A solution of the phase problem. I. Latent lattices and their optimisation

Abstract

In two preceding papers by Jagodzinski (Phys. Stat. Sol.(a) 146 (1994) 477; Acta Crystallogr. A52 (1996) 675), a new method of phase determination has been described. Using latent lattices of point scatterers for a single kind of atoms, the structure is generated by shift vectors, displacing the atoms from the origins of the latent lattice into their correct positions. The mathematical treatment of this procedure resulted in very complex formulae for structure factors E′(h), normalised such that E′(0) = 1. The three components of the displacement vectors define a new real and reciprocal shift space. The method is briefly described in the first three sections of this paper. There is an important difference between one and higher dimensions: the number of latent lattices with different shift functions is infinite, but only very few of them are optimal. Phases and amplitudes of the Fourier coefficients of the shift function are strongly correlated. This property results from the transformation of diffraction data and the prohibition of trespassing of points, caused by large Fourier coefficients of the shift function. In addition to the well known influence of strong reflections, the approximate [unk]-symmetries of the diffraction data and the areas of minimum intensity play an outstanding role for the selection of the optimal latent lattice. Its corresponding shift function has small displacements and small amplitudes of the Fourier coefficients. In the case of complicated structures, these properties demand a careful computer evaluation of the intensity distribution in reciprocal space, in order to determine the optimal latent lattice of a given structure. – The unit cell of the structure contains M (= number of atoms) cells of the latent lattice. As long as M is a product of three integers M_xM_yM_z, there is no difficulty in finding the optimal latent lattice in diffraction. Each set of directions has to be tested in reciprocal space. In all other cases difficulties arise which may only be overcome with the aid of latent sublattices (smaller cell, more atoms in the structure), or latent superlattices (larger cell, less atoms in the structure). Both cases are discussed, and it is shown that the existence of the various types of latent lattices may be found in the normal diffraction pattern, applying the criteria derived in this paper. The meaning of latent sub- and superlattices is discussed from the structural point of view, and phase relations, caused by the prohibition of trespassing of points, are derived. The great power of the new method will be discussed in a later paper, where differences of shift functions are introduced. Their important consequences for phase determination will be demonstrated with the aid of certain systems of linear equations.