Abstract: A set of x-ray powder diffraction data of the high-T_c superconductor Bi₂Sr₂Ca₁Cu₂O_y (Bi-2212) was simulated based on the experimental single-crystal diffraction data by merging together reflections with diffraction angles (2$\theta$) closer to each other than 0.04 degrees. There are three types of overlapping in the powder diffraction data, i.e. (i) overlapping of main reflections; (ii) overlapping of satellite reflections and (iii) overlapping of main and satellite reflections. The third type of overlapping was first separated into main and satellite components according to the ratio between the average intensity of that of types (i) and (ii). Then the overlapped reflections of main reflections and those of the satellites were uniformly partitioned. Heavy-atom sites in the basic/average structure were found using the uniformly decomposed main reflections by the conventional direct method. Phases of the satellites were derived by the multidimensional direct method. The resultant four-dimensional Fourier maps revealed correctly the essential feature of the modulation. No assumption on either the basic structure or the modulation is needed.

Key words: powder diffraction analysis, incommensurate modulated structures, direct methods