Structure factors, projections, inverse Fourier transforms and crystal symmetry

The formulae for Fourier and inverse Fourier transforms have been generalized to take into account crystal symmetry using group theory. When the Fourier components (structure factors) are restricted to a given reciprocal plane, the two-dimensional inverse Fourier transform yields the projection of the charge/spin density in the unit cell in direct space, parallel to the axis normal to that plane. The formulae above are further generalized to this two-dimensional case. The latter case is central to a polarized neutron diffraction data analysis using maximum entropy.