# Refraction distortion

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 , leaving unaffected. The amount of the shift is given by:

Where is the incident angle, and is the critical angle of the film.

## 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 (), the thin film (), and the substrate (). For an incident angle of , one computes a refraction of:

That is, the direct beam shifts by . For a given , one can convert into scattering angle:

The scattered ray refracts as it exits from the film:

If , then scattering is above the horizon (GISAXS); if , then it is sub-horizon scattering (GTSAXS). For GISAXS, the final scattering angle is:

- 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

## 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