The use of phase relationships between quartets of reflexions

Updating direct methods

The standard method of joint probability distribution functions, so crucial for the development of direct methods, has been revisited and updated. It consists of three steps: identification of the reflections which may contribute to the estimation of a given structure invariant or seminvariant, calculation of the corresponding joint probability distribution, and derivation of the conditional distribution of the invariant or seminvariant phase given the values of some diffracted amplitudes. In this article the conditional distributions are derived directly without passing through the second step. A good feature of direct methods is that they may work in the absence of any prior information: that is also their weakness. Different types of prior information have been taken into consideration: interatomic distances, interatomic vectors, Patterson peaks, structural model. The method of directly deriving the conditional distributions has been applied to those cases. Some new formulas have been obtained estimating two-, three- and four-phase invariants. Special attention has been dedicated to the practical aspects of the new formulas, in order to simplify their possible use in direct phasing procedures.

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.

Advances and some recent applications of the origin-free modulus sum function

Since its discovery, the direct methods origin-free modulus sum function [Rius, J., Acta Cryst A49 (1993) 406—409] has been responsible for the solution of a number of difficult crystal structures of minerals and other inorganic compounds from powder diffraction data. This is principally due to the efficiency, robustness and simplicity of implementation of this phase refinement function. The first part of the contribution describes some recent examples on the application of the origin-free modulus sum function to complex structures. In the second part, a powerful variant of this function is introduced which discriminates even better the correct solutions from the wrong ones. This is illustrated with its application to single-crystal data of three selected organic structures. One of these test structures contains 317-atom molecules and is regarded as one of the most difficult structures to be solved with reciprocal space direct methods. This variant could also be useful for those phase refinement strategies based on alternating reciprocal- and real-space procedures, provided that the weak reflections are known.