The small-angle approximation of X-ray and neutron scatter from rigid rods of non-uniform cross section of finite length

The theory of small-angle neutron or X-ray scattering from a solution consisting of rods that are uncorrelated in position and orientation and its use in determining the mass per unit length and cross-sectional radius of gyration is extended to rods of finite length and non-uniform cross-sectional structure. The case of a rod made up of identical motifs spaced in a regularly repeating axial structure is considered first. Analysis of the small-angle scatter by a modified Guinier plot gives the mean mass per unit length of the motifs. The apparent squared cross-sectional radius of gyration is the weight average. These results are then generalized to the case where variable randomly distributed structural elements are present including variable spacing between the motifs making up the rod. In this way expressions are obtained that describe the scatter from rods with structural contributions from random thermal fluctuations, bound ligands and intrinsic structural heterogeneity. It is shown that, in general, systematic errors are introduced in the analysis of rods of finite length having variable structural components. However, if all the motifs have the same mass, and if the variance in their spacing is small, such errors are not important provided that the rods are of sufficient length.