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Sayre Equation
In crystallography, the Sayre equation, named after David Sayre who introduced it in 1952, is a mathematical relationship that allows one to calculate probable values for the phases of some diffracted beams. It is used when employing direct methods to solve a structure. Its formulation is the following: F_ = \sum_ F_F_ which states how the structure factor for a beam can be calculated as the sum of the products of pairs of structure factors whose indices sum to the desired values of h,k,l. Since weak diffracted beams will contribute a little to the sum, this method can be a powerful way of finding the phase of related beams, if some of the initial phases are already known by other methods. In particular, for three such related beams in a centrosymmetric structure, the phases can only be 0 or \pi and the Sayre equation reduces to the triplet relationship: S_ \approx S_ S_ where the S indicates the sign of the structure factor (positive if the phase is 0 and negative if it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The word "crystallography" is derived from the Greek word κρύσταλλος (''krystallos'') "clear ice, rock-crystal", with its meaning extending to all solids with some degree of transparency, and γράφειν (''graphein'') "to write". In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming that 2014 would be the International Year of Crystallography. denote a direction vector (in real space). * Coordinates in ''angle brackets'' or ''chevrons'' such as <100> denote a ''family'' of directions which are related by symmetry operations. In the cubic crystal system for example, would mean 00 10 01/nowiki> or the negative of any of those directions. * Miller indices in ''parentheses'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Sayre
David Sayre (March 2, 1924 – February 23, 2012) was an American scientist, credited with the early development of direct methods for protein crystallography and of diffraction microscopy (also called coherent diffraction imaging). While working at IBM he was part of the initial team of ten programmers who created FORTRAN, and later suggested the use of electron beam lithography for the fabrication of X-ray Fresnel zone plates. The International Union of Crystallography awarded Sayre the Ewald Prize in 2008 for the "unique breadth of his contributions to crystallography, which range from seminal contributions to the solving of the phase problem to the complex physics of imaging generic objects by X-ray diffraction and microscopy(...)". Life and career Sayre was born in New York City. He completed his bachelor's degree in physics at Yale University at the age of 19. After working at the MIT Radiation Laboratory, he earned his MS degree at Auburn University in 1948. In 194 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Problem
In physics, the phase problem is the problem of loss of information concerning the phase that can occur when making a physical measurement. The name comes from the field of X-ray crystallography, where the phase problem has to be solved for the determination of a structure from diffraction data. The phase problem is also met in the fields of imaging and signal processing. Various approaches of phase retrieval have been developed over the years. Overview Light detectors, such as photographic plates or CCDs, measure only the intensity of the light that hits them. This measurement is incomplete (even when neglecting other degrees of freedom such as polarization and angle of incidence) because a light wave has not only an amplitude (related to the intensity), but also a phase (related to the direction), and polarization which are systematically lost in a measurement. In diffraction or microscopy experiments, the phase part of the wave often contains valuable information on the stud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Methods (crystallography)
In crystallography, direct methods are a family of methods for estimating the phases of the Fourier transform of the scattering density from the corresponding magnitudes. The methods generally exploit constraints or statistical correlations between the phases of different Fourier components that result from the fact that the scattering density must be a positive real number. In two dimensions, it is relatively easy to solve the phase problem directly, but not so in three dimensions. The key step was taken by Hauptman and Karle, who developed a practical method to employ the Sayre equation for which they were awarded the 1985 Nobel prize in Chemistry. The Nobel Prize citation was "for their outstanding achievements in the development of direct methods for the determination of crystal structures." At present, direct methods are the preferred method for phasing crystals of small molecules having up to 1000 atoms in the asymmetric unit. However, they are generally not feasible by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structure Factor
In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation of scattering patterns (interference patterns) obtained in X-ray, electron and neutron diffraction experiments. Confusingly, there are two different mathematical expressions in use, both called 'structure factor'. One is usually written S(\mathbf); it is more generally valid, and relates the observed diffracted intensity per atom to that produced by a single scattering unit. The other is usually written F or F_ and is only valid for systems with long-range positional order — crystals. This expression relates the amplitude and phase of the beam diffracted by the (hk\ell) planes of the crystal ((hk\ell) are the Miller indices of the planes) to that produced by a single scattering unit at the vertices of the primitive unit cell. F_ is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centrosymmetric
In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point groups are also said to have ''inversion'' symmetry. Point reflection is a similar term used in geometry. Crystals with an inversion center cannot display certain properties, such as the piezoelectric effect. The following space groups have inversion symmetry: the triclinic space group 2, the monoclinic 10-15, the orthorhombic 47-74, the tetragonal 83-88 and 123-142, the trigonal 147, 148 and 162-167, the hexagonal 175, 176 and 191-194, the cubic 200-206 and 221-230. Point groups lacking an inversion center (non-centrosymmetric) can be ''polar'', ''chiral'', both, or neither. A ''polar'' point group is one whose symmetry operations leave more than one common point unmoved. A polar point group has no unique origin because each of those unmoved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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