The Phase Problem in Structure Analysis and its One-Dimensional Solution
Abstract
In a preceding paper, the author has developed a new method of phase determination using latent lattices. The structure is generated by a shift function which displaces the atoms from their idealized position into the correct one. As a consequence, the number of phases to be determined in reciprocal shift space is reduced to m/2 (M = number of atoms in the unit cell). The application of the new method is subject to a transformation of conventional Fourier coefficients to those of the shift function. This is greatly facilitated by some known conventional phases. In the case of little phase information, a rough proposal may suffice to start the solution of certain sets of linear equations. Each atomic distance in the unit defines a system of M - 1 independent equations. In spite of their interdependence, each system yields just one solution of the phase problem. Two different methods may be applied: (i) (M - 1)/2 exact independent pieces of information on distances may be used to evaluate the corresponding system of linear equations; or (ii) a rough model of the structure with averaged atomic distances or coordinates may serve to apply certain sets of `joker' equations, providing displacements of atoms into the correct direction. In this way, a refined model can be proposed for a repeated use of the joker equations. At the end of this process, a conventional refinement of the structure should be performed. The principles of the new method are explained with the aid of a simple one-dimensional model structure containing one kind of atom only.