Reflection of waves from interfaces

1 Reflection of waves from interfaces 3.1.1 Neutron and x-ray reflectivity 

Much of the simple theory of reflection is presented in optics textbooks a classic and clear presentation is to be found in Bom and Wolf (1975). The general theory of the reflection of waves, both electromagnetic and particle, is dealt with in more detail in the book by Lekner (1987). Reviews of x-ray and neutron reflectivity, including their application to polymers, are available (Russell 1990, Thomas 1995). In this section we concentrate on neutron reflectivity, primarily because of the advantages offered by deuterium labelling, though as x-ray reflectivity can also provide useful information, contrasts between reflection of neutrons and x-rays will be pointed out where necessary. 

We consider a plane wave travelling in medium 0 incident on the smooth surface of medium 1 (figure 3.1) the associated wave-vectors in each medium are and and the refractive index at the boundary is given by = ki/ko. This refractive index can commonly be written as 

Here Vp is the electron radius (2.82 X 10 cm) and is the sum of the electrons in the species. The absorption coefficient, C, for x-rays is p/ A7t), where p is the mass absorption coefficient for the unit (Creagh and Hubbell 1992). NA is Avogadro s number in these formulae. 

At the interface between the two media the grazing angle of incidence, da, is related to the angle of refraction, d, by 

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