(3+1)-Dimensional Patterson and Fourier methods for the determination of one-dimensionally modulated structures

Patterson and Fourier methods applied to one-dimensionally modulated structures in the (3 + 1)- dimensional space R_{3 + 1} can be very helpful tools for the calculation of starting parameters of the atomic modulation functions. The characteristics of the (3 + 1)-dimensional Patterson function [(3 + 1)-PF] are discussed for some typical modulation waves from a geometrical point of view as well as with the aid of known modulated phases. The influence of series termination errors, resulting from incomplete data sets, is demonstrated. The (3 + 1)-PF, in any case, yields sufficient basic information even if first-order satellites only are accessible. Of course, it is necessary to include higher orders if one wants to learn something about the shape of the modulation wave. A comparison is made with the Patterson methods used for the solution of modulated structures until now, and it is shown that they are special cases of the (3 + 1)-PF. Some applications are given for Fourier methods in R_{3 + 1}, for example, to detect fluctuations of the phase or the amplitude of the modulation wave.