Patterson methods

Flarker D 1936 The application of the three-dimensional Patterson method and the crystal structures of proustite, Ag,AsS, and pyrargyrite, Ag,SnS, J. Chem. Phys. 4 381-90... 

Stmcture determination of unknown crystals by electron diffraction was performed by several research groups, on Al-Fe alloys by Gjonnes et al. (1998), on metal-cluster compounds by Weirich et al. (2000) and on zeolites by Wagner et al. (1999). Selected area electron diffraction or electron diffraction collected by a precession technique were used and the structure factor phases were deduced by direct methods, Patterson method or from convergent beam electron diffraction. 

Key words phase-problem, erystallographie image processing, Patterson method, direet methods, strueture determination, eleetron diffraction... 

If the Patterson method cannot be applied because the structure has no or too many heavy atoms, it is possible to use another approach for phase determination, the so-called direct methods. By the term direct methods is meant that class of methods which exploits relationships among the structure factors in order to go directly from the observed magnitudes E to the needed phases < ) (Herbert A. Hauptman, Nobel lecture, 9. Dec., 1985). The direct method approach for solving the phase problem uses probability... 

One way to solve such stmctures is to collect electron dififraction patterns from difierent zone axes of the crystal to get an essentially complete 3D electron diffraction data set. Direct methods or the Patterson method can then be applied to phase the data, similar to what is done in X-ray diffraction (Gjonnes et al., 1998 Gemmi et al., 2000 Wagner et al., 1999). 

The range of values of the atomic scattering factors for electrons is less than that of X-rays. The Patterson method works optimally when... 

By far, the most common procedure for the determination of heavy-atom positions is the difference Patterson method it is often used in combination with the difference Fourier technique to locate sites in second and third derivatives. 

The traditional approach for structure solution follows a close analogy to the analysis of single-crystal XRD data, in that the intensities 1(H) of individual reflections are extracted directly from the powder XRD pattern and are then used in the types of structure solution calculation (e.g. direct methods, Patterson methods or the recently developed charge-flipping methodology [32-34]) that are used for single-crystal XRD data. As discussed above, however, peak overlap in the powder XRD pattern can limit the reliability of the extracted intensities, and uncertainties in the intensities can lead to difficulties in subsequent attempts to solve the structure. As noted above, such problems may be particularly severe in cases of large unit cells and low symmetry, as encountered for most molecular solids. In spite of these intrinsic difficulties, however, there have been several reported successes in the application of traditional techniques for structure solution of molecular solids from powder XRD data. 

So the disparity between intensities of Friedel pairs in the anomalous scattering data set establishes their phases in the nonanomalous scattering data set. The reflection whose phase has been established here corresponds to the vector Fhp in Eq. (6.9). Thus the amplitudes and phases of two of the three vectors in the Eq. (6.9) are known (l)FHp is known from the anomalous scattering computation just shown, and (2) FH is known from calculating the heavy-atom structure factors after locating the heavy atom by Patterson methods. The vector Fp, then, is simply the vector difference establish-... 

As I discussed in Section III.C, Patterson methods do not allow us to distinguish between enantiomeric arrangements of heavy atoms, and phases derived from heavy-atom positions of the wrong hand are incorrect. When high-resolution data are available for the heavy-atom derivative, phases and... 

Patterson methods have also been successfiilly used for structure solution from powder diffraction data. By taking advantage of the Patterson function Fi, usefiil information about the crystal structure can be deduced. Compared to Direct methods, Patterson techniques are more suitable for powder diffraction data with lower resolution, and peak overlap causing significant difficulties. The Patterson function can be calculated by using the equation... 

Modifications of the original Patterson methods have been designed to improve on the success rates for powder diffraction applications. Maximum entropy methods based on Patterson maps have also been developed. ... 

Structure solution Ag and As atoms with Patterson method... 

A total of 1463 raw intensity data were collected. Inspection of the azimuthal scan data showed a variation of /mia//max = 0.82 for the average curve. An empirical correction based on the observed variation was applied to the data as a first approximation. The structure was solved by Patterson methods in space group P. Refinement and elucidation of additional atoms proceeded via standard least-squares and Fourier techniques. Examination of the triclinic model demonstrated the correct monoclinic space group, P2Jc (the apparent absence of hOl, l 2n has been found), and refinement continued in (hat group... 

Patterson methods Map of interatomic vectors. Analyses complicated unless few atoms or heavy atom. Used to locate heavy atom in isomorphous replacement for macromolecules. Can also be used for small molecules even if they contain no heavy atom. 

Isomorphous replacement Intensity dijferences for isomorphous crystals. Best if several derivatives are made. Replaced atom located by Patterson methods. General method for macromolecules. 

Patterson methods Methods for analyzing the Patterson map to obtain the final structure. 

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