International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.2, pp. 752-773
https://doi.org/10.1107/97809553602060000618

## Chapter 9.2. Layer stacking

S. Ďurovič,a P. Krishnab and D. Pandeyc

aDepartment of Theoretical Chemistry, Slovak Academy of Sciences, Dúbravská cesta, 842 36 Bratislava, Slovakia,bRajghat Education Center, Krishnamurti Foundation India, Rajghat Fort, Varanasi 221001, India, and cPhysics Department, Banaras Hindu University, Varanasi 221005, India

The first section of this chapter describes layer stacking in close-packed structures. The structures of SiC and ZnS, CdI2 and GaSe are used as examples. Stacking faults, voids in structures and disorder are discussed. Layer stacking in general polytypic structures is described in the second section of the chapter. The notion of polytypism is explained and symmetry aspects of polytypism are considered. Three examples illustrating the three main methods of analysis of polytypism are given.

### 9.2.1. Layer stacking in close-packed structures

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D. Pandeyc and P. Krishnab

The crystal structures of a large number of materials can be described in terms of stacking of layers of atoms. This chapter provides a brief account of layer stacking in materials with structures based on the geometrical principle of close packing of equal spheres.

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#### 9.2.1.1.1. Close-packed layer

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In a close-packed layer of spheres, each sphere is in contact with six other spheres as shown in Fig. 9.2.1.1 . This is the highest number of nearest neighbours for a layer of identical spheres and therefore yields the highest packing density. A single close-packed layer of spheres has two-, three- and sixfold axes of rotation normal to its plane. This is depicted in Fig. 9.2.1.2(a), where the size of the spheres is reduced for clarity. There are three symmetry planes with indices , , and (11.0) defined with respect to the smallest two-dimensional hexagonal unit cell shown in Fig. 9.2.1.2(b). The point-group symmetry of this layer is 6mm and it has a hexagonal lattice. As such, a layer with such an arrangement of spheres is often called a hexagonal close-packed layer. We shall designate the positions of spheres in the layer shown in Fig. 9.2.1.1 by the letter A'. This A layer has two types of triangular interstices, one with the apex angle up and the other with the apex angle down . All interstices of one kind are related by the same hexagonal lattice as that for the A layer. Let the positions of layers with centres of spheres above the centres of the and interstices be designated as B' and C', respectively. In the cell of the A layer shown in Fig. 9.2.1.1 (a = b = diameter of the sphere and γ = 120°), the three positions A, B, and C on projection have coordinates (0, 0), , and , respectively.

 Figure 9.2.1.1 | top | pdf |The close packing of equal spheres in a plane.
 Figure 9.2.1.2 | top | pdf |(a) Symmetry axes of a single close-packed layer of spheres and (b) the minimum symmetry of a three-dimensional close packing of spheres.

#### 9.2.1.1.2. Close-packed structures

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A three-dimensional close-packed structure results from stacking the hexagonal close-packed layers in the A, B, or C position with the restriction that no two successive layers are in identical positions. Thus, any sequence of the letters A, B, and C, with no two successive letters alike, represents a possible manner of stacking the hexagonal close-packed layers. There are thus infinite possibilities for close-packed layer stackings. The identity period n of these layer stackings is determined by the number of layers after which the stacking sequence starts repeating itself. Since there are two possible positions for a new layer on the top of the preceding layer, the total number of possible layer stackings with a repeat period of n is 2n−1.

In all the close-packed layer stackings, each sphere is surrounded by 12 other spheres. However, it is touched by all 12 spheres only if the axial ratio h/a is , where h is the separation between two close-packed layers and a is the diameter of the spheres (Verma & Krishna, 1966). Deviations from the ideal value of the axial ratio are common, especially in hexagonal metals (Cottrell, 1967). The arrangement of spheres described above provides the highest packing density of 0.7405 in the ideal case for an infinite lattice (Azaroff, 1960). There are, however, other arrangements of a finite number of equal spheres that have a higher packing density (Boerdijk, 1952).

#### 9.2.1.1.3. Notations for close-packed structures

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In the Ramsdell notation, close-packed structures are designated as nX, where n is the identity period and X stands for the lattice type, which, as shown later, can be hexagonal (H), rhombohedral (R), or in one special case cubic (C) (Ramsdell, 1947).

In the Zhdanov notation, use is made of the stacking offset vector s and its opposite −s, which cause, respectively, a cyclic or anticyclic shift of layers in the same plane. The vector s can be either , , or . Zhdanov (1945) suggested summing the number of consecutive offsets of each kind and designating them by numeral figures. Successive numbers in the Zhdanov symbol have opposite signs. The rhombohedral stackings have three identical sets of Zhdanov symbols in an identity period. It is usually sufficient to write only one set.

Yet another notation advanced, amongst others, by Jagodzinski (1949a) makes use of configurational symbols for each layer. A layer is designated by the symbol h or c according as its neighbouring layers are alike or different. Letter k' in place of c' is also used in the literature.

Some of the common close-packed structures observed in metals are listed in Table 9.2.1.1 in terms of all the notations.

 Table 9.2.1.1| top | pdf | Common close-packed metallic structures
Stacking sequenceIdentity periodRamsdell notationZhdanov notationJagodzinski notationPrototype
AB, A 2 2H 11 h Mg
ABC, A 3 3C c Cu
ABCB, A 4 4H 22 hc La
ABCBCACAB, A 9 9R 21 hhc Sm

#### 9.2.1.2. Structure of compounds based on close-packed layer stackings

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Frequently, the positions of one kind of atom or ion in inorganic compounds, such as SiC, ZnS, CdI2, and GaSe, correspond approximately to those of equal spheres in a close packing, with the other atoms being distributed in the voids. All such structures will also be referred to as close-packed structures though they may not be ideally close packed. In the close-packed compounds, the size and coordination number of the smaller atom/ion may require that its close-packed neighbours in the neighbouring layers do not touch each other.

#### 9.2.1.2.1. Voids in close packing

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Three-dimensional close packings of spheres have two kinds of voids (Azaroff, 1960):

• (i) If the triangular interstices in a close-packed layer have spheres directly over them, the resulting voids are called tetrahedral voids because the four spheres surrounding the void are arranged at the corners of a regular tetrahedron (Figs. 9.2.1.3a,b). If R denotes the radius of the four spheres surrounding a tetrahedral void, the radius of the sphere that would just fit into this void is given by 0.225R (Verma & Krishna, 1966). The centre of the tetrahedral void is located at a distance 3h/4 from the centre of the sphere on top of it.

 Figure 9.2.1.3 | top | pdf |Voids in a close packing: (a) tetrahedral void; (b) tetrahedron formed by the centres of spheres; (c) octahedral void; (d) octahedron formed by the centres of spheres.
• (ii) If the triangular interstices pointing up in one close-packed layer are covered by triangular interstices pointing down in the adjacent layer, the resulting voids are called octahedral voids (Figs. 9.2.1.3c,d) since the six spheres surrounding each such void lie at the corners of a regular octahedron. The radius of the sphere that would just fit into an octahedral void is given by 0.414R (Verma & Krishna, 1966). The centre of this void is located half way between the two layers of spheres.

While there are twice as many tetrahedral voids as the spheres in close packing, the number of octahedral voids is equal to the number of spheres (Krishna & Pandey, 1981).

#### 9.2.1.2.2. Structures of SiC and ZnS

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SiC has a binary tetrahedral structure in which Si and C layers are stacked alternately, each carbon layer occupying half the tetrahedral voids between successive close-packed silicon layers. One can regard the structure as consisting of two identical interpenetrating close packings, one of Si and the other of C, with the latter displaced relative to the former along the stacking axis through one fourth of the layer spacing. Since the positions of C atoms are fixed relative to the positions of layers of Si atoms, it is customary to use the letters A, B, and C as representing Si–C double layers in the close packing. To be more exact, the three kinds of layers need to be written as Aα, Bβ, and Cγ where Roman and Greek letters denote the positions of Si and C atoms, respectively. Fig. 9.2.1.4 depicts the structure of SiC-6H, which is the most common modification.

 Figure 9.2.1.4 | top | pdf |Tetrahedral arrangement of Si and C atoms in the SiC-6H structure.

A large number of crystallographically different modifications of SiC, called polytypes, has been discovered in commercial crystals grown above 2273 K (Verma & Krishna, 1966; Pandey & Krishna, 1982a). Table 9.2.1.2 lists those polytypes whose structures have been worked out. All these polytypes have a = b = 3.078 Å and c = n × 2.518 Å, where n is the number of Si–C double layers in the hexagonal cell. The 3C and 2H modifications, which normally result below 2273 K, are known to undergo solid-state structural transformation to 6H (Jagodzinski, 1972; Krishna & Marshall, 1971a, Krishna & Marshall, 1971b) through a non-random insertion of stacking faults (Pandey, Lele & Krishna, 1980a, Pandey, Lele & Krishna, 1980b, Pandey, Lele & Krishna, 1980c; Kabra, Pandey & Lele, 1986). The lattice parameters and the average thickness of the Si–C double layers vary slightly with the structure, as is evident from the h/a ratios of 0.8205 (Adamsky & Merz, 1959), 0.8179, and 0.8165 (Taylor & Jones, 1960) for the 2H, 6H, and 3C structures, respectively. Even in the same structure, crystal-structure refinement has revealed variation in the thickness of Si–C double layers depending on their environment (de Mesquita, 1967).

 Table 9.2.1.2| top | pdf | List of SiC polytypes with known structures in order of increasing periodicity(after Pandey & Krishna, 1982a)
PolytypeStructure (Zhdanov sequence)PolytypeStructure (Zhdanov sequence)
2H 11 57H (23)93333
3C 57R (33)234
4H 22 69R1 (33)332
6H 33 69R2 33322334
8H 44 75R2 (32)3(23)2
10H 3322 81H (33)535(33)634
14H (22)233 84R (33)3(32)2
15R 23 87R (33)432
16H1 (33)222 90R (23)43322
18H (22)333 93R (33)434
19H (23)322 96R1 (33)33434
20H (22)344 99R (33)43222
21H 333534 105R (33)532
21H2 (33)263 111R (33)534
21R 34 120R (22)523222333
24R 35 123R (33)632
27H (33)2(23)3 126R (33)22353433223
27R 2223 129R (33)634
33R 3332 125R 32(33)223(33)323
33H (33)2353334 141R (33)732
34H (33)42332 147R (3332)432
36H1 (33)232(33)234 150R1 (23)332(23)3322332
36H2 (33)43234 150R2 (23)2(3223)4
39H (33)232(33)3(32)2 159R (33)832
39R 3334 168R (23) 1033
40H (33)52332 174R (33)66(33)54
45R (23)232 189R (34)843
51R1 (33)232 267R (23)1722
51R2 (22)323 273R (23)1733
54H (33)6323334 393R (33)2132

The structure of ZnS is analogous to that of SiC. Like the latter, ZnS crystals grown from the vapour phase also display a large variety of polytype structures (Steinberger, 1983). ZnS crystals that occur as minerals usually correspond to the wurtzite and the sphalerite modifications. The structural transformation between the 2H and 3C structures of ZnS is known to be martensitic in nature (Sebastian, Pandey & Krishna, 1982; Pandey & Lele, 1986b). The h/a ratio for ZnS-2H is 0.818, which is somewhat different from the ideal value (Verma & Krishna, 1966). The structure of the stackings in polytypic AgI is analogous to those in SiC and ZnS (Prager, 1983).

#### 9.2.1.2.3. Structure of CdI2

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The structure of cadmium iodide consists of a close packing of the I ions with the Cd ions distributed amongst half the octahedral voids. Thus, the Cd and I layers are not stacked alternately; there is one Cd layer after every two I layers as shown in Fig. 9.2.1.5 . The structure actually consists of molecular sheets (called minimal sandwiches) with a layer of Cd ions sandwiched between two close-packed layers of I ions. The bonding within the minimal sandwich is ionic in character and is much stronger than the bonding between successive sandwiches, which is of van der Waals type. The importance of polarization energy for the stability of such structures has recently been emphasized by Bertaut (1978). It is because of the weak van der Waals bonding between the successive minimal sandwiches that the material possesses the easy cleavage characteristic of a layer structure. In describing the layer stackings in the CdI2 structure, it is customary to use Roman letters to denote the I positions and Greek letters for the Cd positions. The two most common modifications of CdI2 are 4H and 2H with layer stackings and , respectively. In addition, this material also displays a number of polytype modifications of large repeat periods (Trigunayat & Verma, 1976; Pandey & Krishna, 1982a). From the structure of CdI2, it follows that the identity period of all such modifications must consist of an even number of I layers. The h/a ratio in all these modifications of CdI2 is 0.805, which is very different from the ideal value (Verma & Krishna, 1966). The structure of PbI2, which also displays a large number of polytypes, is analogous to CdI2 with one important difference. Here, the distances between two I layers with and without an intervening Pb layer are quite different (Trigunayat & Verma, 1976).

 Figure 9.2.1.5 | top | pdf |The layer structure of CdI2: small circles represent Cd ions and larger ones I ions (after Wells, 1945).

#### 9.2.1.2.4. Structure of GaSe

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The crystal structure of GaSe consists of four-layered slabs, each of which contains two close-packed layers of Ga (denoted by symbols A, B, C) and Se (denoted by symbols α,β,γ) each in the sequence Se–Ga–Ga–Se (Terhell, 1983). The Se atoms sit on the corners of a trigonal prism while each Ga atom is tetrahedrally coordinated by three Se and one Ga atoms. If the Se layers are of A type, then the stacking sequence of the four layers in the slab can be written as or . There are thus six possible sequences for the unit slab. These unit slabs can be stacked in the manner described for equal spheres. Thus, for example, the 2H structure can have three different layer stackings: , and . Periodicities containing up to 21 unit slabs have been reported for GaSe (see Terhell, 1983). The bonding between the layers of a slab is predominantly covalent while that between two adjacent slabs is of the van der Waals type, which imparts cleavage characteristics to this material.

#### 9.2.1.3. Symmetry of close-packed layer stackings of equal spheres

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It can be seen from Fig. 9.2.1.2(a) that a stacking of two or more layers in the close-packed manner still possesses all three symmetry planes but the twofold axes disappear while the sixfold axes coincide with the threefold axes (Verma & Krishna, 1966). The lowest symmetry of a completely arbitrary periodic stacking sequence of close-packed layers is shown in Fig. 9.2.1.2(b). Structures resulting from such stackings therefore belong to the trigonal system. Even though a pure sixfold axis of rotation is not possible, close-packed structures belonging to the hexagonal system can result by virtue of at least one of the three symmetry axes parallel to [00.1] being a 63 axis (Verma & Krishna, 1966). This is possible if the layers in the unit cell are stacked in special ways. For example, a 6H stacking sequence has a 63 axis through 0, 0. It follows that, for an nH structure belonging to the hexagonal system, n must be even. A packing with n odd will therefore necessarily belong to the trigonal system and can have either a hexagonal or a rhombohedral lattice (Verma & Krishna, 1966).

Other symmetries that can arise by restricting the arbitrariness of the stacking sequence in the identity period are: (i) a centre of symmetry at the centre of either the spheres or the octahedral voids; and (ii) a mirror plane perpendicular to [00.1]. Since there must be two centres of symmetry in the unit cell, the centrosymmetric arrangements may possess both centres either at sphere centres/octahedral void centres or one centre each at the centres of spheres and octahedral voids (Patterson & Kasper, 1959).

#### 9.2.1.4. Possible lattice types

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Close packings of equal spheres can belong to the trigonal, hexagonal, or cubic crystal systems. Structures belonging to the hexagonal system necessarily have a hexagonal lattice, i.e. a lattice in which we can choose a primitive unit cell with , α = β = 90°, and γ = 120°. In the primitive unit cell of the hexagonal close-packed structure shown in Fig. 9.2.1.6 , there are two spheres associated with each lattice point, one at 0, 0, 0 and the other at , , . Structures belonging to the trigonal system can have either a hexagonal or a rhombohedral lattice. By a rhombohedral lattice is meant a lattice in which we can choose a primitive unit cell with a = b = c, α = β = γ °. Both types of lattice can be referred to either hexagonal or rhombohedral axes, the unit cell being non-primitive when a hexagonal lattice is referred to rhombohedral axes and vice versa (Buerger, 1953). In close-packed structures, it is generally convenient to refer both hexagonal and rhombohedral lattices to hexagonal axes. Fig. 9.2.1.7 shows a rhombohedral lattice in which the primitive cell is defined by the rhombohedral axes ; but a non-primitive hexagonal unit cell can be chosen by adopting the axes A1, A2,C. The latter has lattice points at 0,0,0; ; and . If this rhombohedral lattice is rotated through 60° around [00.1], the hexagonal unit cell will then be centred at and . These two settings are crystallographically equivalent for close packing of equal spheres. They represent twin arrangements when both occur in the same crystal. The hexagonal unit cell of an nR structure is made up of three elementary stacking sequences of n/3 layers that are related to each other either by an anticyclic shift of layers (obverse setting) or by a cyclic shift of layers (reverse setting) in the direction of z increasing (Verma & Krishna, 1966). Evidently, n must be a multiple of 3 for nR structures.

 Figure 9.2.1.6 | top | pdf |The primitive unit cell of the 2H close packing.
 Figure 9.2.1.7 | top | pdf |A rhombohedral lattice (a1, a2, a3) referred to hexagonal axes (A1, A2, C)(after Buerger, 1953).

In the special case of the close packing [with the ideal axial ratio of ], the primitive rhombohedral unit cell has α = β = γ = 60°, which enhances the symmetry and enables the choice of a face-centred cubic unit cell. The relationship between the face-centred cubic and the rhombohedral unit cell is shown in Fig. 9.2.1.8 . The threefold axis of the rhombohedral unit cell coincides with one of the directions of the cubic unit cell. The close-packed layers are thus parallel to the {111} planes in the cubic close packing.

 Figure 9.2.1.8 | top | pdf |The relationship between the f.c.c. and the primitive rhombohedral unit cell of the c.c.p. structure.

#### 9.2.1.5. Possible space groups

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It was shown by Belov (1947) that consistent combinations of the possible symmetry elements in close packing of equal spheres can give rise to eight possible space groups: P3m1, , , , , R3m, , and Fm3m. The last space group corresponds to the special case of cubic close packing . The tetrahedral arrangement of Si and C in SiC does not permit either a centre of symmetry or a plane of symmetry (m) perpendicular to [00.1]. SiC structures can therefore have only four possible space groups P3m1, R3m1, , and . CdI2 structures can have a centre of symmetry on octahedral voids, but cannot have a symmetry plane perpendicular to [00.1]. CdI2 can therefore have five possible space groups: P3m1, , R3m, , and . Cubic symmetry is not possible in CdI2 on account of the presence of Cd atoms, the sequence representing a 6R structure.

#### 9.2.1.6. Crystallographic uses of Zhdanov symbols

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From the Zhdanov symbols of a close-packed structure, it is possible to derive information about the symmetry and lattice type (Verma & Krishna, 1966). Let n+ and n be the number of positive and negative numerals in the Zhdanov sequence of a given structure. The lattice is rhombohedral if n+n = ±1mod3, otherwise it is hexagonal. The + sign corresponds to the reverse setting and − to the obverse setting of the rhombohedral lattice. Since this criterion is sufficient for the identification of a rhombohedral structure, the practice of writing three units of identical Zhdanov symbols has been abandoned in recent years (Pandey & Krishna, 1982a). Thus the 15R polytype of SiC is written as (23) rather than (23)3.

As described in detail by Verma & Krishna (1966), if the Zhdanov symbol consists of an odd set of numbers repeated twice, e.g. (22), (33), (221221) etc., the structure can be shown to possess a 63 axis. For the centre of symmetry at the centre of a sphere or an octahedral void, the Zhdanov symbol will consist of a symmetrical arrangement of numbers of like signs surrounding a single even or odd Zhdanov number, respectively. Thus, the structures (2)32(4)23 and (3)32(5)23 have centres of symmetry of the two types in the numbers within parentheses. For structures with a symmetry plane perpendicular to [00.1], the Zhdanov symbols consist of a symmetrical arrangement of a set of numbers of opposite signs about the space between two succession numbers. Thus, a stacking |522|225| has mirror planes at positions indicated by the vertical lines.

The use of abridged symbols to describe crystal structures has sometimes led to confusion in deciding the crystallographic equivalence of two polytype structures. For example, the structures (13) and (31) are identical for SiC but not for CdI2 (Jain & Trigunayat, 1977a, Jain & Trigunayat, 1977b).

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#### 9.2.1.7.1. General considerations

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The different layer stackings (polytypes) of the same material have identical a and b parameters of the direct lattice. The a*b* reciprocal-lattice net is therefore also the same and is shown in Fig. 9.2.1.9 . The reciprocal lattices of these polytypes differ only along the c* axis, which is perpendicular to the layers. It is evident from Fig. 9.2.1.9 that for each reciprocal-lattice row parallel to c* there are five others with the same value of the radial coordinate ξ. For example, the rows 10.l, 01.l, , , , and all have ξ = |a*|. Owing to symmetry considerations, it is sufficient to record any one of them on X-ray diffraction photographs. The reciprocal-lattice rows hk.l can be classified into two categories according as hk = 0 mod3 or . Since the atoms in an nH or nR structure lie on three symmetry axes , , and , the structure factor Fhkl can be split into three parts: where , , , and , , are the z coordinates of atoms at A, B, and C sites, respectively. For hk = 0mod3, which is zero except when . Hence, the reflections 00.l, 11.l, 30.l, etc., for which hk = 0mod3, will be extinguished except when . Thus, only those hk.l reciprocal-lattice rows for which hk 0mod3 carry information about the stacking sequence and contain in general reflections with , n − 1, etc. It is sufficient to record any one such row, usually the 10.l row with ξ = |a*|, on an oscillation, Weissenberg, or precession photograph to obtain information about the lattice type, identity period, space group, and hence the complete structure (Verma & Krishna, 1966).

 Figure 9.2.1.9 | top | pdf |The a*−b* reciprocal-lattice net for close-packed layer stackings.

#### 9.2.1.7.2. Determination of the lattice type

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When the structure has a hexagonal lattice, the positions of spots are symmetrical about the zero layer line on the c-axis oscillation photograph. However, the intensities of the reflections on the two sides of the zero layer line are the same only if the structure possesses a 63 axis, and not for the trigonal system. An apparent mirror symmetry perpendicular to the c axis results from the combination of the axis with the centre of symmetry introduced by X-ray diffraction. For a structure with a rhombohedral lattice, the positions of X-ray diffraction spots are not symmetrical about the zero layer line because the hexagonal unit cell is non-primitive causing the reflections hkl to be absent when . For the 10.l row, this means that the permitted reflections will have l = 3n + 1, which implies above the zero layer line 10.1, 10.4, 10.7, etc. reflections and below the zero layer line , , , etc. The zero layer line will therefore divide the distance between the nearest spots on either side (namely 10.1 and ) approximately in the ratio 1:2. This enables a quick identification of a rhombohedral lattice. It is also possible to identify rhombohedral lattices by the appearance of an apparent doubling' of spots along the Bernal row lines on a rotation photograph. This is because of the threefold symmetry which makes reciprocal-lattice rows such as 10.l, , and identical with each other but different from the other identical set, , , and . The extinction condition for the second set requires l = 3n − 1, i.e. l = 2, 5, 8, and , etc., which is different from that for the first set. Consequently, on the rotation photograph, reciprocal-lattice rows with ξ = |a*| will have spots for l = 3n±1 causing the apparent doubling'.

In crystals of layer structures, such as CdI2, where a-axis oscillation photographs are normally taken, the identification of the rhombohedral lattice is performed by checking for the non-coincidence of the diffraction spots with those for the 2H or 4H structures. In an alternative method, one compares the positions of spots in two rows of the type 10.l and 20.l. This can conveniently be done by taking a Weissenberg photograph (Chadha, 1977).

#### 9.2.1.7.3. Determination of the identity period

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The number of layers, n, in the hexagonal unit cell can be found by determining the c parameter from the c-axis rotation or oscillation photographs and dividing this by the layer spacing h for that compound which can be found from reflections with hk = 0mod3. The density of reciprocal-lattice points along rows parallel to c* depends on the periodicity along the c axis. The larger the identity period along c, the more closely spaced are the diffraction spots along c*. In situations where there are not many structural extinctions, n can be determined by counting the number of spacings after which the sequence of relative intensities begins to repeat along the 10.l row of spots on an oscillation or Weissenberg photograph (Krishna & Verma, 1963). If the structure contains a random stacking disorder of close-packed layers (stacking faults), this will effectively make the c parameter infinite and lead to the production of characteristic continuous diffuse streaks along reciprocal-lattice rows parallel to c* for reflections with hk 0mod3 (Wilson, 1942). It is therefore difficult to distinguish by X-ray diffraction between structures of very large unresolvable periodicities and those with random stacking faults. Lattice resolution in the electron microscope has been used in recent years to identify such structures (Dubey, Singh & Van Tendeloo, 1977). A better resolution of diffraction spots along the 10.l reciprocal-lattice row can be obtained by using the Laue method. Standard charts for rapid identification of SiC polytypes from Laue films are available in the literature (Mitchell, 1953). Identity periods as large as 594 layers have been resolved by this method (Honjo, Miyake & Tomita, 1950). Synchrotron radiation has been used for taking Laue photographs of ZnS polytypes (Steinberger, Bordas & Kalman, 1977).

#### 9.2.1.7.4. Determination of the stacking sequence of layers

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For an nH or 3nR polytype, the n close-packed layers in the unit cell can be stacked in possible ways, all of which cannot be considered for ultimate intensity calculations. A variety of considerations has therefore been used for restricting the number of trial structures. To begin with, symmetry and space-group considerations discussed in Subsection 9.2.1.4 and 9.2.1.5 can considerably reduce the number of trial structures.

When the short-period structures act as basic structures' for the generation of long-period polytypes, the number of trial structures is considerably reduced since the crystallographic unit cells of the latter will contain several units of the small-period structures with faults between or at the end of such units. The basic structure of an unknown polytype can be guessed by noting the intensities of 10.l reflections that are maximum near the positions corresponding to the basic structure. If the unknown polytype belongs to a well known structure series, such as (33)n32 and (33)n34 based on SiC-6H, empirical rules framed by Mitchell (1953) and Krishna & Verma (1962) can allow the direct identification of the layer-stacking sequence without elaborate intensity calculations.

It is possible to restrict the number of probable structures for an unknown polytype on the basis of the faulted matrix model of polytypism for the origin of polytype structures (for details see Pandey & Krishna, 1983). The most probable series of structures as predicted on the basis of this model for SiC contains the numbers 2, 3, 4, 5 and 6 in their Zhdanov sequence (Pandey & Krishna, 1975, Pandey & Krishna, 1976a). For CdI2 and PbI2 polytypes, the possible Zhdanov numbers are 1, 2 and 3 (Pandey & Krishna, 1983; Pandey, 1985). On the basis of the faulted matrix model, it is not only possible to restrict the numbers occurring in the Zhdanov sequence but also to restrict drastically the number of trial structures for a new polytype.

Structure determination of ZnS polytypes is more difficult since they are not based on any simple polytype and any number can appear in the Zhadanov sequence. It has been observed that the birefringence of polytype structures in ZnS varies linearly with the percentage hexagonality (Brafman & Steinberger, 1966), which in turn is related to the number of reversals in the stacking sequence, i.e. the number of numbers in the Zhdanov sequence. This drastically reduces the number of trial structures for ZnS (Brafman, Alexander & Steinberger, 1967).

Singh and his co-workers have successfully used lattice imaging in conjunction with X-ray diffraction for determining the structures of long-period polytypes of SiC that are not based on a simple basic structure. After recording X-ray diffraction patterns, single crystals of these polytypes were crushed to yield electron-beam-transparent flakes. The one- and two-dimensional lattice images were used to propose the possible structures for the polytypes. Usually this approach leads to a very few possibilities and the correct structure is easily determined by comparing the observed and calculated X-ray intensities for the proposed structures (Dubey & Singh, 1978; Rai, Singh, Dubey & Singh, 1986).

Direct methods for the structure determination of polytypes from X-ray data have also been suggested by several workers (Tokonami & Hosoya, 1965; Dornberger-Schiff & Farkas-Jahnke, 1970; Farkas-Jahnke & Dornberger-Schiff, 1969) and have been reviewed by Farkas-Jahnke (1983). These have been used to derive the structures of ZnS, SiC, and TiS1.7 polytypes. These methods are extremely sensitive to experimental errors in the intensities.

#### 9.2.1.8. Stacking faults in close-packed structures

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The two alternative positions for the stacking of successive close-packed layers give rise to the possibility of occurrence of faults where the stacking rule is broken without violating the law of close packing. Such faults are frequently observed in crystals of polytypic materials as well as close-packed martensites of cobalt, noble-metal-based and certain iron-based alloys (Andrade, Chandrasekaran & Delaey, 1984; Kabra, Pandey & Lele, 1988a; Nishiyama, 1978; Pandey, 1988).

The classical method of classifying stacking faults in 2H and 3C structures as growth and deformation types, depending on whether the fault has resulted as an accident during growth or by shear through the vector s, leads to considerable ambiguities since the same fault configuration can result from more than one physical process. For a detailed account of the limitations of the notations based on the process of formation, the reader is referred to the articles by Pandey (1984a) and Pandey & Krishna (1982b).

Frank (1951) has classified stacking faults as intrinsic or extrinsic purely on geometrical considerations. In intrinsic faults, the perfect stacking sequence on each side of the fault extends right up to the contact plane of the two crystal halves while in extrinsic faults the contact plane does not belong to the stacking sequence on either side of it. In intrinsic faults, the contact plane may be an atomic or non-atomic plane whereas in extrinsic faults the contact plane is always an atomic plane. Instead of contact plane, one can use the concept of fault plane defined with respect to the initial stacking sequence. This system of classification is preferable to that based on the process of formation. However, the terms intrinsic and extrinsic have been used in the literature in a very restricted sense by associating these with the precipitation of vacancies and interstitials, respectively (see, for example, Weertman & Weertman, 1984). While the precipitation of vacancies may lead to intrinsic fault configuration, this is by no means the only process by which intrinsic faults can result. For example, there are geometrically 18 possible intrinsic fault configurations in the 6H (33) structure (Pandey & Krishna, 1975) but only two of these can result from the precipitation of vacancies. Similarly, layer-displacement faults involved in SiC transformations are extrinsic type but do not result from the precipitation of interstitials (see Pandey, Lele & Krishna, 1980a, b, c; Kabra, Pandey & Lele, 1986). It is therefore desirable not to associate the geometrical notation of Frank with any particular process of formation.

The intrinsic–extrinsic scheme of classification of faults when used in conjunction with the concept of assigning subscripts to different close-packed layers (Prasad & Lele, 1971; Pandey & Krishna, 1976b) can provide a very compact and unique way of representing intrinsic fault configurations even in long-period structures (Pandey, 1984b). We shall briefly explain this notation in relation to one hexagonal (6H) and one rhombohedral (9R) structure.

In the 6H (ABCACB, or hkkhkk) structure, six kinds of layers that can be assigned subscripts 0, 1, 2, 3, 4, and 5 need to be distinguished (Pandey, 1984b). Choosing the 0-type layer in h' configuration such that the layer next to it is related through the shift vector +s (which causes cyclic shift), the perfect 6H structure can be written as There are six crystallographically equivalent ways of writing this structure with the first layer in position A: (i) ; (ii) ; (iii) ; (iv) ; (v) ; and (vi) . Similarly, there are six ways of writing the 6H structure with the starting layer in position B or C. Since an intrinsic fault marks the beginning of a fresh 6H sequence, there can be 36 possible intrinsic fault configurations in the 6H structure. All these intrinsic fault configurations can be described by symbols like , where r and s stand for the subscript of the layer on the left- and right-hand sides of the fault plane while I represents intrinsic. Knowing the two symbols (r and s), one can write down the complete ABC stacking sequence. It may be noted that, of the 36 possible intrinsic fault configurations, only 14 are crystallographically indistinguishable (for details, see Pandey, 1984b). This notation can be used for any hexagonal polytype and requires only the identification of various layer types in the structure. For rhombohedral polytypes, one must consider the layer types in both the obverse and the reverse settings. For example, six layer types need to be distinguished in the 9R (hhk) structure:

Obverse:

Reverse:

In the obverse setting, we choose the origin layer (0 type) in the h configuration such that the next layer is cyclically shifted whereas in the reverse setting the origin layer ( type) in the h configuration is related to the next layer through an anticyclic shift. Tables 9.2.1.3 and 9.2.1.4 list the crystallographically unique intrinsic fault configurations in the 6H and 9R structures.

 Table 9.2.1.3| top | pdf | Intrinsic fault configurations in the 6H (A0B1C2A3C4B5;. . .) structure
Fault configuration
ABC sequence
Subscript
notation
A B C A C B A0C0 A B C B A C I0, 0
A B C A C B A0C1 A B A C B C I0, 1
A B C A C B A0C2 A C B A B C I0, 2
A B C A C B A0C3 B A C A B C I0, 3
A B C A C B A0C4 B A B C A C I0, 4
A B C A C B A0C5 B C A B A C I0, 5
A B C A C B A B1 A0 B C A C B A I1, 0
A B C A C B A B1 A1 B C B A C A I1, 1
A B C A C B A B1 A2B A C B C A I1, 2
A B C A C B A B1 A3C B A B C A I1, 3
A B C A C B A B1 A4C B C A B A I1, 4
A B C A C B A B1 A5C A B C B A I1, 5
A B C A C B A B C2B0 C A B A C B I2, 0
A B C A C B A B C2B1C A C B A B I2, 1
A B C A C B A B C2B2C B A C A B I2, 2
A B C A C B A B C2B3 A C B C A B I2, 3
A B C A C B A B C2B4 A C A B C B I2, 4
A B C A C B A B C2 B5 A B C A C B I2, 5

Notes:

 (1) Vertical dots represent the location of the fault plane with respect to the initial stacking sequence on the left-hand side. (2) I0,1 and I2,3, I0,2 and I1,3, I1,1 and I2,2, and I1,4 and I2,5 are crystallographically equivalent.

 Table 9.2.1.4| top | pdf | Intrinsic fault configurations in the 9R (A0B1A2C0A1C2B0C1B2;. . .) structure
Fault configuration
ABC sequence
Subscript
notation
A B A C A C B C B A0C0 A C B C B A B A I0, 0
A B A C A C B C B A0C1 B A B A C A C B I0, 1
A B A C A C B C B A0C2 B C B A B A C A I0, 2
A B A C A C B C B A0 B C A C A B A B
A B A C A C B C B A0 A B A B C B C A
A B A C A C B C B A0 A C A B A B C B
A B A C A C B C B A B1C0 A C B C B A B A I1, 0
A B A C A C B C B A B1C1 B A B A C A C B I1, 1
A B A C A C B C B A B1C2 B C B A B A C A I1, 2
A B A C A C B C B A B1 B C A C A B A B
A B A C A C B C B A B1 A B A B C B C A
A B A C A C B C B A B1 A C A B A B C B
A B A C A C B C B A B A2B0 C B A B A C A C I2, 0
A B A C A C B C B A B A2B1 A C A C B C B A I2, 1
A B A C A C B C B A B A2B2 A B A C A C B C I2, 2
A B A C A C B C B A B A2 A B C B C A C A
A B A C A C B C B A B A2 C A C A B A B C
A B A C A C B C B A B A2 C B C A C A B A

Note: and , and , and , and I1,2 and I2,0 are crystallographically equivalent.

#### 9.2.1.8.1. Structure determination of one-dimensionally disordered crystals

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Statistical distribution of stacking faults in close-packed structures introduces disorder along the stacking axis of the close-packed layers. As a result, one observes on a single-crystal diffraction pattern not only normal Bragg scattering near the nodes of the reciprocal lattice of the average structure but also continuous diffuse scattering between the nodes owing to the incomplete destructive interference of scattered rays. Just like the extra polytype reflections, the diffuse streaks are also confined to only those rows for which hk 0mod3. A complete description of the real structure of such one-dimensionally disordered polytypes requires knowledge of the average structure as well as a statistical specification of the fluctuations due to stacking faults in the electron-density distribution of the average structure. This cannot be accomplished by the usual consideration of the normal Bragg reflections alone but requires a careful analysis of the diffuse intensity distribution as well (Pandey, Kabra & Lele, 1986).

The first step in the structure determination of one-dimensionally disordered structures is the specification of the geometry of stacking faults and their distribution, both of which require postulation of the physical processes responsible for their formation. An entirely random distribution of faults may result during the layer-by-layer growth of a crystal (Wilson, 1942) or during plastic deformation (Paterson, 1952). On the other hand, when faults bring about the change in the stacking sequence of layers during solid-state transformations, their distribution is non-random (Pandey, Lele & Krishna, 1980a, b, c; Pandey & Lele, 1986a, b; Kabra, Pandey & Lele, 1986). Unlike growth faults, which are accidentally introduced in a sequential fashion from one end of the stack of layers to the other during the actual crystal growth, stacking faults involved in solid-state transformations are introduced in a random space and time sequence (Kabra, Pandey & Lele, 1988b). Since the pioneering work of Wilson (1942), several different techniques have been advanced for the calculation of intensity distributions along diffuse streaks making use of Markovian chains, random walk, stochastic matrices, and the Paterson function for random and non-random distributions of stacking faults on the assumption that these are introduced in a sequential fashion (Hendricks & Teller, 1942; Jagodzinski, 1949a, b; Kakinoki & Komura, 1954; Johnson, 1963; Prasad & Lele, 1971; Cowley, 1976; Pandey, Lele & Krishna, 1980a, b). The limitations of these methods for situations where non-randomly distributed faults are introduced in the random space and time sequence have led to the use of Monte Carlo techniques for the numerical calculation of pair correlations whose Fourier transforms directly yield the intensity distributions (Kabra & Pandey, 1988).

The correctness of the proposed model for disorder can be verified by comparing the theoretically calculated intensity distributions with those experimentally observed. This step is in principle analogous to the comparison of the observed Bragg intensities with those calculated for a proposed structure in the structure determination of regularly ordered layer stackings. This comparison cannot, however, be performed in a straightforward manner for one-dimensionally disordered crystals due to special problems in the measurement of diffuse intensities using a single-crystal diffractometer, stemming from incident-beam divergence, finite size of the detector slit, and multiple scattering. The problems due to incident-beam divergence in the measurement of the diffuse intensity distributions were first pointed out by Pandey & Krishna (1977) and suitable correction factors have recently been derived by Pandey, Prasad, Lele & Gauthier (1987). A satisfactory solution to the problem of structure determination of one-dimensionally disordered stackings must await proper understanding of all other factors that may influence the true diffraction profiles.

### 9.2.2. Layer stacking in general polytypic structures

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S. Ďuroviča

#### 9.2.2.1. The notion of polytypism

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The common property of the structures described in Section 9.2.1 was the stacking ambiguity of adjacent layer-like 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:

 (1) all three types originate simultaneously in the same melt and seemingly also under the same, or nearly the same, conditions; (2) they can be related in a simple way to the same axial ratio (each within an individual primary series); (3) any two types (I and II, II and III) have certain faces in common but, except the basal face, there is no face occurring simultaneously in all three types; (4) the crystals belonging to different, but also to all three, types often form intergrowths with parallel axes; (5) any of the three types exhibits a typical X-ray diffraction pattern and thus also an individual molecular or atomic structure.

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. CdI2 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 nearest-neighbour 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 mixed-layer and turbostratic structures as non-polytypic. 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 block-like 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 two-dimensional 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 Ad-Hoc 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 mixed-layer phyllosilicates. However, the sequence and stacking of layers in a polytype are always subject to well-defined 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 Ad-Hoc 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 two-dimensional 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,1belonging 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 Dornberger-Schiff (1964), Dornberger-Schiff (1966), Dornberger-Schiff (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.

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#### 9.2.2.2.1. Close packing of spheres

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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'.

#### 9.2.2.2.2. Polytype families and OD groupoid families

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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.

#### 9.2.2.2.3. MDO polytypes

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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).

#### 9.2.2.2.4. Some geometrical properties of OD structures

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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).

#### 9.2.2.2.5. Diffraction pattern – structure analysis

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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.

#### 9.2.2.2.6. The vicinity condition

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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):

 (α) VC layers are either geometrically equivalent or, if not, they are relatively few in kind; (β) translation groups of all VC layers are either identical or they have a common subgroup; (γ) equivalent sides of equivalent layers are faced by equivalent sides of adjacent layers so that the resulting pairs are equivalent [for a more detailed specification and explanation see Dornberger-Schiff (1979)].

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).

#### 9.2.2.2.7. Categories of OD structures

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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 c0/2, 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.

#### 9.2.2.2.8. Desymmetrization of OD structures

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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.

#### 9.2.2.2.9. Concluding remarks

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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:

Polytypism is a special case of polymorphism, such that the individual polymorphs (called polytypes) may be regarded as arising through different modes of stacking layer-like 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 & 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.

#### 9.2.2.3. Examples of some polytypic structures

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The three examples below illustrate the three main methods of analysis of polytypism indicated in 9.2.2.2.5.

#### 9.2.2.3.1. Hydrous phyllosilicates

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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 co-workers (for details and references see Bailey, 1980, 1988a; Brindley, 1980), and in the former USSR, represented by Zvyagin and his co-workers (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 single-crystal as well as powder X-ray 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 (Dornberger-Schiff & Ďurovič, 1975a, b), Mg-vermiculite (Weiss & Ďurovič, 1980), the mica group (Dornberger-Schiff, 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, Dornberger-Schiff & 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 & Dornberger-Schiff, 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.

#### 9.2.2.3.1.1. General geometry

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Tetrahedral and octahedral sheets are the fundamental, two-dimensionally periodic structural units, common to all hydrous phyllosilicates. Any tetrahedral sheet consists of (Si,Al,Fe3+,Ti4+)O4 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 c-centred 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 Mg2+, Al3+, Fe2+, Fe3+, but also Li+, Mn2+ 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: homo-octahedral [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 ;6meso-octahedral [two octahedral sites are occupied by the same kind of crystallo­chemical entity, the third by a different one, in an ordered way; symmetry ; and hetero-octahedral [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).

 Figure 9.2.2.6 | top | pdf |(a) Tetrahedral sheet in a normal projection. Open circles: basal oxygen atoms, circles with black dots: apical oxygen atoms and tetrahedral cations. Hexagonal and orthohexagonal basis vectors and symbolic figures (ditrigons) for pictorial representation of these sheets are also shown. (b) Octahedral sheet. Open and shaded circles belong to the lower and the upper oxygen atomic planes, respectively; small triangles denote octahedral sites. Triangular stars on the right are the symbolic figures for pictorial representation of these sheets: the two triangles correspond to the lower and upper basis of any octahedron, respectively.

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).

 Figure 9.2.2.7 | top | pdf |Two possible combinations of one tetrahedral and one octahedral sheet (a) by shared apical O atoms, (b) by hydrogen bonds (side projection).
 Figure 9.2.2.8 | top | pdf |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 homo-octahedral sheets) of the nine vectors shown in Fig. 9.2.2.9.

 Figure 9.2.2.9 | top | pdf |The nine possible displacements in the structures of polytypes of phyllosilicates. The individual vectors are designated by their conventional numerical characters and the signs +, −. The zero displacement <*> is not indicated. The relations of these vectors to the basis vectors a1, a2 or a, b are evident.

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 homo-octahedral 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 NOc = 18 (h centring of Oc). For Figs. 9.2.2.10(b) and (c), Z = 3. Similar relations can be derived for meso- and hetero-octahedral sheets as well as for the pair (Tet; Tet) in the talc–pyrophyllite group.

 Figure 9.2.2.10 | top | pdf |The NFZ relations (a) for the pair tetrahedral sheet–homo-octahedral sheet (with shared apical O atoms), (b), (c) for the pair homo-octahedral sheet–tetrahedral sheet (by hydrogen bonds). The sheets are represented by their symbolic figures; some relevant symmetry elements are also indicated. One of the possible positions (labelled 1) is drawn by full, the other two (2, 3) by broken lines.

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 kort 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:

• The serpentine–kaolin group: The general structural principle is shown in Fig. 9.2.2.11 . The structures belong to category II (9.2.2.2.7.2). In the homo-octahedral family, there are 12 non-equivalent (16 non-congruent) MDO polytypes (any two polytypes belonging to an enantiomorphous pair are equivalent but not congruent); in the meso-octahedral family, there are 36 non-equivalent (52 non-congruent) MDO polytypes. These sets are identical with the sets of standard or regular polytypes derived by Bailey (for references see Bailey, 1980) (trioctahedral) and by Zvyagin (1967) (dioctahedral and trioctahedral). The individual polytypes can be ranged into four groups (subfamilies, which are individual OD groupoid families), each with a characteristic superposition structure.

 Figure 9.2.2.11 | top | pdf |Stereopair showing the sequence of sheets in the structures of the serpentine–kaolin group (kaolinite-1A, courtesy Zoltai & Stout, 1985),
• The mica group: The general structural principle is shown in Fig. 9.2.2.12 . The structures belong to category IV. There are 6 non-equivalent (8 non-congruent) homo-octahedral MDO polytypes, 14 (22) meso-octahedral, and 36 (60) hetero-octahedral MDO polytypes. The homo-octahedral MDO polytypes are identical with those derived by Smith & Yoder (1956); meso-octahedral MDO polytypes include also those with non-centrosymmetric 2:1 layers (Tet; Oc; Tet); some of these have also been derived by Zvyagin et al. (1979). The individual polytypes can be ranged into two groups (subfamilies). For complex polytypes and growth mechanisms, see Baronnet (1975, Baronnet, 1986).

 Figure 9.2.2.12 | top | pdf |Stereopair showing the sequence of sheets in the structures of the mica group (muscovite-2M1, courtesy of Zoltai & Stout, 1985).
• The talc–pyrophyllite group: The general structural principle is shown in Fig. 9.2.2.13 . The structures belong to category I. There are 10 (12) MDO polytypes in the talc family (homo-octahedral) and 22 (30) MDO polytypes in the pyrophyllite family (meso-octahedral); some of these have been derived also by Zvyagin et al. (1979). The structures can be ranged into two groups (subfamilies). For more details, see also Evans & Guggenheim (1988).

 Figure 9.2.2.13 | top | pdf |Stereopair showing the sequence of sheets in the structures of the talc–pyrophyllite group (pyrophyllite-2M, courtesy of Zoltai & Stout, 1985).
• The chlorite–vermiculite group: There are two kinds of octahedral sheets in these structures: the Oc sandwiched between two Tet and the interlayer (Fig. 9.2.2.14 ). The structures belong to category IV. Any Oc can be independently homo-, meso-, or hetero-octahedral, and thus, theoretically, there are nine families here. Although vermiculites have a crystal chemistry different from chlorites, they can be, from the symmetry point of view, treated together. There are 20 (24) homo-homo-octahedral, 44 (60) homo-meso-octahedral and 164 (256) meso-meso-octahedral MDO polytypes (the first prefix refers to the 2:1 layer, the second to the interlayer); the other families have not yet been treated. Some of these polytypes have also been derived by other authors (for references, see Bailey, 1980; Zvyagin et al., 1979).

 Figure 9.2.2.14 | top | pdf |Stereopair showing the sequence of sheets in the structures of the chlorite–vermiculite group (chlorite-1M, courtesy of Zoltai & Stout, 1985).

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.

#### 9.2.2.3.1.2. Diffraction pattern and identification of individual polytypes

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Owing to the trigonal symmetry of the basic structural units and their stacking mode, the single-crystal 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:

• (1) Diffractions 00l (or 000l), always sharp and common to all polytypes of a family including all its subfamilies. They are indicative of the mineral group, but useless for the identification of polytypes.

 Figure 9.2.2.15 | top | pdf |Clinographic projection of the general scheme of a single-crystal diffraction pattern of hydrous phyllosilicates. Family diffractions are indicated by open circles and correspond in this case to a rhombohedral superposition structure. Only the part with l ≥ 0 is shown.
 Figure 9.2.2.16 | top | pdf |Normal projection of the general scheme of a single-crystal diffraction pattern of hydrous phyllosilicates. Rows of family diffractions are indicated by open circles; the h, k indices refer to hexagonal (below) and orthogonal (above) axial systems.
• (2) The remaining diffractions with kort ≡ 0 (mod 3), always sharp and common to all polytypes of the same subfamily.

• (3) All other diffractions: sharp only for periodic polytypes, otherwise present on diffuse rods parallel to c*. These are characteristic of individual polytypes. Diffractions 0kl – if sharp – are common to all polytypes of the family with the same bc projection.

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 kort ≡ 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.

#### 9.2.2.3.2. Stibivanite Sb2VO5

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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 SbO2–O–VO–O–SbO2 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.

 Figure 9.2.2.17 | top | pdf |The structure of stibivanite-2M. 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 VO5 square pyramids (Pyr) parallel to bc, with layer symmetry alternating with sheets containing chains of distorted SbO3 tetrahedron-like 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.

 Figure 9.2.2.18 | top | pdf |The two kinds of OD layers in the stibivanite family (after Merlino et al., 1989).
 Figure 9.2.2.19 | top | pdf |The structure of stibivanite-2O (after Merlino et al., 1989).

The polytypism of stibivanite is reflected in its OD character: the two kinds of sheets Pyr and Tet correspond to two kinds of non-polar 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).

#### 9.2.2.3.3. γ-Hg3S2Cl2

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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 (Dornberger-Schiff, 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 .

 Figure 9.2.2.20 | top | pdf |The structural principle of γ-Hg3S2Cl2. 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, MDO1, is periodic after two layers and has symmetry A2/m. The second, MDO2, is periodic after four layers, has symmetry F2/m and basis vectors 2a, b, c. Domains of MDO1 were present in all crystals (the system of diffuse maxima, mentioned above), but some of them also contained small proportions of MDO2.

Later, single crystals of pure periodic MDO1 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.

#### 9.2.2.3.4. Remarks for authors

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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, non-space-group 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.

#### 9.2.2.4. List of some polytypic structures

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A few examples of some less-common polytypic structures published up to 1994 are listed below:

 Minerals: McGillite (Iijima, 1982), tridymite (Wennemer & Thompson, 1984), pyrosmalite (Takéuchi, Ozawa & Takahata, 1983), zirconolite (White, Segall, Hutchison & Barry, 1984), Ti-biotite (Zhukhlistov, Zvyagin & Pavlishin, 1990), diamond (Phelps, Howard & Smith, 1993), scholzite (Taxer, 1992), fiedlerite (Merlino, Pasero & Perchiazzi, 1994), penkvilskite (Merlino, Pasero, Artioli & Khomyakov, 1994), lengebachite – non-commensurate structure (Makovický, Leonardsen & Moelo, 1994). Inorganic compounds: Borates with general formula RAl3(BO3)4, where R = Y, Nd, Gd (Belokoneva & Timchenko, 1983), BaCrO3 (Chamberland, 1983), hexacyanocomplexes of transition metals (Jagner, 1985), chromium iron carbides (Kowalski, 1985), Fe1−xS (Kuban, 1985), hexagonal copper(I) ferrite (Effenberger, 1991), PbS·18TiS2 – a modulated structure (van Smaalen & de Boer, 1992), α-LiNH4SO4 (Tomaszewski, 1992), fullerene C60 (de Boer, van Smaalen, Petříček, Dušek, Verheijen & Meijer, 1994). Organic compounds: Oxalates (Fichtner-Schmittler, 1979), primetine (Jarchow & Schmalle, 1985), 2-hydroxy-4-methoxy-2H-1,4-benzoxazin-3-one, C9H9NO4 (Kutschabsky, Kretschmer, Schrauber, Dathe & Schneider, 1986), carbam­azepine dihydrate (Reck & Dietz, 1986), piroxicam (Reck, Dietz, Laban, Günther, Bannier & Höhne, 1988), E-octa­decanoic acid (Kaneko, Sakashita, Kobayashi, Kitagawa, Matsuura & Suzuki, 1994). Metals, intermetallic compounds and alloys: Alloys are treated in a monograph by Nikolin (1984); a lecture note by Amelinckx (1986) gives details of high-resolution electron microscopy and examples of the investigation of some alloys but also of other polytypic structures. Special papers: Li metal (Schwarz & Blaschko, 1990), Zr(FeCr)2 Laves phases (Burany & Northwood, 1991), doped Co–W and Co–Mo alloys (Nikolin, Babkevich, Izdkovskaya & Petrova, 1993). Further data are given in articles by Bailey et al. (1977), Dornberger-Schiff (1979), Zvyagin (1988), Baronnet (1992) and Zorkii & Nesterova (1993).

High-resolution 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 La4Ti3O12–LaTiO3 (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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