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. 4.3, pp. 540546
https://doi.org/10.1107/97809553602060000775 Chapter 4.3. Diffuse scattering in electron diffractionDiffuse scattering in electron diffraction arises from inelastic scattering due to electronic excitations, from thermal diffuse scattering and from scattering from crystal defects or disorder. Thermal diffuse scattering and scattering from defects or disorder may be treated using the kinematical approximation. The most prominent contribution to the diffuse background in electron diffraction, however, is the inelastic scattering at low angles. This scattering can only be described using the kinematical approximation if the crystal is very thin and composed of light elements. For other specimens, Bragg scattering and multiple diffuse scattering have to be taken into account using a dynamical scattering model. Multislice calculations for diffraction and imaging, and the qualitative interpretation of the diffuse scattering of electrons, are also discussed. 
The origins of diffuse scattering in electrondiffraction patterns are the same as in the Xray case: inelastic scattering due to electronic excitations, thermal diffuse scattering (TDS) from atomic motions, scattering from crystal defects or disorder. For diffraction by crystals, the diffuse scattering can formally be described in terms of a nonperiodic deviation from the periodic, average crystal potential, :where may have a static component from disorder in addition to timedependent fluctuations of the electron distribution or atomic positions.
In the kinematical case, the diffuse scattering can be treated separately. The intensity as a function of the scattering variable and energy transfer is then given by the Fourier transform of and may also be written as the Fourier transform of a correlation function representing fluctuations in space and time (see Cowley, 1981). When the energy transfers are small – as with TDS – and hence not measured, the observed intensity corresponds to an integral over :and alsowhere the brackets may indicate a time average, an expectation value, or a spatial average over the periodicity of the lattice in the case of static deviations from a periodic structure.
The considerations of TDS and static defects and disorder of Chapters 4.1 and 4.2 thus may be applied directly to electron diffraction in the kinematical approximation when the differences in experimental conditions and diffraction geometry are taken into account.
The most prominent contribution to the diffuse background in electron diffraction, however, is the inelastic scattering at low angles arising mainly from the excitation of outer electrons. This is quite different from the Xray case where the inelastic (`incoherent') scattering, , goes to zero at small angles and increases to a value proportional to Z for high values of . The difference is due to the Coulomb nature of electron scattering, which leads to the kinematical intensity expression , emphasizing the smallangle region. At high angles, the inelastic scattering from an atom is then proportional to , which is considerably less than the corresponding elastic scattering which approaches (Section 2.5.2 ) (see Fig. 4.3.1.1).

Comparison between the kinematical inelastic scattering (full line) and elastic scattering (broken) for electrons and Xrays. Values for silicon [Freeman (1960) and IT C (2004)]. 
The kinematical description can be used for electron scattering only when the crystal is very thin (10 nm or less) and composed of light atoms. For heavy atoms such as Au or Pb, crystals of thickness 1 nm or more in principal orientations show strong deviations from kinematical behaviour. With increasing thickness, dynamical scattering effects first modify the sharp Bragg reflections and then have increasingly significant effects on the diffuse scattering. Bragg scattering of the diffuse scattering produces Kikuchi lines and other effects. Multiple diffuse scattering broadens the distribution and smears out detail. As the thickness increases further, the diffuse scattering increases and the Bragg beams are reduced in intensity until there is only a diffuse `channelling pattern' where the features depend in only a very indirect way on the incidentbeam direction or on the sources of the diffuse scattering (Uyeda & Nonoyama, 1968).
The multiplescattering effects make the quantitative interpretation of diffuse scattering more difficult and complicate the extraction of particular components, e.g. disorder scattering. Much of the multiple scattering involves inelastic scattering processes. However, electrons that have lost energy of the order of 1 eV or more can be subtracted experimentally by use of electron energy filters (Krahl et al., 1990; Krivanek et al., 1992) which are commercially available. Measurement can be made also of the complete scattering function , but such studies have been rare. Another significant improvement to quantitative measurement of diffuse electron scattering is offered by new recording devices: slowscan chargecoupledevice cameras (Krivanek & Mooney, 1993) and imaging plates (Mori et al., 1990).
There are some advantages in the use of electrons which make it uniquely valuable for particular applications.
These experimental and theoretical aspects of electron diffraction have influenced the ways in which it has been applied in studies of diffuse scattering.
In general, we may distinguish three different approaches to the interpretation of diffuse scattering:
Owing to the difficulties of separating the different components in the diffuse scattering, most work on diffuse scattering of electrons has followed one or both of the two last approaches, although Pattersontype interpretation, based upon kinematical scattering including some dynamical corrections, has also been tried.
In the kinematical approximation, a general expression which includes inelastic scattering can be written in the form quoted by Van Hove (1954)for the intensity of scattering as function of energy transfer and momentum transfer from a system of Z identical particles, . Here m and h have their usual meanings; and k, and are wavevectors and energies before and after the scattering between object states and n; are weights of the initial states; W(u) is a form factor (squared) for the individual particle.
In equation (4.3.2.1), u is essentially momentum transfer. When the energy transfer is small , we can still write , then the sum over final states n is readily performed and an expression of the Waller–Hartree type is obtained for the total inelastic scattering as a function of angle:whereand where the oneelectron f's for Hartree–Fock orbitals, , have been calculated by Freeman (1959, 1960) for atoms up to . The last sum is over electrons with the same spin only.
The Waller–Hartree formula may be a very good approximation for Compton scattering of Xrays, where most of the scattering occurs at high angles and multiple scattering is no problem. With electrons, it has several deficiencies. It does not take into account the electronic structure of the solid, which is most important at low values of u. It does not include the energy distribution of the scattering. It does not give a finite cross section at zero angle, if u is interpreted as an angle. In order to remedy this, we should go back to equation (4.3.1.2) and decompose u into two components, one tangential part which is associated with angle in the usual way and one normal component along the beam direction, , which may be related to the excitation energy by the expression . This will introduce a factor in the intensity at small angles, often written as , with estimated from ionization energies etc. (Strictly speaking, is not a constant, not even for scattering from one shell. It is a weighted average which will vary with u.)
Calculations beyond this simple adjustment of the Waller–Hartreetype expression are few. Plasmon scattering has been treated on the basis of a nearly free electron model by Ferrel (1957):where m, v are relativistic mass and velocity of the incident electron, N is the density of the valence electrons and their dielectric constant. Upon integration over :where takes account of the cutoff angle . Innershell excitations have been studied because of their importance to spectroscopy. The most realistic calculations may be those of Leapman et al. (1980) where oneelectron wavefunctions are determined for the excited states in order to obtain `generalized oscillator strengths' which may then be used to modify equation (4.3.1.2).
At high energies and high momentum transfer, the scattering will approach that of free electrons, i.e. a maximum at the socalled Bethe ridge, .
A complete and detailed picture of inelastic scattering of electrons as a function of energy and angle (or scattering variable) is lacking, and may possibly be the least known area of diffraction by solids. It is further complicated by the dynamical scattering, which involves the incident and diffracted electrons and also the ejected atomic electron (see e.g. Maslen & Rossouw, 1984a,b).
Kinematical expressions for TDS or defect and disorder scattering according to equation (4.3.1.3) can be obtained by inserting the appropriate atomic scattering factors in place of the Xray scattering factors in Chapter 4.1. The complications introduced by dynamical diffraction are considerable (see Section 4.3.4). In the most general case, a complete specification of the disordered structure may be needed. However, for thin specimens, approximate treatments of the deviations from kinematical scattering may lead to relatively simple forms. Two such cases are treated in this section, both relying on the smallangle nature of electron scattering. The first is based upon the phaseobject approximation, which applies to small angles and thin specimens.
The amplitude at the exit surface of a specimen can always be written as a sum of a periodic and a nonperiodic part, and may in analogy with the kinematical case [equation (4.3.1.1)] be writtenwhere r is a vector in two dimensions. The intensities can be separated in the same way [cf. equation (4.3.1.3)].
When the phaseobject approximation applies [Section 2.5.2.4(b) ]Then the Bragg reflections are given by Fourier transform of the periodic part, viz:note that an absorption function is introduced.
The diffuse scattering derives fromso that
Thus, the kinematical diffusescattering amplitude is convoluted with the amplitude function for the average structure, i.e. the set of sharp Bragg beams. When the direct beam, , is relatively strong, the kinematical diffuse scattering will be modified to only a limited extent by convolution with the Bragg reflections. To the extent that the diffuse scattering is periodic in reciprocal space, the effect will be to modify the intensity by a slowly varying function. Thus the shapes of local diffuse maxima will not be greatly affected.
The electronmicroscope image contrast derived from the diffuse scattering will be obtained by inserting equation (4.3.3.4) in the appropriate intensity expressions of Section 4.3.8 of IT C (2004).
Another approach may be used for extended crystal defects in thin films, e.g. faults normal or nearnormal to the film surface. Often, an average periodic structure may not readily be defined, as in the case of a set of incommensurate stacking faults. Kinematically, the projection of the structure in the simplest case may be described by convoluting the projection of a unitcell structure with a nonperiodic set of delta functions which constitute a distribution function:Then the diffractionpattern intensity is
Here, is the scattering amplitude of the unit whereas the function , where , gives the configuration of spots, streaks or other diffraction maxima corresponding to the faulted structure (see e.g. Marks, 1985).
In the projection (column) approximation to dynamical scattering, the wavefunction at the exit surface may be given by an expression identical to (4.3.3.6), but with a wavefunction, , for the unit in place of the projected potential, .
An intensity expression of the same form as (4.3.3.7) then applies, with a dynamical scattering amplitude for the scattering unit substituted for the kinematical amplitude .which in the simplest case describes a diffraction pattern with the same features as in the kinematical case. Note that may have different symmetries when the incident beam is tilted away from a zone axis, leading to diffuse streaks etc. appearing also in positions where the kinematical diffuse scattering is zero. More complicated cases have been considered by Cowley (1976a) who applied this type of analysis to the case of nonperiodic faulting in magnesium fluorogermanate (Cowley, 1976b).
The distribution of diffuse scattering is modified by higherorder terms in essentially two ways: Bragg scattering of the incident and diffuse beams or multiple diffuse scattering, or by a combination.
Theoretical treatment of the Bragg scattering effects in diffuse scattering has been given by many authors, starting with Kainuma's (1955) work on Kikuchiline contrast (Howie, 1963; Fujimoto & Kainuma, 1963; Gjønnes, 1966; Rez et al., 1977; Maslen & Rossouw, 1984a,b; Wang, 1995; Allen et al., 1997). Mathematical formalism may vary but the physical pictures and results are essentially the same. They may be discussed with reference to a Bornseries expansion, i.e. by introducing the potential in the integral equation, as a sum of a periodic and a nonperiodic part [cf. equation (4.3.1.1)] and arranging the terms by orders of .
Some of the higherorder terms contributing to the Bragg scattering can be included by adding the essentially imaginary term to the static potential .
Theoretical treatments have mostly been limited to the firstorder diffuse scattering. With the usual approximation to forward scattering, the expression for the amplitude of diffuse scattering in a direction can be written asand read (from right to left): , Bragg scattering of the incident beam above the level ; , diffuse scattering within a thin layer through the Fourier components of the nonperiodic potential ; , Bragg scattering between diffuse beams in the lower part of the crystal. It is commonly assumed that diffuse scattering at different levels can be treated as independent (Gjønnes, 1966), then the intensity expression becomeswhere (1) and (2) refer to the regions above and below the diffusescattering layer. This expression can be manipulated further, e.g. by introducing Blochwave expansion of the scattering matrices, vizwhich may be interpreted as scattering by between Bloch waves belonging to the same branch (intraband scattering) or different branches (interband scattering). Another alternative is to evaluate the scattering matrices by multislice calculations (Section 4.3.5).
Expressions such as (4.3.4.2) contain a large number of terms. Unless very detailed calculations relating to a precisely defined model are to be carried out, attention should be focused on the most important terms.
In Kikuchiline contrast, the scattering in the upper part of the crystal is usually not considered and frequently the angular variation of the is also neglected. In diffraction contrast from smallangle inelastic scattering, it may be sufficient to consider the intraband terms [ in (4.3.4.3)].
In studies of diffusescattering distribution, the factor will produce two types of terms: Those with result only in a redistribution of intensity between corresponding points in the Brillouin zones, with the same total intensity. Those with lead to enhancement or reduction of the total diffuse intensity and hence absorption from the Bragg beam and enhanced/reduced intensity of secondary radiation, i.e. anomalous absorption and channelling effects. They arise through interference between different Fourier components of the diffuse scattering and carry information about position of the sources of diffuse scattering, referred to the projected unit cell. This is exploited in channelling experiments, where beam direction is used to determine atom reaction (Taftø & Spence, 1982; Taftø & Lehmpfuhl, 1982).
Gjønnes & Høier (1971) expressed this information in terms of the Fourier transform R of the generalized or Kikuchiline form factor;includes information both about correlations between sources of diffuse scattering and about their position in the projected unit cell. It is seen that represents the kinematical intensity, hence is the Patterson function. The integral of in the plane gives the anomalous absorption (Yoshioka, 1957) which is related to the distribution of scattering centres across the unit cell.
The scattering factor can be calculated for different modes. For oneelectron excitations as an extension of the Waller–Hartree expression (Gjønnes, 1962; Whelan, 1965):where are the oneelectron amplitudes (Freeman, 1959).
A similar expression for scattering by phonons is obtained in terms of the scattering factors for the branch j, wavevector g and a polarization vector (see Chapter 4.1 ):independent phonons being assumed.
For scattering from substitutional order in a binary alloy with ordering on one site only, we obtain simplywhere are atomic scattering factors. It is seen that then does not contain any new information; the location of the site involved in the ordering is known.
When several sites are involved in the ordering, the dynamical scattering factor becomes less trivial, since scattering factors for the different ordering parameters (for different sites) will include a factor (see Andersson et al., 1974).
From the above expressions, it is found that the Bragg scattering will affect diffuse scattering from different sources differently: Diffuse scattering from substitutional order will usually be enhanced at low and intermediate angles, whereas scattering from thermal and electronic fluctuations will be reduced at low angles and enhanced at higher angles. This may be used to study substitutional order and displacement order (size effect) separately (Andersson, 1979).
The use of such expressions for quantitative or semiquantitative interpretation raises several problems. The Bragg scattering effects occur in all diffuse components, in particular the inelastic scattering, which thus may no longer be represented by a smooth, monotonic background. It is best to eliminate this experimentally. When this cannot be done, the experiment should be arranged so as to minimize Kikuchiline excess/deficient terms, by aligning the incident beam along a not too dense zone. In this way, one may optimize the diffusescattering information and minimize the dynamical corrections, which then are used partly as guides to conditions, partly as refinement in calculations.
The multiple scattering of the background remains as the most serious problem. Theoretical expressions for multiple scattering in the absence of Bragg scattering have been available for some time (Moliere, 1948), as a sum of convolution integralswhere etc., and is normalized.
A complete description of multiple scattering in the presence of Bragg scattering should include Bragg scattering between diffuse scattering at all levels , etc. This quickly becomes unwieldy. Fortunately, the experimental patterns seem to indicate that this is not necessary: The Kikuchiline contrast does not appear to be very sensitive to the exact Bragg condition of the incident beam. Høier (1973) therefore introduced Bragg scattering only in the last part of the crystal, i.e. between the level and the final thickness z for ntimes scattering. He thus obtained the formula:where are normalized scattering factors for nthorder multiple diffuse scattering and are multiplescattering coefficients which include absorption.
When the thickness is increased, the variation of with angle becomes slower, and an expression for intensity of the channelling pattern is obtained (Gjønnes & Taftø, 1976):Another approach is the use of a modified diffusion equation (Ohtsuki et al., 1976).
These expressions seem to reproduce the development of the general background with thickness over a wide range of thicknesses. It may thus appear that the contribution to the diffuse background from known sources can be treated adequately – and that such a procedure must be included together with adequate filtering of the inelastic component in order to improve the quantitative interpretation of diffuse scattering.
The description of dynamical diffraction in terms of the progression of a wave through successive thin slices of a crystal (Chapter 5.2 ) forms the basis for the multislice method for the calculation of electrondiffraction patterns and electronmicroscope images [see Section 4.3.6.1 in IT C (2004)]. This method can be applied directly to the calculations of diffuse scattering in electron diffraction due to thermal motion and positional disorder and for calculating the images of defects in crystals.
It is essentially an amplitude calculation based on the formulation of equation (4.3.4.1) [or (4.3.4.2)] for firstorder diffuse scattering. The Bragg scattering in the first part of the crystal is calculated using a standard multislice method for the set of beams h. In the nth slice of the crystal, a diffusescattering amplitude is convoluted with the incident set of Bragg beams. For each u, propagation of the set of beams is then calculated through the remaining slices of the crystal. The intensities for the exit wave at the set of points are then calculated by adding either amplitudes or intensities. Amplitudes are added if there is correlation between the defects in successive slices. Intensities are added if there are no such correlations. The process is repeated for all u values to obtain a complete mapping of the diffuse scattering.
Calculations have been made in this way, for example, for shortrange order in alloys [Fisher (1969); see also Cowley (1981) ch. 17] and also for TDS on the assumption of both correlated and uncorrelated atomic motions (Doyle, 1969). The effects of the correlations were shown to be small.
This computing method is not practical for electronmicroscope images in which individual defects are to be imaged. The perturbations of the exit wavefunction due to individual defects (vacancies, replaced atoms, displaced atoms) or small groups of defects may then be calculated with arbitrary accuracy by use of the `periodic continuation' form of the multislice computer programs in which an artificial, large, superlattice unit cell is assumed [Section 4.3.6.1 in IT C (2004)]. The corresponding images and microdiffraction patterns from the individual defects or clusters may then be calculated (Fields & Cowley, 1978). A more recent discussion of the image calculations, particularly in relation to thermal diffuse scattering, is given by Cowley (1988).
In order to calculate the diffusescattering distributions from disordered systems or from a crystal with atoms in thermal motion by the multislice method with periodic continuation, it would be necessary to calculate for a number of different defect configurations sufficiently large to provide an adequate representation of the statistics of the disordered system. However, it has been shown by Cowley & Fields (1979) that, if the singlediffusescattering approximation is made, the perturbations of the exit wave due to individual defects are characteristic of the defect type and of the slice number and may be added, so that a considerable simplification of the computing process is possible. Methods for calculating diffuse scattering in electrondiffraction patterns using the multislice approach are described by Tanaka & Cowley (1987) and Cowley (1989). Loane et al. (1991) introduced the concept of `frozen phonons' for multislice calculations of thermal scattering.
Quantitative interpretation of the intensity of diffuse scattering by calculation of e.g. shortrangeorder parameters has been the exception. Most studies have been directed to qualitative features and their variation with composition, treatment etc. Many features in the scattering which pass unrecognized in extensive Xray or neutron investigations will be observed readily with electrons, frequently inviting other ways of interpretation.
Most such studies have been concerned with substitutional disorder, but the extensive investigations of thermal streaks by Honjo and coworkers should be mentioned (Honjo et al., 1964). Diffuse spots and streaks from disorder have been observed from a wide range of substances. The most frequent may be streaks due to planar faults, one of the most common objects studied by electron microscopy. Diffraction patterns are usually sufficient to determine the orientation and the fault vector; the positions and distribution of faults are more easily seen by darkfield microscopy, whereas the detailed atomic arrangement is best studied by highresolution imaging of the structure [Section 4.3.8 in IT C (2004)].
This combination of diffraction and different imaging techniques cannot be applied in the same way to the study of the essentially threedimensional substitutional local order. Considerable effort has therefore been made to interpret the details of diffuse scattering, leaving the determination of the shortrangeorder (SRO) parameters usually to Xray or neutron studies.
Frequently, characteristic shapes or splitting of the diffuse spots from e.g. binary alloys are observed. They reflect order extending over many atomic distances, and have been assumed to arise from forces other than the nearneighbour pair forces invoked in the theory of local order. A relationship between the diffusescattering distribution and the Fourier transform of the effective atompairinteraction potential is given by the ordering theory of Clapp & Moss (1968). An interpretation in terms of longrange forces carried by the conduction electrons was proposed by Krivoglaz (1969). Extensive studies of alloy systems (Ohshima & Watanabe, 1973) show that the separations, m, observed in split diffuse spots from many alloys follow the predicted variation with the electron/atom ratio :where m is measured along the [110] direction in units of and t is a truncation factor for the Fermi surface.
A similarity between the location of diffuse maxima and the shape of the Fermi surface has been noted also for other structures, notably some defect rocksalttype structures. Although this may offer a clue to the forces involved in the ordering, it entails no description of the local structure. Several attempts have been made to formulate principles for building the disordered structure, from small ordered domains embedded in less ordered regions (Hashimoto, 1974), by a network of antiphase boundaries, or by building the structure from clusters with the average composition and coordination (De Ridder et al., 1976). Evidence for such models may be sought by computer simulations, in the details of the SRO scattering as seen in electron diffraction, or in images.
The cluster model is most directly tied to the location of diffuse scattering, noting that a relation between order parameters derived from clusters consistent with the ordered state can be used to predict the position of diffuse scattering in the form of surfaces in reciprocal space, e.g. the relation for ordering of octahedral clusters in the rocksalttype structure (Sauvage & Parthé, 1974).
Some of the models imply local fluctuations in order which may be observable either by diffraction from very small regions or by imaging. Microdiffraction studies (Tanaka & Cowley, 1985) do indeed show that spots from 1–1.5 nm regions in disordered LiFeO_{2} appear on the locus of diffuse maxima observed in diffraction from larger areas.
Imaging of local variations in the SRO structure has been pursued with different techniques (De Ridder et al., 1976; Tanaka & Cowley, 1985; De Meulenaare et al., 1998), viz: dark field using diffuse spots only; bright field with the central spot plus diffuse spots; lattice image. With domains of about 3 nm or more, highresolution images seem to give clear indication of their presence and form. For smaller ordered regions, the interpretation becomes increasingly complex: Since the domains will then usually not extend through the thickness of the foil, they cannot be imaged separately. Since imagecontrast calculations essentially demand complete specification of the local structure, a model beyond the statistical description must be constructed in order to be compared with observations. On the other hand, these models of the local structure should be consistent with the statistics derived from diffraction patterns collected from a larger volume.
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