International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C. ch. 9.2, pp. 752760

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.
In a closepacked 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 closepacked 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 twodimensional hexagonal unit cell shown in Fig. 9.2.1.2(b). The pointgroup 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 closepacked 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.
A threedimensional closepacked structure results from stacking the hexagonal closepacked 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 closepacked layers. There are thus infinite possibilities for closepacked 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 2^{n−1}.
In all the closepacked 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 closepacked 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).
In the Ramsdell notation, closepacked 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 closepacked structures observed in metals are listed in Table 9.2.1.1 in terms of all the notations.

Frequently, the positions of one kind of atom or ion in inorganic compounds, such as SiC, ZnS, CdI_{2}, 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 closepacked structures though they may not be ideally close packed. In the closepacked compounds, the size and coordination number of the smaller atom/ion may require that its closepacked neighbours in the neighbouring layers do not touch each other.
Threedimensional close packings of spheres have two kinds of voids (Azaroff, 1960):
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).
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 closepacked 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 SiC6H, which is the most common modification.
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 solidstate structural transformation to 6H (Jagodzinski, 1972; Krishna & Marshall, 1971a, Krishna & Marshall, 1971b) through a nonrandom 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, crystalstructure refinement has revealed variation in the thickness of Si–C double layers depending on their environment (de Mesquita, 1967).

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 ZnS2H 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).
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 closepacked 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 CdI_{2} 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 CdI_{2} 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 CdI_{2}, 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 CdI_{2} is 0.805, which is very different from the ideal value (Verma & Krishna, 1966). The structure of PbI_{2}, which also displays a large number of polytypes, is analogous to CdI_{2} with one important difference. Here, the distances between two I layers with and without an intervening Pb layer are quite different (Trigunayat & Verma, 1976).
The crystal structure of GaSe consists of fourlayered slabs, each of which contains two closepacked 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.
It can be seen from Fig. 9.2.1.2(a) that a stacking of two or more layers in the closepacked 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 closepacked 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, closepacked 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 6_{3} 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 6_{3} 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).
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 closepacked 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 nonprimitive when a hexagonal lattice is referred to rhombohedral axes and vice versa (Buerger, 1953). In closepacked 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 nonprimitive hexagonal unit cell can be chosen by adopting the axes A_{1}, A_{2},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.

A rhombohedral lattice (a_{1}, a_{2}, a_{3}) referred to hexagonal axes (A_{1}, A_{2}, 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 facecentred cubic unit cell. The relationship between the facecentred 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 closepacked layers are thus parallel to the {111} planes in the cubic close packing.
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 . CdI_{2} structures can have a centre of symmetry on octahedral voids, but cannot have a symmetry plane perpendicular to [00.1]. CdI_{2} can therefore have five possible space groups: P3m1, , R3m, , and . Cubic symmetry is not possible in CdI_{2} on account of the presence of Cd atoms, the sequence representing a 6R structure.
From the Zhdanov symbols of a closepacked 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 6_{3} 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 522225 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 CdI_{2} (Jain & Trigunayat, 1977a, Jain & Trigunayat, 1977b).
The different layer stackings (polytypes) of the same material have identical a and b parameters of the direct lattice. The a*b* reciprocallattice 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 reciprocallattice 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 Xray diffraction photographs. The reciprocallattice rows hk.l can be classified into two categories according as h − k = 0 mod3 or . Since the atoms in an nH or nR structure lie on three symmetry axes , , and , the structure factor F_{hkl} can be split into three parts: where , , , and , , are the z coordinates of atoms at A, B, and C sites, respectively. For h − k = 0mod3, which is zero except when . Hence, the reflections 00.l, 11.l, 30.l, etc., for which h − k = 0mod3, will be extinguished except when . Thus, only those hk.l reciprocallattice rows for which h − k 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).
When the structure has a hexagonal lattice, the positions of spots are symmetrical about the zero layer line on the caxis 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 6_{3} 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 Xray diffraction. For a structure with a rhombohedral lattice, the positions of Xray diffraction spots are not symmetrical about the zero layer line because the hexagonal unit cell is nonprimitive 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 reciprocallattice 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, reciprocallattice rows with ξ = a* will have spots for l = 3n±1 causing the apparent `doubling'.
In crystals of layer structures, such as CdI_{2}, where aaxis oscillation photographs are normally taken, the identification of the rhombohedral lattice is performed by checking for the noncoincidence 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).
The number of layers, n, in the hexagonal unit cell can be found by determining the c parameter from the caxis rotation or oscillation photographs and dividing this by the layer spacing h for that compound which can be found from reflections with h − k = 0mod3. The density of reciprocallattice 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 closepacked layers (stacking faults), this will effectively make the c parameter infinite and lead to the production of characteristic continuous diffuse streaks along reciprocallattice rows parallel to c* for reflections with h − k 0mod3 (Wilson, 1942). It is therefore difficult to distinguish by Xray 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 reciprocallattice 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).
For an nH or 3nR polytype, the n closepacked 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 spacegroup considerations discussed in Subsection 9.2.1.4 and 9.2.1.5 can considerably reduce the number of trial structures.
When the shortperiod structures act as `basic structures' for the generation of longperiod polytypes, the number of trial structures is considerably reduced since the crystallographic unit cells of the latter will contain several units of the smallperiod 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)_{n}32 and (33)_{n}34 based on SiC6H, empirical rules framed by Mitchell (1953) and Krishna & Verma (1962) can allow the direct identification of the layerstacking 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 CdI_{2} and PbI_{2} 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 coworkers have successfully used lattice imaging in conjunction with Xray diffraction for determining the structures of longperiod polytypes of SiC that are not based on a simple basic structure. After recording Xray diffraction patterns, single crystals of these polytypes were crushed to yield electronbeamtransparent flakes. The one and twodimensional 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 Xray intensities for the proposed structures (Dubey & Singh, 1978; Rai, Singh, Dubey & Singh, 1986).
Direct methods for the structure determination of polytypes from Xray data have also been suggested by several workers (Tokonami & Hosoya, 1965; DornbergerSchiff & FarkasJahnke, 1970; FarkasJahnke & DornbergerSchiff, 1969) and have been reviewed by FarkasJahnke (1983). These have been used to derive the structures of ZnS, SiC, and TiS_{1.7} polytypes. These methods are extremely sensitive to experimental errors in the intensities.
The two alternative positions for the stacking of successive closepacked 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 closepacked martensites of cobalt, noblemetalbased and certain ironbased 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 nonatomic 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, layerdisplacement 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 closepacked layers (Prasad & Lele, 1971; Pandey & Krishna, 1976b) can provide a very compact and unique way of representing intrinsic fault configurations even in longperiod 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 0type 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 righthand 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:
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.


Statistical distribution of stacking faults in closepacked structures introduces disorder along the stacking axis of the closepacked layers. As a result, one observes on a singlecrystal 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 h − k 0mod3. A complete description of the real structure of such onedimensionally disordered polytypes requires knowledge of the average structure as well as a statistical specification of the fluctuations due to stacking faults in the electrondensity 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 onedimensionally 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 layerbylayer 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 solidstate transformations, their distribution is nonrandom (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 solidstate 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 nonrandom 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 nonrandomly 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 onedimensionally disordered crystals due to special problems in the measurement of diffuse intensities using a singlecrystal diffractometer, stemming from incidentbeam divergence, finite size of the detector slit, and multiple scattering. The problems due to incidentbeam 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 onedimensionally disordered stackings must await proper understanding of all other factors that may influence the true diffraction profiles.
References
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