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Measurement Techniques, Vol. 60, No. 7, October, 2017
0543-1972/17/6007-0649 ©2017 Springer Science+Business Media New York 649
All-Russia Research Institute of Metrological Service (VNIIMS), Moscow, Russia; National Research Center Kurchatov Institute, Moscow,
Russia; e-mail: khkon@vniims.ru. Translated from Izmeritel’naya Tekhnika, No. 7, pp. 3–7, July, 2017. Original article submitted May 30, 2017.
FUNDAMENTAL PROBLEMS IN METROLOGY
AN ALTERNATIVE SET OF DEFINING CONSTANTS
FOR USE IN REDEFINING THE FOUR UNITS
OF THE INTERNATIONAL SYSTEM OF UNITS
V. V. Khruschov UDC 53.081:531.7
Different sets of constants the fi xed values of which may be selected for new defi nitions of the four units
(kilogram, mole, ampere, and kelvin) of the International System of Units are discussed. The concept of the
“order of a constant” in a given system of units is proposed. Criteria for arriving at an optimal selection of
defi ning constants as well as a set of constants consisting of Planck’s constant h, Avogadro’s constant NA,
Boltzmann’s constant k, and the magnetic permeability of a vacuum (magnetic constant) μ0 are considered.
The proposed set is an alternative to the base set consisting of h, e, k, and NA.
Keywords: redefi nition of the units of the International System of Units, dimension of a physical quantity,
defi ning constant, Planck’s constant, Avogadro’s constant, Boltzmann’s constant, magnetic permeability of
a vacuum, magnetic constant.
Existing systems of the units of the physical quantities are a constituent part of the necessary set of measuring instru-
ments used to conduct scientifi c studies. Modern technologies, the development of industry, and international trade presup-
pose a corresponding scientifi c level of the units of the physical quantities. The International Prototype of the Kilogram pro-
duced from a platinum-iridium compound continues to be used today, despite the fact that it was found to be temporarily
unstable at a level of 5·10–10 kg/yr [1, 2]. This is not acceptable when performing precision measurements and the subsequent
use of the results of these measurements over a long period of time. A redefi nition of the four base units of the International
System of Units (SI) (review of SI), which is now at the stage of preparation, will solve this problem [3, 4]. The planned re-
view of the SI is based on a proposal that involves redefi ning the four units of measurement by establishing exact values of
selected physical constants, following the method used to redefi ne the ampere in 1946 and that used to redefi ne the meter in
1983 [5–7]. A new term, defi ning constant (DC), is used in the ninth edition of the SI Brochure [4] in place of physical con-
stants with fi xed values. The main obstacle to making a transition to the new defi nitions is the insuffi cient level of precision
of the experimental values of the proposed DCs.
A proposal to redefi ne a number of the units of the SI appeared in 1999 [8] and a more complete set of proposals in
2006 [6]. Several different sets of DCs were considered. In particular, it was suggested that the values of Planck’s constant h,
the elementary charge e, Boltzmann’s constant k, and Avogadro’s constant NA should be established precisely without any
experimental uncertainty in order to redefi ne the kilogram, ampere, kelvin, and mole, respectively [4, 6, 7]. This set of DCs is
now considered the most preferable. The International System of Units with new defi nitions of the four base units was pre-
sented in [4], i.e., a Modifi ed System of Units (MSI) or New System of Units is realized on the basis of this set.
DOI 10.1007/s11018-017-1250-z
650
The existing situation with a transition to the MSI is refl ected in Resolution 1 of the 24th CGPM in 2011, that is, to
“continue work on improving the statements of the defi nitions of the base units of the SI based on the fundamental constants,
which possess suffi cient scientifi c rigor and clarity, in order to enable the simplest possible understanding of the constants by
users.” Nevertheless, in 2014 the 25th CGPM, noting the progress that had been achieved following the 24th CGPM, recom-
mended that Resolution 1 of the 24th CGPM continue to be implemented in view of the insuffi cient precision of the measured
values of the DCs. It was proposed that the MSI should be adopted at the 26th CGPM in 2018 [3].
The objective of the present study is to present a critical discussion of the proposed new defi nitions of the four base
units of the SI in light of recent theoretical and experimental results [9–13]. A new concept, that of the order of a constant in
a given system of units, is introduced and a set of criteria that may be used for selecting an appropriate set of DCs is proposed.
An alternative set of DCs, comprising h, k, NA, and the magnetic permeability of a vacuum (magnetic constant) μ0, is pro-
posed. The advantages of this set by comparison with the base set of h, e, k, and NA, which is the one that is currently the
selected set, are noted. It should be expected that a review of these questions will be useful for selecting an optimal set of DCs
and the subsequent use of this set.
The article proposes a new classifi cation of the DCs in accordance with their dimensions relative to the base units
based on the newly introduced concept of the order of a constant and discusses criteria for arriving at a preferential selection
of DCs. Different sets of DCs that have been proposed for a redefi nition of the four units of the SI are then analyzed. New
defi nitions of the kilogram, mole, ampere, and kelvin based on fi xed values of the h, μ0, k, and NA (system D) are considered
next. The use of the concept of the order of a DC is considered separately and criteria for selecting DCs are described. The
advantages and drawbacks of system D relative to other sets are noted.
Classifi cation of the defi ning constants and criteria for their preferential selection. In [4], the Modifi ed System
of Units is presented on the basis of fi xed values of h, e, k, and NA. The general concept of a DC is used in place of more
specialized concepts, such as physical constant, natural constant, International Prototype of the Kilogram, physical parameter,
etc. The set of constants is divided into four classes: the fundamental physical constants, special atomic constants, conversion
factors, and technical constants. Such a classifi cation has a number of drawbacks, however. For example, the speed of light in
a vacuum c is a fundamental physical constant, though once its value is established, it becomes a conversion factor in the
sense used within the framework of the MSI [4]. Once NA is established, it also becomes a conversion factor between the unit
of the amount of a substance and the unit used to count the number of objects [4, 14], though it is also a natural constant
(Avogadro’s law). It is known that the number of constants varies together with the number of base units [15]. Moreover,
within the framework of a given system of units, the number of constants depends on the physical theory or model employed.
For example, within the framework of nonrelativistic classical mechanics there are no constants c or h, whereas they are pres-
ent in relativistic quantum mechanics. In [16], a vector space is associated with the indicators of the degrees of units for the
dimensions of quantities, and a criterion for determining a complete set of constants is also proposed. Different classifi cations
of the physical constants based on their role in the description of physical phenomena are considered in [17].
Below, a new characteristic for constants defi ned in some system of units is proposed. A constant that does not de-
pend on the base units of a given system is said to be a zero-order constant. A constant that does depend on a single base unit
is called a fi rst-order constant and so on. Such a classifi cation of the constants in accordance with their orders is used below
for selecting a set of DCs.
At fi rst glance, it would appear that the value of a physical constant cannot be established in view of the average value
and uncertainty of the result of measurements which are assigned to the value of a physical constant. However, in view of the
relationship between a physical constant and a unit of measurement, the value of a physical constant may be established precisely
if, at the same time, there is no substantial increase in uncertainty in the realization of a redefi ned unit and if, moreover, this leads
to additional advantages in the measurement of other quantities. Such a physical quantity with fi xed value will be a DC for the
given unit of measurement. It would desirable for a set of DCs to be consistent and minimal and not lead to any contradictions
when several DCs are used simultaneously. Of course, the status of any selected DC is not absolute and may vary in a general-
ization of the particular theory in use or with an increase in the precision of measurements of certain physical constants.
In any case, the requirement imposed on the MSI, that it not worsen the situation that now exists for users of the SI,
is critical. This means that there may exist a defi nite succession or continuity relative to the existing SI. Succession
651
presupposes the use of current revisions of the base units and their values in effect at the present time in order to preserve the
vast set of accumulated measurement data without having to perform any corrections.
It is also obvious that a review of the SI must not reduce the stability of the realization of the units. For example, the new
prototypes of the unit of mass approved by the International Bureau of Weights and Measures must possess a temporary instability
that is less than 5·10–10 a year at any time and in any place. Recall that when this condition was not observed for the International
Prototype of the Kilogram, this led to a scheduled transition to the MSI [1, 2]. In order that no contradictions arise between the
system or set of DCs and existing physical theories, the number of DCs must be minimal and the relationships between the DCs
and the units of measurement must be as simple as possible. It is desirable that these requirements be observed in order to success-
fully teach the foundations of metrology in universities and colleges, and this may be achieved if the base unit of measurement and
the corresponding DC are of the same dimension or if the relation between their dimensions is the simplest possible. For example,
the dimension of the speed of light and that of the meter and, similarly, the dimension of the charge of an electron and that the
ampere each differ only in terms of the defi ning degree of time. The dimension of the kilogram coincides with the dimension of
the atomic unit of mass mu = m12C/12, where m12C is the mass of 12C. In terms of the concept of the orders of constants introduced
above, a criterion necessary for this purpose may be stated as follows: the order of a defi ning constant must be the least possible.
Thus, we wish to propose the following criterion for selecting an optimal set of DCs for new defi nitions of the units
[18–20]: (a) a succession between the old and new defi nitions; (b) stability in the transition of the units of measurement; (c) mini-
mal number of DCs; (d) least possible order of a DC.
Analysis of sets of defi ning constants proposed for a redefi nition of the four base units of the International
System. Discussions on the advantages and drawbacks of the proposed new defi nitions of the four base units of the SI are
continuing today [9, 19, 20–22]. The set of DCs consisting of h, e, k, and NA is the preferred variant. Other variants have also
been proposed [8, 9, 19, 20]. For example, in the opinion of the authors of [9], the attained level of the data of CODATA-2014
[13] determines a new context for selecting an optimal set of DCs by comparison with what had been selected in 2007 at the
23rd CGPM [6, 23]. Five possible variants of the SI with different sets of DCs, including the existing set, are considered in
[9]. In the existing SI, the defi nitions of the kilogram, ampere, kelvin, and mole are based on fi xed values of the mass of the
International Prototype of the Kilogram m(K), μ0, the triple point of water Ttpw, and the molar mass of 12C M(12C). In [9],
the effective SI is denoted System A. Other variants of the MSI are based on fi xed values of sets of constants containing h, e,
k, NA, m0, and μ0. Recalling that in all these variants the values of k and NA are fi xed, these variants may be distinguished by
selecting two DCs from among h, e, mu, and μ0. Thus, system B contains h and e; system C, e and mu; system D, h and μ0;
and system E, mu and μ0. Note that the pair h and mu is excluded in [9], since it leads to a redefi nition of the second, while the
pair e and μ0 is excluded because of the lack of any advantage by comparison with variants B, C, D, and E. Let us consider
in more detail the three systems B, C, and E (system D will be described subsequently). System B is a variant of the MSI pro-
posed in [6] and noted in Resolution 1 of the 24th CGPM as a base system. System B assures null uncertainty of the Josephson
constant KJ and the von Khitzing constant RK, which are used as practical electromagnetic standards, i.e., a current problem
of electrical metrology, namely, the existence of fi xed values of KJ and RK adopted in 1990 and consistent with experimental
data within the limits of their uncertainty, is solved in this variant. However, this does not correspond to the actual situation
today, i.e., the values of 1990 must be replaced by new values on the basis of recent CODATA data for h and e published at
the time of a revision based on their known relation to KJ and RK.
Note that for system B the two constants μ0 and Mu (molal mass constant) must vary as functions of their current
experimental data. For example, a dependence for μ0 may be represented as 4π(1 + δ)·10–7 N/A2, where δ = α/α2018 – 1 with
α2018 the best attained experimental value of α at the moment of redefi nition [9]. Values of ur for δ and α are equal, provided
that the value of α2018 is exact. Thus, if the deviations of μ0 and Mu from the values of 4π·10–7 N/A2 and 1 g·mol–1 are suffi -
ciently small, there are basically no problems. But the factor (1 + δ) is hard to explain when introducing μ0 into textbooks and
in lectures on electromagnetism, i.e., the presence of this factor does not agree with Resolution 1 of the 24th CGPM that de-
mands that the MSI must be easily understood by users in general and must be consistent with scientifi c rigor and clarity [3].
This situation is similar to the problem of a varying molal mass constant Mu within the framework of systems B and D.
The new defi nitions of the kilogram and mole in systems E and C may be implemented either by a watt balance or
by silicon spheres. The equivalence of these methods follows from the well-known relationship
652
NAh = Ar(e)Mucα2/(2R∞),
where Ar(e) = me/mu is the relative atomic mass of an electron; Mu = M(12C)/12, where Mu = NAmu coincides with
Mu0 = 1 g·mol–1 in the current SI, and R∞ is the Rydberg constant. The value of Planck’s molal constant is known with
ur = 4.5·10–10 [13], so that there is no loss in precision when determining the value of NA by means of h and conversely. There
are advantages to the use of systems E and C in defi ning the kilogram and the mole [24] that had not been previously noted
due to the high level of uncertainty for the values of RK and KJ, leading to a decrease in the precision of electromagnetic
measurements. System C is considered in detail in [20, 22, 25–27]. It is an intermediate system between system B [6] and
system E [9] and is optimal relative to the proposed criteria.
New defi nitions of the kilogram, ampere, kelvin, and mole in system D. Each variant set of DCs has advantages and
drawbacks which were best considered and discussed in the preparation of the resolution of the 26th CGPM. In the present study,
we will consider in detail an alternative set of DCs used for the new defi nitions of the four units of the SI. This set (system D)
consists of h, μ0, k, and NA and was fi rst proposed by the Working Group on the Base Units of the SI and Fundamental Constants
of the Academy of Sciences in Paris (it was suggested that Planck’s charge be assigned the fi xed value qP = (2ε0hc)1/2) [30].
In system D, the new defi nitions of the mole and kelvin are based on fi xed values of NA and k as in systems B, C,
and E, the new defi nition of the kilogram, on a fi xed value of h as in system B, and the new defi nition of the ampere on a fi xed
value of μ0 as in the existing SI. Because of the lack of a fi xed value of the charge of an electron e in system D, there are
experimental uncertainties for RK and KJ, though they are very small (~10–10) and may be ignored. For exact electromagnetic
measurements, it is only necessary that the values of RK and KJ be stable and known with uncertainties less than some limiting
value. The fi ne structure constant α plays a central role in the determination of the uncertainty of the values of RK and KJ. The
value of α was recently confi rmed by the result of an exact experiment based on the yield of an atom (measurement of h/mu
(97Rb) [28]) that was independent of quantum electrodynamics. Improvements in methods of determining α will continue and
these will lead to a high degree of precision in the determination of the two constants RK and KJ.
Let us now turn to a discussion of the advantage of the fi xed value of μ0 in system D, μ0 = 4π·10–7 N/A2. First, set-
ting a fi xed value for μ0 is compatible with setting a fi xed value of c, a step that had been taken in in 1983. Since the dielectric
permittivity ε0 and the magnetic permeability μ0 of a vacuum obey the relationship ε0μ0c2 = 1, ε0 is also fi xed and equal to
8.854187817... F·m–1. Recalling that α = e2/(2ε0hc), we obtain ur(e2) = ur(α). Note that for the purpose of estimating the
uncertainties of different constants in system D, the results of spectroscopic measurements depend only on the units of length
and time. For example, it is suffi cient to express mu and e in terms of the fi xed constants h and μ0 and the measurable constants
α, Ar(e), and R∞. A fi xed value of μ0 presupposes that the vacuum impedance Z0 is also fi xed. Thus, not only may RK be de-
termined by a direct comparison to Z0 by means of an experiment with a computable capacitor, but KJ may also be determined
using a watt scale, without having to verify whether the formulas relating e, h, and α are valid. Thus, system D is more appro-
priate for assuring scientifi c and technological progress.
Orders of defi ning constants and use of criteria for selecting systems B, C, D, and E. Let us fi nd the orders of
the DCs h, e, k, NA, mu, and μ0 which have been proposed for a redefi nition of the four units of the SI. This is needed in order
to use the criteria for selecting the DCs. The dimensions of these constants relative to the base physical quantities of the SI are
as follows:
[h] = ML2T–1, [e] = TI, [NA] = N–1, [k] = ML2T–2 θ–1, [mu] = M, [μ0] = LMT–2I–2.
We may now calculate the orders of these constants in accordance with the defi nition of the order of a constant:
O(h) = 3, O(e) = 2, O(NA) = 1, O(k) = 4, O(mu) = 1, O(μ0) = 4,
where O(·) is the order of a DC.
Moreover, the order of a system of constants
S = {c1, c2, c3, …}
653
may be defi ned in the following way:
O(S) = ΣO(c1).
Then, O(B) = 10, O(C) = 8, O(D) = 12, and O(E) = 10.
In systems B, C, D, and E, ku and NA are used for new defi nitions of the units of temperature and quantity of matter,
respectively. It remains for us to select two constants for the new defi nitions of the units of mass and current intensity. If cri-
terion d is used directly, we obtain the constants mu and e. In this case, system C proposed in [8] and reviewed in detail for the
redefi nition of the kilogram, mole, ampere, and kelvin in [20, 22, 25, 26] must be selected. Implementation of the new defi ni-
tions must be performed on the basis of criterion b.
Above, the most optimal system C for redefi nition of the four base units of the SI by means of the criteria considered
earlier was selected unambiguously. However, there are other critieria that could be used to select the DCs. For example, the
authors of the ninth edition of the Brochure of the SI [4] selected system B, using the fact that system B possesses great ad-
vantages for electromagnetic measurements, an important reason for selecting the constants h, e, k, and NA. In turn, other sets
possess their own advantages, for example, system D, which is based on the set h, μ0, k, and NA. The advantage of this set lies
in the stability of the values of h and μ0, which refl ects the stability of the properties of a vacuum relative to quantum phe-
nomena. At the same time, establishing fi xed values of e and mu refl ects the stability of the properties of the atomic particles
related to mass and electromagnetic fi elds. There exist two systems, B and C, which each contain e, and the two systems D
and E which each contain μ0. It is known that the value of e depends on the interaction constant or energy of a process, for
example, α–1 ~ 137 at low energies, but α–1 ~ 128 at the scale of the mass Z of the boson [28]. From this point of view, it is
preferable to use system D or system E for the new defi nitions of the four base units of the SI.
Conclusion. The proposed new classifi cation of the DCs based on the new concept of the order of a constant in a
given system of units, together with the criteria that have been presented here, make it possible to select DCs for the redefi ni-
tion of the units of the SI. However, the criteria that have been considered here are not the only ones possible. There are other
arguments regarding the properties of the new set of DCs. In the present study, we have considered an alternative system of
constants for use in the redefi nition of the four base units of the SI (system D), consisting of h, μ0, k, and NA. This system
refl ects the stability of the properties of a vacuum in which quantum phenomena occur. This is compatible with the fi xed value
of c established in 1983 and leads to a fi xed value of ε0 and of the impedance of a vacuum Z0.
However, system B, which is now considered the most preferable system, has advantages when electromagnetic
measurements have to be performed. In this case, the practical units Ω90 and V90 (or, henceforth, Ω2018 and V2018) turn into
units of the MSI. Nevertheless, note that from the theoretical point of view, establishing a fi xed value of e has drawbacks
because of the variation in α as the energy of a process varies as well as because of possible space-time variations [29]. To
this, we might add the fact that the practice of establishing fi xed values of KJ and RK for conducting practical electromagnetic
measurements with the existing degree of precision in the measurement of e and α as well as with an increase in precision
may be preserved in systems D, C, and E. A further increase in the precision with which α is measured makes it possible,
moreover, to verify existing theories and to fi nd new variants of the fundamental interactions. Therefore, system D which has
been considered here is the best variant of the MSI, despite the fact that it was not selected by the CCU [30].
The goal of obtaining exact values of Planck’s and Avogadro’s constants with relative uncertainty ur = 2·10–8 was
achieved in 2015 thanks to the results of experiments carried out by NMIJ, BIPM, PTB, INRIM, NIST, and NRC [10–12].
The goal had been formulated in the resolutions of the 24th and 25th CGPM [3]. The adoption of new defi nitions of the four
base units of the SI can now be expected in 2018 [3, 4]. Nevertheless, there is still time to consider the achievements and
drawbacks of the proposed defi nitions of the four base units in light of new theoretical and experimental results.
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