Difference between revisions of "Refraction distortion"
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==Refraction Correction== | ==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 (<math>\scriptstyle n_a</math>), the thin film (<math>\scriptstyle n_f</math>), and the substrate (<math>\scriptstyle n_s</math>). | + | 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 (<math>\scriptstyle n_a</math>), the thin film (<math>\scriptstyle n_f</math>), and the substrate (<math>\scriptstyle n_s</math>). For an incident angle of <math>\scriptstyle \alpha_i</math>, one computes a refraction of: |
+ | :<math> | ||
+ | \alpha_{ie} = \cos^{-1} \left( \frac{n_a}{n_f}\cos(\alpha_i) \right ) | ||
+ | </math> | ||
+ | That is, the direct beam shifts by <math>\scriptstyle \alpha_i - \alpha_{ie}</math>. For a given <math>\scriptstyle q_z</math>, one can convert into scattering angle: | ||
+ | :<math> | ||
+ | 2\theta_B = 2 \sin^{-1} \left( \frac{q_z}{2 k} \right) | ||
+ | </math> | ||
+ | The scattered ray refracts as it exits from the film: | ||
+ | :<math> | ||
+ | \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} | ||
+ | </math> | ||
+ | If <math>\scriptstyle \alpha_e>0</math>, then scattering is above the [[horizon]] ([[GISAXS]]); if <math>\scriptstyle \alpha_e<0</math>, then it is sub-horizon scattering ([[GTSAXS]]). For GISAXS, the final scattering angle is: | ||
+ | :<math> | ||
+ | \Delta \alpha_f = \alpha_f | ||
+ | <math> | ||
+ | |||
* Byeongdu Lee, Insun Park, Jinhwan Yoon, Soojin Park, Jehan Kim, Kwang-Woo Kim, Taihyun Chang, and Moonhor Ree [http://pubs.acs.org/doi/abs/10.1021/ma047562d Structural Analysis of Block Copolymer Thin Films with Grazing Incidence Small-Angle X-ray Scattering] ''Macromolecules'' '''2005''', 38 (10), 4311-4323. [http://dx.doi.org/10.1021/ma047562d doi: 10.1021/ma047562d] | * Byeongdu Lee, Insun Park, Jinhwan Yoon, Soojin Park, Jehan Kim, Kwang-Woo Kim, Taihyun Chang, and Moonhor Ree [http://pubs.acs.org/doi/abs/10.1021/ma047562d Structural Analysis of Block Copolymer Thin Films with Grazing Incidence Small-Angle X-ray Scattering] ''Macromolecules'' '''2005''', 38 (10), 4311-4323. [http://dx.doi.org/10.1021/ma047562d doi: 10.1021/ma047562d] |
Revision as of 14:40, 4 November 2015
![](/images/thumb/3/35/GISAXS_refraction_distortion.png/300px-GISAXS_refraction_distortion.png)
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.
![](/images/thumb/f/f2/Lu_GTSAXS-Figure5.png/450px-Lu_GTSAXS-Figure5.png)
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:
- <math>
\Delta \alpha_f = \alpha_f <math>
- 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