Search

The cross-correlation function between the target and a model electron density, denoted as the C map, has been crystallographically characterized. In particular, a study of its interatomic vectors and of their relation with the Patterson vectors has been undertaken. Since the C map is not available during the phasing process, the C′ map, its centric modification, is considered. It may be computed at any stage of the phasing process and shows properties that are very useful for the crystal structure determination process. It has been combined with the implication transformation method and with vector-superposition techniques for performing the Patterson deconvolution and obtaining an initial model for dual-space recycling. While Patterson methods are traditionally considered to be more efficient for structures containing heavy atoms, the C map extends their potential to light-atom structures ( i.e. containing atoms not heavier than O). [ABSTRACT FROM AUTHOR]

The direct methods origin-free modulus sum function [Rius (1993). Acta Cryst. A 49, 406-409] includes in its definition the structure factor G(Φ) of the squared crystal structure expressed in terms of Φ, the set of ϕ phases of the normalized structure factors E's of the crystal structure of unit-cell volume V. Here the simpler sum function variant = ∑ H E− H∫ VδP,Δ(Φ)exp( i2π Hr)d V extended over all H reflections is introduced which involves no G's and in which the δP,Δ function corresponds to δP = FT−1{()exp[ iϕ H(Φ)]} (where FT = Fourier transform) with all values smaller than Δ = 2.5σP equated to zero ( is the variance of δP calculable from the experimental intensities). The new phase estimates are obtained by Fourier transforming δP,Δ. This iterative phasing method (δ recycling) only requires calculation of Fourier transforms at two stages. Since δM≃δP/2, similar arguments are valid for δM = FT−1[( E H−〈 E〉)exp( iϕ H)] from which the corresponding phasing function follows. [ABSTRACT FROM AUTHOR]

A fast Fourier transform algorithm is introduced into the method recently defined for calculating powder diffraction patterns by means of the Debye scattering equation (DSE) [Thomas (2010). Acta Cryst. A 66, 64-77]. For this purpose, conventionally used histograms of interatomic distances are replaced by compound transmittance functions. These may be Fourier transformed to partial diffraction patterns, which sum to give the complete diffraction pattern. They also lead to an alternative analytical expression for the DSE sum, which reveals its convergence behaviour. A means of embedding the DSE approach within the reciprocal-lattice-structure-factor method is indicated, with interpolation methods for deriving the peak profiles of nanocrystalline materials outlined. Efficient calculation of transmittance functions for larger crystallites requires the Patterson group symmetry of the crystals to be taken into account, as shown for α- and β-quartz. The capability of the transmittance functions to accommodate stacking disorder is demonstrated by reference to kaolinite, with a fully analytical treatment of disorder described. Areas of future work brought about by these developments are discussed, specifically the handling of anisotropic atomic displacement parameters, inverse Fourier transformation and the incorporation of instrumental (diffractometer) parameters. [ABSTRACT FROM AUTHOR]

The article discusses the study on the use of Patterson-function direct methods (DM) in determining the structure of organic compounds from powder diffraction data. It says that the method is based on the structure factors (Φ) maximization of the sum function of Sp. It cites the optimization of Sp sum with Patterson-function tangent formula (TF) by the Sliding-Fast Fourier Transform (S-FFT) algorithm. It shows that Patterson-function DM can deal with powder data.

The article presents a study on the use of Patterson vectors molecular-replacement (PVMR) method in tracking helix fragments. It states that PVMR is calculated by equating a search model in a real space with Patterson cross and self vectors. It mentions that the equal matches of Patterson vectors are added to get the signal-to-noise ratio. Moreover, findings of the study show that PVMR is suitable in setting small helix fragments as structural root for molecular substitution.

The article offers information on a study which aims to investigate the intensity difference in Friedel opposites on the analysis of small-molecule single-crystal structure using the Patterson functions. It says that Patterson functions give clear view with regards to resonant scattering of the crystal structures and on the difference on the model and observed intensities. The study used least square refinement on the space group determination in the crystal structure.

The maximum-likelihood method is applied to direct methods to derive a more general probability density function of the triple-phase sums which is capable of predicting negative values. This study also proves that maximization of the origin-free modulus sum function S yields, within the limitations imposed by the assumed approximations, the maximum-likelihood estimates of the phases. It thus represents the formal theoretical justification of the S function that was initially derived from Patterson-function arguments [Rius (1993). ]. [ABSTRACT FROM AUTHOR]

Two phasing equations based on the Fourier syntheses δP = T−1[( E2−〈 E2〉)exp( iϕ)] and δM = T−1[( E−〈 E〉)exp( iϕ)] were recently described [Rius (2012). Acta Cryst. A 68, 77-81] ( E is the quasi-normalized structure factor and 〈 E〉 is the average over all reflections). These equations were found by comparison with the direct methods origin-free modulus sum function and constitute the core of the `δ recycling' phasing procedure. The derivation of these phasing equations from the minimization of a residual ( RP) between two differently calculated density functions (one of them including the positivity constraint) is shown. [ABSTRACT FROM AUTHOR]