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case  crystal  crystals  difference electron  difference  dimensional  electron  faulting  new  phase  problem  properties  structure  unit cell 
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Preview: Acta Crystallographica Section A

Acta Crystallographica Section A

Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination

Published: 2017-11-01


{\bb Z}-module defects in crystals


An analysis is presented of the new types of defects that can appear in crystalline structures where the positions of the atoms and the unit cell belong to the same {\bb Z}-module, i.e. are irrational projections of an N > 3-dimensional (N-D) lattice Λ as in the case of quasicrystals. Beyond coherent irrationally oriented twins already discussed in a previous paper [Quiquandon et al. (2016). Acta Cryst. A72, 55–61], new two-dimensional translational defects are expected, the translation vectors of which, being projections of nodes of Λ, have irrational coordinates with respect to the unit-cell reference frame. Partial dislocations, called here module dislocations, are the linear defects bounding these translation faults. A specific case arises when the Burgers vector B is the projection of a non-zero vector of Λ that is perpendicular to the physical space. This new kind of dislocation is called a scalar dislocation since, because its Burgers vector in physical space is zero, it generates no displacement field and has no interaction with external stress fields and other dislocations.

The phase problem for two-dimensional crystals. I. Theory


Properties of the phase problem for two-dimensional crystals are examined. This problem is relevant to protein structure determination using diffraction from two-dimensional crystals that has been proposed using new X-ray free-electron laser sources. The problem is shown to be better determined than for conventional three-dimensional crystallography, but there are still a large number of solutions in the absence of additional a priori information. Molecular envelope information reduces the size of the solution set, and for an envelope that deviates sufficiently from the unit cell a unique solution is possible. The effects of various molecular surface features and incomplete data on uniqueness and prospects for ab initio phasing are assessed. Simulations of phase retrieval for two-dimensional crystal data are described in the second paper in this series.

Extrinsic faulting in 3C close-packed crystal structures: computational mechanics analysis


Extrinsic faulting has been discussed previously within the so-called difference method and random walk calculation. In this contribution it is revisited under the framework of computational mechanics, which allows expressions to be derived for the statistical complexity, entropy density and excess entropy as a function of faulting probability. The approach allows one to compare the disordering process of an extrinsic fault with other faulting types. The ∊-machine description of the faulting mechanics is presented. Several useful analytical expressions such as probability of consecutive symbols in the Hägg coding are presented, as well as hexagonality. The analytical expression for the pairwise correlation function of the layers is derived and compared with results previously reported. The effect of faulting on the interference function is discussed in relation to the diffraction pattern.

About difference electron densities and their properties


Difference electron densities do not play a central role in modern phase refinement approaches, essentially because of the explosive success of the EDM (electron-density modification) techniques, mainly based on observed electron-density syntheses. Difference densities however have been recently rediscovered in connection with the VLD (Vive la Difference) approach, because they are a strong support for strengthening EDM approaches and for ab initio crystal structure solution. In this paper the properties of the most documented difference electron densities, here denoted as F − Fp, mF − Fp and mF − DFp syntheses, are studied. In addition, a fourth new difference synthesis, here denoted as {\overline F_q} synthesis, is proposed. It comes from the study of the same joint probability distribution function from which the VLD approach arose. The properties of the {\overline F_q} syntheses are studied and compared with those of the other three syntheses. The results suggest that the {\overline F_q} difference may be a useful tool for making modern phase refinement procedures more efficient.

A note on X-ray spherical wavefields in the Laue case for perfect crystals


Spherical wavefields in the Laue case are obtained when the real part of the crystal structure factor is zero or small compared with the imaginary part. The results are the same as in the conventional case where the real part is large. Through this work, it is shown that virtual wavefields on the vacuum side should be taken into account to explain the obtained results. It is also shown that, to obtain the virtual wavefields, a modification of the conventional spherical wave theory is necessary.

The Rome de Lisle problem


The `Rome de Lisle problem' on the vertex and edge truncations has been formulated and solved for all crystal closed simple forms (two, eight, five and 15 for orthorhombic, trigonal + hexagonal, tetragonal and cubic syngonies, respectively). The collections of simple forms obtained are enumerated and considered as special combinations of simple forms in symmetry classes.