History and progress in the accuratedetermination of the Avogadro constant

Abstract

The Avogadro constant, NA, is a fundamental physicalconstant that relates any quantity at the atomic scale to itscorresponding macroscopic scale. Inspired by the kinetic gastheory Avogadro proposed his hypothesis in 1811, in order todescribe chemical reactions as an atomic process between atomsor molecules. Starting from his pioneering findings, the determination of this large number has fascinated generations ofscientists up to this day. The review of methods aimed atfinding a value for NA starts with the calculations madeby Loschmidt (1865; NA ≈72×1023 mol-1) who evaluated the number of molecules in a given gasvolume, derived from estimates of molecular diameters and themean free path length. Consideration of Brownian motion led tosome more accurate determinations of NA around thebeginning of the 20th century (Perrin (1908); NA≈6.7×1023 mol-1). Other methodsdeveloped in the following years are based on Millikan's oildrop experiment (1917, NA≈6.064(6)×1023 mol-1), on the counting of alpha particles emitted fromradium or uranium (Rutherford (1909); NA≈6.16×1023 mol-1) and on investigations of molecularmonolayers on liquids (Nuoy (1924); NA≈6.004×1023 mol-1).A modern method to derive NA from the density, the relativeatomic mass, and the unit cell length was introduced by Braggin 1913. It makes use of the diffraction of x-rays by theinteratomic spacings of a crystal lattice and its periodicarrangement. The accuracy of this method is extremely affectedby the fact that the lattice scale of the structurally imperfectlattice can be calibrated only approximately in SI units. Dataof NA were, therefore, found to be in disagreement withother fundamental constants (Bearden (1931); NA≈6.019(3)×1023 mol-1). A break though was achievedwith perfect crystals of silicon and x-ray interferometrymaking available very precise data of atomic distances, expressed in SI units (Bonse and Hart 1965).Today, metrology has re-discovered the Avogadro constant anduses it as one of several possible routes to a re-definition ofthe kilogram because the old platinum iridium artefact exhibitslong-term stability problems. This application of the Avogadroconstant presupposes a final measurement uncertainty of about1×10-8, a challenge for the experimental determination of the quantities involved, i.e. macroscopicdensity, isotopic composition, and unit cell volume of asilicon crystal. Many years of research work were centred onthe problem of how far the perfection of a real crystal is awayfrom 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 aninvariant quantity when the influence of residual defects, for example impurities, is taken into account. Up to a relative measurementuncertainty of a few parts in 107 it has recently beenshown that the molar volume, the ratio of molar mass to density, is constant, too. The combination of data from severalindependent measurements of the unit cell and the molar volumeshas led to a value for the Avogadro constant ofNA = 6.022 1335(30)×1023 mol-1 (De Bièvre et al 2001)recommended by the national metrology institutes involved inthis research project (Becker 2001).Prominent examples of the significance of the research workreviewed here are the use of NA as an input independent ofother data, for the adjustment of a consistent set offundamental constants, and the accompanying outstandingexperimental developments acting as spin-offs in the field oftechnology to make macroscopic dimensions traceable to the atomic scale.