A review of some elements for the history of mechanical twinning centred on its German origins until Otto Mügge’s K ₁ and K ₂ invariant plane notation

Abstract

There is a specific nomenclature for the so-called mechanical twins or twins obtained by mechanical deformation. This nomenclature, (K ₁, η ₁, K ₂, η ₂), is due to Otto Mügge in 1889. The German mineralogist and crystallographer Mügge (1858–1932) rationalised twinning observations made by himself and by other German scientists since 1859 such as Friedrich Pfaff, Heinrich Wilhelm Dove, Eduard Reusch and Heinrich Baumhauer, not to forget a formal contribution by William Thomson and Peter Guthrie Tait noticed by Theodor Liebisch who informed Mügge about it, and further classifications by Arrien Johnsen, one of Mügge’s pupils. The presentation of this scientific development also mentions the medieval alchemist Pseudo-Geber and the Renaissance metallurgist Vannoccio Biringuccio who ‘heard’ the cry of the tin now known to be due to deformation twinning, the first models of progressive twinning involving some local atomic rearrangement, with Woldemar Voigt in 1898, Yacov Il’ich Frenkel and Tatyana Kontorova in 1938 and the first drawings of a twin(ning) dislocation by Vladimirskii in 1947 and by Charles Frank and Jan van der Merwe two years later.

Appendices

Appendix 1

Pseudo-Geber and Vannocio Biringuccio. Creak or cry tests on tin: From written knowledge, the first scientists who intentionally produced mechanical twinning, even if they didn’t know it was twinning, are the pseudo-Geber and Vannoccio Biringuccio with their creak or cry tests on tin,²² between his teeth with Biringuccio for whom it was a test of purity.²³ Twins or twin crystals, however, were not yet conceived in those times,²⁴ let alone twinning in a metal, so that these mentions are only of anecdotical value. These observations by Pseudo-Geber and by Biringuccio did not have any influence on the advancement of mechanical twinning.

Pseudo-Geber is an anonymous European alchemist born in the thirteenth century once confused with Jabir ibn Hayyan, an eigth century Islamic alchemist. In his summa perfectionis (Geberi regis Arabum Summa Perfectionis ministerii), probably written in the thirteenth century, one finds: “I intimate to the sons of learning that tin is a metallic substance which is white, but not pure white; it has a little ring, and emits a creaking sound [sonans parum stridorem];” as translated by Joseph William Mellor [63] and see [62] for both a latin version and a commented translation; Pseudo-Geber also tried to draw conclusions from experiments since he could not be aware that the state of knowledge was too crude in his time to lead to good conclusions. He noted that “After its double calcination, it [tin] does not creak [post vero illius duplicem calcinationem non stridet]”, but believed that the substance causing the creak in tin was mercury [62].

Vannoccio Biringuccio (1480–c.1539) is better known than Pseudo-Geber and he can be considered as a father of the foundry industry, thanks to his famous book De la pirotechnia published a little after his death in 1540 and where he wrote, speaking about tin: “That metal is known to be purer… if when some thin part of it is bent or squeezed by the teeth it gives its natural cracking noise, like that which water makes when it is frozen by cold [Cogno∫ce∫∫i que∫to tanto e∫∫er piu puro,… piegandolo, in qualche parte ∫ottile, o col dente ∫tringendolo, ∫i ∫ente un natural ∫uo ∫tridore, come fa l’acqua dal freddo gelata ²⁵]” as translated by Martha Teach Gnudi and Cyril Stanley Smith [64]. Oddly, C.S. Smith, albeit a great metallurgist and historian of science, makes no footnote about the cry of tin.

If the creaks emitted by tin or zinc under bending are not a sign of purity, it is a sign of cristallinity as it appeared to the German metallurgist Kalisher in 1881: “The assumption that the “cry” of tin [das „Schreien“des Zinns] is a consequence of the crystalline structure finds a new confirmation with zinc” [65]. Mellor’s interpretation of the phenomenon in 1927 was not yet exactly correct, as a kind of internal friction between crystals in polycrystalline tin: “The crystalline structure of tin is further evidenced by the fact noted by Geber in (…); this is supposed to be produced by the grinding of the crystals against one another during the bending of the metal” [63]. Also in 1927, Champion Herbert Mathewson and Albert J. Phillips noticed that “Although satisfactory commercial fine-grained soft strip does not emit a perceptible zinc cry when bent, somewhat coarser-grained strip does, and apparently the intensity of the cry depends on the size of the crystals and might even be recognized in very fine-grained material by using a suitable audiometric method” [66].

Mathewson and G.H. Edmunds probably were the first to attribute Biringuccio’s experience to twinning in 1928, at least implicitly, in their paper about the production of Neumann bands in silicon ferrite, which they clearly recognised as twinning lamellae: “Neumann bands were produced in this material by bending, stretching, rolling and hammering. Their appearance was accompanied by crackling sound similar to the tin or zinc “cry” [67]. (It is amusing to see Mathewson and Edmunds speaking of Mügge’s Einfache Schiebung, not knowing it was a German translation of the English ‘simple shear’ introduced by Thomson and Tait, see the second section about Thomson–Tait, Liebisch and Mügge, and Appendix 3).

The acoustic emission from solids under deformation is nowadays a well-recognised experimental technique, see for instance [68].

Appendix 2

Vocabulary Problems. Twin as a group of two or as a part of a group of two: ‘Twin’ came to be an ambiguous noun: a twin, or twin crystal, or mascle, as originally defined by Romé de L’Isle in 1783 designed a whole, a special group of two crystals: “When, in a given crystal, there is one or several entering angles, one must conclude that this is not an elementary crystal but a group of two or several crystals, or even two inverted halves of a similar crystal. This crystal is then named a MACLE”, see [1]. That definition of a twin as a group of two crystals was repeated by William Hallowes Miller in his 1839 Treatise on Crystallography: “A TWIN crystal is composed of two crystals joined together (in such a manner that…²⁶)”. Yet, for veins which were observed in calcite and determined to be in twinned orientation with respect to the matrix (see Brewster and Reusch for instance), the noun ‘twin crystal’ or ‘twin’ was given to these veins themselves, as a synecdoche. The same synecdoche is used for the more or less extended lenticular veins observed in deformed metals for instance.

One must thus take care that the word ‘twin’ may designate a group of two crystals or just one part of such a group, specially when that part clearly looks embedded in the other part which can be considered as the matrix or parent part. Charles Frank, inspired by the Greek mythology proposed to name Castor and Pollux the two parts which do look equal and tightly related at the atomic level (see [43]). In most articles dealing with disconnections, one finds one crystal labelled with the Greek letter λ and the other with the Greek letter μ. The λ part is normally drawn with white atoms, since λ is the first letter of leucos which means white in Greek, with the μ part normally drawn with black atoms since μ is the first letter of melanos meaning black in Greek [Bob Pond, personal communication].

Our familiar symmetric tilt grain boundaries (GBs) have been first described by René-Just Haüy and by Auguste Bravais as 180° twist GBs: half-turned crystals with respect to an axis perpendicular to the boundary surface: twins by hemitropy.

Appendix 3

Otto Mügge’s original texts in 1889: Otto Mügge [2], p. 281: “Es sind zweierlei Deformationen beobachtet. Beides sind „einfache Schiebungen”; und zwar führt die eine, im folgenden als α bezeichnet, die Krystalle in Zwillingsstellung nach {010} über, die zweite, im folgenden mit β bezeichnet, bewirkt Zwillingsbildung nach [010]”. Which I translate as: “There are two types of observed deformations [with the triclinic salt Mügge studied]. Both are “simple shears”, to wit: the first one, referred as α in the following, brings the crystal in twinning position along the crystal in twinning position along {010}, the second, referred as β in the following, causes twinning by [010]”.

Otto Mügge [2], p. 286: “Die im Vorstehenden beschriebenen Deformationen sind offenbar homogene, und zwar einfache Schiebungen. Bei der ersten (α), deren allgemeine Verhältnisse wir zunächst untersuchen wollen, ist {010} die rationale Gleitfläche: wir bezeichnen dieselbe allgemein mit k₁ = {k₁₁, k₁₂, k₁₃}; sie ist zugleich die erste Kreisschnittsebene und Zwillingsfläche. Für einfache Schiebungen ist weiter charakteristisch eine andere, zweite Kreisschnittsebene genannte Ebene; ihr allgemeines Zeichen sei k₂ = {k₂₁, k₂₂, k₂₃}”. I again freely translate as: “The previously mentioned deformations are obviously homogeneous, namely simple shears. For the first one, (α), the general relations of which we want to examine first, the rational gliding surface is {101}: we designate it in general as k₁ = {k₁₁,k₁₂,k₁₃}. It is at the same time the first circle cutting plane (circular cross-sectional plane) and twin surface. For simple shears there is still another, characteristic circle cutting plane, called second cutting plane. Let its general description be k₂ = {k₂₁, k₂₂, k₂₃}”.

Appendix 4

Mügge’s (K ₁,η ₁,K ₂,η ₂) description of mechanical twins and Friedel’s classification of twins: The aim of Georges Friedel’s classification of twins was to classify all observed twins whatever their origins, during crystal growth or after crystal growth. Otto Mügge’s (K ₁, η ₁, K ₂, η ₂) only deals with mechanical twins, twins obtained by mechanical deformation of crystals. It is clear, as already noticed by Robert Cahn [14], that Friedel’s ‘twins by merohedry’, for which the lattice has the same orientation for the two crystals (only the atomic motif is changed from one crystal to the other, and the boundary does not need to be planar), are not mechanical twins. Only ‘twins by reticular (i.e. lattice) merohedry’ can be formally described by a (K ₁, η ₁, K ₂, η ₂) set, even if they have not been obtained by deformation, such as annealing twins in metals (i.e. twins observed after annealing a polycrystalline metal). Georges Friedel was concerned about finding a common lattice between the two crystals of the bicrystals, the crystal lattice itself for ‘twins by merohedry’ or a small multiple lattice of the two disoriented crystal lattices, with as small a multiplicity (the twin index) as possible in mineral twins (see [1]). Otto Mügge was concerned with a proper description of the mechanical twins, in terms of simple shear thanks to Thomson, Tait, and Liebisch. We now know that, except for cubic crystals, many mechanical twins actually do not have any meaningful coincidence lattice. For instance, for mechanical twins observed in hexagonal metals, a ‘twin index’ could only be defined for approached values of the physical ratio c/a which differs from one hexagonal metal to another when the (K ₁, η ₁, K ₂, η ₂) description remains the same. It is probably pointless to discuss from the point of view of mechanical twinning the twins which correspond to approximate (pseudo, see end of footnote 23) merohedry or to approximate (pseudo) reticular merohedry within the angular limits of tolerance defined by Georges Friedel, viz. around five or six degrees, nor all the other classes of twins which have been defined since (see [69]).

Appendix 5

Tentative illustration of the meaning of the K ₁, K ₂, η ₁, and η ₂ symbols: This appendix is intended to illustrate for a modern non-specialist reader the concepts and the K ₁, K ₂, η ₁, and η ₂ symbols introduced in the present paper. It is illustrated with a realist, simplified, special case, trying to be both not too complicated and not too simple, clearly an impossible task and I apologise on both sides. It should also be kept in mind that although Otto Mügge knew his crystals were made of atoms, he only knew of lattice and crystalline symmetries and did not know the positions of the atoms within the motifs of his complicated minerals (with chemical units such as BaCdCl₄·4H₂O or K₂(Cd,Mn)(SO₄)₂·H₂O).

Before the bicrystal state schematically shown in white and black in Fig. 5, what originally existed, as in the top of Fig. 2, was a monocrystal (white circles, corresponding to atoms or simply to lattice nodes). What exists at a subsequent time is a bicrystal with unchanged white atoms below a K ₁ plane and moved atoms above it. These moved atoms are shown in black. They are in symmetrical positions with respect to the lower white atoms with respect to K ₁. The drawing specifically corresponds to one ($1\bar{1}0$) plane of a {111} twin in an fcc crystal (such a twin is known to have a very small interfacial energy in copper for instance. It could occur as a growth twin or as an annealing twin. We do not assume such an origin here). One can obviously not believe that the positions of the black atoms are the result of a physical half-turn (hemitropic) rotation of the upper white atoms with respect to a well-chosen axis perpendicular to K ₁ (viz. the prolongation in the upper part of the [111] vector drawn in Fig. 5) even if it is true from a pure geometrical point of view. One can try to imagine a more plausible physical way, in terms of a homogeneous (simple) shear motion along x, with s _x (y) = s·y, where y is the distance to K ₁, to be applied to every white i atom having y _i  > 0. Several possibilities still exist. The one which would transform the set of K _2b (half-)planes ((110) planes) into K ^′_2b (half-)planes would imply much larger s_b·y _i atomic motions than the one which transforms the set of K _2a (half-)planes [($11\bar{1}$) planes] into K ^’₂ (half-)planes (with the fcc {111} twin choice, one has s _a = 1/√2 versus 2√2 for s _b). One thus chooses K ₂≡K _2a, and the horizontal arrows show the corresponding shear atomic motions. The plane of shear is the (x, y) plane of the sheet [here a ($1\bar{1}0$) plane]. One then notes η ₁, the shear direction vector. It belongs to K ₁ and the plane of shear (here η ₁∝[$11\bar{2}$]). One also defines a vector η ₂ which belongs to K ₂ and the plane of shear and which may, by conventional choice, reflect or not the periodicity of the lattice (here η ₂∝[112]). We thus have the (K ₁, η ₁, K ₂, η ₂) set that defines a twinning mode. Yet, even such a simple shear motion is unlikely: depending on how the deformation stress is applied, the real atomic motion may occur only progressively from the top, such as shown in Fig. 2 and certainly locally and progressively from left to right, pushing rightwards the upper part of the crystal, with a twinning dislocation such as schematically shown in Fig. 4. Some atomic shuffles also certainly occur at the atomic level (see [56] and [18] with illustrations from atomic simulations).

Figure 5

Tentative illustration of a possibly mechanical twin and its twinning elements. The white (open) circles below the trace of the K ₁ plane, together with the black (filled) circles may correspond to the atoms of a ($1\bar{1}0$) plane of a {111} twin in an elemental face-centred cubic crystal such as copper. See text in Appendix 5 for more comments

The phrase “invariant” planes corresponds to the fact that the set of all planes parallel to K ₁ are not distorted by the simple s _x (y) = s·y shear. They are just translated. The set of all (upper half-)planes parallel to K ₂ are not distorted either by the simple shear, in the sense that they are just rotated into their corresponding K ^′₂ planes (Fig. 5 shows two such K ₂ planes). Any other plane will be distorted.

Figure 6, slightly modified from the 1954 book by Eric Ogilvie Hall (better known for sharing his name with Norman Petch for a famous relationship in plasticity) [10] shows the relations of a sphere and its half upper part homogeneously sheared into a half-ellipsoid. The ‘planes of the circular sections’, or Kreisschnittebenen, are clearly visible. The bicrystal with its nodes and atoms are not visible.

Figure 6

“Invariant” plane relations between a sphere and its half upper part homogeneously sheared into a half-ellipsoid (from Hall 1954 [10])