On the application of phase relationships to complex structures. XXIV. The Sayre tangent formula

Application of Patterson-function direct methods to materials characterization

The aim of this article is a general description of the so-called Patterson-function direct methods (PFDM), from their origin to their present state. It covers a 20-year period of methodological contributions to crystal structure solution, most of them published in Acta Crystallographica Section A. The common feature of these variants of direct methods is the introduction of the experimental intensities in the form of the Fourier coefficients of origin-free Patterson-type functions, which allows the active use of both strong and weak reflections. The different optimization algorithms are discussed and their performances compared. This review focuses not only on those PFDM applications related to powder diffraction data but also on some recent results obtained with electron diffraction tomography data.

A robust tangent procedure

Third-generation direct methods programs are based on a phasing algorithm (e.g.the tangent or the parameter shift method) and on dual space refinement techniques. The two spaces may be alternated during the phasing procedure or used in a sequential way: for example, first phase and after extend and refine. The tangent approach inSIR2011belongs to the second category: phases are first estimated by the tangent formula, then their extension and refinement is performed in direct spaceviaelectron density modification techniques. In this article a number of new algorithms are described, with the aim of improving theSIR2011tangent step and allowing more efficient phase extension and refinement. New criteria were chosen for defining the number of reflections to phase, for modifying the tangent weighting scheme, and for fixing the active use of the psi-0 triplets and of the quartet invariants. Each tangent trial may now be submitted to the RELAX procedure, a tool for moving to the correct position a well oriented but misplaced structural model. The resulting procedure shows surprising efficiency, testified by a wide set of applications. The experimental results have been compared with the tangent and VLD (vive la différence) approaches implemented inSIR2011.

Proteins and direct methods

Conventional direct methods, which work so well for small structures, are less successful for macromolecules. Where it has been demonstrated that a solution might be found using direct methods it is then found that the usual figures of merit are unable to distinguish the few good sets of phases from the large number of sets generated. The reasons for the difficulties with very large structures are considered from a first-principles approach taking into account both the factors of having a large number of atoms and low resolution data. A proposal is made for trying to recognize good phase sets by taking a large structure as a sum of a number of smaller structures for each of which a conventional figure of merit can be applied.

Ab initio direct phasing of proteins: the limits

The role of directed methods in the protein field is analyzed. A criterion is formulated which suggests the necessary conditions for the success or the failure of the ab initio direct procedures. The distribution of quantityz= 〈α〉/σαis central in our considerations. Most of the experimental protein data do not satisfy such a criterion, therefore their ab-initio solution is a quite improbable event.

High-resolution electron microscopy and its applications

A review of research on high-resolution electron microscopy (HREM) carried out at the Institute of Physics, the Chinese Academy of Sciences, is presented. Apart from the direct observation of crystal and quasicrystal defects for some alloys, oxides, minerals, etc., and the structure determination for some minute crystals, an approximate image-contrast theory named pseudo-weak-phase object approximation (PWPOA), which shows the image contrast change with crystal thickness, is described. Within the framework of PWPOA, the image contrast of lithium ions in the crystal of R-Li2Ti3O7 has been observed. The usefulness of diffraction analysis techniques such as the direct method and Patterson method in HREM is discussed. Image deconvolution and resolution enhancement for weak-phase objects by use of the direct method are illustrated. In addition, preliminary results of image restoration for thick crystals are given.