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# 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, Qz.

## Application

$Q_z=\frac{4\pi}{\lambda}\sin\theta$

Where θ is the angle of incidence/reflection of the incident radiation 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 (dn), scattering length density (ρn) and roughness (σn,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:

$k_n=\sqrt{{k_0}^2-4\pi({\rho}_n-{\rho}_0)}$

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

$r_{n,n+1} = \frac{k_{n}-k_{n+1}}{k_{n}+k_{n+1}}$

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.

$r_{n,n+1} = \frac{k_{n}-k_{n+1}}{k_{n}+k_{n+1}}\exp(-2k_{n}k_{n+1}{\sigma_{n,n+1}}^2)$

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

βn = kndn

A characteristic matrix,cn is then calculated for each layer.

$c_{n}=\left[\begin{array}{cc} \exp\left(\beta_{n}\right) & r_{n}\exp\left(\beta_{n}\right)\\ r_{n}\exp\left(-\beta_{n}\right) & \exp\left(-\beta_{n}\right)\end{array}\right]$

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

$M=\prod_{0}^{n}c_{n}$

$R=\left|\frac{M_{00}}{M_{10}}\right|^{2}$

## References

OS Heavens. Optical Properties of Thin Films. Butterworth, London, 1955