Real-space crystallographic methods

Despite such restrictions, real space crystallographic methods based on genetic algorithms, Monte-Carlo methods, or simulated annealing techniques have proved to be powerful means for structure solutions from X-ray powder patterns. Provided with the unit cell, the composition and configuration of the asymmetric unit, and sufficiently texture-free diffraction data, refinable structure models can be obtained within minutes on a personal computer, even for molecules with multiple internal degrees of freedom. The resulting structure models are then refined by Rietveld techniques, which use the whole profile of the X-ray diffraction pattern for refinement . ... 

Real-space averaging A computational method for improvement of phases, when there are two or more identical chemical units in the crystallographic asymmetric unit. The electron densities of these identical units are averaged. Then a new set of phases is computed by Fourier transformation of the averaged structure, and with these a new map is synthesized with the observed F values. By iteration of this procedure, the electron density is improved. 

In reconstruction of the three-dimensional distribution of molecular densities, there are two main approaches. The first is a real space approach and the second a Fourier space approach that is analogous to the crystallographic method. In real space, the back projection technique is used to reverse the operation of obtaining a projection. A projection simply represents the total sum of all densities of the three-dimensional object in a single plane (somewhat like a medical X-ray). To restore the densities of the three-dimensional object the densities of the projections must be extended in the reverse of the projecting direction. There are several algorithms that perform this procedure. 

Diffraction methods depend on interference effects, and therefore obtaining structural information from the observed patterns also involves Fourier transformations. For crystal lattices a three-dimensional FT is used to convert between the recorded diffraction scattering pattern (in reciprocal space) and the crystallographic lattice (in real space) (Sections 3.4 and 10.2). Similarly, in gas-phase electron diffraction a one-dimensional FT converts between the (reciprocal space) diffraction data and the (real space) radial distribution curves, which are one-dimensional plots of increasing distances separating pairs of atoms in the stmcture. 

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