Reciprocal Lattice (reciprocal + lattice)

Distribution by Scientific Domains


Selected Abstracts


The Bravais polar lattice as a didactic tool for diffraction beginners

JOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 5-2 2010
Massimo Nespolo
When undergraduate students discover crystallography for the first time, they are usually already familiar with the phenomenon of diffraction as the `bending' of waves around small obstacles. The special (periodic) nature of crystals acting as `diffraction gratings' that produce interference of diffracted waves is typically rationalized in terms of the reciprocal lattice of the crystal. The concept of the reciprocal lattice, however, remains somewhat abstract for beginners, until they perform a diffraction experiment. It can be made more easily understandable through an intermediate step, namely its ancestor, the Bravais polar lattice. By means of a short historical trip through pre-X-ray crystallography, a generalized introduction to the notion of the dual lattice is given, of which the reciprocal lattice is the most common but by no means the only example, and it is shown how the use of the Bravais polar lattice can ease the introduction of the reciprocal lattice. [source]


Automatic indexing of area-detector data of periodic and aperiodic crystals

JOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 2 2002
Katrin Pilz
An autoindexing procedure is described that produces the indexing of diffraction data of aperiodic crystals. The procedure has been designed for indexing the data obtained with an area detector, but it can also be applied to data obtained with a single-point detector. The essential step in the indexing process is the ability to discriminate between reflections that fit to a reciprocal lattice, the satellite reflections and possible reflections that do not belong to this indexing. To achieve this goal, the refinement of the orientation matrix and the diffractometer parameters is made an intrinsic part of the process of indexing. The proposed autoindexing procedure has been implemented in a computer program called BAYINDEX. Successful application to data sets of three different one-dimensionally modulated structures, one two-dimensionally modulated structure and a periodic crystal is presented. Very good agreement between experimental and theoretical reflection positions is found. The indexing produced by BAYINDEX can serve as the basis for integration routines. [source]


Photonic crystals comprising selectively grown flat-topped and sharp-tipped GaN pyramids

PHYSICA STATUS SOLIDI (C) - CURRENT TOPICS IN SOLID STATE PHYSICS, Issue 1 2007
D. Coquillat
Abstract Selective area growth offers the promise of producing III-nitride photonic crystals (PhCs) without dry etching and concomitant surface damage. Two PhC structures were studied, one comprising an array of flat-topped GaN micropyramids with micrometre-scale periodicity, and the other made up of sharp-tipped pyramids with sub-micrometre periodicity. Angularly resolved reflection and transmission measurements revealed the presence of sharp resonances associated with resonant Bloch modes. As a result, the photonic band structure was determined along symmetry directions of the reciprocal lattice for the two PhC structures. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


Molecular replacement: the probabilistic approach of the program REMO09 and its applications

ACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2009
Rocco Caliandro
The method of joint probability distribution functions has been applied to molecular replacement techniques. The rotational search is performed by rotating the reciprocal lattice of the protein with respect to the calculated transform of the model structure; the translation search is performed by fast Fourier transform. Several cases of prior information are studied, both for the rotation and for the translation step: e.g. the conditional probability density for the rotation or the translation of a monomer is found both for ab initio and when the rotation and/or the translation values of other monomers are given. The new approach has been implemented in the program REMO09, which is part of the package for global phasing IL MILIONE [Burla, Caliandro, Camalli, Cascarano, De Caro, Giacovazzo, Polidori, Siliqi & Spagna (2007). J. Appl. Cryst.40, 609,613]. A large set of test structures has been used for checking the efficiency of the new algorithms, which proved to be significantly robust in finding the correct solutions and in discriminating them from noise. An important design concept is the high degree of automatism: REMO09 is often capable of providing a reliable model of the target structure without any user intervention. [source]


Identification of two allotwins of mica polytypes in reciprocal space through the minimal rhombus unit

ACTA CRYSTALLOGRAPHICA SECTION B, Issue 4 2000
Massimo Nespolo
The X-ray investigation (precession method) of the Ruiz Peak oxybiotite, which is well known for the occurrence of a large number of polytypes and twins, revealed two complex diffraction patterns, which cannot be identified as long-period polytypes. These patterns are analysed in terms of the minimal rhombus, a geometrical asymmetric unit in reciprocal space which permits the decomposition of the composite reciprocal lattice of a twin or allotwin into the reciprocal lattices of the individuals. Both the recorded patterns correspond to a 1M,2M1 allotwin: the relative rotation between the individuals is 120° in one case and 60° in the other. The geometrical criteria for evaluating the presence of twinning or allotwinning are analysed through these two natural examples. [source]


Autoindexing the diffraction patterns from crystals with a pseudotranslation

ACTA CRYSTALLOGRAPHICA SECTION D, Issue 6 2009
Nicholas K. Sauter
Rotation photographs can be readily indexed if enough candidate Bragg spots are identified to properly sample the reciprocal lattice. However, while automatic indexing algorithms are widely used for macromolecular data processing, they can produce incorrect results in special situations where a subset of Bragg spots is systematically overlooked. This is a potential outcome in cases where a noncrystallographic translational symmetry operator closely mimics an exact crystallographic translation. In these cases, a visual inspection of the diffraction image will reveal alternating strong and weak reflections. However, reliable detection of the weak-intensity reflections by software requires a systematic search for a diffraction signal targeted at specific reciprocal-space locations calculated a priori by considering all possible pseudotranslations. Care must be exercised to distinguish between true lattice diffraction and spurious signals contributed by neighboring overlapping Bragg spots, non-Bragg diffraction and noise. Such procedures have been implemented within the autoindexing program LABELIT and applied to known cases from publicly available data sets. Routine use of this type of signal search adds only a few seconds to the typical run time for autoindexing. The program can be downloaded from http://cci.lbl.gov/labelit. [source]


Identification of two allotwins of mica polytypes in reciprocal space through the minimal rhombus unit

ACTA CRYSTALLOGRAPHICA SECTION B, Issue 4 2000
Massimo Nespolo
The X-ray investigation (precession method) of the Ruiz Peak oxybiotite, which is well known for the occurrence of a large number of polytypes and twins, revealed two complex diffraction patterns, which cannot be identified as long-period polytypes. These patterns are analysed in terms of the minimal rhombus, a geometrical asymmetric unit in reciprocal space which permits the decomposition of the composite reciprocal lattice of a twin or allotwin into the reciprocal lattices of the individuals. Both the recorded patterns correspond to a 1M,2M1 allotwin: the relative rotation between the individuals is 120° in one case and 60° in the other. The geometrical criteria for evaluating the presence of twinning or allotwinning are analysed through these two natural examples. [source]


Imperfect pseudo-merohedral twinning in crystals of fungal fatty acid synthase

ACTA CRYSTALLOGRAPHICA SECTION D, Issue 2 2009
Simon Jenni
The recent high-resolution structures of fungal fatty acid synthase (FAS) have provided new insights into the principles of fatty acid biosynthesis by large multifunctional enzymes. The crystallographic phase problem for the 2.6,MDa fungal FAS was initially solved to 5,Ĺ resolution using two crystal forms from Thermomyces lanuginosus. Monoclinic crystals in space group P21 were obtained from orthorhombic crystals in space group P212121 by dehydration. Here, it is shown how this space-group transition induced imperfect pseudo-merohedral twinning in the monoclinic crystal, giving rise to a Moiré pattern-like interference of the two twin-related reciprocal lattices. The strategy for processing the twinned diffraction images and obtaining a quantitative analysis is presented. The twinning is also related to the packing of the molecules in the two crystal forms, which was derived from self-rotation function analysis and molecular-replacement solutions using a low-resolution electron microscopy map as a search model. [source]