Phasing, anomalous dispersion

Fig. 1. (a) Phase matched second harmonic generation (2cJ = 0.49 fiTo) at cj = 0.98 where = refractive index by ordinary rays and = by extraordinary rays, (b) Hypothetical anomalous dispersion phase matching at 850 nm in similar a crystal having a Lorent2ian absorption centered at 650... 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

Single crystals of ferroelectric TGS grown from solution generally contain domains of the two enantiomorphous phases (110). This juxtaposition of enantiomorphous domains may be explained in terms of the minor deviations of the crystal structure from a centrosymmetric arrangement. The relative concentrations of the two enantiomorphous phases in a single crystal may be determined by means of anomalous dispersion of X-rays (111,112). 

Pahler, A., Smith, J. L. and Hendrickson, W. A. (1990). A probability representation for phase information from multiwavelength anomalous dispersion. Acta CrystaT logr. A 46, 537-540. 

The a vaiues are a measure of the electron-density variation in the protein and solvent regions, and the ratio of these numbers is a measure of the contrast between the two regions. Since anomalous dispersion data were used to phase the maps, the map for the correct hand will show greater contrast. In this case, the original direct-methods sites give rise to greater contrast thereby indicating that these sites do correspond to the correct enantiomorph. 

The anomalous components of the total scattering are wavelength dependent and the use of radiation close to an absorption edge may increase or optimise the contribution due to the anomalously scattering atoms. Ramaseshan (1962) pointed out that data collected at multiple wavelengths optimising the anomalous dispersion effects would improve the quality of phase determination. 

A further improvement and more freedom in the choice of laser wavelengths can be expected with the use of dye vapors. In liquids, the phase-matching concentration is set by the requirement that the anomalous dispersion of the dye compensates for the normal dispersion of the solvent. The latter is a new parameter that can be varied at will in the gas phase by changing the nature and partial pressure of the buffer gas. The broader resonances of dyes as opposed to metal vapors, which are sometimes used for this purpose, is an advantage for tunable frequency tripling of dye lasers. Another advantage results from the possibility of working at much lower temperatures than with metal vapors. 

Chemistry of Anomalous-Dispersion Phase-Matched Second Harmonic Generation... 

Our recent report(1) on ADPM SHG (Anomalous-Dispersion Phase-Matched Second-Harmonic Generation) addressed the physics of ADPM, a... 

The most demanding element of macromolecular crystallography (except, perhaps, for dealing with macromolecules that resist crystallization) is the so-called phase problem, that of determining the phase angle ahkl for each reflection. In the remainder of this chapter, I will discuss some of the common methods for overcoming this obstacle. These include the heavy-atom method (also called isomorphous replacement), anomalous scattering (also called anomalous dispersion), and molecular replacement. Each of these techniques yield only estimates of phases, which must be improved before an interpretable electron-density map can be obtained. In addition, these techniques usually yield estimates for a limited number of the phases, so phase determination must be extended to include as many reflections as possible. In Chapter 7,1 will discuss methods of phase improvement and phase extension, which ultimately result in accurate phases and an interpretable electron-density map. 

A second means of obtaining phases from heavy-atom derivatives takes advantage of the heavy atom s capacity to absorb X rays of specified wavelength. As a result of this absorption, Friedel s law (Chapter 4, Section III.G) does not hold, and the reflections hkl and —h — k—l are not equal in intensity. This inequality of symmetry-related reflections is called anomalous scattering or anomalous dispersion. 

In words, the desired electron-density function is a Fourier series in which term hkl has amplitude IFobsl, which equals (7/, /)1/2, the square root of the measured intensity Ihkl from the native data set. The phase ot hkl of the same term is calculated from heavy-atom, anomalous dispersion, or molecular replacement data, as described in Chapter 6. The term is weighted by the factor whU, which will be near 1.0 if ct hkl is among the most highly reliable phases, or smaller if the phase is questionable. This Fourier series is called an Fobs or Fo synthesis (and the map an Fo map) because the amplitude of each term hkl is iFobsl for reflection hkl. 

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