Section 9.2.2.2. Symmetry aspects 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 so-called 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 (Dornberger-Schiff, 1964, pp. 32 ff.); in our case, Z = 12/6 = 2 (Fig. 9.2.2.1 ). This is the so-called 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 next-but-one layer, we infer that all structures built according to these principles are also energetically equivalent and thus equally likely to appear.

Figure 9.2.2.1| top | pdf | Symmetry interpretation of close packings of equal spheres. The layer group of a single layer, the subgroup of its τ operations, and the number of asymmetric units N per unit mesh of the former, are given at the top right. The τ operations that have a continuation for the pair of adjacent layers, the layer group of the pair, and the value of F are indicated at the bottom right.

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 two-dimensionally 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 nearest-neighbour 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 operations³ describing the respective symmetries, irrespective of the crystallochemical content. These coincidence operations can be total or partial (local) and their set constitutes a groupoid (Dornberger-Schiff, 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 non-periodic 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 build-up 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 longer-range interactions are also considered, then it becomes evident that layer triples such as ABA and ABC in close-packed structures are, in general, energetically non-equivalent because they are also geometrically non-equivalent. 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 non-periodicity.

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 layer-by-layer 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 (Dornberger-Schiff, 1982; Dornberger-Schiff & 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 (left-hand 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.

Figure 9.2.2.2| top | pdf | Schematic representation of three structures belonging to the OD groupoid family P(1)m1|1, y = 0.25 (left). The layers are perpendicular to the plane of the drawing and their constituent atomic configurations are represented by isosceles triangles with symmetry [.m.]. All structures are related to a common orthogonal four-layer cell with a = 4a0. The hk0 nets in reciprocal space corresponding to these structures are shown on the right and the diffraction indices refer also to the common cell. Family diffractions common to all members of this family (k = 2) and the characteristic diffractions for individual polytypes (k = 2k + 1) are indicated by open and solid circles, respectively.

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 = a₀+ b/4 is the total operation (|a₀| is the distance between adjacent layers). It has basis vectors a₁ = a₀ + b/4, b₁= b, c₁= 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 a₂ = 2a₀, b₂ = b, c₂= 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 a₃ = 4a₀, b₃ = b, c₃= 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 space-group 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 non-periodic members of a family, a space-group 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 (Dornberger-Schiff, 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 one-line 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)m1|1, y = 0.25; for the family of close packings of equal spheres: P(6/m)mm|1, 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 (Dornberger-Schiff, Ďurovič & Zvyagin, 1982; Guinier et al., 1984).

Let us now consider schematic diffraction patterns of the three structures on the right-hand 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 so-called superposition structure with basis vectors A = 2a₀, 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 non-periodic 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 (Dornberger-Schiff, 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 non-periodic 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 non-space-group absences exhibited by many polytypic structures. They can be used for the determination of the OD groupoid family (Dornberger-Schiff & 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). High-resolution electron microscopy can also be successfully applied – see Subsection 9.2.2.4.

A polytype family contains periodic as well as non-periodic members. The latter are as important as the former, since the very fact that they can be non-periodic carries important crystallochemical information. Non-periodic polytypes do not comply with the classical definition of crystals, but we believe that this definition should be generalized to include rather than exclude non-periodic polytypes from the world of crystals (Dornberger-Schiff & Grell, 1982b). The OD theory places them, together with the periodic ones, in the hierarchy of the so-called VC structures. The reason for this is that all periodic structures, even the non-polytypic ones, can be thought of as consisting of disjunct, two-dimensionally periodic slabs, the VC layers, which are stacked together according to three rules called the vicinity condition (VC) (Dornberger-Schiff, 1964, pp. 29 ff., Dornberger-Schiff, 1979; Dornberger-Schiff & Fichtner, 1972):

If the stacking of VC layers is unambiguous, traditional three-dimensionally 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 half-atoms 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 two-dimensionally 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.

9.2.2.2.7.1. OD structures of equivalent layers

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If the OD layers are equivalent then they are either all polar or all non-polar in the stacking direction. Any two adjacent polar layers can be related either by τ operations only, or by ρ operations only. For non-polar 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 (Dornberger-Schiff, 1964, pp. 24 ff.):

Figure 9.2.2.3| top | pdf | Schematic examples of the three categories of OD structures consisting of equivalent layers (perpendicular to the plane of the drawing): (a) category I – OD layers non-polar in the stacking direction; (b) category II – polar OD layers, all with the same sense of polarity; (c) category III – polar OD layers with regularly alternating sense of polarity. The position of ρ planes is indicated.

Category II is the simplest: the OD layers are polar and all with the same sense of polarity (they are τ-equivalent); 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 τ-equivalent points in two adjacent layers on the stacking direction (perpendicular to the layer planes) is the repeat unit and it is denoted by c₀, a₀, or b₀ 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 non-polar and they can thus exhibit any of the 63 non-polar 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, α-wollastonite (Yamanaka & Mori, 1981), β-wollastonite (Ito, Sadanaga, Takéuchi & Tokonami, 1969), K₃[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 even-numbered layers can be mutually τ-equivalent and the same holds for odd-numbered 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), γ-Hg₃S₂Cl₂ (Ďurovič, 1968), and 2,2-aziridinedicarboxamide (Fichtner & Grell, 1984).

9.2.2.2.7.2. OD structures with more than one kind of layer

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If an OD structure consists of N > 1 kinds of OD layers, then it can be shown (Dornberger-Schiff, 1964, pp. 64 ff.) that it can fall into one of four categories, according to the polarity or non-polarity 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

Figure 9.2.2.4| top | pdf | Schematic examples of the four categories of OD structures consisting of more than one kind of layer (perpendicular to the plane of the drawing). Equivalent OD layers are represented by equivalent symbolic figures. (a) Category I – three kinds of OD layers: one kind (L2+5n) is non-polar, the remaining two are polar. One and only one kind of non-polar layer is possible in this category. (b) Category II – three kinds of polar OD layers; their triples are polar and retain their sense of polarity in the stacking direction. (c) Category III – three kinds of polar OD layers; their triples are polar and regularly change their sense of polarity in the stacking direction. (d) Category IV – three kinds of OD layers: two kinds are non-polar (L1 + 4n and L3 + 4n), one kind is polar. Two and only two kinds of non-polar layers are possible in this category. The position of ρ planes is indicated.

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; c₀ is the projection on this direction of the shortest vector between two τ-equivalent 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 N-tuples 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. c₀ is the distance between the nearest two equivalent ρ planes; a slab with this thickness contains 2 × N non-equivalent OD layers. No representative of this category is known to date.

The structures of category I contain one, and only one, kind of non-polar 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 non-polar 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. c₀ is the distance between the nearest two equivalent ρ planes and a slab with this thickness contains 2 × (N − 1 ) non-equivalent polar OD layers plus one entire non-polar layer. Examples are the MX₂ compounds (CdI₂, MoS₂, etc.) and the talc–pyrophyllite group.

The structures of category IV contain two, and only two, kinds of non-polar 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 non-polar OD layers; the origin can be placed in any one of them. c₀ is chosen as in categories I and III. A slab with this thickness contains 2 × (N − 2) non-equivalent polar layers plus the two non-polar layers. Examples are micas, chlorites, vermiculites, etc.

OD structures containing N > 1 kinds of layers need special symbols for their OD groupoid families (Grell & Dornberger-Schiff, 1982).

A slab of thickness c₀ containing the N non-equivalent 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).

The hierarchy of VC structures is shown in Fig. 9.2.2.5 .

Figure 9.2.2.5| top | pdf | Hierarchy of VC structures indicating the position of OD structures within it.

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 NaCl₆, 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 degree-of-idealization 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 co-exist. 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 Hg₃S₂Cl₂, 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:

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 & Dornberger-Schiff, 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 three-dimensionally periodic framework of certain atoms into which one-dimensionally periodic chains and aperiodic finite configurations of the remaining atoms, respectively, `fit' in two equivalent ways.