New phasing method based on the principle of minimum charge

Abstract

A new method of the phase determination in X-ray crystallography is proposed. The method is based on the so-called "minimum charge" principle, recently suggested by Elser. The electron density function $\rho$ is sought in the form $\rho({\bf x})=|\psi({\bf x})|^2$, where $\psi$ is an $n{-component}$ real function. The norm $\int|\psi({\bf x})|^2 d{\bf x}$ is minimized under the constraint imposed by the measured data on the amplitudes of Fourier harmonics of $\rho$. Compared to 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 is successfully tested on synthetic data for Fibonacci chain and the octagonal tiling. In the latter case the reconstructed density map shows the shape of the atomic surface, despite relatively low data resolution.