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. 492539
doi: 10.1107/97809553602060000774 Chapter 4.2. Disorder diffuse scattering of Xrays and neutrons^{a}Department für Geo und Umweltwissenschaften, Sektion Kristallographie, LudwigMaximilians Universität, Theresienstrasse 41, 80333 München, Germany The scope of this chapter is outlined in Section 4.2.1. The next section, Section 4.2.2, presents a summary of basic scattering theory containing general formulae for scattering cross sections. Section 4.2.3 contains a general treatment of disorder diffuse scattering and describes the basic mathematics and general qualitative formulation of disorder problems. Occupational disorder, displacive disorder, domain disorder and lamellar disorder are discussed. Section 4.2.4 gives a rough guideline for solving a disorder problem. Section 4.2.5 describes quantitative interpretation and covers quantitative solutions and methods for treating disorder problems in layered structures (onedimensional stacking disorder), structures with chainlike elements (twodimensional disorder), defect structures, shortrange ordering, clustering, molecular crystals and orientational disorder. A short section about disorder diffuse scattering from aperiodic crystals is given in Section 4.2.6. Computer simulations and modelling of disordered crystals are covered in Section 4.2.7 and the last section, Section 4.2.8, describes some general experimental aspects of singlecrystal and powder Xray and neutron diffraction by disordered crystals. 
Diffuse scattering of Xrays, neutrons and other particles is an accompanying effect in all diffraction experiments aimed at structure analysis with the aid of elastic scattering. In this case, the momentum exchange of the scattered photon or particle includes the crystal as a whole; the energy transfer involved becomes negligibly small and need not be considered in diffraction theory. Static distortions as a consequence of structural changes cause typical elastic diffuse scattering. Many structural phenomena and processes contribute to diffuse scattering, and a general theory has to include all of them. Hence the exact treatment of diffuse scattering becomes very complex.
Inelastic scattering is due to dynamical fluctuations or ionization processes and may become observable as a `diffuse' contribution in a diffraction pattern. A separation of elastic from inelastic diffuse scattering is generally possible, but difficulties may result from small energy exchanges that cannot be resolved for experimental reasons. The latter is true for scattering of Xrays by phonons, which have energies of the order of 10^{−2}–10^{−3} eV, values which are considerably smaller than 10 keV, a typical value for Xray quanta. Another equivalent explanation, frequently forwarded in the literature, is the high speed of Xray photons, such that the rather slow motion of atoms cannot be `observed' by them during diffraction. Hence, all movements appear as static displacement waves of atoms, and temperature diffuse scattering is pseudoelastic for Xrays. This is not true in the case of thermal neutrons, which have energies comparable to those of phonons. Phononrelated or thermal diffuse scattering is discussed separately in Chapter 4.1 , i.e. the present chapter is mainly concerned with the elastic (or pseudoelastic other than thermal) part of diffuse scattering. A particularly important aspect concerns diffuse scattering related to phase transitions, in particular the critical diffuse scattering observed at or close to the transition temperature. In simple cases, a satisfactory description may be given with the aid of a `soft phonon', which freezes at the critical temperature, thus generating typical temperaturedependent diffuse scattering. If the geometry of the lattice is maintained during the transformation (i.e. there is no breakdown into crystallites of different cell geometry), the diffuse scattering is very similar to diffraction phenomena described in this chapter. Sometimes, however, very complicated interim stages (ordered or disordered) are observed, demanding a complicated theory for their full explanation (see, e.g., Dorner & Comes, 1977).
Obviously, there is a close relationship between thermodynamics and diffuse scattering in disordered systems representing a stable or metastable thermal equilibrium. From the thermodynamical point of view, the system is then characterized by its grand partition function, which is intimately related to the correlation functions used in the interpretation of diffuse scattering. The latter is nothing other than a kind of `partial partition function' where two atoms, or two cell occupancies, are fixed such that the sum of all partial partition functions represents the grand partition function. This fact yields the useful correlation between thermodynamics and diffuse scattering mentioned above, which may well be used for a determination of thermodynamical properties of the crystal. This important subject shall not be included here for the following reason: real threedimensional crystals generally exhibit diffuse scattering by defects and/or disordering effects that are not in thermal equilibrium. They are created during crystal growth, or are frozenin defects formed at higher temperatures. Hence a thermodynamical interpretation of diffraction data needs a careful study of diffuse scattering as a function of temperature or some other thermodynamical parameters. This can be done in very rare cases only, so the omission of this subject seems justified.
As shown in this chapter, electrondensity fluctuations and distribution functions of defects play an important role in the complete interpretation of diffraction patterns. Both quantities may best be studied in the lowangle scattering range. Hence many problems cannot be solved without a detailed interpretation of lowangle diffraction (also called smallangle scattering).
Disorder phenomena in magnetic structures are also not specifically discussed here. Magnetic diffuse neutron scattering and special experimental techniques constitute a large subject by themselves. Many aspects, however, may be analysed along similar lines to those given here.
Glasses, liquids or liquid crystals show typical diffuse diffraction phenomena. Particlesize effects and strains have an important influence on the diffuse scattering. The same is true for dislocations and point defects such as interstitials or vacancies. These defects are mainly described by their strain field, which influences the intensities of sharp reflections like an artificial temperature factor: the Bragg peaks diminish in intensity while the diffuse scattering increases predominantly close to them. These phenomena are less important from a structural point of view, at least in the case of metals or other simple structures. This statement is true as long as the structure of the `kernel' of defects may be neglected when compared with the influence of the strain field. Whether dislocations in more complicated structures meet this condition is not yet known.
Commensurate and incommensurate modulated structures and quasicrystals frequently show a typical diffuse scattering, a satisfactory explanation of which demands extensive experimental and theoretical study. A reliable structure determination becomes very difficult in cases where the interpretation of diffuse scattering has not been incorporated. Many erroneous structural conclusions have been published in the past. The solution of problems of this kind needs careful thermodynamical consideration as to whether a plausible explanation of the structural data can be given.
For all of the reasons mentioned above, this article cannot be complete. It is hoped, however, that it will provide a useful guide for those who need a full understanding of the crystal chemistry of a given structure.
The study of disorder in crystals by diffusescattering techniques can be performed with Xrays, neutrons or electrons. Each of these methods has its own advantages (and disadvantages) and they often can (or have to) be used in a complementary way (cf. Chapter 4.3 of this volume). Electron diffraction and microscopy are usually restricted to relatively small regions in space and thus supply information on a local scale, i.e. local defect structures. Moreover, electronmicroscopy investigations are carried out on thin samples (films), where the disorder could be different from the bulk, and, in addition, could be affected by the high heat load deposited by the impinging electron beam. Xrays and neutrons sample larger crystal volumes and thus provide thermodynamically more important information on averages of the disorder. These methods are also better suited to the analysis of longrange correlated cooperative disorder phenomena. On the other hand, electron microscopy and diffraction often allow more direct access to disorder and can therefore provide valuable information about the underlying model, which can then be used for a successful interpretation of Xray and neutron diffuse scattering. Basic aspects of electron diffraction and microscopy in structure determination are treated in Chapter 2.5 of this volume.
There is no comprehensive treatment of all aspects of diffuse scattering. Essential parts are treated in the textbooks by James (1954), Wilson (1962), Wooster (1962), Krivoglaz (1969, 1996a,b), Schwartz & Cohen (1977), Billinge & Thorpe (1998), Schweika (1998), Nield & Keen (2001), Fultz & Howe (2002), Frey (2003) and Welberry (2004); handbook articles have been written by Jagodzinski (1963, 1964a,b, 1987), Schulz (1982), Welberry (1985) and Moss et al. (2003), and a series of relevant papers has been collected by Collongues et al. (1977).
Finally, we mention that different symbols and `languages' are used in the various diffraction methods. Quite a few of the new symbols in use are not really necessary, but some are caused by differences in the experimental techniques. For example, the neutron scattering length b may usually be equated with the atomic form factor f in Xray diffraction. The differential cross section introduced in neutron diffraction represents the intensity scattered into an angular range dΩ and an energy range dE. The `scattering law' in neutron work corresponds to the square of an (extended) structure factor; the `static structure factor', a term used by neutron diffractionists, is nothing other than the conventional Patterson function. The complicated resolution functions in neutron work correspond to the well known Lorentz factors in Xray diffraction. These have to be derived in order to include all the techniques used in diffusescattering work. In this article we try to preserve the most common symbols. In particular, the scattering vector will be denoted as H, which is more commonly used in crystallography than .
Diffuse scattering results from deviations from the identity of translationally invariant scattering objects and from longrange correlations in space and time. Fluctuations of scattering amplitudes and/or phase shifts of the scattered wavetrains reduce the maximum capacity of interference (leading to Bragg reflections) and are responsible for the diffuse scattering, i.e. scattering parts that are not located in distinct spots in reciprocal space. Unfortunately, the terms `coherent' and `incoherent' scattering used in this context are not uniquely defined in the literature. Since all scattering processes are correlated in space and time, there is no incoherent scattering at all in its strict sense. A similar relationship exists for `elastic' and `inelastic' scattering. Here pure inelastic scattering would take place if the momentum and the energy were transferred to a single scatterer; on the other hand, an elastic scattering process would demand a uniform exchange of momentum and energy with the whole crystal. Obviously, both cases are idealized and the truth lies somewhere in between. In spite of this, many authors use the term `incoherent' systematically for the diffuse scattering away from the Bragg peaks when all diffuse maxima or minima are due to structure factors of molecules or atoms only. Although this definition is unequivocal as such, it is physically incorrect. Other authors use the term `coherent' for Bragg scattering only; all diffuse contributions are then called `incoherent'. This definition is clear and unique since it considers space and time, but it does not differentiate between incoherent and inelastic. In the case of neutron scattering, both terms are essential and neither can be abandoned.
In neutron diffraction, the term `incoherent scattering' refers to scattering by uncorrelated nuclear spin orientations or by a random distribution of isotopes of the same element. Hence another definition of `incoherence' is proposed for scattering processes that are uncorrelated in space and time. In fact, there may be correlations between the spins via their magnetic field, but the correlation length in space (and time) may be very small, such that the scattering process appears to be incoherent. Even in these cases, the nuclei contribute to coherent (average structure) and incoherent scattering (diffuse background). Hence the scattering process cannot really be understood by assuming nuclei that scatter independently. For this reason, it seems to be useful to restrict the term `incoherent' to cases where a random contribution to scattering is realized or, in other words, a continuous function exists in reciprocal space. This corresponds to a δ function in real fourdimensional space. The randomness may be attributed to a nucleus (neutron diffraction) or an atom (molecule). It follows from this definition that the scattering need not be constant, but may be modulated by structure factors of molecules. In this sense we shall use the term `incoherent', remembering that it is incorrect from a physical point of view.
As mentioned in Chapter 4.1 , the theory of thermal neutron scattering must be treated quantum mechanically. In principle, this is also true in the Xray case. In the classical limit, however, the final expressions have a simple physical interpretation. Generally, the quantummechanical nature of the scattering function of thermal neutrons is negligible at higher temperatures and in those cases where energy or momentum transfers are not too large. In almost all disorder problems this classical interpretation is sufficient for the interpretation of diffusescattering phenomena. This is not quite true in the case of orientational disorder (plastic crystals) where H atoms are involved.
The basic formulae given below are valid in either the Xray or the neutron case: the atomic form factor f replaces the coherent scattering length b_{coh} (abbreviated as b). The formulation in the frame of the van Hove correlation function G(r, t) (classical interpretation, coherent part) corresponds to a treatment by a fourdimensional Patterson function P(r, t).
The basic equations for the differential cross sections arewhere N is the number of scattering nuclei of same chemical species; , are the wavevectors after/before scattering and .
The integrations over space may be replaced by summations in disordered crystals, except in cases where structural elements exhibit a liquidlike behaviour. Then the van Hove correlation functions are gives the probability that if there is an atom j at at time zero, there is an arbitrary atom j′ at at an arbitrary time t, while refers to the same atom j at at time t.
Equations (4.2.2.1) may be rewritten using the fourdimensional Fourier transforms of G and , respectively:Incoherent scattering cross sections [(4.2.2.3b), (4.2.2.4b)] refer to one and the same particle (at different times). In particular, plastic crystals (see Section 4.2.5.5) may be studied by means of this incoherent scattering. It should be emphasized, however, that for reasons of intensity only disordered crystals with strong incoherent scatterers can be investigated by this technique. In practice, mostly samples that contain H atoms have been investigated. This topic will not be treated further in this article (see, e.g., Springer, 1972; Lechner & Riekel, 1983). The following considerations are restricted to coherent scattering only.
Essentially the same formalism as given by equations (4.2.2.1a)–(4.2.2.4a) may be described using a generalized Patterson function, which is more familiar to crystallographers:where ρ is the scattering density and τ denotes the time of observation. The only difference between and is the inclusion of the scattering weight (f or b) in . is an extension of the usual spatial Patterson function . Similarly, S is replaced by another function,which is the Fourier transform of . One difficulty arises from neglecting the time of observation. Just as is always proportional to the scattering volume V in the framework of kinematical theory or within Born's first approximation [cf. equation (4.2.2.1a)], so is proportional to volume and observation time. In general, one does not make S proportional to τ, but one normalizes S to be independent of τ as : . Averaging over time τ gives therefore
Special cases (see, e.g., Cowley, 1981) are:
In most practical cases, averaging over time is equivalent to averaging over space: the total diffracted intensity may be regarded as the sum of the intensities from a large number of independent regions due to the limited coherence of a beam. At any time these regions take all possible configurations. Therefore, this sum of intensities is equivalent to the sum of intensities from any one region at different times,From the basic formulae one also derives the well known results for Xray or neutron scattering by a periodic arrangement of particles in space [cf. equation (4.1.3.2) of Chapter 4.1 ]: denotes the Fourier transform of one cell (structure factor); G is the reciprocallattice vector and W is the Debye–Waller factor; the f's are assumed to be real.
The evaluation of the intensity expressions (4.2.2.6), (4.2.2.8) or (4.2.2.9), (4.2.2.9a) for a disordered crystal must be performed in terms of statistical relationships between scattering factors and/or atomic positions.
From these basic scattering formulae some conclusions on the relative merits of Xrays and neutrons can be drawn. Both Xrays and thermal neutrons possess wavelengths of the order of interatomic distances and are thus well suited to the study of the atomic structure of condensed matter. Besides this, there are fundamental differences that make one or the other the radiation of choice for a particular problem or enable them to be used with advantage in a complementary manner. These differences are well documented in a number of textbooks on diffraction and are widely exploited in the determination of average structures. One fundamental difference is related to the interaction with matter: while Xrays are scattered by the electrons, neutrons are scattered by the atomic nuclei (we are disregarding magnetic interactions in this chapter), i.e. Xrays probe the distribution of charges, which (in particular for light atoms) may not coincide with that of the nuclei. This fact forms the basis for the well known X − N technique in the analysis of Bragg intensities. No such studies of disorder diffuse scattering have been performed up to now, but may be a challenge for the future. As a consequence of the different size of the scattering objects, the scattering amplitude falls off with increasing scattering vector for Xrays (the form factor), while it is constant for neutrons. Since for a complete interpretation of diffuse scattering high Q values are generally required, neutrons have their advantages in this respect. The scattering intensity varies as Z^{2} (Z = number of electrons) for Xrays and moreorless erratically for neutrons, for which it also depends on the specific isotope. This imposes problems for Xrays when trying to detect light elements (in particular H and O) in the presence of heavy elements or discriminating neighbouring elements in the periodic table (e.g. Al/Si/Mg, Fe/Co/Ni). On the other hand, these differences in scattering power can be used to identify the atomic species taking part in the disorder by comparing the intensity distributions observed with both methods. The contrast can be further enhanced by marking selected atoms by isotope substitution for neutrons or by using anomalous dispersion for Xrays.
In many practical cases, the complementarities of the two methods can be exploited by using them simultaneously. Some illustrative examples can be found in Boysen & Frey (1998).
Another important difference is in the energies given by the dispersion laws and for Xrays and neutrons, respectively, where c is the velocity of light, m is the neutron mass and k = 2π/λ is the wavevector. For a typical wavelength, λ =1.54 Å (Cu Kα) (k = 4.08 Å^{−1}), E_{x} = 8 keV and E_{n} = 35 meV, i.e. the energies differ by almost six orders of magnitude. This means that neutron energies are similar to those of typical collective lattice excitations, which may therefore be determined more easily by neutron spectroscopy. The basic question of whether the underlying disorder is of static or dynamic origin can be answered using neutrons alone, e.g. by comparing the `integral' (= elastic + inelastic) scattered intensity (i.e. without energy analysis, using a twoaxis diffractometer) with the purely elastic intensity by placing an analyser crystal set to zero energy transfer in the diffracted beam (using a tripleaxis diffractometer) or using timeofflight methods. The high energy resolution of Mössbauer radiation can also be used for this purpose. Owing to the very low available intensities, however, this technique has only been applied occasionally. In the elastic mode, diffracted neutron intensities are also usually rather weak. The decision can, however, also be made simply by comparing the positions of the intensity maxima in reciprocal space. This follows from a consideration of the relative changes of the momentum transfer (the wavevector) in the two cases. For example, for an energy change ΔE = E_{f} − E_{i} (the subscripts refer to final and incident energies) of 1 meV, one obtains a change in the scattered wavevector Δk_{f} of 2 × 10^{−6} Å^{−1} for Xrays and 0.06 Å^{−1} for neutrons. In a twoaxis experiment where the detector does not discriminate energies, i.e. without prior knowledge of the real energies, the distribution of diffuse intensities has to be drawn as if the scattering had been elastic (ΔE = 0). In other words, the signal appears at a position displaced by Δk_{f}. From the estimations above it is clear that only in the neutron case can a measurable effect be expected.
Other specific features of neutrons include their magnetic moment (magnetic scattering is still a domain of neutrons, although progress is now being made with synchrotron radiation) and (nuclear) spin and isotope incoherent scattering processes (which allow the determination of the selfcorrelation functions). It is possible to separate the spinincoherent part and distinguish between nuclear and magnetic scattering using polarization analysis. For most elements scattering and absorption is much weaker for neutrons, allowing larger samples to be analysed and sample environments (furnaces, pressure cells, electric and magnetic fields, special atmospheres etc.) to be handled much more easily. Differences in the experimental techniques for Xray and neutron scattering are discussed in Section 4.2.8.
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).
In general, the structure determination of a disordered crystal should start in the usual way by solving the average structure. The effectiveness of this procedure strongly depends on the distribution of integrated intensities of sharp and diffuse reflections. In cases where the integrated intensities of Bragg peaks are predominant, the maximum information can be drawn from the averaged structure. The observations of fractional occupations of lattice sites, split positions and anomalous large and anharmonic displacement parameters are indications of the disorder involved. Since these aspects of disorder phenomena in the averaged structure may be interpreted very easily, a detailed discussion of this matter is not given here (see any modern textbook of Xray crystallography). Therefore, the anomalies of the average structure can give valuable hints on the underlying disorder and, vice versa, can be used to check the final disorder model derived from the diffuse scattering.
Difficulties may arise from the intensity integration, which should be carried out very carefully to separate the Bragg peaks from the diffuse contributions, e.g. by using a highresolution diffraction method. The importance of this may be understood from the following argument. The averaged structure is determined by the coherent superposition of different structure factors. This interpretation is true if there is a strictly periodic subcell with longrange order that allows for a clear separation of sharp and diffuse scattering. There are important cases, however, where this procedure cannot be applied without loss of information.

The integrated intensity within a Brillouin zone of any structure is independent of atomic positions if the atomic form factors remain unchanged by structural fluctuations. Small deviations of atomic form factors owing to electrondensity changes of valence electrons are neglected. Consequently, the integrated diffuse intensities remain unchanged if the average structure is not altered by the degree of order. The latter condition is obeyed in cases where a geometrical longrange order of the lattice is independent of the degree of order, and no longrange order in the structure exists. This law is extremely useful for the interpretation of diffuse scattering. Unfortunately, intensity integration over coinciding sharp and diffuse maxima does not necessarily lead to a structure determination of the corresponding undistorted structure. This integration may be useful for antiphase domains without major structural changes at the boundaries. In all other cases, the deviations of domains (or clusters) from the averaged structure determine the intensities of maxima, which are no longer correlated with those of the average structure.
If the integrated intensity of diffuse scattering is comparable with, or even larger than, those of the Bragg peaks, it is useful to begin the interpretation with a careful statistical study of the diffuse intensities. Intensity statistics can be applied in a way similar to the intensity statistics in classical structure determination. The following rules are briefly discussed in order to enable a semiquantitative interpretation of the essential features of disorder.

Although it is highly improbable that exactly the same diffraction picture will be found, the use of an atlas of optical transforms (Wooster, 1962; Harburn et al., 1975; Welberry & Withers, 1987) may be helpful at the beginning of any study of diffuse scattering. Alternatively, computer simulations may be helpful, as discussed in Section 4.2.7. The most important step is the separation of the distribution function from the molecular scattering. Since this information may be derived from a careful comparison of lowangle diffraction with the remaining sharp reflections, this task is not too difficult. If the influence of the distribution function is unknown, the reader is strongly advised to disregard the immediate neighbourhood of Bragg peaks in the first step of the interpretation. Obviously information may be lost in this way but, as has been shown in the past, much confusion caused by an attempt to interpret the scattering near the Bragg peaks with specific structural properties of a cluster or molecular model is avoided. The inclusion of this part of diffuse scattering can be made after a complete interpretation of the change of the influence of the distribution function on diffraction in the wideangle region.
In these sections, quantitative interpretations of the elastic part of Xray and neutron diffuse scattering are outlined. Although similar relations are valid for the magnetic scattering of neutrons, this particular topic is excluded. Obviously, all disorder phenomena are strongly temperaturedependent if thermal equilibrium is reached. Consequently, the interpretation of diffuse scattering should include a statistical thermodynamical treatment. Unfortunately, no quantitative theory for the interpretation of structural phenomena is so far available: all quantitative solutions introduce formal order parameters such as correlation functions or distributions of defects. At low temperatures (i.e. with a low concentration of defects) the distribution function plays the dominant role in diffuse scattering. With increasing temperature the number of defects increases with corresponding strong interactions between them. Therefore, correlations become increasingly important, and phase transformations of first or higher order may occur which need a separate theoretical treatment. In many cases, large fluctuations of structural properties occur that are closely related to the dynamical properties of the crystal. Theoretical approximations are possible, but their presentation is far beyond the scope of this article. Hence we restrict ourselves to formal parameters in the following.
Point defects or limited structural units, such as molecules, clusters of finite size etc., may only be observed in diffraction if there is a sufficiently large number of defects. This statement is no longer true in highresolution electron diffraction, where single defects may be observed either by diffraction or by optical imaging if their contrast is high enough. Hence electron microscopy and diffraction are valuable methods in the interpretation of disorder phenomena.
The arrangement of a finite assembly of structural defects is described by its structure and its threedimensional (3D) distribution function. Structures with a strict 1D periodicity (chainlike structures) need a 2D distribution function, while for structures with a 2D periodicity (layers) a 1D distribution function is sufficient. Since the distribution function is the dominant factor in statistics with correlations between defects, we define the dimensionality of disorder as that of the corresponding distribution function. This definition is more effective in diffraction problems because the dimension of the disorder problem determines the dimension of the diffuse scattering: 1D diffuse streaks, 2D diffuse layers or a general 3D diffuse scattering.
Strictly speaking, completely random distributions cannot be realized, as shown in Section 4.2.3. They occur approximately if the following conditions are satisfied:

As already mentioned, disorder phenomena may be observed in thermal equilibrium. Two completely different cases have to be considered:

In many cases, the defects do not occur in thermal equilibrium. Nevertheless, their diffuse scattering is temperaturedependent because of the anomalous thermal movements at the boundary of the defect. Hence the observation of a temperaturedependent behaviour of diffuse scattering cannot be taken as a definite criterion of thermal equilibrium without further careful interpretation.
Ordering of defects may take place in a very anisotropic manner. This is demonstrated by the huge number of examples of 1D disorder. As shown by Jagodzinski (1963), this type of disorder cannot occur in thermal equilibrium for the infinite crystal. This type of disorder is generally formed during crystal growth or mechanical deformation. Similar arguments may be applied to 2D disorder. This is a further reason why the Ising model can hardly ever be used to obtain interaction energies of structural defects. From these remarks it becomes clear that order parameters are formal parameters without strict thermodynamical meaning.
The following section is organized as follows: first we discuss the simple case of 1D disorder where reliable solutions of the diffraction problem are available. Intensity calculations for diffuse scattering from 2D disorder by chainlike structures follow. Finally, the 3D case is treated, where formal solutions of the diffraction problem have been tried and applied successfully to metallic systems to some extent. A short concluding section concerns the special phenomenon of orientational disorder.
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 onedimensional violation of the Laue conditions, and will be called onedimensional 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 nextnearest 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 onedimensional correlation functions, which may be determined uniquely if a single independent correlation function is active. According to equation (4.2.3.50), Fourier transformation yields the information required. In all other cases, a specific model has to be suggested for a full interpretation of diffuse streaks. It is worth noting that disorder parameters can be defined uniquely only if the diffraction pattern allows for a differentiation between longrange and shortrange order. This question can be answered at least partly by studying the line width of sharp reflections at 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 onedimensional. Sometimes the problem may be treated independently if the streaks are sharp and no interference effects are 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 onedimensional disorder which can 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 onedimensional problem, some remarks should be made about the usefulness of quantitative disorder parameters. It is well known from statistical thermodynamics that a onedimensional system cannot show longrange order above T = 0 K. Obviously, this statement is in contradiction with many experimental observations of longrange order 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 twodimensional Ising model it is known that such a system shows longrange 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 threedimensional crystal can be constructed by adding layer after layer. Since each layer has a typical twodimensional free energy, the full statistics of the threedimensional crystal may be calculated by introducing a specific free energy for the various stackings of layers. Obviously, this additional energy also has to include terms describing potential and entropic energies. They may be formally developed into contributions of next, nextbutone etc. nearest neighbours. 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 (this is very rare but is possible over a small temperature range during 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. 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 occur 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 stackingfault probabilities, with the aim of interpreting the chemical bonding, seems to be irrelevant in most cases.
The quantitative solution of the diffraction problem of onedimensional disorder follows a method similar to the Ising model. As long as nextnearest neighbours alone are considered, the solution is very simple if only two possibilities for the 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 and a posteriori probabilities , respectively. In the onedimensional Ising model with two spins and the interaction energies , defining the pair probability asthe full symmetry is and .
ConsequentlyThe scattered intensity is given by where , N is the number of unit cells in the c direction and depends on , , which are the eigenvalues of the matrix From the characteristic equation one has describes a sharp Bragg reflection (from the average structure) which need not be calculated. Its intensity is simply proportional to . The second characteristic value yields a diffuse reflection in the same position if the sign is positive and in a position displaced by in reciprocal space if the sign is negative . Because of the symmetry conditions only is needed; it may be determined with the aid of the boundary conditions and the general relationThe final solution of our problem yields simply The calculation of the scattered intensity is now performed with the general formula Evaluation of this expression yields Since the characteristic solutions of the problem are real,The particlesize effect has been neglected in (4.2.5.5). This result confirms the fact mentioned above that the sharp Bragg peaks are determined by the averaged structure factor and the diffuse ones by its meansquare deviation.
For the following reason there are no examples of quantitative applications: two different structures generally have different lattice constants, so the original assumption of an undisturbed lattice geometry is not 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 longperiod superstructure cannot be explained in terms of nextnearestneighbour interactions. In principle, however, the diffraction effects are similar: instead of the diffuse peak as described by the second term in equation (4.2.5.5), satellites of first and second order etc. accompanying the Bragg peaks are observed. The study of this phenomenon (Korekawa & Jagodzinski, 1967) has in the meantime found a quantitative interpretation (Burandt et al., 1992; Kalning et al., 1997).
Obviously, the symmetry relation used in the formulae discussed above is only valid if the structures described by the are related by symmetries such as translations, rotations or combinations of the two. The type of symmetry has an important influence on the diffraction pattern.

From an historical point of view, stacking disorder in closepacked systems is most important. The three relevant positions of ordered layers are represented by the atomic coordinates in the hexagonal setting of the unit cell, or simply by the figures 1, 2, 3 in the same sequence. Structure factors refer to the corresponding positions of the same layer: hence According to the above discussion, the indices 1, 2, 3 define the reciprocallattice rows exhibiting sharp reflections only, as long as the distances between the layers are exactly equal. The symmetry conditions caused by the translation are normallyFor the case of close packing of spheres and some other problems any configuration of m layers determining the a posteriori probability , has a symmetrical counterpart where μ is replaced by (if ).
In this particular case, and equivalent relations generated by translation.
Nearestneighbour interactions do not lead to an ordered structure if the principle of close packing is obeyed (no pairs are in equal positions) (Hendricks & Teller, 1942; Wilson, 1942). Extension of the interactions to nextbutone 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: Since three of them are equivalent by translation, only four representatives have to be introduced: 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 and are introduced. In a similar way, 121 may be followed by 212 or 213 etc. For these two groups the probabilities and 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 a representative cluster of each group the one having the number 1 at the centre is chosen, e.g. 312 is the 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 closepacked metal structures or in compounds like ZnS, SiC and others. For the calculation of intensities and are needed.
According to the following scheme of sequences, any sequence of pairs is correlated with the same phase factor for 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 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
There are three solutions of the diffraction problem:

In order to calculate the intensities, one has to reconsider the symmetry of the clusters, which is different to 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 and .
The symmetry conditions of the new clusters may be determined using `probability trees' as described by Wilson (1942) and Jagodzinski (1949b). For example: , , , 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 reciprocallattice rows become diffuse if and the diffuseness increases with l. Similar behaviour results for the remaining reciprocallattice rows.
The final solution of the diffraction problem results in the following general intensity formula:Here and 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 (4.2.5.11) 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 semiquantitative interpretation of the sharp and diffuse scattering in any case, without performing the timeconsuming 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 and may be found by evaluating the line width and the position of diffuse reflections. Once this matrix has been found, a semiquantitative 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 closepacked structures with three equivalent translation vectors has been applied very frequently, even to systems that do not obey the principle of closepacking. The first quantitative explanation was published by Halla et al. (1953). It was shown there that single crystals of C_{18}H_{24} 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, CdI_{2}, CdS_{2}, mica and many other compounds. Quantitative treatments are less abundant [e.g. CdI_{2} (Martorana et al., 1986); MX_{3} 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 . This procedure is valuable if the number of stacking faults rather than a 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.
A different approach to the analysis of planar disorder in closepacked structures is given by Varn et al. (2002). With simplifying assumptions, e.g. that spacings between identical defectfree layers are independent of local stacking arrangements, average correlations between them are extracted from experimental diffractograms via Fourier analysis. Spatial patterns of the layers are then constructed by a socalled epsilon machine, which reproduces the correlations with minimal states of the stacking process. The basic statistical description of the ensemble of spatial patterns produces the stacking distribution. With this technique, stacking sequences are generated which are compared with the correlation factors and the diffractograms. The authors report an improved matching of calculated and experimental data for ZnS.
In this section, disorder phenomena that are related to chainlike structural elements in crystals are considered. This topic includes the socalled `1D crystals', where translational symmetry (in direct space) exists in one direction only – crystals in which highly anisotropic binding forces are responsible for chainlike atomic groups, e.g. compounds that exhibit a well ordered 3D framework structure with tunnels in a unique direction in which atoms, ions or molecules are embedded. Examples are compounds with platinum, iodine or mercury chains, urea inclusion compounds with columnar structures (organic or inorganic), 1D ionic conductors, polymers etc. Diffusescattering studies of 1D conductors have been carried out in connection with investigations of stability/instability problems, incommensurate structures, phase transitions, dynamic precursor effects etc. These areas are not treated here. For general reading about diffuse scattering in connection with these topics see, e.g., Comes & Shirane (1979) and references therein. Also excluded are specific problems related to polymers or liquid crystals (mesophases) (see Chapter 4.4 ) and magnetic structures with chainlike spin arrangements.
Trivial diffuse scattering occurs as 1D Bragg scattering (diffuse layers) by internally ordered chains. Diffuse phenomena in reciprocal space are due to `longitudinal' disordering within the chains (along the unique direction) as well as to `transverse' correlations between different chains over a restricted volume. Only static aspects are considered; diffuse scattering resulting from collective excitations or diffusionlike phenomena which are of inelastic or quasielastic origin are not treated here.
As found in any elementary textbook of diffraction, the simplest result of scattering by a chain with period c,is described by one of the Laue equations: which gives broadened profiles for small N. In the context of phase transitions the Ornstein–Zernike correlation function is frequently used, i.e. (4.2.5.13) is replaced by a Lorentzian: where ξ denotes the correlation length.
In the limiting case , (4.2.5.13) becomes The scattering by a real chain a(r) consisting of molecules with structure factor is therefore determined by The Patterson function iswhere the index l denotes the only relevant position L = l (the subscript M is omitted).
The intensity is concentrated in diffuse layers perpendicular to from which the structural information may be extracted. Projections are:Obviously the z parameters can be determined by scanning along a meridian (00L) through the diffuse sheets (e.g. by a diffractometer recording). Owing to intersection of the Ewald sphere with the set of planes, the meridian cannot be recorded on one photograph; successive equiinclination photographs are necessary. Only in the case of large c spacings is the meridian well approximated in one photograph.
There are many examples where a tendency to cylindrical symmetry exists: chains with pfold rotational or screw symmetry around the preferred direction or assemblies of chains (or domains) with a statistical orientational distribution around the texture axis. In this context, it should be mentioned that symmetry operations with rotational parts belonging to the 1D rod groups actually occur, i.e. not only p = 2, 3, 4, 6.
In all these cases a treatment in the frame of cylindrical coordinates is advantageous (see, e.g., Vainshtein, 1966): The integrals may be evaluated by the use of Bessel functions: .
The 2D problem is treated first; an extension to the general case is easily made afterwards.
Along the theory of Fourier series one hasor withIf contributions to anomalous scattering are neglected a(r, ψ) is a real function: Analogously, one has is a complex function; are the Fourier coefficients that are to be evaluated from the : The formulae may be used for calculation of the diffuse intensity distribution within a diffuse sheet, in particular when the chain molecule is projected along the unique axis [cf. equation (4.2.5.18)].
Special cases are:
The general 3D expressions valid for extended chains with period c [equation (4.2.5.12)] are found in an analogous way, Using a series expansion analogous to (4.2.5.23) and (4.2.5.24), one hasIn practice the integrals are often replaced by discrete summation of j atoms at positions , , : or Intensity in the lth diffuse layer is given by

Formulae concerning the reverse method (Fourier synthesis) are not given here (see, e.g., Vainshtein, 1966). There is usually no practical use for this in diffusescattering work because it is very difficult to separate out a single component . Every diffuse layer is affected by all components . There is a chance of doing so only if one diffuse layer corresponds predominantly to one Bessel function.
Deviations from strict periodicities in the z direction within one chain may be due to loss of translational symmetry of the centres of the molecules along z and/or due to varying orientations of the molecules with respect to different axes, such as azimuthal misorientation, tilting with respect to the z axis or combinations of both types. As in 3D crystals, there may or may not exist 1D structures in an averaged sense.

In real cases there are quite strong correlations between different chains, at least within small domains. Deviations from a strict (3D) order of chainlike structural elements are due to several reasons: the shape and structure of the chains, varying binding forces, and thermodynamical or kinetic considerations.
Many types of disorder occur. (1) Relative shifts parallel to the common axis while projections along this axis give a perfect 2D ordered net (`axial disorder'). (2) Relative fluctuations of the distances between the chains (perpendicular to the unique axis) with shortrange order along the transverse a and/or b directions. The net of projected chains down to the ab plane is distorted (`net distortions'). Disorder of types (1) and (2) is sometimes correlated owing to nonuniform cross sections of the chains. (3) Turns, twists and torsions of chains or parts of chains. This azimuthal type of disorder may be treated in a similar way to the case of azimuthal disorder of singlechain molecules. Correlations between axial shifts and torsions produce `screw shifts' (helical structures). Torsion of chain parts may be of dynamic origin (such as rotational vibrations). (4) Tilting or bending of the chains in a uniform or nonuniform way (`conforming/nonconforming' disorder). Many of these types and a variety of combinations between them are found in polymer and liquid crystals, and are therefore treated separately. Only some simple basic ideas are discussed here in brief.
For the sake of simplicity, the paracrystal concept in combination with Gaussians is used again. Distribution functions are given by convolution products of nextnearestneighbour distribution functions. As long as averaged lattice directions and lattice constants in a plane perpendicular to the chain axis exist, only two functions, and , are needed to describe the arrangement of nextnearest chains. Longitudinal disorder is treated as before by a third distribution function . The phenomena of chain bending or tilting may be incorporated by an x and y dependence of . Any general fluctuation in the spatial arrangement of chains is given by (mfold, pfold, qfold selfconvolution of , respectively.) are called fundamental functions. If an averaged lattice cannot be defined, more fundamental functions are needed to account for correlations between them.
By Fourier transformations, the interference function is given by If Gaussian functions are assumed, simple pictures are derived. For exampledescribes the distribution of neighbours in the x direction (with a mean distance ). Parameters , and concern axial, radial and tangential fluctuations, respectively. Pure axial distribution along c is given by projection of on the z axis; pure net distortions are given by projection on the xy plane. If the chainlike structure is neglected, the interference function describes a set of diffuse planes perpendicular to with a mean distance of . These diffuse layers broaden along H with and decrease in intensity along K and L monotonically. There is an ellipsoidshaped region in reciprocal space defined by main axes of length with a limiting surface given by , beyond which the diffuse intensity is completely smeared out. The influence of may be discussed in an analogous way.
If the chainlike arrangement parallel to c [equation (4.2.5.12)] is taken into consideration,the set of planes perpendicular to (and/or ) is subdivided in the L direction by a set of planes located at [equation (4.2.5.15)].
Longitudinal disorder is given by [equation (4.2.5.48), ] and leads to two intersecting sets of broadened diffuse layer systems.
Particular cases like pure axial distributions , pure tangential distributions (net distortions: ), uniform bending of chains or combinations of these effects are discussed in the monograph by Vainshtein (1966).
In this section general formulae for diffuse scattering will be derived that may best be applied to crystals with a well ordered average structure, characterized by (almost) sharp Bragg peaks. Textbooks and review articles concerning defects and local ordering are by Krivoglaz (1969, 1996a,b), Dederichs (1973), Peisl (1975), Schwartz & Cohen (1977), Schmatz (1973, 1983), Bauer (1979), Kitaigorodsky (1984) and Schweika (1998). A series of interesting papers on local order is given by Young (1975) and also by Cowley et al. (1979). Expressions for polycrystalline samples are given by Warren (1969) and Fender (1973).
Two general methods may be applied: (a) the average difference cluster method, where a representative cluster of scattering differences between the average structure and the cluster is used; and (b) the method of shortrangeorder correlation functions, where formal parameters are introduced.
The two methods are equivalent in principle. The cluster method is generally more convenient in cases where a single average cluster is a good approximation. This holds for small concentrations of clusters. In the literature this problem is treated in terms of fluctuations of the distribution functions as will be discussed below (Section 4.2.5.4.6). The most convenient way to derive the distribution function correctly from experimental data is the use of lowangle scattering, which generally shows one or more clear maxima caused by partly periodic properties of the distribution function. For the deconvolution of the distribution function, achieved by Fourier transformation of the corrected diffuse lowangle scattering, the reader is referred to the relevant literature. However, deconvolutions are not unique and some reasonable assumptions are necessary for a final solution.
The method of shortrangeorder parameters is optimal in cases where isolated clusters are not realized and the correlations do not extend to long distances. Otherwise periodic solutions are more convenient in most cases.
In any case, the first step towards the solution of the diffraction problem is the accurate determination of the average structure. As described in Section 4.2.4, important information on fractional occupations, interstitials and displacements of atoms (shown by unusual thermal parameters) may be derived. Unfortunately, all defects contribute to diffuse scattering; hence one has to start with the assumption that the disorder to be interpreted is predominant. Fractional occupancy of certain lattice sites by two or more kinds of atoms plays an important role in the literature, especially in metallic or ionic structures. Since vacancies may be treated as atoms with zero scattering amplitude, structures containing vacancies may be formally treated as multicomponent systems.
Since the solution of the diffraction problem should not be restricted to metallic systems with a simple (primitive) structure, we have to consider the structure of the unit cell – as given by the average structure – and the propagation of order according to the translation group separately. In simple metallic systems this difference is immaterial. It is well known that the thermodynamic problem of propagation of order in a threedimensional crystal can hardly ever be solved analytically in a general way. Some solutions have been published with the aid of the Ising model using nextnearestneighbour interactions. They are excellent for an understanding of the principles of order–disorder phenomena, but they can rarely be applied quantitatively in practical problems. Hence, methods have been developed to derive the propagation of order from the diffraction pattern by means of Fourier transformation. This method has been described qualitatively in Section 4.2.3.1 and will be used here for a quantitative application. In a first approximation, the assumption of a small number of different configurations of the unit cell is made, represented by the corresponding number of structure factors. Displacements of atoms caused by the configurations of the neighbouring cells are excluded. This problem will be treated subsequently.
The finite number of structures of the unit cell in the disordered crystal is given by Note that is defined in real space and gives the position vector of site j; if in the νth structure factor the site j is occupied by an atom of kind μ, and 0 elsewhere.
In order to apply the laws of Fourier transformation adequately, it is useful to introduce the distribution function of ,with if the cell has the structure and elsewhere.
In the definitions given above are numbers (scalars) assigned to the cell. Since all these are occupied we have where is the lattice in real space.
The structure of the disordered crystal is given by consists of points, where is the total (large) number of unit cells and denotes the a priori probability (concentration) of the νth cell occupation.
It is now useful to introduce with Introducing (4.2.5.58) into (4.2.5.57) givesSimilarlyUsing (4.2.5.60) it follows from (4.2.5.58) that Comparison with (4.2.5.59) yields Fourier transformation of (4.2.5.61) gives with The expression for the scattered intensity is therefore Because of the multiplication by L(H), the third term in (4.2.5.62) contributes to sharp reflections only. Since they are correctly given by the second term in (4.2.5.62), the third term vanishes. Hence the diffuse part is given by For a better understanding of the behaviour of diffuse scattering it is useful to return to real space: and with (4.2.5.58): Evaluation of this equation for a single term yields Since l(r) is a periodic function of points, all convolution products with l(r) are also periodic. For the final evaluation the decrease of the number of overlapping points (maximum N) in the convolution products with increasing displacements of the functions is neglected (it is assumed that there is no particlesize effect). Then (4.2.5.66) becomes If the first term in (4.2.5.67) is considered, the convolution of the two functions for a given distance n counts the number of coincidences of the function with . This quantity is given by , where is the probability of a pair occupation in the r direction.
Equation (4.2.5.67) then readswith . The function is usually called the paircorrelation function in physics.
The following relations hold: Functions normalized to unity are also in use. Obviously the following relation is valid: .
Henceis unity for . This property is especially convenient in binary systems.
With (4.2.5.68), equation (4.2.5.64) becomes and Fourier transformation yields It may be concluded from equations (4.2.5.69) that all functions may be expressed by in the case of two structure factors , . Then all are symmetric in r; the same is true for the . Consequently, the diffuse reflections described by (4.2.5.71) are all symmetric. The position of the diffuse peak depends strongly on the behaviour of ; in the case of cluster formation Bragg peaks and diffuse peaks coincide. Diffuse superstructure reflections are observed if the show some damped periodicities.
It should be emphasized that the condition may be violated for if more than two cell occupations are involved. As shown below, the possibly asymmetric functions may be split into symmetric and antisymmetric parts. From equation (4.2.3.8) it follows that the Fourier transform of the antisymmetric part of is also antisymmetric. Hence, the convolution in the two terms in square brackets in (4.2.5.71) yields an antisymmetric contribution to each diffuse peak, generated by the convolution with the reciprocal lattice L(h).
Obviously, equation (4.2.5.71) may also be applied to primitive lattices occupied by two or more kinds of atoms. Then the structure factors are merely replaced by the atomic scattering factors and the are equivalent to the concentrations of atoms . In terms of the (Warren shortrangeorder parameters) equation (4.2.5.71) reads In the simplest case of a binary system A, B,[The exponential in (4.2.5.71b) may even be replaced by a cosine term owing to the centrosymmetry of this particular case.]
It should be mentioned that the formulations of the problem in terms of pair probabilities, paircorrelation functions, shortrangeorder parameters or concentration waves (Krivoglaz, 1969, 1996a,b) are equivalent. The Patterson function may also be used when using continuous electron (or nuclear) density functions where site occupancies are implied (Cowley, 1981).
As shown above in the case of random distributions, all are zero except for . Consequently, may be replaced by According to (4.2.5.59) and (4.2.5.61), the diffuse scattering can be given by the Fourier transformation of or with (4.2.5.72) Fourier transformation gives This is the most general form of any diffuse scattering by systems ordered randomly (`Laue scattering'). Occasionally it is called `incoherent scattering' (see Section 4.2.2).
The diffuse scattering of a disordered binary system without displacements of the atoms has already been discussed in Section 4.2.5.4.1. It could be shown that all distribution functions are mutually dependent and may be replaced by a single function [cf. (4.2.5.69)]. In that case was valid for all. This condition, however, may be violated in multicomponent systems. If a tendency towards an order in a ternary system is assumed, for example, is apparently different from . In this particular case it is useful to introduce and their Fourier transforms , respectively.
The asymmetric correlation functions are therefore expressed by Consequently, (4.2.5.70) and (4.2.5.71) may be separated according to the symmetric and antisymmetric contributions. The final result isObviously, antisymmetric contributions to line profiles will only occur if structure factors of acentric cell occupations are involved. This important property may be used to draw conclusions with respect to the structure factors involved in the statistics. It should be mentioned here that the Fourier transform of the antisymmetric function is imaginary and antisymmetric. Since the last term in (4.2.5.74) is also imaginary, the product of the two factors in brackets is real, as it should be.
Even small displacements may have an important influence on the problem of propagation of order. Therefore, no structural treatments other than the introduction of formal parameters (e.g. Landau's theory) have been published in the literature. Most of the examples with really reliable results refer to binary systems and even these represent very crude approximations, as will be shown below. For this reason we shall restrict ourselves here to binary systems, although general formulae where displacements are included may be developed in a formal way.
Two kinds of atoms, and , are considered. Obviously, the position of any given atom is determined by its surroundings. Their extension depends on the forces acting on the atom under consideration. These may be very weak in the case of metals (repulsive forces, the `size effect'), but longrange effects have to be expected in ionic crystals. For the development of formulae authors have assumed that small displacements may be assigned to the paircorrelation functions by adding a phase factor , which is then expanded in the usual way: The displacements and correlation probabilities are separable if the change of atomic scattering factors in the angular range considered may be neglected. The formulae in use are given in the next section. As shown below, this method represents nothing other than a kind of average over certain sets of displacements. For this purpose, the correct solution of the problem has to be discussed. In the simplest model the displacements are due to nextnearest neighbours only. It is further assumed that the configurations rather than the displacements determine the position of the central atom and a general displacement of the centre of the first shell does not occur (there is no influence of a strain field). Obviously, the formal correlation function of pairs is not independent of displacements. This difficulty may be avoided either by assuming that the paircorrelation function has already been separated from the diffraction data, or by theoretical calculations of the correlation function (the meanfield method) (Moss, 1966; de Fontaine, 1972, 1973). The validity of this procedure is subject to the condition that the displacements have no influence on the correlation functions themselves.
The observation of a periodic average structure justifies the definition of a periodic array of origins, which normally depends on the degree of order. Local deviations of origins may be due to fluctuations in the degree of order and due to the surrounding atoms of a given site. For example, a b.c.c. lattice with eight nearest neighbours is considered. It is assumed that only these have an influence on the position of the central atom owing to different forces of the various configurations. With two kinds of atoms, there are 2^{9} = 512 possible configurations of the cluster (the central atom plus 8 neighbours). Symmetry considerations reduce this number to 28. Each is characterized by a displacement vector. Hence their a priori probabilities and the propagation of 28 different configurations have to be determined. Since each atom has to be considered as the centre once, this problem may be treated by introducing 28 different atomic scattering factors as determined from the displacements: . The diffraction problem has to be solved with the aid of the propagation of order of overlapping clusters. This is demonstrated by a twodimensional model with four nearest neighbours (Fig. 4.2.5.1). Here the central and the neighbouring cluster (full and broken lines) overlap with two sites in, for example, the x direction. Hence only neighbouring clusters with the same overlapping pairs are admitted. These restrictions introduce severe difficulties into the problem of propagation of cluster ordering which determines the displacement field. Since it was assumed that the problem of pair correlation had been solved, the cluster probabilities may be derived by calculating Only nextnearest neighbours have to be included in the product. This must be performed for the central cluster and for the reference cluster at , because all are characterized by different displacements. So far, possible displacement of the centre has not been considered; this may also be influenced by the problem of propagation of cluster ordering. These displacement factors should best be attached to the function describing the propagation of order which determines, in principle, the local fluctuations of the lattice constants (the strain field etc.). This may be understood by considering a binary system with a high degree of order but with atoms of different size. Large fluctuations of lattice constants are involved in the case of exsolution of the two components because of their different lattice parameters, but they become small in the case of superstructure formation where a description in terms of antiphase domains is reasonable (with equal lattice constants). This example demonstrates the mutual dependence of ordering and displacements, which is mostly neglected in the literature.
The method of assigning phase factors to the paircorrelation function is now discussed. Paircorrelation functions average over all pairs of clusters having the same central atom. An analogous argument holds for displacements: using pair correlations for the determination of displacements means nothing other than averaging over all displacements caused by various clusters around the same central atom. There remains the general strain field due to the propagation of order, whereas actual displacements of atoms are realized by fluctuations of configurations. Since large fluctuations of this type occur in highly disordered crystals, the displacements become increasingly irrelevant. Hence the formal addition of displacement factors to the paircorrelation function does not yield much information about the structural basis of the displacements. This situation corresponds exactly to the relationship between a Patterson function and a real structure: the structure has to be found which explains the quite complicated function completely, and its unique solution is rather difficult. We state these points here because in most publications related to this subject these considerations are not taken into account adequately. Displacements usually give rise to antisymmetric contributions to diffuse reflections. As pointed out above, the influence of displacements has to be considered as phase factors which may be attached either to the structure factors or to the Fourier transforms of the correlation functions in equation (4.2.5.71). As has been mentioned in the context of equation (4.2.5.74), antisymmetric contributions will occur if acentric structure factors are involved. Apparently, this condition is met by the phase factors of displacements. In consequence, antisymmetric contributions to diffuse reflections may also originate from the displacements. This fact can also be demonstrated if the assignment of phase factors to the Fourier transforms of the correlation functions is advantageous. In this case, equations (4.2.5.69a,b) are no longer valid because the functions become complex. The most important change is the relation corresponding to (4.2.5.69): Strictly speaking, we have to replace the a priori probabilities by complex numbers , which are determined by the position of the central atom. In this way, all correlations between displacements may be included with the aid of the clusters mentioned above. To a rough approximation it may be assumed that no correlations of this kind exist. In this case, the complex factors may be assigned to the structure factors involved. Averaging over all displacements results in diffraction effects that are very similar to a static Debye–Waller factor for all structure factors. On the other hand, the thermal motion of atoms is treated similarly. Obviously, both factors affect the sharp Bragg peaks. Hence this factor can easily be determined by the average structure, which contains a Debye–Waller factor including static and thermal displacements. It should be pointed out, however, that these static displacements cause elastic diffuse scattering, which cannot be separated by inelastic neutronscattering techniques.
A careful study of the real and imaginary parts of and and their Fourier transforms results, after some calculations, in the following relation for diffuse scattering: It should be noted that all contributions are real. This follows from the properties of Fourier transforms of symmetric and antisymmetric functions. All are antisymmetric; hence they generate antisymmetric contributions to the line profiles. In contrast to equation (4.2.5.75), the real and the imaginary parts of the structure factors contribute to the asymmetry of the line profiles.
In substitutional binary systems (with a primitive cell with only one sublattice) the Borie–Sparks method is widely used (Sparks & Borie, 1966; Borie & Sparks, 1971). The method is formulated in the shortrangeorderparameter formalism. The diffuse scattering may be separated into two parts (a) owing to shortrange order and (b) owing to static displacements.
Corresponding to the expansion (4.2.5.75), , where is given by equation (4.2.5.71b) and the correction terms and relate to the linear and the quadratic term in (4.2.5.75). The intensity expression will be split into terms of A–A, A–B, … pairs. More explicitly and with the following abbreviationsone finds (where the shorthand notation is selfexplanatory): With further abbreviations If the are independent of in a range of measurement which is better fulfilled with neutrons than with Xrays (see below), γ, δ, are the coefficients of the Fourier series: The functions Q, R, S are then periodic in reciprocal space.
The double sums over n, n′ may be replaced by where m, n, p are the coordinates of the interatomic vectors and becomes The intensity is therefore modulated sinusoidally and increases with scattering angle. The modulation gives rise to an asymmetry in the intensity around a Bragg peak. Similar considerations for reveal an intensity contribution times a sum over cosine terms which is symmetric around the Bragg peaks (i = 1, 2, 3 with h_{1} = h, h_{2} = k, h_{3} = l). This term shows a quite analogous influence of local static displacements and thermal movements: an increase of diffuse intensity around the Bragg peaks and a reduction of Bragg intensities, which is not discussed here. The second contribution has no analogue owing to the nonvanishing average displacement. The various diffuse intensity contributions may be separated by symmetry considerations. Once they are separated, the single coefficients may be determined by Fourier inversion. Owing to the symmetry constraints, there are relations between the displacements and, in turn, between the γ and Q components. The same is true for the δ, and R, S components. Consequently, there are symmetry conditions for the individual contributions of the diffuse intensity which may be used to distinguish them. In general, the total diffuse intensity may be split into only a few independent terms. The single components of Q, R, S may be expressed separately by combinations of diffuse intensities that are measured in definite selected volumes in reciprocal space. Only a minimum volume must be explored in order to reveal the behaviour over the whole reciprocal space. This minimum repeat volume is different for the single components: , Q, R, S or combinations of them.
The Borie–Sparks method has been applied very frequently to binary and even ternary systems; some improvements have been communicated by Bardhan & Cohen (1976). The diffuse scattering of the historically important metallic compound Cu_{3}Au was been studied by Cowley (1950a,b) and the paircorrelation parameters could be determined. The typical fourfold splitting was found by Moss (1966) and explained in terms of atomic displacements. The same splitting has been found for many similar compounds such as Cu_{3}Pd (Ohshima et al., 1976), Au_{3}Cu (Bessière et al., 1983) and Ag_{1−x}Mg_{x} (x = 0.15–0.20) (Ohshima & Harada, 1986). Similar paircorrelation functions have been determined. In order to demonstrate the disorder parameters in terms of structural models, computer programs were used (e.g. Gehlen & Cohen, 1965). A similar microdomain model was proposed by Hashimoto (1974, 1981, 1983, 1987). According to approximations made in the theoretical derivation, the evaluation of diffuse scattering is generally restricted to an area in reciprocal space where the influence of displacements is of the same order of magnitude as that of the paircorrelation function. The agreement between calculation and measurement is fairly good, but it should be remembered that the amount and quality of the experimental information used is low. No residual factors are so far available; these would give an idea of the reliability of the results.
The more general case of a multicomponent system with several atoms per lattice point was treated similarly by Hayakawa & Cohen (1975). Sources of error in the determination of the shortrangeorder coefficients are discussed by Gragg et al. (1973). In general, the assumption of constant produces an incomplete separation of the order and displacementdependent components of diffuse scattering. By an alternative method, by separation of the form factors from the Q, R, S functions and solving a large array of linear relationships by leastsquares methods, the accuracy of the separation of the various contributions is improved (Tibbals, 1975; Georgopoulos & Cohen, 1977; Wu et al., 1983). The method does not work for neutron diffraction. The case of planar shortrange order with corresponding diffuse intensity along rods in reciprocal space may also be treated along the Borie & Sparks method (Ohshima & Moss, 1983).
Multiwavelength methods taking advantage of the variation of the structure factor near an absorption edge (anomalous dispersion) are discussed by Cenedese et al. (1984). The same authors show that in some cases the neutron method allows contrast variation by using samples with different isotope substitution.
In general, Xray and neutron methods are complementary. The neutron method is helpful in the cases of hydrides, oxides, carbides, Al–Mg distributions etc. In favourable cases it is possible to suppress (nuclear) Bragg scattering of neutrons when isotopes are used so that for all equivalent positions. Another way to separate Bragg peaks is to record the diffuse intensity, if possible, at low values. This can be achieved either by measurement at low angles or by using long wavelengths. For reasons of absorption the latter are the domain of neutron scattering. Bragg scattering is ruled out by exceeding the Bragg cutoff. In this way, `diffuse' background owing to multiple Bragg scattering is avoided. Other diffusescattering contributions that increase with the value [thermal diffuse scattering (TDS) and scattering due to longrange static displacements] are thus also minimized. Neutrons are preferable in cases where Xrays show only a small scattering contrast: (heavy) metal lattice distortions, Huang scattering and so on should be measured at large values of . TDS can be separated by purely elastic neutron methods within the limits given by the energy resolution of an instrument. This technique is of particular importance at higher temperatures where TDS becomes remarkably strong. Neutron scattering is a good tool only in cases where (isotope/spin) incoherent scattering is not too strong. In the case of magnetic materials, confusion with paramagnetic diffuse scattering could occur. This is also important when electrons are trapped by defects which themselves act as paramagnetic centres.
In the general formula for diffuse scattering from random distributions equation (4.2.3.23a,b) may be used. Here describes the sharp Bragg maxima, while represents the contribution to diffuse scattering. Correlation effects can also be taken into account by using clusters of sufficient size if their distribution may be considered as random to a good approximation. The diffuse intensity is then given bywhere represents the difference structure factor of the νth cluster and is its a priori probability. Obviously equation (4.2.5.82) is of some use in only two cases. (1) The number of clusters is sufficiently small and meets the condition of a nearly random distribution. In principle, the structure may then be determined with the aid of refinement methods according to equation (4.2.5.82). Since the second term is assumed to be known from the average structure, the first term may be evaluated by introducing as many parameters as there are clusters involved. A special computer program for incoherent refinement has to be used if more than one representative cluster has to be introduced. In the case of more clusters constraints are necessary. (2) The number of clusters with similar structures is not limited. It may be assumed that their size distribution may be expressed by fluctuations using well known analytical expressions, e.g. Gaussians or Lorentzians. The distribution is still assumed to be random.
An early application of the cluster method was the calculation of the diffuse intensity of Guinier–Preston zones, where a single cluster is sufficient (see, e.g., Gerold, 1954; Bubeck & Gerold, 1984). Unfortunately, no refinements of cluster structures have so far been published. The full theory of the cluster method was outlined by Jagodzinski & Haefner (1967).
A different approach to analysing extended defects in crystals is simply to construct analytically disorder models that include chemical substitution and atomic displacements around an atomic defect as consequence of the surrounding local distortions. This method works if the number of `shells' of the first, second, … neighbours is limited and if short or longrange correlations between different clusters are neglected. After subtracting the underlying average structure and the calculation of the Fourier transform of the difference structure ΔF, the intensities can be fitted to the observed diffuse diffraction pattern and the parameters of the disorder model (occupancies and displacements) can be determined. For examples of this method see Neder et al. (1990) or KaiserBischoff et al. (2005). Shortrange correlations may simply be included by approximating the distribution function by summing over a finite number of defects (clusters) (Goff et al., 1999).
Molecular crystals show in principle disorder phenomena similar to those discussed in previous sections (i.e. substitutional or displacement disorder). Here we have to replace the structure factors used in the previous sections by the molecular structure factors in their various orientations. These are usually rapidly varying functions in reciprocal space which may obscure the disorder diffuse scattering. Disorder in molecular crystals is treated by Guinier (1963), Amorós & Amorós (1968), Flack (1970), Epstein et al. (1982), Welberry & Siripitayananon (1986, 1987) and others.
A particular type of disorder is very common in molecular and also in ionic crystals: the centres of masses of molecules or ionic complexes form a perfect 3D lattice but their orientations are disordered. Sometimes these solids are called plastic crystals. In comparison, the liquidcrystalline state is characterized by an orientational order in the absence of longrange positional order of the centres of the molecules. A clearcut separation is not possible in cases where translational symmetry occurs in a low dimension, e.g. in sheets or parallel to a few directions in crystal space. For discussion of these mesophases see Chapter 4.4 .
An orientationally disordered crystal may be pictured by freezing molecules in different sites in one of several orientations. Local correlations between neighbouring molecules and correlations between position and orientation may be responsible for orientational shortrange order. Thermal reorientations of the molecules are often related to an orientationally disordered crystal. Thermal vibrations of the centres of masses of the molecules, librational or rotational excitations around one or more axes of the molecules, jumps between different equilibrium positions or diffusionlike phenomena are responsible for diffuse scattering of dynamic origin. As mentioned above, the complexity of molecular structures and the associated large number of thermal modes complicate a separation from static disorder effects.
In general, high Debye–Waller factors are typical for scattering by orientationally disordered crystals. Consequently only a few Bragg reflections are observable. A large amount of structural information is stored in the diffuse background. It has to be analysed with respect to an incoherent and coherent part, elastic, quasielastic or inelastic nature, shortrange correlations within one and the same molecule and between orientations of different molecules, and cross correlations between positional and orientational disorder scattering. Combined Xray and neutron methods are therefore highly recommended.

The preceding sections of this chapter have either been related to disorder phenomena in conventional crystals defined by the presence of 3D translational symmetry, i.e. by a 3D lattice function, at least in the averaged sense, or to solids with crystalline order in only lower (2 or 1) dimensions where the ordering principle along `nonperiodic' directions shows a gradual transition from long to shortrange order, in particular to a liquidlike behaviour and, finally, to an (almost) random ordering behaviour of structural units. Socalled aperiodic crystals do not fit this treatment because aperiodicity denotes a different type of order, which is nonperiodic in 3D space but where translational order may be restored in higherdimensional (n > 3) direct space. Three types of solids are commonly included in the class of aperiodic structures: (i) incommensurately modulated structures (IMSs), where structural parameters of any kind (coordinates, occupancies, orientations of extended molecules or subunits in a structure, atomic displacement parameters) deviate periodically from the average (`basic') structure and where the modulation period is incommensurate compared to that of the basic structure; (ii) composite structures (CSs), which consist of two (or more) intergrown incommensurate substructures with mutual interactions giving rise to mutual (incommensurate) modulations; and (iii) quasicrystals (QCs), which might simply be viewed as made up by n (≥ 2) different tiles – analogous to the elementary cell in crystals – which are arranged according to specific matching rules and which are decorated by atoms or atomic clusters. Most common are icosahedral (itype) quasicrystals with 3D aperiodic order (3D `tiles') and decagonal quasicrystals (dphases) with 2D aperiodicity and one unique axis along which the QC is periodically ordered. The basic `crystallography' of aperiodic crystals is given in Chapter 4.6 of this volume; see also the review articles by Janssen & Janner (1987) and van Smaalen (2004) (and further references therein). We only point out some of the aspects here to provide the background for a short discussion about disorder diffuse scattering in aperiodic crystals.
As outlined in Section 4.6.1 , a ddimensional (dD) ideal aperiodic crystal can be defined as a dD irrational section of an nD (`hyper')crystal with nD crystal symmetry. Corresponding to the section of the nD hypercrystal with the dD (d = 1, 2, 3) direct physical (= `external' or `parallel') space we have a projection of the (weighted) nD reciprocal hyperlattice onto the dD reciprocal physical space. The occurrence of Bragg reflections as a signature of an ordered (in at least an averaged sense) aperiodic crystal form a countable dense pattern of the projected reciprocal (hyper)lattice vectors. Disorder phenomena, i.e. deviations from the periodicity in higherdimensional direct (hyper)space, are thus related to diffuse phenomena in reciprocal (hyper)space projected down to the reciprocal physical space. Therefore only fluctuations from the aperiodic order infer diffuse scattering in its true sense, which might be hard to discern in a dense pattern of discrete (Bragg) reflections. As a consequence, in the higherdimensional description of aperiodic crystals the atoms must be replaced by `hyperatoms' or `atomic surfaces', which are extended along (n − d) dimensions. If, for example, an IMS exhibits a modulation in only one direction (1D IMS), we have n = 2 and a 1D atomic surface (which is a continuous modulation function extended along the 1D `internal' or `perpendicular' subspace). The physical subspace is spanned by 1 (and +2 `unaffected') dimensions. In a second example, a decagonal QC with aperiodic order in two dimensions is described by n = 4 (plus the remaining coordinate along the unaffected periodic direction) and d = 2, i.e. by 2D atomic surfaces. The atomic surfaces are, as shown in Chapter 4.6 , continuous ndimensional objects for IMSs and CSs, and discrete ones in the case of QCs. In addition to the `hyperlattice' fluctuations within the physical (parallel) subspace we have therefore an additional quality of disorder phenomena related to positions and shape and size fluctuations of the atomic surfaces within the perpendicular subspace. Therefore, one has to consider disordering effects in aperiodic crystals related to fluctuations in the (external) physical subspace as well as in the internal subspace, but one should bear in mind that the two types are often coupled. Disordering due to displacements along directions within the physical subspace are – in the context of aperiodic crystals – commonly termed `phononlike', in contrast to `phasonlike' disorder related to displacements parallel to directions within the internal subspace. The term phason originates from a particular type of (dynamical) fluctuations of an IMS (see below).
Quite generally, small domains with periodic or aperiodic structures give rise to broadening of reflections. Diffuse satellite scattering is due to the limited coherence length of a modulation wave within a crystal or due to shortrange order of twin or nanodomains (with an `internal' modulated structure) embedded in a periodic matrix structure. This type of diffuse scattering can be treated by the rules outlined in Section 4.2.3.5. Structural fluctuations with limited correlation lengths are, for example, responsible for diffuse scattering in 1D organic conductors (Pouget, 2004). In some cases, modulated structures are intermediate phases within a limited temperature range between a high and lowtemperature phase (e.g. quartz) where the structural change is driven by a dynamical instability (a soft mode). In addition to (inelastic) softmode diffuse scattering, dynamic fluctuations of phase and amplitude of the modulation wave give rise to diffuse intensity. In particular, the phase fluctuations, phasons, give rise to lowfrequency scattering which might be observable as an additional diffuse contribution around condensing reflections.
In the case of CSs, diffuse scattering relates to interactions between the component substructures which are responsible for mutual modulations: each substructure becomes modulated with the period of the other. If one of the substructures is lowdimensional, for example chainlike structural elements embedded in tubes of a host structure, one observes diffuse planes if direct interchain correlations are mostly absent. The diffuse planes are, however, not only due to the included subsystem, but also due to corresponding modulations of the matrix structure. On the other hand, maxima superimposed on the diffuse planes reflect the influence of the matrix structure on the chain system. The thickness of the diffuse planes depends on the degree of short or longrange order along the unique direction (cf. Section 4.2.5.3). This might be either a consequence of faults in the surrounding matrix which interrupt the (longitudinal) coherence of the chains (Rosshirt et al., 1991) or, intrinsically, due to the degree of misfit between the periods of the mutually incommensurate structures of the host and guest structures. If this misfit becomes large, e.g. as a consequence of different thermal expansion coefficients of the two substructures, the modulation is only preserved within small domains and one observes a series of diffuse satellite planes (Weber et al., 2000). Equivalent considerations relate to composite systems made up from stacks of planar molecules (e.g. van Smaalen et al., 1998) or an intergrowth of layerlike substructures where the modulation is mainly due to a onedimensional stacking along the normal to the layers. Correspondingly, quite extended diffuse streaks occur at positions of the satellites in reciprocal space.
A further discussion of the more complicated disorder diffuse scattering of higher (than one) dimensionally modulated IMSs and CSs and the corresponding diffuse patterns in reciprocal space is beyond the scope of this chapter. We refer to Section 4.6.3 and the references cited therein. For examples, see also Petricek et al. (1991).
As in the case of conventional crystals, there is no unique theory of diffuse scattering by quasicrystals. Chemical disorder, phonon and phasonlike displacive disorder, topological glasslike disorder and domain disorder exist. The term domain covers those with an aperiodic structure as well as periodic approximant domains, which are also known as approximant phases (cf. Section 4.6.3 ). Approximant phases exhibit local atomic clusters which do not differ significantly from those of related aperiodic QCs. In dphases, disorder diffuse scattering occurs which is related to the periodic direction. Reviews of disorder diffuse scattering from quasicrystals are given, e.g., by Steurer & Frey (1998) and Estermann & Steurer (1998).
Chemical disorder. Many of the quasicrystals are ternary intermetallic phases consisting of atomic clusters where two components are transition metals (TMs) with only a small difference in Z (the number of electrons). Most of them exhibit a certain amount of substitutional (chemical) disorder, in particular with respect to the distribution of the TMs. This behaviour is equally true in the approximant phases. Chemical shortrange order between the TMs, if any, is largely uninvestigated because conventional Xray diffraction and electronmicroscopy methods are not sensitive enough to provide significant contrast between the TMs. If the Xray formfactor difference Δf is small, Xray patterns do not show diffuse scattering, whereas an analogous neutron pattern could reveal a diffuse component due to the different scattering contrast Δb of the TMs (where the b's are the neutron scattering lengths). In practice, the chemical disorder phenomena might be more complex as the compositional stability of the QCs is often rather extended and the atomic distribution in a sample is not always structurally homogeneous. Depending on the crystalgrowth process, microstructures may occur with locally coexisting small QC domains with fluctuating chemical content and, moreover, coexisting QC and approximant domains. Chemical disorder might also be caused by phasonic disorder.
Phonontype (static or dynamic) displacements, fluctuations or straining relate to the physical subspace coordinates giving rise to continuous local distortions of the atomic structure. Phasontype disorder describes, as indicated above, discontinuous atomic jumps between different sites in an aperiodic structure. Qualitatively, dominant phonon or phasonrelated scattering may be separated by analysing the dependence of the diffracted intensities on the components of the scattering vector in the external and the internal subspace, H_{e} (Q_{e}) and H_{i} (Q_{i}), respectively (cf. above and also Section 4.6.3 ). There are different kinds of phasonlike disorder, including random phason fluctuations, phasontype modulations and phason straining, and also shape and size fluctuations of the hyperatomic surfaces. If displacing an atomic surface (hyperatom) parallel to an internal space component by any kind of phason fluctuation which is equivalent to an infinitesimal rotation of the external (physical) space, it might happen that the hyperatom no longer intersects the physical space. Then an empty site is created, which is compensated by a (real) atom `appearing' at a different position in physical space. In addition, there might even be a change of atomic species as a hyperatom may be chemically different at different sites of the atomic surface. Randomly distributed phason strains are responsible for Braggpeak broadening and Huangtype diffuse scattering close to Bragg peaks. There are well developed theories based on the elastic theory of icosahedral (e.g. Jaric & Nelson, 1988) or decagonal (Lei et al., 1999) quasicrystals that also include phonon–phason coupling. An example of the quantitative analysis of diffuse scattering by an iphase (Al–Pd–Mn) is given by de Boissieu et al. (1995) and by a dphase (Al–Ni–Fe) by Weidner et al. (2004). Dislocations in quasicrystals have partly phonon and partly phasonlike character; a discussion of the specific dislocationrelated diffuse scattering will not be given here (cf. Section 4.6.3 ). Arcs and rings of localized diffuse scattering are observed in various iphases and could be modelled in terms of some shortrange `glasslike' ordering of icosahedral clusters (Goldman et al., 1988; Gibbons & Kelton, 1993). Phasonrelated diffuse scattering phenomena are discussed in more detail in Section 4.6.3 . Depending on the exact stoichiometry and the growth conditions, dphases very often show intergrown domain structures where the internal atomic structure of an individual domain varies between a (periodic) approximant structure, more or less strained aperiodic domains, or transient aperiodic variants (Frey & Weidner, 2002). Apart from finite size effects of domains of any kind which cause peak broadening, there are complex diffraction patterns of satellite reflections, diffuse maxima and diffuse streaking in dAl–Ni–Co and other dphases (Weidner et al., 2001). In various dphases one also observes, in addition to the Bragg layers, prominent diffuse layers perpendicular to the unique `periodic' axis. They correspond to nfold (n = 2, 4, 8) superperiods along this direction. In different dphases there is a gradual change from almost completely diffuse planes to layers of superstructure reflections (satellites). The picture of a kind of stacking disorder of aperiodic layers does not match such observations. An explanation is rather due to 1D columns of atomic icosahedral clusters along the unique direction (Steurer & Frey, 1998). The temperature behaviour of these diffuse layers was studied by in situ neutron diffraction (Frey & Weidner, 2003), which shed some light on the complicated order/disorder phase transitions in dAl–Ni–Co.
The various analytical expressions given in Section 4.2.5 are mostly rather complex, so their application is often restricted to relatively simple systems. Even in these cases analytical solutions are often not available. For larger displacements the approximations that use an expansion of the exponentials [e.g. equation (4.2.3.27)] are no longer valid. Hence there is a need for alternative approaches to tackling more complicated disorder models. In the past, optical transforms (e.g. Harburn et al., 1974, 1975) and the videographic method (Rahman, 1993) were developed for this purpose. In the first method, a 2D mask with holes with sizes that represent the scattering power of the atoms is generated, which is then subjected to coherent light (from a laser) to produce a diffraction image. Problems arise for strong scatterers requiring very large holes. These problems were overcome in the second method, where the mask is replaced by a computer image with intensities proportional to the scattering power for each pixel. With the advent of more and more powerful computers these methods are now replaced by complete computer simulations, both to set up the disorder model and to calculate the diffraction pattern.
Having established a disordered crystal with the types and positions of all atoms involved (a configuration), e.g. by using one of the methods described below, computer programs employing fastFouriertransform techniques can be used to calculate the diffraction pattern, which may be compared with the observation. It has to be borne in mind, however, that there is still a large gap between a real crystal with its ~10^{23} atoms and the one simulated by the computer with only several thousand atoms. This means that very long range correlations can not be included and have to be treated in an average manner. Furthermore, the limited size of the simulated crystal leads to termination effects, giving rise to considerable noise in the calculated diffraction pattern. Butler & Welberry (1992) have introduced a technique to avoid this problem in their program DIFFUSE by dividing the simulated crystal into smaller `lots'. For each lot the intensity is calculated and then the intensities of all lots are summed up incoherently, which finally results in a smooth intensity distribution. Note, however, that in this case longrange correlations are restricted to even smaller values. To overcome this problem Boysen (1997) has proposed a method for suppressing the subsidiary maxima by multiplying the scattering density of the model crystal by a suitably designed weighting function simulating the effect of the instrumental resolution function.
A very versatile computer program, DISCUS, which allows not only the calculation of the scattering intensities but also allows the model structures to be built up in various ways, has been designed by Proffen & Neder (1999). It contains modules for reverse Monte Carlo (RMC, see below) simulation, the calculation of powder patterns, RMCtype refinement of pairdistribution functions (PDFs, see below) and many other useful tools for analysing disorder diffuse scattering.
Other computer programs have been developed to calculate diffuse scattering, some for specific tasks, such as SERENA (Micu & Smith, 1995), which uses a collection of atomic configurations calculated from a molecular dynamics simulation of molecular crystals.
Several well established methods can be used to create the simulated crystal on the computer. With such a crystal at hand, it is possible to calculate various thermodynamical properties and study the effect of specific parameters of the underlying model. By Fourier transformation, one obtains the diffraction pattern of the total scattering, i.e. the diffuse intensities and the Bragg peaks. The latter may lead to difficulties due to the termination effects mentioned in Section 4.2.7.2. These may be circumvented by excluding regions around the Bragg peaks (note, however, that in this case valuable information about the disorder may be lost), by subtracting the average structure or by using the approximation of Boysen (1997) mentioned above. A major advantage of such modelling procedures is that realistic physical models are introduced at the beginning, providing further insight into the pair interactions of the system, which can only be obtained a posteriori from the correlation parameters or fluctuation wave amplitudes derived from one of the methods described in Section 4.2.5.
Molecular dynamics (MD) techniques have been developed to study the dynamics of a system. They may also be used to study static disorder problems (by taking time averages or snapshots), but they are particularly useful in the case of dynamic disorder, e.g. diffusing atoms in superionic conductors. The principle is to set up a certain configuration of atoms with assumed interatomic potentials Φ_{ij}(r_{ij}) and subject them to Newton's equation of motion,where the force F_{i}(t) is calculated from the gradient of Φ. The equations are solved approximately by replacing the differential dt by a small but finite time step Δt to find new positions r_{i}(t + Δt). This is repeated until an equilibrium configuration is found. MD techniques are quite useful if only shortrange interactions are effective, even allowing the transfer of potential parameters between different systems, but are less reliable in the presence of significant longrange interactions.
Monte Carlo (MC) methods appear to be more suited to the study of static disorder and many examples of their application can be found in the literature. Different variants allowing simulations and refinements have been applied:

All of the different modelling techniques mentioned in this section have their specific merits and limitations and have contributed much to our understanding of disorder in crystalline materials following the interpretation of the corresponding diffuse scattering. It should be borne in mind, however, that application of these methods is still far from being routine work and it requires a lot of intuition to ensure that the final model is physically and chemically reasonable. In particular, it must always be ensured that the average structure remains consistent with that derived from the Bragg reflections alone. This may be done by keeping the Bragg reflections, i.e. by analysing the total scattering, or by designing special algorithms, e.g. by swapping two atoms at the same time (Proffen & Welberry, 1997). Moreover, possible traps and corrections like local minima, termination errors, instrumental resolution, statistical noise, inelasticity etc. must be carefully considered. All this means that the analytical methods outlined in Section 4.2.5 keep their value and should be preferred wherever possible.
The newly emerging technique of using full quantummechanical ab initio calculations for structure predictions along with MD simulations may also be applied to disordered systems. The limited currently available computer power, however, restricts this possibility to rather simple systems and small simulation box sizes, but, with the expected further increase of computer capacities, this may open up new perspectives for the future.
Singlecrystal and powder diffractometry are used in diffuse scattering work. Conventional and more sophisticated special techniques and instruments are now available at synchrotron facilities and modern neutron reactor and spallation sources. The full merit of the dedicated machines may be assessed by inspecting the corresponding handbooks, which are available upon request from the facilities. In the following, some common important aspects that should be considered when planning and performing a diffusescattering experiment are summarized and a short overview of the techniques is given. Methodological aspects of diffuse scattering at low angles, i.e. smallanglescattering techniques, and highresolution singlecrystal diffractometers are excluded. Instruments of the latter type are used when diffuse intensities beneath Bragg reflections or reflection profiles and tails must be analysed to study longrange distortion fields around single defects or small defect aggregates. In the case of small defect concentrations, the crystal structure remains almost perfect and the dynamical theory of diffraction is more appropriate. This topic is beyond the scope of this chapter.
In general, diffuse scattering is weak in comparison with Bragg scattering, and is anisotropically and inhomogeneously distributed in reciprocal space. The origin may be a static phenomenon or a dynamic process, giving rise to elastic or inelastic (quasielastic) diffuse scattering, respectively. If the disorder problem relates to more than one structural element, different parts of the diffuse scattering may show different behaviour in reciprocal space and/or on an energy scale. Therefore, before starting an experiment, some principal aspects should be considered: Is there need for Xray and/or neutron methods? What is the optimum wavelength or energy (band), or does a `white' technique offer advantages? Can focusing techniques be used without too strong a loss of resolution and what are the best scanning procedures? How can the background be minimized? Has the detector a low intrinsic noise and a high dynamic range?
On undertaking an investigation of a disorder problem by an analysis of the diffuse scattering, an overall picture should first be recorded by Xray diffraction. Several sections through reciprocal space help to define the problem. For this purpose `oldfashioned' film methods may be used, where the classical film is now commonly replaced by an imaging plate (IP) or a chargecoupled device (CCD) camera (see below). Clearly, short crystaltodetector distances provide larger sections and avoid long exposure times, but suffer from spatial resolution. Distorted sections through the reciprocal lattice, such as produced by the Weissenberg method, may be transformed into a form suitable for easy interpretation (Welberry, 1983). The transformation of diffuse data measured using an IP or CCD requires suitable software for the specific type of detector and is not always routinely available (Estermann & Steurer, 1998).
Standingcrystal techniques in combination with monochromatic radiation, usually called monochromatic Laue techniques (see, e.g., Flack, 1970), save exposure time, which is particularly interesting for `inhouse' laboratory diffuse Xray work. The Noromosic technique (Jagodzinski & Korekawa, 1973) is characterized by a convergent monochromatic beam which simulates an oscillation photograph over a small angular range. Heavily overexposed images, with respect to Bragg scattering, allow for sampling of diffuse intensity if a crystal is oriented in such a way that there is a well defined section between the Ewald sphere and the diffuse phenomenon under consideration. By combining single Noromosic photographs, Weissenberg patterns can be simulated. This relatively tedious method is often unavoidable because the heavily overexposed Bragg peaks obscure weak diffuse phenomena. Furthermore, standing pictures at distinct crystal settings in comparison with conventional continuous recording are frequently sufficient in diffuse scattering work and save time. Longexposure Weissenberg photographs are therefore not equivalent to a smaller set of standing photographs. In this context it should be mentioned that a layerline screen has not only the simple function of a selecting diaphragm, but the gap width determines the resolution volume within which diffuse intensity is collected (Welberry, 1983). For further discussion of questions of resolution see below. A comparison of Weissenberg and diffractometer methods for the measurement of diffuse scattering is given by Welberry & Glazer (1985). It should be pointed out, however, that diffractometer methods at synchrotron sources become more widely used if only very small (micrometresized) singlecrystal specimens have to be used to study a disorder problem.
The basic arguments for using neutrondiffraction methods were given in Section 4.2.2.2: (i) the different interactions of Xrays and neutrons with matter; (ii) the lower absorption of neutrons, in particular when using longer (> 0.15 nm) wavelengths; and (iii) the matching of the energy of thermal neutrons with that of the phonons that contribute to the TDS background, and, in consequence, to separate it by a `purely' elastic measurement. A comparative consideration of synchrotron and neutronrelated diffuse work on disordered alloys is given by Schweika (1998). Specific aspects of neutron diffraction and instruments are discussed at the end of this section.
Intensities: As mentioned above, diffuse intensities are usually weaker by several orders of magnitude than Bragg data. Therefore intense radiation sources are needed. Even a modern Xray tube is a stronger source, defined by the flux density from the anode (number of photons cm^{−2} s^{−1}), than modern neutron sources. For this reason most experimental work which can be performed with Xrays should be. In home laboratories the intense characteristic spectrum of an Xray tube is commonly used. At synchrotron storage rings any wavelength from a certain range can be selected. The extremely high brilliance (number of quanta cm^{−2}, sr^{−1}, s^{−1} and wavelength interval) of modern synchrotron sources is, however, unnecessary in the case of slowly varying diffuse phenomena. In these cases, an experimental setup at a laboratory rotating anode is competitive and often even superior if specimens with sufficient size are available. Various aspects of diffuse Xray work at a synchrotron facility are discussed by Matsubara & Georgopoulos (1985), Oshima & Harada (1986) and Ohshima et al. (1986). Diffuse neutrondiffraction work can only be performed on a highflux reactor or on a powerful spallation source. Highly efficient monochromator systems are needed when using a crystal diffractometer. Timeofflight (TOF) neutron diffractometers at pulsed (spallation) neutron sources are equivalent to conventional diffractometers at reactors (Windsor, 1982). The merits of diffuse neutron work at pulsed sources have been discussed by Nield & Keen (2001).
Wavelength: The choice of an optimum wavelength is important with respect to the problem to be solved. For example, point defects cause diffuse scattering to fall off with increasing scattering vector; shortrange ordering between clusters causes broad peaks corresponding to large d spacings; latticerelaxation processes induce a broadening of the interferences; static modulation waves with long periods give rise to satellite scattering close to Bragg peaks. In all these cases, a long wavelength is preferable due to the higher resolution. The use of a long wavelength is also profitable when the main diffuse contributions can be recorded within an Ewald sphere as small as the Bragg cutoff of the sample at H ≃ 1/(2d_{max}). `Contamination' by Bragg scattering can thus be avoided. This is also advantageous from a different point of view: because the contribution of thermal diffuse scattering increases with increasing scattering vector H, the relative amount of this component becomes negligibly small within the first reciprocal cell. However, one has to take care with the absorption of longwavelength neutrons.
On the other hand, short wavelengths are needed where atomic displacements play the dominant role. If diffuse peaks in large portions of the reciprocal space, or diffuse streaks or planes, have to be recorded up to high values of the scattering vector in order to decide between different structural disorder models, hard Xrays or hot neutrons are needed. For example, highenergy Xrays (65 keV) provided at a synchrotron source were used by Welberry et al. (2003) to study diffuse diffraction from ceramic materials and allowed studies with better d resolution.
The λ^{3} dependence of the scattered intensity, in the framework of the kinematical theory, is a crucial point for exposure or dataacquisition times. Moreover, the accuracy with which an experiment can be carried out suffers from a short wavelength: generally, momentum as well as energy resolution are lower. For a quantitative estimate detailed considerations of resolution in reciprocal space (and energy) are needed.
A specific wavelength aspect concerns the method of (Xray) anomalous dispersion, which may also be used in diffusescattering work. It allows the contrast and identification of certain elements in a disordered structure. Even small concentrations of impurity atoms or defects as low as 10^{−6} can be determined by this method if the impurity atoms are located at specific sites, e.g. in domain boundaries, or if certain other defect structures exist with characteristic diffuse scattering in reciprocal space. The (weak) diffuse scattering can then be contrasted by tuning the wavelength across an absorption edge of the particular atomic species. To avoid strong fluorescence background scattering such an experiment is usually performed at different wavelengths (energies) at the `low absorption side' close to an edge. The merit of this method was demonstrated by Berthold & Jagodzinski (1990), who analysed diffuse streaks due to boundaries between lamellar domains in an albite feldspar. Similar element contrast can be achieved in neutron diffuse work if using specimens with different isotopes, e.g. H–D exchange.
Monochromacy: In classical crystal diffractometry, monochromatic radiation is used in order to eliminate broadening effects due to the wavelength distribution. Focusing monochromators and other focusing devices (guides, mirrors) help to overcome the lack of luminosity. A focusing technique is very helpful for deciding between geometrical broadening and `true' diffuseness. In a method that is used with some success, the sample is placed in a monochromatic divergent beam with its selected axis lying in the scattering plane of the monochromator (Jagodzinski, 1968). The specimen is fully embedded in the incident beam, which is focused onto a 2D detector. Using this procedure the influence of the sample size is suppressed in one dimension. In whitebeam (neutron) timeofflight diffractometry the time resolution is the counterpart to the wavelength resolution. This is discussed in some detail in the textbook of Keen & Nield (2004).
Detectors: Valuable developments with a view to diffusescattering work are multidetectors (see, e.g., Haubold, 1975) and positionsensitive detectors for Xrays (Arndt, 1986a,b) and neutrons (Convert & Forsyth, 1983). A classical `2D area detector' is the photographic film – nowadays of less importance – which has to be `read out' by microdensitometer scanning. Important progress in recording diffuse Xray data has been made by the availability of multiwire detectors, IPs and CCD cameras (Estermann & Steurer, 1998). For 2D positionsensitive (proportional) counters problems may arise from inhomogeneities of the wire array as well as the limited dynamic range when a Bragg reflection is accidentally recorded. IP systems have the major advantage of a larger dynamic range, 10^{5}–10^{6}, compared to 10^{2}–10^{3} for an Xray film. IPs can also be used in diffuse experiments carried out with hard (65 keV) Xrays (Welberry et al., 2003). IPs are also available for neutron work, where the necessary transformation of the detected neutrons into light signals is provided by special neutronabsorbing converter foils (Niimura et al., 2003). The problem of an intrinsic sensitivity to gamma radiation may be overcome by protection with a thin sheet of lead. IPs allow data collection either in plane geometry or in simple rotation or Weissenberg geometry, both in combination with low and hightemperature devices. CCD detectors are well suited for Xray diffuse scattering, and when used in combination with converter foils they can also be used for the detection of neutrons. A basic prerequisite is a low intrinsic noise, which can be achieved by cooling the CCD with liquid nitrogen. With these detectors, extended diffuse data sets can be collected by rotating the sample in distinct narrow steps around a spindle axis over several χ settings and subsequent oscillations over a small angular range. Thus large parts of the reciprocal space can be recorded. An example is given by Campbell et al. (2004), who studied subtle defect structures in the microporous framework material mordenite. Linear positionsensitive detectors are mainly used in powder work, but can also be used for recording diffuse scattering by single crystals. By combining a linear positionsensitive detector and the TOF method, a whole area in reciprocal space is accessible simultaneously (Niimura et al., 1982; Niimura, 1986).
Diffuse data recorded with IPs or CCDs are commonly evaluated by commercial software supplied by the manufacturer. These software packages include corrections for extracted Bragg data, but no special software tools for treating extended (diffuse) data are commonly available. One particular problem relates to the background definition for diffuse IP data. Even after subtraction of electronic noise, there remains considerable uncertainty about the amount of true background scattering from the sample as recorded by an IP scanner. The definitions of errors and error maps remain doubtful as long as the true conversion rate between the captured neutron or photon versus the recorded optical signal is unknown.
Absorption: Special attention must be paid to absorption phenomena, in particular when (in the Xray case) an absorption edge of an element of the sample is close to the wavelength. Then strong fluorescence scattering may completely obscure weak diffusescattering phenomena. In comparison with Xrays, the generally lower absorption coefficients of neutrons make absolute measurements easier. This also allows the use of larger sample volumes, which is not true in the Xray case. Moreover, the question of sample environment is less serious in the neutron case than in the Xray case. However, the availability of hard Xrays at a synchrotron source (Butler et al., 2000; Welberry et al., 2003) makes the Xray absorption problem less serious: irregularly shaped specimens without special surface treatment could be used. There is also no need for complicated absorption corrections and the separation of fluorescence background is rather simple.
Extinction: An extinction problem does not generally exist in diffusescattering work.
Background: An essential prerequisite for a diffusescattering experiment is the careful suppression of background scattering. Incoherent Xray scattering by a sample gives continuous blackening in the case of fluorescence, or scattering at high 2θ angles owing to Compton scattering or `incoherent' inelastic effects. Protecting the image plate with a thin Al or Ni foil is of some help against fluorescence, but also attenuates the diffuse intensity. Obviously, energydispersive counter methods are highly efficient in this case (see below). Air scattering produces a background at low 2θ angles which may easily be avoided by special slit systems and evacuation of the camera.
In Xray or neutron diffractometer measurements, incoherent and multiple scattering contribute to a background which varies only slowly with 2θ and can be subtracted by linear interpolation or fitting a smooth curve, or can even be calculated quantitatively and then subtracted. In neutron diffraction there are rare cases when monoisotopic and `zeronuclearspin' samples are available and, consequently, the corresponding incoherent scattering part vanishes completely. In some cases, a separation of coherent and incoherent neutron scattering is possible by polarization analysis (Gerlach et al., 1982). An `empty' scan can take care of instrumental background contributions. Evacuation or controlledatmosphere studies need a chamber, which may give rise to spurious scattering. This can be avoided if no part of the vacuum chamber is hit by the primary beam. The problem is less serious in neutron work. The mounting of the specimen, e.g., on a silica fibre with cement, poorly aligned collimators and beam catchers are further sources. Sometimes a specimen has to be enclosed in a capillary, which will always be hit by the incident beam. Careful and tedious experimental work is necessary in the case of low and hightemperature (or pressure) investigations, which have to be carried out in many disorder problems. While the experimental situation is again less serious in neutron scattering, there are problems with scattering from walls and containers in Xray work. Because TDS dominates at high temperatures and in the presence of a static disorder problem, a quantitative separation can rarely be carried out in the case of high experimental background. Calculation and subtraction of the TDS is possible in principle, but difficult in practice. If the disorder problem in which one is interested in does not change with temperature, a lowtemperature experiment can be carried out. Another way to get rid off TDS, at least partly, is by using a neutron diffractometer with an additional analyser set to zeroenergy transfer.
Resolution: A quantitative analysis of diffusescattering data is essential for reaching a definite decision about a disorder model, but – in many cases – it is cumbersome. By comparing the calculated and corrected experimental data the magnitudes of the parameters of the structural disorder model may be derived. A careful analysis of the data requires, therefore, after separation of the background (see above), corrections for polarization (in the Xray case), absorption (in conventional Xray work) and resolution. Detailed considerations of instrumental resolution are necessary; the resolution depends, in addition to other factors, on the scattering angle and implies intensity corrections analogous to the Lorentz factor used in structure analysis from sharp Bragg reflections.
Resolution is conveniently described by the function , which is defined as the probability of detecting a photon or neutron with momentum transfer when the instrument is set to measure . This function R depends on the instrumental parameters (such as the collimations, the mosaic spread of monochromator and the scattering angle) and the spectral width of the source. Fig. 4.2.8.1 shows a schematic sketch of a diffractometer setting. Detailed considerations of resolution volume in Xray and neutron diffractometry are given by Sparks & Borie (1966) and by Cooper & Nathans (1968a,b), respectively. If a tripleaxis (neutron) instrument is used, for example in a purely elastic configuration, the set of instrumental parameters includes the mosaicity of the analyser and the collimations between the analyser and detector. In general, Gaussians are assumed to parameterize the mosaic distributions and transmission functions. Sophisticated resolutioncorrection programs are usually provided at any standard instrument for carrying out experiments at synchrotron and neutron facilities.
The general assumption of Gaussians is not too serious in the Xray case (Iizumi, 1973). Restrictions are due to absorption, which makes the profiles asymmetric. Boxlike functions are considered to be better for the spectral distribution or for large apertures (Boysen & Adlhart, 1987). These questions are treated in some detail by Klug & Alexander (1954). The main features, however, may also be derived by the Gaussian approximation. In practice, the function R may be obtained either by calculation from the known instrumental parameters or by measuring Bragg peaks from a perfect unstrained crystal. In the latter case, the intensity profile is given solely by the resolution function. Normalization with the Bragg intensities is also useful in order to place the diffusescattering intensity on an absolute scale.
In conventional singlecrystal diffractometry the measured intensity is given by the convolution product of with R, where describes the scattering cross section for the disorder problem. In a more accurate form the mosaicity of the sample has to be included: . The mosaic block distribution around a most probable vector is described by : .
In (4.2.8.1) all factors independent of 2θ are neglected. All intensity expressions have to be calculated from equations (4.2.8.1) or (4.2.8.1a).
The intensity variation of diffuse peaks with 2θ measured with a single detector was studied in detail by Yessik et al. (1973). In principle, all special cases are included there. In practice, however, some important simplifications can be made if is either very broad or very sharp compared with R, i.e. for Bragg peaks, sharp streaks, `thin' diffuse layers or extended 3D diffuse peaks (Boysen & Adlhart, 1987).
In the latter case, the cross section may be treated as nearly constant over the resolution volume so that the corresponding `Lorentz' factor is independent of 2θ: For a diffuse plane within the scattering plane with very small thickness and slowly varying cross section within the plane, one derives for a point measurement in the planeexhibiting an explicit dependence on θ (, and determine an effective vertical divergence before the sample, the divergence before the detector and the vertical mosaic spread of the sample, respectively).
In the case of relaxed vertical collimations i.e. again independent of θ.
Scanning across the diffuse layer in a direction perpendicular to it one obtains an integrated intensity which is also independent of 2θ. This is even true if approximations other than Gaussians are used.
If, on the other hand, an equivalent diffuse plane is positioned perpendicular to the scattering plane, the equivalent expression for of a point measurement is given by where ψ gives the angle between the vector and the line of intersection between the diffuse plane and the scattering plane. The coefficients , , , and η are either instrumental parameters or functions of them, defining horizontal collimations and mosaic spreads. In the case of a sharp Xray line (produced, for example, by filtering) the last two terms in equation (4.2.8.4) vanish.
The use of integrated intensities from individual scans perpendicular to the diffuse plane, now carried out within the scattering plane, again gives a Lorentz factor independent of 2θ.
In the third fundamental special case, diffuse streaking along one reciprocal direction within the scattering plane (with a narrow cross section and slowly varying intensity along the streak), the Lorentz factor for a point measurement may be expressed by the product where ψ now defines the angle between the streak and . The integrated intensity taken from an H scan perpendicular to the streak has to be corrected by a Lorentz factor which is equal to [equation (4.2.8.3)]. In the case of a diffuse streak perpendicular to the scattering plane, a relatively complicated equation holds for the corresponding Lorentz factor (Boysen & Adlhart, 1987). Again, simpler expressions hold for integrated intensities from H scans perpendicular to the streaks. Such scans may be performed in the radial direction (corresponding to a θ–2θ scan),or perpendicular to the radial direction (within the scattering plane) (corresponding to an ω scan), Note that only the radial scan yields a simple θ dependence .
From these considerations it is recommended that integrated intensities from scans perpendicular to a diffuse plane or a diffuse streak should be used in order to extract the disorder cross sections. For other scan directions, which make an angle α with the intersection line (diffuse plane) or with a streak, the L factors are simply and , respectively.
One point should be emphasized: since in a usual experiment with a single counter the integration is performed over an angle via a general scan, an additional correction factor arises: β is the angle between and the scan direction and defines the coupling ratio between the rotation of the crystal around a vertical axis and the rotation of the detector shaft. The socalled 1:2 and ωscan techniques are most frequently used, where and 90°, respectively.
Whitebeam techniques: Techniques for the measurement of diffuse scattering using a white spectrum are common in neutron diffraction. Owing to the relatively low velocity of thermal or cold neutrons, TOF methods in combination with timeresolving detector systems placed at a fixed angle 2θ allow for a simultaneous recording along a radial direction through the origin of reciprocal space (see, e.g., Turberfield, 1970; Bauer et al., 1975). The scan range is limited by the Ewald spheres corresponding to and , respectively. With several such detector systems placed at different angles or a 2D detector several scans may be carried out simultaneously during one neutron pulse.
An analogue of neutron TOF diffractometry in the Xray case is a combination of a white source of Xrays and an energydispersive detector. This technique, which has been known in principle for a long time, suffered from relatively weak white sources. With the development of highpower Xray generators and synchrotron sources this method has now become highly interesting. Its use in diffusescattering work (in particular, the effects on resolution) is discussed by Harada et al. (1984).
Some examples of neutron instruments dedicated to diffuse scattering are the diffractometers D7 at the Institut Laue–Langevin (ILL, Grenoble), DNS at the Forschungsneutronenquelle Heinz MaierLiebnitz (FRMII, Munich), G4–4 at the Laboratoire Léon Brillouin (LLB, Saclay) or SXD at ISIS, Rutherford Appleton Laboratory (RAL, UK). The instrument DNS uses a polarization analysis for diffuse scattering (Schweika & Böni, 2001; Schweika, 2003) and the SXD diffractometer is a TOF instrument using a (pulsed) white beam (Keen & Nield, 2004). All these instruments are equipped with banks of detectors. The singlecrystal diffractometer D19 at the ILL, equipped with a multiwire area detector, is also suitable for collecting diffuse data. The flatcone machine E2 at the Hahn–Meitner Institut (HMI, Berlin) is equipped with a bank of area detectors, has an option to record higherorder layers and can also be operated in an `elastic' mode with a multicrystal analyser. The instrument D10 at the ILL is a versatile instrument which can be operated as a lowbackground twoaxis or threeaxis diffractometer with several further options. Neutron diffractometers that have recently become operational are BIX3 at the Japan Atomic Energy Research Institute (JAERI, Japan), LADI at the ILL, where neutronsensitive IPs are used for macromolecular work (for a comparison see Niimura et al., 2003), and RESI at the Forschungsneutronenquelle Heinz MaierLiebnitz (FRMII, Munich) for common solidstate investigations. Information about all these instruments can be found in the respective handbooks or on the websites of the facilities.
The diffuse background in powder diagrams also contains valuable information about disorder. Only in very simple cases can a model be deduced from a powder pattern alone, but a refinement of a known disorder model can favourably be carried out, e.g. the temperature dependence may be studied. On account of the intensity integration, the ratio of diffuse intensity to Bragg intensity is enhanced in a powder pattern. Moreover, a powder pattern contains, in principle, all the information about the sample and might thus reveal more than singlecrystal work. However, in powderdiffractometer experiments preferred orientations and textures could lead to a complete misidentification of the problem. Singlecrystal experiments are generally preferable in this respect. Nevertheless, highresolution powder investigations may give quick supporting information, e.g. about superlattice peaks, split reflections, lattice strains, domainsize effects, latticeconstant changes related to a disorder effect etc.
Evaluation of diffusescattering data from powder diffraction follows the same theoretical formulae developed for the determination of the radial distribution function for glasses and liquids. The final formula for random distributions may be given as (Fender, 1973) represents the number of atoms at distance from the origin. An equivalent expression for a substitutional binary alloy is
A quantitative calculation of a diffuse background is also helpful in combination with Rietveld's method (1969) for refining an averaged structure by fitting Bragg data. In particular, for highly anisotropic diffuse phenomena characteristic asymmetric line shapes occur.
The calculation of these line shapes is treated in the literature, mostly neglecting the instrumental resolution (see, e.g., Warren, 1941; Wilson, 1949; Jones, 1949; and de CourvilleBrenasin et al., 1981). This is not justified if the variation of the diffuse intensity becomes comparable with that of the resolution function, as is often the case in neutron diffraction. The instrumental resolution may be incorporated using the resolution function of a powder instrument (Caglioti et al., 1958). A detailed analysis of diffuse peaks is given by Yessik et al. (1973) and the equivalent considerations for diffuse planes and streaks are discussed by Boysen (1985). The case of 3D random disorder (incoherent neutron scattering, monotonous Laue scattering, averaged TDS, multiple scattering or shortrangeorder modulations) is treated by Sabine & Clarke (1977).
In polycrystalline samples the cross section has to be averaged over all orientations: where n_{c} is number of crystallites in the sample; this averaged cross section enters the relevant expressions for the convolution product with the resolution function.
A general intensity expression may be written as (Boysen, 1985)P denotes a scaling factor that depends on the instrumental luminosity, T is the shortest distance to the origin of the reciprocal lattice, is the corresponding symmetryinduced multiplicity, contains the structure factor of the structural units and the type of disorder, and describes the characteristic modulation of the diffuse phenomenon of dimension n in the powder pattern. These expressions are given below with the assumption of Gaussian line shapes of width D for the narrow extension(s). The formulae depend on a factor , where describes the dependence of the Bragg peaks on the instrumental parameters U, V and W (see Caglioti et al., 1958), and .

As mentioned in Section 4.2.7.3.2, the atomic pairdistribution function (PDF), which is classically used for the analysis of the atomic distributions in liquids, melts or amorphous samples, can also be used to gain an understanding of disorder in crystals. The PDF is the Fourier transform of the total scattering. The measurement of total scattering is basically similar to recording Xray or neutron powder patterns. The success of the method depends, however, decisively on various factors: (i) The availability of a large data set, i.e. reliable intensities up to high H values, in order to get rid of truncation ripples, which heavily influence the interpretation. Currently, values of H_{max} of more than 7 Å^{−1} can be achieved either with synchrotron Xrays [at the European Synchrotron Radiation Facility (ESRF) or Cornell High Energy Synchrotron Source (CHESS)] or with neutrons from reactors (e.g. instrument D4 at the ILL) and spallation sources [at ISIS or at LANSCE (instrument NPDF)]. (ii) High H resolution. (iii) High intensities, in particular at high H values. (iv) Low background of any kind which does not originate from the sample. Highquality intensities are therefore to be extracted from the raw data by taking care of an adequate absorption correction, correction of multiplescattering effects, separation of inelastically scattered radiation (e.g. Compton scattering) and careful subtraction of `diffuse' background which is not the `true' diffuse scattering from the sample. These conditions are in practice rather demanding. A further detailed discussion is beyond the scope of this chapter, but a more thorough discussion is given, e.g., by Egami (2004), and some examples are given by Egami & Billinge (2003).
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