(1) Comparison with Xray diffraction. The relations of realspace and reciprocalspace functions are analogous to those for Xray diffraction [see equations (2.5.2.2), (2.5.2.10) and (2.5.2.15)]. For diffraction by crystals where the integral of (2.5.2.18) and the summation of (2.5.2.19) are taken over one unit cell of volume (see Dawson et al., 1974).
Important differences from the Xray case arise because
(a) the wavelength is relatively small so that the Ewaldsphere curvature is small in the reciprocalspace region of appreciable scattering amplitude;
(b) the dimensions of the singlecrystal regions giving appreciable scattering amplitudes are small so that the `shape transform' regions of scattering power around the reciprocallattice points are relatively large;
(c) the spread of wavelengths is small (10^{−5} or less, with no whiteradiation background) and the degree of collimation is better (10^{−4} to 10^{−6}) than for conventional Xray sources.
As a consequence of these factors, singlecrystal diffraction patterns may show many simultaneous reflections, representing almostplanar sections of reciprocal space, and may show fine structure or intensity variations reflecting the crystal dimensions and shape.

(2) Kinematical diffractionpattern intensities are calculated in a manner analogous to that for Xrays except that
(a) no polarization factor is included because of the smallangle scattering conditions;
(b) integration over regions of scattering power around reciprocallattice points cannot be assumed unless appropriate experimental conditions are ensured.
For a thin, flat, lamellar crystal of thickness H, the observed intensity is where is the excitation error for the h reflection and Ω is the unitcell volume.
For a singlecrystal diffraction pattern obtained by rotating a crystal or from a uniformly bent crystal or for a mosaic crystal with a uniform distribution of orientations, the intensity is where is the crystal volume and is the latticeplane spacing. For a polycrystalline sample of randomly oriented small crystals, the intensity per unit length of the diffraction ring is where is the multiplicity factor for the h reflection and L is the camera length, or the distance from the specimen to the detector plane. The special cases of `oblique texture' patterns from powder patterns having preferred orientations are treated in IT C (2004, Section 4.3.5
).
(3) Twobeam dynamical diffraction formulae: complex potentials including absorption. In the twobeam dynamical diffraction approximation, the intensities of the directly transmitted and diffracted beams for transmission through a crystal of thickness H, in the absence of absorption, are where is the extinction distance, , and where is the deviation from the Bragg angle.
For the case that , with the incident beam at the Bragg angle, this reduces to the simple Pendellösung expression
The effects on the elastic Bragg scattering amplitudes of the inelastic or diffuse scattering may be introduced by adding an outofphase component to the structure amplitudes, so that for a centrosymmetric crystal, becomes complex by addition of an imaginary component. Alternatively, an absorption function , having Fourier coefficients , may be postulated so that is replaced by . The are known as phenomenological absorption coefficients and their validity in manybeam diffraction has been demonstrated by, for example, Rez (1978).
The magnitudes depend on the nature of the experiment and the extent to which the various inelastically or diffusely scattered electrons are included in the measurements being made. If measurements are made of purely elastic scattering intensities for Bragg reflections or of image intensity variations due to the interaction of the sharp Bragg reflections only, the main contributions to the absorption coefficients are as follows (Radi, 1970):
(a) from plasmon and singleelectron excitations, is of the order of 0.1 V_{0} and , for , is negligibly small;
(b) from thermal diffuse scattering; is of the order of 0.1 V_{h} and decreasing more slowly than with scattering angle.
Including absorption effects in (2.5.2.26) for the case gives The Borrmann effect is not very pronounced for electrons because , but can be important for the imaging of defects in thick crystals (Hirsch et al., 1965; Hashimoto et al., 1961).
Attempts to obtain analytical solutions for the dynamical diffraction equations for more than two beams have met with few successes. There are some situations of high symmetry, with incident beams in exact zoneaxis orientations, for which the manybeam solution can closely approach equivalent two or threebeam behaviour (Fukuhara, 1966). Explicit solutions for the threebeam case, which displays some aspects of manybeam character, have been obtained (Gjønnes & Høier, 1971; Hurley & Moodie, 1980).


