International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 4.2, pp. 495507

Any structure analysis of disordered structures should start with a qualitative interpretation of diffuse scattering. This may be achieved with the aid of Fourier transforms and convolutions. A thorough mathematical treatment of Fourier transforms is given in Chapter 1.3 of this volume; here we give a simple short overview of the practical use of Fourier transforms, convolutions, their algebraic operations and examples of functions which are frequently used in diffraction physics (see, e.g., Patterson, 1959; Jagodzinski, 1987). For simplicity, the following modified notation is used in this section: functions in real space are represented by lowercase letters, e.g. a(r), b(r), … except for F(r) [also, more frequently, denoted as ρ(r)] and P(r), which are used as general symbols for a structure and the Patterson function, respectively; functions in reciprocal space are represented by capital letters A(H), B(H); and r and H are general vectors in real and reciprocal space, respectively. H, K, L denote continuous variables in reciprocal space; integer values are given by the commonly used symbols h, k, l. is the scalar product H · r; dr and dH indicate integrations in three dimensions in real and reciprocal space, respectively. Even for Xrays, the electron density will generally be replaced by the scattering potential a(r). Consequently, anomalous contributions to scattering may be included if complex functions a(r) are admitted. In the neutron case a(r) refers to a quasipotential. Using this notation we obtain the scattered amplitude(constant factors are omitted).
a(r) and A(H) are reversibly and uniquely determined by Fourier transformation. Consequently, equations (4.2.3.1) may simply be replaced by , where the doubleheaded arrow represents the two integrations given by (4.2.3.1) and means: A(H) is the Fourier transform of a(r) and vice versa. The following relations may easily be derived from (4.2.3.1): where β is a scalar quantity.
On the other hand, the multiplication of two functions does not yield a relation of similar symmetrical simplicity: (the laws of convolution and multiplication).
For simplicity, the complete convolution integral is abbreviated as . Since ,and vice versa. The convolution operation is commutative in either space.
The distribution law is valid for the convolution as well: The associative law of multiplication does not hold if mixed products (convolution and multiplication) are used: From equations (4.2.3.1) one has(the law of displacements).
Since symmetry operations are well known to crystallographers in reciprocal space as well, the law of inversion is only mentioned here: Consequently, if , then . In order to calculate the intensity, the complex conjugate is needed: Equations (4.2.3.9) yield the relationship (`Friedel's law') if a(r) is a real function. The multiplication of a function with its conjugate is given bywith Note that is not valid if a(r) is complex. Consequently . This is shown by evaluating ,Equation (4.2.3.11) is very useful for the determination of the contribution of anomalous scattering to diffuse reflections.
Most of the diffusediffraction phenomena observed may be interpreted qualitatively or even semiquantitatively in a very simple manner using a limited number of important Fourier transforms, which are given below.
From these basic concepts the generally adopted method in a disorder problem is to try to separate the scattering intensity into two parts, namely one part from an average periodic structure where formulae (4.2.2.10), (4.2.2.11) apply and a second part resulting from fluctuations from this average (see, e.g., Schwartz & Cohen, 1977). One may write this formally aswhere is defined to be timeindependent and periodic in space and . is the density of the average unit cell obtained by the projection of all unit cells into a single one. Fourier transformation gives the average amplitude,where is the usual structure factor (4.2.2.11). The difference structure Δρ leads to the difference amplitude,Because cross terms vanish by definition, the Patterson function isFourier transformation givesSince is periodic, the first term in (4.2.3.23a) describes Bragg scattering,where plays the normal role of a structure factor of one cell of the averaged structure. The second term corresponds to diffuse scattering,In many cases, diffuse interferences are centred exactly at the positions of the Bragg reflections. It is then a serious experimental problem to decide whether the observed intensity distribution is due to Bragg scattering obscured by crystalsize limitations or due to other scattering phenomena.
If disordering is exclusively timedependent, represents the time average, whereas gives the pure elastic scattering part [cf. (4.2.2.8)] and ΔF refers to inelastic scattering only.
Diffuse scattering may be classified in various ways which may be related to specific aspects of the intensity distribution, e.g. according to the type of disorder: substitutional (or density or chemical) or displacive. The general expression (4.2.3.25), which may be rewritten aswhere , contains the difference structure factor ΔF, which may formally be written aswhere (Δf) denotes a fluctuation of the scattering density (the form factor in the case of Xrays and the scattering length in the case of neutrons; note that this may include vacancies) and Δr denotes a fluctuation of the position, i.e. they refer to substitutional and displacive disorder, respectively. Although the two types often occur together in real crystals, they may be discriminated through their different dependence on the modulus of the scattering vector H. This may be seen by considering the diffuse scattering of completely random fluctuations, i.e. without any correlations. For substitutional disorder one easily derives from (4.2.3.26)while for displacive fluctuations with small amplitudes Δr the exponential may be expanded:leading toandHence in the substitutional case the diffuse intensity is constant throughout reciprocal space, while in the displacive case it increases with H^{2}, i.e. it is zero near the origin. Any correlations will modulate this intensity, but will not change this general behaviour.
A second classification is related to the dimensionality of the disorder: onedimensional disorder between twodimensionally ordered objects (planes) leading to diffuse streaks perpendicular to the planes, twodimensional disorder between onedimensionally ordered objects (chains) leading to diffuse planes and general threedimensional disorder.
These different types of disorder will be further discussed separately in the next paragraphs.
From the derivation in Section 4.2.3.3, one may note that information about an averaged disordered structure is contained in the Bragg scattering governed bywhere T_{i}(H) is the Debye–Waller factor. Backtransformation to real space givesi.e. each average position is convoluted with p.d.f._{i}(r − r_{i}), the probability density function, which is the Fourier transform of T_{i}(H). Any disorder model derived from the diffuse scattering must therefore comply with the p.d.f., i.e. this may and should be used to validate these models.
As mentioned above, a completely random distribution of chemical species leads to a uniform distribution of diffuse intensity, which is also called monotonic Laue scattering. For example, for a binary alloy with scattering densities f_{1} and f_{2} and concentrations c_{1} and c_{2} this is simply given by . Several authors use this, i.e. the intensity of a random distribution of occupancies, to define a socalled Laue unit, and therewith to put the general diffuse scattering on a relative scale.
Any deviations from the monotonic Laue scattering may be due either to the scattering factor of the objects (e.g. molecules) or to shortrangeorder correlations. For example, for a simple defect pair with distance R we haveor for more general shortrange orderwhere α_{n} are Warren–Cowley shortrangeorder parameters, as will be discussed in more detail in Section 4.2.5.4.
For a periodic modulation where G is a reciprocallattice vector.
Hence a harmonic density modulation of a structure in real space leads to pairs of satellites in reciprocal space. Each main reflection is accompanied by a pair of satellites in the directions with phases . The reciprocal lattice may then be written in the following form:where . Fourier transformation yields Equation (4.2.3.34) describes the lattice modulated by a harmonic density wave. Since phases cannot be determined by intensity measurements, there is no possibility of obtaining any information on the phase relative to the sublattice. From (4.2.3.34) it is obvious that the use of higher orders of harmonics does not change the situation. If is not rational, such that no (n = integer) coincides with a main reflection in reciprocal space, the modulated structure is incommensurate with the basic lattice and the phase of the density wave becomes meaningless. The same is true for the relative phases of the various orders of harmonic modulations of the density. This uncertainty even remains valid for commensurate density modulations of the sublattice, because coinciding higherorder harmonics in reciprocal space cause the same difficulty; higherorder coefficients cannot uniquely be separated from lower ones, consequently structure determination becomes impossible unless phasedetermination methods are applied. Fortunately, density modulations of pure harmonic character are impossible for chemical reasons; they may be approximated by disorder phenomena for the averaged structure only. If diffuse scattering is taken into account, the situation is changed considerably: A careful study of the diffuse scattering alone, although difficult in principle, will yield the necessary information about the relative phases of density waves (Korekawa, 1967).
Displacement modulations are more complicated, even in a primitive structure. The Fourier transform of a longitudinal or a transverse displacement wave has to be calculated and this procedure does not result in a function of similar simplicity. Formally, a periodic modulation leads towhere a is the amplitude of the displacement wave with and e is the polarization vector: . Equation (4.2.3.35) denotes a set of satellites whose amplitudes are described by Bessel functions of νth order, where ν represents the order of the satellites. The intensity of the satellites increases with the magnitude of the product . This means that a single harmonic displacement causes an infinite number of satellites. They may be unobservable at low diffraction angles as long as the amplitudes are small. If the displacement modulation is incommensurate there are no coincidences with reflections of the sublattice. Consequently, the reciprocal space is completely covered with an infinite number of satellites, or, in other words, with diffuse scattering. This is a clear indication that incommensurate displacement modulations belong to the category of disordered structures. Statistical fluctuations of amplitudes of the displacement waves cause additional diffuse scattering, regardless of whether the period is commensurate or incommensurate (Overhauser, 1971; Axe, 1980). Fluctuations of `phases' (i.e. periods) cause a broadening of satellites in reciprocal space but no change in their integrated intensities as long as the changes are not correlated with fluctuation periods. The broadening of satellite reflections increases with the order of the satellites and . Obviously, there is no fundamental difference in the calculation of diffuse scattering with an ordered supercell of sufficient size.
Many disorder problems may be treated qualitatively in terms of coarsened structures that are made up of clusters or domainlike order extending along one or more directions in space. Common crystals are made up of `mosaic individuals' which are separated irregularly by unspecified defects such as dislocations, smallangle boundaries, microstrain fields and other defects. This `real' crystallinity is not covered by the term `disorder'. Clustering is, in a structural sense, not a very well defined term, but refers to general agglomerations of atoms, vacancies, defects or atomic groups due to preferred chemical bonding or due to some kind of exsolution processes. In general, the term `cluster' is used to describe some inhomogeneity in a basic matrix structure. The term `domain' usually implies either a spatially varying structure forming separate blocks, such as occurring in twin domains, or a spatial variation of a physical property (e.g. magnetic moment), which may be visualized by different configurations (see, e.g., Frey, 1997). Structural domains may be chemically homogeneous, as is the case in twin domain structures, or heterogeneous, which occurs, e.g., in feldspars with their complicated Ca/Na and Al/Si distributions. A definition of a domain structure may be given by symmetry arguments or, equivalently, by the orderparameter concept. Individual domains of the coarsened structure may be derived from a, possibly hypothetical, highsymmetry aristophase obeying the concept of symmetry groups. A lower symmetry of the domain is either due to loss of a pointgroup symmetry element (twin domains) or due to loss of a translational symmetry element of the aristophase, giving rise to the formation of outofphase domains. The special case of antiphase domains is related to a violation of a translational vector t in the aristophase by regular or irregular insertion of lattice displacements ½t. Shear domains, which are related by other fractional parts of t, may be explained by cooperative gliding of structural building blocks, for example coordination polyhedra. Colour (blackandwhite) symmetry has to be used for magnetic domain structures and may also be used for chemical domain ordering. While preserving the same lattice, the disordered (usually hightemperature) phase, which is specified by grey, decomposes into black and white domains, possibly embedded in the grey matrix. However, the term `atomic cluster' is frequently used in this context. Domains can exhibit a new order by themselves, thus creating new symmetries of the superstructure. The boundaries between different domains are apparently essential and may even be used for a definition of a domain. This does not mean that the boundaries or domain walls are simple atomic planes rather than extended intermediate structural states which mellow the transition from one domain to the next one. Thus extended walls may carry a `gradient' structure between neighbouring differently oriented domains and may be treated as new domains with their own structure. If misfits at the planes of coincidence are accompanied by remarkable straining, an array of dislocations may destroy the exact symmetry relation between the individuals. There is a stepwise transition from fully coherent domains to fully incoherent crystal parts. As long as coherency between the individuals is preserved, domain structures can be treated simply by means of Fourier transforms and characteristic features of the disorder problem may be extracted from diffuse patterns.
Quite generally, the scattering density for a general arrangement of domains may be written as (Boysen, 1995; Frey, 1997)where ρ_{i} = ρ_{i}(r) is the structure of the unit cell of domain i, l_{i}(r) is the lattice function, is the shape function (which is unity in the region of domain i with size j and zero elsewhere) and is the distribution function ( are the centres of the domains). Note again that domain walls may be included in this formulation as separate `domains'. Fourier transformation yields the scattering amplitudewhere F_{i} = F_{i}(H) is the usual structure factor, L_{i}(H) is the lattice function in reciprocal space, and are the Fourier transforms of the shape and distribution function, respectively, and the intensity isThe first term represents sharp or diffuse reflections that are modified by the convolution with the Fourier transforms of the shape and distribution function (more correctly, with the Fourier transforms of the corresponding Patterson functions), while the second (cross) term generally leads to smaller additional changes. It should be emphasized that this separation does not correspond to the separation into Bragg and diffuse scattering. Although a further mathematical treatment of this very general expression does not seem to be easy, some qualitative or even semiquantitative conclusions may be drawn:
Example: Clusters in a periodic lattice (low concentrations)
The exsolution of clusters of equal sizes is considered. The lattice of the host is undistorted, structure F_{1}, and the clusters have the same lattice but a different structure, F_{2}. A schematic drawing is shown in Fig. 4.2.3.1. Two different structures are introduced:Their Fourier transforms are the structure factors , of the matrix and the exsolved clusters, respectively. The boxes in Fig. 4.2.3.1 indicate the clusters, which may be represented by box functions in the simplest case. It should be pointed out, however, that a more complicated shape means nothing other than a replacement of by another shape function and its Fourier transform . The distribution of clusters is represented by where m refers to the centres of the box functions (the crosses in Fig. 4.2.3.1). The problem is therefore defined byThe incorrect addition of to the areas of clusters is compensated by subtracting the same contribution from the second term in equation (4.2.3.42a). In order to determine the diffuse scattering, the Fourier transformation of (4.2.3.42a) is performed:The intensity is given byEvaluation of equation (4.2.3.42c) yields three terms (where c.c. means complex conjugate):The first two terms represent modulated lattices [multiplication of by ]. Consequently, they cannot contribute to diffuse scattering, which is completely determined by the third term. Fourier transformation of this term giveswhere , and . According to equation (4.2.3.15) and its subsequent discussion, the convolution of the two expressions in square brackets was replaced by l(r)t(r), where t(r) represents the `pyramid' of nfold height discussed above and n is the number of unit cells within b(r). is the Patterson function of the distribution function d(r). Its usefulness may be recognized by considering the two possible extreme solutions, namely the random and the strictly periodic distribution.
If no fluctuations of domain sizes are admitted, the minimum distance between two neighbouring domains is equal to the length of the domain in the corresponding direction. This means that the distribution function cannot be completely random. In one dimension, the solution of a random distribution of particles of a given size on a finite length shows that the distribution functions exhibit periodicities that depend on the average free volume of one particle (Zernike & Prins, 1927). Although the problem is more complicated in three dimensions, there should be no fundamental difference in the exact solutions.
On the other hand, it may be shown that the convolution of a pseudorandom distribution may be obtained if the average free volume is large. This is shown in Fig. 4.2.3.2(a) for the particular case of a cluster smaller than one unit cell. A strictly periodic distribution function (a superstructure) may result, however, if the volume of the domain and the average free volume are equal. Obviously, the practical solution for the selfconvolution of the distribution function (which is the Patterson function) lies somewhere in between, as shown in Fig. 4.2.3.2(b). If a harmonic periodicity damped by a Gaussian is assumed, this selfconvolution of the distribution in real space may be considered to consist of two parts, as shown in Figs. 4.2.3.2(c), (d). Note that the two different solutions result in completely different diffraction patterns:

Figs. 4.2.3.2(e), (f) show the different diffraction patterns of the diffuse scattering that is concentrated around the Bragg maxima. Although the discussion of the diffuse scattering was restricted to the case of identical domains, the introduction of a distribution of domain sizes does not influence the diffraction pattern essentially, as long as the fluctuation of sizes is small compared with the average volume of domain sizes and no strong correlation exists between domains of any size (a sizeindependent random distribution).
A complete qualitative discussion of the diffraction pattern may be carried out by investigating the Fourier transform of (4.2.3.43a):The first factor in (4.2.3.43b) describes the particlesize effect of a domain containing the influence of a surrounding strain field and the new structure of the domains precipitated from the bulk. D(H) has its characteristic variation near the Bragg peaks (Figs. 4.2.3.2e,f) and is less important in between. For domain structure determination, intensities near the Bragg peaks should be avoided. Note that equation (4.2.3.43b) may be used for measurements using anomalous scattering in both the centric and the acentric case.
Solution of the diffraction problem. In equation (4.2.3.43b) is replaced by its average,where represents the a priori probability of a domain of type μ. This replacement becomes increasingly important if small clusters (domains) have to be considered. Applications of the formulae to Guinier–Preston zones are given by Guinier (1942) and Gerold (1954); a similar application to clusters of vacancies in spinels with an excess of Al_{2}O_{3} was outlined by Jagodzinski & Haefner (1967).
Although refinement procedures are possible in principle, the number of parameters entering the diffraction problem becomes increasingly large if more clusters or domains (of different sizes) have to be introduced. Another difficulty results from the large number of diffraction data which must be collected to perform a reliable structure determination. There is no need to calculate the first two terms in equation (4.2.3.42c) which contribute to the sharp Bragg peaks only, because their intensity is simply described by the averaged structure factor . These terms may therefore be replaced by withwhere is the a priori probability of the structure factor . It should be emphasized here that (4.2.3.43c) is independent of the distribution function d(r) or its Fourier transform D(H).
While a number of solutions to the diffraction problem may be found in the literature for onedimensional (1D) disorder (1D distribution functions) (see, e.g., Jagodzinski, 1949a,b,c; Cowley, 1976a,b; Adlhart, 1981; Pflanz & Moritz, 1992), these become increasingly more complicated with increasing dimensionality. These types are therefore discussed separately in the following.
This lamellar (1D) type of disorder is common in many crystals for energetic reasons. For example, most of the important rockforming minerals exhibit such disorder behaviour. Therefore we give some extended introduction to this field. To illustrate the application of the formalism outlined above, we start with a simple example of domains of an identical structure ρ(r) displaced relative to each other by an arbitrary fault vector (Boysen et al., 1991). If t(r) describes the regions of one domain, then 1 − t(r) describes those of the displaced domains and the complete structure may be writtenIntroducing t′(r) = 2t(r) − 1, this may be rewritten asFourier transformation yieldsThe first term describes sharp reflections due to the multiplication with the lattice function L(H), while the second term gives diffuse reflections due to the convolution with T′(H). The corresponding intensities areIn the general case, all reflections consist of a superposition of sharp and diffuse intensities. Note that the well known separation of sharp and diffuse intensities for antiphase domains is obtained if equals ½ times a lattice vector.

The subject of twodimensional disorder refers to predominantly onedimensional structural elements, e.g. extended macromolecules and chain or columnlike structural units. A short introduction to this subject and some examples taken from inorganic structures are given in Section 4.2.5.3. Most important in this context, however, would be a treatment of disorder diffuse scattering of polymer/fibre structures. These are subjects in their own right and are treated in Chapter 4.5 of this volume and in Chapter 19.5 of Volume F.
The solution of threedimensional disorder problems is generally more demanding, although it may start with the formulation given above. Various algorithms have been developed to tackle these problems at least approximately, most of them restricted to particular models. Both realspace (cluster) and reciprocalspace (fluctuation wave) methods are employed and will be briefly addressed in Section 4.2.5.4. The more recently developed approaches using computer simulations are described in Section 4.2.7.
Here we give only some general remarks on order–disorder problems.
Correlation functions in three dimensions may have very complicated periodicities; hence careful study is necessary to establish whether or not they may be interpreted in terms of a superlattice. If so, extinction rules have to be determined in order to obtain information on the superspace group. In the literature these are often called modulated structures (see Section 4.2.6) because a sublattice, as determined by the basic lattice, and a superlattice may well be defined in reciprocal space: reflections of a sublattice including (000) are formally described by a multiplication by a lattice having larger lattice constants (the superlattice) in reciprocal space; in real space this means a convolution with the Fourier transform of this lattice (the sublattice). In this way, the averaged structure is generated in each of the subcells (the superposition or `projection' of all subcells into a single one). Obviously, the Patterson function of the averaged structure contains little information in the case of small subcells. Hence it is advisable to include the diffuse scattering of the superlattice reflections at the beginning of any structure determination.
N subcells in real space are assumed, each of them representing a kind of a complicated `atom' that may be equal by translation or other symmetry operation. Once a superspace group has been determined, the usual extinction rules of space groups may be applied, remembering that the `atoms' themselves may have systematic extinctions. Major difficulties arise from the existence of different symmetries of the subgroup and the supergroup. Since the symmetry of the supercell is lower in general, all missing symmetry elements may cause domains corresponding to the missing symmetry element: translations cause antiphase domains in their generalized sense; other symmetry elements cause twins generated by rotations, mirror planes or the centre of symmetry. If all these domains are small enough to be detected by a careful study of line profiles, using diffraction methods with a high resolution power, the structural study may be facilitated by introducing scaling factors for groups of reflections affected by the possible domain structures.
If disorder problems involving completely different structures (exsolutions etc.) are excluded, in general the symmetry of the diffusescattering pattern is the same as that of the Bragg peaks, i.e. it corresponds to the point group of the space group of the average structure. Only under specific directional growth conditions are deviations from this rule conceivable (although seemingly not very common). On the other hand, the symmetry of the underlying disorder model in direct space may be lower than that of the space group of the average structure (it is usually a subgroup of the space group of the average structure). The overall symmetry in reciprocal space is then restored by employing the missing (pointgroup) symmetry elements, which have therefore to be used in the calculation of the full diffraction pattern.
From these arguments, some specific disorder models may be classified according to the irreducible representations of the space group of the average structure. While for general wavevectors in the Brillouin zone no further restrictions appear, for highsymmetry directions consideration of the irreducible representations of the little cogroup of the wavevector can help to identify the different symmetries of the disorder model. This becomes particularly evident when the modulationwave approach is used as shown e.g. by Welberry & Withers (1990) and Welberry & Butler (1994). Of particular value are observed extinction rules, which may be calculated by grouptheoretical methods as developed by PerezMato et al. (1998) for the extinctions occurring in inelastic neutronscattering experiments. In favourable cases, the analysis of such extinctions alone can lead to a unique determination of the disorder model (see, e.g., Aroyo et al., 2002).
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