Analytical treatment of the resolution function for small-angle scattering

Analytical expression for the resolution function for small-angle scattering in pinhole geometry are derived. The contributions to the resolution function due to wavelength spread, finite collimation and detector resolution are determined separately using Gaussian functions to approximate the contributions. A general resolution function is derived which is the result of the combined effect of the three contributions. An azimuthal-integrated resolution function, which can be applied to scattering from a material with a circular symmetric scattering cross section, is calculated. This resolution function contains in addition a contribution from the averaging procedure itself. The analytical results are compared with the results of computer simulations. The comparison shows that Gaussian functions give a good description of the resolution function and that the widths agree with those calculated by the analytical expressions. The resolution function is applied in the analysis of two experimental examples: neutron scattering from latex particles [Wignall, Christen & Ramakrishnan (1988). J. Appl. Cryst. 21, 438-451] and neutron scattering from lamellar structures of bilayer lipid membranes (Mortensen, Pfeiffer, Sackmann & Knoll, unpublished). The analytical expressions for the resolution function allow a least-squares analysis to be performed and excellent agreement between experimental and theoretical scattering patterns are obtained.