Any distribution is completely defined by its moments. It is shown tha
t a process of phase refinement can be carried out, based on Fourier t
ransforms, which modifies the moments of electron density, separately
in the protein and solvent regions, towards target values. Tests have
been carried out on two moderate-sized proteins with 800-900 atoms in
the asymmetric unit, one containing heavy atoms and the other not. It
has been found that refinement using the third moment about zero in th
e protein region is most effective and that refinement with higher mom
ents, or in the solvent region, adds nothing useful. Two kinds of weig
hts are necessary in the method. One is for giving a weighted mixture
of new phase indications with original phase estimates from, say, mult
iple isomorphous replacement. The other weights are applied to the Fou
rier coefficients of density maps to give the best possible signal:noi
se ratio. These weights have been explored empirically and the best on
es found are described. It is concluded that since the moments method,
which changes phases in reciprocal space, is independent of other his
togram-matching procedures, which change density in real space, it has
something to offer in a refinement package containing several procedu
res.