X-ray phase determination by the principle of minimum charge

Abstract

When the charge density in a crystal or a quasicrystal is reconstructed from its Fourier modes, the global minimum value of the density is sensitively dependent on the relative phases of the modes. The set of phases that maximizes the value of the global minimum, corresponds, by positivity of the density, to the charge density having the minimum total charge that is consistent with the measured Fourier amplitudes. Phases which minimize the total charge (i.e. the average charge density) also have the property that the lowest minima of the charge density become exactly degenerate and proliferate within the unit cell. The large number of degenerate minima have the effect that density maxima are forced to occupy ever smaller regions of the unit cell. Thus by minimizing charge, the atomicity of the charge density is enhanced as well. Charge minimization applied to simulated crystalline and quasicrystalline diffraction data successfully reproduces the correct phases starting from random initial values.