Space of symmetry matrices with elements 0, ±1 and complete geometric description; its properties and application

A space of 480 matrices with cyclic consecutive powers, determinant 1, elements from {0, ±1} and new geometric characterization is analyzed and offered as the conceptual and practical framework for dealing with the `arithmetic symmetry' of reduced lattices.

Direct methods: a paradox with regard to the convergence of random phase trials toward solutions

A large variance in the `shake-and-bake' minimized target function R(φ_h) can identify large subsets of phases whose mean phase error is considerably smaller than the average during unsuccessful phasing trials when the overall phase error occasionally dips below 75 or 80°. Efforts to exploit this observation to promote convergence to the solution, however, are generally not successful.

Mathematical aspects of molecular replacement. I. Algebraic properties of motion spaces

Search spaces in the method of molecular replacement are shown to be coset spaces of the Lie group of rigid-body motions by the chiral space group of a crystal. The resulting `motion space' can be endowed with a quasigroup operation that has interesting properties which are explored here.

About the hybrid Fourier syntheses: a probabilistic approach

The method of joint probability distribution functions has been applied to formulate a probabilistic analysis of the hybrid Fourier syntheses.

Iteratively reweighted least squares in crystal structure refinements

An attempt to implement a robust regression procedure in a least-squares multipole crystal structure refinement is presented and discussed, together with some explanatory case studies.

Anisotropy of extinction: extrapolation to the kinematical limit by γ-ray diffraction

A direct experimental approach to the problem of anisotropic extinction is presented.

Radial integrals for the magnetic form factor of 5d transition elements

The radial integrals, 〈j_L〉, where L = 0, 2, 4, for several electronic configurations in the 5d electrons of transition metal atoms and ions are calculated. The resultant values are fitted to Gaussian analytical expressions with four exponential terms, and the coefficients of the expressions are tabulated.

Bayesian algorithms for recovering structure from single-particle diffraction snapshots of unknown orientation: a comparison

X-ray free-electron lasers are being used to determine the three-dimensional structure of objects from random snapshots. The two apparently very different Bayesian algorithms capable of performing this at ultra-low signal are fundamentally the same.

Atomic resolution imaging using the real-space distribution of electrons scattered by a crystalline material

An atomic resolution incoherent imaging mode in scanning transmission electron microscopy using detectors in real space, achieving a resolution comparable to conventional HAADF-STEM with good contrast, is presented. Simulations based on a newly proposed imaging mechanism shows excellent qualitative agreement with the experiments.