Analysis of randomly oriented structures by grazing-incidence small-angle neutron scattering

A formalism is presented which allows the quantitative evaluation of data from grazing-incidence small-angle neutron and X-ray scattering - GISANS and GISAXS - in the framework of the distorted wave Born approximation. While several aspects have been reported previously, this formalism combines solutions for scattering intensities in both reflection and transmission hemispheres, taking into account instrumental resolution effects. This formalism is applied to the case of GISANS from self-organized diblock copolymers, ordered in perpendicular lamellar structures on an Si wafer in randomly oriented short-range-ordered regions. The periodicity of D = 85 (9) nm found for deuterated polystyrene-polybutadiene of molecular weight  M_w = 165 kg mol^-1 and a molecular weight fraction of the deuterated polystyrene block of 52% is consistent with atomic force microscopy and specular neutron reflectivity results.