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Phil. Trans. R. Soc. A (2011) 369, 3925–3935
doi:10.1098/rsta.2011.0222
REVIEW
The Avogadro constant: determining the
number of atoms in a single-crystal 28Si sphere
B Y P ETER B ECKER
AND
H ORST B ETTIN*
Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100,
38116 Braunschweig, Germany
The Avogadro constant, the number of entities in an amount of substance of one mole,
links the atomic and the macroscopic properties of matter. Since the molar Planck
constant—the product of the Planck constant and the Avogadro constant—is very well
known via the measurement of the Rydberg constant, the Avogadro constant is also
closely related to the Planck constant. In addition, its accurate determination is of
paramount importance for a new definition of the kilogram in terms of a fundamental
constant. Here, we describe a new and unique approach to determine the Avogadro
constant from the number of atoms in 1 kg single-crystal spheres that are highly enriched
with the 28 Si isotope. This approach has enabled us to apply isotope dilution mass
spectroscopy to determine the molar mass of the silicon crystal with unprecedented
accuracy. The value obtained, NA = 6.022 140 82(18) × 1023 mol−1 , is now the most
accurate input datum for a new definition of the kilogram.
Keywords: fundamental constants; new kilogram definition; enriched silicon crystal
1. Introduction
Exploring the extent and precision to which our theoretical models and
measurement techniques are valid across the different realms of physics is of
utmost interest. Accurate measurements of the fundamental constants of physics
are a way of carrying out such investigations and testing the limits of our
knowledge and technologies. In these tests, the measurement of the Avogadro
constant, NA , holds a prominent position and appears both as an input and
an output of an overall least-squares adjustment of the fundamental constants
because it connects microphysics and macrophysics.
The Avogadro constant NA is the number of atoms or molecules in a mole
of a pure substance, for instance, the number of atoms (unbound, at rest
and in their ground state) in 12 g of the carbon isotope 12 C. Therefore, NA
expresses the mass of 12 C in kilograms according to M (12 C) = NA m(12 C), where
M (12 C) = 12 g mol−1 and m(12 C) are the molar mass and atomic mass of 12 C,
respectively. Many different measurements of the Avogadro constant, from that of
*Author for correspondence (horst.bettin@ptb.de).
One contribution of 15 to a Discussion Meeting Issue ‘The new SI based on fundamental constants’.
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P. Becker and H. Bettin
Loschmidt to that of Perrin, supported the Maxwell and Boltzmann descriptions
of matter in terms of atoms [1]. The Avogadro constant NA establishes detailed
balances in chemical reactions. The Avogadro constant is a scale factor to convert
atomic quantities and macroscopic quantities also as regards electromagnetism
and thermodynamics, i.e. it connects the electron charge e with a macroscopically
measurable electrical charge via F = NA e, where F is the Faraday constant, and it
also connects statistical mechanics with thermodynamics via R = NA kB , where R
and kB are the universal gas constant and the Boltzmann constant, respectively.
The molar Planck constant, NA h, is very well known via the measurement of
the Rydberg constant, R∞ = a2 M (e− )c/(2NA h), where a is the fine-structure
constant, M (e− ) is the molar mass of the electron, c is the speed of light and h
is the Planck constant. Therefore, an accurate measurement of NA also provides
an accurate determination of the Planck constant, and vice versa.
As a forthcoming new definition of the kilogram will most probably be based
on the Planck constant [2], a precise determination of the Avogadro constant is
of paramount importance as well, since presently it is the only alternative way
to obtain an independent value for the Planck constant via the molar Planck
constant. Today, the kilogram is the only base unit still defined by a material
prototype as stated by the 1st General Conference on Weights and Measures in
1889. The mass of the international prototype expressed in terms of the SI unit
is invariable by definition, but since 1889 its absolute mass is suspected to have
drifted by about 50 mg, or 5 × 10−8 in relative terms.
While the uncertainty of the mass of the international prototype is zero by
convention, any mise en pratique of a new definition will fix an uncertainty to
the kilogram. In order to ensure continuity to mass metrology, it has been agreed
that the relative uncertainty of any new realization must not exceed 2 × 10−8 .
Presently two different experiments have the potential to achieve this challenging
goal. One is the watt-balance experiment, which was first proposed in 1975 by
Kibble [3]. It aims to measure the Planck constant by the virtual comparison
of mechanical power with electrical power. The result is a measurement of the
m(K)c 2 /h ratio, where m(K) is the mass of the international prototype. The
other experiment, whose basic principles are described by Becker [1], was outlined
by Zosi [4] in 1983 and requires counting the atoms in 1 kg nearly perfect singlecrystal silicon spheres by determining NA . In this method, crystallization acts
as a ‘low-noise amplifier’ making the lattice parameter accessible to macroscopic
measurements, thus avoiding single atom counting. Silicon is used because it is
one of the best-known materials and, owing to the needs of the semiconductor
industry, it can be grown into high-purity, large and almost perfect single
crystals.
Since 1998, a relative 1.2 × 10−6 discrepancy has been observed when
comparing the results of these two different experiments through the molar Planck
constant [5]. Subsequently, it was conjectured that this discrepancy originated
through the difficulty of accurately determining the isotopic composition of
a natural silicon crystal, a key measurement for NA determination. To solve
this problem, we started a research project to repeat the measurement by
using a silicon crystal highly enriched with the 28 Si isotope. In this way, the
difficult absolute calibration of the mass spectrometer with the required small
uncertainty could be overcome by applying isotope dilution mass spectrometry
combined with multi-collector inductively coupled plasma mass spectrometry.
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Review. The Avogadro constant
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Figure 1. The float-zone 28 Si crystal. To determine its density, two spheres were manufactured from
the two bulges. To determine the lattice parameter, an X-ray interferometer crystal (XINT) was
cut from the material between these spheres. (Online version in colour.)
In his pioneering work on the Avogadro constant determination, Deslattes also
foresaw in 1974 the need for enriched silicon to improve mass measurement
uncertainties [6].
The project started in 2004 with the isotope enrichment being undertaken.
Subsequently, a polycrystal was grown by chemical vapour deposition [7] and,
in 2007, the 5 kg 28 Si boule shown in figure 1 was grown. As an unexpected
by-product, the availability of highly enriched and highly pure 28 Si single
crystals, as well as of highly enriched 29 Si and 30 Si crystals, brought about
physical and technological investigations in the fields of quantum computing and
semiconductor spectroscopy [8].
2. Principle of the measurement
Atoms were counted by exploiting their ordered arrangement in a crystal.
Therefore, the crystal and the unit-cell volumes having been measured, the count
requires their ratio to be calculated, the number of atoms per unit cell being
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3928
P. Becker and H. Bettin
Table 1. Point-defect concentration in the AVO28-S5 and AVO28-S8 spheres and in the XINT
crystal.
defect
unit
AVO28-S5
AVO28-S8
XINT
carbon
oxygen
boron
vacancy
1015 cm−3
1015 cm−3
1015 cm−3
1015 cm−3
0.40(5)
0.283(63)
0.011(4)
0.33(11)
1.93(19)
0.415(91)
0.031(18)
0.33(11)
1.07(10)
0.369(33)
0.004(1)
0.33(11)
known. Ideally, the crystal must be free of imperfections, monoisotopic (or the
isotopic composition must be determined) and chemically pure. The count gives
the Avogadro constant via
NA =
nVmol
nM
=
,
3
a0
r0 a03
(2.1)
where n = 8 is the number of atoms per unit cell, Vmol and a03 are the molar
and unit-cell volumes, M is the molar mass and r0 is the density. We selected
a spherical crystal shape to trace back the volume determination to diameter
measurements and to make possible an accurate geometrical, chemical and
physical characterization of the surface. Hence, two spheres, AVO28-S5 and
AVO28-S8, were taken at 229 and 367 mm distances, respectively, from the seed
crystal position and shaped as quasi-perfect spheres. Their masses and volumes
were accurately measured to obtain their densities.
(a) Isotope enrichment, crystal growth and crystal purity
The isotope enrichment was carried out at the Central Design Bureau of
Machine Building in Saint Petersburg, Russia, by centrifugation of SiF4 gas. After
conversion of the enriched gas into SiH4 , a polycrystal was grown by chemical
vapour deposition at the Institute of Chemistry of High-Purity Substances
of the Russian Academy of Sciences in Nizhny Novgorod, Russia. The 5 kg
crystal was grown and purified by multiple float-zone crystallizations at the
Leibniz-Institut für Kristallzüchtung in Berlin, Germany. The concentrations
of the residual impurities (carbon, oxygen and boron) were determined at
the Physikalisch-Technische Bundesanstalt (PTB), Braunschweig, Germany, by
optical spectroscopy. The concentration of vacancy-related defects was deduced
from positron lifetime spectroscopy at the Halle University, Germany. Results are
shown in table 1.
(b) Lattice parameter
To measure the lattice parameter, the Istituto Nazionale di Ricerca
Metrologica, Torino, Italy, upgraded a combined X-ray and optical interferometer
to expand the measurement capabilities to many centimetres and to achieve
a relative uncertainty approaching 10−9 . To exploit this capability, PTB
manufactured an X-ray interferometer crystal (XINT) with an unusually long
5 cm analyser crystal. The sample used for the XINT was taken from a point
in the boule located between the two spheres, and the lattice parameter was
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Review. The Avogadro constant
(d220 – 192 014 000 am) (am)
718
716
10 8d220
714
712
710
708
0
10
20
30
40
50
x (mm)
Figure 2. Lattice parameter. Map of the {220} lattice-plane spacing. The bars give the standard
deviation.
measured at a distance of 306.5 mm from the seed crystal. To demonstrate the
crystal homogeneity, the National Institute of Standards and Technology (NIST),
Gaithersburg, MD, USA, measured the lattice parameter of crystal samples, taken
above and below the two spheres, by means of a comparison with that of a natural
Si crystal calibrated by X-ray interferometry. The National Metrology Institute of
Japan (NMIJ) demonstrated the crystal perfection by strain topography carried
out by means of a self-referenced X-ray diffractometer at the Photon Factory of
the High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki,
Japan. The mean spacing of the interferometer diffracting plane,
d220 (XINT) = 192.014 712 67(67) pm,
(2.2)
was measured at 20◦ C and 0 Pa by combined X-ray and optical interferometry
(figure 2).
(c) Surface
Silicon is covered with an oxide surface layer. For the determination of the
oxide layer mass and thickness (table 2), synchrotron radiation-based X-ray
reflectometry at specific sphere surface points was selected to calibrate a
subsequent complete thickness mapping by spectroscopic ellipsometry [9]. To
increase the contrast between the optical constants of the silicon and the oxide,
and to increase the range of incidence angles, photon energies around the oxygen
K absorption edge at 543 eV were used. However, additional measurements by
X-ray photoelectron spectroscopy and X-ray fluorescence revealed unexpected
surface contamination by copper and nickel. From near-edge X-ray absorption
fine-structure measurements, these contaminants were found to be present as
silicides, heavily affecting the optical constants of the surface layers. Therefore,
the oxide thickness determination by X-ray reflectometry on the sphere was
replaced by X-ray fluorescence measurements with an excitation energy of 680 eV,
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P. Becker and H. Bettin
(a)
(b)
6.0
5.5
4.5
4.0
3.5
ds.e.(nm)
5.0
3.0
2.5
2.0
Figure 3. Topographic maps of the SiO2 thickness. (a) AVO28-S5, (b) AVO28-S8. The rainbow
colour code ranges from 2.0 nm (blue) to 4.5 nm (yellow). (Online version in colour.)
Table 2. Mass and thickness of the total surface layer and mass correction due to the point defects
of the AVO28-S5 and AVO28-S8 spheres.
surface layer mass
surface layer thickness
mass correction
unit
AVO28-S5
AVO28-S8
mg
nm
mg
222.1(14.5)
2.88(33)
8.1(2.4)
213.6(14.4)
2.69(32)
24.3(3.3)
where the oxygen K fluorescence intensity from the sphere surface was compared
with that from flat samples for which the oxide layer thickness was determined
by X-ray reflectometry.
The total surface layer was modelled, from top to bottom, as follows: a
carbonaceous and an adsorbed water layer, a fictive layer of Cu and Ni silicides
and an SiO2 layer [10]. From this model, the SiO2 thickness was re-evaluated from
the ellipsometric data, showing excellent agreement with the X-ray reflectometry
data. The mass deposition of the carbon, copper and nickel contaminants was
obtained from X-ray fluorescence measurements. The stoichiometries of the
oxide and the thickness of a possible SiO interface were investigated by X-ray
photoelectron spectroscopy. These measurements also confirmed the amount
of SiO to be below the detection limit of approximately 0.05 nm, which is in
agreement with the literature [10]. Since the contribution of this intermediate
layer, estimated on the basis of the present detection limit, is 10 times smaller
than the contribution of any other layer, this has not been included in the model.
Data for chemisorbed water on silicon were taken from the literature [11]. Figure 3
shows the mapping of the surface layer thickness, obtained by spectroscopic
ellipsometry with a spatial resolution of 1 mm. In table 2, the mass and thickness
of the two sphere surface layers are given.
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mass difference to weighted mean (µg)
30
20
AVO28-S8
AVO28-S5
weighted
mean
weighted
mean
10
0
BIPM
–10
BIPM
BIPM
PTB
BIPM
–20
NMIJ
–30
PTB
10–8 kg
NMIJ
–40
Figure 4. Mass of the silicon spheres. The AVO28-S5 and AVO28-S8 spheres were weighed in
vacuum by the BIPM, NMIJ and PTB. The absolute mass difference between the two spheres is
mS5 − mS8 = 23.042 mg. The bars give the standard uncertainties.
(d) Mass
Mass comparisons of the two spheres with Pt–Ir kilogram standards were
carried out in vacuum by the Bureau International des Poids et Mesures
(BIPM), Sèvres, France, NMIJ and PTB. Owing to air–vacuum transfer, a
sorption correction, which was measured by means of sorption artefacts, had
to be considered for the Pt–Ir standards. The mass determinations are shown
in figure 4; they are in excellent agreement and demonstrate a measurement
accuracy of about 5 mg. Corrections for the surface layers and for the crystal
point defects—the mass correction in table 2—have to be considered.
(e) Volume
The spheres were shaped and optically polished by the Australian Centre for
Precision Optics, Lindfield, NSW, Australia, and their volumes were determined
via diameter measurements. The NMIJ measured sets of diameters by means
of a Saunders-type interferometer. PTB used a spherical Fizeau interferometer,
which allowed about 105 diameters to be measured and a complete topographical
mapping to be achieved. Tunable diode lasers traced back to the frequency
standards were applied for phase-shifting techniques. Each sphere is placed
between the end-mirrors (plane, in one interferometer, and spherical, in the
other) of a Fizeau cavity, and the distances between the mirrors and each
sphere, as well as the cavity length, were measured. Since the sphere is almost
perfect, its volume is the same as that of a mathematical sphere having the
same mean diameter. Hence, a number of diameters were measured and averaged
[12]. Figure 5 shows the deviations from a constant diameter in orthographic
projections. For the volumes, the measured diameters were corrected for phase
shifts in beam reflections at the sphere surface, as well for beam retardation
through the surface layer.
(f ) Molar mass
The amount-of-substance fractions of the Si isotopes were measured by the
Institute for Reference Materials and Measurements (IRMM), Geel, Belgium,
via gas mass spectrometry of the SiF4 gas, and by PTB, via isotope dilution
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P. Becker and H. Bettin
(a)
(b)
+ 37 nm
– 63 nm
Figure 5. Diameter topographies of the silicon spheres. Peak-to-valley distances are (a) 98 nm
(AVO28-S5) and (b) 90 nm (AVO28-S8). (Online version in colour.)
Table 3. NA determination. Lattice parameter, volume and density are measured at 20.0 ◦ C
and 0 Pa.
quantity
unit
AVO28-S5
AVO28-S8
M
a
V
m
r = m/V
NA
g mol−1
pm
cm 3
g
kg m−3
1023 mol−1
27.976 970 26(22)
543.099 624 0(19)
431.059 061(13)
1000.087 558(15)
2320.070 841(76)
6.022 140 95(21)
27.976 970 29(23)
543.099 618 5(20)
431.049 111(10)
1000.064 541(15)
2320.070 998(64)
6.022 140 73(19)
combined with multi-collector inductively coupled plasma mass spectrometry. At
IRMM, the spectrometer was calibrated by using synthetic mixtures of enriched
Si isotopes. The natural Si contamination of the solutions used to convert the
samples into SiF4 was analysed at the University of Warsaw, Poland, by graphitefurnace atomic absorption spectroscopy. The isotope fractions were measured also
at the Institute of Mineral Resources of the Chinese Academy of Science (via gas
mass spectrometry, but using a different preparation of the SiF4 gas based on
fluorination by BrF5 ) and at the Institute for Physics of Microstructures of the
Russian Academy of Sciences (by means of a secondary ion mass spectrometer
using a time-of-flight mass analyser).
The PTB measured only the amount-of-substance fractions of the 29 Si and
30
Si isotopes, both forming a virtual two-isotope element within the matrix of
all the isotopes. To recover the unknown 28 Si fraction, the crystal samples were
blended with a spike, a crystal highly enriched with 30 Si. In addition to the
masses of the blended samples, the isotope ratio x30 /x29 , between the 30 Si and
29
Si fractions, was measured in the samples, spike and blends; but the ratio
x28 /x29 , between the 28 Si and 29 Si fractions, needed to be measured only in
the spike. The amount-of-substance fraction of the 28 Si isotope was obtained
indirectly. The spectrometer was calibrated online by using synthetic mixtures
of natural Si and two crystals enriched with the 29 Si and 30 Si isotopes. Natural
Si contamination, memory effects due to previous measurements and offsets were
corrected online by sandwiching each measurement of a sample, spike, blend or
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Review. The Avogadro constant
Table 4. Uncertainty budget of the Avogadro constant determination. The main contributions are
due at present to the surface characterization and the volume determination.
quantity
relative uncertainty (10−9 )
contribution (%)
molar mass
lattice parameter
surface
sphere volume
sphere mass
point defects
total
7
11
14
23
3
4
30
5
13
22
57
1
2
100
mixture with blank—aqueous NaOH solutions—measurements. From samples cut
in the vicinity of the spheres, an average molar mass was calculated and given
in table 3.
3. Results and outlook
Fifty years ago, Egidi [13] thought about realizing an atomic mass standard. In
1965, Bonse & Hart [14] realized the first X-ray interferometer, thus paving the
way to the accomplishment of this dream, and Deslattes [6] soon completed the
first NA determination by counting the atoms in a natural silicon crystal. For
the time being, we have concluded the project by carrying out a very accurate
NA measurement by using a silicon crystal highly isotopically enriched.
The measured values of the quantities necessary to determine the Avogadro
constant NA are summarized in table 3. The two values of the Avogadro constant
NA based on the two spheres differ only by 37(35) × 109 NA , thus confirming the
crystal homogeneity. By averaging these values, the final value of the Avogadro
constant is
(3.1)
NA = 6.022 140 82(18) × 1023 g mol−1 ,
with a relative uncertainty of 3.0 × 10−8 . The majority of the material reported
here can also be found in more detail in Andreas et al. [15].
The measurement uncertainty is 1.5 times higher than that targeted for a
kilogram redefinition, but ‘close to the finish line of the marathon effort to tie
the kilo to a constant of nature’ [16]. The measurement accuracy seems to be
limited by the performance of all the working apparatus. In fact, we have not
detected the effects of crystal imperfections with respect to the measurement
uncertainties reached so far. One of the main contributions to the uncertainty
budget (table 4) is due to the distortions of the optical wave fronts in the
interferometric measurement of the sphere diameters. Another is because of
the metallic contamination of the oxide layer, having an unknown influence on
the layer optical constants. To achieve the desired uncertainty, we are planning
to use improved optical interferometers, at present under test. Investigations
are also under way to eliminate the contamination from the sphere surfaces,
without jeopardizing the excellent roundness and nano-roughness of the spheres,
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3934
P. Becker and H. Bettin
6.022143
CODATA 2006
10–23 NA (mol–1)
6.022142
this paper
6.022141
NPL 1990
NIST 1998
NIST 2007
6.022140
6.022139
10–7 NA
METAS 2010
NPL 2010
6.022138
Figure 6. Avogadro constant determinations. Comparison between the most accurate NA values at
present available. The bars give the standard uncertainty.
which are both essential to accurate volume measurements. Investigations are in
progress to identify the source of natural Si contamination occurring in gas mass
spectrometry; repetitions of the molar mass measurement will be carried out in
additional laboratories.
For the first time, precise values of the Planck constant derived from different
experiments can be compared. This comparison is a test of the consistency
of atomic physics. A parallel experiment, having the purpose of measuring
NA h by absolute nuclear spectroscopy, is aiming at extending this test to
nuclear physics [17]. Figure 6 shows our result compared with those of the
most accurate measurements so far carried out: the watt-balance experiments
of the NIST (USA) [18], the National Physical Laboratory (NPL, UK [19],
I. A. Robinson 2010, private communication) and the Bundesamt für Metrologie
(METAS, Switzerland) [20]. The values of the Planck constant measured by
these experiments were converted into the corresponding NA values by NA h =
3.990 312 682 1(57) × 10−10 J s mol−1 , which has a relative uncertainty of 1.4 ×
10−9 [21].
By significantly reducing existing discrepancies, the present result leads to
a set of numerical values for the fundamental physical constants with better
consistency compared with previous sets. The result is also a significant step
towards demonstrating a successful mise en pratique of a kilogram definition
based on a fixed value of the Avogadro constant or of the Planck constant. The
agreement between the different realizations is not yet as good as it is required
to retire (for the time being) the Pt–Ir kilogram prototype, but, considering
the capabilities already developed and the envisaged improvements, it seems to
be realistic that the targeted uncertainty may be achieved in the foreseeable
future [22].
We wish to thank A. K. Kaliteevski and his colleagues at the Central Design Bureau of Machine
Building and the Institute of Chemistry of High-Purity Substances for their dedication and
the punctual delivery of the enriched material, the directors of the participating metrology
institutes for their advice and financial support, and our colleagues within the International
Avogadro Cooperation (IAC) for their daily work. This research received funds from the European
Community’s 7th Framework Programme ERA-NET Plus (grant 217257) and the International
Science and Technology Center (grant 2630).
Phil. Trans. R. Soc. A (2011)
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3935
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