Rough and diffuse interfaces

Interfaces are never perfect, and the scattering length densities will never take the form of the step functions shown in Figures xx--CrossReference--reflectivityfig:singleint--xx(b), Figure xx--CrossReference--reflectivityfig:twointerfaceint--xx(b) and (c) and Figure xx--CrossReference--reflectivityfig:SL--xx. In reality, the scattering length density will vary more continuously with depth either due to some level of grading across the interface, or due to roughness at the interface. Specular reflectivity cannot distinguish between interdiffusion and roughness, and in either case the scattering length variation must be accounted for in a valid reflectivity model.

In principle, the methods in the sections The recursive method and The characteristic matrix method can explicitly be used. There is no upper or lower limit on the thicknesses, \(d_j\), of each layer required for these methods. Interfaces can be modeled by introducing many thin layers with smoothly varying scattering length densities. This procedure is impractical, however, particularly if data are to be fitted.

A simpler method of accounting for interface width was proposed by Névot and Croce and extended by Cowley and Ryan. A model for the sample is constructed, much as has been the case in the previous examples. Each interface in the sample is assumed to have a Gaussian profile in \(z\). Applying an appropriate Fourier transform over the interface modifies the Fresnel coefficients to:

\begin{equation} \begin{array}{ccc} r_{j-1,j} & = & \frac{k_{\left(j-1\right)z}-k_{jz}}{k_{\left(j-1\right)z}+k_{jz}}\exp\left(-2k_{\left(j-1\right)z}k_{jz}\left<\sigma_j\right>^2\right) \\ & \approx & \frac{k_{\left(j-1\right)z}-k_{jz}}{k_{\left(j-1\right)z}+k_{jz}}\exp\left(-0.5\left[q_z\left<\sigma_j\right>\right]^2\right), \end{array} \label{reflectivityeq:Ruff} \end{equation}

where \(\sigma_j\) defines the width of the Gaussian profile for the \(j\)th interface.

Equation \eqref{reflectivityeq:Ruff} can be substituted directly for the Fresnel coefficients in equations \eqref{reflectivityeq:Mj} and \eqref{reflectivityeq:Cj} to include roughness in the calculations.