One-dimensional disorder of ordered layers

Jagodzinski, H.; Frey, F.

doi:10.1107/97809553602060000564

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 421-425 | 1 | 2 |

Section 4.2.4.2. One-dimensional disorder of ordered layers

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. One-dimensional disorder of ordered layers

| top | pdf |

As has been pointed out above, it is often useful to start the interpretation of diffuse scattering by checking the diffraction pattern with respect to the dimensionality of the disorder concerned. Since each disordered direction in the crystal demands a violation of the corresponding Laue condition, this question may easily be answered by looking at the diffuse scattering. Diffuse streaks in reciprocal space are due to a one-dimensional violation of the Laue conditions, and will be called one-dimensional disorder. This kind of order is typical for layer structures, but it is frequently observed in cases where several sequences of layers do not differ in the interactions of next-nearest neighbours. Typical examples are structures which may be described in terms of close packing, e.g. hexagonal and cubic close packing.

For a quantitative interpretation of diffuse streaks we need one-dimensional correlation functions, which may uniquely be determined if a single independent correlation function is active. According to equation (4.2.3.28) Fourier transformation yields the information required. In all other cases a specific model has to be suggested for a full interpretation of diffuse streaks. Another comment seems to be necessary: disorder parameters can be defined uniquely only if the diffraction pattern allows for a differentiation between long-range and short-range order. This question can at least partly be answered by studying the line width of sharp reflections with a very good resolution. Since integrated intensities of sharp reflections have to be separated from the diffuse scattering, this question is of outstanding importance in most cases. Inclusion of diffuse parts in the diffraction pattern during intensity integration of sharp reflections may lead to serious errors in the interpretation of the average structure.

The existence of diffuse streaks in more than one direction of reciprocal space means that the diffraction problem is no longer one-dimensional. Sometimes the problem may be treated independently, if the streaks are sharp, and no interference effects may be observed in the diffraction pattern in areas where the diffuse streaks do overlap. In all other cases there are correlations between the various directions of one-dimensional disorder which may be determined with the aid of a model covering more than one of the pertinent directions of disorder.

Before starting the discussion of the quantitative solution of the one-dimensional problem, some remarks should be made on the usefulness of quantitative disorder parameters. It is well known from statistical thermodynamics that a one-dimensional system cannot show long-range order above $[T = 0\hbox{ K}]$ . Obviously, this statement is in contradiction with many experimental observations where long-range order is realized even in layer structures. The reason for this behaviour is given by the following arguments which are valid for any structure. Let us assume a structure with strong interactions at least in two directions. From the theoretical treatment of the two-dimensional Ising model it is known that such a system shows long-range order below a critical temperature $[T_{c}]$ . This statement is true even if the layer is finite, although the strict thermodynamic behaviour is not really critical in the thermodynamical sense. A three-dimensional crystal can be constructed by adding layer after layer. Since each layer has a typical two-dimensional free energy, the full statistics of the three-dimensional crystal may be calculated by introducing a specific free energy for the various stackings of layers. Obviously, this additional energy has to include terms describing potential and entropic energies as well. They may be formally developed into contributions of next, overnext etc. nearest neighbours. Apparently, the contribution to entropy must include configurational and vibrational parts which are strongly coupled. As long as the layers are finite, there is a finite probability of a fault in the stacking sequence of layers which approaches zero with increasing extension of the layers. Consequently, the free energy of a change in the favourite stacking sequence becomes infinite quadratically with the size of the layer. Therefore, the crystal should be either completely ordered or disordered; the latter case can only be realized if the free energies of one or more stacking sequences are exactly equal (very rare, but possible over a small temperature range of phase transformations). An additional positive entropy associated with a deviation from the periodic stacking sequence may lead to a kind of competition between entropy and potential energy, in such a way that periodic sequences of faults result. Obviously, this situation occurs in the transition range of two structures differing only in their stacking sequence. On the other hand, one must assume that defects in the stacking sequence may be realized if the size of the layers is small. This situation occurs during crystal growth, but one should remember that the number of stacking defects should decrease with increasing size of the growing crystal. Apparently, this rearrangement of layers may be suppressed as a consequence of relaxation effects. The growth process itself may influence the propagation of stacking defects and, consequently, the determination of stacking-fault probabilities, aiming at an interpretation of the chemical bonding seems to be irrelevant in most cases.

The quantitative solution of the diffraction problem of one-dimensional disorder follows a method similar to the Ising model. As long as next-nearest neighbours alone are considered, the solution is very simple only if two possibilities of structure factors are to be taken into account. Introducing the probability of equal pairs 1 and 2, α, one arrives at the known solution for the a priori probability $[p_{\mu}]$ and a posteriori probabilities $[p_{\mu \mu'}({\bf m})]$ , respectively. In the one-dimensional Ising model with two spins and the interaction energies $[(U \mp \Delta U) / k_{B}T]$ , defining the pair probability $[[\alpha = p_{11} ({\bf 1})]]$ $[\alpha = {\exp \{\pm \Delta U / k_{B}T\}\over [\exp \{+ \Delta U / k_{B}T\} + \exp \{- \Delta U / k_{B}T\}]}]$ the full symmetry is $[p_{1} = p_{2} = {1\over 2}]$ , and $[p_{11} ({\bf m}) = p_{22} ({\bf m})]$ .

Consequently: $[p_{12} ({\bf m}) = p_{22} ({\bf m}) = 1 - p_{11} ({\bf m}).]$ The scattered intensity is given by $[{I} ({\bf H}) = L (h, k) \textstyle\sum\limits_{{\bf m}} \langle FF_{m}^{+}\rangle (N - |m|) \exp \{- 2\pi iml\}, \eqno(4.2.4.1)]$ where $[{\bf m} = m{\bf c}]$ , $[N = \hbox{number of unit cells}]$ in the c direction and $[\langle FF^{+}\rangle]$ depends on $[\lambda_{1}]$ , $[\lambda_{2}]$ which are the eigenvalues of the matrix $[\pmatrix{\alpha &1 - \alpha\cr 1 - \alpha &\alpha\cr}.]$ From the characteristic equation $[\lambda^{2} - 2\alpha \lambda - 1 + 2\alpha = 0 \eqno(4.2.4.2)]$ one has $[\lambda_{1} = 1\hbox{;} \quad \lambda_{2} = 2\alpha - 1. \eqno(4.2.4.2a)]$ $[\lambda_{1}]$ describes a sharp Bragg reflection (average structure) which need not be calculated. Its intensity is simply proportional to $[\langle F({\bf H})\rangle]$ . The second characteristic value yields a diffuse reflection in the same position if the sign is positive $[(\alpha\gt 0.5)]$ , and in a position displaced by $[{1\over 2}]$ in reciprocal space if the sign is negative $[(\alpha\lt 0.5)]$ . Because of the symmetry conditions $[p_{11} ({\bf m})]$ only is needed; it may be determined with the aid of the boundary conditions $[p_{11} ({\bf 0}) = 1, \quad p_{11} ({\bf 1}) = \alpha,]$ and the general relation that $[p_{\mu \mu'} ({\bf m})]$ is given by $[p_{\mu \mu'} ({\bf m}) = c'_{\mu \mu'} \lambda_{1}^{m} + c''_{\mu \mu'} \lambda_{2}^{m}.]$ The final solution of our problem yields simply: $[\eqalign{p_{11} ({\bf m}) &= {\textstyle{1\over 2}} + {\textstyle{1\over 2}} \lambda_{2}^{m} = p_{22} ({\bf m}),\cr p_{12} ({\bf m}) &= {\textstyle{1\over 2}} - {\textstyle{1\over 2}} \lambda_{2}^{m} = p_{21} ({\bf m}).}]$ The calculation of the scattered intensity is now performed with the general formula $[\eqalignno{I({\bf H}) &= L(h, k) \textstyle\sum\limits_{m} \textstyle\sum\limits_{\mu, \, \mu'} p_{\mu} p_{\mu \mu'} ({\bf m}) F_{\mu} F_{\mu'}^{+} (N - |m|) &\cr &\quad \times \exp \{- 2 \pi iml\}. &(4.2.4.3)}]$ Evaluation of this expression yields $[\eqalignno{I({\bf H}) &= L(h, k) \textstyle\sum\limits_{m} (N - |m|) \exp \{- 2 \pi iml\} &\cr &\quad \times [|{\textstyle{1\over 2}} (F_{1} + F_{2})|^{2} \lambda_{1}^{m} + |{\textstyle{1\over 2}} (F_{1} - F_{2})|^{2} \lambda_{2}^{m}]. &(4.2.4.4)}]$ Since the characteristic solutions of the problem are real: $[\eqalignno{I({\bf H}) &= L({\bf h}) |(F_{1} + F_{2})/2|^{2} &\cr &\quad + L(h, k) |(F_{1} - F_{2})/2|^{2} &\cr &\quad \times {1 - |\lambda_{2}|^{2}\over 1 - 2 \lambda_{2} \cos 2 \pi l + |\lambda_{2}|^{2}}. &(4.2.4.5)}]$ This particle size effect has been neglected in (4.2.4.5). This result confirms the fact mentioned above that the sharp Bragg peaks are determined by the averaged structure factor and the diffuse one by its mean-square deviation.

For the following reason there are no examples for quantitative applications: two different structures generally have different lattice constants; hence the original assumption of an undisturbed lattice geometry is no longer valid. The only case known to the authors is the typical lamellar structure of plagioclases, reported by Jagodzinski & Korekawa (1965). The authors interpret the well known `Schiller effect' as a consequence of optical diffraction. Hence, the size of the lamellae is of the order of 2000 Å. This long-period superstructure cannot be explained in terms of next-nearest-neighbour interactions. In principle, however, the diffraction effects are similar: instead of the diffuse peak as described by the second term in equation (4.2.4.5), satellites of first and second order, etc. accompanying the Bragg peaks are observed. The study of this phenomenon (Korekawa & Jagodzinski, 1967) has not so far resulted in a quantitative interpretation.

Obviously, the symmetry relation used in the formulae discussed above is only valid if the structures described by the $[F_{\mu}]$ are related by symmetries such as translations, rotations or combinations of both. The type of symmetry has an important influence on the diffraction pattern.

(1) Translation parallel to the ordered layers

If the translation vector between the two layers in question is such that $[2 \Delta {\bf r}]$ is a translation vector parallel to the layer, there are two relevant structure factors $[F_{1}, F_{2} = F_{1} \exp (2 \pi i{\bf H} \cdot \Delta {\bf r}).]$ $[{\bf H} \cdot \Delta {\bf r}]$ may be either an integer or an integer $[+\ {\textstyle{1\over 2}}]$ . Since any integer may be neglected because of the translation symmetry parallel to the layer, we have $[F_{1} = F_{2}]$ in the former case, and $[F_{1} = - F_{2}]$ in the latter. As a consequence either the sharp reflections given in equation (4.2.4.4) vanish, or the same is true for the diffuse ones. Hence, the reciprocal lattice may be described in terms of two kinds of lattice rows, sharp and diffuse, parallel to the reciprocal coordinate l.

Disorder of this type is observed very frequently. One of the first examples was wollastonite, CaSiO₃, published by Jefferey (1953). Here the reflections with are sharp Bragg peaks without any diffuse scattering. Diffuse streaks parallel to (h00), however, are detected for . In the light of the preceding discussion, the translation vector is $[{1\over 2}{\bf b}]$ , and the plane of ordered direction (plane of intergrowth of the two domains) is (100). Hence the displacement is parallel to the said plane. Since the intensity of the diffuse lines does not vary according to the structure factor involved, the disorder cannot be random. The maxima observed are approximately in the position of a superstructure, generated by large domains without faults in the stacking sequence, mutually displaced by $[{1\over 2}{\bf b}]$ (antiphase domains). This complicated ordering behaviour is typical for 1D order and may easily be explained by the above-mentioned fact that an infinitely extended interface between two domains causes an infinite unfavourable energy (Jagodzinski, 1964b, p. 188). Hence, a growing crystal should become increasingly ordered. This consideration explains why the agreement between a 1D disorder theory and experiment is often so poor.

Examples where more than one single displacement vector occur are common. If these are symmetrically equivalent all symmetries have to be considered. The most important cases of displacements differing only by translation are the well known close-packed structures (see below). A very instructive example is the mineral maucherite (approximately Ni₄As₃). According to Jagodzinski & Laves (1947) the structure has the following disorder parameters: interface (001), displacement vectors [000], $[[{1\over 2}00]]$ , $[[0{1\over 2}0]]$ , $[[{1\over 2}{1\over 2}0]]$ . From equation (4.2.4.5) we obtain: $[\langle F({\bf H})\rangle = [1 + \exp \{\pi ih\} + \exp \{\pi ik\} + \exp \{\pi i(h + k)\}]/4.]$ Hence there are sharp reflections for $[h, k = \hbox{ even}]$ , and diffuse ones otherwise. Further conclusions may be drawn from the average structure.
(2) Translation perpendicular to the ordered layers

If the translation is c/2 the structure factors are: $[\eqalign{F_{2} &= F_{1} \exp \{2 \pi il\}\cr F_{1} &= F_{2} \hbox{ for } l = \hbox{even}\cr F_{1} &= - F_{2} \hbox{ for } l = \hbox{odd}.}]$ There are sharp and diffuse reflections on all reciprocal-lattice rows discussed above.

Since the sharp and diffuse reflections occur on the same reciprocal line there is a completely different behaviour compared with the preceding case. In general, a component of any displacement vector perpendicular to the interface gives rise to a change in chemical composition as shown in the next example: in a binary system consisting of A and B atoms with a tendency towards an alternating arrangement of A and B layers, any fault in the sequence BABAB|BABAB|B increases the number of B atoms (or A atoms). Generally such kinds of defects will lead to an interface with a different lattice constant, at least in the direction perpendicular to the interface. Consequently the exact displacement vectors of $[{{1} \over {2}}]$ , $[{{1} \over {3}}]$ , $[{{1} \over {4}}]$ are rare. Since ordered structures should be realized in the 1D case, incommensurate superstructures will occur which are very abundant during ordering processes. An interesting example has been reported and interpreted by Cowley (1976a,b) where the displacement vector has a translational period of $[{{1} \over {4}}]$ perpendicular to the plane of intergrowth. Reflections 00l and 22l with are sharp, all remaining reflections more or less diffuse. Since the maxima $[(\bar{1}11), (\bar{1}33)]$ show a systematically different behaviour, there is also a displacement component parallel to the plane of intergrowth in question. The semi-quantitative interpretation has been given in his paper.
(3) Rotations

The discussion concerning layers related by a twofold rotation parallel to c may easily be made by simply considering their structure factors. Since the layers do not obey the twofold symmetry their structure factors are generally different; unless they equalize accidentally there are sharp and diffuse reflections according to the values of $[\langle F\rangle]$ and $[\Delta F]$ , respectively. Obviously, $[F_{1} = F_{2}]$ is valid only if ; consequently there is just one reciprocal-lattice row free from diffuse scattering.
(4) Asymmetric case

In the asymmetric case the symmetry conditions used above are no longer valid: $[p_{1} \neq p_{2}, p_{12} \neq p_{21}, p_{11} \neq p_{22}.]$ But there is one condition which may be derived from the invariance of the numbers of pairs in the relevant and its opposite direction: $[p_{\mu} p_{\mu \mu'} ({\bf m}) = p_{\mu'} p_{\mu' \mu} (- {\bf m}).]$ This equation requires that $[p_{\mu \mu'}({\bf m})]$ is not necessarily symmetric in m. The calculation of characteristic values yields $[\lambda_{1} = 1, \qquad \lambda_{2} = (\alpha_{1} + \alpha_{2}) - 1. \eqno(4.2.4.6)]$ The a priori probabilities are now different from $[{{1} \over {2}}]$ , and may be calculated by considering $[p_{\mu \mu'}({\bf m}) \rightarrow p_{\mu'} ({\bf m} \rightarrow \infty)]$ : $[p_{1} = \alpha_{1}/(\alpha_{1} + \alpha_{2})\hbox{;} \qquad p_{2} = \alpha_{2}/(\alpha_{1} + \alpha_{2}).]$ The intensity is given by an expression very similar to (4.2.4.5): $[\eqalignno{{I}({\bf H}) &= L({\bf h})|[\alpha_{1}/(\alpha_{1} + \alpha_{2})] F_{1} + [\alpha_{2}/(\alpha_{1} + \alpha_{2})] F_{2}|^{2} &\cr &\quad+ L(h, k)|[\alpha_{1}/(\alpha_{1} + \alpha_{2})] F_{1} - [\alpha_{2}/(\alpha_{1} + \alpha_{2})] F_{2}|^{2} &\cr &\quad\times (1 - |\lambda_{2}|^{2})/(1 - 2\lambda_{2} \cos 2\pi l + |\lambda_{2}|^{2}). &(4.2.4.7)}]$ Again there are sharp Bragg reflections and diffuse ones in the same positions, or in a displaced position depending on the sign of $[\lambda_{2}]$ .

From a discussion of the next-nearest-neighbour Ising model one may conclude that the detailed study of the qualitative behaviour of sharp and diffuse reflections may give additional information on the symmetry of the layers involved.

In the case of translations between neighbouring layers not fulfilling the condition $[{\bf h} \cdot {\bf r}]$ = integer, where r is parallel to the layer, more than two structure factors have to be taken into account. If $[n{\bf h} \cdot {\bf r}]$ = integer, where n is the smallest integer fulfilling the said condition, n different structure factors have to be considered. The characteristic equation has formally to be derived with the aid of an $[n \times n]$ matrix containing internal symmetries which may be avoided by adding the phase factors $[\varepsilon = \exp \{2 \pi i{\bf H} \cdot {\bf r}/n\}]$ , $[\varepsilon^{+} = \exp \{-2 \pi i{\bf H} \cdot {\bf r}/n\}]$ to the probability of pairs. The procedure is allowed if the displacements r and −r are admitted only for neighbouring layers. The matrix yielding the characteristic values may then be reduced to $[\pmatrix{\alpha_{1}\varepsilon &(1 - \alpha_{1})\varepsilon^{+}\cr (1 - \alpha_{2})\varepsilon &\alpha_{2}\varepsilon^{+}\cr}]$ and yields the characteristic equation $[\lambda^{2} - \lambda (\alpha_{1}\varepsilon + \alpha_{2}\varepsilon^{+}) - 1 + \alpha_{1} + \alpha_{2} = 0. \eqno(4.2.4.8)]$ Equation (4.2.4.8) gives sharp Bragg reflections for $[{\bf H} \cdot {\bf r}/n]$ = integer; the remaining diffuse reflections are displaced corresponding to the phase of the complex characteristic value. Equation (4.2.4.8) has been used in many cases. Qualitative examples are the mixed-layer structures published by Hendricks & Teller (1942). An example of a four-layer-type structure is treated by Dubernat & Pezerat (1974). A first quantitative treatment with good agreement between theory and experimental data (powder diffraction) has been given by Dorner & Jagodzinski (1972) for the binary system TiO₂–SnO₂. In the range of the so-called spinodal decomposition the chemical composition of the two domains and the average lengths of the two types of domains could be determined. Another quantitative application was reported by Jagodzinski & Hellner (1956) for the transformation of $[\hbox{RhSn}_{2}]$ into a very complicated mixed-layer type. A good agreement of measured and calculated diffuse scattering (asymmetric line profiles, displacement of maxima) could be found over a wide angular range of single-crystal diffraction.

4.2.4.2.1. Stacking disorder in close-packed structures

| top | pdf |

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.

References

Conradi, E. & Müller, U. (1986). Fehlordnung bei Verbindungen mit Schichtstrukturen. II. Analyse der Fehlordnung in Wismuttriiodid. Z. Kristallogr. 176, 263–269.Google Scholar

Cowley, J. M. (1976a). Diffraction by crystals with planar faults. I. General theory. Acta Cryst. A32, 83–87.Google Scholar

Cowley, J. M. (1976b). Diffraction by crystals with planar faults. II. Magnesium fluorogermanate. Acta Cryst. A32, 88–91.Google Scholar

Dorner, C. & Jagodzinski, H. (1972). Entmischung im System SnO₂–TiO₂. Krist. Tech. 7, 427–444.Google Scholar

Dubernat, J. & Pezerat, H. (1974). Fautes d'empilement dans les oxalates dihydratés des métaux divalents de la série magnésienne (Mg, Fe, Co, Ni, Zn, Mn). J. Appl. Cryst. 7, 387–393.Google Scholar

Edwards, O. S. & Lipson, H. (1942). Imperfections in the structure of cobalt. I. Experimental work and proposed structure. Proc. R. Soc. London Ser. A, 180, 268–277.Google Scholar

Frey, F. & Boysen, H. (1981). Disorder in cobalt single crystals. Acta Cryst. A37, 819–826.Google Scholar

Frey, F., Jagodzinski, H. & Steger, W. (1986). On the phase transformation zinc blende to wurtzite. Bull. Minéral. Crystallogr. 109, 117–129.Google Scholar

Halla, F., Jagodzinski, H. & Ruston, W. R. (1953). One-dimensional disorder in dodecahydrotriphenylene, C₁₈H₂₄. Acta Cryst. 6, 478–488.Google Scholar

Hendricks, S. B. & Teller, E. (1942). X-ray interference in partially ordered layer lattices. J. Chem. Phys. 10, 147–167.Google Scholar

Jagodzinski, H. (1949a). Eindimensionale Fehlordnung und ihr Einfluß auf die Röntgeninterferenzen. I. Berechnung des Fehlordnungsgrades aus den Röntgeninterferenzen. Acta Cryst. 2, 201–208.Google Scholar

Jagodzinski, H. (1949b). Eindimensionale Fehlordnung und ihr Einfluß auf die Röntgeninterferenzen. II. Berechnung der fehlgeordneten dichtesten Kugelpackungen mit Wechsel wirkungen der Reichweite 3. Acta Cryst. 2, 208–214.Google Scholar

Jagodzinski, H. (1949c). Eindimensionale Fehlordnung und ihr Einfluß auf die Röntgeninterferenzen. III. Vergleich der Berechungen mit experimentellen Ergebnissen. Acta Cryst. 2, 298–304.Google Scholar

Jagodzinski, H. (1954). Der Symmetrieeinfluß auf den allgemeinen Lösungsansatz eindimensionaler Fehlordnungsprobleme. Acta Cryst. 7, 17–25.Google Scholar

Jagodzinski, H. (1964b). Diffuse disorder scattering by crystals. In Advanced methods of crystallography, edited by G. N. Ramachandran, pp. 181–219. London: Academic Press.Google Scholar

Jagodzinski, H. (1972). Transformation from cubic to hexagonal silicon carbide as a solid state reaction. Kristallografiya, 16, 1235–1246. [Translated into English in Sov. Phys. Crystallogr. 16, 1081–1090.]Google Scholar

Jagodzinski, H. & Hellner, E. (1956). Die eindimensionale Phasenumwandlung des RhSn₂. Z. Kristallogr. 107, 124–149.Google Scholar

Jagodzinski, H. & Korekawa, M. (1965). Supersatelliten im Beugungsbild des Labradorits (Ca₂Na)(Si₂Al)₂O₈. Naturwissenschaften, 52, 640–641.Google Scholar

Jagodzinski, H. & Laves, R. (1947). Über die Deutung der Entmischungsvorgänge in Mischkristallen unter besonderer Berücksichtigung der Systeme Aluminium–Kupfer und Aluminium–Silber. Z. Metallkd. 40, 296–305.Google Scholar

Jefferey, J. W. (1953). Unusual X-ray diffraction effects from a crystal of wollastonite. Acta Cryst. 6, 821–826.Google Scholar

Kakinoki, J. & Komura, Y. (1954). Intensity of X-ray diffraction by a one-dimensionally disordered crystal. I. General derivation in the cases of `Reichweite' s = 0 and s = 1. J. Phys. Soc. Jpn, 9, 169–183.Google Scholar

Kakinoki, J. & Komura, Y. (1965). Diffraction by a one-dimensionally disordered crystal. I. The intensity equation. Acta Cryst. 19, 137–147.Google Scholar

Korekawa, M. & Jagodzinski, H. (1967). Die Satelliten-reflexe des Labradorits. Schweiz. Mineral. Petrogr. Mitt. 47, 269–278.Google Scholar

Mardix, S. & Steinberger, I. T. (1970). Tilt and structure transformation in ZnS. J. Appl. Phys. 41, 5339–5341.Google Scholar

Martorana, A., Marigo, A., Toniolo, L. & Zenetti, R. (1986). Stacking faults in the β-form of magnesium dichloride. Z. Kristallogr. 176, 1–12.Google Scholar

Müller, H. (1952). Die eindimensionale Umwandlung Zinkblende-Wurtzit und die dabei auftretenden Anomalien. Neues Jahrb. Mineral. Abh. 84, 43–76.Google Scholar

Pandey, D., Lele, S. & Krishna, P. (1980a). X-ray diffraction from one dimensionally disordered 2H-crystals undergoing solid state transformation to the 6H structure. I. The layer displacement mechanism. Proc. R. Soc. London Ser. A, 369, 435–439.Google Scholar

Pandey, D., Lele, S. & Krishna, P. (1980b). X-ray diffraction from one dimensionally disordered 2H-crystals undergoing solid state transformation to the 6H structure. II. The deformation mechanism. Proc. R. Soc. London Ser. A, 369, 451–461.Google Scholar

Pandey, D., Lele, S. & Krishna, P. (1980c). X-ray diffraction from one dimensionally disordered 2H-crystals undergoing solid state transformation to the 6H structure. III. Comparison with experimental observations on SiC. Proc. R. Soc. London Ser. A, 369, 463–477.Google Scholar

Takaki, Y. & Sakurai, K. (1976). Intensity of X-ray scattering from one-dimensionally disordered crystal having the multilayer average structure. Acta Cryst. A32, 657–663.Google Scholar

Wilson, A. J. C. (1942). Imperfections in the structure of cobalt. II. Mathematical treatment of proposed structure. Proc. R. Soc. London Ser. A, 180, 277–285.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 421-425