Determining the Avogadro constant

The Avogadro constant is accurately determined using X-ray diffraction techniques. Diamond has a face-centred cubic unit cell (figure III below) that is formed on a two-carbon-atom basis, which is shaded yellow in figure I. With reference to figure III, there are 4 x 2 = 8 carbon atoms in each unit cell.

The unit cell of diamond can also be perceived as two interpenetrating face-centred cubic unit cells (figure II) in a way that a line joining the centres of atoms 1, 2, 3 forms a body diagonal of one of the face-centred cubic unit cells (formed by the black atoms). The length of the line joining atoms 1 and 2 is a quarter of that of the body diagonal.

Since the Avogadro constant was previously defined as the number of atoms in 0.012 kilogram of carbon-12, a hypothetical way to precisely evaluate it through X-ray diffraction is to synthesise a perfect sphere of pure carbon-12 that weighs exactly 0.012 kilogram and calculate the ratio of its molar volume Vmol to that of one-eighth of its unit cell (n = 8 carbon atoms in a unit cell),

N_A=\frac{V_{mol}}{a^3/n}=\frac{nM}{\rho\: a^3}\; \; \; \; \; \; \; (70)

where M is the molar mass, ρ is the density and a is the cubic unit cell dimension. The shape of a sphere is preferred, as it is easier to measure the size of the crystal and hence its density.

In reality, it is impossible to carve a perfect sphere out of diamond with minimal defects and the Avogadro constant is instead determined using a sphere that is grown from highly enriched silicon-28, which also has the same unit cell structure as diamond. Furthermore, in some experiments, the interplanar distance between the {220} family of planes d220 is usually measured due to the way the crystal is eventually cut and mounted on the diffractometer. From figure IV above, d220 is equal to a/√8. Therefore, eq70 becomes

N_A=\frac{nM}{\rho(d_{220}\sqrt{8})^3}\; \; \; \; \; \; \; (71)

In an internationally coordinated project called the Avogadro project in 2011, the molar mass (independent of the Avogadro constant) of the enriched silicon-28 crystal was determined with high accuracy via mass spectrometry. The mass of the crystal was calibrated versus the then Pt-Ir kilogram standard in vacuum, while the diameter of the crystal was measured via an interferometer. With the radius of the sphere determined, the volume of the sphere is known and hence its density. Finally, the interplanar distance d220 is measured using X-ray interferometry, a technique that combines the principles of X-ray diffraction and interferometry. The value of the Avogadro constant was presented by the project team as 6.02214078×1023 mol-1. In 2017, the value was refined to 6.02214076×1023 mol-1 and in Nov 2018, the Avogardo constant was defined as exactly 6.02214076×1023 mol-1.


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