# Abeles

### From Motofit

## Abeles Matrix Formalism

The Abeles matrix method is a computationally fast and easy way to calculate the specular reflectivity from a stratified interface, as a function of the perpendicular momentum transfer, *Q _{z}*.

Where *θ* is the angle of incidence/reflection and *λ* is the wavelength of the radiation.
The measured reflectivity depends on the variation in the scattering length density (SLD)
profile, (*ρ*(*z*)) perpendicular to the interface. Although the scattering length density profile
is normally a continuously varying function, the interfacial structure can often be well ap-
proximated by a slab model in which layers of thickness (*d _{n}*), scattering length density (

*ρ*) and roughness (σ

_{n}_{n,n+1})are sandwiched between the super- and sub-phases. One then uses a refinement procedure to minimise the differences between the theoretical and measured reflectivity curves, by changing the parameters that describe each layer. In this description the interface is split into

*n*layers. Since the incident neutron beam is refracted by each of the layers the wavevector,

*k*, in layer

*n*, is given by:

The Fresnel reflection coefficient between layer *n* and *n+1* is then given by:

Since the interface between each layer is unlikely to be perfectly smooth the roughness/diffuseness of each interface modifies the Fresnel coefficient and is accounted for by an error function, as described by Nevot and Croce.

A phase factor, β is introduced, which accounts for the thickness of each layer.

A characteristic matrix,c_{n} is then calculated for each layer.

The resultant matrix is defined as the product of these characteristic matrices, from which the reflectivity is calculated.