A brief summary of neutron reflectivity principles is given here.
We assume a neutron beam reflecting on a flat surface with an incident angle theta. This surface (see Figure.1) is defined by the interface between the air (n=1) and a medium of refractive indices n. For a wavelength lambda, the wavevector is defined by :
(1)
The wavevector in the medium of indices n is defined by :
(2)
N is the number of atoms per volume unit and b is the neutron coherent scattering length. The product Nb is called the neutron coherent scattering length density. For a homogeneous medium, the refractive indices n is defined by the ratio of the wavevector in the material to the wavevector in the vacuum :
At the interface between the air and a medium of indices n, Descartes'law can be written :
(4)
Total reflection occurs if theta lower than thetac where thetac is defined so as thetan=0, which means :
n is close to 1 so :
From (5) and (6), we obtain :
(7)
which is equivalent to :
where the a parameter is used
to caraterized the material.
In Tableau 1 a few typical neutron coherent scattering length
density are reported.
The projection of the wavevector on the z axes (perpendicular to the surface) is defined as the q variable :
(9)
If the material is made of many layers, each one having an indices
n_{p}., the propagation of a plane wave in the layers
p and p+1 can be written the following way:
where i^{2}=1, and A_{p} and B_{p}
are respectively the incoming and outgoing amplitudes in the layer
p. We have also :
We can write the continuity conditions at the interface p/p+1:
The reflectivity in z_{p/p+1} is defined as being the ration of the intensity of the reflected beam by layer p+1 to the intensity arriving in layer p and is written:
where u(z_{p/p+1}) and u'(z_{p/p+1}) are functions of z_{p/p+1} and q_{p+1}. 
With this equation, you can calculate the reflectivity at the
last interface (last layer/bulk), and then recursively calculate
the reflectivity at each interfaces, and so obtain the reflectivity
of the air/first layer interface.
Medium p and p+1 are replaced by air and a subtrat of indices n. If there is no interfacial roughnesses, reflectivity of such a system is called Fresnel's reflectivity (R_{F}). In the bulk, B_{p+1}=B_{s}=0 in equation (10) . No intensity comes from z=infini.
Fresnel's reflectivity can be written :
or :
where q_{c} et q_{s} are defined from equation (11) :
(15)
(16)
We consider the system represented on Figure 3. The 3 media are : air (n=1), a homogeneous layer of indices n_{1}, thickness d and scattering length density Nb_{1} , and the substrat of indices n_{s}, scattering length density Nb_{s} and infinit thickness. The interface air/layer is at z=0, the interface layer/substrat is at z=d.
After a few calculation, we obtain:
From this equation, the curve of R as a function of q presents oscillations with 2q_{1}d=m x 2p,. This is the Bragg's relation corrected from the critical angle (m is an integer). The oscillations are called Kiessig fringes.
A system can be describe by a scattering length density profile
as a function of the distance to the surface z : Nb(z). Up to
now we have assumed that the interface between the layer p and
the layer p+1 was perfect, and presented an abrupt change from
(Nb)_{p} to (Nb)_{p+1} (step function).
In the reality, interfaces are not abrupt because of two main
reasons :




Specular reflectivity cannot distinguish between roughness and interdiffusion if the size of the imperfection of the surface is smaller than the coherence length of the neutron wave (which is of the order of the micron in the reflectivity conditions). Both interfaces are represented by the following error function :
where p/p+1 is the interface between two successive layers.
The curve representing this error function has an inflexion point
in z_{p/p+1}. s_{p/p+1}
is given by the inverse of the slope of the tangent to the curve
in z_{p/p+1}. The thickness of the interface is
given by 2s_{p/p+1}.
It can be shown that introducing equation (18) in the reflectivity calculation lead to multiply the calculated reflectivity R for a perfect interface between p and p+1 by a DebyeWaller factor of the form :
(19)
with q_{p} and q_{p+1} defined by
equation (11).
Another method is also used. In that case, the roughness profile is discretized in many perfect thin layers with a small Nb variation between each layer. In that case however, the needed computation time is much more important.
Références