Volume 30, pages 469-482, 1945

     In recent papers^1,2,3 some observations on the occurrence of twinning have been made and the question of what is responsible for the formation of twins has been raised. The writer's own views on this question have had a limited circulation. In view of the general interest in the subject, it seems desirable to give them wider publication.

     The cause of twinning is easily approached through energy considerations. A stable crystal represents the lowest free energy state of that collection of atoms; this follows from the very notion of stability. The pattern preferably assumed by atoms assembling themselves as a crystal is thus the configuration of least free energy. Any departure from this pattern represents an increase in the energy of the collection of atoms. Since the pattern of atoms in the twin boundary represents a departure from the rest of the crystal pattern, it is obvious at the outset that the twin boundary region is a locus of higher energy than the rest of the crystal. What permits these atoms to assume this configuration of higher energy? To answer this question is to assign a cause to twin formation.

     RELATION OF TWINS TO OTHER KINDS OF INTERGROWTHS

     The energy in the twin boundary is stored as surface or interfacial energy. This kind of energy also exists in the mutual boundaries of other kinds of intergrowths. The writer has pointed out⁴ that the locations of the interfaces of unmixing intergrowths is conditioned by requirements of minimum interfacial energy, and that low energy is made possible by the existence of a plane of atoms in common between the two unmixing phases. This provides a gradation in structure rather than a complete discontinuity; consequently the additional energy in the junction plane is not far above that of a plane in either crystal. This near-minimum energy requirement naturally occurs in all kinds of intergrowths and it conditions the mutual orientations of crystals in such intergrowths.⁵

     Evidently an atom can participate in a twin boundary structure and contribute a low energy to the summation provided it can merely satisfy its immediate coordination requirements. In the comparatively few cases in which the attention has been called to the relation between a structure and the transition structure inferred for the twin boundary,^6,7,8,9,10 It has been evident that the twin exists by virtue of the fact that the atoms in the structure permit more than one linking of themselves consistent with their immediate coordination requirements, giving rise to two different configurations; one of these configurations represents the structure and its continuation, the other represents a transition to the twin configuration.

     It must be inferred from this that the existence of a twin is permitted by the nature of the structure itself. If the structure is of such a nature that, in detail, it permits a continuation of itself in alternative twin junction configuration without involving violation of the immediate coordination requirements of its atoms, the junction has low energy and the twin is energetically possible.

     This theory of the energy permissivity of twins emphasizes the characteristics of the structure itself, in contrast with the French theory of twinning,^1,3,11 which, ignoring the structure itself, merely requires certain special dimensional qualities in the identity periods of the structure. Furthermore, the French theory of twinning is strictly empirical, while the theory presented here is a rational one. Apparently the two theories are in conflict, but some relations between them will be considered further on in this paper.

     TWIN FORMATION

  INTRODUCTION

     There appear to be at least three distinct devices which are responsible for bringing about the high energy states of twin boundaries. Corresponding with these there may be recognized several genetic types of twins: growth twins, transformation twins, and gliding twins. In the following sections, some aspects of the genesis of these twins are considered.

 GROWTH TWINS

     Geometrical aspects. - To understand how twins develop during the growth of the crystal, one must be acquainted with the mechanism of crystal growth, for twins develop as accidents of the growth process.

     The general rule of crystal growth is that the newly arriving atom (or cluster of atoms) takes up a position such that the part of its normal coordination which it assumes on contact with the surface is a maximum as compared with what it would be on other parts of the surface. Thus, if the atom is the first one to arrive at a plane surface, all positions where it could continue the pattern are the same, and it can therefore assume any one of these identical positions. On the other hand, the next atom to arrive which can also coordinate with the first atom must necessarily assume a position so that it does coordinate with it; in this way, the atom improves its coordination and so loses more energy than by occupying a position on the plane and away from the first atom. In a like manner, each newly arriving atom locates itself in a place touching one or more of the atoms already located on the old surface, and the process of crystal growth consists of spreading of the new layer to cover the old surface. Under near-equilibrium conditions, a second new layer does not start to form until the first new layer is completed. During the process and until a layer is completed, there is always a corner at the base of the new surface to which atoms or clusters of atoms are especially attracted because of the greatest energy loss in the form of greatest coordination offered there.

     The above considerations are primarily concerned with the addition of atoms to the continuation of the normal pattern of the crystal. An atom can also fall into a position which does not continue the pattern, yet which satisfies its first coordination sphere. Should the atom fall into such a position, its energy would be an approximation to minimum energy, according to the discussion under "Structural Requirements for Low Energy." Since it does not have actually the lowest energy, however, it has a comparatively high fugacity, and it normally would lose its place for a more suitable one by a mechanism of thermal agitation and atomic bombardment.

     What factors are responsible for maintaining an atom in a position of subminimum energy? A possible factor might be the simultaneous arrival at neighboring positions of a pair of atoms or more which can mutually coordinate, or, what amounts to substantially the same thing, the arrival in the twin position of a group or cluster of atoms already coordinated. The entire cluster has a smaller fugacity than a single atom, and an atom bound within the cluster by coordination also has a lower fugacity than any single atom arriving alone at a subminimum energy location. If the arrival of the cluster at the subminimum location is immediately followed by the arrival of other atoms or clusters to continue the twin pattern, a twin may persist.

     We now turn to the empirical rules of twinning of the French School. Bravais¹² and Mallard¹³ observed that twins appear to be possible if the lattice of a crystal has a higher dimensional symmetry or pseudo symmetry than the crystal. These additional symmetry elements are the ones which relate the individuals of the twin and constitute the elements of the twin law. Friedel¹¹ generalized this purely empirical rule and held that twins are possible even if a multiple cell (of small multiplicity) has a pseudo symmetry beyond the symmetry of the crystal. The occurrence of twinning, according to Friedel, is more probable the lower the "index" (which is related to the multiplicity) of the pseudo cell and the smaller the obliquity of the row in this cell to an ideally symmetrical row.

FIG 1

     It should be possible to interpret the findings of the empirical French School on grounds of energy and with the aid of the theory of crystal growth by spreading of one layer over another. A stage in the normal growth of the crystal is represented in Fig. 1A. Here the representation of the crystal is by means of cells in accordance with the French School, not by means of structure. The crystal plane, PP, which is growing, is being covered by the new layer, LL. The cliff at the right shows how the simple cell is related to Friedel's dimensionally more symmetrical multiple cell. Now if, at any time, such a cliff is momentarily established, say, due to rapid growth, it becomes geometrically possible for several atoms (or a cluster) to add themselves to both surfaces of the reentrant at the foot of the cliff. The two surfaces of the reentrant provide a greater-than-normal coordination for a large cluster, especially as compared with the plane PP alone. Thus, should a cluster arrive in twin position, it might be able to retain this subminimum energy attachment merely because it has a multiple grip on the crystal which acts as a considerable energy barrier to removal. Once established, the pattern of the twin continues.

     If this view is correct, then the interpretation of Friedel's rules must be as follows: the geometry of the lattice aids twin formation because a cluster, represented by points abcde, Figs. 1A and 1B, arriving at the crystal, can superpose itself in twinned orientation on the untwinned. crystal, a'b'c'd'e', since a small cluster is sufficiently elastic. Thus e₁ coincides with e' and c with c'. Friedel's angle of obliquity is then half the three-dimensional angle e'ae_l. Such strain is directly proportional to the tangent of the angle. (A continued coincidence of the two twinned parts of the crystal is unlikely since it involves the work of shearing the entire crystal through the angle of obliquity.) The significance of Friedel's low multiplicity of the multiple cell would be that as many points as possible, such as e and c, should be capable of joining the crystal and its twinned structure.

     At this point it is necessary to amend the French theory.* Its rules provide, perhaps, necessary conditions for certain growth twins to occur, but they are definitely not sufficient conditions. The French theory, as it stands, provides only an incomplete picture of permissivity of twinning. To amend it we add the requirement of low energy junctions from twinned to untwinned structure, at least at the point of inception of the twin. This implies, in general, that the structure must be continuous, at least in the nearer coordination spheres of the atoms in the junction structures. It should be emphasized that this requirement is not only rational, but it has been found to hold in such twins as have been examined with regard to structure.

     Environment factors favoring genesis of growth twins. - In addition to the geometrical factors permitting growth twinning, just discussed, certain external and internal environmental factors encourage the formation of structures of subminimum energy. Among these are supersaturation and minuteness of crystal.

     It is evident that if the rate of addition of atoms to the surface of a crystal is sufficiently high, the probability will be increased that an atom arriving at a twin position will not be relocated before it is joined by a second atom. Once a second atom joins the first, the pair has a very low energy because each member has its coordination increased by the other. Thus, once started, the twinned layer may persist and grow.

     The probability of twin formation increases with the rate of addition of atoms or clusters to the crystal, and thus, within limits at least, increases with the degree of supersaturation, suggesting the genetic name supersaturation twin. While supersaturation may cause twinning at any time during the history of growth of the crystal, the most obvious time when the degree of supersaturation is very high is just before nuclei start to form. Once nuclei appear, the immediate regions surrounding them become cleared of a high degree of supersaturation, and a normal growth rate is established consistent with the rate of diffusion. For this reason, the condition causing supersaturation twins is most likely to arise just once - as the crystal nucleus forms - and then the conditions leading to this type of twinning do not usually occur again. Nuclei supersaturation twins are therefore characteristically simple pairs (such as Carlsbad, Baveno, and Manebach twins in feldspars). This condition is enhanced by the factor discussed in the next paragraph. Supersaturation twins can, however, recur. Suppose, for example, that feldspar crystals, forming in a magma, should sink to a lower region where the magma is supersaturated. The conditions there are ripe for the formation of supersaturation twins.

     Special conditions obtain during the formation and early growth of a crystal nucleus which are particularly favorable to twin formation. When a crystal consists of merely two or three layers, the energy of an atom on its surface is not the same as that of the atom on the surface of an indefinitely large crystal. The energy summation consists of only a few terms rather than an infinite number and may be too small to closely approximate the converging value of the series. Conditions are thus different; they may be less unfavorable, if not positively favorable, to the formation of twins. As the crystal grows, the energy sum changes, or may even fluctuate during growth, until the crystal is large enough so that the energy sum has substantially converged. If growth now continues under equilibrium conditions, a twinned crystal has a higher energy and is subject to dissolution and reprecipitation on an untwinned crystal. If there is supersaturation, however, the chances are increased that the twinned crystals will grow to sufficiently large size that they will not readily redissolve.

 TRANSFORMATION TWINS

     It can be theoretically demonstrated that a pair of crystals connected by a high-low transformation possess related symmetries. The low temperature form has a symmetry which is a subgroup of the symmetry of the high temperature form. Now, in the transformation from the high form to the low form, the general structure remains the same, but some of its symmetry operations are suppressed. The low form merely forms at the expense of the high form by loss of symmetry. This can, in general, take place in such a way that the low form has more than one orientation with respect to the high form. This is indicated by the following analysis: the low form contains fewer coincidence operations than the high form. It can be transformed into itself in n ways, where n is the number of operations in its symmetry group. Since the group of the high form contains the group of the low form as a subgroup, the high form can be transformed into itself in mn ways. This means that the low form can be oriented in n indistinguishable ways in the structure while the high form can be oriented in mn indistinguishable ways in the structure. For any given orientation of the low form, there are m possible orientations of the high form. That is, there are m mutually different orientations of high and low form.

     High-low transformations take place as soon as the temperature of a region has reached the transformation level, and is propagated as a wave as rapidly as heat can be transmitted. Unless the crystal is extremely small, transformation nuclei may appear in various parts of the crystal spontaneously, particularly at the edges, and the transformation spreads from these centers. With falling temperature, the chance that any center shall assume the same orientation as a neighbor is 1/m. In general, then, a transformation from the high to the low form consists of a spontaneous formation of nuclei in m different orientations and these nuclei grow until they make contact with one another. From the mechanism of formation of these crystals, their contacts are often, though not necessarily, irregular. All individuals of the aggregate are either in parallel or different orientation. Those in different orientation are in twinned orientation with respect to one another. This follows from the fact that they could be brought into coincidence by one of the m operations of the high form which vanished in the formation of the low form.

GLIDING TWINS

     It is well known that some crystals can yield to stress plastically by a process known as gliding. Two types of gliding are known, translation-gliding and twin-gliding.¹⁴

     Parting and Replacement. - It has already been pointed out that a twin boundary is the locus of high energy. The work required to disrupt the crystal is partly represented by this increase in energy of the crystal at the boundary. If the boundary is plane, it is the weakest individual plane of all parallel planes, and if tension is applied to the crystal, the crystal breaks at the twin boundary (excluding cleavage). This constitutes parting. Thus twin boundaries are parting planes. (Other causes of "parting" are also recognized.¹⁵) The high energy aspect of the twinning plane also localizes chemical action, so that chemical alteration and replacement tend to occur preferentially at twin boundaries. Such alteration, of course further weakens the resistance of the twin boundary to stresses and enhances the ease of parting.

     Relation of Twinning to Polymorphism. - Anyone who has been concerned with the mechanism of twin-gliding⁶ has noticed that the structure in the twin boundary may be a polymorphic modification of the original structure. Aminoff and Broome⁷ have discussed the case of twinned sphalerite, which has the wurtzite arrangement in the twin boundary. Other examples could easily be cited. Indeed it is true in general that the structure in a twin boundary is at least a possible polymorphous alternative structure. This is because the atoms of most (though not all) polymorphous pairs have identical first ocordination spheres (some polymorphous pairs have different first coordination spheres) but different higher coordination spheres. This is exactly the situation in the twin boundary. In this sense, twin boundaries are cases of two-dimensional polymorphism. Thus the high energy of a twin boundary is related structurally to the high energy of an unstable polymorphous form.

     This suggests the following hypothesis: the possible growth twin laws of a crystal truly unstable under the conditions of its formation can include only such laws as have boundary structures of less stable polymorphic forms, and cannot include twin laws which have boundary structures of the stable modification. (Otherwise, the structure of the stable modification once started, would perpetuate the stable modification and the twin would cease to exist since the intergrowth would be one between stable and unstable modifications.) Should such a twin law be observed in an unstable crystal, the crystal is not a growth twin, or else the modification was not actually unstable under the conditions of formations. The possibilities of twinning in unstable crystals are thus restricted, and, in general, an unstable modification should display fewer twin laws than a stable modification.

     Relation of Twin Formation to Polymorphic Transformation. - Not only is twinning related to polymorphism, but a change from untwinned to twinned condition (as in glide twinning) is related to polymorphic transformation. From a structural point of view, two important kinds of polymorphic transformation can be recognized:

     (1) Transformations involving only a slight displacement of parts of the structure, of the order of heat motion. This kind of transformation may be termed a displacive transformation. It is of the high-low type and is called forth instantly by transgression of the critical temperature (example: low-quartz. to high-quartz).

     (2) Transformations requiring a bit-by-bit tearing down of one structure and a rebuilding of the bits to form a differently linked structure. This may be called a reconstructive transformation. It is of the sluggish variety because it requires a distillation of one structure into the other (example: quartz to cristobalite). Such transformations cannot be effected by simple displacement.

     There are also displacive and reconstructive varieties of twinning. In the displacive variety, the crystal and its twin boundary patterns may be made to transform into one another by simple displacement. Accordingly such twins can be formed by a high-low transformation or by twin-gliding. The possibility of a displacement is the structural equivalent of a low energy trough connecting energy depressions mentioned above.

     Other twinning is of the reconstructive variety, that is, the crystal can be transformed into its twin boundary only by a complete disintegration into bits and then reconstruction of its structure. Obviously, such twinning cannot result from high-low transformation or from the simple shear displacement of twin-gliding. Such twins can only be formed by the original precipitation of the twinned structure, and consequently reconstructive twins are limited to growth twins. This corresponds closely to the requirements for a reconstructive transformation.

     It might be pointed out that the albite twinning of plagioclase is of the displacive type, since the triclinic character of this crystal is due to a collapse of the monoclinic orthoclase structure such that oxygens coordinate with the small sodium atom. No relinking of the structure is required in passing from untwinned to twinned structure. It is thus possible that albite twins can arise by either transformation twinning or glide twinning, as well as by growth twinning. Guided by this reasoning, the writer and his students have experimentally produced polysynthetic twinning in albite by glide twinning.

     Relation of Transformation Twins to Unmixing. - In another place the writer has explained the formation, of segregate phase crystals in unmixing.¹⁶ These are called forth in response to a requirement for ordering in the structure as the temperature falls. Transformation twins are also called forth by falling temperature, and often, if not always, this is equivalent to a requirement for some kind of ordering. Transformation twining is merely a degenerate case of segregate phase formation in which both original disordered and final ordered phases have identical chemical compositions.

     REMARKS ABOUT COMMON STATEMENTS OF THE CAUSE OF TWINNING

     One commonly finds statements equivalent to the following: "Twinning may be regarded as an unsuccessful attempt to establish a symmetry higher than that to which the simple crystal belongs." This kind of statement misses the meaning of symmetry and misses the significance of twinning.

     Symmetry, in essence, is a relation equivalent to the repetition of some geometrical feature. Any two similar things form a symmetrical pair, in a broad sense, since one is the repetition of the other. Crystallographic symmetry is merely a restriction of this notion to cases consistent with lattice translations. There is no virtue in symmetry such that atoms, coming together to form a crystal, should strive to attain it. There is no potential favoring it. Indeed, in the case of twins, the potential favors its absence.

     The writer is indebted to Dr. George Tunnell and Dr. J. D. H. Donnay for kindly reading this paper in manuscript and for offering constructive criticism of certain points raised in it. This should not be construed as necessarily committing them to the writer's views.

    NOTES