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. 752-760

Section 9.2.1. Layer stacking in close-packed structures

D. Pandeyc and P. Krishnab

9.2.1. Layer stacking in close-packed structures

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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.

9.2.1.1. 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[link] . 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),[link] where the size of the spheres is reduced for clarity. There are three symmetry planes with indices [(1\bar2.0)], [(\bar21.0)], and (11.0) defined with respect to the smallest two-dimensional hexagonal unit cell shown in Fig. 9.2.1.2(b)[link]. 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[link] by the letter `A'. This A layer has two types of triangular interstices, one with the apex angle up [(\bigtriangleup)] and the other with the apex angle down [(\bigtriangledown)]. 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 [\bigtriangleup] and [\bigtriangledown] interstices be designated as `B' and `C', respectively. In the cell of the A layer shown in Fig. 9.2.1.1[link] (a = b = diameter of the sphere and γ = 120°), the three positions A, B, and C on projection have coordinates (0, 0), [({1\over3},{2\over3})], and [({2\over3},{1\over3})], respectively.

[Figure 9.2.1.1]

Figure 9.2.1.1 | top | pdf |

The close packing of equal spheres in a plane.

[Figure 9.2.1.2]

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 [\sqrt{2/3}], where h is the separation between two close-packed layers and a is the diameter of the spheres (Verma & Krishna, 1966[link]). Deviations from the ideal value of the axial ratio are common, especially in hexagonal metals (Cottrell, 1967[link]). The arrangement of spheres described above provides the highest packing density of 0.7405 in the ideal case for an infinite lattice (Azaroff, 1960[link]). There are, however, other arrangements of a finite number of equal spheres that have a higher packing density (Boerdijk, 1952[link]).

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[link]).

In the Zhdanov notation, use is made of the stacking offset vector s and its opposite −s, which cause, respectively, a cyclic [(A\rightarrow B\rightarrow C\rightarrow A)] or anticyclic [(A\rightarrow C\rightarrow B\rightarrow A)] shift of layers in the same plane. The vector s can be either [(1/3)[1\bar 100]], [(1/3)[01\bar10]], or [(1/3)[\bar 1010]]. Zhdanov (1945[link]) 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[link]) 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[link] 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[\ldots] 2 2H 11 h Mg
ABC, A[\ldots] 3 3C [\infty] c Cu
ABCB, A[\ldots] 4 4H 22 hc La
ABCBCACAB, A[\ldots] 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[link]):

  • (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[link]). 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[link]). 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]

    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[link]) 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[link]). 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[link]).

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[link] depicts the structure of SiC-6H, which is the most common modification.

[Figure 9.2.1.4]

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[link]; Pandey & Krishna, 1982a[link]). Table 9.2.1.2[link] 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[link]; Krishna & Marshall, 1971a[link], Krishna & Marshall, 1971b[link]) through a non-random insertion of stacking faults (Pandey, Lele & Krishna, 1980a[link], Pandey, Lele & Krishna, 1980b[link], Pandey, Lele & Krishna, 1980c[link]; Kabra, Pandey & Lele, 1986[link]). 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[link]), 0.8179, and 0.8165 (Taylor & Jones, 1960[link]) 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[link]).

Table 9.2.1.2| top | pdf |
List of SiC polytypes with known structures in order of increasing periodicity(after Pandey & Krishna, 1982a[link])

PolytypeStructure (Zhdanov sequence)PolytypeStructure (Zhdanov sequence)
2H 11 57H (23)93333
3C [\infty] 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[link]). ZnS crystals that occur as minerals usually correspond to the wurtzite [(\quad/AB/\ldots)] and the sphalerite [(\quad/ABC/\ldots)] modifications. The structural transformation between the 2H and 3C structures of ZnS is known to be martensitic in nature (Sebastian, Pandey & Krishna, 1982[link]; Pandey & Lele, 1986b[link]). The h/a ratio for ZnS-2H is 0.818, which is somewhat different from the ideal value (Verma & Krishna, 1966[link]). The structure of the stackings in polytypic AgI is analogous to those in SiC and ZnS (Prager, 1983[link]).

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[link] . 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[link]). 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 [A\gamma B\,C\alpha B \ldots] and [A\gamma B\, A\gamma B], respectively. In addition, this material also displays a number of polytype modifications of large repeat periods (Trigunayat & Verma, 1976[link]; Pandey & Krishna, 1982a[link]). 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[link]). 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[link]).

[Figure 9.2.1.5]

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[link]).

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[link]). 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 [A\beta\beta A] or [A\gamma\gamma A]. 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: [/A\beta\beta A\, B\gamma\gamma B/\ldots], [/A\beta\beta A\, B\alpha\alpha B/\ldots] and [/A\beta\beta A\, C\beta\beta C/]. Periodicities containing up to 21 unit slabs have been reported for GaSe (see Terhell, 1983[link]). 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)[link] 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[link]). The lowest symmetry of a completely arbitrary periodic stacking sequence of close-packed layers is shown in Fig. 9.2.1.2(b)[link]. 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[link]). This is possible if the layers in the unit cell are stacked in special ways. For example, a 6H stacking sequence [/ABCACB/\ldots] 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 [nH/nR] with n odd will therefore necessarily belong to the trigonal system and can have either a hexagonal or a rhombohedral lattice (Verma & Krishna, 1966[link]).

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[link]).

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 [a=b\neq c], α = β = 90°, and γ = 120°. In the primitive unit cell of the hexagonal close-packed structure [/AB/\ldots] shown in Fig. 9.2.1.6[link] , there are two spheres associated with each lattice point, one at 0, 0, 0 and the other at [1\over3], [2\over3], [1\over2]. 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 [\neq 90]°. 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[link]). In close-packed structures, it is generally convenient to refer both hexagonal and rhombohedral lattices to hexagonal axes. Fig. 9.2.1.7[link] shows a rhombohedral lattice in which the primitive cell is defined by the rhombohedral axes [a_1, a_2, a_3]; 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; [{2\over3},{1\over3},{1\over3}]; and [{1\over3}, {2\over3}, {2\over3}]. If this rhombohedral lattice is rotated through 60° around [00.1], the hexagonal unit cell will then be centred at [{1\over3}, {2\over3},{1\over3}] and [{2\over3}, {1\over3},{2\over3}]. 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 [A\rightarrow C\rightarrow B\rightarrow A] (obverse setting) or by a cyclic shift of layers [A\rightarrow B\rightarrow C\rightarrow A] (reverse setting) in the direction of z increasing (Verma & Krishna, 1966[link]). Evidently, n must be a multiple of 3 for nR structures.

[Figure 9.2.1.6]

Figure 9.2.1.6 | top | pdf |

The primitive unit cell of the 2H close packing.

[Figure 9.2.1.7]

Figure 9.2.1.7 | top | pdf |

A rhombohedral lattice (a1, a2, a3) referred to hexagonal axes (A1, A2, C)(after Buerger, 1953[link]).

In the special case of the close packing [/ABC/\ldots] [with the ideal axial ratio of [\sqrt{(2/3)}]], 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[link] . The threefold axis of the rhombohedral unit cell coincides with one of the [\langle 111\rangle] 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]

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[link]) that consistent combinations of the possible symmetry elements in close packing of equal spheres can give rise to eight possible space groups: P3m1, [P\bar3m1], [P\bar6m2], [P6_3mc], [P6_3/mmc], R3m, [R\bar 3m], and Fm3m. The last space group corresponds to the special case of cubic close packing [/ABC/\ldots]. The tetrahedral arrangement of Si and C in SiC does not permit either a centre of symmetry [(\bar1)] or a plane of symmetry (m) perpendicular to [00.1]. SiC structures can therefore have only four possible space groups P3m1, R3m1, [P6_3mc], and [F\bar43m]. 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, [P\bar3m], R3m, [R\bar3m], and [P6_3mc]. Cubic symmetry is not possible in CdI2 on account of the presence of Cd atoms, the sequence [/A\gamma BC\beta AB\alpha C/] 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[link]). 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[link]). Thus the 15R polytype of SiC is written as (23) rather than (23)3.

As described in detail by Verma & Krishna (1966[link]), 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[link], Jain & Trigunayat, 1977b[link]).

9.2.1.7. Structure determination of close-packed layer stackings

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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[link] . 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[link] 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, [\bar11.l], [\bar10.l], [0\bar 1.l], and [1\bar1.l] 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 [\pm1\,{\rm mod}\,3]. Since the atoms in an nH or nR structure lie on three symmetry axes [A:[00.1]_{00}], [B:[00.1]_{{1\over3},-{1\over3}}], and [C:[00.1]_{-{1\over3},{1\over3}}], the structure factor Fhkl can be split into three parts: [F_{hkl}=P+Q\exp[2\pi i(h-k)/3]+R\exp[-2\pi i(h-k)/3],]where [P=\sum_{z_A}\exp(2\pi ilz_A/n)], [Q=\sum_{z_B}\exp(2\pi ilz_B/n)], [R=\sum_{z_C}\exp(2\pi i l z_C/n)], and [z_A/n], [z_B/n], [z_C/n] are the z coordinates of atoms at A, B, and C sites, respectively. For hk = 0mod3, [F_{hkl}=P+Q+R=\sum^{n=-1}_{z=0}\exp(2\pi ilz/n),]which is zero except when [l=0,n,2n,\ldots]. Hence, the reflections 00.l, 11.l, 30.l, etc., for which hk = 0mod3, will be extinguished except when [l=0,n,2n,\ldots]. Thus, only those hk.l reciprocal-lattice rows for which hk [\neq] 0mod3 carry information about the stacking sequence and contain in general reflections with [l=0,1,2,\ldots], 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[link]).

[Figure 9.2.1.9]

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 [6_3] 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 [-h+k+l\neq3n] [(\pm n=0,1,2,\ldots)]. 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 [10.\bar2], [10.\bar5], [10.\bar8], etc. The zero layer line will therefore divide the distance between the nearest spots on either side (namely 10.1 and [10.\bar2]) 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, [\bar11.l], and [0\bar1.l] identical with each other but different from the other identical set, [01.l], [\bar10.l], and [1\bar1.l]. The extinction condition for the second set requires l = 3n − 1, i.e. l = 2, 5, 8, and [\bar1,\bar4,\bar7], 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[link]).

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[link]). If the structure contains a random stacking disorder of close-packed layers (stacking faults), this will effectively make the c parameter infinite [({\bf c}^*\rightarrow0)] and lead to the production of characteristic continuous diffuse streaks along reciprocal-lattice rows parallel to c* for reflections with hk [\neq] 0mod3 (Wilson, 1942[link]). 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[link]). 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[link]). Identity periods as large as 594 layers have been resolved by this method (Honjo, Miyake & Tomita, 1950[link]). Synchrotron radiation has been used for taking Laue photographs of ZnS polytypes (Steinberger, Bordas & Kalman, 1977[link]).

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 [2^{n-1}] 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[link] and 9.2.1.5[link] 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[link]) and Krishna & Verma (1962[link]) 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[link]). 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[link], Pandey & Krishna, 1976a[link]). For CdI2 and PbI2 polytypes, the possible Zhdanov numbers are 1, 2 and 3 (Pandey & Krishna, 1983[link]; Pandey, 1985[link]). 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[link]), 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[link]).

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[link]; Rai, Singh, Dubey & Singh, 1986[link]).

Direct methods for the structure determination of polytypes from X-ray data have also been suggested by several workers (Tokonami & Hosoya, 1965[link]; Dornberger-Schiff & Farkas-Jahnke, 1970[link]; Farkas-Jahnke & Dornberger-Schiff, 1969[link]) and have been reviewed by Farkas-Jahnke (1983[link]). 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[link]; Kabra, Pandey & Lele, 1988a[link]; Nishiyama, 1978[link]; Pandey, 1988[link]).

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[link]) and Pandey & Krishna (1982b[link]).

Frank (1951[link]) 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[link]). 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[link]) 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[link], b[link], c[link]; Kabra, Pandey & Lele, 1986[link]). 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[link]; Pandey & Krishna, 1976b[link]) can provide a very compact and unique way of representing intrinsic fault configurations even in long-period structures (Pandey, 1984b[link]). We shall briefly explain this notation in relation to one hexagonal (6H) and one rhombohedral (9R) structure.

In the 6H (ABCACB,[\ldots] 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[link]). 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 [A\rightarrow B\rightarrow C\rightarrow A] shift), the perfect 6H structure can be written as [Scheme cbch9.2scheme1.tif] There are six crystallographically equivalent ways of writing this structure with the first layer in position A: (i) [A_0B_1C_2A_3C_4B_5]; (ii) [A_1B_2C_3B_4A_5C_0]; (iii) [A_2B_3A_4C_5B_0C_1]; (iv) [A_3C_4B_5A_0B_1C_2]; (v) [A_4C_5B_0C_1A_2B_3]; and (vi) [A_5C_0A_1B_2C_3B_4]. 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 [(ABCACB,\ldots)] structure. All these intrinsic fault configurations can be described by symbols like [I_{r,s}], 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[link]). 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: [Scheme cbch9.2scheme2.tif]

Reverse: [Scheme cbch9.2scheme3.tif]

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 ([{\bar0}] type) in the h configuration is related to the next layer through an anticyclic shift. Tables 9.2.1.3[link] and 9.2.1.4[link] 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
[\ldots]A B C A C B A0[\vdots]C0 A B C B A C [\ldots] I0, 0
[\ldots]A B C A C B A0[\vdots]C1 A B A C B C [\ldots] I0, 1
[\ldots]A B C A C B A0[\vdots]C2 A C B A B C [\ldots] I0, 2
[\ldots]A B C A C B A0[\vdots]C3 B A C A B C [\ldots] I0, 3
[\ldots]A B C A C B A0[\vdots]C4 B A B C A C [\ldots] I0, 4
[\ldots]A B C A C B A0[\vdots]C5 B C A B A C [\ldots] I0, 5
[\ldots]A B C A C B A B1[\vdots] A0 B C A C B A [\ldots] I1, 0
[\ldots]A B C A C B A B1[\vdots] A1 B C B A C A [\ldots] I1, 1
[\ldots]A B C A C B A B1[\vdots] A2B A C B C A [\ldots] I1, 2
[\ldots]A B C A C B A B1[\vdots] A3C B A B C A [\ldots] I1, 3
[\ldots]A B C A C B A B1[\vdots] A4C B C A B A [\ldots] I1, 4
[\ldots]A B C A C B A B1[\vdots] A5C A B C B A [\ldots] I1, 5
[\ldots]A B C A C B A B C2[\vdots]B0 C A B A C B [\ldots] I2, 0
[\ldots]A B C A C B A B C2[\vdots]B1C A C B A B [\ldots] I2, 1
[\ldots]A B C A C B A B C2[\vdots]B2C B A C A B [\ldots] I2, 2
[\ldots]A B C A C B A B C2[\vdots]B3 A C B C A B [\ldots] I2, 3
[\ldots]A B C A C B A B C2[\vdots]B4 A C A B C B [\ldots] I2, 4
[\ldots]A B C A C B A B C2[\vdots] B5 A B C A C B[\ldots] 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
[\ldots]A B A C A C B C B A0[\vdots]C0 A C B C B A B A[\ldots] I0, 0
[\ldots]A B A C A C B C B A0[\vdots]C1 B A B A C A C B [\ldots] I0, 1
[\ldots]A B A C A C B C B A0[\vdots]C2 B C B A B A C A[\ldots] I0, 2
[\ldots] A B A C A C B C B A0[\vdots][C_{\bar 0}] B C A C A B A B[\ldots] [I_{0, {\bar 0}}]
[\ldots]A B A C A C B C B A0[\vdots][C_{\bar 1}] A B A B C B C A[\ldots] [I_{0, {\bar 1}}]
[\ldots]A B A C A C B C B A0[\vdots][C_{\bar 2}] A C A B A B C B[\ldots] [I_{0, {\bar 2}}]
[\ldots]A B A C A C B C B A B1[\vdots]C0 A C B C B A B A[\ldots] I1, 0
[\ldots]A B A C A C B C B A B1[\vdots]C1 B A B A C A C B[\ldots] I1, 1
[\ldots]A B A C A C B C B A B1[\vdots]C2 B C B A B A C A[\ldots] I1, 2
[\ldots]A B A C A C B C B A B1[\vdots][C_{\bar 0}] B C A C A B A B[\ldots] [I_{1, {\bar 0}}]
[\ldots]A B A C A C B C B A B1[\vdots][C_{\bar 1}] A B A B C B C A[\ldots] [I_{1, {\bar 1}}]
[\ldots]A B A C A C B C B A B1[\vdots][C_{\bar 2}] A C A B A B C B[\ldots] [I_{1, {\bar 2}}]
[\ldots]A B A C A C B C B A B A2[\vdots]B0 C B A B A C A C[\ldots] I2, 0
[\ldots]A B A C A C B C B A B A2[\vdots]B1 A C A C B C B A[\ldots] I2, 1
[\ldots]A B A C A C B C B A B A2[\vdots]B2 A B A C A C B C[\ldots] I2, 2
[\ldots]A B A C A C B C B A B A2[\vdots][B_{\bar 0}] A B C B C A C A[\ldots] [I_{2, {\bar 0}}]
[\ldots]A B A C A C B C B A B A2[\vdots][B_{\bar 1}] C A C A B A B C[\ldots] [I_{2, {\bar 1}}]
[\ldots]A B A C A C B C B A B A2[\vdots][B_{\bar 2}] C B C A C A B A[\ldots] [I_{2, {\bar 2}}]

Note: [I_{0,\bar0}] and [I_{1,\bar1}], [I_{0,\bar1}] and [I_{1,\bar2}], [I_{0,\bar2}] and [I_{2,\bar1}], 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 [\neq] 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[link]).

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[link]) or during plastic deformation (Paterson, 1952[link]). 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[link], b[link], c[link]; Pandey & Lele, 1986a[link], b[link]; Kabra, Pandey & Lele, 1986[link]). 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[link]). Since the pioneering work of Wilson (1942[link]), 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[link]; Jagodzinski, 1949a[link], b[link]; Kakinoki & Komura, 1954[link]; Johnson, 1963[link]; Prasad & Lele, 1971[link]; Cowley, 1976[link]; Pandey, Lele & Krishna, 1980a[link], b[link]). 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[link]).

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[link]) and suitable correction factors have recently been derived by Pandey, Prasad, Lele & Gauthier (1987[link]). 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.

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