The Avogadro Project
The kilogram is the only remaining fundamental unit within the
International System of Units (or SI, from Système International
d'Unités in French) which is defined in terms of an
artifact. This mass standard comes in the form of a Pt-Ir cylinder
kept in the International Bureau of Weights and Measures (or BIPM,
from Bureau International des Poids et Mesures in French)
situated in Paris. The proposition of the Avogadro Project is
to redefine the kilogram in terms of the Avogadro constant.
 By
definition, an Avogadro number of Carbon-12 atoms weigh exactly
12 grams. As such, the kilogram could bedefined as the mass of
1000/12 * Avogadro's number of Carbon-12 atoms. The Avogadro constant
itself is obtained from the ratio of the molar mass to the mass
of an atom. For a crystalline structure such as silicon, the atomic
volume is obtained from the lattice parameter and the number of
atoms per unit cell. The atomic mass is then the product of the
volume and density.
The Avogadro Project involves an international collaboration
between laboratories in Germany, Italy, Belgium, Japan, Australia
and USA. Currently the Avogadro constant is known to an uncertainty
of approximately 0.1 ppm. It is hoped that the uncertainty will
be reduced to 0.01 ppm after a further five years.
In determining the Avogadro constant, the preferred method has
been to use one of the high-precision spheres fabricated here
at the ACPO. These come in the form of a highly polished 1 kg
single crystal silicon sphere, fabricated with a roundness in
range of 60 nm. Silicon is used because of its well known crystal
structure, stability and its relative ease of use. The volume
is determined from the measurement of the silicon sphere's diameter
and roundness. Accurate measurement of the mass then allows the
density to be derived.
 The
nominal diameter of a 1 kg Si sphere is 93.6 mm. In order to obtain
an accuracy of 0.01 ppm in volume, the diameter must be known
to a range of 0.6 nm. In other words, within one atom spacing.
Such high accuracy requires specialised equipment and one such
procedure is by optical interferometry using a precision etalon
through a stabilised laser light. The measurements are sensitive
to many parameters, particularly to those of temperature and pressure.
An instability within the range of 2 mK would be sufficient to
cause the silicon to expand by more than the allowable uncertainty.
The refractive index of air (and hence the wavelength of the light)
is sensitive to the surrounding air pressure. It is therefore
necessary to carry out the measurements in a controlled environment.
High purity silicon boules have been produced especially for
this project by Wacker in Germany. The silicon is produced by
the float zone process and a very small quantity of nitrogen is
introduced to minimise defects, but at a concentration sufficiently
low as to not affect the molar mass. The determination of the
molar mass is conducted though mass spectroscopy.
Corrections must be applied for surface impurities such as oxides
and absorbed water. Typically, silicon has an oxide layer 3 to
4 nm thick, which is a mixture of SiO and SiO2. It
is also possible for the surface to absorb some monolayers of
water. Since much of the absorbed water is removed in a vacuum,
a number of the key measurements are made in a vacuum environment.
A further correction must then be applied for the difference in
bulk modulus between the air and vacuum.
Before a permanent and absolute definition of the kilogram is
introduced, the relative stability of the silicon sphere and the
existing Pt/Ir aftifact will have to be monitored. The kilogram
can then be defined in terms of a specific number of Carbon-12
atoms.
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UNCERTAINTIES IN THE SILICON KILOGRAM
The limiting factors currently are:
- The variability from sample to sample of the isotopic
abundances M(Si)
- The content of impurities and vacancies (n)
- Realisation of accurate density standards (m,V)
The list below first gives the current estimated
uncertainty, followed by the ultimate uncertainties which
can be expected if existing methods are pursued to their
limits.
- Molar Mass: 0.2 ppm to ultimately 0.05ppm.
- Atoms per unit cell: 0.2 ppm to ultimately 0.01 ppm
- Mass: 0.05 ppm to ultimately 0.01 ppm
- Volume: 0.08 ppm to ultimately 0.02 ppm
- Lattice parameter: 0.05 ppm to ultimately 0.01 ppm
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The roundness delta of the finished sphere (being held
above) is about 50 nm on a 93.6 mm diameter. It is believed
to be the roundest object in the world.
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