Anomalous dispersion corrections

Crystals of the compound of empirical formula FiiPtXe are orthorhombic with unit-cell dimensions a = 8-16, h = 16-81. c = 5-73 K, V = 785-4 A . The unit cell volume is consistent with Z = 4, since with 44 fluorine atoms in the unit cell the volume per fluorine atom has its usual value of 18 A. Successful refinement of the structure is proceeding in space group Pmnb (No. 62). Three-dimensional intensity data were collected with Mo-radiation on a G.E. spectrogoniometer equipped with a scintillation counter. For the subsequent structure analysis 565 observed reflexions were used. The platinum and xenon positions were determined from a three-dimensional Patterson map, and the fluorine atom positions from subsequent electron-density maps. Block diagonal least-squares refinement has led to an f -value of 0-15. Further refinements which take account of imaginary terms in the anomalous dispersion corrections are in progress. 

Scattering factors for neutral F, Xe, and Pd were taken from Doyle and Turner" and Cromer and Waber." Cromer and Liberman s values" of Af and A/" were used for anomalous dispersion corrections. ... 

Three-dimensional Patterson functions showed the positions of the Xe and As atoms. Subsequent least-squares rerinements, Fourier calculations, and difference Fourier calculations revealed the fluorine atom positions. Least-squares refinements, in which the function EwdfJ -was minimized, converged rapidly to the final structures. Scattering factors of Doyle and Turner were used, and anomalous dispersion corrections were applied. The resulting R factors are given in Table I. 

Section A3.2 gives the anomalous dispersion corrections (f and f", in e ) tabulated as a function of wavelength for a few elements, as an illustration, namely the K edges of selenium and bromine as well as the L edges of platinum and mercury. 

Table A3.2. Tabulated values of anomalous dispersion corrections for selenium, bromine, platinum and mercury.

In Eqs. (2.4) and (2.5), f and f are the real and imaginary parts of the anomalous dispersion correction to the atomic scattering factor, which can be important when the X-ray energy is close to an absorption edge. Z is the number of electrons. Below, for simplicity, such effects will be assumed to be included in the electron densities, when necessary. 

The parameters were then further refined by four successive least-squares procedures, as described by Hughes (1941). Only hk() data were used. The form factor for zinc was taken to be 2-4 times the average of the form factors for magnesium and aluminum. The values of the form factor for zinc used in making the average was corrected for the anomalous dispersion expected for copper Kot radiation. The customary Lorentz, polarization, temperature, and absorption factors were used. A preliminary combined scale, temperature, and absorption factor was evaluated graph-... 

From this equation it can be seen that the depth of penetration depends on the angle of incidence of the infrared radiation, the refractive indices of the ATR element and the sample, and the wavelength of the radiation. As a consequence of lower penetration at higher wavenumber (shorter wavelength), bands are relatively weaker compared to a transmission spectrum, but surface specificity is higher. It has to be kept in mind that the refractive index of a medium may change in the vicinity of an absorption band. This is especially the case for strong bands for which this variation (anomalous dispersion) can distort the band shape and shift the peak maxima, but mathematical models can be applied that correct for this effect, and these are made available as software commands by some instrument manufacturers. 

Atomic scattering factors corrected for anomalous dispersions were obtained from Cromer and Waber and Cromer and Liberman... 

Even if anomalous dispersion data are involved in the substructure determination process, it is the magnitude of the anomalous differences that are used, and the substructures of the biologically occurring macromolecule and its enantiomer are both consistent with the data. The probability that the substructure obtained by direct methods can be developed into a protein model with L-amino acids and right-handed a helices is 50%. Therefore, before proceeding further, other information must be used to determine the correct hand. 

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. 

In contrast to/), the correction terms Af and Af" depend only slightly on the diffraction angle 0. Since/) decreases strongly at higher values of sin 0/A, the relative contribution of anomalous dispersion becomes more significant at higher diffraction angles. 

Table 1. Correction Terms Af and Af" for Anomalous Dispersion of Some Elements, Which Commonly Occur in Organic Compounds, at Three X-ray Wavelengths25...

Before this information can be used, the data set has to undergo some routine corrections, this process is known as data reduction. The Lorentz correction, L, relates to the geometry of the collection mode the polarization correction, p, allows for the fact that the nonpolarized X-ray beam, may become partly polarized on reflection, and an absorption correction is often applied to data, particularly for inorganic structures, because the heavier atoms absorb some of the X-ray beam, rather than just scatter it. Corrections can also be made for anomalous dispersion, which affects the scattering power of an atom when the wavelength of the incident X-ray is close to its absorption edge. These corrections are applied to the scattering factor, 4 of the atom. 

On both sides of the resonance region n increases with increasing frequency, which is called normal dispersion. Only in the immediate vicinity of the resonance frequency does n decrease with frequency, so-called anomalous dispersion. Such a reversal of dispersion, if it occurred in transparent regions, would provide a much-needed material for designing color-corrected lenses. Unfortunately, anomalous dispersion occurs only in regions of high absorption where no appreciable light is transmitted. 

Solution and Refinement of the Structure. Scattering factors for the hydrogen (14) and nonhydrogen (15) atoms are those used previously. Anomalous dispersion terms (16) were included in Fc for rhodium and phosphorus atoms. For the processing of the data and solution and refinement of the structure procedures and computer programs standard in this laboratory were used. (See, e.g., Ref. 17).) Trial absorption corrections calculated for a random selection of reflections gave transmission factors in the range 0.71 to 0.73 therefore a full absorption correction was considered to be unnecessary. 

The difference Fourier synthesis, phased by the P, Cl, N, C and O atoms, revealed the hydrogen atoms with their expected locations. Thus, the final refinement could be performed on the entire set of atoms including hydrogens with fixed isotropic thermal parameter factor, BH = 4 A2. Final R and S values are 0.022 and 0.948, respectively. The last difference Fourier map showed no values to be greater than 0.3 eA 3 (Table 15). Atomic scattering factors were corrected for anomalous dispersion from Cromer and WaberS8). 

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