Determining skewness in atomic probability density functions for non-centrosymmetric structures

Skewness in atomic probability density functions can be represented by odd-order cumulants in the Edgeworth expansion about a Gaussian distribution, or by odd-order quasi-moments in the Gram-Charlier expansion. In the case of the Edgeworth expansion it is known that the absolute values of some odd-order cumulants cannot be determined from Bragg reflection data for non-centrosymmetric structures - because these cumulants affect only the phases of the calculated structure factors and not their magnitudes. It is shown that, in general, this problem is imposed by the form of the Edgeworth expansion and can be avoided by using the Gram-Charlier expansion instead. An example is given of the refinement of third-order quasi-moments for the non-centrosymmetric phase of PbTiO3, using neutron-diffraction data collected at the Institut Laue-Langevin, Grenoble.