The Current SI Seen From the Perspective of the Proposed New SI.

A revised International System of Units (SI) proposed by the International Committee for Weights and Measures is under consideration by the General Conference on Weights and Measures for eventual adoption. Widely recognized as a significant advance for both metrology and science, it is defined via statements that explicitly fix the numerical values of a selected set of seven reference constants when the values of these constants are expressed in certain specified units. At first sight this approach to defining a system of units appears to be quite different from that used to define the current SI. However, by showing how the definitions of the seven base units of the current SI also fix the numerical values of a set of seven reference constants (broadly interpreted) when the values of these constants are expressed in their coherent SI units, and how the definition of the current SI can be recast into the same form as that of the revised SI under consideration, we show that the revision is not as radical a departure from the current SI as it might initially seem.

1. Introduction

Although it is assumed that the reader is familiar with the current International System of Units (SI), for convenience, Sec. 4, Appendix A, which is based on Refs. [1, 2], provides a brief review of the current SI and the international bodies responsible for it, namely, the CGPM, CIPM, BIPM, and CCU (General Conference on Weights and Measures, International Committee for Weights and Measures, International Bureau of Weights and Measures, and Consultative Committee on Units, respectively). However, the actual definitions of the current SI base units, second s, meter m, kilogram kg, ampere A, kelvin K, mole mol, and candela cd, are not included in Appendix A, because they are given and used in Secs. 2.1–2.7 of the main text. This order of base units is employed throughout the paper so that no unit definition depends on another unit whose definition appears later in the sequence, rather than the traditional order m, kg, s, A, K, mol, and cd.

1.1 The New SI

Over at least the last dozen years there has been extensive discussion in various international and national forums concerning the possible redefinition of the SI in order to meet the challenges of the 21st century [3-9]. The culmination of this discussion is Draft Resolution A considered by the 24th CGPM at its October 2011 meeting [10], which was prepared by the CIPM upon the recommendation of its CCU. Reference [11] reviews and discusses in some detail the development of and the basis for the proposals in Draft Resolution A and summarizes the many benefits of the “New SI” as it has come to be called. Indeed, a broad consensus now exists in both the international metrological and scientific communities in support of the adoption of the New SI [12]. Section 5, Appendix B gives its definition as presently envisioned, the formal adoption of which could be at the 25th meeting of the CGPM in 2014.

As can be seen from Appendix B, the New SI is based on the set of seven reference constants Δν (133Cs)hfs, the ground state hyperfine splitting frequency of the cesium 133 atom; c, the speed of light in vacuum; h, the Planck constant; e, the elementary charge (charge of a proton); k, the Boltzmann constant; NA, the Avogadro constant; and Kcd, the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz. The principal purpose of this paper is to show fully and clearly that the current SI can also be understood as being based on a set of seven reference constants and that its current definition can be recast, as demonstrated in Sec. 6, Appendix C, into the same form of wording as the New SI as given in Appendix B. It is thus shown that the New SI, although widely recognized as a major advance, is not the radical departure from the current SI as it might first appear. (Appendices are used to keep the main text concise and focused on the paper’s principal purpose.)

1.2 Quantity, Value of a Quantity, and Numerical Value of a Quantity

As a prelude, we briefly recall the relationship between a quantity, its value, and its numerical value when the value of the quantity is expressed as a number times a unit. In general, the value of a well-defined quantity A is expressed as where {A}[] is the numerical value of A when the value of A is expressed in the unit [A]. Equation (1) can be rearranged to read which means that if A can be assumed to be an invariant quantity, then the unit [A] can be defined as the quotient of the invariant quantity A and a fixed (i.e., exact) numerical value. Simply put, by fixing the numerical value of an invariant quantity such as the property of an atom or a fundamental constant, we define its unit.

To illustrate, let us assume that the height of the Washington Monument hWM (an obelisk about 169 m tall on the National Mall in Washington D.C. built to commemorate George Washington) is an invariant quantity and that we wish to define a new unit of length for measuring the height of tall buildings called the “washington,” symbol Wa. We could do this by fixing the numerical value of hWM to be, for example, exactly 10 when the value of hWM is expressed in the unit Wa. It then follows from Eq. (2) that the washington is defined by Wa = hWM/10, and from Eq. (1) hWM = 10 Wa. (Since hWM ≈ 169 m, 1 Wa ≈ 16.9 m.)

2. Base-Unit Definitions and the Seven Reference Constants of the Current SI

With Sec. 1.2 in mind, we turn our attention to the identification of the set of reference constants on which the current SI is based, by which we mean the set of seven constants (broadly interpreted) whose numerical values are fixed by the seven current SI base-unit definitions. In terms of the discussion in Sec. 1.2, these constants correspond to the quantity A on the left-hand-side of Eq. (1) and in the numerator of the right-hand-side of Eq. (2). Our procedure is to first give the current SI definition of the base unit in bold type, and then immediately following the definition the derivation of the associated reference constant. The definitions of the base units are taken from Refs. [1, 2], but for simplicity we do not include any accompanying statements meant to clarify a definition or to indicate how the definition should be realized in practice.

2.1 Second and Δν (133Cs)hfs

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

Since the period T of a sinusoidal electromagnetic wave is related to its frequency ν by the equation T = 1/ν, if the symbol for the ground state hyperfine splitting transition frequency of the cesium 133 atom is taken to be Δν (133Cs)hfs, then it follows from the definition of the second that or since for a periodic signal s−1 = Hz, and thus

We see that Eq. (3a) defines s, Eq. (3b) is identical in form to Eq. (2) and defines Hz, and from Eq. (3c), which is identical in form to Eq. (1), that the reference constant whose numerical value is fixed when it is expressed in its SI unit Hz is Δν (133Cs)hfs.

2.2 Meter and c

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

Since c is the symbol for the speed of light in vacuum, the length of path l travelled by light in vacuum in a time t is given by l = ct. It therefore follows from the definition of the meter that or and thus

In this case we see that although Eq. (4a) formally defines m, it also depends on the definition of s, that Eq. (4b) defines the unit m s−1, and from Eq. (4c) that the reference constant whose numerical value is fixed when it is expressed in its SI unit m s−1 is c. It is important to recognize that the formal definition of the meter does not actually define m but the unit m s−1, and thus without a definition of the second the meter is not completely defined by its formal definition.

2.3 Kilogram and m( )

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

The international prototype of the kilogram (IPK) is denoted by the symbol , hence the symbol for its mass is simply m( ). It therefore follows from the definition of the kilogram that or and thus

Here we see that Eq. (5a) formally defines kg, that Eq. (5b) also defines the unit kg, and from Eq. (5c) that the reference constant whose numerical value is fixed when it is expressed in its SI unit kg is m( ). Of course, the mass of the IPK is not an invariant of nature and hence not a constant in the sense of Δν (133Cs)hfs and c. Indeed, one of the principal motivations for the New SI is the fact that the kilogram is the only SI base unit whose definition is still based on a material artifact.

2.4 Ampere and μ0

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 × 10

The expression from electromagnetic theory for the force F per length l between two straight parallel conductors a distance d apart in vacuum, of infinite length and negligible cross section, and carrying currents I1 and I2 is where μ0 is the magnetic constant (also called “permeability of vacuum”). Applying this expression to the definition of the ampere, we have F = 2 × 10−7 N, l = 1 m, I1 = I2 = 1 A, and d = 1 m, which leads to or and thus

In this case we see that although Eq. (6a) formally defines A, it also depends on the definitions of s, m, and kg since N = s−2 m kg, that Eq. (6b) defines the unit N A−2 = s−2 m kg A−2, and from Eq. (6c) that the reference constant whose numerical value is fixed when it is expressed in its SI unit N A−2 = s−2 m kg A−2 is μ0. (The value of μ0 is often expressed in the equivalent unit henry per meter, H m−1, where H = s−2 m2 kg A−2.) As in the case of the meter, it is important to recognize that the formal definition of the ampere does not actually define A but the unit N A−2 = s−2 m kg A−2, and thus without definitions of the second, meter, and kilogram the ampere is not completely defined by its formal definition.

2.5 Kelvin and TTPW

The kelvin, unit of thermodynamic temperature, is the fraction 1 / 273.16 of the thermodynamic temperature of the triple point of water.

If TTPW is taken as the symbol for the thermodynamic temperature of the triple point of water, in principle an invariant quantity, it follows directly from the definition of the kelvin that and hence

Here we see that Eq. (7a) formally defines K, and from Eq. (7b) that the reference constant whose numerical value is fixed when it is expressed in its SI unit K is TTPW.

The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is “mol.”

When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

2.6 Mole and M(12C)

The molar mass of an entity X, symbol M(X), is the mass of X divided by the amount of substance n of X; its coherent SI derived unit is kilogram per mol, kg mol−1. If the entity X is the carbon 12 atom, symbol 12C, then it follows from the definition of the mole that the molar mass of carbon 12 is 0.012 kg divided by one mole, M(12C) = 0.012 kg/(1 mol), hence or and thus

In this case we see that although Eq. (8a) formally defines mol, it also depends on the definition of kg, that Eq. (8b) defines the unit kg mol−1, and that from Eq. (8c) the reference constant whose numerical value is fixed when it is expressed in its SI unit kg mol−1 is M(12C). Again, as in the case of the meter, it is important to recognize that the formal definition of the mole does not actually define mol but the unit kg mol−1, and thus without a definition of the kilogram the mole is not completely defined by its formal definition.

2.7 Candela and Kcd

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 10

The luminous intensity of monochromatic radiation in a given direction, Iv, is related to the radiant intensity of the radiation, Ie, and the spectral luminous efficacy of the radiation at the frequency of the radiation, K, by the equation Iv = IeK. Applying this relation to the definition of the candela with Iv = 1 cd and Ie = (1/683) W sr−1, and denoting K at the frequency 540 × 1012 Hz by the name luminous efficacy with quantity symbol Kcd, yields or since lm = cd sr, and thus

Here we see that although Eq. (9a) formally defines cd, it also depends on the definitions of s, m, and kg since W = s−3 m2 kg and sr = m2 m−2 = 1, that Eq. (9b) defines the unit lm W−1 = s3 m−2 kg−1 cd sr since lm = cd sr, and from Eq. (9c) that the reference constant whose numerical value is fixed when it is expressed in its SI unit lm W−1 = s3 m−2 kg−1 cd sr is Kcd. Once again it is important to recognize that the formal definition of the candela does not actually define cd but the unit lm W−1 = s3 m−2 kg−1 cd sr, and thus without definitions of the second, meter, and kilogram, the candela is not completely defined by its formal definition.

2.8. Explicit-Unit vs. Explicit-Constant Definitions

The definitions of the current SI base units in Secs. 2.1–2.7 are now called “explicit unit,” because the exact value of the constant to which the unit is linked is specified indirectly by explicitly defining the unit in terms of a quantity of the same kind as the unit [4, 5]. They should be compared with those for the same units as given in Part II of the definition of the New SI in Sec. 5, Appendix B. These are of a form that explicitly indicates the constant to which the unit is linked. This form is called “explicit-constant,” because the unit is defined indirectly by specifying explicitly an exact numerical value for a constant of nature [4, 5]. The formulation of the definition of the New SI as given in Appendix B, which begins with an overall scaling statement followed by base-unit definitions in explicit-constant form, has the great advantage that the foundation of the New SI is uncovered for all to see. Moreover, the explicit-constant form has the additional advantage that each base unit can be defined in the same way, thereby increasing the uniformity of their definitions and one would expect their understandability.

3. Conclusion

In summary, we have shown explicitly in Secs. 2.1–2.7 that in the current SI the ensemble of base units, s, m, kg, A, K, mol, and cd, are actually defined by fixing the numerical values of the set of seven reference constants Δν (133Cs)hfs, c, m( ), μ0, TTPW, M(12C), and Kcd when these constants are expressed in the units s, m s−1, kg, N A−2, K, mol−1, and lm W−1, respectively, and as a result, the values of these seven constants in SI units are exactly known.

Equally as important, we have shown in Sec. 6, Appendix C how the definition of the current SI can be recast into a fully equivalent form that is identical to the two-part form proposed for defining the New SI by the CIPM and which is under consideration by the CGPM for eventual adoption. Thus, putting aside the difference in the wordings of the definitions, the principal metrological difference between the current SI and the New SI is that the group of four reference constants m( ), μ0, TTPW, and M(12C) in the current SI is replaced in the New SI by the group h, e, k, and NA.

Of course, the difference in wordings is highly significant, because the New SI wording makes eminently clear the constants and their values in terms of which it is defined; this is certainly not the case for the wording of the current SI. Indeed, as discussed in Sec. 7, Appendix D, the Part I simple but clear form of definition of the New SI can be viewed as a more general, easier to understand, and transparent way to formulate a system of units.

As indicated in Sec. 4, Appendix A, the CGPM formally established the SI in 1960, some 50 years ago. Nevertheless, it was not until the publication by the BIPM of the 7th edition of the SI Brochure in 1998 that the effect of the base-unit definitions on the values of certain constants was begun to be widely noted. In particular, in this edition the formal definitions of the meter and ampere were followed by the statements “Note that the effect of this definition is to fix the speed of light at exactly 299 292 458 m s−1,” and “Note that the effect of this definition is to fix the permeability of vacuum at exactly 4π × 10−7 H m−1,” respectively. It was not until the 8th edition of the SI brochure published in 2006 [1] that a similar explanatory statement followed each formal base-unit definition. Moreover, to the best of the author’s knowledge, there has never been a published paper prior to this one that explicitly shows how these definitions lead to exact values of Δν (133Cs)hfs, c, m( ), μ0, TTPW, M(12C), and Kcd.

One may well wonder why this useful way of viewing the SI did not gain prominence sooner. Although we have no real answer to this question, a motivating factor for the inclusion of the two explanatory statements in the 7th edition of the SI Brochure was likely the discussion by Ulrich Feller of the national metrology institute of Switzerland at the 21st meeting of the CIPM’s Consultative Committee on Electricity (CCE) of document CCE/97-3, which he had submitted to the meeting [13, 14]. In it, Feller points out the dependence of five of the SI base units on fundamental constants.

Table 1

Concise summary of the current SI base quantities and units (adapted from Refs. [1, 2] and unchanged in the New SI)

Base quantity	Base unit
Name	Name	Symbol
time	second	s
length	meter	m
mass	kilogram	kg
electric current	ampere	A
thermodynamic temperature	kelvin	K
amount of substance	mole	mol
luminous intensity	candela	cd

Table 2

Concise summary of the coherent derived units in the current SI with special names and symbols that appear in this paper (adapted from Refs. [1, 2] and unchanged in the New SI)

Derived quantity	SI coherent derived unit		Expressed in terms of other SI units	Expressed in terms of SI base units
	Name	Symbol
solid angle	steradian	sr		m2 m−2 = 1
frequency	hertz	Hz		s−1
force	newton	N		s−2 m kg
energy	joule	J	Nm	S−2 m2 kg
power	watt	W	Js−1	s−3 m2 kg
electric charge	coulomb	C		s A
voltage	volt	V	WA−1	s−3 m2 kg A−1
electric resistance	ohm	Ω	VA−1	s−3 m2 kg A−2
inductance	henry	H	V s A−1	s−2 m2 kg A−2
luminous flux	lumen	lm	cd sr	m2 m−2 cd = cd

Table 3

The New SI as defined in Part I of Sec. 5, Appendix B viewed as an alternative formulation of a system of units. The words “per” and “by” are to be interpreted to mean “divided by” and “times,” respectively. Column one gives the set of seven reference constants of the New SI, columns two and three the kind of quantity and its commonly used symbol of which each reference constant is a particular example, and columns four and five the unit and its symbol defined by assigning an exact numerical value to the corresponding reference constant. The last two columns show how the kind of quantity of which each reference constant is a particular example can be expressed in terms of other quantities of the system of quantities. (N is the quantity symbol for cycles, which is a quantity of dimension one and can be a decimal number; its unit is the number one [1, 2]. t, l, I, and m are quantity symbols for time, length (distance), electric current, and mass, respectively)

Reference constant	Kind of quantity and its symbol		Unit and its symbol fixed by reference constant		Quantity in terms of other quantities and their symbols
Δν (133Cs)hfs	frequency	ν	hertz	Hz	cycles per time	N/t
c	velocity	υ	meter per second	m s−1	length per time	l/t
h	action	S	joule second	J s	energy by time	Et
e	charge	Q	coulomb	C	current by time	It
k	(change in) energy per (change in) thermodynamic temperature	E/T	joule per kelvin	J K−1	mass by velocity squared per thermodynamic temperature	mυ2/T
NA	reciprocal amount of substance	n−1	reciprocal mole	mol−1
Kcd	luminous flux per power	Φv/P	candela steradian per watt	cd sr W−1	luminous flux per energy per time	Φv/(E/t)