# Refraction distortion

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Illustration of GISAXS refraction distortion. The reciprocal-space scattering is a hexagonal array of peaks. However, these peaks are both shifted, and compressed/stretched along qz, due to refraction. This effect is especially pronounced near the Yoneda (orange line).

In GISAXS, GIWAXS, and other grazing-incidence techniques, the refractive index difference between the film and the ambient causes the incident and scattered x-ray beams to be refracted. This extent of refraction depends on the incident and exit angles. Thus, the data that appears on an area detector in a grazing-incidence experiment is non-linearly distorted. This makes data interpretation more problematic.

## Mathematics

The GISAXS refraction distortion shifts the data along ${\displaystyle \scriptstyle q_{z}}$, leaving ${\displaystyle \scriptstyle q_{x}}$ unaffected. The amount of the shift is given by:

{\displaystyle {\begin{alignedat}{2}\Delta q_{z}&=q_{z}-q_{z}^{\prime }\\q'_{z}&=k_{0}\left({\sqrt {\sin \alpha _{i}^{2}-\sin \alpha _{ct}^{2}}}+{\sqrt {\sin \alpha _{f}^{2}-\sin \alpha _{ct}^{2}}}\right)\\&=k_{0}\left({\sqrt {\sin \alpha _{i}^{2}-\sin \alpha _{ct}^{2}}}+{\sqrt {\left({\frac {q_{z}}{k_{0}}}-\sin \alpha _{i}\right)^{2}-\sin \alpha _{ct}^{2}}}\right)\end{alignedat}}}

Where ${\displaystyle \scriptstyle \alpha _{i}}$ is the incident angle, and ${\displaystyle \scriptstyle \alpha _{ct}}$ is the critical angle of the film.

Figure from Lu. et al. (doi: 10.1107/S0021889812047887 J. of Appl. Cryst. 2013, 46, 165) showing the amount of distortion along qz, for different conditions.

## Refraction Correction

When computing theoretical scattering patterns, one must account for the refraction correction. The correction is essentially an application of Snell's law, where one using the x-ray refractive index for ambient (${\displaystyle \scriptstyle n_{a}}$), the thin film (${\displaystyle \scriptstyle n_{f}}$), and the substrate (${\displaystyle \scriptstyle n_{s}}$). For an incident angle of ${\displaystyle \scriptstyle \alpha _{i}}$, one computes a refraction of:

${\displaystyle \alpha _{ie}=\cos ^{-1}\left({\frac {n_{a}}{n_{f}}}\cos(\alpha _{i})\right)}$

That is, the direct beam shifts by ${\displaystyle \scriptstyle \alpha _{i}-\alpha _{ie}}$. For a given ${\displaystyle \scriptstyle q_{z}}$, one can convert into scattering angle:

${\displaystyle 2\theta _{B}=2\sin ^{-1}\left({\frac {q_{z}}{2k}}\right)}$

The scattered ray refracts as it exits from the film:

{\displaystyle {\begin{alignedat}{2}\alpha _{s}&=2\theta _{B}-\alpha _{ie}\\\alpha _{e}&=\cos ^{-1}\left({\frac {n_{f}}{n_{a}}}\cos(\alpha _{s})\right)\end{alignedat}}}

If ${\displaystyle \scriptstyle \alpha _{e}>0}$, then scattering is above the horizon (GISAXS); if ${\displaystyle \scriptstyle \alpha _{e}<0}$, then it is sub-horizon scattering (GTSAXS). For GISAXS, the final scattering angle is:

{\displaystyle {\begin{alignedat}{2}\Delta \alpha _{s}&=\alpha _{e}-\alpha _{s}\\2\theta _{Bf}&=\alpha _{i}+\alpha _{e}\\q_{z}\prime &=2k\sin(2\theta _{B}/2)\end{alignedat}}}