Anomalous dispersion measurements

The experimental method used to determine the chirality or absolute structure of a molecule or crystal structure involves the use of the anomalous dispersion of X-rays by one or more atoms in the structure. We will now describe this effect and how Bijvoet used it to determine the absolute configuration of (-l-)-tartaric acid from the differences in the intensities of the hkl and iM Bragg reflections. 

The scattering factor for the anomalously scattering atom, j, is complex and is expressed as  

If an atom absorbs X radiation in this way (even moderately), the result will be a phase change in the X rays scattered by that atom, relative to the X rays scattered by the other atoms in the structure (which do not scatter anomalously). This phase change is equivalent to a path length 

FIGURE 14.18. A/ and A/ versus atomic number for Cu Ka radiation. Note how large A/ may be. 

FIGURE 14.19. The phase difference depends on the absolute structure, (a) In normal diffraction, with no anomalous dispersion, for the hkl Bragg reflection, the path difference (PD) = 2p. (b) For the hkl Bragg reflection, the path difference also equals 2p in normal diffraction. Therefore I hkl) = I (hkl). (c) If there is an anomalous scattering atom in the crystal structure, the PD = 2p + q for thre hkl Bragg reflection, and (d) the PD — 2p - q for the hkl Bragg reflection. Therefore I hkl) I(hki). 

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Anomalous scattering can also be used directly if the protein is small and a suitable anomalous scatterer can be used. The three-dimensional structure of the small protein, crambin, was determined by W ayne A. Hendrickson and Martha Teeter by the use of anomalous dispersion measurements. This protein contains 45 amino acid residues and diffracts to 0.88 A resolution. It crystallizes with 72 water and four ethanol molecules per protein molecule. Since there is a sulfur atom in the protein molecule, the use of its anomalous scattering was made. The nearest absorption edge of sulfur lies at 5.02 A, but for Cu Ka radiation, wavelength 1.5418 A, values of A/ and A/" for sulfur are 0.3 and 0.557, respectively. Friedel-related pairs of reflections were measured to 1.5 A resolution, and sulfur atom positions were computed from difference Patterson maps. The structure is now fully refined and a portion of an a helix was shown in Figure 12.27 (Chapter 12). 

Anomalous dispersion measurements have proved to be very useful as an aid in phase determination in protein structure determination. 

The unit cell of a protein is assigned with respect to a right-handed system of axes. Once a heavy atom has been located, its phase angle may be +a or —a for FH, since it is not known whether the interpreted peak in the Patterson map is from atom 1 to atom 2 or atom 2 to atom 1. Several methods have been developed to remove this ambiguity of which the most decisive are those that involve the preparation of a derivative containing both heavy atoms and/or anomalous dispersion measurements (5). 

This is very crude measurement. Nevertheless, it is sufficient to discriminate against some of the dispersion formulae that have been proposed. This matter is considered in the next paper where it is shown that the formulae which are satisfactory in that they put the anomalous dispersion at the observed frequencies yield values for (jun 1) which differ from that given by experiment, by several times the margin indicated in (8). 

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 preceding characterization of anomalously dispersive transit pulses aroused considerable interest, both from a theoretical and an experimental viewpoint. Attention focused on the latter was stimulated by the possibility of using the log-log display technique to identify ft in cases where the dispersion was such as to obscure any change of gradient in conventionally displayed transit pulses. However, it became necessary to question the validity of such measurements of ft under conditions where individual carrier transit times vary over such a wide range. 

According to F. P. le Roux, like all vapours with a large selective absorption, iodine has an anomalous dispersion since it increases with a fall of temp., being about 0 06 from A. Hurion s measurements—approximately as large a negative number as glass is positive. The atomic refraction of solid iodine is 24-5 by the //.-formula, and 14-12 by the /t2-formula. 

The principle problem with diffuse reflectance is that the specular component of the reflected radiation, that which does not penetrate the sample, is measured along with the diffuse reflected light which penetrates the sample. Generally, the change in specular reflection with frequency is small except in regions of strong absorption bands where the anomalous dispersion leads to Reststrahlen bands in the specular reflection spectrum. When the Reststrahlen bands are observed, the absorption bands can appear inverted at their center. This effect makes quantitative measurements on samples with strong absorptivity very difficult. 

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. 

Bishop s attention turned to accurate calculations of electrical and magnetic properties, especially those of importance in nonlinear optics. Since most experiments in this field measure ratios, not absolute values, it is necessary to have a calculated value. Universally, Bishop s helium nonlinear optical properties are used. In the same field, he was the first to seriously investigate the effects of electric fields on vibrational motions, with a much-quoted paper.65 His theory and formulation has now been added to two widely used computational packages HONDO and SPECTROS. He has also derived a rigorous formula to account for the frequency dependence (dispersion) in nonlinear optical properties.66 He used this theory to demonstrate that the anomalous dispersion in neon, found experimentally, is an artifact of the measurements. 

In recent papers we have shown that small-angle X-ray scattering (SAXS) is a highly suitable method to investigate stiff-chain polyelectrolytes [71]. In particular, it has been demonstrated there that the effect of anomalous dispersion [72] can be applied to discern the contribution of the counterions to the measured scattering intensity I(q). Here the main points of this analysis that is based on earlier work by small-angle neutron scattering (SANS [73-76]) and by SAXS [77, 78] are presented and discussed. 

The measured intensity at a wavelength with anomalous dispersion terms f and f" is given by... 

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