arXiv.org > physics > arXiv:1601.00857v.3 A critical review of the CCU 2016 Draft of the Brochure for the “New SI” definition using (fundamental) constants Franco Pavese formerly National Research Council, Istituto di Metrologia “G.Colonnetti” (from 2006, Istituto Nazionale di Ricerca Metrologica), Torino, Italy e-mail:
[email protected] (Submitted: v.1 on 30 Dec 2015; v.2 21 January 2016; v.3 24 February 2016) Abstract This note discusses the role of constants (called “fundamental” until 2015) in the proposed “New SI” formulation of the definition of the International System of Units, namely in the present official documents and in some relevant literature. The meaning assigned to the use of constants is found substantially different even among the advocates of the proposal. Then, this note focus on how the stipulation of their numerical values is obtained, according to the specific features of measurement units, arising some special problems, not all cleared yet, that the stipulation of a fundamental constant can place when it becomes part of the definition of the New SI and of a base unit of it. Some reasons are discussed why it is urgent that these basic issues are clarified. In the Appendix, a specific critical review is reported of the text of Sections 1 and 2 of the CCU 2016 Draft, the last implementation of the New SI for the 9th Brochure on the SI, published on the BIPM website on January 4, 2016, and now under examination by the SI top stakeholders. Though this draft is advanced with respect to the CCU 2013 Draft, it is shown how the description of the complex structure of the new definition may still be difficult to be clearly caught even by informed readers, while most users might be lost about their tasks needed to implement the new definition.
Introduction At the 2014 meeting of the Conférence Générale des Poids et Measures (CGPM), Resolution 1 [1] concerning the International System of Units (SI) “encouraged continued effort” toward a suitable proposal “that could enable the planned revision of the SI to be adopted by the CGPM” in 2018. Drafts of the needed official documents to that effect have been provided by the Comité Consultatif des Unités (CCU) of the Bureau International des Poids et Measures (BIPM) (last the 2006 Draft of the 9th Brochure of the SI [2]), and they are available, together with an illustration, on the website of the BIPM [3]. However, some discrepancies in the present intention of using fundamental constants in the definition of the system of measurement units are found among the advocates of the new definition, namely in the present official documents and in some relevant literature. The present “SI consists of a set of base units, prefixes and derived units, as described in these pages. The SI base units are a choice of seven well-defined units, which by convention are regarded as independent: the metre, the kilogram, the second, the ampere, the kelvin, the mole, and the candela. Derived units are formed by combining the base units according to the algebraic relations linking the corresponding quantities.” (8th Ed. SI Brochure) [4]. “Les unités choisies doivent être accessibles à tous, supposées constantes dans le temps et l’espace, et faciles à réaliser avec une exactitude élevée” (from the official French text) (ibidem) [4]. Each of the base units is individually defined. The present definitions are of different types: (a) Make use of an artefact: for mass “The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram”; (b) Make use of a physical state or condition: e.g., for temperature “The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water”. Similarly for time, amount of substance and luminous intensity; (c) Define a realisation of the unit, e.g. for length: “The metre is the length of the path travelled
arXiv.org > physics > arXiv:1601.00857v.3 by light in vacuum during a time interval of 1/299 792 45 8 of a second”. Similar approach for the electrical current. Actually, also the base unit types (a) and (b) indirectly imply a ‘definitional method’ to be realised with appropriate procedures (not to be confused with the “mises en pratique” [5] that are approximations of the definition) —according to the meaning of term 2.6 of VIM [6]. The use of the fundamental constants in the New SI BIPM official documents In [2] one finds the basic statement reported here as Eqs. A1 in the Appendix to this Note (the speed of light in vacuum, c, is often referred to as c0; the Boltzmann constant, k, is often referred to as kB). It is said to be the definition of the New SI. That can be called a ‘group-definition’ [7] and looks a self-contained and sufficient new definition of the SI, as also acknowledged in [8], and it represents a turning point with respect to the present one. In addition, the former base units remain and are the same, but the rational of their definition is completely different from the present one. In fact, until now the core of the definitions of the base units consisted in the specification of a ‘definitional method’ for the realisation of each base unit: initially by using artefacts (e.g., for mass, length), then using also invariant states of the nature (e.g., for temperature); further, more recently, by adopting a “mise en pratique” of the definitions of the base units. As a consequence, any National Measurement Institute (NMI) wanting to realise a base unit in an independent way was required to realise the definitional procedure. [9] In some cases, that was a strong limitation to the users, also considering the fact that new methods were developed in the mean time, often superior to the definitional one. The addition of a mise en pratique [5] for each of the base units was adopted by the Comité International des Poids et Mesures (CIPM), an executive committee of the BIPM, to extend the flexibility of the NMI in realising the standards. However, though already a good advancement, these methods were constrained to a lower-rank status with respect to the current base unit definitions. [10] Since more than a decade, a search for a more advanced solution accelerated, especially due to the impelling problems with the mass standard and with some electrical standards. In the panorama of science, a single frame has basically emerged in the last decades, of which indirectly also metrology benefited: the experimental studies on the numerical values of fundamental constants of physics and chemistry, whose uncertainties has dramatically decreased in many cases, with a much larger number of experiments available and wider in methodologies than in the past. The main reason for these studies was not, at least initially, strictly metrological, though they used the superior tools of top metrology to advance in the check for the consistency of the basic models of the physical/chemical world. Should the base units be independent to a sufficient degree—as initially assumed by the historical promoters of the MKSA system and then of the SI (“Les grandeurs de base sont, par convention, considérées comme indépendantes” Section 1.2 in [4], 8th Ed.)—their mutual consistency would be intrinsical. At present, instead, four of the seven base units are not independent: the unit of length involves the second; the unit of amount of substance involves the kilogram; the unit of electric current involves the newton and the metre; the unit of luminous intensity involves the hertz, the watt and the steradian. Considering, in addition, that the numerical values of the constants are determined by the used measurement units, the feedback of the above results to the metrological field was that a corresponding degree of metrological compatibility [6] of the base units has been established. This degree has further been refined, with respect to the normal statistical analysis of the experimental data, by the studies of the CODATA Group on fundamental constants using the
arXiv.org > physics > arXiv:1601.00857v.3 Least Squares Adjustment method. [11, 12] The new situation seemed therefore to offer the tools for a new definition combining a sounder basis for the definitions, by using only invariant states or constants [3] that can be considered as ‘natural quantities’, with the possibility for the users of a more flexible implementation in realising the standards. In fact, this choice also avoids the indication of any definitional method. It consists, as already indicated, in taking the ‘best’ numerical value today available of each constant as a satisfactory one, and in stipulating it, i.e., in fixing it exact by convention (reference value)—thus also zeroing its non-uniqueness. See hereinafter about stipulation. On the other hand, the group-definition of the SI, as spelled out in Eqs. 1 of the Appendix, does not tell anything explicit about the definition of individual base units or about their realisation. For that reason, one still finds in [3] (under “What”) the following complement to the groupdefinition: “The SI may alternatively be defined by statements that explicitly define seven individual base units: the second, metre, kilogram, ampere, kelvin, mole, and candela. These correspond to the seven base quantities time, length, mass, electric current, thermodynamic temperature, amount of substance, and luminous intensity. All other units are then obtained as products of powers of the seven base units, which involve no numerical factors; these are called coherent derived units” (emphases added).
That requirement has been considered by the CCU, since its 2013 Draft, as the basis for an ‘individual base unit’ definition. The use of the fundamental constants in the New SI relevant literature In a recent paper [13], one of the leaders of the CODATA Group on fundamental constants that advocates the New SI definition, illustrated a perspective that is common to other authors and is closer to the original intention of the physical Community. As shown in Table 1, the fundamental constants can be taken as representing base quantities different from the present ones. Table 1. Constants, present and New SI base quantities and their units, base quantities in [13]. Constant a Δν(133Cs)hfs c0 h e kB NA Kcd a
b
Present dimensions s–1 m s–1 J s = kg m2 s–1 C=As J K–1 = kg m2 s–2 K–1 mol–1 lm W = cd kg–1 m–2 s3
Present and New SI base quantity (BIPM [3]) Time Length Mass Electrical current Thermodynamic temperature Amount of substance
Present and New SI base units (BIPM [3] ) second metre kilogram ampere kelvin
New SI base quantity (and unit) (Newell [13]) Frequency (Hz) Velocity b (m s–1) Action (J s) Electric charge (C) –1 Heat capacity (J K )
mole
Luminous intensity
candela
Amount of substance (reciprocal) Luminous intensity
Five are fundamental constants, in the first and last row they are ‘constants’ only in the broad sense. Actually, it is “speed”, a non-vectorial quantity.
This position is amply justified from the physics viewpoint, but, as clearly evident from Table 1, it would imply a change in the base units too. That would be a very intricate affair, because the second, the metre, the kilogram, the ampere, the kelvin and the mol would become derived units, i.e., to be expressed in terms of the new base ones: e.g., for the simplest case of the electrical current, the present unit would be A = C Hz.
arXiv.org > physics > arXiv:1601.00857v.3 Are the two approaches equivalent with each other? The present ‘group definition’ reported in the BIPM website (see Appendix) is not the more exact translation of the idea behind Newell’s meaning (also shared by others), i.e. that the physical quantity closest to the meaning of the corresponding fundamental constant becomes the new base unit. On the other hand, the latter would fulfil anew the original intention of the SI founders, to have independent base units (leaving aside the unchanged luminous intensity and its unit, the candela). This would also be appropriate in avoiding the use of inverse quantities, a conceptual difficulty. However, in that case it would be wise to consider a full change of the name of the System of Measurement Units, from SI to another name, in order to avoid confusion, as it was done when changing from the CGS to the SI systems; or, to reserve the name SI to the conventional system using the present base units. One can appreciate the possible advantages of the above possibility, by looking at the difficulties found by the CCU in implementing the wish in [3] to adopt also (or alternatively) single-unit definitions. The current CCU implementation [2] in Chapter 2.2 uses the following way, somewhat different from that of the 2013 Draft, reported immediately after in italics (see Appendix where each single definition is analysed): “The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency ΔνCs, the unperturbed ground-state hyperfine splitting frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s–1 for periodic phenomena. – “The second, symbol s, is the SI unit of time; its magnitude is set by fixing the numerical value of the unperturbed ground state hyperfine splitting frequency of the caesium 133 atom to be exactly 9 192 631 770 when it is expressed in the SI unit s–1, which for periodic phenomena is equal to Hz.” The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit m s–1, where the second is defined in terms of the caesium frequency ΔνCs. – The metre, symbol m, is the SI unit of length; its magnitude is set by fixing the numerical value of the speed of light in vacuum to be exactly 299 792 458 when it is expressed in the SI unit for speed m s–1. The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 040 ×10 –34 when expressed in the unit J s, which is equal to kg m2 s–1, where the metre and the second are defined in terms of c and ΔνCs. – The kilogram, symbol kg, is the SI unit of mass; its magnitude is set by fixing the numerical value of the Planck constant to be exactly 6.626 069 57 10−34 when it is expressed in the SI unit for action J s = kg m2 s−1. The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602 176 620 8 ×10–19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of ΔνCs. – The ampere, symbol A, is the SI unit of electric current; its magnitude is set by fixing the numerical value of the elementary charge to be exactly 1.602 176 565 10−19 when it is expressed in the SI unit for electric charge C = A s. The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380 648 52 ×10–23 when expressed in the unit J K–1, which is equal to kg m2 s–2 K–1, where the kilogram, metre and second are defined in terms of h, c and ΔνCs. – The kelvin, symbol K, is the SI unit of thermodynamic temperature; its magnitude is set by fixing the numerical value of the Boltzmann constant to be exactly 1.380 648 8 10−23 when it is expressed in the SI unit for energy per thermodynamic temperature J K–1 = kg m2 s2 K–1.
arXiv.org > physics > arXiv:1601.00857v.3 The mole, symbol mol, is the SI unit of amount of substance of a specified elementary entity, which may be an atom, molecule, ion, electron, any other particle or a specified group of such particles. It is defined by taking the fixed numerical value of the Avogadro constant NA to be 6.022 140 857 ×1023 when expressed in the unit mol–1. – The mole, symbol mol, is the SI unit of amount of substance of a specified elementary entity, which may be an atom, molecule, ion, electron, any other particle or a specified group of such particles; its magnitude is set by fixing the numerical value of the Avogadro constant to be exactly 6.022 141 29 1023 when it is expressed in the SI unit mol–1.
The candela, symbol cd, is the SI unit of luminous intensity in a given direction. It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540 ×1012 Hz, Kcd, to be 683 when expressed in the unit lm W–1, which is equal to cd sr W–1, or kg–1 m–2 s3 cd sr, where the kilogram, metre and second are defined in terms of h, c and ΔνCs.” – The candela, symbol cd, is the unit of luminous intensity in a given direction; its magnitude is set by fixing the numerical value of the luminous efficacy of monochromatic radiation of frequency 540 1012 Hz to be exactly 683 when it is expressed in the SI unit kg−1 m−2 s3 cd sr = lm W−1 = cd sr W−1. For five base units the relevant constant is a single of the fundamental ones listed in the groupdefinition of the SI (c0 having already been used since 1983 for the metre), included in the CODATA analysis. For the second and the candela the fixed numerical value is that of an invariant physical state, not presently included in the CODATA analysis. Why the above formulations are not satisfactory? Each of them (in either the 2013 and the 2016 versions) only uses explicitly one of the relevant constants for the definition of some of the single base units. See below the base units expressed as combinations on the proposed constants (dimensional expressions), originating from Table 1, column 2: [c0/ ΔνCs ] = [metre]; [h ΔνCs / c02] = [kilogram]; [h ΔνCs /kB] = [kelvin]; [e ΔνCs] = [ampere]; [1/NA] = [mole].
(1)
(NA depends only on the unit mol—but see [9]; ΔνCs for the second, and Kcd for the candela are not “fundamental” constants) The constant indicated in each base unit definition [2] is only the ‘dominant’ one, in the sense illustrated as follows, by using the original pre-stipulation uncertain numbers (CODATA 2010 data): for the kelvin, the contributions to the uncertainty of ΔνCs and h are not significant (utot = 4.16·10–7 taking them into account, compared with 4.15·10–7 for kB only). Similarly, considering the relevant constants, for the kilogram (utot = 3.01·10–48 instead of 2.97·10–48 for h only) and the ampere (utot = 3.23·10–12 instead of 3.22·10–12 for e only), a bit less for the candela (utot = 1.68·10–18 instead of 1.58·10–18 for Kcd only) and for the metre (utot = 9.64·10–11 instead of 9.53·10–11for c0 only). However, that cannot be considered a sufficiently rational reason for the exclusive adoption in the spelling of the definitions of the ‘dominant’ constant—see, on the other hand, in the Appendix for the circularity induced by the 2016 version and for some peculiar features of each definition. This difficulty is totally missing in Newell’s scenario, where new base units are adopted: this is one reason why the two scenarios are not equivalent. In the present BIPM’s scenario, as said, the ‘group definition’ of the New SI does not tell anything explicit about (do not spell out) the definition of the individual base units. The literal meaning of the wording of a single-unit definition derived from the group-definition, should in fact be spelled out as follows: According to the new definition, each base unit is ‘such that’ the chosen stipulated numerical value(s) of the relevant constant(s) apply. From there, a second basic difference between the two scenarios derives.
arXiv.org > physics > arXiv:1601.00857v.3 Basically, the BIPM’s New SI definition is explicitly based on the following reasoning: (i) With the dramatic decrease of the experimental uncertainty of these constants (or others that might possibly later be preferred, like mu [14]), the scientific community is in the position to consider having today a sufficiently accurate check of the degree of mutual consistency of the present base units, thanks, and only thanks, to the CODATA studies; (ii) This is made possible, in the experimental determination of the numerical values of interrelated constants, by the use of different methods where the base units play a different role: they would reveal inconsistencies in the set of units, should metrologically incompatible results be obtained—something probably occurring at present for electrical quantities [15]. Actually, also the numerical value of constants not in itself related to others, like NA, is often obtained through an algebraic combination of other interrelated constants, e.g., in that case NA = R kB–1. The cross-combination of different constants of the selected group (see Eq. 1) increases the confidence in the consistency check. (iii) Consequently, it is considered appropriate to retain those numerical values, and stipulate them as exact, which implicitly means retaining the present state-of-the-art degree of consistency of the present base units. It would instead be quite difficult, and indirect, to transfer this property to a different set of base units [14]—apart from the obvious discomfort for the users. An understandable consequence of this choice is that mutual consistency becomes the single and only criterion established for the new definition using the same base units presently used: the realisation of a base unit is correct if it provides a numerical value that is compatible with the set of stipulated numerical values of the relevant constants of the group—actually it should mean ‘exactly the stipulated value(s)’, but this cannot be proved, being the realisations experimental. However, instead of considering the interrelationship of the constants as a problem to avoid, this choice uses it by privileging the resulting possibility of a sound consistency check. This choice is not compatible with Newell’s one: the difference is the same that exists between a cause (the present base units determining the numerical values of the constants) and an effect (these same resulting numerical values of the constants be taken, as a group, to fix the set of the same base units). Notice, however, that the BIPM concept here is not the same that is presently spelled out in [2, 3] (but that is often claimed for), where the constants appear to be chosen for ‘fundamental’ physical reasons as in Newell’s approach: indeed, the constants are not explicitly used in the definition for their physical meaning, instead—repetita juvant—for their interrelated properties that allows them to be fundamental for a sound consistency check of the set of base units—and for avoiding a reference to any specific substance. In addition, the definition does not indicate any specific method or procedure to ensure the fulfilment of the above criterion, because the latter does not refer to any specific definition of individual base constant: any set of methods or procedures will be appropriate until they have the potential to fulfil the criterion. In all instances, they are not anymore the responsibility of the CGPM to be promulgated, but of the Comité International des Poids et Measures (CIPM) to later approve their choice on input from the individual Comités Consultatifs. Difficulties in linking the group-definition to the conventional SI In [2] it is stated: the “description in terms of base and derived units is maintained in the present [new] definition of the SI, but has been reformulated as a consequence of adoption of the defining constants”. However, no specific guidance is provided to link the group-definition with the singleunit definitions: the single-unit definitions listed above are not sufficient without a specific illustration. [2] One of the problems is how to correctly understand the numerical values in these definitions. A numerical value must be associated to a ‘measurement unit’ (term 1.9 in [6]) (‘unit’ in short) to fix the exact measure of the magnitude (see term 1.1 in [6]) intended for that ‘kind of quantity’
arXiv.org > physics > arXiv:1601.00857v.3 (term 1.2 in [6]), because a unit should be required to be univocal—as traditionally is considered necessary, irrespective of the precision of its realisation in practice. The way by which this effect is reached may vary depending on the nature of the unit and on the type of scale. For example, let us consider first the case when an artefact materialises the unit (type (a) standard in the above Introduction and Type 1 in [16, 17]), like at present is the case of the mass unit and of the object that is assigned the value 1 kg exactly. It is the ‘realisation’ of the present-SI unit “kilogram”. One has to make a clear distinction between the uniqueness of the measurand and possible changes of such an object: the first may be time independent, the second can only be time dependent. The latter changes are the object of studies and international concern. Only the uniqueness will be discussed here. Such a material solid object was supposed to have an unequivocal firm uniqueness. However, in modern times, the question arose about the definition of the interface between the solid artefact and the environment: in fact, in the case of mass measurements, a vast literature exists about the pros and cons of performing measurements in vacuum, where it is assumed that, in principle, the interface should be more precisely defined. In the same field of mass standards, similar studies, much more refined, were possible in the case of the “Avogadro Project”, where the artefact is a silicon object of spherical shape. The non-uniqueness in these cases comes from the differences in composition and thickness of the interfacing layer(s), having an effect different from zero, in addition to a time-dependent variability, on the actual magnitude of the object. Also in the case of the previous length standard, a graduated metal bar, a non-uniqueness problem arose when the width and shape of the graduations introduced non-uniqueness in the measurement of the distance between two graduations. In both cases, a stipulation of an exact value was the only way to remove the non-uniqueness. With it, the distinction comes between the exact conventional value and any experimental realisation of it—determination of the magnitude of the object—that is affected by an uncertainty and needs data analysis. In the case of units defined by means of physical-chemical ‘states’ or ‘parameters’ or ‘conditions’ (type (b) standards in the above Introduction and Type 2 in [16, 17]), let us consider the unit for temperature. This case is different from the previous ones also because the former were concerning an “extensive” quantity—a term today considered sometimes outdated—and a corresponding type of measurement scale, while temperature is an “intensive” quantity requiring a different type of scale. Limiting here the issue to the non-uniqueness of the quantity, one intends, in principle to take advantage of a state of the matter that should have sufficient theoretical bases to ideally be considered a unique one. In the case of the temperature scale with unit kelvin, such a state is presently the “triple point” of a specific substance, water. Is it true? Term 2.11 in [6] indicates in Note 1: “…owing to the inherently incomplete amount of detail in the definition of a quantity, there is not a single true quantity value but rather a set of true quantity values consistent with the definition” (emphasis added). With the refinement of the uncertainty of the realisations, irrelevant details (properties) of a substance may become relevant, inducing a non-uniqueness: with water, in the last couple of decades the specification of a conventional isotopic composition was deemed to become necessary—and was formalised in 2005 [18]; note that, contrarily to chemical impurities, samples of different isotopic compositions result in different substances (substance nonuniqueness). The need for additional “details” in the definition also concerns the unit of time, where, for the second, the frequency of “the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom” [19] was chosen, but later specifying “this definition refers to a caesium atom at rest at a temperature of 0 K” and, further, “it is understood that the Cs atom at a temperature of T = 0 K is unperturbed by black-body radiation”. In general, this was the fate of all units based on these types of definition: an affordable but undesirable need of ‘evolution’ in time of those definitions to avoid significant non-uniqueness, though not necessarily
arXiv.org > physics > arXiv:1601.00857v.3 needing to change the stipulated numerical value, simply attributed to the stipulated state. It is a fact that the definition of any unit needs the specification of at least a numerical value, in order to indicate the condition that the chosen property must comply with for the value 1 (or unitary difference) be assigned in the proper scale. It is deemed to be independent of realisation uncertainty, irrespective to the type of data treatment. How this requirement transfers to the case where the unit definition is based on constants (type (c) in the above Introduction) is not a trivial process, and is illustrated in the next Sections. Recommended vs. stipulated numerical values The recommendation of numerical values is common practice in many fields: in most cases it is the result of a critical evaluation of sets of published experimental values, or, in the case of ‘reference materials’, the result of specific experimental work. The recommended values are always decorated with an evaluation of the associated uncertainty, most usually the standard deviation of the observed dispersion or resulting from an uncertainty budget. This is also the case for the numerical values of the fundamental constants that outcome from the CODATA evaluation [11, 20], or of the atomic ‘weights’ that outcome from the AMDC evaluation [21, 22], both based on the LSA method, but in basically different circumstances. In the latter case, a second set of recommended values have been made available in 2014, based on a distinct study [23]. IUPAC CIAAW provides recommended values for both atomic “weights” and isotopic abundances [24]. However, the other studies provide the numerical values in the form 123.456(78) (form A), where the digits in parentheses are the uncertainty (standard deviation) applied to the last two digits (and the ‘8’ is a mere ‘guard digit’ included to avoid summing up of rounding errors in complex calculations). CIAAW, since 2013 compilation, decided to indicate instead, for some atoms, an interval of possible variation, in the form, e.g. for TSAW2013 bromine, [79.901, 79.907] (form B); similarly for the abundance: e.g. 18 O, [0.001 87, 0.002 22]. This representation may require the use of available appropriate statistical tools, e.g. see [25, 26], where today the preferred representation is not the form B, but the more robust form C [0.002 045; 0.000 017.5] (middle value; semi-interval width) is preferred—which may be deceiving since its meaning is not that of form A, but the one of form B. On the contrary, stipulation means that a numerical value is chosen by convention and taken as exact. In principle, a conventional value can be chosen for mere convenience, as, for example, the year 1 of our era, or 0 °C for the melting temperature of ice for the Celsius scale. However, in other cases, and normally in metrology, the choice is constrained. A constraint is usually arising from the statistical data analysis provided in form A, where the choice is not necessarily univocal. In fact, there are examples in the literature (including [2]) of stipulation of the form A as 123.46 or even 123.456, i.e. including one or both the uncertain digits, while an exact number should be rounded to 123.5. If form C is used, notice that the middle value does not necessarily always coincide with the ‘mean’ value of form A. Thus, stipulation does not mean simply dropping the uncertainty indication. In addition, a further constraint may arise from non-statistical reasons, like the strict metrological requirement to ensure the continuity of a measurement unit through changes of its definition, a quantity whose value—not the numerical value—must be invariant through a change of the unit definition [27] (this is now explicitly indicated also in the CCU 2016 Draft [2]). However, for uncertain numbers of experimental origin, the original variability represents a due constraint, whose boundary is not fixed or ‘objective’, but is narrower or wider depending on the width of the confidence interval, or degree of belief, that is considered appropriate for the specific purpose.
arXiv.org > physics > arXiv:1601.00857v.3 Stipulating a fundamental constant It is not redundant to recall that, while the value of a constant is unique, but unknown, the numerical value of it, to be known, must be obtained experimentally (no true value knowable, at least with top accuracy). The latter thus depends on all the relevant units that have been used in its measurement: these are the ones concerning all the quantities included in the measurement model, i.e., according to the traditional distinction, concerning not only the “input quantities” [28] but also the “corrections” [6,17, 28, 29] that significantly influence the resulting numerical value. It means that all measurements should be considered as ‘indirect’ [6]: however, the resulting precision of different measurement methods may be differently affected by the same influence quantities, i.e. by the same units. This is the case also of the measured numerical values of any fundamental constant. Since no one is exactly known (not even the dimensionless—or of dimension 1—ones, like the fine-structure constant α), the stipulation maintains one degree of freedom in setting the numerical values of all: the CODATA used it to fix the numerical value of the ‘magnetic (or permeability) constant’ µ0— for the situation after the stipulation of c0 see [20]. All the recommended values depend on this choice. Notice that µ0 is not dimensionless, µ0/2π = 2·10–7 m kg s–2 A–2, meaning that some units are constrained by this choice; it would probably be more neutral to use a dimensionless constant. Stipulating the numerical values of a set of constants is considered by the CCU directly fixing the base units, according to their proposal of the single-unit definitions: the “… magnitude [of the unit] is set by fixing the numerical value of the [single relevant constant] to be exactly [numerical value] when it is expressed in the SI unit for [kind-of-quantity] [new SI units]” (in the 2013 Draft); or “It is defined by taking the fixed numerical value of the [single relevant constant] to be [numerical value] when expressed in the unit [SI units from group-definition], which is equal to [SI base units], where the [other base units involved] are defined in terms of [other relevant constants]” (in 2016 Draft [2]). However, due to the effective interrelations shown in Eq. (1) above, the stipulation involves, directly or indirectly through the respective measurement equations, all the base units, meaning that it establishes only an indirect constraint on the base units. It may be true that, having all the constants of the set a stipulated value, a corresponding reference value for each unit exists, though unknowable, and it is not a stipulated one. In fact, because the relationships between constants and base unit are algebraic, the result of such a numerical operation is not a stipulated number, but, as shown in Eq. (2) below (derived from Eq. (1) above), is a rational number, sometimes a real number: c0*/ ΔνCs* = 3.261 225 561 106 45… ×10–2 m; h* ΔνCs*/ c0*2 = 6.777 265 181 006 83… ×10–41 kg; h* ΔνCs*/kB* = 4.411 764 153 914 89… ×10–1 K; e* ΔνCs* = 1.472 821 956 237 59… ×10–9 A; 1/NA* = 1.660 539 000 996 49… ×10–24 mol;
(2)
where the asterisk indicates the stipulated values (here from CODATA 2010 recommended values). The stipulation of the numerical values of the constants forming the algebraic expression does not automatically generate a stipulated numerical value in the definition of the units: in particular, the use of the inverse of a constant in the new definitions places the same problem. If stipulation is deemed necessary, a distinct decision should be taken with a separate specific stipulation (number rounding or truncation). The only way to do it is to resort to the original uncertain numbers coming into the expression and obtain, using the normal rules of uncertainty propagation, the corresponding uncertainty of the result of the expression, from where one has the rule for rounding (or truncating it). The correct number of digits of the result of the expression only comes from the correct number
arXiv.org > physics > arXiv:1601.00857v.3 of digits of each single stipulated number coming in it, which is established by its original uncertainty. This is apparently not universally accepted yet: for more details see [30]. In all instances, the previous procedure only supplies in itself the result of the operation as an uncertain number; e.g., for the metre c0*/ ΔνCs* = 3.261 225 5611(95) ×10–2 m. Does the implicit relationship between the constants and the base units still dictate, as the only correct procedure, to perform a type A rounding of that numerical value? Some consequences of stipulation The stipulation fixes an exact value for each unit. Does, in the New SI, each base unit retain the mandate to avoid non-uniqueness? This is not a purely mathematical issue, and it is particularly obscure in this case. As said, the New SI group-definition does not indicate any definitional method for its implementation, e.g., not anymore for c0, while the present definition does (see definition type (c) in the above Introduction); nor do the single-unit definitions. The only requirement that both definitions place, as discussed above, is: each base unit is ‘such that’ the chosen stipulated numerical value(s) of the relevant constant(s) apply, i.e. any method is acceptable until it has the potential to fulfil the above requirement. However, no method is able to comply exactly, for two main reasons: (i)
(ii)
The first reason is not of statistical nature. No pair of two distinct experimental methods implementing a definition can be able to generate exactly the same outcome: they should have the same model equation. The New SI moves the non-uniqueness problem from the definition of the measurand to the implementation of different realisations; Secondly, should a result have exactly the stipulated value, this could only happen by chance, as the value of the position parameter of an experimentally-generated distribution of values, with its probability determined by the dispersion distribution.
In general, the continuity requirement can only be approximated, and, at best, one can require that the results are “metrologically compatible” (term 2.47 in [6]) with the stipulated value [27]. In statistical terms, this would correspond to say that they would not be significantly different from the stipulated value. Figure 1 qualitatively illustrates this fact. The upper part shows the changes in the base unit definitions since 1950; until 2018 the width of the box schematises the realisation uncertainty in a qualitative way; the vertical box at 2018 indicates the expected CODATA uncertainty level before the SI change; the subsequent line indicates the degree of continuity, assumed to remain within that uncertainty; for electrical current the present inconsistency (significant step change) is shown qualitatively. Later in time inconsistent results could show up (broken line). The lower part takes, as an example, the time evolution for kB (depending on kg, m, s and K). The 1970-2014 changes of the CODATA recommended values [31] (see comments in [12]) are shown on the left. The vertical bar in 2015-18 is the foreseen experimental uncertainty (k = 1). If the base units will not show significant discontinuities in 2018 (surely not for the constants c0, ΔνCs and Kcd), the numerical value of kB is expected to remain the same within that uncertainty, to possibly change in an unforeseeable future (broken thick line). Without evidence of continuity, the value would step-change since 2018 (light gray line).
arXiv.org > physics > arXiv:1601.00857v.3
Fig. 1. Evolution of continuity of the SI base units in time. The lower part takes kB as an example.
How can a user perceive these facts? It depends on its level in the metrological hierarchy and on the type of implementation chosen. As to the level, a full discussion can be found in [32]. For top users (the National Metrology Institutes), the NMIs must demonstrate to have their own realisations done of the definition of the SI and of the base units, otherwise their country would depend on another country. At present, it means having realised to ‘definitional method’ of each unit, an affordable task for the vast majority of the countries. For the New SI the situation is not clear yet: in [2] it is stated: “The definitions of the SI units as decided by the CGPM represent the highest reference level for measurement traceability to the SI”. However, information in [3] indicates that the NMIs are preparing only “mise en pratique” [5] methods; and, in [2] the following statements are found: “A variety of experimental methods generally described by the Consultative Committees may be used to realize the definitions. Descriptions of realizations are also called ‘mises en pratique’ ”. However, with the present SI, the latter are not at the first step, but at the second of the metrological ladder, approximations of the definition as their name indicates, thus they would not fit the purpose in the New SI. Actually, a big question mark remains unexplained at present: the fact that the different implementation models allowed by the new definition can be subdivided in two categories: (i) Those providing a direct use of the relevant constant(s) (e.g., the watt balance for the kilogram, acoustic gas thermometry for the kelvin), i.e., having the constant in their model equation; (ii) Providing indirect consistency with the relevant constant(s) (e.g., with the aid of a single crystal of highly enriched 28Si for the kilogram, the ITS-90 Scale for the kelvin). Are categories (i) and (ii) both realisation methods? Of the constants’ values? Can all still be called "mise en pratique”? Are the realisations of the base units at a lower level with respect to the realisation of the constants?
arXiv.org > physics > arXiv:1601.00857v.3 In all instances, the measurements used to obtain the numerical values later stipulated in the New SI have been performed with (some of) the methods (i) only. After stipulation, any user can take advantage of the very same methods because they have demonstrated their capability to reproduce the stipulated value of the relevant constant. A realisation must always have an uncertainty associated: this is, at best, the uncertainty associated to the value of the constant (e.g., by CODATA) before stipulation. In other words, the same methods used for the initial studies become the only ones that the NMIs can use to get the top of the metrological ladder with the top accuracy. However, as a user, the method is always employed in a blind way, i.e., by assigning the stipulated value of the constant(s), in the model equation. No user, individually, is able to check if that assumption is correct. A similar blind situation already occurs at present, e.g., for the definition of the kelvin—the temperature of the triple point of water being 273.16 K exactly. Everybody freezing a sample of water at its triple point should assign to it that numerical value on the SI unit kelvin as the resulting temperature (with the associated estimate of the realisation uncertainty). This does not mean that one is forbidden from assigning that same sample a temperature numerical value different from the stipulated one, but this can be done only by: a) comparing, in a distinct experiment called inter-comparison, that sample with another one; b) taking note of the difference in the measured numerical values (if any) of temperature of the different samples, and of the associated uncertainties; c) using the other sample as a reference standard, thus assigning it the SI value 273.16 K; d) summing up the noted difference to the SI value, to obtain the SI value of the original sample—and, deciding if the latter numerical value is “metrologically compatible” [6] with the reference one or not. This is a simple experiment providing a different numerical value for the state to a specific sample, which can be done any time after the initial stipulation: however, it requires a reference standard to which the stipulated numerical value has been assigned. All the other values are relative to this choice (notice the similarity with the LSA requirement). Differently to the case of the use of a ‘reference material’, here the reference standard must be arbitrarily designated. The situation is somewhat different in the New SI. Also in this case, blindness can only be removed by comparison. If the comparison is a non-hierarchical one, the above steps a) to d) apply, and the choice of a reference, if required, is arbitrary and purely informal. However, differently in the case of the constants, for an initial period of time laboratories will exist in the world that had provided the data used for the stipulation: a comparison with one of them of a new realisation would provide the difference of the latter result with the pre-stipulation constant value. However, this would mean that the assessment of the new realisation depends on another laboratory, breaking the independence required for the top step of the metrological ladder. As a consequence, in order to avoid creating a hierarchy between countries [30], everybody proving to have realised one of the above methods must be considered as having complied with the requirement of the top step; the uncertainty arises from its uncertainty budget, with the lower bound set by the pre-stipulation value uncertainty. In that sense, those methods are called ‘primary methods’ by somebody. However, the New SI not making use of definitional methods, also other methods can be used, which are the indirect ones of the type (ii) above. They must have been formally compared with the ‘primary’ ones, so that the degree of “metrological compatibility” [6] or their ‘equivalence’ has been estimated. Strictly speaking, this should mean that also these methods should be considered, within that degree, a proxy of methods (i) for the demonstration of the fulfilment of the New SI condition—to realise the stipulated values—i.e., to implement the correct base units. If this is true, they should not be considered as methods of the mise en pratique level, but may be used as surrogates of the ‘primary’ ones.
arXiv.org > physics > arXiv:1601.00857v.3 In all instances, the fulfilment of the condition concerning the value of the constants does not represent the realisation of any specific base unit. For example, for the kelvin, the ‘dominant’ constant is the Boltzmann one, and the main method used for assessing its numerical value is acoustic gas thermometry (AGT), a relatively recent type of accurate thermodynamic thermometry. The main temperature standard presently used, presently as a mise en pratique of the kelvin, is instead the ITS-90, an empirical scale whose relationship to thermodynamic temperature was measured over decades with several different methods, where the AGT only covers a small part of the temperature range: thus, strictly speaking, a full degree of metrological compatibility of the ITS90 with the ‘primary’ method, and thus with kB, is not yet proved. Final considerations Not all these intricate issues are clarified yet in [2]. The New SI proposal has been considered like a Kuhn’s “scientific revolution” [33]. However, no achievement can be considered the “ultimate” one, as wished in the title of [7]: no achievement nor model can be considered ‘definitive’, no decision can be ‘once and for all’. A Kuhn’s revolution is only the next; there is not a final one. One should stay with Popper’s principle of falsification as a necessary condition for the scientific frame. In metrology, a decision in 2018 could even be considered ‘sufficient forever’ for its regulatory purpose. In ‘general’ science, on the contrary, it never does. At least, the scientists, and the users, should be correctly informed in a comprehensive and consistent way about the intended meaning of having the New SI based on fundamental constants, and on what the claim is based that the new definition basically does not imply any change for most users’ standards. A new basic advancement is still deemed necessary before any decision is taken in 2018 or later, starting from a sound conceptual foundation of the logical process bringing to the novel structure for the proposed New SI and from a sound definition of ‘top hierarchical level’ and, in general, of the new metrological ladder. APPENDIX: Critical review of the text of the CCU of the 9th Brochure of the New SI, draft 2016 On January 4, 2016, the BIPM published on its website [2] the adjournment of the Draft of the SI brochure concerning the New SI, prepared by the Comité Consultatif des Unités (CCU) and dated December 11, 2015. It seems useful to perform a specific critical review of Sections 1 and 2 of it, containing most of the novelties, to show the differences with respect to the 2013 Draft, no more available on the BIPM website. The new CCU draft shows a useful rearrangement of the material in the Brochure, with respect to the 2013 Draft, placing in the first ten pages the basic illustration of the New SI. The impression that a generic reader can get from it is that the new text, though clearer than that of the previous draft, is not quite informative about the reasons of the use of the constants (no more indicated as “fundamental”)—note that the above text of this Note, maintain the spelling valid until 2013 to allow the reader to appreciate the differences with respect to the 2016 Draft. A generic user can be totally lost about the differences and novelties in the implementation of the New SI required to the users. Even an informed reader, does not find in the new text what is supposed to be: not the synthetic official document defining the New SI, but a ‘brochure’ of explanations and of justifications of the new definition, about the status of these constants with respect to the new definition of the base quantities of the present-SI definition and about the entirely new twofold structure of the New SI.
arXiv.org > physics > arXiv:1601.00857v.3 Why the constants and meaning of “unit” in the New SI The Introduction in [1] states: “The definition of the SI units is established in terms of a set of seven defining constants. From the units of these defining constants the complete system of units can be derived. These seven defining constants are the most fundamental feature of the definition of the entire system of units” (emphasis added). The second sentence is circular if the constants are expressed in the units of the system they are supposed to be the definition of. In addition, the expression “the units of these defining constants” is ambiguous: does it refer to the present SI units (e.g., present-SI “velocity” for c0), or to the present set of base units (e.g., present-SI “length” and “time” for c0), or to a New SI unit for c0? In addition, the third sentence is a statement about the prevalence of the definition of the group of constants (called the group-definition hereinabove). In fact, the period then states: “The specific constants have been identified as the best choice reflecting the previous definition of the SI based on seven base units and the progress in science” (emphasis added), putting the base units as the alternative way used by the previous SI definition. This fact is confirmed by the last sentence in Section 1.1: the “description in terms of base and derived units is maintained in the present [New] SI definition, but has been reformulated as a consequence of adoption of the defining constants” (emphasis added). According to this approach, Section 1.1 “Motivation for the use of defining constants to define the SI” [2] states: “the realizations are separated conceptually from the definitions” (emphasis added), a basic statement implicitly meaning that the definition does not indicate any ‘definitional method’ [2, 34] like the previous SI, where instead in practice “the definition and the realization” were “equivalent”. The text also adds “the units can … be realized independently at any place and at any time” (emphasis added), which is ambiguous since it does not tell if, for “units”, the (previous SI) base units or the “New SI” units are referred to. In addition, it does look more a motto than a consequence of the above choice: with the exception of a single unit (kilogram) based on an artefact, the above property already applies to any present-SI base unit. The lack of ‘definitional methods’, indicated as the effect of using constants in the definition, looks one of the few—important—effective reasons for the latter choice. However, the implementation problems are actually moved, from the realization by definitional methods to the relationship between definition and realizations, and to their link to the previous SI. SI Group definition Status of the constants in the New SI One may look [2] at Section 1.2 “Implementation of the SI” for the status of the constants, depending on which the status of the “realizations” should come. The first sentence, “the definitions of the SI units as decided by the CGPM represent the highest reference level for measurement traceability to the SI”, is unequivocal in stating that the top of the metrological traceability chain (hierachical ladder) is the “definitions of the SI units”, and that this is the task of the Conférence Générale des Poids et Mesures (CGPM). Thus, realizations not directly referable to the definitions of the units are on a lower step of the hierarchical ladder. It is the task of the Consultative Committees (CC) that “provide the frame for establishing the equivalence of the realizations”, i.e., to decide the hierarchical level of different types of realizations. For understanding what is meant for “definitions of the SI units”, the reader has to look at Section 2 “The International System of Units”—notice, not ‘of Quantities’— and then to Section 2.1 “Definition of the [New] SI”:
arXiv.org > physics > arXiv:1601.00857v.3 “The International System of Units, the SI, is the system of units in which • the unperturbed ground state hyperfine splitting frequency of the caesium 133 atom ΔνCs is 9 192 631 770 Hz, • the speed of light in vacuum c is 299 792 458 m/s, • the Planck constant h is 6.626 070 040 •10 −34 J s, • the elementary charge e is 1.602 176 620 8 •10 −19 C, (A1) • the Boltzmann constant k is 1.380 648 52 •10−23 J/K, • the Avogadro constant NA is 6.022 140 857 •1023 mol–1, • the luminous efficacy Kcd of monochromatic radiation of frequency 540 •1012 hertz is 683 lm/W The numerical values of the seven defining constants [”fundamental” is now dropped] have no uncertainty. “
The above is called “group definition” in hereinbefore or “Global Constant Definition” GCD” in [35]. As to the only possible explanation for it, one finds the statement: “the definitions below specify the exact numerical value of each constant when its value is expressed in the corresponding SI unit”. Here the reader is lost, since it looks that, for “SI units”, one should intend present-SI units, not even only the base ones. In fact, as made explicit in Table 1 of Section 2.1 “The seven defining constants of the SI, and the seven corresponding units they define”, the New SI units are (in parenthesis the quantity): (frequency) Hz = s–1, (speed) m s–1, (action) J s = kg m2 s–1, (electrical charge) C = A s, (temperature) J K–1, (amount of substance) mol–1, (luminous efficacy) lm W–1 = cd sr W–1: but what is the meaning? (New base) units, as proposed, e.g., by Newell [6, 2] as to the left of the dimensional equation, or expressions of present-SI units as to the right? In all instances, if Eqs. 1 is the definition of the New SI, what are the units indicated there? In addition, concerning the stipulation of the numerical value of a constant, the sentence continues by saying: “By fixing the exact numerical value the unit becomes defined, since the product of the numerical value {Q} and the unit [Q] has to equal the value Q of the constant, which is postulated to be invariant” (emphasis added), arising from the indication “Q = {Q} [Q]. Here, Q denotes the value of the constant, and {Q} denotes its numerical value when it is expressed in the unit [Q]”. That looks basically incorrect, because the expression “the value of the constant” is equivalent to the expression “the true value of the constant”, both in the GUM [28], where simply specifying “true” is considered unnecessary, and in the VIM [6]. It is truly “invariant” with respect to the choice of the measurement unit, but is unknowable, as all the “true” values are. Thus, a numerical value of a constant can only be known approximately (“When a measurement result of a quantity is reported, the estimated value of the measurand (the quantity being measured [not the VIM definition]), and the uncertainty associated with that value are necessary. Both are expressed in the same unit”) That is true even for an exactly-defined unit [Q]: Q ≈ ({Q}, u{Q}) [Q]. No component of u{Q} can be attributed to a u[Q]; maybe Q ≈ E({Q}), because different statistical treatments can be used to obtain E({Q}). Apart from that (or within its uncertainty) the numerical value {Q} is strictly a consequence of the choice of [Q], whence the invariance of {Q} [Q]—incidentally a logical operation. Stipulation is not a necessary operation to fix [Q]; instead, it is needed to fix, when deemed necessary, the numerical estimate of Q: Q*[Q]–1 = {Q}*, where the asterisk indicates a stipulated value (exact by convention). Continuity requirement through definition changes So far, the users remain helpless about the status of the constants, because: (a) they are expressed in terms of units that should not be the ones defining the New SI; (b) a numerical value has been introduced in the new ‘group definition’ of Eqs. (A1), whose origin is said to be the following “The numerical values of the defining constants have been chosen to be consistent with the earlier
arXiv.org > physics > arXiv:1601.00857v.3 definitions [of what?] insofar as advances in science and knowledge allow” (without mentioning their CODATA origin with its associated uncertainty, [11] nor making reference to the issue of the ‘consistency’ of the set). This is justified by the cryptic sentence: “Preserving continuity is an essential feature of any changes to the International System of Units, which has always been assured in all changes to the definitions”. The continuity concept is now eventually introduced, according to a discussion going on at least since 2013 [34, 20], but the implications of its importance are omitted. Together with the lack of definitional methods, the continuity requirement is probably the only other basic feature of the New SI strategy: the reader should consult [20] for a more comprehensive illustration. In short, one can summarise here the rational of the concept of continuity and the way it works in the New SI as follows: •
• • • • •
To ensure the continuity of the units in the simplest way, the New SI is based on the same seven base units on which the present SI is founded, though differently defined—it might be considered not necessary (see Newell [13]), but the evidence of continuity would become implicit; At the time of the change, the old and the new units have the same nominal magnitude, within the current state-of-the-art uncertainty, so the old (stable) and the new standards realising them are interchangeable in measurement within the current uncertainty; The numerical values that are used in the New SI proposal come from the use of the present units; The New SI flips the definition by saying that ‘the units will be such that those numbers apply’; For doing so, those numbers are assumed to be stipulated in the new definitions (“apply exactly”); However, the ‘units such that those numbers apply’ are also the present ones.
Exactness of the continuity requirement. The basic condition (implicitly) set by the groupdefinition, the continuity requirement as defined hereinbefore, is affected by limitations, as any issue subjected to experimental verification. In fact, also this condition cannot be set exactly, because it is affected by the same inexactness referred to the determination of the numerical values of the constants, corresponding to the uncertainty associated by the CODATA to the adjusted values used in the CCU draft. The latter is the state-of-the-art continuity that can be stated—within a stated level of confidence (or degree of believe)—at the moment when the new unit definitions come into effect. This limitation cannot be removed by stipulation. One might believe that the continuity could be stated more precisely by comparing the numerical values stipulated in the present-SI definitions with the numerical values of the same quantities under the new definitions. This is not possible, because the latter values can only remain, initially, exactly the same as before, except that an uncertainty will be associated to them arising from the new definitions—e.g., the value of the triple point of water remains 273.16 K, but with an associated uncertainty equal to the relative uncertainty arising from the current determinations of the constants involved by the definition of the kelvin, about 10–6, corresponding to ≈ 2.5 · 10–4 K (the best experimental uncertainty of the realisation of that physical state being ≈ 3 ·10–5 K). The consequence is that the circularity induced by stipulation is removed by the continuity condition, because: • The new definition, by not fixing any definitional method, leaves freedom about the realisation methods; • The only condition set for the latter is to potentially enable one to get the stipulated values of the constants, so acting as a ‘continuing reference for the realization of the units’; • By definition, the present definitional method for each previous unit remains one of the possible methods of realising the corresponding new unit, within their current uncertainties;
arXiv.org > physics > arXiv:1601.00857v.3 •
The present (stable) standards realising the old definitions, can still be correctly used also under the new definitions, at least for an initial period of time. This period of time holds until one might obtain sufficient evidence of the fact that significant drifts in time of the values of some of the realisations of the old units have occurred.
SI base unit definitions What about the present base units? As recalled above, the 2016 Brochure [2] tells “description [of the new system of units] in terms of base and derived units is maintained in the present-SI definition, but has been reformulated as a consequence of adoption of the defining constants”. Thus, for the New SI it is not anymore “the definition”, but is “a description” that is maintained. Does it mean a difference in status? No more but still yes The readers find the “description” of the single base units on Section 2.2 “Definitions of the SI units” of [2]. First one has to note that this is the only place where the “definitions” of the SI units are spelled out. This has a two-fold possible meaning: (a) The SI units are not the ones indicated in Table 1. This looks in contradiction with the indication in its caption “… and the seven corresponding units they define”. (b) The definition of the SI units is provided only here. Thus the group-definition in Eqs. 1 are not the definitions of the SI units, but the definitions of the seven “traditional” (i.e. presentSI) base unit “follow from the definition of the seven defining constants”. A review of the definitions has to be different for each of the base units. •
“The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency Δν Cs, the unperturbed ground-state hyperfine splitting frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s–1 for periodic phenomena.”
Notice that the constraint in the meaning of the unit hertz means, in principle, that the time is not defined for non-periodic phenomena. •
“The mole, symbol mol, is the SI unit of amount of substance of a specified elementary entity, which may be an atom, molecule, ion, electron, any other particle or a specified group of such particles. It is defined by taking the fixed numerical value of the Avogadro constant NA to be 6.022 140 857 •1023 when expressed in the unit mol–1.”
Notice that the definition looks limited to “particles”, which is a term different from “entity”. In Appendix I Part 3 it is stated “However, because of recent technological advances, this number [“number of atoms in 12 grams of carbon 12”] is now known with such precision that a simpler and more universal definition of the mole has become possible, namely, by specifying exactly the number of entities in one mole of any substance, thus specifying exactly the value of the Avogadro constant.” (emphasis added): it does not reflect the fact that 1 mol, instead, can be said to be ‘defined by taking the fixed numerical value of the reciprocal Avogadro constant NA to be 1/6.022 140 857 •10–23 when expressed in the unit mol’. The latter definition would also be consistent with the fact that NA is an integer number, while the quantity of substance can be a real number, so fractions of the mole are allowed: on the contrary, in the proposed definition, how could be that an integer can be “expressed in the unit mol–1”, where mol–1 only sparingly can express an integer? In addition, it is unclear how to spell out an integer number written as 6.022 140 857 •1023 [30]. •
“The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit m s–1, where the second is defined in terms of the caesium frequency Δν Cs.”
arXiv.org > physics > arXiv:1601.00857v.3 Notice that the metre definition actually involves s–1 = Hz. Thus, the constraint indicated above for the second does not apply here. In addition, the explicit meaning of this definition is that ‘… it also takes the fixed numerical value of the caesium frequency ΔνCs to be 9 192 631 770’. •
“The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 040 •10–34 when expressed in the unit J s, which is equal to kg m2 s–1, where the metre and the second are defined in terms of c and ΔνCs.”
The unit symbol J, introduced here, and without spelling its name, Joule, is unknown according to the logical sequence of the text in the 2016 draft of the 9th Brochure, because it is not a base unit. In addition, the explicit meaning of this definition is that ‘… it also takes the fixed numerical value of the speed of light in vacuum c to be 299 792 458, the latter taking the fixed numerical value of the caesium frequency ΔνCs to be 9 192 631 770’. •
“The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602 176 620 8 •10–19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of ΔνCs.”
The unit symbol C, introduced here, and without spelling its name, Coulomb, is unknown according to the logical sequence of the text in the 2016 draft of the 9th Brochure, because it is not a base unit. In addition, the explicit meaning of this definition is that ‘… it also takes the fixed numerical value of the caesium frequency ΔνCs to be 9 192 631 770’. •
“The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380 648 52 •10–23 when expressed in the unit J K–1, which is equal to kg m2 s-2 K-1, where the kilogram, metre and second are defined in terms of h, c and ΔνCs.”
The unit symbol J, introduced here, and without spelling its name, Joule, is unknown according to the logical sequence of the text in the 2016 draft of the 9th Brochure, because it is not a base unit. In addition, the explicit meaning of this definition is that ‘… it also takes the fixed numerical value of
the Planck constant h to be 6.626 070 040 •10–34, the latter taking the fixed numerical value of speed of light in vacuum c to be 299 792 458, the latter taking the fixed numerical value of the caesium frequency ΔνCs to be 9 192 631 770’.
As to the degree Celsius, having the New SI eliminated to constraint for the numerical value of the triple point of water to be 273.16 K exactly, the equation “t/°C = T/K – 273.15°” cannot hold anymore if the numerical value 273.15 is kept exact, because of the statement “The unit of Celsius temperature is the degree Celsius, symbol °C, which is by definition equal in magnitude to the unit kelvin” (emphasis added). If the kelvin temperature value of the triple point of water is no more 273.16 exactly, also the numerical value 273.15 cannot be exact. In turn, should instead the latter be taken exact, the unit degree Celsius cannot be anymore by definition equal in magnitude to the unit kelvin. • “The candela, symbol cd, is the SI unit of luminous intensity in a given direction. It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540 •1012 Hz, Kcd, to be 683 when expressed in the unit lm W–1, which is equal to cd sr W-1, or kg-1 m-2 s3 cd sr, where the kilogram, metre and second are defined in terms of h, c and ΔνCs.”
The unit symbol W, introduced here, and without spelling its name, watt, is unknown according to the logical sequence of the text in the 2016 draft of the 9th Brochure, because it is not a base unit. In addition, the explicit meaning of this definition is that ‘… it also takes the fixed numerical value of
the Planck constant h to be 6.626 070 040 •10–34, the latter taking the fixed numerical value of speed of light in vacuum c to be 299 792 458, the latter taking the fixed numerical value of the caesium frequency ΔνCs to be 9 192 631 770’. Notice that the steradian, symbol sr, is one of the units with “special names”, a set
distinct from that of the derived units (Table 4 in [2]).
arXiv.org > physics > arXiv:1601.00857v.3 In other words, five of the seven base units significantly depend not only on the (numerical value of the) constant explicitly indicated in the definition, but also to up to three others. This is due to the multidimensionality of some of the constants. The reason of this implicit definition and the implications of this fact are not clarified nor commented in the new draft of the Brochure. It is important to notice a circular reasoning between the “group” definition and the “single unit” definition. In the former group definition, the single units are used, which are defined only in the latter definitions, but the latter, for defining the base units, refer to the constants in the group definition. For avoiding the newly introduced circularity, the 2016 definitions should not indicate the constants, but rather an expression like ‘units of [quantity name related to the constant]’, where for ‘quantity name related to the constant’ is: frequency for Hz = s–1, speed for m s–1, action for J s = kg m2 s–1, electrical charge for C = A s, temperature for J K–1, amount of substance for mol–1, luminous efficacy for lm W–1 = cd sr W–1, as listed above just after Eqs. 1. For example, instead of “when expressed in the unit J s, which is equal to kg m2 s–1, where the metre and the second are defined in terms of c and ΔνCs” the following text should be used: ‘when expressed in unit of action, kg m2 s-2 K-1’.
This would also make clear the relations with the use of the present-SI base units. [35] Realisations and “Mise en pratique” of the unit definitions As said herein above, the new SI “definition” is said to consist of the “definition” of seven constants, but it can be “described” by means of the same present-SI “base units”, differently—but equivalently—defined. Section 1.2 “Implementation of the SI” of the 2016 Brochure [2], states “The definitions of the SI units as decided by the CGPM represent the highest reference level for measurement traceability to the SI”, meaning that the latter level is to be that of definition of the constants. Then, Section 2.1 states: “The metrology institutes [NMI] across the world establish the practical realizations of the definitions in order to allow for traceability of measurements to the SI” (emphasis added: see below about the use of the specification “practical”). It follows that, to get “the highest reference level for measurement traceability”, any NMI should demonstrate to have its own reference to the relevant constants. This means having realised the definition of the SI. Since, contrarily to the realization of the present-SI, the NMIs have not anymore in the New SI the task to realize the definitional method stated or implied by each unit definition—because the New SI does not have such a category of methods—they have to perform a realization of the definition of the constants. That means that they have to demonstrate that they can assume, as the correct numerical values of those constants, the stipulated ones. A variety of methods exist to this effect, which can also change in time: “It is not limited by today’s science or technology but future developments may lead to different ways of realizing units to higher accuracy. Defined in this way, there is, in principle, no limit to the accuracy with which a unit might be realized”. However, the latter sentence is taken from the second period of Section 2.2.2 entitled “Practical realization of SI units”. The word “practical” looks misused, or at least ambiguous and unnecessary as any user reading the first period of Section 2.2.2 can easily understand: “The highest-level experimental methods used for the realization of units using the equations of physics are known as primary method. The essential characteristic of a primary method is that it allows a quantity to be measured in a particular unit by using only measurements of quantities that do not involve that unit. In the present formulation of the SI the basis of the definitions is different from that used previously, so new
arXiv.org > physics > arXiv:1601.00857v.3 methods may be used for the practical realization of SI units” (emphases added). In the New SI there are only realizations tout-court based on experimental methods. Instead, the term “practical” recalls too easily the kind of realizations previously called “mises en pratique”, though the latter term is not anymore explicitly used in the 2016 draft of the Brochure, except that it is recalled in its Introduction: “A variety of experimental methods generally described by the Consultative Committees may be used to realize the definitions. Descriptions of realizations are also called ‘mises en pratique’ ”. The latter sentence is inaccurate, since the BIPM definition valid for the present SI is: “A ‘mise en pratique’ for the definition of a unit is a set of instructions that allows the definition to be realized in practice at the highest level. The ‘mise en pratique’ should describe the primary realizations based on top-level primary methods” (emphases added). (from the report of the 97th meeting of the CIPM (2008) [5]). Because of that, the last sentence in Section 2.1 stating “The Consultative Committees [CC] provide the frame for establishing the equivalence of the realizations in order to harmonize the traceability world-wide” (emphasis added) can too easily be misunderstood as expressing the need for the CCs to establish new “mises en pratique”, as actually they did so far. See also the BIPM website, where Appendix 2 to 2013 Draft of the Brochure was dealing with the “mises en pratique”. On the other hand, the information still available on the BIPM website [36] contradicts the 2016 Brochure, where it reports, after the sentence “The CIPM Consultative Committees are preparing draft mises en pratique for the future new definitions of the units” (emphasis added), the “mise en pratique” of four of the seven base units (ampere, kilogram, mole and kelvin). In addition, the 2016 Draft contradicts itself when, at the end of Section 2.2.2, one is asked to consult Appendix I, Part 1 “for more complete explanation of the realization of SI units”, while this is only dealing with “the historical development of the realization of SI units”. The above description, not only is extremely confusing, but is still also incorrect, as discussed in [27] for two main reasons: (A) In the present SI, all realizations different from those indicated in the unit definition using the ‘definitional method’ where necessarily part of the “mise en pratique” of each base unit. This was then the task of the CCs since the introduction of this category of realizations, which were called “practical” just to distinguish them from the “definitional” one of each base unit. This is not anymore the situation in the New SI, as amply illustrated hereinbefore. (B) In the New SI, not all the realisation methods are of the same, but of two different types: (i) those called “primary methods” in [2] because the model includes the relevant constants among the input quantities (“using the equations of physics”), (ii) those that cannot be labelled ‘primary’ because they can only indirectly be related (traceable) to the realization of constants. Only this type might remain labelled realizations of a “mise en pratique”. Both types are now supposed to be the task of the CCs, though they necessarily bring to two different levels of the metrological hierarchical ladder. The CCU 2016 draft of the 9th Brochure can be considered an advancement in correctness with respect to the 2013 draft, by having incorporated several comments and reasoning also consistently suggested in published and in submitted papers. However, there are still many issues that currently have not yet been spelled out correctly, and a great many issues where the spelling is ambiguous (possibly reflecting unresolved problems within the CCU). A third basic metrological principle, in addition to the lack of definitional methods and to the continuity of the units through the changes in definition— the internal degree of consistency of the units of a set of dimensionally interrelated quantities—is not tackled at all. In general, a reader—also an informed one—finds him/herself helpless about the actual intention of the CCU, and about the way to understand the New SI, especially in respect to the implementation needed to the users. Many of the “illustrations” contained in this Draft rather place new doubts.
arXiv.org > physics > arXiv:1601.00857v.3
References 1. CGPM Resolution 1, 2014, http://www.bipm.org/en/cgpm-2014/ 2. CCU, New SI, Draft of 9th edition of the SI Brochure, issued on Dec 11 2015, in Draft Documents on http://www.bipm.org/en/measurement-units/new_si/ (January 4, 2016) 3. http://www.bipm.org/en/measurement-units/new-si/ (consulted on January 20, 2016) 4. http://www.bipm.org/en/publications/si-brochure/ 5. http://www.bipm.org/en/measurement-units/new-si/mise-en-pratique.html 6. BIPM (2008). International Vocabulary of Metrology—Basic and General Concepts and Associated Terms (VIM) 3rd edn. http://www.bipm.org/en/publications/guides/ 7. Knotts S, Mohr P J, Phillips W D, An introduction to the New SI, arXiv:1503.03496v2, 23 July 2015 8. Quinn T J, From Artefacts to Atoms: The BIPM and the Search for Ultimate Measurement Standards, Oxford University Press, Oxford, UK, 2011 9. Schwartz R, Richard Ph, Ehrlich C, Miki Y, OIML-Bulletin-New-SI-Draft-v4-20120831_Draft.doc, 2012, available online 10. Pavese F., Accred. Qual. Assur. 19 (2014) 307–314 11. CODATA http://physics.nist.gov/cuu/Constants/index.html , http://www.bipm.org/extra/codata/ 12. Pavese F., arXiv: 1512.03668 December 2015 13. Newell D.B., A more fundamental International System of Units, Physics Today 67(7), 35 (2014); doi: 10.1063/PT.3.2448. View online: http://dx.doi.org/10.1063/PT.3.2448 14. Fletcher N., Davis R.S., Stock M. and Milton M.J.T., Modernizing the SI – implications of recent progress with the fundamental constants, arXiv 1510.08324 (October 15, 2015) 15. Bronnikov K.A., Ivashchuk V.D., Kalinin M.I., Khruschov V.V. v, Kononogov S.A. and Melnikov V.N., On new definitions of SI base units. Why is the ‘atomic’ kilogram preferable, Measurement Techniques (english), May 2015, DOI 10.1007/s11018-015-0674-6; arXiv:1410.7906v1 29 October 2014 16. Pavese F., Ciarlini P., Classes of intercomparisons and the case of temperature standards, PTB Berichte, PTB-IT-10 (2003), 63-76. Pavese F., Dependence of the treatment of systematic error in interlaboratory comparisons on different classes of standards, ACQUAL 15 (2010) 305-315. 17. Pavese F., Mathematical and statistical tools in metrological measurement, 2013, Chapter in Physical Methods, Instruments and Measurements, [Ed. UNESCO-EOLSS Joint Committee], in Encyclopedia of Life Support Systems(EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Oxford, UK, http://www.eolss.net 18. CIPM, Report of the 94th meeting (October 2005) http://www.bipm.org/jsp/en/CIPMRecommendations.jsp 19. CGPM, 13th Meeting (1967); then CIPM, 1997; then CCTF, 1999. http://www.bipm.org/ 20. Pavese F., Problems in implementing some of the new measurement unit definitions of the SI, IMEKO TC7, Genova, September 2013, Journal of Physics: Conference Series 459 (2013) 012042 doi:10.1088/1742-6596/459/1/012042, Measurement 46 (2013) 3725–3729 21. Wapstra A. H., Audib G., Thibault C, The Ame2003 atomic mass evaluation: (I). Evaluation of input data. Adjustment procedures. Nuclear Physics A 2003; 729:129–336; Wapstra A. H., Audi G., Thibault C, (II). Tables, graphs and references, Nuclear Physics A 2003; 729:337-676 22. Audi G., Wang M., Wapstra A. H., Kondev F. G., McCormick M., Xu X., Pfeiffer B. Atomic weights and isotopic compositions. Chinese Physics C 2012; 36:1287–1602. 23. Pfeiffer B., Venkataramaniah K., Czoka U., Scheidenberger C., Atomic mass compilation 2012, Atomic Data and Nuclear Data Tables 100 (2014) 403–535 24. http://www.ciaaw.org/atomic-weights.htm 25. Ferson S., Kreinovich V., Hajagos J., Oberkampf W., Ginzburg L, Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty, SANDIA Report, SAND2007-0939, May 2007. 26. Kreinovich V., Interval computations and interval statistical techniques: estimating uncertainty of the results of data processing and indirect measurements, in Advanced mathematical and computational tools in metrology and testing X, F. Pavese, W. Bremser, A.G. Chunovkina, N. Fischer, A.B. Forbes (eds),. World Scientific Publishing Co, Singapore, 2015:38–49. 27. Pavese F., Problems in implementing new measurement unit definitions of the SI by using fundamental constants, IMEKO World Congress 2015, Praha, September 2015, http://www.imeko.org/indix.php/proceedings, also submitted to a Journal.
arXiv.org > physics > arXiv:1601.00857v.3 28. Guide to the Expression of Uncertainty in Measurement (GUM) 2nd edn 1995 (BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML) ISBN 92-67-10188-9 (electronic version JCGM100:2008 from BIPM/JCGM at http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf 29. Pavese F., On the difference of meanings of ‘zero correction’: zero value vs. no correction, and the associated uncertainty, AMCTM IX, F. Pavese, M. Baer, J.-R. Filtz, A. B. Forbes, L. Pendrill and K. Shirono, Eds., Series on Advances in Mathematics for Applied Sciences vol 84, World Scientific, Singapore, 2012, pp. 297-309. Pavese F., Why should correction values be better known than the measurand true value?, Journal of Physics 459 (2013) 012036 doi:10.1088/1742-6596/459/1/012036 30. Pavese F., Rounding and notation, namely when using stipulations in the definition of measurement units, Measurement 46 (2013) n.9, ISSN 0263–2241, p. 3725–3729 31. P.J. Mohr, D.B. Newell, B.N. Taylor (2015) CODATA recommended values of the fundamental constants 2014, arXiv:1507.07956v1 32. Pavese F., How much does the SI, namely the proposed ‘‘New SI’’, conform to principles of the Metre Treaty?, Accred. Qual. Assur. 19 (2014) 307–314. 33. Pietsch W., A revolution without tooth and claw—Redefining the physical base units, Studies in History and Philosophy of Science 46 (2014) 85–93. 34. Pavese F., A digest for scientists and practitioners on pros and cons of the proposed new definition of the International System of measurement units (SI) 2015, submitted to a Journal 35. Mari L. and Pavese F., Again on the New SI (“Ancora sul nuovo SI”), in Italian, Tutto_Misure 18 (2016) in press. Mari L. and Pavese F., A conceptual roadmap for setting up a system of units in the “New SI” context, submitted to IMEKO TC7 Workshop, Berkeley, USA, August 2016. 36. http://www.bipm.org/en/measurement-units/new-si/#communication
