Phase refinement

The United States refining industry is undergoing a restructuring phase. Refiners will continue to be under pressure and only the most efficient and profitable operations are going to survive. The survivors will be those who have some niche in the market place, have the versatility to handle low-cost crude, meet product demand, and conform to environmental regulations. 

At liquid-liquid equilibrium, the composition of the two phases (refined phase and extracted phase ) can be determined from the following equations... 

Delarue, M. and Qrland, H. (2000). General formalism for phase combination and phase refinement a statistical thermodynamics approach in reciprocal space. Actu Crystallogr. A 56, 562-574. 

Debaerdemaeker, T. and Woohson, M. M. (1983). On the application of phase relationships to complex structures. XXII. Techniques for random phase refinement. Acta Crystallogr. A 39, 193-196. 

DeTitta, G. T., Weeks, C. M., Thuman, P, Miller, R. and Hauptman, H. A. (1994). Structure solution by minimal-function phase refinement and Fourier filtering. 

J. -X. and Zheng, C.-D. (2000). Aflexible and efficient procedure for the solution and phase refinement of protein structures. Acta Crystallogr. D 56,1137-1147. 

Unfortunately, the relations between the electron density, the restraints we have discussed here, and the structure factors are non-linear. Thus, the only strategy we can adopt is to use the approximate phases we start out with and improve these iteratively. Even this is not straightforward, mainly because Eq. 1 is expensive to compute. However, there exists a powerful and straightforward procedure that is used in virtually all phase refinement programs Eourier cycling. 

Figure 10.1 A flow chart demonstrating phase refinement in practice with the iterative recycling between real and reciprocal space.

Figure 10.6 Histogram matching. In (a) are shown a histogram from a first map in phase refinement (dashed line) and a theoretical protein histogram (solid line). In (b), the protein histogram (dashed line) and a sharp solvent histogram (solid line) are shown.

Abrahams, J. P. (1997). Bias reduction in phase refinement by modified interference functions introducing the correction. Acta Crystallogr. D 53, 371-376. 

Once the position and orientation of the particle are known, phase refinement and extension can... 

In the MEM/Rietveld analysis, each of the observed structure factors of intrinsically overlapped reflections (for instance, 333 and 511 in a cubic system) can be deduced by the structure model based on a free atom model in the Rietveld refinement. In such a case, the obtained MEM charge density will be partially affected by the free atom model used. In order to reduce such a bias, the observed structure factors should be refined based on the deduced structure factors from the obtained MEM charge density. The detail of the process is described in the review article [9,22-24]. In addition, the phased values of structure factors based on the structure model used in Rietveld analysis are used in the MEM analysis. Thus, the phase refinement is also done for the noncentrosymmetric case as P2, of Sc C82 crystal by the iteration of MEM analysis. The detail of the process is also described elsewhere [25]. All of the charge densities shown in this article are obtained through these procedures. 

A rough outline of the organization for the integration project - for example, the establishment of an integration board and an integration office - should be drawn up as early as possible in the design phase. Refinement of the projects structure - that is, its separation into smaller projects and sub-projects - and the establishment of controls do not occur until the beginning of the second phase. 

Once diffraction data have been gathered the next stage in the structure analysis is usually focused on seeking a definition of the orientation of the particle and, often, its position in the crystal cell. This information is essential for both phase refinement and the analysis of isomorphous replacement experiments. Because of the inevitable noncrystallographic redundancy (a minimum of 5-fold) and the fixed relationship between the various icosahedral symmetry axes it is often possible to solve the orientation problem by analysis of the diffraction data in the absence of a model, usually by use of a self-rotation function. 

Raghavan, N. V., and Tulinsky, A. The structure of a-chymotrypsin. II. Fourier phase refinement and extension of the dimeric structure at 1.8 A resolution by density modification. Acta Cryst. B35, 1776-1785 (1979). 

Tulinsky, A. Phase refinement/extension by density modification. Methods in Enzymology 115, 77-89 (1985). 

Langs, D. A. Unique solutions for the SIR and SAS phase problems and the use of partial structural information in phase refinement. Acta Cryst. A42, 362-368 (1986). 

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