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}