Reflection and transmission of x rays by graded interfaces

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Abstract
The reflection and transmission of x rays (and neutrons) by graded interfaces and surfaces of arbitrary profile are calculated by generalizing the approximation introduced by Nevot and Croce. The results apply also to the specular scattering by Gaussian and non-Gaussian random rough surfaces in the limit of short lateral correlation length. Two alternative approximation schemes are given: one involves the self-consistent calculation of the reflected field while the other imposes self-consistency for the transmitted field. The two alternatives give identical results for the reflection coefficient but not for the transmission coefficient. A third and better approximation is given by the geometric mean of the transmission coefficients obtained by the two alternatives above. Comparison with exact solutions for the special Epstein profile helps ascertain the accuracy of the approximations. The well-known peak of the transmitted amplitude for incidence at the critical angle is appreciably enhanced by grading the interface, and/or by roughness.