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. 758-760

Section 9.2.1.8. Stacking faults in close-packed structures

D. Pandeyc and P. Krishnab

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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