Direct methods in superspace. I. Preliminary theory and test on the determination of incommensurate modulated structures

The validity of the Sayre equation [Sayre (1952). Acta Cryst. 5, 60-65] for (3 + n)-dimensional periodic structures is examined. A practical procedure is proposed for the determination of incommensurate modulated structures; this is an extension of the direct method previously proposed for solving superstructures [Fan Hai-fu, He Lao, Qian Jin-zi & Liu Shi- xiang (1978). Acta Phys. Sin. 27, 554-558]. With the newly proposed method, the phase problem for the main as well as the satellite reflections can be solved directly without making particular assumptions about the modulation. A known incommensurate modulated structure, γ-Na₂CO₃, was used in the test. Satisfactory results were obtained.