The phase problem in the analysis of X-ray diffraction data in terms of electron-density distributions

The refinement of electron-density distributions for noncentrosymmetric crystals from X-ray diffraction data may lead to a very good fit between model and data but to totally meaningless electron densities. This is to a large extent because varying certain parameters, or combination of parameters, in the model mainly leads to a change in the phases of the structure factors. A formal analysis of why this happens, when using multipole models, is given as well as specific examples using real data: the contributions of odd-order multipoles, which are invariant under crystal-class symmetry operations, are poorly determined. The importance of applying constraints on the models is stressed. The conclusions of this analysis can be carried over to refinements of anharmonic atomic vibrations.