Qu­antitative crystal structure descriptors from multiplicative congruential generators

Certain number-theoretic relations, known as multiplicative congruential generators, are developed into quantitative crystal structure descriptors, facilitating a linearization procedure that eventually allows for a concise, fully reconstructable representation of structural information. Potential applications are related to the computational storage, retrieval and analysis of crystal structures, their algorithmic generation in the first place, and their combinatorial enumeration and classification.

Lattice refinement strategies

This article quantitatively reconciles crystallographic and mechanics approaches to lattice refinement as part of X-ray diffraction procedures. It also derives finite deformation relations of reciprocal-lattice vectors, and demonstrates infinitesimal strain descriptions, commonly employed in stress analysis, to be correct with an error at quadratic order in strain. This error may be important to consider for high-precision or high-strain studies.

New consistency tests for high-accuracy measurements of X-ray mass attenuation coefficients by the X-ray extended-range technique

A new method – the multiple independent foil technique – provides data consistency tests and extends the X-ray extended-range technique for measuring X-ray mass attenuation coefficients and X-ray absorption spectroscopy.

Image formation in the scanning transmission electron microscope using object-conjugate detectors

The use of object-conjugate detectors in STEM encompasses the three-dimensional imaging technique of SCEM and the recently developed atomic resolution technique of R-STEM. Image formation in this optical geometry is analysed using geometric optics and wave optics, and the relationship between these theories is examined using Wigner distributions. It is shown that geometric optics accurately describes R-STEM and is also applicable to SCEM in a restricted sense.

Mathematical aspects of molecular replacement. II. Geometry of motion spaces

The geometric structure of fundamental domains and coset spaces constructed from the continuous group of rigid-body motions and any chiral space group are explored. Almost-uniform sampling methods for these spaces are developed for use in the molecular-replacement method.

The superstructure determination of displacive distortions via symmetry-mode analysis

A structural parameter set consisting of group-theoretically derived symmetry modes facilitates an unbiased determination of a displacively distorted structure. The space-group symmetry is identified from the active symmetry modes rather than being assumed.

The global long-range order of quasi-periodic patterns in Islamic architecture

The first global structural model for describing the long-range translational and orientational order of quasi-crystalline formations in Islamic architecture is presented. It is shown that ancient designers, using simple consecutive geometry, have resolved the complicated long-range principles of quasi-periodic formations.

Covariance and correlation estimation in electron-density maps

The covariance between two points of an electron-density map has been calculated, no matter the correlation between the model and target structures.

Phase retrieval in protein crystallography

An oversampling method based on the hybrid input–output algorithm is proposed to retrieve the phases of reflections in crystallography from the low-resolution envelopes of the proteins.

Higher-dimensional crystallography of N-fold quasiperiodic tilings

Crystallography and periodic average structures of two-dimensional quasiperiodic tilings with N-fold symmetry (N = 7, 8, 9, 10, 11, 12, 13, 15) were studied using the higher-dimensional approach.

Computation of small-angle scattering profiles with three-dimensional Zernike polynomials

Three-dimensional Zernike polynomials are employed to efficiently compute small-angle X-ray scattering (SAXS) profiles. Three-dimensional Zernike polynomials faithfully reproduce three-dimensional objects with complex surfaces, including surface-bound solvent. The resulting SAXS profiles are in good agreement with existing computational procedures.

Totally unimodular nets

A method for generating a special class of nets from finite graphs is described.

Frequency distribution of the reduced unit cells of centred lattices from the Protein Data Bank

This study presents the frequency distribution of the reduced unit cells of all centred lattice entries of the Protein Data Bank in four unit-cell-dimension-based groups and seven interaxial-angle-based subgroups.

A novel iterative solution to the phase problem

An iterative phase solution based on the global minimization is proposed.

On the significance of Bragg reflections

The statistical descriptor, W, recently introduced by Henn & Meindl [Acta Cryst. (2010), A66, 676–684] is modified to include the scale factors introduced in data processing, and the effects of using empirical error models on the final significance distribution are discussed.

Response to On the significance of Bragg reflections by Jørgensen et al.

A response is given to the article On the significance of Bragg reflections by Jørgensen et al. [Acta Cryst. (2012). A68, 301–303].

Herbert A. Hauptman (1917–2011)

Obituary for Herbert A. Hauptman.

Notes for authors 2012