ON MECHANICAL TWINNING OF CRYSTALS

D. B. Gogoberidze, Tbilisi

“Mechanical twinning” is the name given to a deformation in which one part of a crystal passes into a position symmetrical with respect to some plane of another part of it[^1]. In this case the plane of symmetry is called the twinning plane, and the direction of the vector parallel to the displacement of the twinned part is called the direction of twinning. Finally, the line lying in the twinning plane and perpendicular to the direction of twinning is called the twinning axis.

It was usually considered that the twinning plane must be a rational crystallographic plane, but recent investigations have cast doubt on this assertion in some cases[^2].

The formation of twins during the growth of crystals is a widely known phenomenon. Almost all minerals form twins on growth that have long been known to mineralogists and crystallographers. Metallurgists and crystal physicists have also drawn attention to another circumstance: it has turned out that a whole series of crystals, under mechanical action, gives twins very similar to growth twins. The classic object in this respect is calcite (Iceland spar), studied as early as by Baumhauer, Mügge, and others[^3]. They found that when forces along certain directions are applied, calcite crystals pass into the twinned position and that the twins obtained in this way are similar to growth twins.

Metallurgists also observed that some metals, under plastic deformation, show a characteristic parallel striation (called Neumann lines, after the investigator who discovered it), caused by the formation within the mother crystal of a parallel system (or systems) of twins.

It proved of interest that crystals of all metals in which Neumann lines are observed readily form growth twins during recrystallization.

At present deformation twins are known in the following metals (Table 1).

The plane \(k_1\) is the twinning plane, and \(k'_2\) is the plane perpendicular in the normal crystal to the twinning plane, while in the twin it is rotated toward it through the corresponding angle.

Let us denote by \(2\varphi\) the angle formed by the planes \(k_1\) and \(k_2\) in the twinned crystal. It can be calculated from the following obvious relation:

\[ \operatorname{tg} 2\varphi = \frac{2}{S}. \tag{1} \]

The magnitude of the displacement \(S\) given in Table 1 is the distance through which a point moves that is at a distance equal to unity from the twinning plane in a crystal not yet twinned.

These quantities completely determine the character of twinning. In fact, the magnitude \(S\) gives the magnitude of the displacement (in units of the lattice constant) for a point separated from the twinning plane by one constant, i.e. it characterizes the magnitude of the displacement of atoms at the vertices of the cell. The indices of the planes \(k_1\) and \(k_2\) are precisely the quantities that are usually determined experimentally.

TABLE 1⁴

The geometry of twinning has in general been studied satisfactorily. At the same time, however, a fact that is interesting and has not yet found a satisfactory explanation is the circumstance that deformation twins are completely analogous to growth twins.

On closer examination, however, not everything is as simple as Elam thinks and as follows from the table cited above. As early as Zacharias⁵, in studying deformed zinc crystal pairs, some of which had well-developed faces, observed that the twins were not always parallel to one another. Measurements we have made show that the angles between such twins, obtained by Zacharias, are very small (of the order of \(2^\circ\)). Even in the old work of Wassermann and Schmid⁶ on X-ray photographs of deformed zinc one may notice a characteristic splitting of lines, to which the authors did not pay attention; according to our calculations, these correspond in magnitude to the rotations observed by Zacharias. Thus one may think that already in zinc the picture is far from so simple, and that the twinning planes can sometimes have not altogether simple crystallographic indices, as Schmid and Elam think. Even more clearly this is indicated by similar studies of deformation phenomena in rock salt, carried out by Brilliantov and Obreimov². They undoubtedly showed that in rock salt, under deformation, twins are formed, and these twins are not parallel to one another. The angle that these twins form with one another is of the order of \(1\)—\(2^\circ\); the same angles they form with the plane \((110)\). In Fig. 1 a Laue diagram of such a twin is given. In the X-ray photograph the double spots are clearly visible. The spots corresponding to the normal crystal are more intense than the spots corresponding to the twin. In the figure, weak bridges connecting the spots corresponding to the normal crystal and to the twin are clearly visible. The presence of these bridges indicates the existence of some disturbances in the lattice, the character of which is not yet clear. The characteristic eight-rayed figure around the central spot points to the same thing. In Fig. 2 an analogous Laue diagram of a twin of rock salt is shown, but taken in such a way that the beam of X-rays is parallel to the twinning axis.

In the microphotograph (Fig. 3) the non-parallelism in rock salt of individual twins to one another is clearly visible. It turned out that during displacement

of the X-ray beam relative to the crystal, the reflection from one and the same plane shifts, and at different places we see reflections displaced by different angles. Fig. 4 shows such a photograph. It was obtained in the following way: the sharpest reflection is selected, and the crystal is oriented in such a manner that, besides it, as few reflections as possible would be visible on the screen. Then all the reflections except one are covered, and the crystal and the plate are moved synchronously perpendicular to one another and to the X-ray beam. In this case, if we have an ideal crystal, we should obtain, from the selected reflection, a straight line parallel to the straight line left by the central beam. If, however, we have a crystal of complex structure, then we should have a series of spots. As is seen in the photograph, we obtain a series of spots separated by distances corresponding to the angle of rotation of the reflecting plane. Careful consideration of Fig. 4 shows that the angles of rotation have different magnitudes at different points of the crystal, and that the angles formed by the individual twins with one another have approximately the same magnitude. Thus, one may consider it proven experimentally that in rock salt there are deformation twins having an irrational* crystallographic plane as the twinning plane and not parallel to one another.

In connection with this, it is interesting to note that we observe the same phenomenon also in the growth of rock salt. We have in mind the so-called mosaic structure. Everyone who has had rock-salt crystals in his hands has undoubtedly noticed that the cleavage planes are often covered with a characteristic striation. This striation is formed by planes slightly inclined to one another. It is easy to observe that in this case the crystal consists, as it were, of individual blocks bounded by planes close to the plane of the rhombic dodecahedron but not coinciding with it. At the same time, the angles of inclination of these planes with respect to one another and to the plane of the rhombic dodecahedron are somewhat larger than those usually observed in rock salt in deformation twins, but are of approximately the same order of magnitude. X-ray diagrams from crystals showing split growth give a pattern characteristic of twins**. Thus, there is reason to think that, just as in many metals we encounter deformation twins alongside growth twins, so also in rock salt, alongside deformation twins, we encounter growth twins (split growth).

It is extraordinarily interesting to know what explains the presence of such a parallelism between growth and plastic deformation. To explain this we shall have to examine more deeply the mechanism of both these phenomena. It is well known that, during deformation of rock salt along slip planes (twinning planes), a rather large quantity of heat is released. As shown by G. Haber’s experiments on rock salt, the amount of heat released during deformation is equal to 85 to 95% of all the work of deformation. This heat is mainly released along a narrow region on the twinning plane; therefore a considerable rise in temperature should be expected here. According to experiments and somewhat rough calculations by A. Stepanov⁹, the temperature increase should be so great that in this place melting and even evaporation occur.

These calculations proceed from the fact that the thickness of the layer in which heat is released is of the order of \(10^{-5}\)—\(10^{-6}\) cm. It is difficult to say whether these data are not underestimated, and consequently whether the temperature in the thin layer is not overestimated, but in any case there is no doubt that along certain narrow regions such intense heating occurs that the body either softens or recrystallizes. Thus

* By an irrational crystallographic plane we mean a plane that has no simple crystallographic indices, i.e., a plane whose indices, though integral, are large numbers.

** An investigation of the mosaic structure is being conducted by us at present, and detailed results will be published in the near future.

To the article by D. B. Gogoberidze

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

To the article by D. B. Gogoberidze

Fig. 5a.

Fig. 5b.

ON THE MECHANICAL TWINNING OF CRYSTALS

It is clear that, in the formation of deformation twins, laws must operate that are very similar to those which determine the slip of crystals.

It is clear that everything said above applies to twinning along irrational planes, and that twinning along rational planes must differ very substantially from it. Indeed, in twinning along rational planes, the geometrical conditions determine the direction of twinning much more strictly, and at the same time the disturbances in the lattice are considerably smaller than in twinning along irrational planes.

It is therefore highly important to study in detail some example of such twinning and to establish that there really are cases in which twinning proceeds along rational planes. This latter proposition was all the more in need of verification because the older works did not distinguish between these two kinds of twinning, and consequently it was impossible to be certain whether any such twinning existed at all. To clarify these questions it was quite natural to turn to the classical object, after the works of Mügge and Baumhauer—the calcite (Iceland spar). Such a study was begun by us and has already yielded a number of results.^10

The first question requiring clarification was the question of the parallelism or non-parallelism of the twinning planes in calcite. Even the first microphotographs of twins made by us showed their complete parallelism. In specimens of calcite in which alternating lamellae of the twin and of the normal crystal are present (the so-called polysynthetic twin), the complete parallelism of the twins to one another is clearly visible. This did not yet prove the rationality of the twinning planes, but indicated only their parallelism. In addition, it was of substantial interest to check whether twinning occurs by a series of successive shears, as Schmidt^11 and Mügge think, or by a pure rotation of the lattice.

In Fig. 5 (a and b) photographs of calcite twins are shown, taken under illumination by a sliding beam of light (dark field, and only certain planes reflect light into the microscope). The photographs (a and b) were taken with the stage carrying the specimen rotated by 180°; these photographs are related to one another almost as a positive to a negative. These photographs undoubtedly prove that, in addition to the parallelism of the twins, we also have complete symmetry between the lamellae of the twin and of the parallel crystal.

This contradicts Schmidt’s views and shows that the model of twinning proposed by him is not correct.^11 As will be noted below, this assertion follows still more clearly from the Laue diagrams of calcite (Fig. 8).

However, already in these photographs, cracks and irregularities are noticeable here and there along the twinning planes. We succeeded in establishing that characteristic triangular cracks are often located along the twinning planes. These cracks are superficial in character and apparently do not extend into the interior of the crystal. Moreover, we sometimes succeeded in observing a separation of finely crystalline substance on the surface of the crystal. This finely crystalline substance is located on the surface of the crystal along the twinning planes. This separation is readily visible to the naked eye and can be photographed (Fig. 6). It is easy to establish that what we have here is indeed a finely crystalline powder of calcite. In fact, with very long exposures we succeeded in obtaining from it a Debye–Scherrer diagram, identical to the usual Debye diagram of calcite (without traces of texture). The superficial character of the separation of this finely crystalline powder can easily be established. It can be removed by dissolution in a weak solution of HCl or simply by mechanical chipping. By chipping or surface dissolution one can obtain crystals (polysynthetic twins), which are just as transparent and entirely similar to normal crystals. Often, however, it is possible to obtain equally clean twins without any additional manipulations, but simply by careful twinning of good crystals.

It was of considerable interest to investigate the question of whether lattice destructions occur in these crystals. To study this question we carried out experiments on the comparative solubility of twinned and untwinned crystals. In doing so we selected the purest twinned crystals possible, without surface cracks and always with only one system of twins. The point is that in Mogg’s earlier experiments, in which the comparative solubility of normal and twinned crystals was studied, on the contrary, crystals with two systems of twins were selected (the presence in a crystal of two intersecting systems of twins, by a purely geometrical condition, leads to the formation of channels in the crystal, and consequently to an unaccounted increase in the surface of dissolution). Moreover, in Mogg’s work there is no indication of how carefully crystals without cracks and without surface defects were selected. We, as has been said, selected crystals with all possible care. Carefully selected crystals were examined under a microscope in order to establish the surface area and were weighed. Simultaneously, control crystals crystallized out from the same sample as the twinned crystals. The crystals were then immersed in a weak solution. The solution was chosen of such a concentration that within twenty-four hours the loss in weight would be about 1/2 the weight of the crystals (10–15 mg). After dissolution the crystals were removed from the acid, washed in water, and dried in a thermostat. After this, a second weighing was carried out and the loss of weight per unit surface was determined. The accuracy of such experiments did not exceed 5%. Within the limits of this accuracy it may be asserted that we observed no difference in the solubility of normal and twinned crystals. This conclusion is the mean of various series of experiments, but even the results of individual series show no such difference. This undoubtedly indicates that lattice disturbances, if they exist, are not very great. For a more detailed elucidation of this question we turned to X-ray investigation. We again selected, with the greatest care, the purest twinned crystals. In doing so, we always chose crystals in which there was only one system of twins. This was done in order to facilitate the interpretation of the results obtained and to study the phenomenon under the simplest possible conditions. Everyone is undoubtedly well acquainted with the Laue pattern of a normal calcite crystal (Fig. 7). A change in the Laue pattern makes it possible conveniently to judge the character of the lattice changes. Figure 8 shows the Laue pattern of a polysynthetic twin. It is clearly seen on it that it is a superposition of two Laue patterns of the type of Fig. 7. On the gnomonic projection it is clearly seen that the angle between the two crystals giving these diffraction spots corresponds precisely to the goniometrically measured angle between the normal crystals and the twin. Thus, owing to the sharpness of the spots, one may be certain that the angles of rotation of relatively normal crystals are constant and equal to the angles measured goniometrically.

Further, from consideration of the Laue pattern in Fig. 8 it follows beyond doubt that no substantial disturbances of the lattice in the crystal occur during twinning. In fact, sharp spots without any traces of blurring or tails indicate an undisturbed state of the lattice. In this respect twins in calcite differ substantially from twins in NaCl (Figs. 1 and 2), where narrow streaks joining the spots corresponding to the normal and twinned crystal are clearly visible in the Laue pattern and testify to disturbances in the lattice. This is also extremely clearly seen from consideration of X-ray photographs of calcite taken along the plane of twinning in such a way that the beam of X-rays traveled in the direction of twinning*. Here too we see, in contrast to what occurs in rock salt, sharply delineated spots without any traces of their blurring.

It is known that in earlier works on the study of twinning in zinc

* A certain asymmetry of the pattern is caused by a slight inaccuracy of orientation.

To the article by D. B. Gogoberidze

Fig. 6.

Fig. 7.

Fig. 8.

Schmidt and Wassermann observed asterism phenomena. On the basis of our photographs and those of Brilliantov we believe that the asterism phenomena observed by them were caused not by twinning, but by certain secondary phenomena—perhaps bending of planes, and possibly by other causes as well. For a more thorough check of our conclusions we made control photographs of growth twins of calcite. In growth twins it was possible to be certain that there were no lattice disturbances. The Lauegrams that we took from growth twins showed complete analogy with the Lauegrams taken from deformation twins. It is not necessary, however, to think that in twinning along rational planes no disturbances whatever occur in the crystal. It is enough to construct a scheme of twinning,* in order to convince oneself that even in an ideal model lattice disturbances are inevitable, albeit of the second order (counting the forces between particles at adjacent lattice angles as forces of the first order, and the forces through a node as forces of the second order). These disturbances, in order of magnitude, may be comparable with the disturbances that occur in the surface layer of the crystal edge. If this is indeed correct, then it should be expected that, in twinning along rational planes, secondary cleavage will occur along the twinning planes. It should be noted that, for example, rock salt has no such cleavage, i.e., in twinning along rational planes this phenomenon is absent.

As for calcite, the question of the existence of such cleavage was not entirely clear. In the literature there were both assertions and denials of the presence of such cleavage.

In order to check this question, we carried out experiments on splitting crystals along the twinning planes.¹⁰ These experiments were completely successful. We succeeded in obtaining a perfectly good split. The split plane is a completely perfect plane and differs in no way in appearance from ordinary cleavage planes. It must be added, however, that we were unable to obtain a clean split along the entire twinning plane, but only along a certain part of it; in the remaining parts the split appeared as a stepped surface bounded by ordinary cleavage planes. Apparently, it was this circumstance that misled previous investigators.

Thus it may be asserted that twinning of calcite crystals occurs along rational parallel planes and is not accompanied by lattice disturbance, with the exception of the formation of planes of secondary cleavage.

LITERATURE

1. E. Schmidt u. W. Boas, Kristallplastizität, Berlin, Springer, 1935.
2. Brilliantow a. Obreimow, Phys. Z. Sowetun., 1935; Gogoberidze, Phys. Z. Sowetun., 1935.
3. Mügge, Neues Jahrbuch, 1889.
4. Elam, Distortion of metal crystals, Oxford 1935.
5. Zacharias, Z. Kristallogr., 86, 1932.
6. Schmidt u. Wassermann, Z. Physik, 48, 370, 1928.
7. Gogoberidze, Phys. Z. Sowetun., 8, 208, 1935.
8. Stepanow, Phys. Z. Sowetun., 1933.
9. Garber, Phys. Z. Sowetun, 1935.
10. Gogoberidze und Ananiaschwili, Phys. Z. Sowetun., 7, 543, 1935.
    Gogoberidze and Nakashidze. Transactions of Tbilisi State University, issue 2.
11. E. Schmidt, Festigkeit und Elastizität von Metallkristallen. Metallwirtschaft. 7, 1011, 1928; Kristallplastizität, p. 75, fig. 49.

* I.e., to draw the arrangement of atoms in the lattice of the twinned crystal and trace how this position has changed in comparison with the normal crystal.