Phase problem direct methods for solving

The use of blank-disc CBED patterns for solving crystal stmctures by electron diffraction (in conjunction with direct methods for the phase problem) would seem to have many advantages ... 

Hauptman, H. (in press) The phase problem of X-ray crystallography. In Direct Methods for Solving Macromolecular Structures, Fortier, S. (Ed.), Kluwer, Dordrecht. 

The conunonly used methods for solving the phase problem required for stmcture solutions are the direct methods and Patterson maps. Direct methods use relationships between phases such as triplets (0 = 4>h + 4>k + -h-k The probabihty of 0 >= 0 increases with the magnitude of the product of the normalized stmcture factors of the three reflections involved. Once these triplets associated with high certainty are identified based on diffraction intensities, they are used to assign new phases based on a set of known phases. Since the number of phase relationships is large the problem is over determined. Another approach is based on the Sayre equation, which is derived based on the relationship between the electron density and its square ... 

In this paper the authors produced a class of non-linear explicit second-order methods for solving one-dimensional periodic initial value problems (IVPs). These methods are P-stable (based on the definition given by Lambert and Watson ) and they have phase-lag of high-order. The authors introduce a special vector operation such that the obtained methods to be extended to be vector-applicable directly. With this extension the produced methods can be applied to multi-dimensional problems. 

In small-molecule crystallography the phase problem was solved by so-called direct methods (recognized by the award of a Nobel Prize in chemistry to Jerome Karle, US Naval Research Laboratory, Washington, DC, and Herbert Hauptman, the Medical Foundation, Buffalo). For larger molecules, protein aystallographers have stayed at the laboratory bench using a method pioneered by Max Perutz and John Kendrew and their co-workers to circumvent the phase problem. This method, called multiple isomorphous replacement... 

Among various methods to synthesize nanometer-sized particles [1-3], the liquid-phase reduction method as the novel synthesis method of metallic nanoparticles is one of the easiest procedures, since nanoparticles can be directly obtained from various precursor compounds soluble in a solvent [4], It has been reported that the synthesis of Ni nanoparticles with a diameter from 5 to lOnm and an amorphous-like structure by using this method and the promotion effect of Zn addition to Ni nanoparticles on the catalytic activity for 1-octene hydrogenation [4]. However, unsupported particles were found rather unstable because of its high surface activity to cause tremendous aggregation [5]. In order to solve this problem, their selective deposition onto support particles, such as metal oxides, has been investigated, and also their catalytic activities have been studied. 

A well-known method for peak tracking is based on the phase-system switching idea, " which was developed to solve problems of mobile phase incompatibility in LC/MS target compound analysis. An analytical column is usually connected to a trapping column in tandem mode. A switching valve is placed after the UV detector, and the flow of nonvolatile eluents is directed through the trapping column to waste. When the peak of interest elutes from the analytical column it is... 

If the Patterson method cannot be applied because the structure has no or too many heavy atoms, it is possible to use another approach for phase determination, the so-called direct methods. By the term direct methods is meant that class of methods which exploits relationships among the structure factors in order to go directly from the observed magnitudes E to the needed phases < ) (Herbert A. Hauptman, Nobel lecture, 9. Dec., 1985). The direct method approach for solving the phase problem uses probability... 

The electron density is everywhere positive (a probability can not be negative ). Nevertheless, that the positivity of the density function is not a necessary prerequisite for solving the phase problem via direct methods was recently shown [20]... 

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