The use of Sayre's equation with constraints for the direct determination of phases

Sayre's equation and density modification may be combined into a single procedure for the direct determination of phases or for the determination of electron density. The constraints on either the phases or the density are expressed in terms of a system of non-linear simultaneous equations with twice as many equations as unknowns. A method of solving these equations is described which combines the conjugate gradient technique with the use of fast Fourier transforms (FFT's). This gives the possibility of obtaining a least-squares solution of the equations for tens of thousands of unknowns using only minutes of computer time. It is trivial to add different types of electron-density constraint to the system. The use of the complete diffraction pattern in the calculations makes this method more powerful than conventional direct methods of phase determination. Examples of phase determination and refinement are given, together with an indication of the high quality of the resulting maps.