A review of some elements in the history of grain boundaries, centered on Georges Friedel, the coincident ‘site’ lattice and the twin index

• Published in: Journal of materials science Vol. 46; no. 12; pp. 4116 - 4134
• Main Author: Hardouin Duparc, O. B. M.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-06-2011; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Characterization and Evaluation of Materials; Chemistry and Materials Science; Classical Mechanics; Crystal lattices; Crystal structure; Crystallography and Scattering Methods; Crystals; Density; Engineering Sciences; Grain boundaries; Iib 2010; Lattice sites; Lattices; Liquid crystals; Materials Science; Mathematical models; Mechanics; Metallurgy; Mineralogy; Nodes; Organic chemistry; Polymer Sciences; Readers; Solid Mechanics; Structure; Symbols; Twins; Unit cell; Twin Plane; Symmetric Tilt; Bravais Lattice; Iron Carbide; Coincident Site Lattice
• ISSN: 0022-2461; 1573-4803
• DOI: 10.1007/s10853-011-5367-1

I trace the origin of the inverse density of coincident lattice sites to Georges Friedel in 1904 (Études sur les groupements cristallins). Georges Friedel (1865–1933), son of the Chemist and Mineralogist Charles Friedel, called this parameter the twin (macle) index and defined it as 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’ is our Coincident Site Lattice. Georges Friedel introduced the Σ symbol in 1920 (Contribution à l’étude géométrique des macles) as the ratio of the volume of a (not necessarily primitive) multiple cell to the volume of the primitive cell. G. Friedel provides his reader with several formulae which, in the cubic case, give Σ = 
h
2
 + 
k
2
 + 
l
2
(
h
,
k
and
l
being the indices of the twin plane) and a twin index
I
equal to Σ if Σ is odd, equal to Σ/2 if Σ is even. All these definitions and formulae are included in the 1926 version of his celebrated textbook ‘Leçons de Cristallographie’. Georges Friedel was also concerned with the ‘material lattice’ (the crystal structure) behind the mathematical lattice, but besides his contributions to the study of liquid crystals, Georges Friedel was mainly interested in Mineralogy and not in Metallurgy. This may explain why Walter Rosenhain apparently never knew of Friedel’s work and why Kronberg and Wilson had to re-discover the importance of the density of coincidence sites, at the atomistic level, in 1949 in copper. Georges Friedel’s grandson, Jacques Friedel, made the first numerical estimate of interface energies using interatomic potentials that same year but only published these results in 1953. Knowledge of these past events may help us to better understand the present theories and, hopefully, to develop our future understanding more efficiently.