Phasing via SAD/MAD data: the method of the joint probability distribution functions

The method of the joint probability distribution functions is applied to derive a probabilistic formula which is able to phase reflections in the MAD case accurately, under the condition that the anomalous-scatterer substructure has been defined previously. The mathematical approach takes into account both measurement and model errors, which are treated as primitive random variables, as well as the atomic positions defining the unknown part of the crystal structure. The probabilistic formula has the classical tangent expression. All the parameters influencing the phase estimation are immediately interpretable in terms of experimental quantities: i.e. anomalous and dispersive differences, magnitude of the errors and normalized structure-factor moduli. The formula has been applied to several practical cases: a procedure has also been designed which is able to refine the phases and lead to easily interpretable electron-density maps.