Stacking disorder in close-packed structures

Jagodzinski, H.; Frey, F.

doi:10.1107/97809553602060000564

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 423-425 | 1 | 2 |

Section 4.2.4.2.1. Stacking disorder in close-packed structures

H. Jagodzinski^a and F. Frey^b

^aInstitut für Kristallographie und Mineralogie, Universität, Theresienstrasse 41, D-8000 München 2, Germany, and ^bInstitut für Kristallographie und Mineralogie, Universität, Theresienstrasse 41, D-8000 München 2, Germany

4.2.4.2.1. Stacking disorder in close-packed structures

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From an historical point of view stacking disorder in close-packed systems is most important. The three relevant positions of ordered layers are represented by the atomic coordinates $[|0, 0|, |{{1} \over {3}}, {{2} \over {3}}|, |{{2} \over {3}}, {{1} \over {3}}|]$ in the hexagonal setting of the unit cell, or simply by the figures 1, 2, 3 in the same sequence. Structure factors $[F_{1}, F_{2}, F_{3}]$ refer to the corresponding positions of the same layer: $[\eqalign{F_{2} &= F_{1} \exp \{2 \pi i(h - k)/3\},\cr F_{3} &= F_{1} \exp \{- 2 \pi i(h - k)/3\},}]$ hence $[F_{1} = F_{2} = F_{3}\quad \hbox{if } h - k \equiv 0\ (\hbox{mod } 3).]$ According to the above discussion the said indices define the reciprocal-lattice rows exhibiting sharp reflections only, as long as the distances between the layers are exactly equal. The symmetry conditions caused by the translation are normally: $[\eqalign{&p_{1} = p_{2} = p_{3},\qquad p_{11} = p_{22} = p_{33}, \cr &p_{12} = p_{23} = p_{31},\quad p_{13} = p_{21} = p_{32}.}]$ For the case of close packing of spheres and some other problems any configuration of m layers determining the a posteriori probability $[p_{\mu \mu'} ({\bf m}), \mu = \mu']$ , has a symmetrical counterpart where μ is replaced by $[\mu' + 1]$ (if $[\mu' = 3, \mu' + 1 = 1]$ ).

In this particular case $[p_{12}({\bf m}) = p_{13}({\bf m})]$ , and equivalent relations generated by translation.

Nearest-neighbour interactions do not lead to an ordered structure if the principle of close packing is obeyed (no pairs in equal positions) (Hendricks & Teller, 1942; Wilson, 1942). Extension of the interactions to next-but-one or more neighbours may be carried out by introducing the method of matrix multiplication developed by Kakinoki & Komura (1954, 1965), or the method of overlapping clusters (Jagodzinski, 1954). The latter procedure is outlined in the case of interactions between four layers. A given set of three layers may occur in the following 12 combinations: $[\eqalignno{&123, 231, 312\hbox{;}\quad 132, 213, 321\hbox{;}\cr &121, 232, 313\hbox{;}\quad 131, 212, 323.}]$ Since three of them are equivalent by translation, only four representatives have to be introduced: $[123\hbox{;} \quad 132\hbox{;} \quad 121\hbox{;} \quad 131.]$ In the following the new indices 1, 2, 3, 4 are used for these four representatives for the sake of simplicity.

In order to construct the statistics layer by layer the next layer must belong to a triplet starting with the same two symbols with which the preceding one ended, e.g. 123 can only be followed by 231, or 232. In a similar way 132 can only be followed by 321 or 323. Since both cases are symmetrically equivalent the probabilities $[\alpha_{1}]$ and $[1 - \alpha_{1}]$ are introduced. In a similar way 121 may be followed by 212 or 213 etc. For these two groups the probabilities $[\alpha_{2}]$ and $[1 - \alpha_{2}]$ are defined. The different translations of groups are considered by introducing the phase factors as described above. Hence, the matrix for the characteristic equation may be set up as follows. As representative cluster of each group is chosen that one having the number 1 at the centre, e.g. 312 is representative for the group 123, 231, 312; in a similar way 213, 212 and 313 are the remaining representatives. Since this arrangement of three layers is equivalent by translation, it may be assumed that the structure of the central layer is not influenced by the statistics to a first approximation. The same arguments hold for the remaining three groups. On the other hand, the groups 312 and 213 are equivalent by rotation only. Consequently their structure factors may differ if the influence of the two neighbours has to be taken into account. A different situation exists for the groups 212 and 313 which are correlated by a centre of symmetry, which causes different corresponding structure factors. It should be pointed out, however, that the structure factor is invariant as long as there is no influence of neighbouring layers on the structure of the central layer. The latter is often observed in close-packed metal structures, or in compounds like ZnS, SiC and others. For the calculation of intensities $[p_{\mu} p_{\mu \mu'}, F_{\mu} F_{\mu'}^{+}]$ is needed.

According to the following scheme of sequences any sequence of pairs is correlated with the same phase factor for $[FF^{+}]$ due to translation, if both members of the pair belong to the same group. Consequently the phase factor may be attached to the sequence probability such that $[FF^{+}]$ remains unchanged, and the group may be treated as a single element in the statistics. In this way the reduced matrix for the solution of the characteristic equation is given by

$[F_{\mu}]$	$[F_{\mu'}^{+}]$
	(1)	(2)	(3)	(4)
	$[312, 123 (\varepsilon^{+}), 231 (\varepsilon)]$	$[212, 323 (\varepsilon^{+}), 131 (\varepsilon)]$	$[213, 321 (\varepsilon^{+}), 132 (\varepsilon)]$	$[313, 121 (\varepsilon^{+}), 232 (\varepsilon)]$
(1) $[312, 123 (\varepsilon), 231 (\varepsilon^{+})]$	$[\alpha_{1} \varepsilon^{+}]$	0	0	$[(1 - \alpha_{1}) \varepsilon^{+}]$
(2) $[212, 323 (\varepsilon), 131 (\varepsilon^{+}) ]$	$[(1 - \alpha_{2}) \varepsilon^{+} ]$	0	0	$[\alpha_{2} \varepsilon^{+}]$
(3) $[213, 321 (\varepsilon), 132 (\varepsilon^{+})]$	0	$[(1 - \alpha_{2}) \varepsilon]$	$[\alpha_{1} \varepsilon]$	0
(4) $[313, 121 (\varepsilon), 232 (\varepsilon^{+})]$	0	$[\alpha_{2} \varepsilon]$	$[(1 - \alpha_{2})\varepsilon]$	0

There are three solutions of the diffraction problem:

(1) If (mod 3), $[\varepsilon = +1]$ , there are two quadratic equations: $[\eqalign{\lambda^{2} - (\alpha_{1} + \alpha_{2})\lambda - 1 + \alpha_{1} + \alpha_{2} &= 0\cr \lambda^{2} - (\alpha_{1} - \alpha_{2})\lambda + 1 - \alpha_{1} - \alpha_{2} &= 0\cr} \eqno(4.2.4.9)]$ with solutions $[\displaylines{\lambda_{1} = 1, \qquad \lambda_{2} = \alpha_{1} + \alpha_{2} - 1\cr \hfill \lambda_{3/4} = {{\alpha_{1} - \alpha_{2}} \over {2}} \pm \left[{{(\alpha_{1} - \alpha_{2})^{2}} \over {4}} - 1 + \alpha_{1} - \alpha_{2}\right]^{1/2}. \hfill (4.2.4.10)}]$

$[\lambda_{1}]$ and $[\lambda_{2}]$ are identical with the solution of the asymmetric case of two kinds of layers [cf. equation (4.2.4.6)]. They yield sharp reflections for l = integer, and diffuse ones in a position determined by the sequence probabilities $[\alpha_{1}]$ and $[\alpha_{2}]$ (position either l = integer, l = ½ +integer, respectively). The remaining two characteristic values may be given in the form $[\lambda = |\lambda| \exp \{2 \pi i\varphi\}]$ , where φ determines the position of the reflection. If the structure factors of the layers are independent of the cluster, $[\lambda_{2}, \lambda_{3}, \lambda_{4}]$ become irrelevant because of the new identity of the F's (no diffuse scattering). Weak diffuse intensities on the lattice rows (mod 3) may be explained in terms of this influence.
(2) The remaining two solutions for $[\varepsilon = \exp \{\pm 2 \pi i(h - k)/3\}]$ are equivalent, and result in the same characteristic values. They have been discussed explicitly in the literature; the reader is referred to the papers of Jagodzinski (1949a,b,c, 1954).

In order to calculate the intensities one has to reconsider the symmetry of the clusters, which is different from the symmetry of the layers. Fortunately, a threefold rotation axis is invariant against the translations, but this is not true for the remaining symmetry operations in the layer if there are any more. Since we have two pairs of inequivalent clusters, namely 312, 213 and 212, 313, there are only two different a priori probabilities $[p_{1} = p_{3}]$ and $[p_{2} = p_{4} = {{1} \over {2}} (1 - 2p_{1})]$ .

The symmetry conditions of the new clusters may be determined by means of the so-called `probability trees' described by Wilson (1942) and Jagodzinski (1949b, pp. 208–214). For example: $[p_{11} = p_{33}]$ , $[p_{22} = p_{44}]$ , $[p_{13} = p_{31}]$ , $[p_{24} = p_{42}]$ etc.

It should be pointed out that clusters 1 and 3 describe a cubic arrangement of three layers in the case of simple close packing, while clusters 2 and 4 represent the hexagonal close packing. There may be a small change in the lattice constant c perpendicular to the layers. Additional phase factors then have to be introduced in the matrix for the characteristic equation, and a recalculation of the constants is necessary. As a consequence, the reciprocal-lattice rows $[(h - k) \equiv 0\;(\hbox{mod } 3)]$ become diffuse if $[l \neq 0]$ , and the diffuseness increases with l. A similar behaviour results for the remaining reciprocal-lattice rows.

The final solution of the diffraction problem results in the following general intensity formula: $[\eqalignno{I({\bf H}) &= L(h, k) N \textstyle\sum\limits_{\nu} \{A_{\nu}({\bf H}) (1 - |\lambda_{\nu}|^{2}) &\cr &\quad \times [1 - 2|\lambda_{\nu}| \cos 2 \pi (l - \varphi_{\nu}) + |\lambda_{\nu}|^{2}]^{-1} &\cr &\quad - 2 B_{\nu}({\bf H})|\lambda_{\nu}| \sin 2 \pi (l - \varphi_{\nu}) &\cr &\quad \times [1 - 2|\lambda_{\nu}| \cos 2 \pi (l - \varphi_{\nu}) + |\lambda_{\nu}|^{2}]^{-1}\}. &(4.2.4.11)}]$ Here $[A_{\nu}]$ and $[B_{\nu}]$ represent the real and imaginary part of the constants to be calculated with the aid of the boundary conditions of the problem. The first term in equation (4.2.4.11) determines the symmetrical part of a diffuse reflection with respect to the maximum, and is completely responsible for the integrated intensity. The second term causes an antisymmetrical contribution to intensity profiles but does not influence the integrated intensities. These general relations enable a semi-quantitative interpretation of the sharp and diffuse scattering in any case, without performing the time-consuming calculations of the constants which may only be done in more complicated disorder problems with the aid of a computer program evaluating the boundary conditions of the problem.

This can be carried out with the aid of the characteristic values and a linear system of equations (Jagodzinski, 1949a,b,c), or with the aid of matrix formalism (Kakinoki & Komura, 1954; Takaki & Sakurai, 1976). As long as only the line profiles and positions of the reflections are required, these quantities may be determined experimentally and fitted to characteristic values of a matrix. The size of this matrix is given by the number of sharp and diffuse maxima observed, while $[|\lambda_{\nu}|]$ and $[\exp\{2 \pi i \varphi_{\nu}\}]$ may be found by evaluating the line width and the position of diffuse reflections. Once this matrix has been found, a semi-quantitative model of the disorder problem can be given. If a system of sharp reflections is available, the averaged structure can be solved as described in Section 4.2.3.2. The determination of the constants of the diffraction problem is greatly facilitated by considering the intensity modulation of diffuse scattering, which enables a phase determination of structure factors to be made under certain conditions.

The theory of closed-packed structures with three equivalent translation vectors has been applied very frequently, even to systems which do not obey the principle of close-packing. The first quantitative explanation was published by Halla et al. (1953). It was shown there that single crystals of C₁₈H₂₄ from the same synthesis may have a completely different degree of order. This was true even within the same crystal. Similar results were found for C, Si, CdI₂, CdS₂, mica and many other compounds. Quantitative treatments are less abundant [e.g. CdI₂: Martorana et al. (1986); MX₃ structures: Conradi & Müller (1986)]. Special attention has been paid to the quantitative study of polytypic phase transformations in order to gain information about the thermodynamical stability or the mechanism of layer displacements, e.g. Co (Edwards & Lipson, 1942; Frey & Boysen, 1981), SiC (Jagodzinski, 1972; Pandey et al., 1980a,b,c), ZnS (Müller, 1952; Mardix & Steinberger, 1970; Frey et al., 1986) and others.

Certain laws may be derived for the reduced integrated intensities of diffuse reflections. `Reduction' in this context means a division of the diffuse scattering along l by the structure factor, or the difference structure factor if $[\langle F \rangle \neq 0]$ . This procedure is valuable if the number of stacking faults rather than the complete solution of the diffraction problem is required.

The discussion given above has been made under the assumption that the full symmetry of the layers is maintained in the statistics. Obviously, this is not necessarily true if external lower symmetries influence the disorder. An important example is the generation of stacking faults during plastic deformation. Problems of this kind need a complete reconsideration of symmetries. Furthermore, it should be pointed out that a treatment with the aid of an extended Ising model as described above is irrelevant in most cases. Simplified procedures describing the diffuse scattering of intrinsic, extrinsic, twin stacking faults and others have been described in the literature. Since their influence on structure determination can generally be neglected, the reader is referred to the literature for additional information.

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International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 423-425