The small-angle scattering of distorted lamellar structures

The effects of deviations from an ideal lamellar structure (infinite-size clusters of parallel layers of alternating electron densities) on the small-angle scattering curve are treated with the aid of the correlation function. If surrounded by a matrix of the average electron density, reduction of the size of the clusters in the direction of the layer normals leads to a simple modification of the one-dimensional correlation function. Distortions giving rise to structures containing concentric layers have little effect on this function, whereas corrugation of the surfaces causes minor modifications. Second-order defects are shown to reduce the three-dimensional correlation function of the ideal structure γ°(r) according to γ(r) = γ°(r) exp (−2r/d), where d is the `distortion length'. This is the average length of the vectors for which the number of intersections with lamellar interfaces has changed by ±1 as a consequence of the distortions. Calculated diffraction curves show that the effects of reducing the cluster size and of increasing the width β of the lamellar thickness distribution function are very similar. However, changes in d and β affect the scattering curves in a different way, which, other conditions being favourable, may enable these parameters to be determined from observed scattering curves.