Phase determination in x-ray and neutron reflectivity using logarithmic dispersion relations

Logarithmic dispersion relations are shown to be applicable to the determination of the phase of the structure factor of surface layers probed by neutron and x-ray reflectivity. For certain profiles it is shown that the phase {Φ}(q) of the structure factor F(q) is determined entirely by the observed reflectivity R(q) through the modified Hilbert transform {Φ}(q)=2q/{π}F₀^∞ ln[{}F(q'){}/{}F(q){}]/(q²-q'²)dq', where q is the momentum transfer, and F(q) is related to R(q) and the Fresnel reflectivity via R(q)=RF(q){}F(q)².