Vector method, diffraction

MATHEMATICAL METHODS IN PHYSICS AND ENGINEERING, John W. Denman. Algebraically based approach to vectors, mapping, diffraction, other topics in applied math. Also generalized functions, analytic function theory, more. Exercises. 448pp. 5X 85. 65649-7 Pa. 8.95... 

Dolomite is much less prone to electron damage than calcite and so it is easier to determine Burgers vectors using diffraction contrast methods. The vectors for perfect dislocations are 2 110 and 12 01). However, dislocation dissociation is also possible in dolomite when it occurs in the basal planes, pairs of... 

We will use an example as illustration. The dipole moment vector for formamide has been determined both by diffraction and microwave spectroscopy. As the diffraction experiment measures a continuous charge distribution, the moments derived are defined in terms of the method used for space partitioning, and are not necessarily equal. Nevertheless, the results from different techniques (Fig. 7.1) agree quite well. 

Unfortunately, the experimental data cannot be correlated directly with the crystal structure, because some of the information needed is lost in recording the X-ray intensity pattern. Indeed, the diffracted X-ray beam is a vector, and it has both an intensity and a phase angle. The information concerning phase angle is lost during measurement of the X-ray intensity. The phase angle could, perhaps, be recorded by a holographic method, but X-ray lasers do not as yet exist. 

Part II deals, in six chapters, with the principles underlying the progressive stages in the elucidation of internal structure. Chapters VI and VII deal with the principles of structure determination by trial Chapter VIII with the use of physical properties (such as habit, cleavage, and optical, magnetic, pyro- and piezo-electric properties) as auxiliary evidence in structure determination. In Chapter IX are to be found several examples of the derivation of complete structures. Chapter X gives an introductory account of the use of direct and semi-direct methods based on the calculation of electron density distributions and vector distributions from X-ray diffraction data. 

Similar in principle to this is the use of two different X-ray wavelengths on the same crystal, one near to and the other far from the absorption edge of a marker atom, to give different diffracting powers this is the "ideal isomorphism method advocated by Lipson and Cochran (195 ) and Pepinsky (1956). A comparison of the vector maps for the two wavelengths, or a vector map based on the differences of structure amplitudes for corresponding reflections for the two wavelengths, indicates which peaks are due to the marker atoms. 

There are two approaches to the solution of the phase problem that have remained in favor. The first is based on the tremendously important discovery or Patterson in the 1930s ihal the Fourier summation of Eq. 3. with (he experimentally known quantities F2 (htl> replacing F(hkl) leads nol to a map of scattering density, but to a map of all interatomic vectors. The second approach involves the use of so-called direct methods developed principally by Karie and Hauptman of the U.S. Naval Research Laboratory and which led to the award of the 1985 Nobel Prize in Chemistry. Building upon earlier proposals that (he relative intensities of the spots in a diffraction pattern contain information about a crystal phase. Hauptman and Karie developed a mathematical means of extracting the information. A fundamental proposition of (heir direct method is that if thrice intense spots in the pattern have positions whose coordinates add up to zero, their relative phases will cancel out. Compulations done with many triads of spots yield probable phases for a significant number of diffracted waves and further mathematical analysis leads lo a likely solution for the structure of the molecule as a whole. 

The computational labor associated with two-dimensional Fourier syntheses is not too formidable, and two-dimensional Fourier maps can be constructed without machine help. The labor associated with two-dimensional Patterson sysntheses is even less, and a two-dimensional vector map can often be obtained from measured intensities in a few hours. For Fourier and Patterson syntheses in three-dimensions, however, machine help is virtually indispensable. Before application of automatic computers to x-ray diffraction, the main obstacle standing in the way of a structure determination was generally the computational effort involved. In the 1950 8, the use of computers became commonplace, and the main obstacle became the conversion of measured intensities to amplitudes (the so-called phase problem ). There is still no general way of attacking this problem that is applicable in all situations, but enough methods have been developed so that by use of one, or a combination of them, all but very complicated structures may, with time and ingenuity, be determined. 

Figure 7.16. Illustration of Ewald sphere construction, and diffraction from reciprocal lattice points. This holds for both electron and X-ray diffraction methods. The vectors AO, AB, and OB are designated as an incident beam, a diffracted beam, and a diffraction vector, respectively.

The structure of W(CO)3(PPr 3)2(H2) has been established by a variety of methods, including a neutron diffraction study. The hydrogen molecule is present as a ligand with the H-H axis collinear to the P-M-P vector (eq. (2)). 

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