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 q_z, 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:

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

Where $\scriptstyle \alpha _{i}$ 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 q_z, 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:

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

The scattered ray refracts as it exits from the film:

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

If $\scriptstyle \alpha _{e}>0$, 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:

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

• Byeongdu Lee, Insun Park, Jinhwan Yoon, Soojin Park, Jehan Kim, Kwang-Woo Kim, Taihyun Chang, and Moonhor Ree Structural Analysis of Block Copolymer Thin Films with Grazing Incidence Small-Angle X-ray Scattering Macromolecules 2005, 38 (10), 4311-4323. doi: 10.1021/ma047562d
• P. Busch, M. Rauscher, D.-M. Smilgies, D. Posselt and C. M. Papadakis Grazing-incidence small-angle X-ray scattering from thin polymer films with lamellar structures - the scattering cross section in the distorted-wave Born approximation J. Appl. Cryst. 2006, 39, 433-442. doi: 10.1107/S0021889806012337
• Rémi Lazzari, Frédéric Leroy, and Gilles Renaud Grazing-incidence small-angle x-ray scattering from dense packing of islands on surfaces: Development of distorted wave Born approximation and correlation between particle sizes and spacing Phys. Rev. B 2007, 76, 125411. doi: 10.1103/PhysRevB.76.125411
• D. W. Breiby, O. Bunk, J. W. Andreasen, H. T. Lemke and M. M. Nielsen Simulating X-ray diffraction of textured films J. Appl. Cryst. 2008, 41, 262-271. doi: 10.1107/S0021889808001064
• Lu, X.; Yager, K.G.; Johnston, D.; Black, C.T.; Ocko, B.M. Grazing-incidence transmission X-ray scattering: surface scattering in the Born approximation Journal of Applied Crystallography 2013, 46, 165–172. doi: 10.1107/S0021889812047887
• Liu, J.; Yager, K.G. Unwarping GISAXS data IUCrJ 2018, 5, 737–752. doi: 10.1107/S2052252518012058

See Also

• DWBA
• Simulating X-ray diffraction of textured films D. W. Breiby, O. Bunk, J. W. Andreasen, H. T. Lemke and M. M. Nielsen J. Appl. Cryst. 2008, 41, 262-271. doi: 10.1107/S0021889808001064
• Indexation scheme for oriented molecular thin films studied with grazing-incidence reciprocal-space mapping D.-M. Smilgies and D. R. Blasini J. Appl. Cryst. 2007, 40, 716-718. doi: 10.1107/S0021889807023382
• Grazing-incidence small-angle X-ray scattering from thin polymer films with lamellar structures - the scattering cross section in the distorted-wave Born approximation P. Busch, M. Rauscher, D.-M. Smilgies, D. Posselt and C. M. Papadakis J. Appl. Cryst. 2006, 39, 433-442. doi: 10.1107/S0021889806012337