Crystallographic parameters, refined

Toraya s WPPD approach is quite similar to the Rietveld method it requires knowledge of the chemical composition of the individual phases (mass absorption coefficients of phases of the sample), and their unit cell parameters from indexing. The benefit of this method is that it does not require the structural model required by the Rietveld method. Furthermore, if the quality of the crystallographic structure is poor and contains disordered pharmaceutical or poorly refined solvent molecules, quantification by the WPPD approach will be unbiased by an inadequate structural model, in contrast to the Rietveld method. If an appropriate internal standard of known quantity is introduced to the sample, the method can be applied to determine the amorphous phase composition as well as the crystalline components.9 The Rietveld method uses structural-based parameters such as atomic coordinates and atomic site occupancies are required for the calculation of the structure factor, in addition to the parameters refined by the WPPD method of Toraya. The additional complexity of the Rietveld method affords a greater amount of information to be extracted from the data set, due to the increased number of refinable parameters. Furthermore, the method is commonly referred to as a standardless method, since the structural model serves the role of a standard crystalline phase. It is generally best to minimize the effect of preferred orientation through sample preparation. In certain instances models of its influence on the powder pattern can be used to improve the refinement.12... 

As briefly mentioned in the previous chapter, the determination of a crystal structure may be considered complete only when multiple pattern variables and crystallographic parameters of a model have been fully refined against the observed powder diffraction data. Obviously, the refined model should remain reasonable from both physical and chemical standpoints. The refinement technique, most commonly employed today, is based on the idea suggested in the middle 1960 s by Rietveld. The essence of Rietveld s approach is that experimental powder diffraction data are utilized without extraction of the individual integrated intensities or the individual structure factors, and all structural and instrumental parameters are refined by fitting a calculated profile to the observed data. 

The final step in a structure solution is to optimise and refine the experimental diffraction pattern and crystallographic parameters against the observed powder... 

Crystallographic structure refinement is generally understood to be the last step in the determination of a crystal structure by diffraction methods. The usual procedure of a crystal structure analysis includes collection of X-ray or neutron diffraction intensities, data reduction yielding structure factor amplitudes, the solution of the crystallographic phase problem yielding approximate structural parameters and finally refinement of these parameters to obtain a best fit of the observed structure factor amplitudes with... 

M. Body, G. SiUy, C. Legein, J.-Y. Buzare, F. Calvayrac, P. Blaha, Al NMR experiments and quadrupolar parameter ab initio calculations crystallographic structure refinement of P-BajAlFs, Chem. Phys. Lett., 424, 321 326 (2006). 

Table 6.2 Refined crystallographic parameters and reliability factors in Rietveld and MPF analyses for Lao.64(Tio.92Nbo.o8)02.99 [H]...

Table 6.3 Refined crystallographic parameters and reliability factors obtained from Rietveld and MPF analysis for Lao gSro.4Co03 s at 1531.4 K (5 = 0.34(2)) [12]...

Crystallographic refinement is a procedure which iteratively improves the agreement between structure factors derived from X-ray intensities and those derived from a model structure. For macro molecular refinement, the limited diffraction data have to be complemented by additional information in order to improve the parameter-to-observation ratio. This additional information consists of restraints on bond lengths, bond angles, aromatic planes, chiralities, and temperature factors. 

The structure was refined by block-diagonal least squares in which carbon and oxygen atoms were modeled with isotropic and then anisotropic thermal parameters. Although many of the hydrogen atom positions were available from difference electron density maps, they were all placed in ideal locations. Final refinement with all hydrogen atoms fixed converged at crystallographic residuals of R=0.061 and R =0.075. 

The compound [Fe(Cp )2][Ni(a-tpdt)2], although not isostructural with the [Cr(Cp )2] analogue, presents a lower symmetric structure, where one of the crystallographic axis is doubled. This axis doubling is associated with a slight alternation in the D+-A contacts along the chains and consequently also in interchain contacts. Otherwise the network of intermolecular contacts remains similar. There are no full structural refinements for the other compounds of this series but the unit cell parameters of the (M = Fe, M = Au) compound and powder diffraction data of the salt with M = Co, M = Ni indicate that they are isostructural with the M = Fe, M = Ni compound. On the other hand, powder diffraction data on the M = Mn, M = Ni compound indicate that it is isostructural with the M = Cr, M = Ni compound. 

Macromolecular crystallographic refinement is an example of a restrained optimization problem. Standard refinement programs adjust the atomic positions and, typically, also their atomic displacement parameters of a given model with the... 

Figure 11.3 The ratio of the number of reflections to the number of parameters in the XYZB crystallographic refinement of a protein model as a function of the resolution of the data. The data are assumed to be complete and the solvent content to be 50%.

Engh, R. A. and Huber, R. (1991) Accurate bond and angle parameters for x-ray protein structure refinement. Acta Crystallograph. A A47, 392-400. 

The number of reflection intensities measured in a crystallographic experiment is large, and commonly exceeds the number of parameters to be determined. It was first realized by Hughes (1941) that such an overdetermination is ideally suited for the application of the least-squares methods of Gauss (see, e.g., Whittaker and Robinson 1967), in which an error function S, defined as the sum of the squares of discrepancies between observation and calculation, is minimized by adjustment of the parameters of the observational equations. As least-squares methods are computationally convenient, they have largely replaced Fourier techniques in crystal structure refinement. 

Of great interest to the molecular biologist is the relationship of protein form to function. Recent years have shown that although structural information is necessary, some appreciation of the molecular flexibility and dynamics is essential. Classically this information has been derived from the crystallographic atomic thermal parameters and more recently from molecular dynamics simulations (see for example McCammon 1984) which yield independent atomic trajectories. A diaracteristic feature of protein crystals, however, is that their diffraction patterns extend to quite limited resolution even employing SR. This lack of resolution is especially apparent in medium to large proteins where diffraction data may extend to only 2 A or worse, thus limiting any analysis of the protein conformational flexibility from refined atomic thermal parameters. It is precisely these crystals where flexibility is likely to be important in the protein function. 

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