A Method for Obtaining a Possibly Unique (Unambiguous) Crystal Structure Solution using Multiple Resonant Scattering Data

A sequence of steps for determining a crystal structure, possibly without ambiguities, is presented. The prerequisites are: centrosymmetry (at present) and two different anomalous scatterers, a₁, a₂. Their partial structure amplitudes |F_a1(hkl)| and |F_a2(hkl)| are separated by multiple-wavelength measurements (MAD). The core part of the method is a recursive algebraic technique applied to the geometrical part of these structure amplitudes from central reciprocal-lattice rows. At least m + 1 reflections are necessary at each row if 2m atoms of e.g. a₁ are in a unit cell with space group P. For each partial structure of a₁ and a₂ atoms, respectively, the algebra finds all homometric and pseudohomometric solutions and presents the corresponding signs for each F_a(hkl) used. Regions of confidence for atomic coordinates are given. Five reciprocal-lattice rows (or more) suffice for a `tomographic' location of all atoms a₁ and a₂ in three dimensions. The two independently determined partial structures for a₁ and a₂ are then aligned to the same origin and moduli plus signs of the remaining partial structure factors of the non-resonant atoms are determined. Various aspects of the method are discussed by application to the Cu₃SbSe₃ structure, an example exhibiting partial pseudosymmetry.