Application of the maximum-entropy method to powder-diffraction data

This chapter provides a comprehensive overview of the maximum-entropy method (MEM) as it is applied to X-ray powder-diffraction data for computation of an unbiased electron-density map. The MEM requires a strictly positive electron-density map that is described by its values on a fine grid over the unit cell (with grid sizes of approximately 0.04 A). The entropy is defined for such a gridded density and the role of the prior or reference density is discussed. An in-depth presentation is given of the crystallographic MEM equations and the various iterative algorithms for solving these equations. All these considerations apply equally well to single-crystal and X-ray powder-diffraction data. The experimental data are incorporated into the MEM through constraints involving the structure factors. Specific to powder diffraction are the various methods for extracting estimates of the structure-factor amplitudes or group amplitudes, the use of G constraints in addition to F constraints on single-crystal diffraction data and the various methods of estimating the phases of the structure factors. This then leads to the definition of various types of MEM maps that range from maps completely biased by a structure model to ab initio electron-density maps. Irrespective of the type of MEM, series-termination effects are much less prominent in MEM-optimized electron-density maps than in Fourier maps obtained with the same data. Applications of the MEM are discussed concerning its use for improving structure models (e.g. the MEM + Rietveld method), its use for the characterization of disorder and anharmonic motion within crystal structures, its use as part of a protocol for structure solution, its use as an alternative to multipole refinements, and its application to electron densities in superspace for aperiodic crystals.