Exact determination of the neutron reflection amplitude or phase

We show that by using a known reference layer having three tunable values of scattering length density, an exact determination of the complex neutron specular reflection amplitude can be made for any unknown real potential (i.e., in the case where there is no effective absorption). Furthermore, with two such reference layers, one preceeding and the other following the unknown potential of interest, the individual elements of the transfer matrix can be obtained to within a single algebraic sign. These straightforward yet remarkable general results are valid even in the dynamical regime where the conventional Born approximation fails. Thus, it becomes feasible to consider direct solutions of the inverse scattering problem which yield the unknown scattering length density profile normal to the reflecting surface directly and unambiguously from the reflection amplitude.