Analysis of neutron and x-ray reflectivity data by constrained least-squares methods

Two methods for the determination of scattering length density profiles from specular reflectivity data are discussed. For either method kinematical or dynamical theory can be used to calculate the reflectivity. In the first method the scattering density is parametrized using cubic splines. The coefficients in the series are determined by constrained nonlinear least-squares methods, in which the smoothest solution that agrees with the data is chosen. In the second approach the profile is expressed as a series of sine and cosine terms. A smoothness constraint is used which reduces the coefficients of the higher harmonics in the Fourier series. The two methods for analyzing reflectivity data have been applied to sets of simulated data based on examples from the literature. The two methods work equally well in most situations, and they are able to recover the original profiles. In general, the method using splines as the basis functions is better suited for aperiodic than for periodic structures, whereas the sine/cosine basis is well suited for periodic and nearly periodic structures.