Difference between revisions of "Refraction distortion"
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+ | [[Image:GISAXS refraction distortion.png|right|thumb|300px|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<sub>z</sub>'', 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 [[scattering|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 [[Tutorial:What_to_do_with_data|data interpretation]] more problematic. | In [[GISAXS]], [[GIWAXS]], and other grazing-incidence techniques, the [[refractive index]] difference between the film and the ambient causes the incident and [[scattering|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 [[Tutorial:What_to_do_with_data|data interpretation]] more problematic. | ||
+ | |||
+ | ==Mathematics== | ||
+ | The GISAXS refraction distortion shifts the data along <math>\scriptstyle q_z</math>, leaving <math>\scriptstyle q_x</math> unaffected. The amount of the shift is given by: | ||
+ | :<math> | ||
+ | \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} | ||
+ | </math> | ||
+ | Where <math>\scriptstyle \alpha_i</math> is the incident angle, and <math>\scriptstyle \alpha_{ct}</math> is the [[critical angle]] of the film. | ||
+ | |||
+ | [[Image:Lu GTSAXS-Figure5.png|center|thumb|450px|Figure from Lu. et al. ([http://dx.doi.org/10.1107/S0021889812047887 doi: 10.1107/S0021889812047887 ''J. of Appl. Cryst.'' '''2013''', 46, 165]) showing the amount of distortion along ''q<sub>z</sub>'', 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 (<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> | ||
+ | \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} | ||
+ | </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] | ||
+ | * P. Busch, M. Rauscher, D.-M. Smilgies, D. Posselt and C. M. Papadakis [http://scripts.iucr.org/cgi-bin/paper?S0021889806012337 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. [http://dx.doi.org/10.1107/S0021889806012337 doi: 10.1107/S0021889806012337] | ||
+ | * Rémi Lazzari, Frédéric Leroy, and Gilles Renaud [http://journals.aps.org/prb/abstract/10.1103/PhysRevB.76.125411 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. [http://dx.doi.org/10.1103/PhysRevB.76.125411 doi: 10.1103/PhysRevB.76.125411] | ||
+ | * D. W. Breiby, O. Bunk, J. W. Andreasen, H. T. Lemke and M. M. Nielsen [http://scripts.iucr.org/cgi-bin/paper?S0021889808001064 Simulating X-ray diffraction of textured films] ''J. Appl. Cryst.'' 2008, 41, 262-271. [http://dx.doi.org/10.1107/S0021889808001064 doi: 10.1107/S0021889808001064] | ||
+ | * Lu, X.; [[Yager, K.G.]]; Johnston, D.; Black, C.T.; Ocko, B.M. '''[http://scripts.iucr.org/cgi-bin/paper?S0021889812047887 Grazing-incidence transmission X-ray scattering: surface scattering in the Born approximation]''' ''Journal of Applied Crystallography'' '''2013''', 46, 165–172. [http://dx.doi.org/10.1107/S0021889812047887 doi: 10.1107/S0021889812047887] | ||
+ | * Liu, J.; Yager, K.G. [http://journals.iucr.org/m/issues/2018/06/00/hf5364/index.html Unwarping GISAXS data] ''IUCrJ'' '''2018''', 5, 737–752. [http://dx.doi.org/10.1107/S2052252518012058 doi: 10.1107/S2052252518012058] | ||
==See Also== | ==See Also== | ||
* [[DWBA]] | * [[DWBA]] | ||
− | * | + | * [http://scripts.iucr.org/cgi-bin/paper?S0021889808001064 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. [http://dx.doi.org/10.1107/S0021889808001064 doi: 10.1107/S0021889808001064] |
+ | * [http://journals.iucr.org/j/issues/2007/04/00/issconts.html 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. [http://dx.doi.org/10.1107/S0021889807023382 doi: 10.1107/S0021889807023382] | ||
+ | * [http://scripts.iucr.org/cgi-bin/paper?S0021889806012337 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. [http://dx.doi.org/10.1107/S0021889806012337 doi: 10.1107/S0021889806012337] |
Latest revision as of 08:49, 16 April 2019
![](/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:
- 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