1. Introduction

A brief summary of neutron reflectivity principles is given here.

1. Principles
2. Flat surface reflection

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 :

(3)

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 :

(5)

n is close to 1 so :

(6)

From (5) and (6), we obtain :

(7)

which is equivalent to :

(8)

where the a parameter is used to caraterized the material.

In Tableau 1 a few typical neutron coherent scattering length density are reported.

	Ni	Ti	Si	SiO2
density d (g.cm-3)	8,9	4,51	2,33	2,5
Atomic Mass M (g.mol-1)	58,71	47,9	28,09	60,09
N=dNAM-1 (cm-3)	9,13x1022	5,67x1022	5,00x1022	2,51x1022
b (cm-1)	1,03x10-12	-0,3438x10-12	0,41491x10-12	1,58x10-12
Nb (Å-2)	9,4x10-6	-1,95x10-6	2,07x10-6	3,96x10-6

1. Reflection on a stratified medium

Figure.1 : Plane wave reflection on a flat surface.

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:

(10)

where i²=-1, and A_p and B_p are respectively the incoming and outgoing amplitudes in the layer p. We have also :

(11)

We can write the continuity conditions at the interface p/p+1:

The reflectivity in zp/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: (12) where u(zp/p+1) and u'(zp/p+1) are functions of zp/p+1 and qp+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.

1. A few example of reflectivity calculation
2. Ideal interface : the Fresnel reflectivity

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 :

(13)

or :

(14)

where q_c et q_s are defined from equation (11) :

(15)

(16)

1. Homogeneous layer on an infinit substrat

We consider the system represented on Figure 3. The 3 media are : air (n=1), a homogeneous layer of indices n₁, thickness d and scattering length density Nb₁ , 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:

(17)

From this equation, the curve of R as a function of q presents oscillations with 2q₁d=m x 2p,. This is the Bragg's relation corrected from the critical angle (m is an integer). The oscillations are called Kiessig fringes.

1. Roughnesses

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 :

roughness due to abrupt composition variation, but at different z values from one point of the surface to another one.
interdiffusion of the materials between two different layers.

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 :

(18)

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 Debye-Waller 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.

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