Bragg reflections, phases determined

Molecular orientational order in adsorbed monolayers can be inferred indirectly from elastic neutron diffraction experiments if it results in a structural phase transition which alters the translational symmetry of the 2D lattice. In such cases, Bragg reflections appear which are not present in the orientationally disordered state. Experiments of this type have inferred orientational order in monolayers of oxygen (41) and nitrogen (42) adsorbed on graphite. However, these experiments have not observed a sufficient number of Bragg reflections to determine the molecular orientation by comparing relative Bragg peak intensities with a model structure factor. 

Phase problem The problem of determining the phase angle (relative to a chosen origin) that is to be associated with each diffracted wave that is combined to give an electron-density map. The measured intensities of diffracted beams give only the squares of the amplitudes, but the relative phases cannot normally be determined experimentally (see Chapter 8). The determination of the relative phases of the Bragg reflections is crucial to the calculation of the correct electron density map. 

FIGURE 6.18. Overall scheme of a crystal structure determination in one dimension. (a) Atomic structure, (b) Bragg reflections that are measured, (c) phases assigned to give electron-density waves with the correct phases, a(hkl), and amplitudes. I F(hkl) I, (d) the sununation of density waves to give an electron-density map, and (e this electron-density map has peaks at atomic positions (compare with the situation in (a), the true atomic arrangement). 

Currently, techniques for analyzing powder diffraction that involve the analytical dissection of measured Bragg reflection shapes " " can be used to derive a data set similar to that for a single crystal. As a result, crystal structures may be determined, and in some cases, refined anisotropically using powder diffraction data for X rays or neutrons. Powder diffraction methods can be used at a wide variety of temperatures and pressures and are well suited for studies of phase transformations. 

This Chapter is concerned with methods for obtaining the relative phase angles for each Bragg reflection so that the correct electron-density map can be calculated and, from it, the correct molecular structure determined. When scattered light is recombined by a lens, as described in Chapters 3 and 6, the relationships between the phases of the various diffracted beams are preserved. In X-ray diffraction experiments, however, only the intensities of the Bragg reflections are measured, and information on the relative phases is lost. An attempt is maxle to remedy this situation by deriving relative phases by one of the methods to be described in this Chapter. Then Equation 6.3 (Chapter 6) is used to obtain the electron-density map. Peaks in this map represent atomic positions. 

Direct methods, direct phase determination A method of deriving relative phases of diffracted beams by consideration of relationships among the Miller indices and among the structure factor amplitudes of the stronger Bragg reflections. These relationships come from the conditions that the structure is composed of atoms and that the electron-density map must be positive or zero everywhere. Only certain values for the phases are consistent with these conditions. 

Harker-Kasper inequalities Inequalities among structure factors, dependent on the space group, that lead to equations that allow the determination of the relative phases of certain intense Bragg reflections. 

Heavy-atom derivative of a protein The product of soaking a solution of the salt of a metal of high atomic number into a crystal of a protein. If the heavy-atom derivative is to be of use in structure determination, the heavy atom must be substituted in only one or two ordered positions per asymmetric unit. Then the method of isomorphous replacement can be used to determine the relative phase angles of the Bragg reflections. 

When small crystal structures are studied, all Bragg reflection data are used, and relative phase angles are derived by one of the methods described in Chapter 8, and electron-density maps are calculated to the maximum possible resolution that the wavelength of the X rays permit. On the other hand, because isomorphous replacement methods are used to obtain relative phase angles for macromolecular structures, it is usual to calculate electron-density maps at low resolution initially, and to increase the resolution as more phases from isomorphous replacement data become available. Traditionally the structure determination is divided into three resolution shells that correspond to the minima of the radial distribution of intensities. ... 

In any of the reciprocal space methods, which are based exclusively on the use of the observed structure factors, the powder diffraction pattern must be deconvoluted and the integrated intensities of all, or as many as possible, individual Bragg reflections determined with a maximum precision. Only then, Patterson or direct phase angle determination techniques may be employed to create a partial or compete structural model. Theoretical background supporting these two methods was reviewed in section 2.14. 

For the determination of the electron density map of a molecule the amplitudes and phases of the waves scattered by a single crystal are required for a number of Bragg reflections. Common X-ray techniques yield the product (Ah vh) (Ah 1waves scattered by the electrons of the molecule into the Bragg reflection H. Thus, the scattering amplitude Ah is obtained, but the phase information is lost. The solution of this phase problem for protein structure determination is based on Perutz and Kendrew s isomorphous replacement method (108-111). In this procedure Bragg reflections have to be measured at least three times, first on a crystal of native molecules, and then on two crystals, in which reference scatterers (for example Hg atoms) have been substituted at well-defined positions. From the difference of the measured intensities one can calculate the relative phases without ambiguity. 

All methods of deduction of the relative phases for Bragg reflections from a protein crystal depend, at least to some extent, on a Patterson map, commonly designated P(uvw) (46, 47). This map can be used to determine the location of heavy atoms and to compare orientations of structural domains in proteins if there are more than one per asymmetric unit. The Patterson map indicates all the possible relationships (vectors) between atoms in a crystal structure. It is a Fourier synthesis that uses the indices, l, and the square of the structure factor amplitude f(hkl) of each diffracted beam. This map exists in vector space and is described with respect to axes u, v, and w, rather than x,y,z as for electron-density maps. 

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