The origin of the inverse density of coincident lattice sites was traced here to Georges Friedel (1865–1933), in 1904. He called this parameter the twin (macle) index and defined it to be the ratio of the total number of nodes of the primitive lattice to the number of coincident nodes restored by the twin operation. Friedel’s 1904 ‘multiple lattice’ was the present-day coincident site lattice. He introduced the Σ symbol in 1920 as the ratio of the volume of a (not necessarily primitive) multiple cell to the volume of the primitive cell. He provided the reader with several formulae which, in the cubic case, gave Σ = h2 + k2 + l2 (h, k and l being the indices of the twin plane) and a twin index I equal to Σ if Σ was odd and equal to Σ/2 if Σ was even. All of these definitions and formulae were included in the 1926 version of his ‘Leçons de Cristallographie’. He was also concerned with the ‘material lattice’ (the crystal structure) behind the mathematical lattice, but was mainly interested in mineralogy rather than metallurgy. It was concluded that this perhaps explained why Rosenhain appeared never to have heard of Friedel’s work and why Kronberg and Wilson had had to re-discover the importance of the density of coincidence sites, at the atomistic level in 1949. It was also noted that Friedel’s grandson had made the first (1949) numerical estimate of interface energies by using interatomic potentials but did not publish the results until 1953.

A Review of Some Elements in the History of Grain Boundaries, Centered on Georges Friedel, the Coincident ‘Site’ Lattice and the Twin Index. O.B.M.H.Duparc: Journal of Materials Science, 2011, 46[12], 4116-34