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derivative  distribution  edge  electron  magnetic  model  resonant magnetic  resonant  scattering  structure  symmetry  target  tilings 
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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: 2017-05-01


Non-crystallographic symmetry of liquid metal, flat crystallographic faults and polymorph transformation of the M7C3 carbide


The fine lamellar fault structure of primary M7C3 carbide particles in the heat-resistant Fe–Cr–Ni–C alloy of the HP series (0.45C–25Cr–35Ni) in the cast condition has been revealed using transmission electron microscopy. The lamellar fault structure is regarded as an inheritance of the initial melt structure. A one-to-one correspondence between the crystal structures of M7C3, M5C2, M3C and M23C6 carbides and constructions of the projective 103 Desargues configuration and its subconfigurations is shown. Mutual mapping between 103 Desargues subconfigurations determines transformations of the ten-vertex equal-edged triangulated clusters appearing as building units of these carbides.

The phantom derivative method when a structure model is available: about its theoretical basis


This study clarifies why, in the phantom derivative (PhD) approach, randomly created structures can help in refining phases obtained by other methods. For this purpose the joint probability distribution of target, model, ancil and phantom derivative structure factors and its conditional distributions have been studied. Since PhD may use n phantom derivatives, with n ≥ 1, a more general distribution taking into account all the ancil and derivative structure factors has been considered, from which the conditional distribution of the target phase has been derived. The corresponding conclusive formula contains two components. The first is the classical Srinivasan & Ramachandran term, relating the phases of the target structure with the model phases. The second arises from the combination of two correlations: that between model and derivative (the first is a component of the second) and that between derivative and target. The second component mathematically codifies the information on the target phase arising from model and derivative electron-density maps. The result is new, and explains why a random structure, uncorrelated with the target structure, adds useful information on the target phases, provided a model structure is known. Some experimental tests aimed at checking if the second component really provides information on φ (the target phase) were performed; the favourable results confirm the correctness of the theoretical calculations and of the corresponding analysis.

Edge-2-transitive trinodal polyhedra and 2-periodic tilings


All trinodal, edge-2-transitive polyhedra and 2-periodic tilings are enumerated and described. These are of special interest for the design and synthesis of materials such as metal–organic polyhedra and frameworks.

Diffuse single-crystal scattering corrected for molecular form factor effects


This paper shows that chemical short-range order in two-component molecular crystals can be solved directly by separating the influence of the molecular form factor from the diffraction pattern. This novel technique is demonstrated by analysing the diffuse scattering of tris-tert-butyl-1,3,5-benzene tricarboxamide.

Groupoids and labelled quotient graphs: a topological analysis of the modular structure in pyroxenes


The analysis of the modular structure of pyroxenes, recently discussed in Nespolo & Aroyo [Eur. J. Mineral. (2016), 28, 189–203], has been performed on the respective labelled quotient graphs (LQGs). It is shown that the structure and maximum symmetry of the module, i.e. its layer group, can be determined directly from the LQG. Partial symmetry operations between different modules have been associated with automorphisms of the quotient graph that may not be consistent with net voltages over the respective cycles. These operations have been shown to generate the pyroxene groupoid structure.

On the Penrose and Taylor–Socolar hexagonal tilings


The intimate relationship between the Penrose and the Taylor–Socolar tilings is studied, within both the context of double hexagon tiles and the algebraic context of hierarchical inverse sequences of triangular lattices. This unified approach produces both types of tilings together, clarifies their relationship and offers straightforward proofs of their basic properties.

Resonant and non-resonant magnetic scatterings with circularly polarized X-rays: magnetic scattering factor and electron density of gadolinium iron garnet


Theoretical and experimental studies on resonant magnetic scattering have been carried out using circularly polarized X-rays. Polarization dependence of the asymmetric ratio between right- and left-handed polarized scattering amplitudes has revealed that resonant and non-resonant magnetic scatterings can be observed simultaneously. In this study, synchrotron X-ray intensity experiments were conducted on Gd iron garnet (GdIG) at the pre-edge of the Fe K absorption edge. The asymmetric ratio between circular polarizations exhibited three peaks due to the resonant magnetic responses of Fe3+ in the Fe2 (24d) site. These magnetic responses are antiparallel and have apparently higher resolution than dispersive X-ray magnetic circular dichroism at the pre-edge region; this can be mainly attributed to electric dipole transitions from 1s to t2 energy levels of the 3d orbitals in the tetrahedral symmetry in an unstable state quenched at higher temperature. Resonant magnetic scattering factors f′m were estimated from the asymmetric ratios. Fourier analysis was conducted using the intensity differences among circularly polarized X-rays at an energy of E = 7.1085 keV. At this energy level, the characteristic distribution of magnetic electron density was clearly observed. This distribution was interpreted as a spin arrangement of Gd and Fe ions in GdIG. Also observed was the presence of hybridization of the magnetic electron orbitals as well as of the resonant magnetic Fe3+. The proposed method has the potential to be widely used in crystal structure analysis.

Convex polyhedra with minimum and maximum names


The paper reports the combinatorial types of convex n-acra (i.e. n-vertex polyhedra) for which the minimum (min.) and maximum (max.) names are attained. Hence, min. and max. names can be independently found without generating the whole combinatorial variety of convex n-acra (e.g. by the routine recurrence Fedorov algorithm) and calculating names for each n-acron.