Geometric sources of redundancy in intensity data and their use for phase determination

Linear equations are derived in direct space, which express the relation between the electron densities of crystals built from the same molecule, but with different lattices or several identical subunits in their asymmetric units. They are shown to be equivalent to the most general 'molecular-replacement' phase equations in reciprocal space. The solution of these phase equations by the method of successive projections is discussed. This algorithm, best implemented in direct space by averaging operations, is shown to be convergent for over-determined problems, and to be equivalent to a least-squares solution of the phase equations.