Derivation of a new tangent formula from Patterson-function arguments

Most Patterson-search methods are based on functions measuring the coincidence of the observed Patterson function with the model Patterson function calculated as a function of some variables, e.g. the rotation angles in the case of the rotation function. In parallel with such methods, a new phase-refinement function based on the maximization of ZR(Φ) = ΣH(EH − 〈EH〉) GH(Φ) as a function of the collectivity Φ of phases of the reflections with large E values is presented, where EH − 〈EH〉 and GH(Φ) are the Fourier coefficients of two Patterson-type functions, the first with the origin peak removed and the second taking into account the atomicity condition. It will be shown how ZR(Φ) can be maximized by a new tangent formula actively using the small E values and not possessing any weighting factor. The utility of this formula is illustrated on the basis of two representative test examples.