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Title: The Avogadro and the Planck constants for redefinition of the kilogram
Author(s): Mana, G.; Massa, E.
Journal: rivista del nuovo cimento
Year: 2012, Volume: 35, Issue: 7
Publisher: Italian Physical Society
DOI: 10.1393/ncr/i2012-10078-5
Funding programme: EMRP A169: Call 2011 SI Broader Scope
Project title: SIB03: kNOW Realisation of the awaited definition of the kilogram - resolving the
discrepancies
Copyright note: This is an author-created, un-copyedited version of an article accepted for
publication in 'rivista del nuovo cimento'. The Italian Physical Society is not responsible for any
errors or omissions in this version of the manuscript or any version derived from it. The definitive
publisher-authenticated version is available online at http://dx.doi.org/10.1393/ncr/i2012-10078-5
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RIVISTA DEL NUOVO CIMENTO
Vol. ?, N. ?
?
The Avogadro and the Planck constants for redefinition of the
kilogram
G. Mana and E. Massa(∗ )
INRIM – Istituto Nazionale di Ricerca Metrologica, str. delle Cacce 91, 10135 Torino, Italy
(ricevuto ?)
Summary. — The last century developments of physics and technology are challenging the International System of units. Therefore, metrologists are considering
to redefine it by assigning conventional values to a set of fundamental constants;
this will be the greatest change after the introduction of the metric system. All the
units are under scrutiny, but the kilogram and the Avogadro and Planck constants
are in the spotlight.
PACS
PACS
PACS
PACS
PACS
PACS
01.55.+b
06.20.-f
06.20.Jr
06.20.F06.30.Dr
01.52.+r
–
–
–
–
–
–
General physics.
Metrology.
Determination of fundamental constants.
Units and standards.
Mass and density.
National and international laboratory facilities.
1. – Introduction
In the international system of units (Système International d’unités – SI) there are
problems originating in the gap between the unit definitions and the concepts of modern
physics as well as between the unit realizations and technology developments, which
make it possible to realize the units on the basis of fundamental constants [1, 2, 3].
Fundamental constants, like the speed of light in the Einstein relativity, are synthesizers equating seemingly different concepts, such as space and time [4]. Accordingly,
they are conversion factors between measurement units, disappear from theory, and are
of the utmost interest in metrology. An example of a never born constant is the proportionality factor between the inertial and gravitational masses in the Newton’s theory
of gravity; no longer existing constants are the mechanical and electrical equivalents of
heat. The Avogadro constant, NA , is another disappeared constant. It states that atoms
are the constituents of matter, makes the chemists’ amount of substance quantized and
(∗ ) corresponding author: e.massa@inrim.it
c Società Italiana di Fisica
⃝
1
2
G. MANA ETC.
identifiable with atoms and molecules, and is the scale factor between the mole and a
single entity. However, since atoms are now deeply embedded in our vision of reality,
NA is no longer considered in this way. NA now stands for the number of 12 C atoms
(unbound, at rest and in their ground state) per mole, expresses their mass in kilograms
as m(12 C) = M (12 C)/NA , where M (12 C) = 12 g/mol is the carbon 12 molar mass, and
connects the atomic and macroscopic mass scales. It relates microscopic and macroscopic
scales also for electrical and thermal quantities. It scales up the elementary charge to
macroscopically measurable amounts of electricity via F = NA e, where F is the Faraday
constant, and links the kinetic gas-theory to thermodynamics via R = NA kB , where R
and kB are the gas and Boltzmann constants.
Self-consistent values of the constants and of conversion factors of physics and chemistry – based on the least squares adjustment of the available data – are periodically
provided by the Committee on Data for Science and Technology (CODATA). The last
set of values is available on the World Wide Web [5]; the first set was published in 1973
[6] and updatings followed in 1986, 1998, 2002, and 2006 [7, 8, 9, 10]. The origin of these
adjustments was the work carried out by Birge in 1929 [11] and the subsequent work of
Cohen and collaborators in 1955, Bearden and Thomsen in 1957, and Taylor, Parker,
and Langenberg in 1969 [12, 13, 14] (1 ).
These adjustments draw a picture of about one-century evolution of precision measurements, which evolution is challenging the international system. All the units are
challenged, but the kilogram is playing a central role [15]. This is because the Josephson and quantum Hall effects allowed the electrical quantities to be measured in terms
of practical units, indicated by the 90 subscript and defined by conventional values,
483597.9 GHz/V90 and 25812.807 Ω90 , of the Josephson, KJ = 2e/h, and von Klitzing,
RK = h/e2 , constants. However, the relation h = 4/(RK KJ2 ) and the equivalence of
electrical and mechanical energies make the electrical units of energy and mass different
from the corresponding mechanical ones. Further pressure on the international system
arises because, since the international prototype of the kilogram was manufactured in
1889, its mass is suspected to have drifted by several tens of micrograms [16].
To relieve the strain of the international system, a conventional value can be given to
the Planck constant, h. It links energy and frequency via the Planck E = hν equation.
Therefore, it connects energy to time and, because of the Einstein E = mc2 identity, it
links also mass and time. This connection indicates that, so space can be measured in
seconds, provided c = 1, as mass can be measured in hertz, provided c = h = 1. A number
of experiments measured energy or momentum in terms of frequency or wavelength via
the Planck and de Broglie equations. Since, as regards atoms and sub-atomic particles,
molar masses are well known, these experiments deliver accurate values of the molar
Planck constant, NA h. A recent measurement of the Avogadro constant opened the way
to the estimate of h from the results of these experiments [17, 18] and to the realization
of a mass unit on that basis.
This review illustrates the metrologists’ effort to align the international system with
our present view of reality, with particular emphasis to the kilogram redefinition. The
motivations of this realignment lie also within the framework of the interactions between
science and technology. To quote from B. Petley [19], precision measurements push theory
and experiment to the very limits of which they are capable. This is an area where theory
(1 ) References [6, 7, 8, 9, 10], as well as the 1969 adjustment by Taylor, Parker and Langenberg,
can be downloaded free of charge from [5]
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
3
100
HN A - N0 LN0
1
0.01
10-4
10-6
10-8
1850
1900
1950
2000
2050
year
Fig. 1. – Reduction of the NA uncertainty from the Loschmidt’s measurement (1865) to today.
The data are from [10, 23]; the reference value, N0 = 6.02214129(27) × 1023 1/mol is from [5].
and experiment are tested to the limit, where the fallibility is often too apparent, and
where one’s best is only just good enough. Sections 2, 3, and 4 review the measurements
of the Avogadro, molar Planck, and Planck constants describing how the experiments
which delivered the most accurate values of these constants are near to realizing the
kilogram to within the 2 × 10−8 relative accuracy necessary to ensure the continuity of
mass metrology. The concept of mass in classical and modern physics, its links with
the Avogadro and Planck constants, and the impact of the NA and h measurements on
understanding reality are discussed in section 5. The concluding section 6 examines the
possible evolution of the international system of units.
2. – NA measurements
In 1811, to explain the experiments of Gay Lussac on gas reactions like 2H2 + O2 →
2H2 O, Avogadro introduced the concept of molecule – la molécule intégrante de l’eau sera
composée d’une demi-molécule d’oxygène avec une molècule ou, ce qui est la même chose,
deux demi-molécules d’hydrogène – and removed the difficulty of explaining the volume
and density changes [20]. The concept of molecule allowed him to advance the hypothesis
that le nombre des molécules intégrantes dans le gaz quelonques est toujours le même
à volume égal, which hypothesis is named after him. Fifty year later, the Cannizzaro’s
Sunto [21] triggered both the acceptance of the Avogadro’s interpretative model and NA
measurements, which continue today. Reviews of the NA measurements can be found in
[22, 23]. As shown in Fig. 1, in the last century the accuracy of the NA values increased
by about five orders of magnitude.
.
2 1. Early measurements. – A milestone on the way towards the agreement on the
atomic model of matter was the Einstein theory of the Brownian motion and its 1908
confirmation measurement by Perrin, NA = 7.1 × 1023 1/mol. Until then, the NA
4
G. MANA ETC.
measurements were part of the studies aimed at validating the atomic model. The variety
of measurements – about eighty were carried out from 1865 to 1932 by means of more
than twenty methods [24] – testifies the NA relevance as a concept synthesizer and its
widespread presence in the physics straddling the 1900.
The first value (1865, NA = 72×1023 1/mol) was a fallout of Loschmidt’s estimates of
the diameter and mean free path of air molecules [25]. Other estimates were proposed by
Planck (1901, NA = R/kB = 6.17 × 1023 1/mol), from the black-body determination of
kB and the gas constant [26]; Millikan (1917, NA = F/e = 6.064(6) × 1023 1/mol), from
the Faraday constant and his measurement of the electron charge [27]; Rutherford (1908,
NA = 6.2 × 1023 1/mol), from the rate of production of helium by radium decay [28];
du Nouy (1924, NA = 6.003(8) × 1023 1/mol), from the estimate of the size of molecules
in mono-layers films on the surface of water [29]. The numbers in bracket are the value
of the uncertainties referred to the corresponding last digits of the measurement results.
We will use this notation throughout the paper.
.
2 2. X-ray crystal-density measurements. – A breakthrough in the NA measurement
was made by the discovery that X-ray wavelengths can be determined by means of
diffraction by calibrated gratings [30]. Before this discovery, the measurements of X-ray
wavelengths were based on diffraction by crystals according to the Bragg law
(1)
nλ = 2d sin(θB ),
where the diffraction order n is an integer, λ is the X-ray wavelength, θB the diffraction
angle, and d the spacing of diffracting planes. This equation links the X-ray wavelength
to the lattice-plane spacing, which, following Bragg [31], was calculated as
√
(2)
a=
3
qM
,
ρNA
where q is the number of atoms per unit cell, and M/ρ is the molar volume, M and ρ
being the mean molar-mass and density of the crystal.
Grating diffraction allowed NA to be determined by reversion of formulae (1) and (2).
In the years from 1925 to 1965, lattice parameter measurements were carried out via (1),
where the X-ray wavelength λ was calibrated by ruled gratings. The principle of this
calibration is the same as for the diffraction of visible light, but the diffraction angles are
very small – only a fraction of a degree. The limiting factor was now that of measuring
small angles to a high degree of accuracy.
From 1930 onwards, instead of determining X-ray wavelengths from the lattice parameter and the lattice parameter from the NA value and crystal molar volume, as Bragg
did, NA is calculated from the measured values of both the lattice parameter and molar volume. Atoms were counted by exploiting their ordered arrangement in crystals:
crystallization acts as a low-noise amplifier making the lattice parameter accessible to
macroscopic measurements, thus avoiding single atom counting. Provided the crystal
and the unit cell volumes are measured and the number of atoms per unit cell is known,
the counting requires their ratio to be calculated as
(3)
NA =
qM
.
ρa3
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
5
The results of the measurements of Bearden (1931, NA = 6.019(3) × 1023 1/mol
[32]), Stille (1952, NA = 6.0253(15) × 1023 1/mol [33]), and Straumanis (1954, NA =
6.02489(30) × 1023 1/mol [34]) were lower than Millikan’s and it was impossible to attribute the discrepancy to experimental errors. Eventually, an error was discovered in
Millikan’s value of electron charge, owing to a wrong value of the air viscosity.
.
2 3. Counting 28 Si atoms. – In 1963, Egidi thought about realizing an atomic mass
standard by counting, according to (3), the atoms in a crystal scaled so to result outside
in a form of cube with faces parallel to the reticular planes [35]. However, the technology
of his time was unable to do that to the necessary accuracy. This was made possible
in 1965, when Bonse and Hart operated the first X-ray interferometer and paved the
way towards absolute measurements of the atom distance in crystals [36], without any
limitation to the accuracy owing to the calibration of the X-ray wavelength via artificial
.
gratings (section 2 2). Soon Deslattes completed a counting of the atoms in a silicon
crystal and determined the Avogadro constant (1974, NA = 6.0220943(63) × 1023 1/mol
[37]); further measurements soon followed [38, 39, 40, 41, 42].
These measurements were made by means of natural silicon crystals because, owing
to the demands of modern electronics, silicon is grown as high-purity, large, and quasiperfect single crystals. At the end of the last century, the NA measurements came
to a halt because of an unsolvable discrepancy of more than 10−6 NA with respect to
the NA value (1998, NA = 6.02214199(47) × 1023 1/mol) estimated by adjustments of
the fundamental-constant [8]. In addition, insuperable difficulties impaired the efforts to
improve the accuracy of the molar-mass measurement. The last measurement carried out
with the use of a natural Si crystal, NA = 6.0221353(18) × 1023 1/mol, was completed
in 2005 [42].
In 2004, to get around the problem of the molar mass measurement – following an
idea outlined by Zosi in 1983 [22, 43] – an international project (2 ) combined resources
and competence to produce a silicon crystal highly enriched with 28 Si [44, 45]. Isotope
enrichment made it possible accurate molar mass measurements by isotope dilution mass
spectroscopy combined with multicollector inductively coupled plasma mass spectrometry [46, 47]. The enrichment process was undertaken by the Central Design Bureau of
Machine Building in St Petersburg, which enriches uranium for nuclear power plants. The
bureau’s centrifuges enriched a considerable amount of SiF4 gas to more than 99.999%
28
SiF4 . Subsequently, after conversion of the enriched gas into SiH4 , a polycrystal was
grown by chemical vapour deposition by the Institute of Chemistry of High-Purity Substances of the Russian Academy of Sciences in Nizhny Novgorod and a 5 kg 28 Si ingot
was grown and purified by the Leibniz-Institut für Kristallzüchtung in Berlin.
To turn Egidi’s idea into practice, two slices were taken from the enriched ingot
and shaped as quasi-perfect 1 kg spheres by the Australian Centre for Precision Optics
(Fig. 2). The spheres’ composition, mass, volume, density and lattice parameter were
accurately determined and their surfaces were geometrically, chemically, and physically
characterized at the atomic scale.
(2 ) International Avogadro Coordination – IAC: Bureau International des Poids et Mesures
(BIPM), Institute for Reference Material and Measurements - European Commission Joint Research Center (IRMM - Belgium), Istituto Nazionale di Ricerca Metrologica (INRIM - Italy),
National Institute of Standards and Technology (NIST - USA), National Measurement Institute of Japan (NMIJ), National Measurement Laboratory (NML - Australia), National Physical
Laboratory (NPL - UK), Physikalisch-Technische Bundesanstalt (PTB - Germany)
6
G. MANA ETC.
Fig. 2. – Past and future: a Pt-Ir prototype of the kilogram mirrors in a 28 Si sphere. The
cylinder diameter and height are about 39 mm; the sphere diameter is about 93 mm. Courtesy
of the Physikalisch-Technische Bundesanstalt.
The measured value of the Avogadro constant,
(4)
NA = 6.02214082(18) × 1023 mol−1 ,
is the most accurate so far obtained [17, 18]. This result allows the accuracy of the
absolute atomic-mass scale to be improved; for instance, the mass of carbon 12 is
(5)
m(12 C) =
M (12 C)
= 1.992646861(60) × 10−26 kg.
NA
As will be shown in section 3, equation (4) is significant also for the Planck constant
determination. Since the molar mass, density, lattice parameter, and contaminant concentrations were not measured in the same sample, the values used in (3) had to be
obtained by extrapolation. Therefore, the most critical aspects of this NA determination
are the perfection and homogeneity of the 28 Si ingot. At the present level of sensitivity,
no evidence of imperfections or non-homogeneity has emerged from the background noise
of the measurements.
.
2 3.1. Molar mass measurement. In natural silicon, 92.2% of the atoms are 28 Si; the
remaining are 29 Si (about 4.7%) and 30 Si (about 3.1%). The determination of the crystal
molar mass,
(6)
M = M (28 Si) +
30
∑
[ i
]
M ( Si) − M (28 Si) xi ,
i=29
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
7
to within an accuracy better than 10−6 M has so far been impossible. In order to calculate the correction to the 28 Si molar mass indicated in (6), the isotope fractions xi –
actually, the isotope fraction ratios xi /xj – must be accurately determined. To this end,
owing to the high signal-to-noise ratio and to the control capability of the ion-current
measurements, gas mass spectrometry of SiF4 was resorted at the Institute for Reference
Materials and Measurements (Belgium) [48]. Measurements combined high-accuracy
chemical assay with high-accuracy mass spectrometry. The mass spectrometer was used
to measure ion-current ratios, which were calibrated using synthetic mixtures prepared
from quasi-pure silicon isotopes. To calibrate the ion-current ratio measurements and to
convert silicon into SiF4 required complex chemistry. Fractionation effects and accurate
spectrometer calibration revealed insuperable problems [49].
As shown by (6), in an enriched 28 Si crystal, the minority isotopes contribute to the
molar mass only through very small corrective terms [22, 50]. For this reason, measurements of x29 and x30 having a relatively large uncertainty are sufficient and the
spectrometer calibration is no longer a big problem. It was thus possible to resort to a
combination of isotope dilution mass spectrometry and high resolution inductively coupled plasma mass spectrometry. This measurement procedure requires much simpler
chemistry, that is, the sample dissolution in an aqueous solution of NaOH [46]. In order to cope with the calibration of the ion-current ratios measurements and with the
impossibility to detect the 28 Si+ current, because it is five orders of magnitude higher
than those related to the minority isotopes, a new technique based on isotope dilution
was developed. The samples of the 28 Si crystal were blended with samples of a crystal
enriched with 30 Si; whereas the 29 Si+ and 30 Si+ currents were measured for both the
parents and blend, the 28 Si+ current was measured only for the enriched 30 Si sample.
The x28 fraction of the 28 Si crystal was recovered by data analysis [51]. Calibration was
carried out on-line by means of mixtures of natural silicon and crystals enriched with 29 Si
and 30 Si. Contamination by natural silicon, memory effects, and offsets were monitored
and corrected on-line by continuous measurement of the NaOH solutions. The crystal
molar mass was determined with a 8.2 × 10−9 M uncertainty, thus paving the way to a
very accurate NA determination.
.
2 3.2. Density measurement. Density was determined as the ratio between mass and
volume. A spherical crystal-shape was selected because it has no edges that might get
damaged, because its volume can be calculated from diameter measurements, and because accurate geometric, chemical, and physical characterizations of the surface are
possible. To weigh the crystal sphere against Pt-Ir prototypes and, ultimately, against
the international prototype itself, the sphere mass must be 1 kg to within a few tens of a
milligram. The prototypes of the Bureau International des Poids et Mesures, of Japan,
and of Germany were used and the sphere mass was determined, in vacuo, to within an
accuracy of 5 µg [52, 53].
The diameter of 1 kg Si spheres is 93.6 mm. In order to obtain a 10−8 relative accuracy
in volume determination, the mean diameter must be measured to a range of 0.3 nm,
that is, to within an atom spacing. Such high accuracy requires sub-nanometre surface
roughness and a quasi-perfect spherical shape [54]. In practice, anisotropy limited the
perfection of the 28 Si spheres to about 90 nm, though roundness errors of only 30 nm
were apparent. The roundness error shows a die-shaped pattern; it has been explained
in terms of dependence of both the Young’s modulus and the work necessary for atom
removal on crystal orientation [55].
The sphere volumes were measured by the Physikalisch-Technische Bundesanstalt
8
G. MANA ETC.
Fig. 3. – Topography of a 28 Si sphere in Mollweide projection. The colour scale, from blue
(valleys) to red (peaks) spans about 98 nm. The eight valleys are located approximately as the
vertices of a dice. Courtesy of the Physikalisch-Technische Bundesanstalt.
(PTB-Germany) and the National Metrology Institute of Japan (NMIJ) by means of
optical interferometry [56, 57]. The interferometers are Fizeau cavities formed, in the
PTB case, by two confocal spherical mirrors and, in the NMIJ case, by a pair of flat
mirrors. The mirror spacing is firstly measured. Then, the sphere is placed inside the
cavity and the two gaps with the cavity mirrors are measured. The sphere diameter is
obtained by difference. The interference patterns are acquired by high dynamic cameras
allowing application of sophisticated algorithms for image processing. For the spherical
cavity the interference patterns are equal thickness fringes in the full field of view. These
patterns made it possible to determine one diameter for each camera pixel and a complete
topographical mapping of the spheres. Figure 3 shows the roundness errors in Mollweide
projection. The volume is calculated according to
(7)
4πa300
V ≈
3
(
∞ ∑
l
∑
alm
1+3
a00
)
2
+ ... ,
l=1 m=−l
where alm are the coefficients of the
(8)
R(ζ, ϕ) =
∞ ∑
l
∑
alm Ylm (ζ, ϕ),
l=1 m=−l
radius expansion in spherical harmonics, ζ = cos(θ), and θ and ϕ are the inclination and
azimuth. These coefficients are determined by a linear least squares estimate based on
the data set of the diameter values [58, 55, 59].
The main contribution to the uncertainty of volume measurements is due to the wavefront distortions that result from the roundness error. To reduce roundness errors to 20
nm, numerically controlled technologies based on ultra-precision ion beam figuring and
plasma jet machining are being developed by the Leibniz-Institut für Oberflaechenmodifizierung (Germany).
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
9
Fig. 4. – Thickness of the SiO2 surface layer on a 28 Si sphere measured by spectroscopic ellipsometry. The colour code ranges from 2.0 nm (blue) to 4.5 nm (yellow). Courtesy of the
Physikalisch-Technische Bundesanstalt.
The surface of silicon is covered with a thin layer of silicon dioxide which makes the
spheres inert to further oxidation. Since, to determine NA according to (3), only the
silicon atoms must be counted, the sphere surface was characterized from the chemical and physical viewpoints by X-ray reflectometry, X-ray photoelectron spectroscopy,
X-ray fluorescence, near-edge X-ray absorption fine-structure spectroscopy, and spectroscopic ellipsometry to determine contamination, stoichiometry, mass, and thickness of
the oxide layer [60]. Figure 4 shows mapping of the surface layer thickness, obtained by
ellipsometry, with a spatial resolution of 1 mm2 .
The crystal must also be free from imperfections and chemically pure. Consequently,
the 28 Si ingot was purified by the float-zone technique, no dopant was added, and the
pulling speed was so chosen in order to reduce the self-interstitial concentration. The
28
Si ingot is dislocation free and, to apply the relevant corrections, the concentrations of
carbon, oxygen, and boron atoms and vacancies were measured by infrared and positron
life-time spectroscopies [61]. Since, in determining the crystal molar mass, consideration
was given only to the silicon atoms, each measured sphere-mass was corrected for the mass
of the surface layer and the crystal point-defects to calculate the mass of an equivalent
naked sphere having one Si atom at each lattice site.
.
2 3.3. Lattice parameter measurement. X-ray interferometry is the technology that
enabled metrologists to count atoms, thus improving the accuracy of the NA measurement
by several orders of magnitude [62, 63]. As shown in Fig. 5, an X-ray interferometer is
similar to a Mach-Zehnder interferometer of classical optics. It consists of two separate
Si crystals so cut that the (220) diffracting planes are orthogonal to the crystal surfaces.
X rays from a 17 keV Mo Kα source having a (10 × 0.1) mm2 line focus are split by Laue
diffraction by the first two crystal slabs and delivered to the third (called the analyzer),
where they recombine. If the analyzer is moved along a direction orthogonal to the (220)
planes, a periodic variation in the transmitted and reflected X-ray intensities is observed,
the period being the diffracting-plane spacing. The interference originates from both the
phase shift between the beams transmitted and reflected by the analyzer lattice and the
different absorption of the crystal-guided field when the lattice planes are aligned with the
10
G. MANA ETC.
Fig. 5. – Scheme of a combined X-ray and optical interferometer. When the analyzer moves in
the direction indicated by the arrow, X-ray and optical interference fringes are detected whose
periods are 192 pm and 316 nm, respectively.
field nodes or antinodes [64]. In the low absorption case (neutron interferometry, highenergy X rays, and thin crystals), the interference fringes belonging to the transmitted
and reflected beams have opposite phases, as expected from energy conservation. In the
high absorption case (low-energy X rays and thick crystals), the interference fringes are
in phase.
The potential of X-ray interferometry to measure the spacing of the interferometer
diffracting-planes was soon understood [65, 66, 67, 68, 69]. For such measurements, it
is necessary to measure the displacement of the analyzer relative to the interferometer
fixed-crystal and to count simultaneously the number of travelling interference fringes,
that is, to count the number of the passing lattice planes. The carrying out of an absolute
measurement corresponds to measuring the plane spacing in terms of the Bohr frequency
of an atomic transition used to realize the metre according to its definition. Therefore,
the analyzer embeds front and rear mirrors and its displacement is measured by laser
interferometry, where the laser is stabilized against the frequency of one of such Bohr
frequencies.
To operate a combined X-ray and optical interferometer is a very difficult task. It
requires millikelvin temperature control and nanoradian and picometre controls of the
crystal attitude, vibrations, and position. To insulate the apparatus from ground vibration, to eliminate the adverse influence of the refractive index of air, and to ensure
temperature uniformity and stability, the experiment is carried out in a thermo-vacuum
chamber which rests on a pendulum-like antivibration support.
The measurement equation is
(9)
d220 =
nλ
,
2m
where n is the number of X-ray fringes in m optical fringes of λ/2 period. Silicon is
a cubic crystal with eight atoms per face centered unit cell; therefore, the cell edge a
is related to the measured d220 spacing of the planes having Miller indices (2,2,0) by
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
11
√
a = 8d220 . The equivalence between d220 and the period of the X-ray fringes, as well
as the equivalence between λ/2 and the period of the optical fringes, must be examined
with care. Actually, measurement equations are
(10a)
nd220 = ĥ · s + χ(ω, s)
and
(10b)
mλ/2 = k̂ · (s + b × ω) + ς(ω, s),
where k̂ is the propagation vector of the laser beam, s is the displacement of the X-ray
interferometer, b is the offset between the centers of the X-ray and laser beams, ω is the
parasitic rotation of the X-ray interferometer, ĥ is the normal to the lattice planes, and
ς(ω, s) and χ(ω, s) are corrections taking account of aberrations in both the X-ray and
optical interferometers.
Laser stabilization ensures that the beam frequency is known in terms of the frequency
of the hyperfine transition of the 133 Cs ground state which realizes the second. However,
laser stabilization does not ensure the beam wavelength to be known in terms of the
speed of light in vacuo. To obtain this knowledge, wavelength must be traced back to
frequency, but the plane-wave λ = c/ν relationship between wavelength and frequency,
at the 10−9 relative-accuracy required, does not hold exactly. In fact, the beam disperses
outside the region in which it would be expected to remain in plane wave propagation, the
wave fronts bend, and their spacing varies from one point to another and is different from
the wavelength of a plane wave. The actual wavelength must be calculated according to
suitable beam-propagation models, and measurements must be corrected [70].
According to (9), the diffracting-plane spacing is determined by comparing the unknown period of the X-ray fringes against the known period of the optical ones. It is clear
that the larger is the displacement mλ/2 = nd220 , the higher is the measurement resolution. The progress of combined X-ray and optical interferometry involved the continuous
development of more powerful techniques for finer control over experimental conditions.
An increase of the crystal-displacement up to 5 cm has been obtained by means of an
L shaped carriage sliding on a quasi-optical rail [71, 72]. An active tripod with three
piezoelectric legs rests on the carriage. Each leg expands vertically and shears as well
in the transverse directions, thus allowing compensation of the sliding errors and also
electronic positioning of the X-ray interferometer over six degrees of freedom to atomicscale accuracy. Crystal displacement, parasitic rotations, and transverse motions are
sensed via laser interferometry and capacitive transducers. Feedback loops provide picometre positioning, nanoradian alignment, and nanometre straightness of the analyzer
movement.
Since there are five million lattice-planes per millimetre and since, owing to the low
photon flux (about 103 photons mm−2 s−1 ) and the consequently long counting window
(typically, 0.1 s), the time required to detect each X-ray fringe is unacceptable. Therefore,
the measurement starts from the approximation λ/(2d220 ) = 1648.28; next, the fringe
fraction is measured only at the ends of increasing displacements mλ/2, where m = 1, 10,
100, 1000, and 3000. The fraction measurements are made with accuracy sufficient for
predicting the integer number of X-ray fringes in the next displacement. The λ/(2d220 )
ratio is thus updated step by step. The measurement result, d220 (28 Si) = (192014712.67±
0.67) am at 20.0 ◦ C and 0 Pa, has a relative accuracy of 3.5×10−9 ; it is the most accurate
dimensional measurement ever made [73, 74].
12
G. MANA ETC.
109 @d - dH306 mmLD  dH306 mmL
10
5
0
æ
æ
AVO28-S5
AVO28-S8
-5
-10
-15
æ
-20
150
200
250
300
350
400
450
axial position  mm
Fig. 6. – Variation of the (220) lattice-plane spacing along the axis of the 28 Si ingot. The
reference value, d(306 mm) = 192014712.67 am, has been measured at the 306 mm coordinate
by combined X-ray and optical interferometry. The red line is the variation expected from the
gradient of the contaminant concentration, mainly carbon and oxygen. The blue data are the
lattice spacing value in samples located at the 177 mm, 302 mm, and 421 mm coordinates,
measured by a two-crystal diffractometer comparison against a natural silicon crystal whose
lattice parameter was calibrated by combined X-ray and optical interferometry. The mean
values at the AVO28-S5 and AVO28-S8 sphere-centers were obtained by interpolation.
The 28 Si lattice-parameter is greater by 1.9464(67) × 10−6 than the that of natural
silicon; this confirms quantum-mechanics calculations [75]. The parameter dependence
on isotopic composition is an effect of thermodynamics and quantum mechanics. Atom
distances minimize the Gibbs free energy with respect to the cell volume. In addition to
elastic energy, the free energy depends on both the phonon energy and entropy. While
elastic energy sets an equilibrium distance independent of nuclear mass, the phonon energy does not. Anharmonic effects imply a greater equilibrium distance and cause thermal expansion. Since heavier isotopes have smaller phonon energy, they set a smaller
distance. Entropy increases with temperature and has an opposite effect. At zero temperature, only the zero-point phonon energy survives, so that 28 Si has the greater lattice
parameter; this is a pure quantum-mechanical effect. When temperature increases, the
lattice parameter difference decreases, as a consequence of the increasing entropy.
3. – NA h measurements
The Planck constant links energy and momentum to the frequency and the wavelength of the wave-function. The determinations of h match energy (momentum) and
frequency (wavelength) measurements and, according to whether a mechanical, electrical, or thermal system is considered, the measurement result is a value of the h/m, the
h/e, or the h/kB ratio, where m, e, and kB are a mass, the electron charge, and the
Boltzmann constant, respectively. Therefore, to determine the Planck constant, absolute
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
13
measurements of mass, charge, or temperature are necessary.
In this section, we examine the determinations of h/m, where m is the mass of a
particle or of an atom. These measurements are carried out by combining the Planck
and de Broglie equations E = hν and p = h/λ with the Einstein equation E = mc2
and p = mv, where v is velocity. Since, the masses of atoms and sub-atomic particles
are not well known, but their molar masses are, h is determined by application of the
h/m = NA h/M identity, where M is the molar mass and NA h is the molar Planck
constant. An accurate NA determination is thus necessary.
.
3 1. h/m(n) ratio. – The measurement of h/m(n) was carried out at the high-flux
reactor of the Institut Laue-Langevin in 1998 [76, 77]. The m(n)v = h/λ equation was
used by measuring both the wavelength and the velocity – λ and v – of monochromatic neutrons. Monochromaticity was obtained by Bragg-reflection on a silicon crystal,
whereas the neutron velocity was determined by time-of-flight measurements. The lattice
parameter of a series of monochromator crystals was determined by comparison against
the lattice parameter of a reference crystal [76, 78]. In turn, the lattice parameter of
the reference crystal was measured by combined X-ray and optical interferometry [79].
The measurement result can be expressed in terms of the molar Planck constant and the
neutron molar-mass as
(11)
NA h
h
=
= 3.956033285(287) × 10−7 m2 s−1 .
M (n)
m(n)
The neutron molar-mass – M (n) = 1.00866491597(43) g/mol – was determined by comparing its binding energy in deuterium with the molar-mass difference between deuterium
.
and hydrogen atoms [10]. As will be explained in more detail in section 3 4, the relevant
measurement equation is
(12)
[ 1
]
M ( H) + M (n) − M (2 H) c2 = NA hνD ,
where the Bohr frequency νD is measured by nuclear spectroscopy after the capture
of a thermal neutron by a hydrogen nucleus. The M (n) determination requires prior
knowledge of NA h. A vicious circle is avoided by observing that NA hν/c2 is a small
correction to the M (2 H) − M (1 H) difference. Since the uncertainty of the M (n) value
is negligible, the relative uncertainty of the NA h value derived from (11) is the same as
that of the h/m(n) ratio.
.
3 2. h/m(e) ratio. – Historically, the h/m(e) ratio was determined by measuring the
Compton wavelength of the electron, λC (e) = h/[m(e)c], which was itself obtained by
measuring the wavelength of the 511 keV γ rays emitted in the reaction
(13)
e− + e+ → 2γ,
where the electron-positron pair annihilate at rest. Approximately, this is true when the
reaction (13) occurs in a solid, since the annihilation probability is large only when the
relative velocity of the (e− , e+ ) pair is small. In this case, conservation of energy yields
(14)
m(e)c2 = hν = hc/λC (e).
14
G. MANA ETC.
The γ-ray wavelength was measured in X-units by diffraction by calcite crystals. The
most accurate of measurements was carried out in 1964 [80]; re-expressed in units of the
international system, the measured value is λC (e) = 2.426445(37) × 10−12 m [6, 81].
Today, the quotient h/m(e) is obtained from the measurement values of the Rydberg
and the fine-structure constants. The Rydberg constant, R∞ = 10973731.568527(73)
m−1 , is determined by comparing the frequencies of the photons emitted in the transitions
of hydrogen and deuterium with the theoretical expression for atom energy [10]. Since
R∞ is given by
(15)
R∞ =
α2 m(e)c
,
2h
where
(16)
α=
µ0 ce2
= 7.2973525376(50) × 10−3
2h
is the fine structure constant [10], by writing the electron mass m(e) in terms of molar
mass, M (e) = NA m(e), where M (e) = 5.4857990943(23) × 10−4 g/mol [10], one obtains
(17)
NA h
h
α2 c
=
=
= 7.27389504(10) × 10−4 m2 s−1 .
M (e)
m(e)
2R∞
Another way to determine the h/m(e) ratio is via the measurement of the gyromag.
netic ratio of a particle or of a nucleus; this method will be described in section 4 2.3.
.
3 3. h/m(133 Cs) and h/m(87 Rb) ratios. – The h/m(133 Cs) ratio was determined by
atom interferometry via the measurement of the recoil frequency-shift ∆ν of photons
absorbed and emitted by 133 Cs [9, 10, 82]. When an atom at rest adsorbs or emits a
photon, the photon momentum is balanced by recoil of the atom. Therefore, part of the
transition energy is stored in the kinetic energy of the atom and the photon has a higher
(adsorption) or lower (emission) frequency than the ν0 value expected from the energy
difference. Conservation of momentum and energy yields
(18)
m(133 Cs)c2 ∆ν
= hν0
ν0
and, when substituting the molar mass M (133 Cs) = 132.905451932(24) g/mol for the
Cs mass [10] and NA h for the Planck constant, one obtains [82]
133
(19)
NA h
h
=
= 3.002369432(46) × 10−9 m2 s−1 .
M (133 Cs)
m(133 Cs)
In the case of a 87 Rb atom, the recoil velocity vRb was measured and the conservation
of momentum yields m(87 Rb)vRb = h/λ0 , where λ0 is the photon wavelength [10, 83].
By substituting the molar mass M (87 Rb) = 86.909180526(12) g/mol for the 87 Rb mass
[10] and NA h for the Planck constant, one obtains [83]
(20)
NA h
h
=
= 4.5913592729(57) × 10−9 m2 s−1 .
M (87 Rb)
m(87 Rb)
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
15
.
3 4. h/m(12 C) ratio. – The h/m(12 C) ratio is determined by measuring the frequencies
of the γ photons emitted in the cascades from the neutron
capture state to the ground
∑
state in the reaction n + n X → n+1 X∗ → n+1 X +
γ. The measurement is based on
the fact that, in the neutron capture reaction, the daughter isotope is lighter than the
ensemble formed by its parents and the mass defect can be measured by determining
the frequencies of the γ-rays emitted in the decay of the capture state to the ground
state. Frequencies were determined at the Institut Laue-Langevin in terms of the lattice
parameter of a diffracting crystal traced back to optical wavelengths by combined X-ray
and optical interferometry [84]. The measurement [85] is based on the comparison of the
total energy of the emitted γ rays against the mass defect ∆m between the capture and
ground states. Energy and mass are compared via the Einstein’s and Planck’s relations
E = ∆mc2 and E = hν. For example, by combining
∑
[ 32
]
M ( S) + M (n) − M (33 S) c2 = NA h
ν33 S∗ →33 S
(21a)
and
[
(21b)
]
M (1 H) + M (n) − M (2 H) c2 = NA hνD∗ →D ,
the comparison can be expressed as
(21c)
(∑
)
[ 32
]
M ( S) + M (2 H) − M (33 S) − M (1 H) c2 = NA h
ν33 S∗ →33 S − νD∗ →D .
The equation (21c) has been written in terms of the molar mass differences between the
parent atoms because these differences are the quantities actually measured by simultaneous comparisons of the cyclotron frequencies of ions of the initial and final isotopes
confined in a Penning trap [86, 87]. The molar Planck constant values obtained are
NA h = 3.9903165(32) × 10−7 J s mol−1
(22a)
from the n +
28
Si →
29
Si + γ reaction and
NA h = 3.9903118(21) × 10−7 J s mol−1
(22b)
from the n + 32 S → 33 S + γ reaction. The equation (21c) can be also written in terms
of the 12 C molar mass,
(23)
[
]
Ar (32 S) + Ar (2 H) − Ar (33 S) − Ar (1 H) c2
NA h
∑
=
=
,
m(12 C)
M (12 C)
ν33 S∗ →33 S − νD∗ →D
h
where Ar (X) = M (X)/M (12 C) is the relative atomic mass.
The Bohr frequencies ν = c/λ are obtained by wavelength measurements via the
Bragg’s equation λ = 2d sin(θB ), where 2 sin(θB ) is measured with the aid of a two-crystal
spectrometer and the spacing d of the crystal diffracting-planes is measured in terms of
a primary metre realization by combined X-ray and optical interferometry. As shown in
Fig. 7, the γ rays hit the first crystal at Bragg’s angle and are diffracted. By rocking
the second crystal in both the non-dispersive and dispersive geometries, diffraction peaks
16
G. MANA ETC.
Fig. 7. – Scheme of a two-crystal γ-ray spectrometer; γ rays are diffracted by two Si crystals
(Si-1 and Si-2). Only the non-dispersive geometry is shown; by rocking the Si-2 crystal, a
diffraction peak is recorded. An angle interferometer (not shown in the figure) measures the
crystal rotation 2θB via retro-reflection cube corners rigidly fastened to the crystal spindle.
are recorded, whose distance is 2θB and which identify the two configurations satisfying
Bragg’s law. Since the diffraction angle of MeV’s γ rays is less than 5 mrad, in order
to achieve 0.01 parts per million measurement-uncertainty, it is necessary to measure
angles with a resolution better that 50 prad. With an angle interferometer having 0.3 m
baseline (the optical lever of the interferometer), it is necessary to have 15 pm resolution
in the measurement of the differential displacements of the lever ends. Additionally,
since the interferometer calibration angle is about 250 mrad, utmost care must be given
to linearity.
The most accurate measurements of atomic-mass ratios are carried out by comparing
the cyclotron frequencies of ions confined in a Penning’s trap [86, 87]. Ions are trapped
radially by means of a uniform magnetic field and axially by an electric quadrupole field.
They show three motions, an axial oscillation and two circular – magnetron and cyclotron
– motions, and are confined in a small volume, about 1 mm3 , for several weeks. The
mass ratio is determined by measuring the ratio of the free space cyclotron frequencies
(24)
ω=
eB
,
m
where B is the magnetic field and e is the electron charge.
4. – h measurements
In this section we will examine the determination of the h/kB and h/e by electrical metrology. The archetype of the electrical determinations is Millikan’s photoelectric
measurement of the h/e ratio [88], the modern equivalent of which is the measurement
of the Josephson constant by the tunneling of Cooper’s pairs in a Josephson junction.
As regards thermodynamic measurements, their archetype is Planck’s black-body determination of the h/kB ratio [89], the modern equivalent of which is the measurement of
the Boltzmann constant by the power of Johnson noise in a resistor.
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
17
100
Hh - h0 Lh0
1
0.01
10-4
10-6
10-8
1850
1900
1950
2000
2050
year
Fig. 8. – Reduction of the h uncertainty from the Planck’s determination (1901) to today. Up
to about 1940, the measured values overlap in substantial agreement. The data are from [5, 11];
the reference value, h0 = 6.62606957(29) × 1034 J s is from [5].
The present electrical metrology uses practical units defined by conventional values
of the Josephson and the von Klitzing constants, – with
(25a)
KJ = 2e/h
and
(25b)
RK = h/e2 .
Therefore, whenever a measurement equation involves electrical quantities, normalization
is necessary to make the equation independent of them. This is made possible by the
identities e = 2/(KJ RK ) and h = 4/(KJ2 RK ).
Figure 8 shows reduction of the h uncertainty from Planck’s determination to today.
Up to the mid thirties, no significant progress can be seen; the measured values accumulate reciprocally. After that period, similarly to NA , the accuracy increased by six orders
of magnitude. The reason is twofold. On the one side, quoting J. E. Faller, Because we
are aware of earlier results, we tend to look for and find systematic errors which permit
us to correct our result until it stands at least close in the shadow of these measurements.
At this point we stop looking, we fold up our equipment, and publish our new result in
.
substantial agreement with ... [90]. On the other side, as explained in section 3 4, the
measurements of the Planck and the Avogadro constants are closely related and only
after about 1940 the accuracy of the NA measurements was good enough to open the
way to more accurate h determinations.
.
4 1. h/kB determination. – The quotient between the Planck and the Boltzmann
constants can be determined by examining the spectrum of a black body; in this way it
was first measured in the early twentieth century.
18
G. MANA ETC.
.
4 1.1. Black-body radiation. In 1901, Planck derived the
(26)
Leν =
2hν 3
1
,
c3 exp(hν/kB T ) − 1
law for the spectral radiance (the power radiated per unit area, frequency ν, and solid
angle) of a black body in thermal equilibrium at temperature T and introduced the
universal constants h and kB [89]. To derive the values of kB and h, Planck combined
the Wien displacement law,
(27)
λmax = b/T,
with the radiant exitance (the power radiated per unit area obtained by integrating (26)
over a hemisphere and all frequencies),
Me = σT 4 .
(28)
In these equations, b is the Wien displacement-constant, λmax is the solution of
(29a)
dLeν
=
dλ
(
1−
ch
5kB λT
)
(
exp
ch
kB λT
)
− 1 = 0,
and
(29b)
σ=
4
2π 5 kB
3
15h c2
is the Stefan-Boltzmann constant. From the measured values of the Wien displacement,
b = 2.94 × 10−3 m K, and of the Stefan-Boltzmann constant, σ = 7.061−8 W m−2 K−4 ,
Planck obtained kB = 1.346 × 10−23 J/K and h = 6.55 × 10−34 J s.
The Stefan-Boltzmann constant is now measured by using cryogenic radiometers to
compare the radiant power of a black body against an electric power measured in terms
of the W90 practical power-unit. Since (29b) can be written in terms of σ/h, which is
independent of the electrical units, the h/kB ratio can be obtained from the relation
√
(30)
h
=
kB
8π 5
,
15σc2 KJ2 RK
where the identity h = 4/(KJ2 RK ) is used. Unfortunately, the most accurate measurement
result of the Stefan-Boltzmann constant, σ = 5.66959(76) × 10−8 W m−2 K−4 , has a
relative uncertainty of 1.3 × 10−4 .
.
4 1.2. Johnson noise. According to the Nyquist theorem, the mean square-voltage
⟨U ⟩ in a bandwidth ∆ν across the terminals of a resistor of resistance R in thermal
equilibrium at temperature T is
2
(31)
⟨U 2 ⟩ = 4kB T R∆ν.
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
19
In order to determine the Boltzmann constant, since ⟨U 2 ⟩ is measured in terms of the V90
practical voltage-unit, (31) must be written in terms of ⟨U 2 ⟩/(Rh), which is independent
of the electrical units. Hence,
(32)
h
16T R∆ν
= 2
,
kB
KJ RK ⟨U 2 ⟩
where h = 4/(KJ2 RK ) has been used. The National Institute of Standards and Technology
(NIST-USA) completed a measurement by comparing the noise of a resistor at the triple
point of water with a quantum-based voltage reference signal generated by means of
a superconducting Josephson-junction waveform synthesizer [91]. The measured value,
h/kB = 4.799235(58) × 10−11 K s, is the first to achieve a relative uncertainty as small
as 12 × 10−6 .
.
4 2. h/e determination. – The quotient between the Planck constant and the elementary charge was first measured by Millikan. Nowadays, these measurements are
by-products of electrical metrology and are based on the theories of the Josephson and
quantum Hall effects.
.
4 2.1. Photoelectric effect. Quotient h/e can be determined by examining the energy
of photoelectrons; in this way its value was first measured by Millikan in 1916 [88]. In
1905, soon after Planck quantized the absorption and the emission of light by atoms,
Einstein extended the quantization idea to light and introduced the photon concept.
This extension enabled him to predict that the maximum energy, Emax , of the electrons
emitted in the photoelectric effect would be governed by
(33)
Emax = eV0 = hν − ϕ,
where ν is the frequency of the incident light and ϕ the extraction work of the electrons.
Millikan made an experiment to verify Einstein’s theory and measured the voltages V0
required to stop photoelectric currents from sodium and lithium plates moved (inside an
evacuated glass bulb) into the path of monochromatic light at various frequencies. The
slope of the stopping voltage vs. frequency is equal to the h/e ratio and, by using the
value of the elementary charge measured by his oil-drop apparatus, Millikan obtained
h = 6.569 × 10−34 J s (sodium) and h = 6.584 × 10−34 J s (lithium), with 1% estimated
uncertainties.
Until the sixties, the h/e ratio was determined by measuring the shortest X-ray wavelength λmin of the Bremsstrahlung radiation emitted by electron accelerated by a voltage
V (about 8 kV). This measured value follows from the equation
(34)
eV = hν = hc/λmin .
The results were expressed in terms of the measured λmin V product, with the wavelength
measured in X-units and the voltage in terms of the laboratory as-maintained volt.
Expressed in the units of the international system, the most accurate result is h/e =
4.13585(14) × 10−15 J s/C [81, 92]
20
G. MANA ETC.
.
4 2.2. Faraday constant. The quotient h/e can be obtained also from the Faraday and
the Rydberg constants. The Faraday constant F = NA e is the charge of one mole of
electrons. It was measured by comparing the amount of silver m(Ag) dissolved by the
Ag → Ag+ + e− electrolysis of a silver anode owing to the flow of a current I [93]. Hence,
(35)
F =
M (Ag)Q
= 96485.39(13) C90 /mol.
m(Ag)
where M (Ag) is molar mass and Q is the amount of charge.
By writing in (15) the electron mass m(e) in terms of F and molar mass, M (e) =
NA m(e) = F m(e)/e, one obtains
(36)
α2 M (e)c
h
=
.
e
2F R∞
In order to infer the h value from (36), since F is measured in terms of the C90 /mol
practical unit, it is necessary to write the elementary charge as e = 2/(KJ RK ). Hence,
(37)
h=
α2 M (e)c
.
KJ RK F R∞
.
4 2.3. Magnetic resonance. A 1/2 spin particle, when placed in a magnetic field B not
aligned with the magnetic moment, precesses at the frequency νs = γB/(2π), where µ is
the magnetic moment and
(38)
γ=
2µ
,
~
where ~ = h/(2π). The gyromagnetic ratio γ was measured as the quotient between
the spin precession frequency of protons and helions in samples of H2 O and 3 He gas in
vacuo. In actual measurements, the protons are shielded from the magnetic field by the
electrons and the spin precession frequency is shifted by diamagnetic and susceptibility
effects; the relevant shielded quantities γ ′ and µ′ are indicated by a prime.
The ratio of the shielded magnetic moment µ′ to the electron magnetic moment µe is
measured to high accuracy, as the imprecisely-known value of the magnetic field cancels
in calculating the ratio. Furthermore, if expressed in Bohr magnetons, µB = e~/[2m(e)],
the electron magnetic moment is µe = ge µB /2, where m(e) is the electron mass and
ge = 2.00231930436153(53) is the electron g-factor. Hence
(39)
γ′ =
ge e µ′
,
2m(e) µe
where the electron to shielded proton magnetic moment ratio µe /µ′p = 658.2275971(72)
and the electron to shielded helion magnetic moment ratio µe /µ′h = 864.058257(10)
are well known quantities. From this equation, by deriving the electron mass m(e) =
2R∞ h/(α2 c) from (15), we obtain
(40a)
h
ge α2 c µ′
=
e
4R∞ γ ′ µe
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
and, by deriving the electron charge e =
(40b)
21
√
2αh/(µ0 c) from (16), we obtain
h
µ0 cγ ′ e µe
=
.
m(e)
αge µ′
Two methods have been applied to determine the magnetic field. In the low-field
method the field is of the order of 1 mT and it is generated by a solenoid carrying
an electric current I [94, 95]. The magnetic field is calculated from B = µ0 GI, where
µ0 = 4π × 10−7 N/A2 is the permeability of the vacuum and G is the solenoid constant
and has the dimension of reciprocal length. This field causes an energy splitting of the
spin states of 0.2 neV and a spin flip frequency of 50 kHz. In the high-field method the
field is of the order of 0.5 T and it is generated by an electromagnet or a permanent
magnet [96, 97, 95, 98]. The magnetic field B = GF/I is measured in terms of the force
F on a wire of length 1/G carrying the electric current. This field causes an energy
splitting of the spin states of 0.2 µeV and a spin flip frequency of 50 MHz.
In both case, the current is measured in terms of the practical unit of current A90 .
In the low-field measurement,
′
γlo
=
(41)
2πν
µ0 GI
′
′
is inversely proportional to the current. Therefore, eγlo
= 2γlo
/(KJ RK ), where the
identity e = 2/(KJ RK ) has been used, is independent of the electrical units and
(42)
′
NA h
h
2µ0 cγlo
µe
=
=
M (e)
m(e)
KJ RK αge µ′
can be inferred from (40b). In the high-field measurement,
′
γhi
=
(43)
4πνI
eGF
′
′
is directly proportional to the current. Therefore, γhi
/e = γhi
KJ RK /2 is independent of
the electrical units and
(44)
h=
ge α2 c
µ′
′
2KJ RK R∞ γhi µe
can be inferred from (40a). The measurement results are given in table I.
.
4 2.4. Josephson constant. The modern way to measure the h/e ratio is founded on
solid state physics and is based on determination of the Josephson constant. When a
Josephson junction is irradiated with a microwave of frequency ν, the dc voltage across
the junction, VJ , exhibits quantized steps proportional to the microwave frequency, the
proportionality factor being theoretically predicted to be 1/KJ = h/(2e). These steps
are due to the mixing of the applied microwave with the ac Josephson current and occur
when
(45)
2eVj = nhν,
22
G. MANA ETC.
Fig. 9. – Scheme of a watt balance. Static mode (left): the force F acting on the current-carrying
coil is balanced against the weight mg of the test mass. The current I flowing in the coil is
measured in terms of Josephson voltage and quantum Hall resistance. Dynamic mode (right):
the coil is moved at velocity u in the vertical direction through the magnetic field B and the
induced voltage E is measured in terms of Josephson voltage.
where n is an integer. The Josephson constant, KJ , was measured by comparing the
voltage generated by an array of Josephson junctions with voltage measured in units
of the international system by allowing an electrostatic force to balance a gravitational
force [99, 100]. Since the microwave frequency and the voltage across each junction are
typically 10 GHz (with n ≈ 50) and 1 mV, the energy scale of these measurements is 1
meV. By combining the results of the KJ measurements with the fine structure constant
(16) or with the von Kilitzing constant (25b), the Planck constant can be calculated as
(46a)
h=
8α
µ0 cKJ2
h=
4
.
RK KJ2
or
(46b)
.
4 3. h/m(K) determination. – The direct way of access to the h/m(K) ratio, where
m(K) is the mass of a kilogram prototype, is by a watt balance [101, 102, 103, 104, 105,
106, 107]. This experiment compares virtually the mechanical and the electrical powers
produced by the motion of a kilogram prototype in the earth gravitational field and by
the motion of the supporting coil in a magnetic field.
As shown in Fig. 9, the comparison is carried out in two steps. In the first step, a
balance is used, in substitution mode, to compare the prototype weight m(K)gz with
the force generated by interaction between the electrical current in the coil, I, and the
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
23
magnetic field B. Hence,
∫
(47a)
∫
L
2π
∫
r0
m(K)gz = ẑ ·
0
0
rB(r, φ, τ ) × j(r, φ, τ ) dr dφ dτ,
0
where ẑ is the balance measuring-direction, gz is the component of the gravitational field
along ẑ, j(r, φ, τ ) is the current density, r, φ, and τ are pseudo-cylindrical coordinates
along the coil wires, and r0 and L are the coil-cable radius and length. If the current
density is the same everywhere, j dτ = I dτ /(2πr02 ) and (47a) can be rewritten as
∫
L
m(K)gz = I ẑ ·
∫
(47b)
0
1
2πr02
[∫
2π
∫
0
r0
]
rB(r, φ, τ ) dr dφ × dτ
0
L
B(τ ) × dτ = κstatic I,
= I ẑ ·
0
where B(τ ) is the mean field over the cable cross-section and κstatic is a geometric factor.
In the second step, the coil is moved and the induced electromotive force,
∫
(48a)
L
E=
u(r, φ, τ ) × B(r, φ, τ ) · dτ ,
0
is measured at the its ends. It must be noted that, provided the end surfaces τ = 0 and
τ = L are equipotential, E is independent of the integration path. Therefore, we can set
r = 0 and evaluate (48a) on the coil axis. If the u velocity is the same everywhere, (48a)
can be rewritten as
∫
(48b)
E =u·
L
B 0 (τ ) × dτ = κdynamic uz ,
0
where B 0 (τ ) = B(r = 0, φ, τ ), the scalar triple-product identity (a × b) · c = a · (b × c) is
used, uz = uẑ, and κdynamic is a geometric factor. If E is measured when the coil passes
through the same position as when equilibrium was achieved in the static phase, u = uẑ
(that is, the coil velocity is parallel to the balance measuring-direction), and
∫
∫
L
0
L
B 0 (τ ) × dτ ,
B(τ ) × dτ =
(49)
0
the geometric factors in (47b) and (48b) are the same. By eliminating them from these
equations, we obtain the measurement equation
(50)
m(K)gz u = EI,
which, virtually, balances mechanical and electrical powers.
The measurement of the electromotive force is based on the Josephson effect, that is,
E = hν2 /(2e), where ν2 is frequency. The current measurement, I = U/R, is based on
the Josephson effect in conjunction with the quantum Hall effect. Hence, U = hν1 /(2e),
24
G. MANA ETC.
Fig. 10. – INRIM copy of the international prototype of the kilogram. It is a Pt-Ir cylinder
having the diameter equal to the height, to minimize the surface.
R = h/(ne2 ), and I = nν1 e/2, where n is an integer and ν1 is frequency. Eventually, the
h/m(K) ratio is measured in units of the international system from
(51)
h
4gz u
=
.
m(K)
nν1 ν2
All the quantities in the left-hand side of (51) are measured with uncertainties small
enough to give h/m(K) with relative uncertainty of less than 1×10−8 , but in the practical
execution of the measurement, there are a number of other sources of uncertainty, for
example, alignments, unwanted motion, parasitic forces and torques, that must be made
harmless. Detailed descriptions of the watt-balance experiment can be found in [103, 106].
The most accurate value of the Planck constant inferred from these experiments is h =
6.62606891(24) × 10−34 J s [104].
5. – Mass metrology
The physicists’ mass appears firstly in Newton’s Principia. In Definition I – quantitas materiae est mensura ejusdem orta ex illius densitade et magnitudine conjunctim
– Newton grounded mass on the concept of density, which was assumed primitive. As
suggested by additivity and invariance (experimental, rather than theoretical, facts),
Newton identified mass, which would later be the inertia of physicists, with the quantity
of matter, which will later be the chemist’s amount of substance (the quantity used to
specify the amount of a chemical element or a compound). In the following centuries,
the atomic model of matter identified the amount of substance as the number of elemen-
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
25
100
ò
ì
50
à
32
841
7
ì
ò
à
Dm  Μg
ô
æ
0
ì
æ
à
ò
ç
ô
ç
43
1
47
æ
-50
-100
1880
1900
1920
1940
1960
1980
2000
year
Fig. 11. – Mass variation of the prototype témoins 1, 7, 841 , 32, 43, 47 with respect to the mass
of the international prototype of the kilogram. The témoins number 8 was erroneously marked
as 41. This comparison has been carried out only three times, in 1889, 1946, and 1989, on the
occasion of verification of the national prototypes sanctioned by the Convention du Mètre. The
data are from [16].
tary constituents, atoms, molecules, ions, electrons, and so on. Accordingly, the mass of
composite systems is the sum of its constituent masses in a nested process ending with
the elementary particles, the masses of which are model parameters. However, Einstein’s
mass-energy equivalence splits the amount-of-substance and inertia concepts. Mass is
identified with energy – no longer with the amount of substance – and depends on temperature, binding energy, excitation state, etcetera. Additivity and invariance still hold,
but do not apply separately to energy and mass.
Furthermore, in quantum field theories, mass is not a fundamental concept, nor a
fundamental parameter. Most of the mass is the binding energy of the system constituents, the remainder being supposed due to the coupling with a pervasive Higgs field
[108, 109, 110]. From a classical viewpoint, since these constituents relax in stable energy
states with high barriers to excitations of almost all degrees of freedom, we can focus attention only on the few excitable ones and neglect the background. From this viewpoint,
mass is a thermodynamic concept which, by shielding intractable details, allows physics
to be developed.
According to this framework, the unit of mass, the kilogram, is the mass an artefact of
platinum-iridium (see Fig. 10). It is kept at the Bureau International des Poids et Mesures
under the conditions specified by the 1st Conférence Générale des Poids et Mesures in
1889, when it sanctioned: This prototype shall henceforth be considered to be the unit of
mass. In 1901, the 3rd Conférence Générale des Poids et Mesures, confirmed that: The
kilogram is the unit of mass; it is equal to the mass of the international prototype of the
kilogram.
As shown in Fig. 11, after three verifications – carried out in 1889, 1946, and 1989 –
a drift of about 50 µg per century has been observed between the mass of the prototype,
26
G. MANA ETC.
which is 1 kg by definition, and the mass of its témoins and national copies. Though the
prototype is invariant by definition, it seems that it is its mass to be changing, rather
than the mass of all témoins and most of the national copies [16, 53]. This is prompting
metrologists to seek a replacement, for instance, by choosing a conventional h value and,
then, by realizing the kilogram in practice by reversing the experiments measuring it.
If the h value is fixed, the definition of the second, together with the already fixed
value of the speed of light, would fix the mass scale. In fact, the mass difference between
the hyperfine levels of the 133 Cs ground-state whose Bohr frequency ν(133 Cs) defines
the second, will be exactly given by ∆m(133 Cs) = hν(133 Cs)/c2 and, consequently, the
kilogram will be exactly given by a multiple of ∆m(133 Cs). This mass defect is too small
to be detectable in practice, but nuclear mass defects are not and, as described in section
.
3 4, they can be measured in terms of frequency by γ-ray spectroscopy. Atomic masses
can be expressed in frequency units by scaling the measured frequencies according to the
ratio between the mass defects and atomic masses. Hence, a link can be made with the
macroscopic mass scale by realizing an object of which the number of atoms is known
and its mass is 1 kg; this is equivalent to determining NA .
In order to ensure continuity to mass metrology, any change of the kilogram definition
must be invisible to users and all the existing data must keep the same numerical value
when expressed in the new unit. This means that the practical realizations of the kilogram
based on the new definition must differ from the mass of the international prototype by
less than the uncertainty of weighings due to the secular-change of the prototype mass.
Therefore, the Comité International des Poids et Mesures established that the uncertainty
of the new realizations must not exceed 2 × 10−8 kg. Since these realizations will be
obtained by reversing the experiments measuring the Planck and Avogadro constants,
the relative uncertainty of the h and NA measurements must not exceed 2 × 10−8 .
The state-of-the-art of the measurement of the Planck and Avogadro constants is
summarized in table I and Fig. 12. Only the watt-balance experiments access directly
to the Planck constant. The remaining values are obtained by means of relationships
between h and other constants; the h values are calculated by using the measurement
results indicated in the table and the recommended values of the other constants [5].
The energy scale is that of the measurements of the constant indicated, excluded NA .
For example, let
(52)
h=
′
2µ0 cM (e) µe γlo
′
KJ RK αge µ NA
be considered again – see (42). The h values in the first lines are obtained by using the
′
measured γlo
and NA values given in the references and the recommended values of the
′
other constants in the left-hand side of this equation. The energy scale is that of the γlo
measurement. If all the recommended values are used, the calculated h value would have
been identical to the recommended one, as expected because of consistency.
The values obtained from the h/m ratios, where m is the mass of the neutron, electron,
133
Cs, 87 Rb, and 12 C, have been obtained by adoption of the NA value given in (4). The
uncertainty of the values derived from Rydberg constant, h/m(133 Cs), and h/m(87 Rb)
measurements is governed by the NA uncertainty; therefore, these values are correlated.
However, the NA uncertainty affects only marginally the NA values derived from the γlo ,
h/m(n), and h/m(12 C) measurements.
There are both practical and theoretical difficulties in fixing NA and h. The practical
difficulties are illustrated in Fig. 12. Though close to the end of the effort to tie the
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
27
Table I. – Selected measured values of the Planck constant. The values calculated from the
NA h/m(n), NA h/m(e), NA h/m(Cs), NA h/m(Rb), h/∆m(29 Si), and h/∆m(33 S) quotients depend on the same NA value.
measured ratio
laboratory
energy / eV
h/m(e)
h/m(e)
h/m(e)
′
measurement
NA and γlo
NIST 1989, IAC 2011
2 × 10−10
NIM 1995, IAC 2011
2 × 10−10
KRISS/VNIIM 1998, IAC 2011 2 × 10−10
′
γhi
measurement
1034 h / J s
reference
6.62607122(73)
6.6260686(44)
6.6260715(12)
[10, 94]
[10, 95]
[10, 98]
2 × 10−7
2 × 10−7
6.6260729(67)
6.626071(11)
[10, 96]
[10, 95]
10−3
10−3
6.6260684(36)
6.6260670(42)
[99]
[100]
h/e
h/e
NPL 1979
NIM 1995
h/(2e)
h/(2e)
NMI 1989
PTB 1991
h/m(K)
h/m(K)
h/m(K)
h/m(K)
h/m(K)
h/m(K)
watt-balance experiments
NPL 1990
10−3
NIST 1998
10−3
NIST 2007
10−3
METAS 2011
10−3
NPL 2012
10−3
NRC 2012
10−3
6.6260682(13)
6.62606891(58)
6.62606901(34)
6.6260691(20)
6.6260712(13)
6.62607063(43)
[101]
[102]
[104]
[105]
[106]
[107]
h/m(n)
NA and quotient of h and the neutron mass
PTB 1998, IAC 2011
10−2
6.62606887(52)
[18, 77]
6.6260657(88)
[10, 93]
6.62607003(20)
[10, 18]
6.62607000(22)
6.62607011(22)
[18, 82]
[18, 83]
KJ measurement
h/e
h/m(e)
h/m(Cs)
h/m(Rb)
h/e
R∞ and Faraday constant measurement
CODATA 2006, NIST 1980
1
R∞ and NA measurement
CODATA 2006, IAC 2011
1
NA and quotient of h and the mass of an atom
Stanford 2002, IAC 2011
1
LKB 2011, IAC 2011
1
Bremsstrahlung radiation: short wavelength limit
JUH 1964
104
6.62636(22)
h/∆m(29 Si)
h/∆m(33 S)
NA and quotient of h and nuclear mass defects
MIT/ILL 2005, IAC 2011
106
6.6260764(53)
MIT/ILL 2005, IAC 2011
106
6.6260686(34)
h/m(e)
NA and (e− , e+ ) annihilation
AECL 1964
106
6.62644(10)
[81, 92]
[18, 85]
[18, 85]
[6, 81]
NIST: National Institute of Standards and Technology (USA), IAC: International Avogadro
Coordination, NIM: National Institute of Metrology (People’s Republic of China), KRISS: Korea Research Institute of Standards and Science (Republic of Korea), VNIIM: D.I. Mendeleev
All-Russian Institute for Metrology (Russia), NPL: National Physical Laboratory (UK), NMI:
National Metrology Institute (Australia), PTB: Physikalisch Technische Bundesanstalt (Germany), METAS: Federal Office of Metrology (Switzerland), NRC: National Research Council
(Canada), CODATA: Committee on Data for Science and Technology (USA), Stanford: Stanford University (USA), LKB: Laboratoire Kastler-Brossel (France), JUH: John Hopkins University (USA), MIT: Massachusetts Institute of Technology (USA), ILL: Institut Laue-Langevin
(France), AECL: Atomic Energy of Canada Limited (Canada)
28
G. MANA ETC.
0.8
0.6
æ
Γlo
NIST
1989
Γlo hH2eL hH2eL hmK
KRISS NMI PTB NPL
1998 1989 1991 1990
0.4
hmK
NIST
2007
hmK hmK
NPL NRC
2012 2012
R¥
IAC
2011
hmHRbL
LKB
2011
æ
æ
æ
106 Hh - h0 Lh0
æ
æ
æ
0.2
æ
æ
æ
æ
æ
-0.2
-0.4
-0.6
æ
±2‰10-8 h
0.0
Γhi
Γhi
he
NIST NIM NPL
1980 1995 1979
Γlo
NIM
1995
æ
æ
æ
æ
æ
hmK
NIST
1998
hmK
METAS
2011
hmHnL
PTB
1999
hmHCsL
Standford
2002
hDm
ILL
2005
æ
Fig. 12. – Comparison of the determinations of the Planck constant with sufficient accuracy to be
relevant when computing a recommended value by the Task Group on Fundamental Constants
of the Committee on Data for Science and Technology (CODATA). The error bars indicate the
standard deviations. The reference is the CODATA 2010 value, h0 = 6.62606957(29) × 10−34 J
s [5]. The pink line indicates the 2 × 10−8 h uncertainty required to make it possible a kilogram
redefinition based on a conventionally agreed value of h.
kilogram to a constant of nature, the smallest measurement uncertainty is still 1.5 times
higher than that targeted for its redefinition. In addition, though no value is totally out
of scale, the two most accurate values, obtained via the NIST’s watt-balance experiment
and by counting of the 28 Si atoms, do not agree with each another. The most likely
reason is that the uncertainty of at least one of the measured values has been estimated
too low. But, for the time being, there is no indication about which method is at fault.
The theoretical difficulty arises from the fact the h values rely on the correctness
of the interpretative models of the relevant measurements, that is, of the Josephson
and quantum Hall effect, wave-particle duality, atomic and nuclear physics. If these
models are inaccurate – though there is no evidence at present to suggest they are – the
measurements would give inconsistent values. Therefore, the agreement of the measured
h values tests the consistency of these interpretative models. As shown in Fig. 13, the
relevant measurement equations are consistent to within 1 × 10−7 h. The energy scales of
the h measurements span from less than 1 neV (low-field nuclear magnetic resonance), to
1 meV (Josephson and quantum Hall effects), to 0.1 eV (wave-particle duality of thermal
neutrons), to 1 eV (optical spectroscopy and atom interferometry), to 1 MeV (nuclear
spectroscopy). This motivates efforts to reduce the measurement uncertainties and to
increase confidence in the measured values.
6. – The future of the international system
In 1790 a group of experts, appointed by Louis XVI, established a system of measurement units which would subsequently be the international system. They proposed mass
and length units based on fixed values of the length of the Paris meridian and of water
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
29
1.5
106 Hh - h0 Lh0
1.0
0.5
æ
æ
æ
æ
0.0
æ
æ
-0.5
-10
-5
0
5
Log108energy  eVD
Fig. 13. – Planck constant values sorted according the measurement energy. The values obtained
by means of experiments carried out at the same energy scale have been averaged. The data are
from table I. The reference is the CODATA 2010 value, h0 = 6.62606957(29) × 10−34 J s [5].
density (at the melting temperature of ice and at atmospheric pressure). However, after
two platinum artifacts representing the metre and the kilogram were manufactured and
stored in the Archives de la République in 1799, material standards – instead of natural
ones – were sanctioned by the 1st Conférence Générale des Poids et Mesures in 1889.
In the following century unit definitions based on natural constant gradually substituted for the material standards. Nowadays, as shown in table II, the international
system is based on the speed of light in vacuo, the frequency of the photon emitted in
the transition between the hyperfine levels of the 133 Cs ground state, the permeability
of vacuum, the temperature of the triple point of water (having standard isotopic composition), the molar mass of 12 C (at rest, unbound, and in the ground state), and the
spectral luminous efficacy of a 540 × 1012 Hz radiation.
Table II. – Internationally adopted values of the constants at present used to define the base
units of the international system. The kilogram is the only unit defined by fixing a property of
an artefact (see Fig. 10).
unit
symbol
metre
kilogram
second
ampere
kelvin
mole
candela
m
kg
s
A
K
mol
cd
constant
c = 299792458
m(K) = 1
ν(hfs 133 Cs) = 9192631770
µ0 = 4π × 10−7
Ttpw = 273.16
M (12 C) = 12 × 10−3
K(λ555 ) = 683
unit
m/s
kg
Hz
N/A2
K
kg/mol
cd sr/W
30
G. MANA ETC.
Table III. – Fundamental constants that would be fixed to redefine the international system
according to the resolution 1 of the XXIV Conférence Générale des Poids et Mesures. The placeholders indicate the digits to be agreed. Although the definition of the candela is not linked
to a fundamental constant, it may be viewed as being linked to an invariant of nature.
constant
value
133
hyperfine transition of the
Cs ground state
speed of light
Planck constant
electron charge
Boltzmann constant
Avogadro constant
spectral luminous efficacy @ 540 × 1012 Hz
ν(hfs
133
Cs) = 9192631770
c = 299792458
h = 6.626068 × 10−34
e = 1.6021764 × 10−19
kB = 1.38065 × 10−23
NA = 6.022141 × 1023
K(λ555 ) = 683
unit
Hz
m/s
Js
C
J/K
1/mol
cd sr/W
In 1988, the Comité International des Poids et Mesures recommended the use of practical voltage and resistance units based on fixed values of the Josephson and von Klitzing
constants and, in 1989, recommended the use of a practical temperature scale (ITS90)
based on interpolation between the agreed temperatures of a number of phase transitions.
It is to be noted that these recommendation were never sanctioned by the Conférence
Gènèrale des Poids et Mesures. Owing to the drift between the international prototype of
the kilogram and its copies and to bring back thermal and electrical metrology to within
the international system, a reconstruction of the whole system is being considered.
The 24th meeting of the Conférence Générale des Poids et Mesures, held in Paris
from 17 to 21 October 2011, adopted a resolution on the possible future revision of
the international system of units [111]. Final approval will be made by the conference
after the prerequisite conditions necessary to ensure continuity to metrology have been
met. According to this resolution, the measurement units will be implicitly defined by
assigning conventionally agreed values to the constants listed in table III. In this way,
any difference between the base and the derived units will be eliminated. Hence, the
International System of Units, the SI, will be the system of units in which:
• the ground state hyperfine splitting frequency of the caesium 133 atom is exactly
9192631770 hertz,
• the speed of light in vacuum is exactly 299792458 metre per second,
• the Planck constant is exactly 6.626068 × 10−34 joule times second,
• the elementary charge is exactly 1.6021764 × 10−19 coulomb,
• the Boltzmann constant is exactly 1.38065 × 10−23 joule per kelvin,
• the Avogadro constant is exactly 6.022141 × 1023 reciprocal mole,
• the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz is
exactly 683 lumen per watt,
where the placeholders represent additional digits to be adde using values recommended
by the Committee on Data for Science and Technology (CODATA). It follows that the
international system of units will continue to have the present set of seven base units:
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
31
• the kilogram will continue to be the unit of mass, but its magnitude will be
set by fixing the numerical value of the Planck constant to be equal to exactly
6.626068 × 10−34 when it is expressed in the SI unit m2 kg s−1 , which is equal
to J s;
• the ampere will continue to be the unit of electric current, but its magnitude will
be set by fixing the numerical value of the elementary charge to be equal to exactly
1.6021764 × 10−19 when it is expressed in the SI unit A, which is equal to C;
• the kelvin will continue to be the unit of thermodynamic temperature, but its
magnitude will be set by fixing the numerical value of the Boltzmann constant to
be equal to exactly 1.38065 × 10−23 when it is expressed in the SI unit m2 kg
s−2 K−1 , which is equal to J K−1 ;
• the mole will continue to be the unit of amount of substance of a specified elementary entity, which may be an atom, a molecule, an ion, an electron, any other particle or a specified group of such particles, but its magnitude will be set by fixing the
numerical value of the Avogadro constant to be equal to exactly 6.02214 × 1023
when it is expressed in the SI unit mol−1 .
As a consequence of the new definitions,
• the mass of the international prototype of the kilogram will be 1 kg, but with a
relative uncertainty equal to that of the recommended value of the Planck constant
and, subsequently, its value will be determined experimentally;
• the permeability of vacuum µ0 will be 4π × 10−7 H m−1 , but with a relative
uncertainty equal to that of the recommended value of the fine-structure constant
and, subsequently, its value will be determined experimentally;
• the thermodynamic temperature of the triple point of water TTPW will be 273.16
K, but with a relative uncertainty equal to that of the recommended value of the
Boltzmann constant and, subsequently, its value will be determined experimentally;
• the molar mass of carbon 12 M (12 C) will be 0.012 kg mol−1 , but with a relative
uncertainty equal to that of the recommended value of NA h and, subsequently, its
value will be determined experimentally.
As shown in Fig. 14, once the Avogadro, Boltzmann, and Planck constants and the
electron charge have been fixed, mass and electrical metrologies are traced back to time
and frequency measurements via either microscopic or macroscopic paths. This allows
consistency checks of both the measurement technologies and the underlying physical
models to be made. There still remains to make a choice for the unit of time. Is there a
natural way to define the second? With c and h being fixed, a mass, or a mass difference,
fixes the time scale. In the proposed redefinition of the international system, the second
is fixed by the rest-mass difference between the two hyperfine levels of the 133 Cs ground
state. In addition, there is the electron mass. The fixing of the Compton frequency of the
electron νe = m(e)c2 /h would then be an alternative choice, but it is not conceptually
different from fixing ν(133 Cs).
A natural time scale requires that gravitation and general relativity come into play.
In the general relativity, the concept of mass is linked to the Schwarzschild’s solution of
32
G. MANA ETC.
Fig. 14. – Interconnections between the mass-related units. The arrows indicate the interlinks.
The kg, K, and C symbols represent macroscopic (molar) quantities. The mol symbol represents
the corresponding microscopic quantity.
Einstein’s field-equation. By using spherical coordinates, this solution is singular at a
Schwarzschild distance, a way to measure the mass ability to bend space and time,
(53)
rs =
2Gm
,
c2
where G is the Newtonian gravitational constant and, within the Newtonian limit, m is
the mass of the central body. This equation implies that the gravitational constant is
the conversion factor between mass and time. Accordingly, the Compton frequency of a
mass the Compton wavelength of which is equal to a half of the Schwarzschild radius is
√
(54)
νP =
c5
,
hG
which corresponds to the inverse of the Planck time.
Though the technologies so far developed went only near to making the kilogram
redefinition and the restructuring of the international system possible, they are already
capable to monitor the mass stability of the international prototype of the kilogram.
Two way of approach are being investigated. A proposal is to keep several watt-balance
apparatuses operational, to monitor the h/m(K) ratio. Another, is to monitor the mass
difference between the international prototype and natural Si témoins having their surfaces monitored from geometric, physical, and chemical viewpoints to determine mass
changes, if any. By reducing the present 15 µg uncertainty of the measurement of the
mass of the Si-environment interface by a factor three, the supposed 0.5 µg/year mass
drift of the international prototype could be detected in a few tens of years. The monitoring of the prototype stability is a long-term experiment, which should be planned as
soon as possible.
∗ ∗ ∗
THE AVOGADRO AND THE PLANCK CONSTANTS FOR REDEFINITION OF THE KILOGRAM
33
This work was jointly funded by the European Metrology Research Programme (EMRP)
participating countries within the European Association of National Metrology Institutes
(EURAMET) and the European Union. We thank our former colleagues A. Bergamin,
G. Cavagnero, and G. Zosi who pioneered X-ray interferometry at the INRIM. Without
their work our contribution to the kilogram redefinition would not have been possible.
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[1] Bureau International des Poids et Mesures, The International System of Units (SI)
(Stedi Media, Paris) 2006.
[2] Borde C. J., Phil. Trans. R. Soc. A, 363 (2005) 2177.
[3] Mills I. M., Mohr P. J., Quinn T. J., Taylor B. N. and Williams E. R., Metrologia,
43 (2006) 227.
[4] Lévy Le Blond J.-M., Rivista del Nuovo Cimento, 7 (1977) 187.
[5] Mohr P. J., Taylor B. N. and Newell D. B., Rev. Mod. Phys.,
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