Section 4.5.3.3. Crystal structure analysis

Two approaches to crystal structure analysis are generally employed in polymer electron crystallography. As already mentioned, the procedure adapted from fibre X-ray crystallography relies on the construction of a model (Brisse, 1989; Perez & Chanzy, 1989). Conformational searches (Campbell Smith & Arnott, 1978) simultaneously minimize the fit of observed diffraction data to calculated values (the R factor based on structure factors computed via known atomic scattering factors) and a nonbonded atom–atom potential function (Tadokoro, 1979). Reviews of structures solved by this approach have been published (Dorset, 1989, 1995b).

Recently, direct phasing methods of the kind used in X-ray crystallography (Chapter 2.2 and, applied to electron diffraction, Section 2.5.7 ) have also been found to be particularly effective for electron crystallographic structure analyses (Dorset, 1995b). While the Fourier transform of an electron micrograph would be the most easily imagined direct method, yielding crystallographic phases after image analysis (see Section 2.5.5 ), this use of micrographs has been of less importance to polymer crystallography than it has been in the study of globular proteins, even though there is at least one notable example where it has been helpful (Isoda et al., 1983a) for the determination of a structure from X-ray fibre data. On the other hand, high-resolution images of polymer crystals are of considerable use for the characterization of packing defects (Isoda et al., 1983b).

In polymer electron crystallography, the sole reliance on the diffraction intensities for structure analysis has proven, in recent years, to be quite effective. Several direct-methods approaches have been pursued, including the use of probabilistic techniques, either in the symbolic addition procedure, or in more automated procedures involving the tangent formula (see Chapter 2.2 ). The Sayre (1952) equation has been found to be particularly effective, where the correct structure is identified via some figure of merit after algebraic phase values are used to generate multiple solutions (Stanley, 1986). More recently, maximum-entropy and likelihood methods (Gilmore et al., 1993) have also been effective for solving such structures. After the initial atomic model is found, it can be improved by refinement, generally using Fourier techniques. Least-squares refinement can be carried out under most favourable circumstances (Dorset, 1995a), but requires the availability of a sufficient number of diffraction data. Even so, the refinement of thermal parameters must be uncoupled from that of the atomic positions. Also, positional shifts must be dampened (if X-ray crystallographic software is used) to prevent finding a false minimum, especially if the kinematical R factor is used as a figure of merit.