Neutron reflection interferometry: Extraction of the phase in total reflection from stratified media

Lloyd's mirror configuration has been suggested for determination of the reflection phase, in order to obtain information about the reflecting stratification. For thermal neutrons total reflection occurs at small glancing angles. We show that the phase of the reflection amplitude in total reflection is linear in the glancing angle at small angles, and determine the coefficient of proportionality for three solvable variations of the scattering length density. The semiclassical reflection phase is shown to fail at grazing incidence, but, for smooth profiles which are thick compared to the neutron wavelength, may have a useful range of glancing angles smaller than the critical angle.