Improved Calculations for X-Ray Reflectivity on Surfaces and Interfaces

The calculated result for X-ray reflectivity sometimes showed a strange phenomenon in that the amplitude of the oscillation due to interference effects increases in the case of a specific roughness of the surface. In our previous study, we proposed that this strange result originated in a neglect of the effect of a decrease in the intensity of refracted X-rays due to diffuse scattering at a rough surface and interface, and we showed an analysis of the X-ray reflectivity taking into account roughness-induced diffuse scattering. In the present study, we propose more clearly that the strange result has its origin in a currently used equation due to a serious mistake in which the Fresnel transmission coefficient in the reflectivity equation is increased at a rough interface, and the increase in the transmission coefficient completely overpowers any decrease in the value of the reflection coefficient because of a lack of consideration of diffuse scattering. We now present a new accurate formalism that corrects this mistake by using a Fresnel transmission coefficient taking into account diffuse scattering at a rough surface and interface. The calculated reflectivity by this new accurate reflectivity equation should enable the structure of buried interfaces to be analyzed more.