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. 2.5, pp. 393394

The aim of electroncrystallographic data collection is to minimize the effect of dynamical scattering, so that the unitcell potential distribution or its Fourier transform is represented significantly in the recorded signal. It would be a mistake, however, to presume that these data ever conform strictly to the kinematical approximation, for there is always some deviation from this ideal scattering condition that can affect the structure analysis. Despite this fact, some direct phasing procedures have been particularly `robust', even when multiple scattering perturbations to the data are quite obvious (e.g. as evidenced by large crystallographic residuals).
The most effective direct phasing procedures seem to be those based on the triple invariants. These phase relationships will not only include the symbolic addition procedure, as it is normally carried out, but also the tangent formula and the Sayre equation (since it is well known that this convolution can be used to derive the functional form of the threephase invariant). The strict ordering of magnitudes is, therefore, not critically important so long as there are no major changes from large to small values (or vice versa). This was demonstrated in direct phase determinations of simulated nbeam dynamical diffraction data from a sulfurcontaining polymer (Dorset & McCourt, 1992). Nevertheless, there is a point where measured data cannot be used. For example, intensities from ca 100 Åthick epitaxically oriented copper perchlorophthalocyanine crystals become less and less representative of the unitcell transform at lower electronbeam energies (Tivol et al., 1993) and, accordingly, the success of the phase determination is compromised (Dorset, McCourt, Fryer et al., 1994). The similarity between the Sayre convolution and the interactions of structurefactor terms in, e.g., the multislice formulation of nbeam dynamical scattering was noted by Moodie (1965). It is interesting to note that dynamical scattering interactions observed by direct excitation of and triples in convergentbeam diffraction experiments can actually be exploited to determine crystallographic phases to very high precision (Spence & Zuo, 1992, pp. 56–63).
While the evaluation of positive quartet invariant sums (see Chapter 2.2 ) seems to be almost as favourable in the electron diffraction case as is the evaluation of triples, negative quartet invariants seem to be particularly sensitive to dynamical diffraction. If dynamical scattering can be modelled crudely by a convolutional smearing of the diffraction intensities, then the lowest structurefactor amplitudes, and hence the estimates of lowest values, will be the ones most compromised. Since the negative quartet relationships require an accurate prediction of small `crossterm' values, multiple scattering can, therefore, limit the efficacy of this invariant for phase determination. In initial work, negative quartets have been mostly employed in the NQEST figure of merit, and analyses (Dorset, McCourt, Fryer et al., 1994; Dorset & McCourt, 1994a) have shown how the degradation of weak kinematical terms effectively reduced its effectiveness for locating correct structure solutions via the tangent formula, even though the tangent formula itself (based on triple phase estimates) was quite effective for phase determination. Substitution of the minimal function for NQEST seems to have overcome this difficulty. [It should be pointed out, though, that only the triple contribution to is considered.]
Structure refinement is another area where the effects of dynamical scattering are also problematic. For example, in the analysis of the paraelectric thiourea structure (Dorset, 1991b) from published texture diffraction data (Dvoryankin & Vainshtein, 1960), it was virtually impossible to find a chemically reasonable structure geometry by Fourier refinement, even though the direct phase determination itself was quite successful. The best structure was found only when higherangle intensities (i.e. those least affected by dynamical scattering) were used to generate the potential map. Later analyses on heavyatomcontaining organics (Dorset et al., 1992) found that the lowest kinematical Rfactor value did not correspond to the chemically correct structure geometry. This observation was also made in the leastsquares refinement of diketopiperazine (Dorset & McCourt, 1994a). It is obvious that, if a global minimum is sought for the crystallographic residual, then dynamical structure factors, rather than kinematical values, should be compared to the observed values (Dorset et al., 1992). Ways of integrating such calculations into the refinement process have been suggested (Sha et al., 1993). Otherwise one must constrain the refinement to chemically reasonable bonding geometry in a search for a local Rfactor minimum.
Corrections for such deviations from the kinematical approximation are complicated by the presence of other possible data perturbations, especially if microareas are being sampled, e.g. in typical selectedarea diffraction experiments. Significant complications can arise from the diffraction incoherence observed from elastically deformed crystals (Cowley, 1961) as well as secondary scattering (Cowley et al., 1951). These complications were also considered for the larger (e.g. millimetre diameter) areas sampled in an electrondiffraction camera when recording texture diffraction patterns (Turner & Cowley, 1969), but, because of the crystallite distributions, it is sometimes found that the twobeam dynamical approximation is useful (accounting for a number of successful structure analyses carried out in Moscow).
References
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Dorset, D. L. (1991b). Is electron crystallography possible? The direct determination of organic crystal structures. Ultramicroscopy, 38, 23–40.
Dorset, D. L. & McCourt, M. P. (1992). Effect of dynamical scattering on successful direct phase determination in electron crystallography – a model study. Trans. Am. Crystallogr. Assoc. 28, 105–113.
Dorset, D. L. & McCourt, M. P. (1994a). Automated structure analysis in electron crystallography: phase determination with the tangent formula and leastsquares refinement. Acta Cryst. A50, 287–292.
Dorset, D. L., McCourt, M. P., Fryer, J. R., Tivol, W. F. & Turner, J. N. (1994). The tangent formula in electron crystallography: phase determination of copper perchlorophthalocyanine. Microsc. Soc. Am. Bull. 24, 398–404.
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Spence, J. C. H. & Zuo, J. M. (1992). Electron Microdiffraction. New York: Plenum Press.
Tivol, W. F., Dorset, D. L., McCourt, M. P. & Turner, J. N. (1993). Voltagedependent effect on dynamical scattering and the electron diffraction structure analysis of organic crystals: copper perchlorophthalocyanine. Microsc. Soc. Am. Bull. 23, 91–98.
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