Anomalous scattering dispersion methods

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. 

In order to exploit the heavy atom method with crystals of conventional molecules, or to utilize the isomorphous replacement method or anomalous dispersion technique for macro-molecular structure determination, it is necessary to identify the positions, the x, y, z coordinates of the heavy atoms, or anomalously scattering substituents in the crystallographic unit cell. Only in this way can their contribution to the diffraction pattern of the crystal be calculated and employed to generate phase information. Heavy atom coordinates cannot be obtained by biochemical or physical means, but they can be deduced by a rather enigmatic procedure from the observed structure amplitudes, from differences between native and derivative structure amplitudes, or in the case of anomalous scattering, from differences between Friedel mates. 

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. 

Many years ago, Stuhrmann showed that anomalous dispersion can circumvent this problem in studies of polyelectrolytes by SAXS [15,16], This method utilizes the dependence of the scattering factor / if the energy of the incident radiation is near the absorption edge of the counterions [16], Hence, he scattering factor /ion becomes a complex function of the energy E of the incident radiation near the absorption edge of the ions [15,16],... 

Protein structures are now more often determined by using multiple anomalous dispersion (MAD) techniques. In this method, scattering that does not obey the kinematic theory is used. The kinematic theory deals with single scatter-... 

The anomalous dispersion effect is associated with the ejection of photoelectrons from inner shell electrons in an atom. The normal scattering describes the interaction of all the electrons in the atom with the X-ray beam. The radial distribution of the electrons in an atom can be calculated using quantum mechanics, originally by Hartree s self-consistent field method (Hartree 1933). In figure 9.12 this distribution is given for rubidium, which has a K edge at 0.8155 A the mean radius for... 

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