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

  • IIB 2010
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Abstract

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.

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Notes

  1. In French, Georges is spelt with a silent final s.

  2. A process called ‘hydrothermal synthesis’. Robert Bunsen used glass vessels in 1839.

  3. In 1893, in parallel with Henri Moissan, he even thought he might have succeeded in synthesizing diamond.

  4. Charles Friedel had studied at the University and was not a ‘Polytechnicien’. Conversely, the French University always refused to grant Georges Friedel a salary when he taught at Strasbourg University after 1919. This dual French school system still exists today.

  5. Having spent his youth in an apartment in the building of the School of Mines, where his parents lived since his father also was the curator of the mineralogical collection, Georges had first expressed the wish to specialize in a quite different field, namely Naval Architecture, after the École Polytechnique. Yet, being rated first, he was not free to choose.

  6. G. Friedel strongly objected to the inappropriate term ‘liquid crystal’ (Otto Lehmann’s Fliessende Krystalle) but this appellation remained.

  7. First noted by Nicolas Steno, or Nils Stensen [‘Son of Stone’], (1638–1686), an outstanding anatomist who also layed down the fundamental principles of stratigraphy, and thus of geology. Steno later ruined his health in Catholic missionary work. He was beatified in 1988. His ‘preamble about solids naturally contained within solids’ (De solido intra solidum naturaliter contento dissertationis prodromus, that is, an attempt to explain the formation of fossils), published in Florence in 1669, remained largely unknown. To give full support to the law of constancy of inter-facial angles for minerals of the same species, the technical invention of the contact goniometer by Arnould Carangeot, Romé de l’Isle’s assistant, was essential. About goniometers, see [22].

  8. Note the variability of the orthography: crystaux, cristaux, cristallen, Krystalle, Kristalle. The Greek root ‘krustallos’ meant ‘solidified by cold’ (kruos: see cryogenics). In ancient times quartz was believed to be a permanently solidified form of ice. Robert Boyle was the first to use the word crystal in a general sense, not restricting it to rock crystal, in The Sceptical Chymist (1661). The German word Quarz is presumed to be of Slavic origin, although its exact meaning is not known.

  9. With the common etymology of zwei and two, akin to duo, double, duplex, dyad, as well as deux in French. Macle was also used in English, for instance by Lord Kelvin of Largs, in a Robert Boyle Lecture delivered at the Oxford University Junior Scientific Club, on the evening of May 16, 1893 (and reproduced as Appendix H in the Baltimore Lectures): ‘Coming back to quartz, we can now understand perfectly the two kinds of macling which are well known to mineralogists …’.

  10. The French also use the name ‘joint de grains’ = grain boundary (Korngrenze). For the etymology of macle, Webster’s dictionary gives the heraldry term mascle which corresponds to an empty lozenge, from the middle-old Dutch mask and maesche, which also gave mesh (= Maesche = maille). Hence, sometimes, a circumflex on the a: mâcle (for instance Pierre Curie in 1900, [17]). Recent comprehensive French dictionaries like Le Grand Robert (1985) and the Trésor de la langue française (CNRS & INLF Nancy, 1985) give the same etymology. The Trésor writes both macle and mâcle (mâcler).

  11. As a Teacher of Grammar at the Cardinal Lemoine College in Paris, René-Just Haüy got acquainted with Charles Lhomond, a famous grammarian who was very keen in botany. In order to please his Friend, René-Just learned the names of hundreds of plants and this certainly later helped him name crystals and groupings of crystals. René-Just Haüy had no inherent taste for Botany, but his walks with Charles Lhomond in the Jardin du Roy, adjoining their College led him to attend the lectures of Louis-Jean-Marie Daubenton who also taught Mineralogy.

  12. But from a purely aesthetical view point, I am sure that all would have been delighted at the sight of a drawing by Albrecht Dürer in his manual on measurement of lines, areas and solids by compass and ruler [27]. This drawing looks like a (double) five-fold twin, almost quasiperiodic (a size-limited approximant in fact), see [28] and [29]. Figure 2.3, by Eric Lord, Alan Mackay and S. Ranganathan, has a perfect pentagonal shape, much akin to the characteristic hexagonal shape of multiple twinning in aragonite. It is a pity that multiple-twinning led Linus Pauling (1901–1994) to reject the discovery of quasicrystals in 1985 [30]. Yet that challenge was not illogical and certainly stimulated refinement of the arguments in favour (the pigeons pro eventually won versus the contra cat, see [31]). Linus Pauling tried to contact with Danny Shechtman but unfortunately no satisfactory agreement was reached before Pauling’s death (D. Shechtman, personal communication, July 2010).

  13. Even if his father was a proponent of the existence of atoms, as was his mentor Adolphe Wurtz (1817–1884), it was clear that little was known about their actual organization within a crystal until the advent of X-ray diffraction in 1912. It is worth noting that ‘Friedel’s law’ in X-ray crystallography was established by Georges Friedel as early as 1913. [32]. This law states that the intensities of the Laue-Bragg diffractions I(Q) and I(-Q) are equal under normal conditions. That is, the information obtained by diffraction is centro-symmetric and cannot establish whether the real-space atomic distribution is centro-symmetric (with a point of inversion) or not. This also means that diffraction spots appear in pairs which have recently been called ‘Friedel pairs’: Friedel pairs are key to both the efficiency and accuracy of X-ray diffraction contrast tomography, which permits non-destructive mapping of grain shape and crystal orientation in polycrystals. [33].

  14. With the rejection of many chance crystal groupings that may look like twins at first sight but are actually aberrant twins (‘macles aberrantes’). For instance, ‘specimens noted I and III [in quartz] by Zyndel would be [according to Zyndel] twinned according to the La Gardette law (Japanese law). It suffices here to look at the sample with the naked eye, without any measuring and just having its faces glance, to realize that such is not the case’. [34].

  15. Many German scientists also contributed to the descriptions and analyses of twins: Christian Samuel Weiss, Friedrich Mohs, Carl Friedrich Naumann, Gustav Rose, Friedrich Eduard Reusch, Heinrich Adolf Baumhauer, Otto Mügge, Gustav Tschermack, Victor Moritz Goldschmidt, Jakob Beckenkamp, not to name all. Georges Friedel opposed many of them, but not because France had lost in Sedan in 1870: Georges Friedel also opposed his compatriot Frédéric Wallerant [35], and criticized Ernest Mallard when their views differed.

  16. Mero means part in Greek, as opposed to holo meaning whole. Hedra means seat, base or face. Hemi means half, tetartos means four and ogdoas eight. When hedra comes with a prefix it usually looses the aspired h (a special diacritic sign on the Ε in Greek which is then pronounced by rough breathing). This explains the French and German spellings of for instance: polyèdre, holoèdre, polyeder and holoeder. Modern English spelling keeps the h and writes polyhedron and holohedry, etc. French writes mériédrie, although méroédrie would be slightly more correct.

  17. Only the orientation of the atomic unit (basis) within the primitive cells changes. These twins have a twin index (see below) equal to 1.

  18. Auguste Bravais first wrote with his elder brother, Louis: an Essai géométrique sur la symétrie des feuilles curvisériées et rectisériées. This was transmitted to the French Academy of Sciences before 1837, see [36].

  19. ‘But it is clear that this is a purely fictional move’ (‘mais il est clair que c’est là un mouvement purement fictif, et que la coordination moléculaire se fait symétriquement par rapport au plan d’hémitropie’), said Auguste Bravais in 1850 [37]. For instance, the Σ = 3 {111} 〈110〉 θ ~ 70.53° twins in cubic systems can also be described as hemitropic Σ = 3 {111} 〈111〉 (θ = 180°) (as well as Σ = 3 {111} 〈111〉 θ = 60° twist grain boundaries since 〈111〉 are threefold axes in cubic systems).

  20. In 1885 [38], not in 1876 [25].

  21. As is the case for the twins by merohedry described previously.

  22. The primitive lattice point group is not necessarily a subgroup of the coincidence lattice point group, because it may possess symmetry elements that are not shared by the coincident lattice (see [40] and [41]).

  23. These twins are frequent in minerals which have a complicated atomic group associated with each primitive lattice point or node.

  24. With a natural extension to cases of twins formed by pseudo reticular merohedry. See Table 2 for a synthetic presentation of the four classes developed by Friedel.

  25. Friedel noted that the (310) case had never been observed.

  26. A School to whose development he significantly contributed. He became Head of the School from 1907 to 1919, except for the war years.

  27. The Friedels were Alsatian from Strasbourg. As a first name, Friedel is a variant for Gottfried. The German Friede means peace.

  28. See Appendix C for some further considerations on simple analytical properties of the twin index in cubic crystals.

  29. It is remarkable that G. Friedel immediately understood the principles of X-ray diffraction discovered in 1912, and explicited the inversion symmetry limitation (under normal conditions) known as Friedel’s law ([32], see also footnote 13). Donnay and Harker were able to expand Friedel’s 1911 preview for surfaces in 1937 [46]. Friedel’s formalism is limited to a consideration of Bravais lattices and does not incorporate the Schönflies-Fedorov space groups. Denis Gratias, Richard Portier, Robert Pond and others further extended Friedel’s formalism for twins in the 1980’s, but this goes beyond the scope of this article. Also see Appendix D.

  30. In contrast, the French(‘La Gardette’)-Japanese twin in quartz is much rarer and complicated, see [48, 51–54]. There are many other twin laws for quartz. One of these is also known as the Friedel-law, or Friedel-twin, having been found by Charles Friedel in artificial quartz in 1888 [55].

  31. Gregori Aminoff and (his wife) Birgit Broomé proposed several rules in 1935 about the atomic structure of twins in minerals. These rules have been reported by Robert Cahn [39]: 1. When two individuals form a contact twin either one or two layers of the structure at the interface are common to both individuals. 2. The atomic coordination in the transition layer is either (almost) identical with that in the crystal structure or closely related to it. In the latter case, the transition structure is that of a possible polymorphic modification of the structure, or else that of a modification which would be possible for that substance. British physicist by heart, R.W. Cahn was born in Germany and could speak French, see [57]. His 1954 review article is a wealth of informations about twins. .

  32. The order n of a Σ = 3n twin (or any Σ = Σ no twin) should not be confused with the twin index Σ itself. Such a confusion can arise because n has also been used as a symbol for the twin index (by G. Friedel himself, in his textbooks. In 1926 one finds n and I), and Paul Niggli (see Appendix B) translated Friedel’s French word indice by Ordnung in his books in German ([63, 64]).

  33. These twins thus merit study at the atomistic level, and have been investigated by joint numerical and observational studies, see [65, 66]. These twins are called grain boundaries in these studies. See, however, Appendix E.

  34. For instance William Lawrence Bragg, Bragg junior, could derive the fcc structure of copper as soon as 1914 thanks to natural crystal specimens from the Mineral Laboratory at Cambridge [67]. Other metal samples, usually consisting of tiny crystalline grains, had to await the development of the powder diffraction technique in 1917 (Peter Debye and Paul Scherrer in Göttingen, Germany, Albert Hull in Schenectady, in the US).

  35. Compare the following with Ronald King and Bruce Chalmers [68], Chaps 1 and 2 of Donald McLean [1], Ernest Hondros’ ‘enquiry’ in 1995 [69], and David Brandon’s recent perspective [70].

  36. Typical is the first sentence of HC Sorby in his article ‘On the microscopical structure of iron and steel’, published in 1887: ‘It is now more than 20 years since I first commenced to carefully study the microscopical structure of iron and steel, in order, if possible, to throw light on the origin of meteoric iron; but soon found that the results were of even more value in connection with practical metallurgy’. [72]. Most ceramic materials are also manufactured, but rarely from the melt and, at first glance, look more like hard minerals than like soft metals.

  37. Rosenhain had been thinking about the problem for several years, presumably since 1904, as would appear from the discussion of a paper by GD Bengough [82] where Guy Bengough (1876–1945) wrote that ‘the first action of a dilute reagent is to eat into the crystalline boundaries’ so that ‘the deduction may reasonably be drawn that the individual crystals in a pure metal are normally bound to one another by some substance stronger than the crystals themselves, but more easily attacked by etching agents. This substance must surely be no other than Beilby's amorphous material, arranged in a thin, more or less continuous layer round the crystals’. Sir George Beilby (1850–1924) never reacted positively to this hypothesis concerning the structure of grain boundaries.

  38. Although he did teach ferrous metallurgy in the 1890’s at the École des Mines in Saint Étienne.

  39. Speaking of Anderson and Norton, Foley wrote: ‘The authors have apparently driven another spike in the coffin of the general amorphous-metal hypothesis which has been reared a weakling from its inception’.

  40. That is, the (ductile) intergranular fracture of metallic polycrystals at high temperatures versus the (brittle) transgranular fracture at low temperatures (when the amorphous interface can be assumed to be as hard as a glass below its transition temperature).

  41. ‘such positions as will balance the atomic forces’. This over estimation of the amplitude of atomic relaxations probably allowed them to consider that, under stress the interface would become amorphous so that ‘the material will behave in the manner described by Rosenhain in connection with the amorphous cement theory’. Yet, even in the unstressed condition, Rosenhain wrote in the discussion that he could not accept their illustration: ‘I think that it implies an arrangement of atoms in a condition which I think is not one of possible stable equilibrium. It implies atoms being brought in some places too close together and in others too far apart to fulfil what we believe is known of the conditions of atomic linkage that exist in solid metals. Such an arrangement of atoms is certainly improbable and would require proof before one could accept it as a fact’. Recent observations and simulations of asymmetrical grain boundaries in copper show that atomic disorder exists only locally [65, 66]. A true \( (100)_{1} //(\bar{4}30)_{2} \) tilt GB has recently been grown, observed and simulated, in a ceramic material: SrTiO3 [90].

  42. Thum saw the development of slip planes but he unfortunately failed to recognize dislocations (which were probably present) because he had not been expecting to see them. Bubble soap models were rediscovered by Sir W. Lawrence Bragg and John F. Nye in 1947. With respect to Rosenhain’s model, dislocations did not help to simplify the considerations of deformation mechanisms in polycrystalline metals, but they are necessary to explain in details what is observed because they correspond to reality. At about the same time, the early thirties, the neutrino was postulated, not to simplify the existing theories, but to explain the riddle of the observed continuous energy spectrum of nuclear beta electrons.

  43. G. Friedel proposed the term nematics, from the Greek nema meaning thread, ‘because of the linear discontinuities, which are twisted like threads’, (‘à cause des discontinuités, contournées comme des fils’ [21]). These topological defects correspond to disclinations. For discontinuities in smectics, G. Friedel and his son Edmond (see Table 1) noted in 1931 that they must have ‘the form of groups of focal conics’ (‘les discontinuités n’y peuvent apparaître que sous la forme d’un groupe de coniques focales’ [91], See also G. Friedel and François Grandjean in 1910/1911 [92, 93]). François Grandjean (1882-1975) later became a specialist of acarians.

  44. This idea was generalized by Nevill Mott in 1948: ‘If two crystal planes are in contact, but cannot fit owing to different indices of orientation, one may suppose that the surface of contact is divided into islands where the fit is reasonably good, separated by lines near which fit is bad’. [94].

  45. The same reasoning applies for twin orientations near the special {111} twin in elemental fcc crystals, and as well as for the contribution of widely spaced secondary dislocations, so, as noted by Thornton Read and William Shockley, misorientations near energetically favoured twins may have the characteristic |δθ| ln|δθ| additional cusp contribution [96]. This does not work near the structurally favoured {122} twin, the energy of which does not correspond to a cusp.

  46. Jacques Friedel went to Bristol (1949–1952) to work with Nevill Mott and learn more about the electronic interactions in metals. Some of Jacques Friedel’s later contributions in that field eventually led, with François Ducastelle, to the development of the Finnis–Sinclair potentials for transition and noble metals (see [99]). His stay in Bristol also led Jacques Friedel meeting his future wife, a younger sister of Nevill Mott’s wife. He also met there with Charles Frank, who thus learnt of Georges Friedel’s Leçons.

  47. Friedel wrote that what mostly prompted this extention was a ‘beautiful work’ by M. Schaskolsky and A. Schubnikow using in 1933 crystals of alum in an experiment very much akin to the later MgO smoke experiments. This Schubnikow is no one else that Alekseï Vasil’evich Shubnikov (1887–1970) who is famous for the Shubnikov groups, or antisymmetry groups, magnetic groups, coloured groups, which have been used in many fields, including the study of Grain Boundaries (for instance by Yves Le Corre and Hubert Curien in 1958, and by Denis Gratias and Richard Portier, and Demosthenes Vlachavas and Robert Pond in the 1970s and 1980s). Shubnikov had also been a pioneer in the formation and growth of crystals, in Leningrad (St Petersburg) and in Moscow.

  48. Denis Gratias and Richard Portier already complained in 1982 that ‘the terminology to day in common use in grain boundary community is often unfortunate and confuses, for example, lattice nodes and crystal sites’. [117].

  49. E.g. Charles Kittel, Introduction to Solid State Physics, pp. 4–5, Neil Ashcroft and David Mermin, Solid State Physics (1976) p. 75. It was introduced, in German, by Max Born in 1922 [118]. At the Göttingen University Born served as an assistant to David Hilbert, his mentor, who was famous for his ‘basis theorem’ so that Born was certainly aware of the first usage of the term basis.

  50. Although, as we have seen, it is a sublattice from a mathematical point of view.

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Acknowledgements

I thank Professor Jacques Friedel for uncounted mail exchanges. I thank my two anonymous referees for their valuable suggestions. I thank David Brandon for his encouragements and several email exchanges, and Hans Grimmer and Massimo Nespolo for several interesting discussions. I also thank the library department of the École Polytechnique, and the ‘fonds patrimoniaux’ of the École des Mines in Paris.

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  1. Laboratoire des Solides Irradiés, LSI, CEA-X-CNRS, UMR 7642, École Polytechnique, 91128, Palaiseau Cedex, France

    O. B. M. Hardouin Duparc

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Appendices

Appendix A

Georges Friedel and René-Just Haüy and the common lattice

Just after he had published his conceptual discovery of the existence of ‘common lattices’ in twins, Georges Friedel found [101] a preliminary indication of this phenomenon in Haüy’s first Treatise on Mineralogy (1801) when Haüy discusses what he intends to prove by examination of the staurolite twins: ‘In the examination (…) we shall mainly apply ourselves to show: 1°. That each hexagon junction [that is, each mathematical interface which can be drawn at the junction of the branches of the cross in the two varieties of staurolite, see Fig. 1: 90° (Greek cross staurolite) and 60° (Saint Andrews cross staurolite)] is, with respect to one or other of the prisms [that is, the grains], as would be a surface produced by the ‘decrement’-law [that is, the boundary plane is a lattice plane of simple indices with respect to both grains]. 2°. That if one supposes the planes of each prism to be extended into the other prism, these extensions will have positions which can similarly be understood according to a ‘decrement’-law [and this, together with 1° and the fact that this is the same lattice plane for a symmetrical twin, will give the common lattice]’ [102] (see G8 in Appendix G).

Georges Friedel was correct of course. He could have added that Haüy’s statement 2° is only preliminary. In fact Haüy failed to repeat it in his 1822 Treatise of Mineralogy (nor in the accompanying 1822 Treatise of Crystallography).

Only Georges Friedel could have found that preliminary indication, and only he would have written that this was a clear statement (‘exprimé avec la plus grande clareté’) of his own discovery.

In 1905 Georges Friedel gave the precise reference to Haüy: 1801, vol. 2, p. 88. He was not to give it again in his 1926 book (p. 425), where he only referred to ‘Haüy (1801)’, a treatise which comprises four volumes of written text.

Appendix B

The transmission of G. Friedel’s teaching about twins: both ups and downs

The Swiss crystallographer Paul Niggli (1888–1953) was the first to integrate Friedel’s classification of twins into his German textbooks in 1919 and 1924 [63, 64]. With a smaller impact, G. Friedel’s approach appears also in published studies by some German scientists: Ernst Schiebold in 1919 [103] and Friedrich Heide in 1928 [48]. Jakob Beckenkamp in 1923 [104] and Margarette Löffler in 1934 [105] mentioned this approach in their review articles. In English, Joseph Désiré Hubert (José) Donnay, a Belgian born crystallographer and mineralogist, wrote an obituary of G. Friedel in 1934 [13] and used his approach in 1940 [106]. Charles Crussard referenced Friedel’s basic idea in 1945 in a metallurgical study of zinc (hexagonal), containing ‘obliquity’ (pseudo reticular merohedric twins) [107].

To my knowledge, Whitwham, Mouflard & Lacombe were the first to restate Friedel’s Σ with its explicit formula for the case of cubic crystals such as germanium and copper: Σ = h 2 + k 2 + l 2 or {h 2 + k 2 + l 2}/2, in 1951 [62].

At the same time, Jacques Friedel, attending Charles Frank’s lectures in Bristol (see footnote 46), commented on Frank’s description of twins and led him to discover his grandfather’s ‘Leçons de Cristallographie’. The ‘Leçons’ fitted Frank’s own sense of geometry (Jacques Friedel, personal communication, June 2010).

Jacques Friedel and Charles Crussard did not mention the twin index in their 1953 paper [98] because it was unnecessary.

JDH Donnay and his wife Gabrielle Donnay presented Friedel’s possible twin indices in terms of S = ∣hu + kv + lw∣ (Friedel’s original Σ) in a synoptic table in the International Tables for X-Ray Crystallography in 1959 [45], see Table 3.

Besides to his own grandsons, Charles Crussard and Jacques Friedel, Georges Friedel’s book was familiar to all French Metallurgists. These included Paul Lacombe and Claude Goux. The latter, after completing a PhD on grain boundaries thanks to bicrystals of pure aluminium specially prepared in Georges Chaudron’s laboratory at Vitry near Paris [108], established an important research group for both experimental and theoretical work on grain boundaries at the École des Mines in Saint Étienne in the sixties. Srinivasa Ranganathan learned about Friedel’s book through Claude Goux at Saint Étienne (see [109]).

It seems that this memory of Georges Friedel’s work faded in the international metallurgical community. In his 1957 milestone monograph ‘Grain boundaries in Metals’, Donald McLean mentioned the work of Jacques Friedel and Charles Crussard, but not that of Georges Friedel [1]. In contrast the memory of Georges Friedel is very much alive in the mineralogy community (see Appendix F for a tentative explanation of this contrast), with Hans Grimmer, Theo Hahn, Helmut Klapper, Giovanni Ferraris and Massimo Nespolo, for instance ([41, 110–112], being selective). It is interesting to read in volume D of the International Tables of Crystallography that the ‘Σ’ symbol is specially used by metallurgists: ‘The degree of three-dimensional lattice coincidence is defined by the coincidence-site lattice index, twin lattice index, or sublattice index [j], for short: lattice index. This index is often called Σ, especially in metallurgy. It is the volume ratio of the primitive cells of the twin lattice and of the (original) crystal lattice (i.e., 1/j is the ‘degree of dilution’ of the twin lattice with respect to the crystal lattice)’. [41].

A new term has been proposed to designate a specific ‘science of twins’: geminography [111, 112]. This comes from the Latin ‘geminus’ for twin. Gemini: twins, e.g., Castor and Pollux. The term should not be confused with gemology or gemmology, which is the science of gems and comes from the Latin ‘gemma’ meaning either a bud or precious stone. The term geminography appears for the first time in an article in Japanese by Hiroshi Takeda [113]. It has been proposed by José Donnay in a personal communication to Hiroshi Takeda who followed his lectures at the Johns Hopkins University in 1963 (see [112]). Donnay and Takeda published several articles together, including one on Compound tessellations in crystal structures (Acta Cryst. 1965 19:474–476), following Harold Coxeter’s idea of a ‘compound tessellation’ (Configurations and Maps, Rep. Math. Colloquium, 1948/1949 8:18–38). Formulae in this same 1949 report inspired Srinivasa Ranganathan to derive his generative function during his PhD work in Cambridge. This was shortly before David Brandon left Cambridge for the Battelle Memorial Institute in Geneva, Switzerland, to work with Walter Bollmann (1920–2009) [100, 109].

Appendix C

Ranganathan’s generating formula versus Friedel’s approach, for cubic crystals

Friedel’s approach: (hkl) [hkl] 180° (hemitropy)

$$ \Upsigma = \left( {h^{ 2} + k^{ 2} + l^{ 2} } \right)/\beta $$

h, k and l are the (Miller) indices of the boundary plane: three relatively prime integers and β is 1 or 2 according to parity (in order to have Σ odd as it ought to be for cubic crystals)

Comments

One may then want to look for the minimal (〈uvw〉, θ) mathematical representation (among the 24 equivalent representations in cubic crystals), or for a representation more suitable for a given purpose (for instance, the fcc twin Σ = 3 (111) [111] 180° is also (111) [111] 60° (the 〈111〉 axes are threefold axes in cubic crystals) but is better visualized along a 〈110〉 direction if one wants to see the traces of the ABC planes and the ABA fault in the …ABCABACBA… mirror structure. It is then described as a tilt (111) [\( 1\bar{1}0 \)] 2tg−1(1/\( \sqrt 2 \)) (~70.53°).

Ranganathan also noted in 1966 that ‘this approach leaves the question of finding the possible Σ for a given axis undecided’, without the help of a computer.

Ranganathan’s formula [109]: considering a rotation axis 〈uvw〉

$$ \Upsigma = \left( {x^{ 2} + R^{ 2} y^{ 2} } \right)/\alpha R = \sqrt {u^{2} + v^{2} + w^{2} } = {\text{Norm}}\left( { {<}uvw{>} } \right){\text{ tg}}(\theta / 2) = Ry/x $$

x and y are the two relatively prime integers (with no common divisors except 1) and α is a multiple of 2 so as to get Σ odd (as it ought to be for cubic crystals)

Comments

It is then not too difficult to find three relatively prime integers h, k and l which will fit Friedel’s equation and hu + kv + lw = 0 (because the plane contains the rotation axis). The solution may not be unique (thanks to the possible division by α). Ranganathan’s formula is concerned with coincidence, not with the choice of a GB. Now, for a symmetrical tilt GB, what matters most is the knowledge of its (hkl) Miller indices: one needs to know the interfacial plane, just as one needs to know the Miller indices of a surface plane. Σ is a non univocal associated number, and θ is also ambiguous when the rotation axis is a symmetry axis of the structure. One may also be concerned with asymmetrical tilt GBs, best defined as (h 1 h 1 l 1)//(h 2 h 2 l 2) which may have a coincidence index or not (see for instance [65, 66, 114]), or with twist GBs or mixed tilt/twist GBs.

If one has a computer program, one can use Friedel’s approach and the search for the minimal (〈uvw〉,θ) rotation matrix in a systematic way. That approach can also be generalized to non cubic crystals in a brute force way with computers, and tolerance limits in cases of reticular pseudomerohedry (as it is most often the case in non cubic crystals, see Hans Grimmer, David Warrington, Roland Bonnet, George Bleris, Pierre Delavignette, Gérard Nouet, Serge Hagège, Theodoros Karakostas, not to name them all).

One can mention another property of Σ for cubic crystals: considering coincidence rotations, Warrington and Bufalini showed in 1971 that Σ2 is a sum of three squared integers: \( \Upsigma^{ 2} = R_{i1}^{2} + R_{i2}^{2} + R_{i3}^{2} \), i = 1, 2, 3 where R ij are the nine integer elements of the \( {\frac{1}{\Upsigma }}\left( {R_{ij} } \right) \) rotation [115]. Grimmer, Bollmann and Warrington later provided a nice demonstration that Σ is odd for cubic crystals [116]: Let us express \( R_{i1}^{2} + R_{i2}^{2} + R_{i3}^{2} \) in the form 4n + k i where n is an integer and k i  = 0, 1, 2, or 3, i.e., k i is the number of odd integers among R i1 , R i2 , R i3 , i = 1, 2, 3. If Σ is even, then Σ2 is a multiple of 4 and k 1  = k 2  = k 3  = 0, so that all R ij and Σ are even, which contradicts the convention that there is no integral factor common to Σ and the nine R ij . Thus, for cubic crystals, Σ is odd.

Appendix D

‘Lattice sites:’ atomic sites or mathematical lattice sites

This is an important question. Unfortunately, common usage has imposed a confusing terminology. The ‘sites’ of the coincident site lattices (CSLs) are mathematical nodes of Bravais lattices. Yet ‘lattice sites’ correspond to atomic sites (occupied or empty) in many papers, and matter is made of atoms, not mathematical nodes.Footnote 48

Initially, people who believed in the existence of atoms did not know what they look like: neither their size nor how they could be distributed in crystalline matter. For instance physicists thought of salt (NaCl, halite as the mineral, common salt otherwise) as made of tiny ‘molecules’ (NaCl) regularly spaced, so that compression of a crystal of halite would decrease the lattice spacing but not necessarily the size of the ‘molecule’. Georges Friedel strongly opposed such ideas and considered it better, in a first approach, not to include the atoms in his description of twins, although he believed in their existence and knew that the atomic motif played a role in crystal structure: see the 1904 and 1911 quotations reproduced in the section on the material lattice. In consequence, he only considered Bravais lattices, for the sake of simplicity. Friedel did mention Fedorov’s work quite respectfully but wrote he was unsure how it might add to the Bravais’ description. In his 1926 book, he of course acknowledged the Schönflies-Fedorov work.

In the field of crystallography, a ‘node’ is a mathematical point of a Bravais lattice with which an atomic motif is to be associated. The word ‘site’ is normally restricted to the explicit atomic distribution: either a site actually occupied by an atom, or which could be occupied by an atom. It is unfortunately not the case in the CSL where it means a ‘node’.

The word ‘basis’ can be used in its mathematical sense of a (minimal) set of basic elements or vectors which can generate a group or lattice, viz the set of periodically spaced Bravais nodal points. The use of the word ‘basis’ in English in the field of solid state physics, can also mean the atomic motif associated with each Bravais node,Footnote 49 and is therefore unfortunately at odds with the other usage, despite the phonetical resemblance.

In the metallurgical community, when Kronberg and Wilson re-discovered, in 1949 the concept of coincidence sites in grain boundaries, in their study of secondary recrystallization in copper [119], they drew a ‘coincidence plot showing relation between positions of atoms’. Their ‘density of coincidence sites’ was the density of coincident atomic sites, for instance: ‘It is seen that 1/7 of the atoms of the new orientation are in coincidence with atoms of the old orientation, and the positions of these coincidence atoms define a unique equilateral net which is a multiple of the primitive net’. When Ellis and Treuting further considered the atomic relationships in the cubic twinned state, and introduced the ‘Coincidence Site Superlattice’Footnote 50 phrasing as the title of one section in their article [120], they were also explicitly considering atomic lattices. Ellis and Treuting were working on germanium, a semiconductor, their work is rarely cited in the metallurgical literature. At this same time, Jacques Friedel was attending Frank’s lectures in Bristol, and began his discussions with Charles Frank (see Appendix B). It later became apparent that CSLs were dealing with Bravais nodes, not with atomic sites. It is important to keep in mind these semantic difficulties.

At the atomistic level, the atoms at the interface relax in such a way as to minimize stresses (and, more generally, the free energy, see [121]). Their positions and their electronic structures may change. It can be compared to surface reconstruction cases. This relaxation will move the atoms away from their positions expected from a purely geometrical model, often in a non negligible way except in some simple cases such as the coherent Σ = 3 (111) twin in fcc metals. This atomic relaxation may also involve a non negligible global translation of the grains with respect to each other. Such ‘rigid body translations’ were first suspected via atomistic calculations by Michael Weins and coll. [122, 123] and experimentally confirmed in 1974 using displacement fringes by Bob Pond, David Smith and William Clark [124, 125].

Appendix E

‘Twins’ versus ‘grain boundaries’?

This may sound like a decadent scholastic question, but it still provokes lively discussions every now and then. There is no definitive answer. As for notations, it is probably best to define carefully the distinction that one wants to make.

In the first instance, a macle, or a twin, was a grouping of mineral crystals whose macroscopic shapes obviously exhibited some special orientation relationship (that is, they presumably obeyed a physical law, even if the exact nature of that law was difficult to determine). Nothing was known about the internal nature of the interface that separated the two grains. It might be planar or it might be thought of as a random interface so that the two grains seem to penetrate one another and the twin could be designated a ‘penetration [inter-penetrating] twin’ (‘macle par pénétration’). In practice, Mallard distinguished between ‘groupements par pénétration’ (grouping by merohedry or pseudomerohedry), for which he thought the interface could be ‘shapeless’, with full interpenetration, and ‘macles’ for which the interface was planar, as in the Bravais reticular hemitropy, picture. Wallerant objected to this and thought the interface was necessarily shapeless. While Friedel thought the interface was essentially shapeless, but necessarily planar for some pseudo-merohedric twins. He also wrote that the situation was the same for the two reticular cases. Thus, the common assertion that twins, or twin boundaries, are simple (symmetric, even locally, at the atomic level) whereas grain boundaries are more complicated has no historical basis.

A further distinction might be that twins are ‘natural’ and occur in natural minerals, due to ‘natural’ growth mechanisms. Such twin boundaries should presumably, but not necessarily, be simple and of low energy. Industrial processes and intentional experimental growth could generate artificial, and more complex, grain boundaries. This distinction, difficult to maintain in practice, could yet sound reasonable, since it points to the true historical evolution: scientists first started by observations of the available natural specimens before they could master the production and observation of artificial materials.

Appendix F

Mineralogy versus metallurgy, or natural materials versus man-made materials

Aristotle, in his Meteorology, divided the mineral world in two groups: stones and metals (see Eichholz [126]). Stones cannot be melted whereas metals are fusible and malleable (like iron, gold and copper). Aristotle’s physics is common sense physics and has some truth. Metals are rarely found in nature as recognizable crystals, as it is the case for quartz, calcite, fluorite, halite (rock salt), not to mention gemstones like topaz, sapphire, ruby and diamond. In spite of its fooling gold lustre, pyrite is not a metal (it is a semiconductor with a bandgap equal to 0.95 eV. In not that old physics textbooks, semiconductors as silicon or germanium did not exist as such and were simply considered as non metals). Even if metals are minerals, in principle, see for instance books III and IV of Albertus Magnus’ De Mineralibus Libri, respectively, entitled Metals in General and The Metals Individually (see also, of course, Romé de l’Isle and Haüy’s treatises), common sense clearly ‘feels’, still today, the distinction between Mineralogy and Metallurgy. This seems to be true in Occident as well as in Orient: Metal and Earth are two of the five distinct elements of the traditional Chinese system. We of course ought to be faithful to the scientific spirit of Aristotle, viz. observation and interpretation, rather than to his letter, which has been written more than two thousand years ago. Man-made metals and native minerals are nowadays equally available and observable.

Appendix G

Original french texts

  • G1: ‘Quand dans un criftal quelconque, il se trouve un ou plufieurs angles rentrans, on doit en conclure que ce n’eft point un criftal simple, mais un groupe de deux ou de plufieurs criftaux, ou même de deux moitiés retournées d’un même criftal. Ce criftal prend alors le nom de MACLE’ [23].

  • G2: ‘Ce qui importe (…), c’est de savoir quel est (…) le rapport du nombre total des nœuds du réseau simple au nombre des nœuds rétablis. Nous donnerons à ce rapport le nom d’indice de la macle. Il est de 1 pour les macles par mériédrie ou par pseudo-mériédrie’ [39, p. 1090].

  • G3: ‘quel que soit le mécanisme en vertu duquel la continuation d’un réseau multiple suffit à assurer la cohésion entre les motifs multiples diversement orientés, cette cause ne peut agir que si la maille multiple n’est pas trop grande’ [39, p. 1072].

  • G4: ‘Nous trouverons, au cours de l’étude des espèces, des raisons de croire que c’est le réseau matériel plutôt que le réseau cristallin des points analogues qui détermine les macles’ [39, p. 1089].

  • G5: ‘Il en est des macles comme des faces extérieures : étant donné un réseau, nous pouvons prévoir quelles sont les macles possibles et en gros quelles seront les plus fréquentes, mais non dans le détail quelles sont celles qui se produiront dans telles ou telles conditions de cristallisation’. ‘Les propriétés du motif interviennent de nouveau, et cela d’une manière jusqu’ici impossible à prévoir et à expliquer, pour rendre fréquente telle macle, rare ou inconnue telle autre qui semblerait, d’après les conditions réticulaires, devoir se produire’ [42, pp. 260–261].

  • G6: ‘Comment se fait cette solidification? (…) C’est là un fait très général que l’expérience montre vrai pour l’acier. Dans ce cas particulier, ce sont des globulites de fer qui vont se précipiter au sein d’un liquide mère formé essentiellement de carbure de fer et contenant en outre diverses combinaisons du fer avec les métalloïdes; (…) ils se serrent les uns contre les autres (…) et se limitent par des faces de polyèdres. Mais (…) les granulations restent mouillées par leur eau-mère qui se distribue en couches minces dans leurs intervalles capillaires. (…) Finalement, il reste à l’état fluide un mélange plus ou moins complexe, où domine ordinairement le fer carburé, qui se solidifie à son tour dans les joints des globulites polyédrisés et les unit en un seul bloc : c’est le ciment’ [73].

  • G7: ‘Considérons un corps formé de grains cristallins isolés, très petits, empâtés dans un réseau à peu près continu de matière très visqueuse’ [74].

  • G8: ‘Dans l’examen que nous allons faire (…) nous nous attacherons principalement à prouver : 1°. que chacun des hexagones de jonction est situé, par rapport à l’un ou l’autre des prismes, comme le seroit une face produite par une loi de décroissement; 2°. que si l’on suppose les pans de chaque prisme prolongés dans l’intérieur de l’autre prisme, les prolongements auront de même des positions que l’on pourra rapporter à des lois de décroissement’ [102].

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Hardouin Duparc, O.B.M. A review of some elements in the history of grain boundaries, centered on Georges Friedel, the coincident ‘site’ lattice and the twin index. J Mater Sci 46, 4116–4134 (2011). https://doi.org/10.1007/s10853-011-5367-1

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