Limits of phase expansion in direct methods

Phase expansion starting from a few initial phases is investigated with reference to the size of the starting set, phase errors in the starting set, the lower limit of the E value in expansion and the different phase-determining formulae. The results stress the need for a sufficient size of the initial phase set with small phase errors for subsequent application of a phase-expanding procedure. The common basis of phase-expansion procedures is shown to consist of a cyclic modification of the preliminary structure and explains the impossibility of correcting substantial errors in already known phases associated with larger E values by subsequent phase determination for smaller E values. The phenomenon of losing structure information by careless application of the tangent formula and consequently the appearance of partial structures is pointed out. This information-destroying phase expansion is shown not to exist in the `phase-correction' procedure.