If a neutron beam is incident on a flat smooth surface, the neutrons interact at the surface with a potential which depends on the number density of the atoms in the surface and their average coherent scattering length. The potential acts perpendicular to the surface and it changes the normal component of the wave vector *k*_{i} of the incident wave is:

where *k*_{i} is the wave number of the incident neutron and *θ* is the grazing angle of incidence. Total reflection occurs for a critical glancing angle at which *E*_{i⊥} is equal to the surface potential. For elastic scattering the momentum transfer q is normal to the surface and equal to *2k*_{i} sinθ.

When there is a thin uniform layer sandwiched between two bulk phases, the neutron beam is reflected at both the first and the second interfaces, giving rise to interference of the two beams and to the appearance of fringes in the reflectivity profile. The calculation of the reflectivity can be carried out by standard optical methods, using Fresnel coefficients for the transmitted and reflected amplitudes at each interface. The calculation of the reflectivity profile for a system with many layers can be carried out by the so-called optical matrix method.

While *specular reflection*, in which the angle of the incoming beam is equal to the angle of the reflected beam, gives information in a direction perpendicular to the interface, the lateral structure of the interface is probed by *non-specular scattering*.

#### References

B.T.M. Willis and C.J. Carlile, *Experimental Neutron Scattering*, Oxford University Press, 2009.