  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.
