Phase retrieval from the power spectrum of a periodic object

The phase-retrieval problem in crystallography is analyzed for the purpose of studying the application of phase-retrieval algorithms to crystallographic data of power spectra to obtain directly the electronic density distribution in the crystal. The theoretical importance of considering that the periodic object is formed by a finite or by an infinite number of unit cells is shown. A phase-retrieval method that is frequently used in optics, the iterative Fourier transform (IFT) algorithm, has been adapted for this application. To test this method I have performed computer simulations of power spectra that correspond to infinite periodic objects and have obtained satisfactory reconstructions in some interesting cases. From these results I have established some approximate conclusions about the probability of having a unique solution or multiple solutions. The importance of the existence of noncrystallographic symmetry in the unit cell for the resolution of this phase-retrieval problem has been proved, as has the ability of the IFT to incorporate and use this additional information successfully.