# Refraction distortion

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 $\scriptstyle q_z$, leaving $\scriptstyle q_x$ unaffected. The amount of the shift is given by:

\begin{alignat}{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{alignat}

Where $\scriptstyle \alpha_i$ is the incident angle, and $\scriptstyle \alpha_{ct}$ 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 ($\scriptstyle n_a$), the thin film ($\scriptstyle n_f$), and the substrate ($\scriptstyle n_s$). For an incident angle of $\scriptstyle \alpha_i$, one computes a refraction of:

$\alpha_{ie} = \cos^{-1} \left( \frac{n_a}{n_f}\cos(\alpha_i) \right )$

That is, the direct beam shifts by $\scriptstyle \alpha_i - \alpha_{ie}$. For a given $\scriptstyle q_z$, one can convert into scattering angle:

$2\theta_B = 2 \sin^{-1} \left( \frac{q_z}{2 k} \right)$

The scattered ray refracts as it exits from the film:

\begin{alignat}{2} \alpha_{s} & = 2\theta_B - \alpha_{ie} \\ \alpha_{e} & = \cos^{-1} \left( \frac{n_f}{n_a}\cos(\alpha_s) \right) \end{alignat}

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

\begin{alignat}{2} \Delta \alpha_s & = \alpha_e-\alpha_s \\ 2\theta_{Bf} & = \alpha_i + \alpha_e \\ q_z\prime & = 2 k \sin( 2 \theta_B /2 ) \end{alignat}