La coupe du roi and other methods to halve objects

The symmetry theory of halving objects into two identical fragments is developed and illustrated by examples. La coupe du roi is a fascinating way to slice an apple into chiral fragments with the same handedness. It may be applied to any object with point group symmetry 2 or 222 and their supergroups, e.g. to cones, cylinders or spheres. Variants of coupes du roi are presented and applied to a cube. An object is divided into halves by a number of cuts. The important cuts extend to the centre of the object, and these must form a single closed loop. The symmetry of the division into two fragments is given by the point group of the decorated object, i.e. the object including all the cuts. Point groups comprising only rotations result in chiral fragments with the same handedness. Such divisions may illustrate the reaction path for changing the sense of chirality of molecules via a dimeric achiral transition state rather than by deformation. Non-trivial divisions of objects into achiral fragments or into racemic chiral fragments are obtained with point groups, again of the decorated object, comprising roto-inversions X̅, excepting X = 4n + 2. All other point groups either do not lead to half-objects, or they result in a trivial single planar cut parallel to a mirror plane

Stacking disorder: the hexagonal polymorph of tris(bicyclo[2.1.1]hexeno)benzene and related examples

X-ray diffractograms of tris(bicyclo[2.1.1]hexeno)benzene, crystallized at the interface between a benzene solution and a layer of acetonitrile, show hexagonal symmetry and streaks of diffuse scattering along c*. The heavily faulted layer stacking is analyzed qualitatively and quantitatively in terms of a systematic protocol. This protocol requires partitioning the crystal structure into layers in such a way that pairs of adjacent layers may be stacked in different, but geometrically equivalent ways, which are dictated by the layer group symmetry. This approach is shown to provide a consistent alternative for analysis of a number of related cases provided the layers are defined on the basis of geometrical criteria rather than chemical intuitio

On uncertainty estimates of crystallographic quantities including cell-parameter uncertainties

In a recent publication, Haestier [J. Appl. Cryst. (2009), 42, 798-809] has proposed a method to take care of the unit-cell-parameter uncertainties in the calculation of geometric quantities such as interatomic distances and bond angles by modifying the uncertainties of the atomic coordinates. This problem is addressed here with a different approach, which gives additional insight. For the cell-edge uncertainties, Haestier's results are confirmed and their importance is easily appreciated. However, for the cell-angle uncertainties, it is proved that there exists no simple solution short of calculating the derivatives of the quantity of interest with respect to the angles. Simple rules of thumb are presented for guessing the importance of edge-length uncertainties

The success story of crystallography

Diffractionists usually place the birth of crystallography in 1912 with the first X-ray diffraction experiment of Friedrich, Knipping and Laue. This discovery propelled the mathematical branch of mineralogy to global importance and enabled crystal structure determination. Knowledge of the geometrical structure of matter at atomic resolution had revolutionary consequences for all branches of the natural sciences: physics, chemistry, biology, earth sciences and material science. It is scarcely possible for a single person in a single article to trace and appropriately value all of these developments. This article presents the limited, subjective view of its author and a limited selection of references. The bulk of the article covers the history of X-ray structure determination from the NaCl structure to aperiodic structures and macromolecular structures. The theoretical foundations were available by 1920. The subsequent success of crystallography was then due to the development of diffraction equipment, the theory of the solution of the phase problem, symmetry theory and computers. The many structures becoming known called for the development of crystal chemistry and of data banks. Diffuse scattering from disordered structures without and with partial long-range order allows determination of short-range order. Neutron and electron scattering and diffraction are also mentioned.LC

La coupe du roi and other methods to halve objects

The symmetry theory of halving objects into two identical fragments is developed and illustrated by examples. La coupe du roi is a fascinating way to slice an apple into chiral fragments with the same handedness. It may be applied to any object with point group symmetry 2 or 222 and their supergroups, e.g. to cones, cylinders or spheres. Variants of coupes du roi are presented and applied to a cube. An object is divided into halves by a number of cuts. The important cuts extend to the centre of the object, and these must form a single closed loop. The symmetry of the division into two fragments is given by the point group of the decorated object, i.e. the object including all the cuts. Point groups comprising only rotations result in chiral fragments with the same handedness. Such divisions may illustrate the reaction path for changing the sense of chirality of molecules via a dimeric achiral transition state rather than by deformation. Non-trivial divisions of objects into achiral fragments or into racemic chiral fragments are obtained with point groups, again of the decorated object, comprising roto-inversions X, excepting X = 4n + 2. All other point groups either do not lead to half-objects, or they result in a trivial single planar cut parallel to a mirror plane. © 2015 by De Gruyter

The success story of crystallography

Diffractionists usually place the birth of crystallography in 1912 with the first X-ray diffraction experiment of Friedrich, Knipping and Laue. This discovery propelled the mathematical branch of mineralogy to global importance and enabled crystal structure determination. Knowledge of the geometrical structure of matter at atomic resolution had revolutionary consequences for all branches of the natural sciences: physics, chemistry, biology, earth sciences and material science. It is scarcely possible for a single person in a single article to trace and appropriately value all of these developments. This article presents the limited, subjective view of its author and a limited selection of references. The bulk of the article covers the history of X-ray structure determination from the NaCl structure to aperiodic structures and macromolecular structures. The theoretical foundations were available by 1920. The subsequent success of crystallography was then due to the development of diffraction equipment, the theory of the solution of the phase problem, symmetry theory and computers. The many structures becoming known called for the development of crystal chemistry and of data banks. Diffuse scattering from disordered structures without and with partial long-range order allows determination of short-range order. Neutron and electron scattering and diffraction are also mentioned

'Seeing' Atoms: The Crystallographic Revolution

Laue's experiment in 1912 of the diffraction of X-rays by crystals led to one of the most influential discoveries in the history of science: the first determinations of crystal structures, NaCl and diamond in particular, by W. L. Bragg in 1913. For the first time, the visualisation of the structure of matter at the atomic level became possible. X-ray diffraction provided a sort of microscope with atomic resolution, atoms became observable physical objects and their relative positions in space could be seen. All branches of science concerned with matter, solid-state physics, chemistry, materials science, mineralogy and biology, could now be firmly anchored on the spatial arrangement of atoms. During the ensuing 100 years, structure determination by diffraction methods has matured into an indispensable method of chemical analysis. We trace the history of the development of 'small-structure' crystallography (excepting macromolecular structures) in Switzerland. Among the pioneers figure Peter Debye and Paul Scherrer with powder diffraction, and Paul Niggli and his Zurich School with space group symmetry and geometrical crystallography. Diffraction methods were applied early on by chemists at the Universities of Bern and Geneva. By the 1970s, X-ray crystallography was firmly established at most Swiss Universities, directed by full professors. Today, chemical analysis by structure determination is the task of service laboratories. However, the demand of diffraction methods to solve problems in all disciplines of science is still increasing and powerful radiation sources and detectors are being developed in Switzerland and worldwide