Nonlinear neutron and x-ray reflectance for surface films

This paper presents an analysis of the relationship between Parratt's recurrence formula and several existing analytical formulations for the neutron and x-ray reflectance of a surface film. In particular, it was shown that the Born approximation and the weighted-superposition approximation (WSA) for the neutron and x-ray reflectance can both be derived directly from Parratt's recurrence formula. It was also shown that the well-known nonlinear differential equation for the reflectance at an arbitrary point inside a film can be obtained from Parratt's recurrence formula and the WSA was an approximate solution to this equation. Finally, the nonlinearity in the WSA was discussed through a comparison between the Born approximation and the WSA. It was pointed out that the exponential factor in the WSA correctly accounted for both the phase shift effect due to the scattering length density of the film and the amplitude attenuation effect due to evanescence