Direct phase determination

Hendrickson WA, Smith JL, Sheriff S. Direct phase determination based on anomalous scattering. Methods Enzymol 1985 115 44-55. 

Bricogne, G. (1993) Direct phase determination by entropy maximization and likelihood ranking status report and perspectives, Acta Cryst., D49, 37-60. 

Direct methods, direct phase determination A method of deriving relative phases of diffracted beams by consideration of relationships among the Miller indices and among the structure factor amplitudes of the stronger Bragg reflections. These relationships come from the conditions that the structure is composed of atoms and that the electron-density map must be positive or zero everywhere. Only certain values for the phases are consistent with these conditions. 

J. Karle and H. Hauptman, A theory of phase determination for the four types of non-centrosymmetric space groups, Acta Cryst. 9, 635 (1956). Jerome Karle, US crystallographer, and Herbert A. Hauptman, US mathematician, laid a foundation towards the development of modem direct phase determination techniques. They won 1985 Nobel Prize in Chemistry for their outstanding achievements in the development of direct methods for the determination of crystal stmctures - http //www.nobel.se/ chemistry/laureates/1985/. 

Currently, Patterson and direct methods are the most frequently employed classical structure solution approaches. The direct phase determination methods are especially successful in solving structures from single crystal data, but their use in powder diffraction increases progressively as the quality of powder data improves, better deconvolution techniques are developed and more precise individual structure factors become available. 

It is worth noting that practically all non-traditional methods for solving crystal structures have been initially developed for both powder and single crystal diffraction data to manage intrinsic incompleteness or poor quality that cannot be improved experimentally. Despite a variety of structure solution approaches, traditional direct phase determination methods appear to be the most common and successful when powder diffraction data are adequate. Patterson methods also work quite well but they require the presence of a heavy atom and, perhaps, more extensive crystallographic expertise. The non-traditional methods are generally employed when other techniques fail and their use is somewhat restricted by both the complexity and limited availability of computer codes. 

There is a variety of software that can be used to process pseudo-single crystal experimental diffraction data represented in a form of individual structure factors or their squares (consult http //www.iucr.org and/or http //www.ccpl4.ac.uk). The most commonly used software products are SHELXS-90 and SHELXL-97 (author G.M. Sheldrick), which are distributed free for academic use (consult SHELX home page at http //shelx.uni-ac.gwdg.de/SHELX/index.html for details). Unless noted otherwise, processing of the individual structure factor data, direct phase determination, Patterson-, Fourier- and E-map calculations shown in this chapter were performed using WinCSD software, which is available from STOE Cie, Gmbh (http //www.stoe.com/products/index.htm). 

The structure was solved using SHELXS-90 based on the extracted structure amplitudes of 425 possible reflections below 20 = 70°. The direct phase determination with = 1-1 resulted in the E-map containing an acceptable model with three heavy peaks Table 6.38) that were automatically assigned to vanadium. The distances from the vanadium atoms to all but one from the list of nine strongest peaks are a good match for V-O bonds. The only exception is Q4, which is too far from all V atoms and too close to Q6 (1.32 A). The Q4-Q6 distance is not listed in Table 6.38. 

Dorset, D. L. McCourt, M. P., Direct Phase Determination for Polymer Fibre X-ray Data—The Structure of Poly(tetrameth -p-silphenylene siloxane). Polymer 1997,38,1985-1988. 

Patterson synthesis by itself cannot solve even moderately large molecular structures (five or six equally heavy atoms) unless concrete structure information already exists so that the vector space can be systematically searched for a presumed structure or substructure. A search of this kind might be done with a computer, perhaps by the method of pattern-seeking functions, or the convolution molecule method [57]. [75]. These techniques, however, are not so often used today, having been supplanted by direct phase-determination methods (Section 15.2.2.2), which make it possible to reach the goal quickly on the basis of relatively little prior knowledge [76], Patterson synthesis may return to prominence when used to identify sets of starting phases (Section 15.2.2.2). 

The best-known equation for direct phase determination is that of David Sayre 78] ... 

From Equation (35), the normalized structure amplitudes l l were obtained. These provided the basis for direct phase determination. 

The fundamental reason for this situation is that the relationships in direct phase-determination methods are nothing more than probability relations. If these can be turned into true equations, single-crystal X-ray structure analyses would become a routine analytical procedure. H. Haupt-... 

The experiment with 40% was carried out with two different sets of random errors to illustrate the statistical character of the results. We clearly see from Table 3 that direct phase determination up to four phases is possible in principle, but in practice will depend upon the experimental errors. Considering Table 3 we could state that a phase is true when its 4>-value is smaller by a factor of at least 5 than the next hi er -value. 

We condude with a list of possibfe e ierimental errors which prevent direct phase determination by 4>-value alone, as inferred from our experimental evidence. All should be mitigable to a tolerable level taking suhable care, during preparation and X-ray exposure of the membrane sanqdes. 

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