The aperiodic nature of incommensurately modulated structures

The discovery of aperiodic crystals is perhaps one of the most important event which has changed our vision on crystalline architectures since the discovery of diffraction 100years ago. It was the merit of a Dutch crystallographer, P.M. de Wolff, to interpret their diffraction pattern as a three dimensional projection of a higher dimensional reciprocal lattice, idea which led directly to the generalization of the concept of crystal. Aperiodic crystals are currently described as periodic objects in higher-dimensional space, i.e. the superspace and their structures can be described in terms of 3-d cuts. Incommensurate structures, composite structures and quasicrystals all belong to aperiodic structures. Many interesting properties of superspace have been discovered which are also directly applicable to crystals in the conventional sense, i.e. crystals with 3-d periodicity. In particular the concept of structure type can be extended for a better understanding of structure relations. The notion of solid solution has also benefited from superspace considerations. Moreover, superspace is a very powerful tool for a better understanding of structure-property relations in material science, e.g. luminescence properties could be directly associated to the description of structures in superspace. Recently, this concept has been used for the prediction of new structural modifications including polytypes and even polytypic modifications of a well-known pharmaceutical produc

David H. Templeton, 1920-2010

Acta Crystallographica Section A: Foundations of Crystallography 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 by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered

Crystallographic excursion in superspace

After studying more than 100 different samples of calaverite Au1-pAgpTe2 (p 2CO3 resisted any attempt for a precise structural analysis. The appearance of satellite reflections was noted on single crystal diffractograms which led to the generalisation of the concept of crystal. This generalisation consisted in using at least four integers to fully characterise each individual diffractions peaks. The theory of periodic crystals in space of higher dimension, i.e. the superspace was then developed to deal with the new experimental observations. Later, a new class of materials called composite crystals and still later, the discovery of quasicrystals only reinforced the validity of the superspace concept to describe any material requiring more than three integers to index their diffraction pattern.What is the essence of superspace to describe crystalline structures? Any crystal structure requesting more than three integers to index its diffraction pattern can be described as a periodic object in superspace with dimension equal to the number of required integer. The structure observed in our real word is a three dimensional cut of the superspace description. In general this cut is irrational and consequently the crystal is aperiodic. Calaverite and [gamma]-Na2CO3 are examples of aperiodic crystals which includes incommensurately modulated crystals, composite and quasi-crystals. Rational cuts are also possible. In this case, the structure is periodic and is usually called a superstructure

The aperiodic nature of incommensurately modulated structures

The discovery of aperiodic crystals is perhaps one of the most important event which has changed our vision on crystalline architectures since the discovery of diffraction 100 years ago. It was the merit of a Dutch crystallographer, P.M. de Wolff, to interpret their diffraction pattern as a three dimensional projection of a higher dimensional reciprocal lattice, idea which led directly to the generalization of the concept of crystal. Aperiodic crystals are currently described as periodic objects in higher-dimensional space, i.e. the superspace and their structures can be described in terms of 3-d cuts. Incommensurate structures, composite structures and quasicrystals all belong to aperiodic structures. Many interesting properties of superspace have been discovered which are also directly applicable to crystals in the conventional sense, i.e. crystals with 3-d periodicity. In particular the concept of structure type can be extended for a better understanding of structure relations. The notion of solid solution has also benefited from superspace considerations. Moreover, superspace is a very powerful tool for a better understanding of structure–property relations in materials science, e.g. luminescence properties could be directly associated to the description of structures in superspace

SUPERFLIP– a computer program for the solution of crystal structures by charge flipping in arbitrary dimensions

SUPERFLIP is a computer program that can solve crystal structures from diffraction data using the recently developed charge-flipping algorithm. It can solve periodic structures, incommensurately modulated structures and quasicrystals from X-ray and neutron diffraction data. Structure solution from powder diffraction data is supported by combining the charge-flipping algorithm with a histogram-matching procedure. SUPERFLIP is written in Fortran90 and is distributed as a source code and as precompiled binaries. It has been successfully compiled and tested on a variety of operating systems