Refinement of a crystal

The Mossbauer spectrum of Fe3(CO)i2 shows two distinct iron sites, with a relative ratio of 2 1. In fact, this result led to the refinement of a crystal structure that had reported three equivalent iron sites. Although this happened in the early days of Mossbauer spectroscopy, that is, the 1960s, it... 

It is important for the reader to understand that in a least-squares refinement of a crystal structure it is the shifts in parameters that are calculated in order to improve the structure, not the parameters themselves. The preliminary parameters that are shifted to more appropriate values come from the trial structures (see Chapters 8 and 9). 

Correlation between parameters A correlation is a measure of the extent to which two mathematical variables are dependent on each other. In the least-squares refinement of a crystal structure, parameters related by symmetry are completely correlated, and temperature factors and occupancy factors are often highly correlated. 

Refinement of a crystal structure A process of improving the parameters of an approximate (trial) structure until the best fit of calculated structure factor amplitudes to those observed is obtained. The process usually requires many successive stages. 

Figure 8.71 Computer screen of the automated results of the refinement of a crystal structure by the XtaLAB mini . (Used by permission of RIgaku Corporation, www.rlgaku.com.)...

The elementary cell or lattice is the lowest structural level of a crystal. The lattice is characterized by a space symmetry group, atom positions and thermal displacement parameters of the atoms as well as by the position occupancies. In principle, the lattice is the smallest building block for creating an ideal crystal of any size by simple translations, and it is the lattice that is responsible for the fundamental parameter. Therefore, it is extremely important to perform the structure refinement of a crystal obtained, especially if the crystal represents a solid solution compound or demonstrates unusual properties or has unknown oxygen content or is assumed to form a new structure modification. 

Wang, D., Bode, W., Huber, R. Bovine chymotrypsinogen A. X-ray crystal structure analysis and refinement of a new crystal form at 1.8 A resolution. /. Mol. Biol. 

The information obtained from X-ray measurements on the arrangement of the water molecules naturally depends very much on the resolution and state of refinement of the crystal structure investigated. For detailed information on the organization of water molecules in the protein hydration shell at the surface and on the bulk water in the crystals a 1,2 to 1,8 A resolution range is necessary 153>. 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

To a first order approximation, the scattering potential of a crystal may be represented as a sum of contributions from isolated atoms, having charge distributions of spherical symmetry around their nuclei. In a real crystal the charge distribution deviates from the spherical symmetry around the nucleus and the difference reflects the charge redistribution or bonding in the crystal. The problem of experimental measurement of crystal bonding is therefore a problem of structure factor refinement, i.e. accurate determination of the difference between the true crystal structure factors... 

According to the chemical analysis and coordination distances, the Rietveld refinement of the crystal structure at room temperature revealed 1.2 Co2+ atoms per unit cell at the Col and Co2 sites, whereas the 1.4 Ag+ cations are spread over the Co3 site, from now on referred to as Ag5 for clarity, and two new sites, Ag2 and Ag3, located near Co2 in the 10-membered ring (Fig. 3). In addition, for this catalyst the presence of Ag° clusters outside the zeolite structure was recognized by the detection of a strong reflection at about 40° 28. In agreement with the lower Ag content, in Ag2.7Co2.8AF the Ag3 site... 

Refinement takes place by adjusting the model to find closer agreement between the calculated and observed structure factors. For proteins the refinements can yield R-factors in the range of 10-20%. An example taken from reference 10 is instructive. In a refinement of a papain crystal at 1.65-A resolution, 25,000 independent X-ray reflections were measured. Parameters to be refined were the positional parameters (x, y, and z) and one isotropic temperature factor parameter... 

The crystal structure of hydrated NaA a detailed refinement of a pseudosymmet-ric zeolite structure. Z. Kristallogr., 133, 134-149. 

Grundy H. D. and Ito J. (1974). The refinement of the crystal structure of a synthetic non-stoichiometric Sr-feldspar. Amer. Mineral, 68 1319-1326. 

High pressure crystallization process has already been applied In Industry to separate a pure substance from a mixture. In this process, pressure Is used for crystallization. Instead of cooling In the conventional method. Although the pressure Is usually applied In Industry so quickly that It reaches 200 MPa In 10 sec, a sufficient amount of crystals can be grown, which results In attaining the separation and refining of a short cycle within a few minutes (U. 

Pellegrini, M., Gronbech-Jensen, N., Kelly J. A., Pfluegl, G., and Yeates, T. O. (1997) Highly constrained multiple-copy refinement of protein crystal structures. Proteins 29, 426-432. 

Several formulations were proposed [65, 66], but the intuitive notation introduced by Hansen and Coppens [67] afterwards became the most popular. Within this method, the electron density of a crystal is expanded in atomic contributions. The expansion is better understood in terms of rigid pseudoatoms, i.e., atoms that behave stmcturally according to their electron charge distribution and rigidly follow the nuclear motion. A pseudoatom density is expanded according to its electronic stiucture, for simplicity reduced to the core and the valence electron densities (but in principle each atomic shell could be independently refined). Thus,... 

As has become clear in previous sections, atomic thermal parameters refined from X-ray or neutron diffraction data contain information on the thermodynamics of a crystal, because they depend on the atom dynamics. However, as diffracted intensities (in kinematic approximation) provide magnitudes of structure factors, but not their phases, so atomic displacement parameters provide the mean amplitudes of atomic motion but not the phase of atomic displacement (i.e., the relative motion of atoms). This means that vibrational frequencies are not directly available from a model where Uij parameters are refined. However, Biirgi demonstrated [111] that such information is in fact available from sets of (7,yS refined on the same molecular crystals at different temperatures. 

Evidence for a hydride transfer mechanism (Scheme 29) for the PQQ-dependent enzyme methanol dehydrogenase (MDH) was obtained by a theoretical analysis combined with an improved refinement of a 1.9 A resolution crystal structure of MDH from Methylophilus methylotrophus in the presence of CH3OH <2001PNA432>. The alternative mechanism proceeding via a hemiketal intermediate was discounted when the observed tetrahedral configuration of the C-5 atom of PQQ in that crystal structure was shown to be the C-5-reduced form of the cofactor 198, a precursor to the more common reduced form of PQQ 199. 

The object of a crystal-structure determination is to ascertain the position of all of the atoms in the unit cell, or translational building block, of a presumed completely ordered three-dimensional structure. In some cases, additional quantities of physical interest, e.g.. the amplitudes of thermal motion, may also be derived from the experiment. The processes involved in such crystal-structure determinations may he divided conveniently into (I) collection of the data. (2) solution of the phase relations among the scattered x-rays (phase problem)—determination of a correct trial structure, and (3) refinement of this structure. 

The y phase of bismuth molybdate underwent a reversible transformation to the metastable tetragonal y" modification. This metastable modification was observed in the temperature range of 520° to 550°C and underwent an irreversible transformation to the y modification which readily formed at 700°C. The results indicated that the y modification corresponds to that reported by Blasse (83). However, refinement of the crystal data utilizing a single crystal revealed that this y modification was orthorhombic with lattice parameters a = 15.99 A, b = 15.92 A, and c = 17.43 A. An additional observation was the reversible transformation of the y modification to y at 900°C. 

Ballirano, P. and Maras, A. (2002) Refinement of the crystal structure of arsenolite, As203. Zeitschrift fur Kristallo-graphie New Crystal Structures, 217(2), 177-78. 

Further refinement of the crystal structure consisting of double helices is difficult, because the x-ray photograph is not well-defined, and the possibility of a disordered structure must be considered, e.g., right and left-hand helices, and up and down chains. Although there are some unexplained feature of the double helical model, such as the mode of rapid double helix formation during crystallization, the author and his coworkers believe the result to be essentially correct. 

There has been no controversy about the structure of fluorene (31) but its true conformation was in doubt for a number of years. From an early X-ray analysis, Iball (1936a) concluded that the fluorene molecule had a folded conformation and, in a review, Cook and Iball (1936) discussed further evidence for a non-planar conformation, provided by optical activity studies of unsymmetrically substituted fluorene derivatives. Later stereochemical studies (Weisburger et al., 1950) suggested that fluorene had, in fact, a planar conformation. A reinvestigation of the crystal structure by Burns and Iball (1954, 1955) and, independently, by Brown and Bortner (1954) showed that the early X-ray work was in error and confirmed the planar conformation. The refinement of the crystal structure (Burns and Iball, 1954, 1955), by two-dimensional Fourier and least-squares methods, reveals that the maximum deviation of the carbon atoms from the mean molecular plane is 0-030 A, the r.m.s. deviation being 0-017 A. This deviation, 0-017 A, is taken by Burns and Iball to be a measure of the accuracy of their analysis, assuming now that the molecule is strictly planar. 

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