x 10 7 H/m. 7

8 Base units in the present SI Base quantities are conventionally regarded as independent, base units are not: mol m kg cd s K A 8

9 Derived units Derived units are defined d as products of powers of the base units. When the product of powers includes no numerical factor other than one, the derived units are called coherent derived units. Some of the coherent derived units in the SI are given special names 9

10 Some derived quantities Quantity Unit Symbol In terms of other SI units In terms of SI base units frequency hertz Hz s -1 force newton N m kg s -2 energy, work, amount of heat joule J N m m 2 kg s 2 power, radiant flux watt W J/s m 2 kg s 3 electric charge, coulomb C s A amount of electricity it electric potential volt V W/A m 2 kg s 3 A 1 difference, electromotive ti force electric resistance ohm Ω V/A m 2 kg s 3 A 2 10

11 Areas of improvement Structure of the system Stability of the mass unit Uncertainty of fundamental constants Situation of electrical units 11

12 Structure of the system Many 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) 12

13 Stability of the mass unit The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram (Third CGPM, 1901) (By courtesy of the BIPM) 13

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18 G. Girard, Metrologia, 1994, 31 18

19 A provisional solution The CIPM declared that, pending further research, the reference mass of the international prototype is that immediately after cleaning and washing by a specified method (PV, 1989, 57, and PV, 1990, 58, 95-97). (Brochure SI, 8 th ed., 2006) 19

20 Uncertainty of fundamental constants Most fundamental constants t are experimentally determined in terms of SI units. Therefore, their recommended SI values have an associated measurement uncertainty t and are periodically updated. 20

21 Present situation Quantity Symbol Unit Rel. std. unc. u r speed of light in vacuum c,c 0 m s 1 exact mass of the Prototype kg m(k) kg exact frequency of hyperfine splitting of ν (hfs s 1 exact 133 Cs Cs) magnetic constant μ 0 N A 2 exact molar mass of 12 C M( 12 C) g/mol exact temperature of water triple point T tpw K exact spectral luminous efficacy K cd sr/w exact Planck constant h Js 50x elementary charge e C 2.5 x 10 8 electron mass m e kg 5.0 x 10 8 Avogadro constant N A, L mol x 10 8 Boltzmann constant k J K x 10 6

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23 Situation of electrical units The volt and the ohm are nowadays represented in a highly reproducible 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 electromechanical experiments) are affected by much higher uncertainties. 23

24 Early proposal

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26 2007 The 23rd General Conference On the possible redefinition of certain base units of the International System of Units (SI) The 23rd General Conference, (omissis) recommends that t National Metrology Institutes t and the BIPM pursue 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 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 to the 24th General Conference.

27 The essence of the proposal To assign conventionally exact SI values to seven suitable invariants of nature. This implies re-definition iti of the seven base and of any conceivable derived unit in terms of the seven invariants. Which values? The best available (CODATA) at the moment of the redefinition. Which invariants? Debate is still open, but broad consensus exists on a (sub)set of invariants.

28 The proposed invariants 28

29 The future scenario Quantity Symbol Unit Rel. std. unc. u r speed of light in vacuum cc c,c ms 1 0 exact mass of the Prototype kg m(k) kg 2.0 x 10 8 frequency of hyperfine splitting of ν (hfs 133 Cs Cs) s 1 exact magnetic constant μ 0 N A x 10 8 molar mass of 12 C M( ( 12 C) g/mol 5.0 x 10 8 temperature of water triple point T tpw K 1.7 x 10 6 spectral luminous efficacy K cd sr/w exact Planck constant h J s exact elementary charge e C exact electron mass m 1.4x10 9 e kg 10 Avogadro constant N A, L mol -1 exact Boltzmann constant k J K -1 exact 29

30 Resistenze alle ridefinizioni proposte Difficoltà di ordine sperimentale Difficoltà di ordine concettuale 30

31 Difficoltà di ordine sperimentale Determinazione della costante di Planck h (anche attraverso la costante di Avogadro N A ): decisiva per ridefinire il kilogrammo 31

32 Difficoltà di ordine sperimentale Determinazione della costante di Boltzmann k : decisiva per ridefinire il kelvin u 020 = γ kt / m m atomic mass ( 40 Ar) γ = 5/3 for gas of single atoms u 0 speed of sound receiver transducer Obiettivo: Confermare (o contraddire) l'attuale unico dato di riferimento con accuratezza confrontabile (1, ) d Non ancora raggiunto risonanza elettromagnetica risonanza acustica 32

33 Difficoltà di ordine concettuale Resistenza ad abbandonare la logica tradizionale del campione: l'unità è basata sul campione riproducibile con la minima incertezza la definizione dell'unità insegue l'evoluzione tecnologica dei campioni la realizzazione dell'unità non garantisce la coerenza tra le unità a favore di una logica di sistema: ste l'unità è basata su un invariante fondamentale delle teorie scientifiche la definizione è resa obsoleta solo dal decadimento di quelle teorie la realizzazione dell'unità garantisce intrinsecamente la coerenza del sistema la ridefinizione può comportare un'aumento dell'incertezza di realizzazione 33

34 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 campione mantenendo distinte le componenti di incertezza che intervengono separatamente nella pratica metrologica (misurazioni dirette o indirette) u S incertezza del campione (tarature, misurazioni dirette) u S nuova realizzazione dell'unità u SI incertezza di realizzazione (misurazioni indirette) u SI 34

35 Difficoltà di ordine concettuale Acquisire un consenso sul tipo di definizione da adottare per vecchie e nuove definizioni di unità: Definizione del tipo explicit-unit (tradizionale) Definizione del tipo explicit-constant constant (CCU) Definizione del tipo explicit-reference (nuova proposta) 35

36 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/ 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

37 Explicit-constant definitions A 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 metres per second. Claimed advantages: Simplicity... draws attention to the implications of the definition for fundamental physics No reference to, or suggestion of a specific experiment for the realization 37

38 Explicit-constant definitions. Difficulties 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 Mysterious wording All these difficulties stem from a common weakness, intrinsic in the choosen kind of definition (more on this later). 38

39 Explicit-constant definitions. Incompleteness Incompleteness is the main concern. Excluding the second, which is the first definition in the list, and the mole, which is now independent of the kilogram, each definition implies knowledge of at least one of the preceding definitions. Examples: The metre, unit of length, is such that the speed of light in vacuum is equal to exactly metres per second. This definition is not self-consistent, as requires the second to be 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

40 Explicit-constant definitions. Incompleteness - further example The kilogram, unit of mass, is such that the Planck constant is equal to exactly joule second. Also this definition iti is not self-consistent, t as it implies knowledge of the dimension of the joule and the definition of the second. Again, taken alone, the definition establishes a necessary and not sufficient condition. There are 2 kilograms satisfying i the definition, iti depending on how the second and the metre are defined. The change in the order of definitions, intended to avoid defining a unit in terms of another which is defined later, is a necessary but not sufficient measure. The result is still unsatisfactory. 40

41 Explicit-constant definitions. Circularity 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 per second, unit of speed, the time derivative of length. The kilogram, unit of mass, is defined d 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. 41

42 Explicit-constant constant definitions. Bi-univocal association of a unit with a constant t 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 mole and the Avogadro constant, t 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. 42

43 Explicit-constant 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) And, again from CCU/10-3.1, a format common to all definitions: The, unit of, is such that t., is equal to exactly. In 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. 43

44 Explicit-constant 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 value. These sentences can perhaps be grasped intuitively, but would not stand a rational or semantic analysis. Not casually, they cannot be written as explicit mathematical relations. I anticipate considerable problems in teaching. 44

45 Why these difficulties? All the difficulties arise because in the definitions the base units [Q B ] i and the associated constants C i refer to different quantities. For example, the kilogram is defined in terms of an action. It might be said that the international system of units SI (or perhaps the underlying system of quantities ISQ) and the international system of constants are based on two different sets of base quantities. This causes incompleteness and circularity. The obscure wording reflects the complication involved in defining i the unit of a quantity in terms of units of different quantities. 45

46 Is there a possible way out? A further type of unit definition is proposed, which could be called explicit-reference. It derives from a systemic approach, with no necessary association of the unit with a single constant. Also the distinction between base and derived quantities becomes unnecessary and could be dropped. 46

47 Proposal The proposed structure for any unit definition: "The unit of [quantity], [special name], is equal to [numerical coefficient] times [monomial expression of the reference constants]" The monomial expression of the reference constants is a constant quantity homogeneous with the unit being defined. 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 o is systemic (from a set of statements on the reference constants and their fixed values to a system of unit definitions). 47

48 Proposal Set of initial statements (for constants proposed by CCU): the ground state hyperfine splitting frequency of the caesium 133 atom Δν( 133 Cs) hfs is exactly hertz; the speed of light in vacuum c is exactly metres per second; t 626 the Planck constant h is exactly joule second; the elementary charge e is exactly coulomb; the Boltzmann constant k is exactly joule per kelvin; 23 the Avogadro constant N A is exactly per mole; the spectral luminous efficacy K cd of monochromatic radiation of frequency hertz is exactly 683 lumen per watt. 48

49 Proposal The same statements in form of equations are: Dn( 133 Cs) hfs = s 1 c = m s 1 h = m 2 kg s 1 e = A s 2 K 1 k = m 2 kg s 2 K 1 N A = mol 1 K = 683 cd m 2 kg 1 s 3 Whose solution yields the explicit-reference definitions: the unit of time, second, is equal to times 1 / Dn( 133 Cs) hfs ; the unit of length, metre, is equal to times c / Dn( 133 Cs) hfs ; the unit of mass, kilogram, is equal to â times h Dn( 133 Cs) hfs / c 2 ; the unit of electric current, ampere, is equal to â 10 8 times e Dn( 133 Cs) hfs ; the unit of temperature, kelvin, is equal to times h Dn( 133 Cs) hfs / k; the unit of amount of substance, mole, is equal to â times 1 / N A A; the unit of luminous intensity, candela, is equal to â times h K Dn( 133 Cs) hfs 2." 49

50 Proposal 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 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 generalised system, published together with partial solutions (See Cabiati and Bich, Metrologia 46, 2009). 50

51 Comments The wording sounds familiar, corresponding to the most original i definitions iti Each definition is self-consistent (no need for other definitions) Circularity is avoided The relation between the unit and the constants involved in each definition is clearly shown in elementary algebraic terms, independent of any possible physical interpretation The definitions are rational and intuitive at the same time. helps understanding and teaching them This greatly 51

52 Straying on forbidden grass This way of defining cruelly brings into light how all units except the mole depend on Δν( 133 Cs) hfs, definitely not a universal constant, rather an invariant of nature which is ultimately nothing else than a standard of the same kind of the Prototype kilogram (the former natural, the latter artificial). Should a better transition be found, all these definitions would need to be changed accordingly. This is not a proof of the superiority of the explicit constant definitions. On the contrary, the latter simply hides the problem, does not solve it. 52

53 Straying on forbidden grass The natural invariant Δν( 133 Cs) hfs is inadequate as base constant, and would be better placed as a recommended transition frequency in a mise en pratique. An indirect proof of the inadequacy of Δν( 133 Cs) hfs as a base constant is the large value of the numerical coefficient for the kilogram. This value is questioned in some quarters as unphysical. Clearly, itisnotappropriate p toattachaspecial physical meaning to combinations of universal constants and a specific natural invariant. 53

54 Straying on forbidden grass The adequate base constant to complete the transformation of the SI would be the electron mass m e. Including this important constant within the reference set of the SI, the definition of the kilogram would very naturally be The kilogram, unit of mass, is equal to exactly 1, m e. Several other constants would benefit from an exact valueof the electron mass and other unit definitions from the introduction of that base constant. In particular, the Compton wavelength h/(m e c) and Compton frequency m 2 e c /h would become the reference quantities for the length and time units respectively. Contrary to their belief, the time&frequency community would not even notice the change in the definition of the second. But this is a different story 54

55 Sviluppi recenti ( si/) 55

56 Ritocchi ( The metre, m, is the unit of length; its magnitude is set by fixing the numerical value of the speed of light in vacuum to be equal to exactly when it is expressed in the unit m s -1. The kilogram, kg, is the unit of mass; its magnitude is set by fixing the numerical value of the Planck constant to be equal to exactly X x10 34 when it is expressed in the unit s 1 m 2 kg, which is equal to J s. 56

57 Nuovi dati As-stated uncertainties May METAS watt bal Js) 10 6 (h -6,6 3 1 NPL watt bal. 28 Si Avogadro NIST watt bal. Weighted mean

58 e nuove resistenze ( 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 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 At least one of these results should have a relative standard d uncertainty t not larger than 2 parts in 10 8, 2. for each of the relevant constants, values provided by the different experiments be consistent at the 95 % level of confidence, 3. traceability of BIPM prototypes to the international prototype of the kilogram be confirmed, 58

59 that t the CODATA recommended d values be adopted d for the relevant fundamental constants, that 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 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 that the uncertainty component arising from the practical realization of the unit be suitably taken into account. 59

60 Conclusione Tutto è rinviato (almeno) al 2015 Grazie per l attenzione! 60