Solution to the phase problem for specular x-ray or neutron reflectivity from thin films on liquid surfaces

The phase problem for specular x-ray and neutron reflectivity from liquid surfaces and thin films on liquid surfaces can be solved in the distorted-wave Born approximation. The gradient of the scattering-length density (SLD) profile normal to the plane of the surface is bounded in these cases. This provides a powerful constraint allowing the phase problem to be solved with no a priori assumptions via an iterative Fourier refinement procedure applied to the Fresnel-normalized reflectivity. The critical boundary condition can be determined experimentally from the autocorrelation of the gradient profile obtained via an inverse Fourier transform of the Fresnel-normalized reflectivity without phase information. The phase solution and the resulting gradient SLD profile can be shown to be unique, and therefore unambiguously determined, when all of phase space is systematically explored for particular cases, especially for thin films on liquid surfaces. This gradient SLD profile can then be integrated either numerically, or better, analytically to provide the scattering-length density profile itself on an absolute scale.