International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 320321

The concept of using experimental electrondiffraction intensities for quantitative crystal structure analyses has already been presented in Section 2.5.4. Another aspect of quantitative structure analysis, employing highresolution images, has been presented in Sections 2.5.5 and 2.5.6. That is to say, electron micrographs can be regarded as an independent source of crystallographic phases.
Before direct methods (Chapter 2.2 ) were developed as the standard technique for structure determination in smallmolecule Xray crystallography, there were two principal approaches to solving the crystallographic phase problem. First, `trial and error' was used, finding some means to construct a reasonable model for the crystal structure a priori, e.g. by matching symmetry properties shared by the point group of the molecule or atomic cluster and the unitcell space group. Secondly, the autocorrelation function of the crystal, known as the Patterson function (Chapter 2.3 ), was calculated (by the direct Fourier transform of the available intensity data) to locate salient interatomic vectors within the unit cell.
The same techniques had been used for electrondiffraction structure analysis (nowadays known as electron crystallography ). In fact, advocacy of the first method persists. Because of the perturbations of diffracted intensities by multiplebeam dynamical scattering (Chapter 5.2 ), it has often been suggested that trial and error be used to construct the scattering model for the unit crystal in order to test its convergence to observed data after simulation of the scattering events through the crystal. This indirect approach assumes that no information about the crystal structure can be obtained directly from observed intensity data. Under more favourable scattering conditions nearer to the kinematical approximation, i.e. for experimental data from thin crystals made up of light atoms, trial and error modelling, simultaneously minimizing an atom–atom nonbonded potential function with the crystallographic residual, has enjoyed widespread use in electron crystallography, especially for the determination of linear polymer structures (Brisse, 1989; Pérez & Chanzy, 1989).
Interpretation of Patterson maps has also been important for structure analysis in electron crystallography. Applications have been discussed by Vainshtein (1964), Zvyagin (1967) and Dorset (1994a). In face of the dynamical scattering effects for electron scattering from heavyatom crystals realized later (e.g. Cowley & Moodie, 1959), attempts had also been made to modify this autocorrelation function by using a power series in to sharpen the peaks (Cowley, 1956). (Here , replacing the notation for the kinematical electrondiffraction structure factor employed in Section 2.5.4.) More recently, Vincent and coworkers have selected firstorderLauezone data from inorganics to minimize the effect of dynamical scattering on the interpretability of their Patterson maps (Vincent & Exelby, 1991, 1993; Vincent & Midgley, 1994). Vainshtein & Klechkovskaya (1993) have also reported use of the Patterson function to solve the crystal structure of a lead soap from texture electrondiffraction intensity data.
It is apparent that trialanderror techniques are most appropriate for ab initio structure analysis when the underlying crystal structures are reasonably easy to model. The requisite positioning of molecular (or atomic) groups within the unit cell may be facilitated by finding atoms that fit a special symmetry position [see IT A (2005)]. Alternatively, it is helpful to know the molecular orientation within the unit cell (e.g. provided by the Patterson function) to allow the model to be positioned for a conformational or translational search. [Examples would include the polymerstructure analyses cited above, as well as the layerpacking analysis of some phospholipids (Dorset, 1987).] While attempts at ab initio modelling of threedimensional crystal structures, by searching an ndimensional parameter space and seeking a global internal energy minimum, has remained an active research area, most success so far seems to have been realized with the prediction of twodimensional layers (Scaringe, 1992). In general, for complicated unit cells, determination of a structure by trial and error is very difficult unless adequate constraints can be placed on the search.
Although Patterson techniques have been very useful in electron crystallography, there are also inherent difficulties in their use, particularly for locating heavy atoms. As will be appreciated from comparison of scatteringfactor tables for Xrays [IT C (2004) Chapter 6.1 ] with those for electrons, [IT C (2004) Chapter 4.3 ] the relative values of the electron form factors are more compressed with respect to atomic number than are those for Xray scattering. As discussed in Chapter 2.3 , it is desirable that the ratio of summed scattering factor terms, , where Z is the scattering factor value at sin , be near unity. A practical comparison would be the value of r for copper (DLalaninate) solved from electrondiffraction data by Vainshtein et al. (1971). For electron diffraction, compared to the value 2.36 for Xray diffraction. Orientation of salient structural features, such as chains and rings, would be equally useful for lightatom moieties in electron or Xray crystallography with Patterson techniques. As structures become more complicated, interpretation of Patterson maps becomes more and more difficult unless an automated search can be carried out against a known structural fragment (Chapter 2.3 ).
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