International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 9.2, pp. 760773
Section 9.2.2. Layer stacking in general polytypic structures
S. Ďurovič^{a}

The common property of the structures described in Section 9.2.1 was the stacking ambiguity of adjacent layerlike structural units. This has been explained by the geometrical properties of close packing of equal spheres, and the different modifications thus obtained have been called polytypes.
This phenomenon was first recognized by Baumhauer (1912, Baumhauer, 1915) as a result of his investigations of many SiC single crystals by optical goniometry. Among these, he discovered three types and his observations were formulated in five statements:

Baumhauer recognized the special role of these types within modifications of the same substance and called this phenomenon polytypism – a special case of polymorphism. The later determination of the crystal structures of Baumhauer's three types indicated that his results can be interpreted by a family of structures consisting of identical layers with hexagonal symmetry and differing only in their stacking mode.
The stipulation that the individual polytypes grow from the same system and under (nearly) the same conditions influenced for years the investigation of polytypes because it logically led to the question of their growth mechanism.
In the following years, many new polytypic substances have been found. Their crystal structures revealed that polytypism is restricted neither to close packings nor to heterodesmic `layered structures' (e.g. CdI_{2} or GaSe; cf. homodesmic SiC or ZnS; see 9.2.1.2.2 to 9.2.1.2.4), and that the reasons for a stacking ambiguity lie in the crystal chemistry – in all cases the geometric nearestneighbour relations between adjacent layers are preserved. The preservation of the bulk chemical composition was not questioned.
Some discomfort has arisen from refinements of the structures of various phyllosilicates. Here especially the micas exhibit a large variety of isomorphous replacements and it turns out that a certain chemical composition stabilizes certain polytypes, excludes others, and that the layers constituting polytypic structures need not be of the same kind. But subsequently the opinion prevailed that the sequence of individual kinds of layers in polytypes of the same family should remain the same and that the relative positions of adjacent layers cannot be completely random (e.g. Zvyagin, 1988). The postulates declared mixedlayer and turbostratic structures as nonpolytypic. All this led to certain controversies about the notion of polytypism. While Thompson (1981) regards polytypes as `arising through different ways of stacking structurally compatible tabular units [provided that this] should not alter the chemistry of the crystal as a whole', Angel (1986) demands that `polytypism arises from different modes of stacking of one or more structurally compatible modules', dropping thus any chemical constraints and allowing also for rod and blocklike modules.
The present official definition (Guinier et al., 1984) reads:
``An element or compound is polytypic if it occurs in several different structural modifications, each of which may be regarded as built up by stacking layers of (nearly) identical structure and composition, and if the modifications differ only in their stacking sequence. Polytypism is a special case of polymorphism: the twodimensional translations within the layers are (essentially) preserved whereas the lattice spacings normal to the layers vary between polytypes and are indicative of the stacking period. No such restrictions apply to polymorphism.
Comment: The above definition is designed to be sufficiently general to make polytypism a useful concept. There is increasing evidence that some polytypic structures are characterized either by small deviations from stoichiometry or by small amounts of impurities. (In the case of certain minerals like clays, micas and ferrites, deviations in composition up to 0.25 atoms per formula unit are permitted within the same polytypic series: two layer structures that differ by more than this amount should not be called polytypic.) Likewise, layers in different polytypic structures may exhibit slight structural differences and may not be isomorphic in the strict crystallographic sense.
The AdHoc Committee is aware that the definition of polytypism above is probably too wide since it includes, for example, the turbostratic form of graphite as well as mixedlayer phyllosilicates. However, the sequence and stacking of layers in a polytype are always subject to welldefined limitations. On the other hand, a more general definition of polytypism that includes `rod' and `block' polytypes may become necessary in the future.''
This definition was elaborated as a compromise between members of the IUCr AdHoc Committee on the Nomenclature of Disordered, Modulated and Polytype Structures. It is a slightly modified definition proposed by the IMA/IUCr Joint Committee on Nomenclature (Bailey et al., 1977), which was the target of Angel's (Angel, 1986) objections.
The official definition has indeed its shortcomings, but not so much in its restrictiveness concerning the chemical composition and structural rigidity of layers, because this can be overcome by a proper degree of abstraction (see below). More critical is the fact that it is not `geometric' enough. It specifies neither the `layers' (except for their twodimensional periodicity), nor the limitations concerning their sequence and stacking mode, and it does not state the conditions under which a polytype belongs to a family.
Very impressive evidence that even polytypes that are in keeping with the first Baumhauer's statement may not have exactly the same composition and the structure of their constituting layers cannot be identical has been provided by studies on SiC carried out at the Leningrad Electrotechnical Institute (Sorokin, Tairov,Tsvetkov & Chernov, 1982; Tsvetkov, 1982). They indicate also that each periodic polytype is sensu stricto an individual polymorph. Therefore, it appears that the question whether some real polytypes belong to the same family depends mainly on the idealization and/or abstraction level, relevant to a concrete purpose.
This very idealization and/or abstraction process caused the term polytype to become also an abstract notion meaning a structural type with relevant geometrical properties,^{1}belonging to an abstract family whose members consist of layers with identical structure and keep identical bulk composition. Such an abstract notion lies at the root of all systemization and classification schemes of polytypes.
A still higher degree of abstraction has been achieved by DornbergerSchiff (1964), DornbergerSchiff (1966), DornbergerSchiff (1979) who abstracted from chemical composition completely and investigated the manifestation of crystallochemical reasons for polytypism in the symmetry of layers and symmetry relations between layers. Her theory of OD (order–disorder) structures is thus a theory of symmetry of polytypes, playing here a role similar to that of group theory in traditional crystallography. In the next section, a brief account of basic terms, definitions, and logical constructions of OD theory will be given, together with its contribution to a geometrical definition of polytypism.
Polytypism of structures based on close packing of equal spheres (note this idealization) is explained by the fact that the spheres of any layer can be placed either in all the voids of the preceding layer, or in all the voids – not in both because of steric hindrance (Section 9.2.1, Fig. 9.2.1.1).
A closer look reveals that the two voids are geometrically (but not translationally) equivalent. This implies that the two possible pairs of adjacent layers, say AB and AC, are geometrically equivalent too – this equivalence is brought about e.g. by a reflection in any plane perpendicular to the layers and passing through the centres of mutually contacting spheres A: such a reflection transforms the layer A into itself, and B into C, and vice versa. Another important point is that the symmetry proper of any layer is described by the layer group P(6/m)mm,^{2} and that the relative position of any two adjacent layers is such that only some of the 24 symmetry operations of that layer group remain valid for the pair. It is easy to see that 12 out of the total of 24 transformations do not change the z coordinate of any starting point, and that these operations constitute a subgroup of the index [2]. These are the socalled τ operations. The remaining 12 operations change any z into −z, thus turning the layer upside down; they constitute a coset. The latter are called ρ operations. Out of the 12 τ operations, only 6 are valid for the layer pair. One says that only these 6 operations have a continuation in the adjacent layer. Let us denote the general multiplicity of the group of τ operations of a single layer by N, and that of the subgroup of these operations with a continuation in the adjacent layer by F: then the number Z of positions of the adjacent layer leading to geometrically equivalent layer pairs is given by Z = N/F (DornbergerSchiff, 1964, pp. 32 ff.); in our case, Z = 12/6 = 2 (Fig. 9.2.2.1 ). This is the socalled NFZ relation, valid with only minor alterations for all categories of OD structures (9.2.2.2.7). It follows that all conceivable structures based on close packing of equal spheres are built on the same symmetry principle: they consist of equivalent layers (i.e. layers of the same kind) and of equivalent layer pairs, and, in keeping with these stipulations, any layer can be stacked onto its predecessor in two ways. Keeping in mind that the layer pairs that are geometrically equivalent are also energetically equivalent, and neglecting in the first approximation the interactions between a given layer and the nextbutone layer, we infer that all structures built according to these principles are also energetically equivalent and thus equally likely to appear.
It is important to realize that the above symmetry considerations hold not only for close packing of spheres but also for any conceivable structure consisting of twodimensionally periodic layers with symmetry P(6/m)mm and containing pairs of adjacent layers with symmetry P(3)m1. Moreover, the OD theory sets a quantitative stipulation for the relation between any two adjacent layers: they have to remain geometrically equivalent in any polytype belonging to a family. This is far more exact than the description: `the stacking of layers is such that it preserves the nearestneighbour relationships'.
All polytypes of a substance built on the same structural principle are said to belong to the same family. All polytypic structures, even of different substances, built according to the same symmetry principle also belong to a family, but different from the previous one since it includes structures of various polytype families, e.g. SiC, ZnS, AgI, which differ in their composition, lattice dimensions, etc. Such a family has been called an OD groupoid family; its members differ only in the relative distribution of coincidence operations^{3} describing the respective symmetries, irrespective of the crystallochemical content. These coincidence operations can be total or partial (local) and their set constitutes a groupoid (DornbergerSchiff, 1964, pp. 16 ff.; Fichtner, 1965, 1977). Any polytype (abstract) belonging to such a family has its own stacking of layers, and its symmetry can be described by the appropriate individual groupoid. Strictly speaking, these groupoids are the members of an OD groupoid family. Let us recall that any space group consists of total coincidence operations only, which therefore become the symmetry operations for the entire structure.
Any family of polytypes theoretically contains an infinite number of periodic (Ross, Takeda & Wones, 1966; Mogami, Nomura, Miyamoto, Takeda & Sadanaga, 1978; McLarnan, 1981a, b, c) and nonperiodic structures. The periodic polytypes, in turn, can again be subdivided into two groups, the `privileged' polytypes and the remaining ones, and it depends on the approach as to how this is done. Experimentalists single out those polytypes that occur most frequently, and call them basic. Theorists try to predict basic polytypes, e.g. by means of geometrical and/or crystallochemical considerations. Such polytypes have been called simple, standard, or regular. Sometimes the agreement is very good, sometimes not. The OD theory pays special attention to those polytypes in which all layer triples, quadruples, etc., are geometrically equivalent or, at least, which contain the smallest possible number of kinds of these units. They have been called polytypes with maximum degree of order, or MDO polytypes. The general philosophy behind the MDO polytypes is simple: all interatomic bonding forces decrease rapidly with increasing distance. Therefore, the forces between atoms of adjacent layers are decisive for the buildup of a polytype. Since the pairs of adjacent layers remain geometrically equivalent in all polytypes of a given family, these polytypes are in the first approximation also energetically equivalent. However, if the longerrange interactions are also considered, then it becomes evident that layer triples such as ABA and ABC in closepacked structures are, in general, energetically nonequivalent because they are also geometrically nonequivalent. Even though these forces are much weaker than those between adjacent layers, they may not be negligible and, therefore, under given crystallization conditions either one or the other kind of triples becomes energetically more favourable. It will occur again and again in the polytype thus formed, and not intermixed with the other kind. Such structures are – as a rule – sensitive to conditions of crystallization, and small fluctuations of these may reverse the energetical preferences, creating stacking faults and twinnings. This is why many polytypic substances exhibit nonperiodicity.
As regards the close packing of spheres, the well known cubic and hexagonal polytypes ABCABC and ABAB, respectively, are MDO polytypes; the first contains only the triples ABC, the second only the triples ABA. Evidently, the MDO philosophy holds for a layerbylayer rather than for a spiral growth mechanism. Since the symmetry principle of polytypic structures may differ considerably from that of close packing of equal spheres, the OD theory contains exact algorithms for the derivation of MDO polytypes in any category (DornbergerSchiff, 1982; DornbergerSchiff & Grell, 1982a).
As already pointed out, all relevant geometrical properties of a polytype family can be deduced from its symmetry principle. Let us thus consider a hypothetical simple family in which we shall disregard any concrete atomic arrangements and use geometrical figures with the appropriate symmetry instead.
Three periodic polytypes are shown in Fig. 9.2.2.2 (lefthand side). Any member of this family consists of equivalent layers perpendicular to the plane of the drawing, with symmetry P(1)m1. The symmetry of layers is indicated by isosceles triangles with a mirror plane [.m.]. All pairs of adjacent layers are also equivalent, no matter whether a layer is shifted by +b/4 or −b/4 relative to its predecessor, since the reflection across [.m.] transforms any given layer into itself and the adjacent layer from one possible position into the other. These two positions follow also from the NFZ relation: N = 2, F = 1 [the layer group of the pair of adjacent layers is P(1)11] and thus Z = 2.
The layers are all equivalent and accordingly there must also be two coincidence operations transforming any layer into the adjacent one. The first operation is evidently the translation, the second is the glide reflection. If any of these becomes total for the remaining part of the structure, we obtain a polytype with all layer triples equivalent, i.e. a MDO polytype. The polytype (a) (Fig. 9.2.2.2) is one of them: the translation t = a_{0}+ b/4 is the total operation (a_{0} is the distance between adjacent layers). It has basis vectors a_{1} = a_{0} + b/4, b_{1}= b, c_{1}= c, space group P111, Ramsdell symbol 1A,^{4} Hägg symbol +. This polytype also has its enantiomorphous counterpart with Hägg symbol −. In the other polytype (b) (Fig. 9.2.2.2), the glide reflection is the total operation. The basis vectors of the polytype are a_{2} = 2a_{0}, b_{2} = b, c_{2}= c, space group P1a1, Ramsdell symbol 2M, Hägg symbol + −. The equivalence of all layer triples in either of these polytypes is evident. The third polytype (c) (Fig. 9.2.2.2) is not a MDO polytype because it contains two kinds of layer triples, whereas it is possible to construct a polytype of this family containing only a selection of these. The polytype is again monoclinic with basis vectors a_{3} = 4a_{0}, b_{3} = b, c_{3}= c, space group P1a1, Ramsdell symbol 4M, and Hägg symbol +−−+.
Evidently, the partial mirror plane is crucial for the polytypism of this family. And yet the space group of none of its periodic members can contain it – simply because it can never become total. The spacegroup symbols thus leave some of the most important properties of periodic polytypes unnoticed. Moreover, the atomic coordinates of different polytypes expressed in terms of the respective lattice geometries cannot be immediately compared. And, finally, for nonperiodic members of a family, a spacegroup symbol cannot be written at all. This is why the OD theory gives a special symbol indicating the symmetry proper of individual layers (λ symmetry) as well as the coincidence operations transforming a layer into the adjacent one (σ symmetry). The symbol of the OD groupoid family of our hypothetical example thus consists of two lines (DornbergerSchiff, 1964, pp. 41 ff.; Fichtner, 1979a, b): where the unusual subscript 2 indicates that the glide reflection transforms the given layer into the subsequent one.
It is possible to write such a symbol for any OD groupoid family for equivalent layers, and thus also for the close packing of spheres. However, keeping in mind that the number of asymmetric units here is 24 (λ symmetry), one has to indicate also 24 σ operations, which is instructive but unwieldy. This is why Fichtner (1980) proposed simplified oneline symbols, containing full λ symmetry and only the rotational part of any one of the σ operations plus its translational components. Accordingly, the symbol of our hypothetical family reads: P(1)m11, y = 0.25; for the family of close packings of equal spheres: P(6/m)mm1, x = 2/3, y = 1/3 (the layers are in both cases translationally equivalent and the rotational part of a translation is the identity).
An OD groupoid family symbol should not be confused with a polytype symbol, which gives information about the structure of an individual polytype (DornbergerSchiff, Ďurovič & Zvyagin, 1982; Guinier et al., 1984).
Let us now consider schematic diffraction patterns of the three structures on the righthand side of Fig. 9.2.2.2. It can be seen that, while being in general different, they contain a common subset of diffractions with – these, normalized to a constant number of layers, have the same distribution of intensities and monoclinic symmetry. This follows from the fact that they correspond to the socalled superposition structure with basis vectors A = 2a_{0}, B = b/2, C = c, and space group C1m1. It is a fictitious structure that can be obtained from any of the structures in Fig. 9.2.2.2 as a normalized sum of the structure in its given position and in a position shifted by b/2, thus Evidently, this holds for all members of the family, including the nonperiodic ones. In general, the superposition structure is obtained by simultaneous realization of all Z possible positions of all OD layers in any member of the family (DornbergerSchiff, 1964, p. 54). As a consequence, its symmetry can be obtained by completing any of the family groupoids to a group (Fichtner, 1977). This structure is by definition periodic and common to all members of the family. Thus, the corresponding diffractions are also always sharp, common, and characteristic for the family. They are called family diffractions.
Diffractions with are characteristic for individual members of the family. They are sharp for periodic polytypes but appear as diffuse streaks for nonperiodic ones. Owing to the C centring of the superposition structure, only diffractions with = 2n are present. It follows that diffractions are present only for = 2n , which, in an indexing referring to the actual b vector reads: 0kl present only for k = 4n. This is an example of nonspacegroup absences exhibited by many polytypic structures. They can be used for the determination of the OD groupoid family (DornbergerSchiff & Fichtner, 1972).
There is no routine method for the determination of the structural principle of an OD structure. It is easiest when one has at one's disposal many different (at least two) periodic polytypes of the same family with structures solved by current methods. It is then possible to compare these structures, determine equivalent regions in them (Grell, 1984), and analyse partial symmetries. This results in an OD interpretation of the substance and a description of its polytypism.
Sometimes it is possible to arrive at an OD interpretation from one periodic structure, but this necessitates experience in the recognition of the partial symmetry and prediction of potential polytypism (Merlino, Orlandi, Perchiazzi, Basso & Palenzona, 1989).
The determination of the structural principle is complex if only disordered polytypes occur. Then – as a rule – the superposition structure is solved first by current methods. The actual structure of layers and relations between them can then be determined from the intensity distribution along diffuse streaks (for more details and references see Jagodzinski, 1964; Sedlacek, Kuban & Backhaus, 1987a, b; Müller & Conradi, 1986). Highresolution electron microscopy can also be successfully applied – see Subsection 9.2.2.4.
A polytype family contains periodic as well as nonperiodic members. The latter are as important as the former, since the very fact that they can be nonperiodic carries important crystallochemical information. Nonperiodic polytypes do not comply with the classical definition of crystals, but we believe that this definition should be generalized to include rather than exclude nonperiodic polytypes from the world of crystals (DornbergerSchiff & Grell, 1982b). The OD theory places them, together with the periodic ones, in the hierarchy of the socalled VC structures. The reason for this is that all periodic structures, even the nonpolytypic ones, can be thought of as consisting of disjunct, twodimensionally periodic slabs, the VC layers, which are stacked together according to three rules called the vicinity condition (VC) (DornbergerSchiff, 1964, pp. 29 ff., DornbergerSchiff, 1979; DornbergerSchiff & Fichtner, 1972):
If the stacking of VC layers is unambiguous, traditional threedimensionally periodic structures result (fully ordered structures). OD structures are VC structures in which the stacking of VC layers is ambiguous at every layer boundary (Z > 1). The corresponding VC layers then become OD layers. OD layers are, in general, not identical with crystallochemical layers; they may contain halfatoms at their boundaries. In this context, they are analogous with unit cells in traditional crystallography, which may also contain parts of atoms at their boundaries. However, the choice of OD layers is not absolute: it depends on the polytypism, either actually observed or reasonably anticipated, on the degree of symmetry idealization, and other circumstances (Grell, 1984).
Any OD layer is twodimensionally periodic. Thus, a unit mesh can be chosen according to the conventional rules for the corresponding layer group; the corresponding vectors or their linear combinations (Zvyagin & Fichtner, 1986) yield the basis vectors parallel to the layer plane and thus also their lengths as units for fractional atomic coordinates. But, in general, there is no periodicity in the direction perpendicular to the layer plane and it is thus necessary to define the corresponding unit length in some other way. This depends on the symmetry principle of the family in question – or, more narrowly, on the category to which this family belongs.
OD structures can be built of equivalent layers or contain layers of several kinds. The rule (γ) of the VC implies that a projection of any OD structure – periodic or not – on the stacking direction is periodic. This period, called repeat unit, is the required unit length.
If the OD layers are equivalent then they are either all polar or all nonpolar in the stacking direction. Any two adjacent polar layers can be related either by τ operations only, or by ρ operations only. For nonpolar layers, the σ operations are both τ and ρ. Accordingly, there are three categories of OD structures of equivalent layers. They are shown schematically in Fig. 9.2.2.3 ; the character of the corresponding λ and σ operations is as follows (DornbergerSchiff, 1964, pp. 24 ff.):
Category II is the simplest: the OD layers are polar and all with the same sense of polarity (they are τequivalent); our hypothetical example given in 9.2.2.2.4 belongs to this category. The layers can thus exhibit only one of the 17 polar layer groups. The projection of any vector between two τequivalent points in two adjacent layers on the stacking direction (perpendicular to the layer planes) is the repeat unit and it is denoted by c_{0}, a_{0}, or b_{0} depending on whether the basis vectors in the layer plane are ab, bc, or ca, respectively. The choice of origin in the stacking direction is arbitrary but preferably so that the z coordinates of atoms within a layer are positive. Examples are SiC, ZnS, and AgI.
OD layers in category I are nonpolar and they can thus exhibit any of the 63 nonpolar layer groups. Inspection of Fig. 9.2.2.3(a) reveals that the symmetry elements representing the λ–ρ operations (i.e. the operations turning a layer upside down) can lie only in one plane called the layer plane. Similarly, the symmetry elements representing the σ–ρ operations (i.e. the operations converting a layer into the adjacent one) also lie in one plane, located exactly halfway between two nearest layer planes. These two kinds of planes are called ρ planes. The distance between two nearest layer planes is the repeat unit . Examples are close packing of equal spheres, GaSe, αwollastonite (Yamanaka & Mori, 1981), βwollastonite (Ito, Sadanaga, Takéuchi & Tokonami, 1969), K_{3}[M(CN)_{6}] (Jagner, 1985), and many others.
The OD structures belonging to the above two categories contain pairs of adjacent layers, all equivalent. This does not apply for structures of category III, which consist of polar layers that are converted into their neighbours by ρ operations. It is evident (Fig. 9.2.2.3c) that two kinds of pairs of adjacent layers are needed to build any such structure. It follows that only evennumbered layers can be mutually τequivalent and the same holds for oddnumbered layers. There are only σ–ρ planes in these structures, and again they are of two kinds; the origin can be placed in either of them. is the distance between two nearest ρ planes of the same kind, and slabs of this thickness contain two OD layers. There are three examples for this category known to date: foshagite (Gard & Taylor, 1960), γHg_{3}S_{2}Cl_{2} (Ďurovič, 1968), and 2,2aziridinedicarboxamide (Fichtner & Grell, 1984).
If an OD structure consists of N > 1 kinds of OD layers, then it can be shown (DornbergerSchiff, 1964, pp. 64 ff.) that it can fall into one of four categories, according to the polarity or nonpolarity of its constituent layers and their sequence. These are shown schematically in Fig. 9.2.2.4 ; the character of the corresponding λ and σ operations is
Here also category II is the simplest. The structures consist of N kinds of cyclically recurring polar layers whose sense of polarity remains unchanged (Fig. 9.2.2.4b). The choice of origin in the stacking direction is arbitrary; c_{0} is the projection on this direction of the shortest vector between two τequivalent points – a slab of this thickness contains all N OD layers of different kinds. Examples are the structures of the serpentine–kaolin group.
Structures of category III also consist of polar layers but, in contrast to category II, the Ntuples containing all N different OD layers each alternate regularly the sense of their polarity in the stacking direction. Accordingly (Fig. 9.2.2.4c), there are two kinds of σ–ρ planes and two kinds of pairs of equivalent adjacent layers in these structures. The origin can be placed in either of the two ρ planes. c_{0} is the distance between the nearest two equivalent ρ planes; a slab with this thickness contains 2 × N nonequivalent OD layers. No representative of this category is known to date.
The structures of category I contain one, and only one, kind of nonpolar layer, the remaining N − 1 kinds are polar and alternate in their sense of polarity along the stacking direction (Fig. 9.2.2.4a). Again, there are two kinds of ρ planes here, but one is a λ–ρ plane (the layer plane of the nonpolar OD layer), the other is a σ–ρ plane. These structures thus contain only one kind of pair of equivalent adjacent layers. The origin is placed in the λ–ρ plane. c_{0} is the distance between the nearest two equivalent ρ planes and a slab with this thickness contains 2 × (N − 1 ) nonequivalent polar OD layers plus one entire nonpolar layer. Examples are the MX_{2} compounds (CdI_{2}, MoS_{2}, etc.) and the talc–pyrophyllite group.
The structures of category IV contain two, and only two, kinds of nonpolar layers. The remaining N − 2 kinds are polar and alternate in their sense of polarity along the stacking direction (Fig. 9.2.2.4d). Both kinds of ρ planes are λ–ρ planes, identical with the layer planes of the nonpolar OD layers; the origin can be placed in any one of them. c_{0} is chosen as in categories I and III. A slab with this thickness contains 2 × (N − 2) nonequivalent polar layers plus the two nonpolar layers. Examples are micas, chlorites, vermiculites, etc.
OD structures containing N > 1 kinds of layers need special symbols for their OD groupoid families (Grell & DornbergerSchiff, 1982).
A slab of thickness c_{0} containing the N nonequivalent polar OD layers in the sequence as they appear in a given structure of category II represents completely its composition. In the remaining three categories, a slab with thickness c_{0/2}, the polar part of the structure contained between two adjacent ρ planes, suffices. Such slabs are higher structural units for OD structures of more than one kind of layer and have been called OD packets. An OD packet is thus defined as the smallest continuous part of an OD structure that is periodic in two dimensions and which represents its composition completely (Ďurovič, 1974a).
If a fully ordered structure is refined, using the space group determined from the systematic absences in its diffraction pattern and then by using some of its subgroups, serious discrepancies are only rarely encountered. Space groups thus characterize the general symmetry pattern quite well, even in real crystals. However, experience with refined periodic polytypic structures has revealed that there are always significant deviations from the OD symmetry and, moreover, even the atomic coordinates within OD layers in different polytypes of the same family may differ from one another. The OD symmetry thus appears as only an approximation to the actual symmetry pattern of polytypes. This phenomenon was called desymmetrization of OD structures (Ďurovič, 1974b, Ďurovič, 1979).
When trying to understand this phenomenon, let us recall the structure of rock salt. Its symmetry is an expression of the energetically most favourable relative position of Na^{+} and Cl^{−} ions in this structure – the right angles αβγ follow from the symmetry. Since the whole structure is cubic, we cannot expect that the environment of any building unit, e.g. of any octahedron NaCl_{6}, would exercise on it an influence that would decrease its symmetry; the symmetries of these units and of the whole structure are not `antagonistic'.
Not so in OD structures, where any OD layer is by definition situated in a disturbing environment because its symmetry does not conform to that of the entire structure. `Antagonistic' relations between these symmetries are most drastic in pure MDO structures because of the regular sequence of layers. The partial symmetry operations become irrelevant and the OD groupoid degenerates into the corresponding space group.
The more disordered an OD structure is, the smaller become the disturbing effects that the environment exercises on an OD layer. These can be, at least statistically, neutralized by random positions of neighbouring layers so that partial symmetry operations can retain their relevance throughout the structure. This can be expressed in the form of a paradox: the less periodic an OD structure is, the more symmetric it appears.
Despite desymmetrization, the OD theory remains a geometrical theory that can handle properly the general symmetry pattern of polytypes (which group theory cannot). It establishes a symmetry norm with which deviations observed in real polytypes can be compared. Owing to the high abstraction power of OD considerations, systematics of entire families of polytypes at various degreeofidealization levels can be worked out, yielding thus a common point of view for their treatment.
Although very general physical principles (OD philosophy, MDO philosophy) underlie the OD theory, it is mainly a geometrical theory, suitable for a description of the symmetry of polytypes and their families rather than for an explanation of polytypism. It thus does not compete with crystal chemistry, but cooperates with it, in analogy with traditional crystallography, where group theory does not compete with crystal chemistry.
When speaking of polytypes, one should always be aware, whether one has in mind a concrete real polytype – more or less in Baumhauer's sense – or an abstract polytype as a structural type (Subsection 9.2.2.1).
A substance can, in general, exist in the form of various polymorphs and/or polytypes of one or several families. Since polytypes of the same family differ only slightly in their crystal energy (Verma & Krishna, 1966), an entire family can be considered as an energetic analogue to one polymorph. As a rule, polytypes belonging to different families of the same substance do not coexist. Al(OH)_{3} may serve as an example for two different families: the bayerite family, in which the adjacent planes of OH groups are stacked according to the principle of close packing (Zvyagin et al., 1979), and the gibbsite–nordstrandite family in which these groups coincide in the normal projection.^{5} Another example is the phyllosilicates (9.2.2.3.1). The compound Hg_{3}S_{2}Cl_{2}, on the other hand, is known to yield two polymorphs α and β (Carlson, 1967; Frueh & Gray, 1968) and one OD family of γ structures (Ďurovič, 1968).
As far as the definition of layer polytypism is concerned, OD theory can contribute specifications about the layers themselves and the geometrical rules for their stacking within a family (all incorporated in the vicinity condition). A possible definition might then read:
Polytypism is a special case of polymorphism, such that the individual polymorphs (called polytypes) may be regarded as arising through different modes of stacking layerlike structural units. The layers and their stackings are limited by the vicinity condition. All polytypes built on the same structural principle belong to a family; this depends on the degree of a structural and/or compositional idealization.
Geometrical theories concerning rod and block polytypism have not yet been elaborated, the main reason is the difficulty of formulating properly the vicinity condition (Sedlacek, Grell & DornbergerSchiff, private communications). But such structures are known. Examples are the structures of tobermorite (Hamid, 1981) and of manganese(III) hydrogenbis(orthophosphite) dihydrate (Císařová & Novák, 1982). Both structures can be thought of as consisting of a threedimensionally periodic framework of certain atoms into which onedimensionally periodic chains and aperiodic finite configurations of the remaining atoms, respectively, `fit' in two equivalent ways.
The three examples below illustrate the three main methods of analysis of polytypism indicated in 9.2.2.2.5.
The basic concepts were introduced by Pauling (1930a), Pauling (1930b) and confirmed later by the determination of concrete crystal structures. A crystallochemical analysis of these became the basis for generalizations and systemizations. The aim was the understanding of geometrical reasons for the polytypism of these substances as well as the development of identification routines through the derivation of basic polytypes (9.2.2.2.3). Smith & Yoder (1956) succeeded first in deriving the six basic polytypes in the mica family.
Since the 1950's, two main schools have developed: in the USA, represented mainly by Brindley, Bailey, and their coworkers (for details and references see Bailey, 1980, 1988a; Brindley, 1980), and in the former USSR, represented by Zvyagin and his coworkers (for details and references see Zvyagin, 1964, 1967; Zvyagin et al., 1979). Both these schools based their systemizations on idealized structural models corresponding to the ideas of Pauling, with hexagonal symmetry of tetrahedral sheets (see later). The US school uses indicative symbols (Guinier et al., 1984) for the designation of individual polytypes, and singlecrystal as well as powder Xray diffraction methods for their identification, whereas the USSR school uses unitary descriptive symbols for polytypes of all mineral groups and mainly electron diffraction on oblique textures for identification purposes. For the derivation of basic polytypes, both schools use crystallochemical considerations; symmetry principles are applied tacitly rather than explicitly.
In contrast to crystal structures based on close packings, where all relevant details of individual (even multilayer) polytypes can be recognized in the section, the structures of hydrous phyllosilicates are rather complex. For their representation, Figueiredo (1979) used the concept of condensed models.
Since 1970, the OD school has also made its contribution. In a series of articles, basic types of hydrous phyllosilicates have been interpreted as OD structures of N > 1 kinds of layers: the serpentine–kaolin group (DornbergerSchiff & Ďurovič, 1975a, b), Mgvermiculite (Weiss & Ďurovič, 1980), the mica group (DornbergerSchiff, Backhaus & Ďurovič, 1982; Backhaus & Ďurovič, 1984; Ďurovič, Weiss & Backhaus, 1984; Weiss & Wiewióra, 1986), the talc–pyrophyllite group (Ďurovič & Weiss, 1983; Weiss & Ďurovič, 1985a), and the chlorite group (Ďurovič, DornbergerSchiff & Weiss, 1983; Weiss & Ďurovič, 1983). The papers published before 1983 use the Pauling model; the later papers are based on the model of Radoslovich (1961) with trigonal symmetry of tetrahedral sheets. In all cases, MDO polytypes (9.2.2.2.3) have been derived systematically: their sets partially overlap with basic polytypes presented by the US or the USSR schools. The OD models allowed the use of unitary descriptive symbols for individual polytypes from which all the relevant symmetries can be determined (Ďurovič & DornbergerSchiff, 1981) as well as of extended indicative Ramsdell symbols (Weiss & Ďurovič, 1985b). The results, including principles for identification of polytypes, have been summarized by Ďurovič (1981).
The main features of polytypes of basic types of hydrous phyllosilicates, of their diffraction patterns and principles for their identification, are given in the following.
Tetrahedral and octahedral sheets are the fundamental, twodimensionally periodic structural units, common to all hydrous phyllosilicates. Any tetrahedral sheet consists of (Si,Al,Fe^{3+},Ti^{4+})O_{4} tetrahedra joined by their three basal O atoms to form a network with symmetry P(3)1m (Fig. 9.2.2.6a ). The atomic coordinates can be related either to a hexagonal axial system with a primitive unit mesh and basis vectors , , or to an orthohexagonal system with a ccentred unit mesh and basis vectors a, b . Any octahedral sheet consists of M(O,OH)_{6} octahedra with shared edges (Fig. 9.2.2.6b), and with cations M most frequently Mg^{2+}, Al^{3+}, Fe^{2+}, Fe^{3+}, but also Li^{+}, Mn^{2+} 0.9 Å, etc. There are three octahedral sites per unit mesh . Crystallochemical classification distinguishes between two extreme cases: trioctahedral (all three octahedral sites are occupied) and dioctahedral (one site is – even statistically – empty). This classification is based on a bulk chemical composition. A classification from the symmetry point of view distinguishes between three cases: homooctahedral [all three octahedral sites are occupied by the same kind of crystallochemical entity, i.e. either by the same kind of ion or by a statistical average of different kinds of ions including voids; symmetry of such a sheet is ;^{6}mesooctahedral [two octahedral sites are occupied by the same kind of crystallochemical entity, the third by a different one, in an ordered way; symmetry ; and heterooctahedral [each octahedral site is occupied by a different crystallochemical entity in an ordered way; symmetry P(3)12]. The prefixes homo, meso, hetero can be combined with the prefixes tri, di, or used alone (Ďurovič, 1994).
A tetrahedral sheet (Tet) can be combined with an octahedral sheet (Oc) either by a shared plane of apical O atoms (in all groups of hydrous phyllosilicates, Fig. 9.2.2.7a ), or by hydrogen bonds (in the serpentine–kaolin group and in the chlorite group, Fig. 9.2.2.7b). Two tetrahedral sheets can either form a pair anchored by interlayer cations (in the mica group, Fig. 9.2.2.8a ) or an unanchored pair (in the talc–pyrophyllite group, Fig. 9.2.2.8b).

Two possible combinations of one tetrahedral and one octahedral sheet (a) by shared apical O atoms, (b) by hydrogen bonds (side projection). 

Combination of two adjacent tetrahedral sheets (a) in the mica group, (b) in the talc–pyrophyllite group (side projection). 
The ambiguity in the stacking occurs at the centres between adjacent Tet and Oc and between adjacent Tet in the talc–pyrophyllite group. From the solved and refined crystal structures it follows that the displacement of (the origin of) one sheet relative to (the origin of) the adjacent one can only be one (or simultaneously three – for homooctahedral sheets) of the nine vectors shown in Fig. 9.2.2.9.
The number of possible positions of one sheet relative to the adjacent one can be determined by the corresponding NFZ relations (9.2.2.2.1). As an example, the contact (Tet; Oc) by shared apical O atoms, and the contacts (Oc; Tet) by hydrogen bonds, for a homooctahedral case, are illustrated in Figs. 9.2.2.10(a) and 9.2.2.10(b), (c), respectively. The two kinds of sheets are represented by the corresponding symbolic figures indicated in Fig. 9.2.2.6. For Fig. 9.2.2.10(a): the symmetry of Tet is P(3)1m, thus N = 6; the symmetry of Oc is and its position relative to Tet is such that the symmetry of the pair is P(3)1m, thus F = 6 and Z = 1: this stacking is unambiguous.^{7} But, if the sequence of these two sheets is reversed, Z = 3, because N_{Oc} = 18 (h centring of Oc). For Figs. 9.2.2.10(b) and (c), Z = 3. Similar relations can be derived for meso and heterooctahedral sheets as well as for the pair (Tet; Tet) in the talc–pyrophyllite group.
A detailed geometrical analysis shows that the possible positions are always related by vectors ±b/3. This, together with the trigonal symmetry of the individual sheets, leads to the fact that any superposition structure (9.2.2.2.5) is trigonal (also rhombohedral) or hexagonal, and the set of diffractions with k_{ort} 0 mod3) has this symmetry too. This is important for the analysis of diffraction patterns.
Some characteristic features of basic types of hydrous phyllosilicates are as follows:
In order to preserve a unitary system, some monoclinic polytypes necessitate a `third' setting, with the a axis unique. These should not be transformed into the standard second setting.
Owing to the trigonal symmetry of the basic structural units and their stacking mode, the singlecrystal diffraction pattern of hydrous phyllosilicates has a hexagonal geometry and it can be referred to hexagonal or orthohexagonal reciprocal vectors or a*, b*, respectively (Figs. 9.2.2.15 and Fig. 9.2.2.16 ). It contains three types of diffractions:
From descriptive geometry, it is known that two orthogonal projections suffice to characterize unambiguously any structure and, therefore, the superposition structure (which implicitly contains the ac projection) together with the bc projection suffice for an unambiguous characterization of any polytype. It also follows that the diffractions with k_{ort} ≡ 0 (mod 3) together with the 0kl diffractions with k 0 (mod 3) suffice for its determination (except for homometric structures) (Ďurovič, 1981).
From the trigonal or hexagonal symmetry of any superposition structure and from Friedel's law, it follows that the reciprocal rows 20l, 13l, , , , and (Fig. 9.2.2.16) carry the same information. Therefore, for identification purposes, it suffices to calculate the distribution of intensities along the reciprocal rows 20l (superposition structure – subfamily) and 02l (bc projection) for all MDO polytypes 769 . Experience shows (Weiss & Ďurovič, 1980) that a mere visual comparison of calculated and observed intensities along these two rows suffices for an unambiguous identification of a MDO polytype. A similar scheme has been presented by Bailey (1988b).
The above considerations are based on the ideal Radoslovich model. Diffraction patterns of real structures may exhibit deviations owing to the distortion of the ideal lattice geometry and/or symmetry of the structure.
The crystal structure of this mineral has been determined by Szymański (1980). It turned out to be identical with that of the compound of the same composition synthesized earlier (Darriet, Bovin & Galy, 1976). The structure is monoclinic with space group C12/c1, lattice parameters a = 17.989 (6), b = 4.7924 (7), c = 5.500 (2) Å, β = 95.13 (3)°.
Structural units are formed of SbO_{2}–O–VO–O–SbO_{2} extended along a, with adjacent units bonded along c through Sb—O—Sb and V—O—V bonds. Ribbons are thus formed with no bonding along b, and only the Sb—O interactions [2.561 (4) Å] along a (Fig. 9.2.2.17 ). This accounts for the excellent acicular cleavage.

The structure of stibivanite2M. The unit cell is outlined and some relevant symmetry operations are indicated (after Merlino et al., 1989). 
Merlino et al. (1989) recognized in this structure sheets of VO_{5} square pyramids (Pyr) parallel to bc, with layer symmetry alternating with sheets containing chains of distorted SbO_{3} tetrahedronlike pyramids (Tet) with layer symmetry (Fig. 9.2.2.18 ). Owing to the higher symmetry of Pyr, they concluded that there may also exist an alternative attachment of Tet to Pyr, such that the triples (Tet; Pyr; Tet) will exhibit the layer symmetry , and they will be arranged so that another polytype 2O with symmetry (Fig. 9.2.2.19 ) is formed [in the original 2M polytype, the triples (Tet; Pyr; Tet) have the layer symmetry ]. A mineral with such a structure, with lattice parameters a = 17.916 (3), b = 4.700 (1), c = 5.509 (1) Å, has actually been found.
The polytypism of stibivanite is reflected in its OD character: the two kinds of sheets Pyr and Tet correspond to two kinds of nonpolar layers: their relative position is given by the family symbol: the NFZ relations being Both polytypes are slightly desymmetrized. Since the shift between the origins of Pyr and Tet is irrational, stibivanite has no superposition structure and thus the diffraction patterns of its polytypes have no common set of family diffractions. Another remarkable point is that the recognition of these structures as OD structures of layers has nothing to do with their system of chemical bonds (see above).
Among about 40 investigated crystals synthesized by Carlson (1967), not one was periodic. All diffraction patterns exhibited a common set of family diffractions, but the distribution of intensities along diffuse streaks varied from crystal to crystal (Ďurovič, 1968). The maxima on these streaks indicated in some cases a simultaneous presence of domains of three periodic polytypes – one system of diffuse maxima was always present and it was referred to a rectangular cell with a = 9.328 (5), b = 16.28 (1), c = 9.081 (6) Å with monoclinic symmetry. All crystals were more or less twinned, sometimes simulating orthorhombic symmetry in their diffraction patterns.
The superposition structure with A = a, B = b/2, C = 2c and space group Pbmm was solved first. Here, a comparison of the family diffractions with the diffraction pattern of the α modification (Frueh & Gray, 1968) proved decisive. It turned out that only one kind of Hg atom contributed with half weight to the superposition structure. This means that only these atoms repeat in the actual structure with periods b = 2B and c = 2C. All other atoms (in the first approximation) repeat with the periods of the superposition structure and thus do not contribute to the diffuse streaks.
The symmetry of the superposition structures is compatible with the OD groupoid family determined from systematic absences: where the subscripts 1/2 indicate translational components of b/4 (DornbergerSchiff, 1964, pp. 41 ff.).
The solution of the structural principle thus necessitated only the correct location of the `disordered' Hg atom in one of two possible positions indicated by the superposition structure. An analysis of the Fourier transform relevant to the above OD groupoid family showed that the Patterson function calculated with coefficients shows interatomic vectors within any single OD layer, and it turned out that even a first generalized projection of the function yields the necessary `yes/no' answer.
The structural principle is shown in Fig. 9.2.2.20 .

The structural principle of γHg_{3}S_{2}Cl_{2}. The shared corners of the pyramids are occupied by the Hg atoms; unshared corners are occupied by the S atoms. A pair of layers, but only the Cl atoms at their common boundary, are drawn. The two geometrically equivalent arrangements (a) and (b) are shown. 
There are two MDO polytypes in this family. Both are monoclinic (c axis unique) and consist of equivalent triples of OD layers. The first, MDO_{1}, is periodic after two layers and has symmetry A2/m. The second, MDO_{2}, is periodic after four layers, has symmetry F2/m and basis vectors 2a, b, c. Domains of MDO_{1} were present in all crystals (the system of diffuse maxima, mentioned above), but some of them also contained small proportions of MDO_{2}.
Later, single crystals of pure periodic MDO_{1} were also found in specimens prepared hydrothermally (Rabenau, private communication). The results of a structure analysis confirmed the previous results but indicated desymmetrization (Ďurovič, 1979). The atomic coordinates deviated significantly from the ideal OD model; the monoclinic angle became 90.5°. Even the family diffractions, which in the disordered crystals exhibited an orthorhombic symmetry, deviated significantly from it in their positions and intensities.
When encountering a polytypic substance showing disorder, many investigators try to find in their specimens a periodic single crystal suitable for a structure analysis by current methods. So far, no objections. But a common failing is that they often neglect to publish a detailed account of the disorder phenomena observed on their diffraction photographs: presence or absence of diffuse streaks, their position in reciprocal space, positions of diffuse maxima, suspicious twinning, nonspacegroup absences, higher symmetry of certain subsets of diffractions, etc. These data should always be published, even if the author does not interpret them: otherwise they will inevitably be lost. Similarly, preliminary diffraction photographs (even of poor quality) showing these phenomena and the crystals themselves should be preserved – they may be useful for someone in the future.
A few examples of some lesscommon polytypic structures published up to 1994 are listed below:

Highresolution electron microscopy (HREM) applied in the structure analysis of (also disordered) polytypic substances: orientite (Mellini, Merlino & Pasero, 1986), ardennite (Pasero & Reinecke, 1991), pseudowollastonite (Ingrin, 1993), laurionite and paralaurionite (Merlino, Pasero & Perchiazzi, 1993), perovskites in the system La_{4}Ti_{3}O_{12}–LaTiO_{3} (Bontchev, Darriet, Darriet, Weill, Van Tendeloo & Amelinckx, 1993), baumhauerite (Pring & Graeser, 1994), bementite (Heinrich, Eggleton & Guggenheim, 1994), parsettensite (Eggleton & Guggenheim, 1994).
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