Modernizing the SI: implications of recent progress with the fundamental constants

Abstract

Recent proposals to re-define some of the base units of the SI make use of
definitions that refer to fixed numerical values of certain constants. We
review these proposals in the context of the latest results of the
least-squares adjustment of the fundamental constants and against the
background of the difficulty experienced with communicating the changes. We
show that the benefit of a definition of the kilogram made with respect to the
atomic mass constant (mu) may now be significantly stronger than when the
choice was first considered 10 years ago.

Full text

1
Modernizing the SI – implications of recent progress with the fundamental
constants
Nick Fletcher, Richard S. Davis, Michael Stock and Martin J.T. Milton,
Bureau International des Poids et Mesures (BIPM), Pavillon de Breteuil, 92312 Sèvres
CEDEX, France.
e-mail : nick.fletcher@bipm.org
Abstract
Recent proposals to re-define some of the base units of the SI make use of definitions
that refer to fixed numerical values of certain constants. We review these proposals in
the context of the latest results of the least-squares adjustment of the fundamental
constants and against the background of the difficulty experienced with
communicating the changes. We show that the benefit of a definition of the kilogram
made with respect to the atomic mass constant (mu) may now be significantly stronger
than when the choice was first considered 10 years ago.
Introduction
The proposal to re-define four of the base units of the SI with respect to fixed numerical
values of four constants has been the subject of much discussion and many publications.
Although the possibility had been foreseen in publications during the 1990’s [1] they were not
articulated as a complete set of proposals until 2006 [2]. Subsequently, the General
Conference on Weights and Measures, the forum for decision making between the Member
States of the BIPM on all matters of measurement science and measurement units, addressed
the matter at its 23rd meeting in 2007. It recognised the importance of considering such a re-
definition, and invited the NMIs to “come to a view on whether it is possible”. More recently
at its 24th meeting in 2011, it noted that progress had been made towards such a re-definition
and invited a final proposal when the experimental data were sufficiently robust to support
one. The resolution also invited
‘the CIPM to continue its work towards improved formulations for the definitions of
the SI base units in terms of fundamental constants, having as far as possible a more
easily understandable description for users in general, consistent with scientific
rigour and clarity’.
In response to this resolution, we have reviewed some of the possible formulations, for a re-
definition of four base units (the kilogram, ampere, mole and kelvin). These have been
considered previously in an unpublished paper discussed at the CCU in 2007 [3] and partially
in a publication the same year [4].
This paper has been developed at a time when very good technical progress has been reported
towards the adoption of new definitions for the base units of the SI [5]. In particular the latest
least-squares adjustment of the fundamental constants indicates that the measured values of
the Planck constant may have reached the necessary level of agreement. Other activities
relating to the demonstration of the practical realization of the kilogram by the watt balance

2
and the Si-XRCD methods are underway. However, the process of establishing a globally-
agreed formulation and communicating it to users as requested by the CGPM in 2011 is
proving to be difficult. For this reason it is timely to consider possible formulations for a re-
definition in view of the recent results of the CODATA 2014 adjustment of the fundamental
constants, which became available online in June 2015 [6]. (The full paper for the 2014
adjustment is not expected before the end of 2015, but a summary of recent advances is
available [7]).
The previous analysis of possible formulations [3, 4] was based on the CODATA 2006 data
set, and the 2005 proposals were prepared using the CODATA 2002 data. In addition to
reduced uncertainties for most of the constants since those publications, the CODATA 2014
results illustrate a significantly improved level of confidence in the experimentally-
determined value of the fine structure constant (α) than was justified 10 years ago. These
improvements, together with the recognition that abandoning the 1990 conventional values
for of KJ and RK will introduce unavoidable step changes in the realization of the electrical
units [8] at the relative level of −9.8×10−8 and +1.8 × 10−8, provide a different context for the
choice of formulation than was possible in 2007.
In this paper, we review five possible formulations, including the present version of the SI, in
the light of the experimental progress in the determination of the values of the relevant
fundamental constants. In the following section we provide a short review of relevant
improvements in the CODATA 2014. We calculate the uncertainties of a list of constants
subject to four different formulations of the SI base units and discuss their possible impact.
The adjustment of the fundamental constants and the consideration of alternative unit
systems
The CODATA adjustment of the fundamental constants has been published every four years
since 1998 [9]. Each publication has brought together the best available experimental data,
with the best available theory. Table 1 shows the improvements over the last five published
adjustments for a selection of experimental constants of most relevance to the redefinition of
the SI. The values given are relative standard uncertainties, and are plotted in Figure 1. All
values are determined within the formulation of the present SI. (The symbols used in this
table, and throughout the paper, are defined in Annex A).
We use the CODATA 2014 data set to analyse the impact of different formulations on a
redefined SI [3,4]. The discussion of these systems benefits from the insight that any
definition of a base unit can be considered to be a statement that a certain quantity is fixed.
For example, the definitions of the kilogram, the ampere, the kelvin and the mole in the
present SI can be considered to be statements that the mass of the International Prototype of
the Kilogram (m(K)), the magnetic constant (µ0), and the triple point of water (Ttpw), and the
molar mass of 12C (M(12C)) are fixed.
We consider the five formulations listed in Table 2, labelled Systems A to E. System A is the
present SI and is shown as a reference for the comparison. Systems B to E have in common
that the present definitions of the second and the meter are not changed. Fixed numerical
values of kB and NA are used to replace the kelvin and mole definitions. They also all remove
the artefact kilogram definition by giving a fixed numerical value to one other constant. They

3
differ in the choice of the two constants used to define the mass and electrical units from the
possible set {h, e, mu, µ0} 1 2. System B is the revision to the SI proposed by Mills et al. [2]
and noted in Resolution 1 of the 24th CGPM which provides zero uncertainty in the two
constants used for quantum electrical standards. System E provides greater clarity in the
definition of the mass unit by defining it with respect to a fixed value of the atomic mass
constant (mu). Systems C and D provide mixtures of the benefits brought by B and E and are
not considered further because they would provide no compelling advantages for those most
concerned with either electrical or mass measurements.
Uncertainties of a set of fundamental constants in the five formulations
It is not necessary to perform a complete least-squares adjustment of the constants to illustrate
the effects of different choices of constants to define the base units. A simplified approach can
yield valid results [10]. The relative standard uncertainties of a set of fundamental constants in
the systems A to E are shown in Table 3. We illustrate in detail how we derive the
uncertainties on RK, KJ, h and e within System E starting with the relationships:

K
=

(1)

J
=2

(2)
=

(3)

u
=2

r
(e)
(4)
The quantities α and Ar(e) are dimensionless, and thus their uncertainties must be independent
of the choice of unit system. The Rydberg constant (R∞) is derived from spectroscopic
measurements that depend only on the units metre and second, and are unaffected by the
choices we are considering here. To determine the uncertainties of the ‘non-fixed’ constants
in System E, we rearrange (3) and (4) to give expressions for h and e in terms of the fixed
constants mu and µ0, and the three measured constants α and Ar(e) and R∞.
=u(e)
2
(5)
1 From the six possible combinations, we examine only four. The pair h, mu is excluded because fixing the
numerical values of both would redefine the second via the frequency muc2/h; The pair e, µ0 does generate a
complete and valid unit system, but would add no benefits over the four examined here.
2 We do not consider the alternative of fixing me rather than mu since it brings no useful simplification to the
definition of the mole and further removes the kilogram definition from one based on the mass of a single atom
of 12C.

4
=u(e)

∞ (6)
Then we can calculate the two electrical constants as

K
=
2
(7)

J
=2

K
= 4 
u

r
(e)


(8)
The relative uncertainties of System E can be calculated from equations (5) to (8). The
uncertainties within the Systems B, C and D can be derived in a similar way.
Review of the advantages of System E
We note the distribution of uncertainties associated with System E, which gives a fixed
numerical value to mu in combination with maintaining the present numerical value of µ0.
There are clear advantages to this choice for the definitions of the kilogram and mole, but it
was previously considered incompatible with the universal use of quantum Hall and
Josephson standards in electrical metrology because the constants RK and KJ do not have
defined numerical values in this system. In the discussion below, we consider why this
constraint might now be reconsidered, and review the possible benefits of System E.
The needs of mass metrology
A detailed analysis of the options for the definition of the kilogram has been published [4,
11], but only in combination with a fixed value of e. In fact, the choice between h or mu as a
defining constant has no impact on the uncertainties achievable in the dissemination of
macroscopic mass. This is because the ratio h/mu is now known with relative uncertainty
4.5×10−10, which is negligible compared to the uncertainties achievable by either the watt
balance [5] or silicon XRCD [5, 11] routes to the realization of the kilogram at a macroscopic
scale. This was not true based on the 2002 data set, when many of the defining concepts for a
possible SI redefinition were established. Two distinct arguments were made to prefer h over
NA for the kilogram redefinition3 (as first expressed in [1]), that have been superseded by
recent experimental progress. Firstly, the diversity and accuracy of silicon XRCD experiments
has significantly improved and the availability of highly-pure spheres of isotopically enriched
silicon will increase in the near future [12]. This route to the realization of the kilogram after
redefinition will be at least as important as watt balances. Secondly, the improved uncertainty
of h/mu [7, 13] means that both techniques can realize the kilogram equally well regardless of
the choice made for the definition.
3 The early discussion of a formulation of the definition of the kilogram with respect to a fixed
value of NA corresponds to either System C or E discussed here.

5
There is, however, a strong argument in favour of fixing mu. It allows a simple, readily
understandable definition of the unit of mass expressed in terms of a constant which is itself a
mass. The general public is familiar with the atomic theory of matter, and can visualize a
kilogram defined by a specified number of identical atoms. By contrast, when h is introduced
as a ‘defining constant’, there is no straightforward way to present the mass unit; quantum
physics must be invoked at some point, and this puts the base unit definition beyond the scope
of accessible explanations. This disadvantage can certainly be justified if the more complex
definition is necessary to obtain significant benefits for precision measurement. We suggest
that the balance of this consideration should be reviewed given our updated analysis of the
needs of electrical metrology presented below.
We note that the choice of defining constant has little or no impact on the technical
preparations for the redefinition of the kilogram. The CCM/CCU roadmap for implementation
would remain unchanged, the CCM conditions [11] for acceptance are unaffected, and the
mise en pratique could readily be adapted. The only effect is in the communication of the
change, where the simplified definition would be a significant advantage.
The needs of chemical metrology
An aspect of the possible adoption of System B that has created unexpected objections is the
proposal to re-define the mole based on a fixed value of NA hence defining it as a fixed
number of entities rather than as a mass of material equal to the atomic weight. The concerns
have centred on the diminished importance of the molar mass constant (Mu = M(12C)/12 )
resulting from it becoming an experimentally determined quantity; an objection that could be
expressed as a concern about a lack of coherence in the system. Although there are no
practical implications for measurements involving amount of substance or molar quantities,
the introduction of a non-exact molar mass constant Mu is not easy to accommodate in
teaching. The choice of mu as the constant that defines the mass unit removes this difficulty
and immediately allows the molar mass and the Avogadro constants to have fixed values; one
may maintain both the present relation Mu = 1 g mol−1 and have a defined numerical value for
the Avogadro constant, NA, which satisfies the relation NA = Mu/mu. This combination of fixed
constants directly addresses the concerns raised about the mole being defined according to
System B.
The needs of electrical metrology
A re-definition of the SI base units based on System B solves the present problem of electrical
metrology, namely the existence of conventional 1990 units, which are no longer coherent
with their SI counterparts even within the relatively large uncertainties with which the latter
are realized. The conventional values of RK and KJ were within the uncertainties of the
corresponding SI values at the time they were agreed in 1990, but that is not the case today.
The necessary step to resolve this is that the 1990 conventional values are abandoned, and
quantum Hall resistance and Josephson voltage standards return to using the SI values of the
constants RK and KJ respectively. This can be achieved if RK and KJ have defined numerical
values according to System B but this condition, whilst obviously being sufficient, is not
perhaps necessary. What is required is that the values of RK and KJ are known with

6
uncertainty low enough and with a demonstrated stability so that these values can be used
without practical impact on electrical measurements for the foreseeable future. We discuss
whether System E now fulfils this condition.
In System E, the relative uncertainty of RK is the same as that of α (now 2.3 × 10−10) and that
of KJ is essentially equal to the uncertainty of √α (1.2 × 10−10). Our evidence base for
suggesting that these values are adequate for the needs of electrical metrology comes from a
study by a CCEM task group on the impact of moving from the 1990 values of RK and KJ to
the SI values. The required offsets are rather significant; using the CODATA 2014 values, the
relative change in RK is +1.8 × 10−8; the relative change in KJ is −9.8×10−8. These predicted
changes were communicated to the NMIs and industrial users of electrical measurements at
the CPEM and NCSLI conferences in 2014 [8]. The conclusion was that whilst these changes
would be noticeable in some top-level laboratories, they could be accommodated with
minimal practical disruption. The residual uncertainties allocated to RK and KJ in System E
are, respectively 100 and 1000 times smaller than these changes. We therefore argue that the
usual metrological requirement of having a “safe” margin between the definitions of primary
standards and the needs of the most demanding users is fulfilled.
Whilst having fixed numerical values for RK and KJ would clearly be the first preference for
electrical metrology, we argue that the residual uncertainties of System E are in fact small
enough to make the distinction irrelevant. Relative uncertainties of 1.2×10−10 and 2.3×10−10
for KJ and RK respectively can be included by default in the uncertainty budgets of even the
best primary standards without consequence. (We contrast this to the presently applicable
uncertainties linking the 1990 values and the SI, 4×10−7 and 1×10−7 respectively, which must
be omitted by default in any precision electrical work.) The only situations where the System
E uncertainties would be discernible are in direct comparisons of Josephson arrays or
quantum Hall devices. In this case, the uncertainty would be correlated between the two
systems and would not enter the final result of the comparison. We also note that whilst such
tests of the underlying theory at the highest possible accuracy are of great interest, they are
independent of the definitions of the unit system. They are also not representative of real
world measurement needs.
The fine structure constant α plays a central role in determining the uncertainties claimed for
RK and KJ, so it is reasonable to ask how reliable we believe the value of α to be. In System E
future shifts in the CODATA value of α larger than its claimed uncertainty would result in
inconvenient changes to the reference values of RK and KJ used in electrical metrology.
Between CODATA 2006 and 2010, the reported value for α did indeed change by 6.5 times
the 2006 uncertainty (a relative shift of 4.4 × 10−9). Fortunately, we can make the case that
the reliability of experimental determinations of α has now greatly improved. The reason for
the 2006 error is well understood (an error in the theoretical value of the electron magnetic–
moment anomaly, ae, which is no longer reliant on a single calculation). Importantly the value
of α derived using ae has also since been confirmed by the completely independent, non QED
route of atomic recoil experiments (e.g. measurements of h/m(87Rb)) [9, 13]. The reliability of
the measured value of α is key to the acceptability of System E; we can now have good reason
to be confident in the CODATA 2014 uncertainty. We note also that we can expect

7
improvements in α determinations to continue; this is a central and active area of both
experimental and theoretical physics. After redefinition, System E would continue to benefit
from these improvements with even smaller uncertainties on KJ and RK.
Retaining the present exact value of the magnetic constant µ0 ( = 4π×10−7 N/A2)
Discussion of System B has concluded that there are no practical problems with µ0 having an
experimental value [2], especially as the uncertainty, being equal to that of α, is negligible for
any known application. We do not argue otherwise, but nonetheless draw attention to the
advantages of maintaining the present fixed value. In the proposed SI based on System B, we
would need to allow the value of the magnetic constant to change depending on experimental
values and uncertainties. We parameterize this extra dependence of µ0 as
4π × (1+δ) × 10−7 N/A2, where the value of δ = (α/α2018 – 1), α2018 being the best available
experimental value of α at the time of redefinition. The uncertainty of
δ
is the same as the
relative uncertainty of
α
, taking the value of α2018 to be exact. Thus if α = α2018,
δ
= 0 but
u(
δ
) = u(
α
)/α2018. The difficulty is that whilst µ0 needs to be introduced in introductory texts
in electromagnetism, the factor (1+δ) is difficult to explain without an advanced discussion of
unit systems. This is similar to the problem posed by Mu in texts treating molar quantities.
It has been noted that the deviations of µ0 and Mu from the exact values 4π×10−7 N/A2 and
1 g mol−1 will be small enough to be “ignored in practice” [2]. However, the familiar exact
values are no longer strictly valid within an SI based on System B, and the explanation of why
this is so involves a digression into topics beyond any introductory treatment, where both
these constants are needed. This does not seem to be compatible with the desire of the CGPM
for a system of units that is ‘easily understandable for users in general, consistent with
scientific rigour and clarity’. System E maps these problems onto the numerical values for RK
and KJ, which are only visible to readers of the mise en pratique for electrical units – an
audience that is better equipped to understand the subtle compromises required in maintaining
a coherent system of units.
We note also the relation between the SI and Gaussian or Lorentz-Heaviside unit systems.
These cgs systems remain in widespread use [14]. In addition, the cgs-emu system was
important when choosing the value and the unit of µ0 in the MKSA system, which later
became the SI. In such systems that have no base dimension of an electrical nature, the fine
structure constant is given by α = e2/ħc . This is difficult to map on to a system that
simultaneously defines numerical values for c, e (in coulombs) and h. The factors used for
conversion between unit systems [14] would have to be changed to avoid errors in high-
precision calculations carried out in the cgs unit system. An example would be the conversion
factor needed for the unit of charge (the statcoulomb) in the fine-structure constant when it is
written in Gaussian units.
Although, again, there are no practical problems foreseen, an experimental value of µ0 risks
pushing the SI further away from a unit system that can be readily reconciled with theoretical
physics.

8
Summary and conclusions
We have shown the impact of the CODATA 2014 adjustment of the fundamental constants on
four possible formulations for a re-definition of the base units of the SI. We note that the
arguments for and against the different formulations have changed since they were first
considered in 2005 and subsequently proposed in 2006.
In particular, the advantages of System E as presented here are not new and do not
individually make the case for a reconsideration of the plans for the adoption of a “new SI”
based on System B. However, our analysis of the possible impact on electrical metrology
presents a new perspective and we have shown that System E offers some clear advantages
and overcomes some disadvantages that are intrinsic to System B. The advantages of System
E are:
− It uses a definition of the kilogram that refers to a mass (in this case mu). It is therefore
easier to explain than one that refers to a fixed value of h and does not necessitate
recourse to an “explicit constant” formulation of the definitions, with its associated
complexity. There will be no impact on the uncertainty with which the kilogram can
be realised by the watt balance or silicon XRCD methods.
− It will be possible to define the mole with respect to a fixed value of the Avogadro
constant, whilst also fixing the atomic mass constant (mu) which is itself in universal
use as the basis of the scale for atomic masses. Consequently, the molar mass constant
(Mu) will continue to be fixed as it is at present. This is an approach that has been
advocated for many years by practitioners in the chemical measurement community.
− Electrical measurements will be brought fully back within the SI and the magnetic
constant (
µ
0) will continue to be fixed. The straightforward conversion between the SI
formulation of the electrical units and those in the Gaussian system will remain. The
uncertainty in the experimentally determined values in KJ and RK will not place any
limit on the practical use of Josephson voltage standards and quantum Hall resistance
standards for high-accuracy electrical metrology. Future improvements in the
experimental determination of the fine-structure constant, which can be expected to
result from innovative new experimental methods, will in turn feed through to reduced
uncertainties.
In summary, the choice of System E would bring the process of the search for an improved
formulation for the definition of the kilogram and the mole back to the idea published in 1974
[15] and formulated into two definitions in 1999 [1] that:
The kilogram is the mass of 5.018 450 XX ×1025 free 12C atoms at rest and in their
ground state.
The mole is the amount of substance of a system that contains 6.022 140 XX ×1023
specified entities.
In this formulation, the definition of the ampere can remain unchanged whilst allowing
realisation of the electrical units through the electrical quantum metrology effects with a
practical level of uncertainty unaffected by the residual uncertainties of KJ and RK. We believe
that System E will provide an experimentally based, accessible and comprehensible system
that is also coherent and scientifically correct [16].

9
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On the redefinition of the kilogram
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2
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to implementing CIPM recommendation 1 (CI-2005)
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3
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kelvin and mole, 18th CCU (2007) unpublished committee document number
CCU/07-09.
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Mana, G., (2007)
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units
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(2014)
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NCSLI Measure J. Meas. Sci. 9, 30-35 (2014)
9
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constants: 2010
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10
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doi:10.1088/0026-1394/47/3/019

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14
J.D. Jackson
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Third edition, Wiley and Sons, New York, 1998.
15
Deslattes R et al
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16
Martin J T Milton, Jonathan M Williams and Seton J Bennett (2007)
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doi:10.1088/0026-1394/44/5/012
Annex A: Symbols and quantities
α fine structure constant ( = µ0e2c/2h)
KJ Josephson constant ( = 2e/h)
RK Von Klitzing constant ( = h/e2)
h Planck constant
e elementary charge
c speed of light in vacuum
µ0 magnetic constant (also commonly known as permeability of free space)
m(K) mass of the international prototype kilogram
mu atomic mass constant ( = m(12C)/12)
Mu molar mass constant
me mass of the electron
Ar(e) the relative atomic mass of the electron ( = me/mu)
R∞ Rydberg constant ( = α2mec/2h)
kB Boltzmann constant
Ttpw temperature of the triple point of water
NA Avogadro constant

11
Table 1: Relative standard uncertainties for selected constants from successive
CODATA adjustments, as calculated within the present SI. The data are shown in
graphical form in Figure 1.
Date
k
B
h
e
N
A
h/m
u
α
A
r
(e)
R
∞
R
K
K
J
1998
1.7×10−6
7.8×10−8
3.9×10−8
7.8×10−8
7.6×10−9
3.7×10−9
2.1×10−9
7.6×10−12
3.7×10−9
3.9×10−8
2002
1.8×10−6
1.7×10−7
8.5×10−8
1.7×10−7
6.7×10−9
3.3×10−9
4.4×10−10
6.6×10−12
3.3×10−9
8.5×10−8
2006
1.7×10−6
5.0×10−8
2.5×10−8
5.0×10−8
1.4×10−9
6.8×10−10
4.2×10−10
6.6×10−12
6.8×10−10
2.5×10−8
2010
9.1×10−7
4.4×10−8
2.2×10−8
4.4×10−8
7.0×10−10
3.2×10−10
4.0×10−10
5.0×10−12
3.2×10−10
2.2×10−8
2014
5.7×10−7
1.2×10−8
6.1×10−9
1.2×10−8
4.5×10−10
2.3×10−10
2.9×10−11
5.9×10−12
2.3×10−10
6.1×10−9
Table 2: The definition of the five systems considered in this paper. System A
corresponds to the present SI.
System
Quantities with exact numerical values
A
m(K), µ
0
, T
tpw
, and M(12C)
B
h , e , k
B
and N
A
.
C
m
u
, e , k
B
and N
A
D
h , µ
0 ,
k
B
and N
A
E
m
u
, µ
0
, k
B
and N
A

12
Table 3: Relative standards uncertainties calculated for a selection of constants subject
to the differing constraints of the five systems considered here. Calculations use data
from CODATA-2014 [6]. System A corresponds to the present SI.
Quantity System A System B System C System D System E
m(K) 0 1.2×10−8 1.2×10−8 1.2×10−8 1.2×10−8
h 1.2×10
−8
0 4.5×10
−10
0 4.5×10
−10
e 6.1×10
−9
0 0 1.2×10
−10
3.5×10
−10
α
2.3×10
−10
2.3×10
−10
2.3×10
−10
2.3×10
−10
2.3×10
−10
RK 2.3×10
−10
0 4.5×10
−10
2.3×10
−10
2.3×10
−10
KJ 6.1×10
−9
0 4.5×10
−10
1.2×10
−10
1.2×10
−10
µ0 0 2.3×10
−10
6.9×10
−10
0 0
mu 1.2×10
−8
4.5×10
−10
0 4.5×10
−10
0
me 1.2×10
−8
4.5×10
−10
2.9×10
−11
4.5×10
−10
2.9×10
−11
NA 1.2×10
−8
0 0 0 0
M(
12
C) 0 4.5×10
−10
0 4.5×10
−10
0
kB 5.7×10
−7
0 0 0 0
Ttpw 0 5.7×10
−7
5.7×10
−7
5.7×10
−7
5.7×10
−7
Figure 1: Evolution of the relative standard uncertainties of the constants shown in
Table 1.