Structure determination from powder data using anomalous scattering: Difference and partial, Patterson densities, and phases of structure factors

For structure determination from X-ray powder data, anomalous (resonant) scattering can be used to obtain difference Patterson and partial Patterson densities as well as phases of structure factors. Usually tuneable synchrotron radiation is used, and two or more powder patterns near and far the absorption edge of an atom type contained in the crystal are recorded. The algebraic theory together with some novel and efficient approximations is given in detail. Also symmetry restrictions, experimental and scaling procedures, and the use of the Maximum-Entropy method (MEM) are discussed. The application to the structure of an iron garnet Fe2Ca3Ge3O12 is presented. This resulted in the first determination of two partial Patterson densities: of the iron atoms and of germanium atoms. From an MEM calculation of the electron-density using the signed structure factors F(H) together with only the absolute values \F(H)\ of the remaining reflections, all atoms could be localized with good resolution and low background, whereas the ordinary Fourier calculation failed completely to show the oxygen atoms. This result is based on only 34 unique out of 83 total. reflections in 0 less than or equal to sin theta/lambda less than or equal to 0.494 Angstrom(-1). These successful applications open up new possibilities in the structure determination of unknown compounds.