Section 2.4.1.4. Powder-pattern intensities

In the kinematical approximation, the expression for intensities of electron diffraction follows that for X-ray diffraction with the exception that, because only small angles of diffraction are involved, no polarization factor is involved. Following Vainshtein (1964), the intensity per unit length of a powder line is where is the incident-beam intensity, is the structure factor, is the unit-cell volume, V is the sample volume, and M is the multiplicity factor.

The kinematical approximation has limited validity. The deviations from this approximation are given to a first approximation by the two-beam approximation to the dynamical-scattering theory. Because an averaging over all orientations is involved, the many-beam dynamical-diffraction effects are less evident than for single-crystal patterns.

By integrating the two-beam intensity expression over excitation error, Blackman (1939) obtained the expression for the ratio of dynamical to kinematical intensities: where is the zero-order Bessel function, with the interaction constant , and H is the crystal thickness. Careful measurements on ring patterns from thin aluminium films by Horstmann & Meyer (1962) showed agreement with the `Blackman curve' [from equation (2.4.1.4)] to within about 5% with some notable exceptions. Deviations of up to 40 to 50% from the Blackman curve occurred for several reflections, such as 222 and 400, which are second-order reflections from strong inner reflections. A practical algorithm for implementing Blackman corrections has been published by Dvoryankina & Pinsker (1958).

Such deviations result from plural-beam systematic interactions, the coherent multiple scattering between different orders of a strong inner reflection. When the Bragg condition is satisfied for one order, the excitation errors for the other orders are the same for all possible crystal orientations and these other orders contribute systematically to the ring-pattern intensities. A correction for the effects of systematic interactions may be made by use of the Bethe second approximation (Bethe, 1928) (see Chapter 8.8 ).

For non-systematic reflections, corresponding to reciprocal-lattice points not collinear with the origin and the reciprocal-lattice point of interest, the averaging over all crystal orientations ensures that the powder-pattern intensity calculated from the two-beam formula will not be appreciably affected. Appreciable effects from non-systematic interactions may, however, occur when the averaging is over a limited range of crystal orientations, as in the case of strong preferred orientations. It was shown theoretically by Turner & Cowley (1969) and experimentally by Imamov, Pannhorst, Avilov & Pinsker (1976) that appreciable modifications of intensities of oblique-texture patterns may result from non-systematic interactions for particular tilt angles, especially for heavy-atom materials [see also Avilov, Parmon, Semiletov & Sirota (1984)].

The techniques for the measurement of electron diffraction intensities are described in Chapter 7.2 . Most commonly electron diffraction powder patterns are recorded by photographic methods and a microdensitometer is used for quantitative intensity measurement. The Grigson scanning method, using a scintillator and photomultiplier to record intensities as the pattern is scanned over a fine slit, has considerable advantages in terms of linearity and range of the intensity scale (Grigson, 1962). This method also has the advantage that it may readily be combined with an energy filter so that only elastically scattered electrons (or electrons inelastically scattered with a particular energy loss) may be recorded.

Small-angle electron diffraction may give useful information in some cases, but must be interpreted carefully because the features may result from multiple scattering or other artefacts. It may give additional details of periodicity (super-periods) and deviations of the real symmetry from the ideal symmetry suggested by other data. Care must be taken with the interpretation of additional reflections, as they may relate to the structure of small regions that are not typical of the bulk specimens such as are examined by X-ray diffraction.

The techniques for interpretation of electron diffraction powder-pattern intensities follow those for X-ray patterns when the kinematical approximation is valid. For very small crystals, giving very broad rings, it is possible to use the method, commonly applied for diffraction by gases, of performing a Fourier transform to obtain a radial distribution function (Goodman, 1963).