Crystallography is an interdisciplinary science covering a wide area, from biology to earth sciences, mathematics and materials science. Its role is growing, owing to the contribution crystallography can offer to the understanding of such diverse fields as biological structures, high-temperature superconductors, mineral properties, and phase transitions. The book describes both the theoretical bases and applications of different areas interacting with crystallography. As with the first and second editions, it is organized as a collection of chapters written by recognized specialists, with all contributions being harmonized into a unified whole. The main text is devoted to the presentation of basics; the appendices deal with specialist aspects. In this third edition topics have been updated so as to document the present state of the art: emphasis is placed upon areas of current research. To facilitate learning and make teaching more effective, new illustrations have been introduced. As with the second edition, a software package is included via the book's OUP web site: modern graphics will help users to better understand the basics of this science via three-dimensional images, simulation of experiments, and exercises.
Giacovazzo Fundamentals Of Crystallography Pdf 14
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Twinning is not an uncommon effect in crystallography, although it has long been considered to be one of the most serious potential obstacles to structure determination. The use of area detectors has much facilitated the detection of twinning and the treatment of diffraction patterns from twinned crystals. Computer software has also now been developed to such an extent that previously intractable twinning problems have yielded results of comparable precision to those obtained with untwinned samples. Structure determinations from twinned crystals are therefore becoming more common and the aim of this article is to present an introduction to the phenomenon of twinning. An extensive database of papers describing twinning has been assembled by Spek and Lutz (Utecht University, The Netherlands) and is available on the internet at _twin.html . The review by Yeates (1997) will be of particular interest to macromolecular crystallographers.
Fig. 1(b) is similar to a single crystal; Fig. 1(c) resembles a twinned crystal. In Fig. 1(c) there are two domain variants: bricks (which correspond to unit cells) within the same domain are related to each other by translation; bricks in different domains are related by a translation plus a rotation which occurs in the point symmetry of the outline or overall shape of the brick. This extra symmetry operation corresponds in crystallography to the twin law. Had the extra element been chosen to be a mirror plane the mirror image of the words `London Brick' would have appeared in the second domain and it is important to bear this in mind during the analysis of enantiopure crystals of chiral compounds (such as proteins). In protein crystallography the only possible twin laws are rotation axes. The fraction of the bricks in the alternative orientation corresponds to the twin scale factor, which in this example is 0.5.
Common signs of twinning have been listed by Herbst-Irmer & Sheldrick (1998, 2002). Additional signs to those described above include an inability to solve a structure even though the data appear to be of good quality or, if a structure can be solved, a high R factor or a noisy inexplicable difference electron-density map. Twinning also reveals itself in the Patterson function and this is discussed by Dauter (2003) and Yeates (1997). Since reflections from one domain may overlap with systematic absences from another, the observed systematic absences may either not be consistent with any known space group or appear to imply a very rare space group. Low-symmetry trigonal and hexagonal crystal structures appear to be particularly susceptible to twinning. It was pointed out by a referee to this paper that because twinning makes the point symmetry appear higher than it actually is, there seem to be more molecules in the unit cell than is actually the case, so that an unreasonably high packing density can also be taken to be a useful warning sign of twinning in macromolecular crystallography.
Twinning can occur whenever a compound crystallizes in a unit cell with a higher point group than that corresponding to the space group. This can occur for crystal structures in non-centrosymmetric space groups, since all lattices have inversion symmetry. Thus, a crystal of a compound in a space group such as P21 may contain enantiomorphic domains (Flack, 2003). This type of twinning does not occur for an enantiopure compound and it can therefore be ruled out in protein crystallography. The twin law in this case is the inversion operator,
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the mean positions of the atoms in the crystal can be determined, as well as their chemical bonds, their crystallographic disorder, and various other information.
X-ray crystallography is related to several other methods for determining atomic structures. Similar diffraction patterns can be produced by scattering electrons or neutrons, which are likewise interpreted by Fourier transformation. If single crystals of sufficient size cannot be obtained, various other X-ray methods can be applied to obtain less detailed information; such methods include fiber diffraction, powder diffraction and (if the sample is not crystallized) small-angle X-ray scattering (SAXS).If the material under investigation is only available in the form of nanocrystalline powders or suffers from poor crystallinity, the methods of electron crystallography can be applied for determining the atomic structure.
Crystals, though long admired for their regularity and symmetry, were not investigated scientifically until the 17th century. Johannes Kepler hypothesized in his work Strena seu de Nive Sexangula (A New Year's Gift of Hexagonal Snow) (1611) that the hexagonal symmetry of snowflake crystals was due to a regular packing of spherical water particles.[2] The Danish scientist Nicolas Steno (1669) pioneered experimental investigations of crystal symmetry. Steno showed that the angles between the faces are the same in every exemplar of a particular type of crystal.[3] René Just Haüy (1784) discovered that every face of a crystal can be described by simple stacking patterns of blocks of the same shape and size. Hence, William Hallowes Miller in 1839 was able to give each face a unique label of three small integers, the Miller indices which remain in use for identifying crystal faces. Haüy's study led to the idea that crystals are a regular three-dimensional array (a Bravais lattice) of atoms and molecules; a single unit cell is repeated indefinitely along three principal directions. In the 19th century, a complete catalog of the possible symmetries of a crystal was worked out by Johan Hessel,[4] Auguste Bravais,[5] Evgraf Fedorov,[6] Arthur Schönflies[7] and (belatedly) William Barlow (1894). Barlow proposed several crystal structures in the 1880s that were validated later by X-ray crystallography;[8] however, the available data were too scarce in the 1880s to accept his models as conclusive.
Since the 1920s, X-ray diffraction has been the principal method for determining the arrangement of atoms in minerals and metals. The application of X-ray crystallography to mineralogy began with the structure of garnet, which was determined in 1924 by Menzer. A systematic X-ray crystallographic study of the silicates was undertaken in the 1920s. This study showed that, as the Si/O ratio is altered, the silicate crystals exhibit significant changes in their atomic arrangements. Machatschki extended these insights to minerals in which aluminium substitutes for the silicon atoms of the silicates. The first application of X-ray crystallography to metallurgy likewise occurred in the mid-1920s.[71][72][73][74][75][76] Most notably, Linus Pauling's structure of the alloy Mg2Sn[77] led to his theory of the stability and structure of complex ionic crystals.[78]
X-ray crystallography of biological molecules took off with Dorothy Crowfoot Hodgkin, who solved the structures of cholesterol (1937), penicillin (1946) and vitamin B12 (1956), for which she was awarded the Nobel Prize in Chemistry in 1964. In 1969, she succeeded in solving the structure of insulin, on which she worked for over thirty years.[90] 2ff7e9595c
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