One-dimensional mathematical treatment of small-angle X-ray scattering from a system of alternating lamellar phases

Abstract

A theory for the small-angle X-ray scattering (SAXS) from alternating lamellar phases is formulated on the basis of the concept that the scattering is due to positive and negative density deviations from the average density of the system. In contrast to all previous theories, the equation for the scattered intensity satisfies Babinet's reciprocity theorem for crystallinity (in the case of a semicrystalline polymer) or volume fraction (for a block copolymer), when there is no fluctuation in the thickness of the alternating lamellar phases. It is also shown that in order to obtain the correct SAXS intensity distribution for the case where the thickness of the lamellar phases shows no fluctuation, the I_C term, as defined by Hosemann and Blundell, must be included in the summation.