History and progress in the accurate determination of the Avogadro constant

The Avogadro constant, N_A, is a fundamental physical constant that relates any quantity at the atomic scale to its corresponding macroscopic scale. Inspired by the kinetic gas theory Avogadro proposed his hypothesis in 1811, in order to describe chemical reactions as an atomic process between atoms or molecules. Starting from his pioneering findings, the determination of this large number has fascinated generations of scientists up to this day. The review of methods aimed at finding a value for N_A starts with the calculations made by Loschmidt (1865; N_A ≈72×10²³ mol^-1) who evaluated the number of molecules in a given gas volume, derived from estimates of molecular diameters and the mean free path length. Consideration of Brownian motion led to some more accurate determinations of N_A around the beginning of the 20th century (Perrin (1908); N_A≈6.7×10²³ mol^-1). Other methods developed in the following years are based on Millikan's oil drop experiment (1917, N_A≈6.064(6)×10²³ mol^-1), on the counting of alpha particles emitted from radium or uranium (Rutherford (1909); N_A≈6.16×10²³ mol^-1) and on investigations of molecular monolayers on liquids (Nuoy (1924); N_A≈6.004×10²³ mol^-1).

A modern method to derive N_A from the density, the relative atomic mass, and the unit cell length was introduced by Bragg in 1913. It makes use of the diffraction of x-rays by the interatomic spacings of a crystal lattice and its periodic arrangement. The accuracy of this method is extremely affected by the fact that the lattice scale of the structurally imperfect lattice can be calibrated only approximately in SI units. Data of N_A were, therefore, found to be in disagreement with other fundamental constants (Bearden (1931); N_A≈6.019(3)×10²³ mol^-1). A break though was achieved with perfect crystals of silicon and x-ray interferometry making available very precise data of atomic distances, expressed in SI units (Bonse and Hart 1965).

Today, metrology has re-discovered the Avogadro constant and uses it as one of several possible routes to a re-definition of the kilogram because the old platinum iridium artefact exhibits long-term stability problems. This application of the Avogadro constant presupposes a final measurement uncertainty of about 1×10^-8, a challenge for the experimental determination of the quantities involved, i.e. macroscopic density, isotopic composition, and unit cell volume of a silicon crystal. Many years of research work were centred on the problem of how far the perfection of a real crystal is away from the ideal state. At present, it is widely accepted that, in the limits of the desired uncertainty, the lattice parameter, and thus the unit cell volume of silicon, can be seen as an invariant quantity when the influence of residual defects, for example impurities, is taken into account. Up to a relative measurement uncertainty of a few parts in 10⁷ it has recently been shown that the molar volume, the ratio of molar mass to density, is constant, too. The combination of data from several independent measurements of the unit cell and the molar volumes has led to a value for the Avogadro constant of N_A = 6.022 1335(30)×10²³ mol^-1 (De Bièvre et al 2001) recommended by the national metrology institutes involved in this research project (Becker 2001).

Prominent examples of the significance of the research work reviewed here are the use of N_A as an input independent of other data, for the adjustment of a consistent set of fundamental constants, and the accompanying outstanding experimental developments acting as spin-offs in the field of technology to make macroscopic dimensions traceable to the atomic scale.