A new phasing method based on the principle of minimum charge

A new method for phase determination in X-ray crystallography is proposed. The method is based on the so-called 'minimum-charge' principle, recently suggested by Elser [Acta Cryst. (1999), A55, 489-499]. The electron-density function rho is sought in the form rho(x) = /psi(x)/2, where psi is an n-component real function. The norm integral/psi(x)/2dx is minimized under the constraint imposed by the measured data on the amplitudes of Fourier harmonics of rho. Compared with the straightforward implementation of the 'minimum-charge' scheme, the method attenuates the Gibbs phenomenon and is also capable of extrapolation of the diffraction data beyond the set of measured amplitudes. The method is applicable to quasicrystals under the condition that the number of components n of the function psi is bigger than the dimensionality of the 'atomic surface'. It has been successfully tested on synthetic data for a Fibonacci chain and octagonal tiling. In the latter case, the reconstructed density map shows the shape of the atomic surface, despite relatively low resolution data.