|
|
Line 21: |
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| :<math> | | :<math> |
| \begin{alignat}{2} | | \begin{alignat}{2} |
− | \mathbf{v}_1 & = R_x(\theta_g) \mathbf{v}_i \\ | + | \mathbf{v}_2 & = R_x(\theta_g) \mathbf{v}_i \\ |
| & = \begin{bmatrix} | | & = \begin{bmatrix} |
| 1 & 0 & 0 \\ | | 1 & 0 & 0 \\ |
Line 33: |
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| :<math> | | :<math> |
| \begin{alignat}{2} | | \begin{alignat}{2} |
− | \mathbf{v}_f & = R_z(\phi_g) \mathbf{v}_1 \\ | + | \mathbf{v}_f & = R_z(\phi_g) \mathbf{v}_2 \\ |
| & = \begin{bmatrix} | | & = \begin{bmatrix} |
| \cos \phi_g & -\sin \phi_g & 0 \\ | | \cos \phi_g & -\sin \phi_g & 0 \\ |
Revision as of 11:21, 13 January 2016
In wide-angle scattering (WAXS), one cannot simply assume that the detector plane is orthogonal to the incident x-ray beam. Converting from detector pixel coordinates to 3D q-vector is not always trivial, and depends on the experimental geometry.
Area Detector on Goniometer Arm
Consider a 2D (area) detector connected to a goniometer arm. The goniometer has a center of rotation at the center of the sample (i.e. the incident beam passes through this center, and scattered rays originate from this point also). Let
be the in-plane angle of the goniometer arm (rotation about
-axis), and
be the elevation angle (rotation away from
plane and towards
axis).
The final scattering vector depends on:
: Pixel position on detector (horizontal).
: Pixel position on detector (vertical).
: Sample-detector distance.
: Elevation angle of detector.
: In-plane angle of detector.
Note that
and
are defined relative to the direct-beam. That is, for
and
, the direct beam is at position
on the area detector.
Central Point
The point
can be thought of in terms of a vector that points from the source-of-scattering (center of goniometer rotation) to the detector:
![{\displaystyle \mathbf {v} _{i}={\begin{bmatrix}0\\d\\0\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/529d681d5cc76768e53f334ce54b020f91bdddfe)
This vector is then rotated about the
-axis by
:
![{\displaystyle {\begin{alignedat}{2}\mathbf {v} _{2}&=R_{x}(\theta _{g})\mathbf {v} _{i}\\&={\begin{bmatrix}1&0&0\\0&\cos \theta _{g}&-\sin \theta _{g}\\0&\sin \theta _{g}&\cos \theta _{g}\\\end{bmatrix}}{\begin{bmatrix}0\\d\\0\end{bmatrix}}\\&={\begin{bmatrix}0\\d\cos \theta _{g}\\d\sin \theta _{g}\end{bmatrix}}\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/217e0ce320de26d576860ed664a9b19f90932043)
And then rotated about the
-axis by
:
![{\displaystyle {\begin{alignedat}{2}\mathbf {v} _{f}&=R_{z}(\phi _{g})\mathbf {v} _{2}\\&={\begin{bmatrix}\cos \phi _{g}&-\sin \phi _{g}&0\\\sin \phi _{g}&\cos \phi _{g}&0\\0&0&1\\\end{bmatrix}}{\begin{bmatrix}0\\d\cos \theta _{g}\\d\sin \theta _{g}\end{bmatrix}}\\&=d{\begin{bmatrix}-\sin \phi _{g}\cos \theta _{g}\\\cos \phi _{g}\cos \theta _{g}\\\sin \theta _{g}\end{bmatrix}}\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0eb3f5e653009b182cd42739015a5d80a9e27c9)
The point
on the detector probes the total scattering angle
, which is simply the angle between
and
:
![{\displaystyle {\begin{alignedat}{2}\cos \Theta &={\frac {\mathbf {v} _{i}\cdot \mathbf {v} _{f}}{\left\|\mathbf {v} _{i}\right\|\left\|\mathbf {v} _{f}\right\|}}\\&=\cos \phi _{g}\cos \theta _{g}\\2\theta _{s}&=\arccos \left[\cos \phi _{g}\cos \theta _{g}\right]\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16f579eadf3797f5e60e452d03043fd715a20db0)
Thus:
![{\displaystyle {\begin{alignedat}{2}q&={\frac {4\pi }{\lambda }}\sin \left(\theta _{s}\right)\\&=\pm {\frac {4\pi }{\lambda }}{\sqrt {\frac {1-\cos 2\theta _{s}}{2}}}\\&={\frac {4\pi }{\lambda }}{\sqrt {{\frac {1}{2}}\left(1-\cos \phi _{g}\cos \theta _{g}\right)}}\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e24357bc94eb153969537c9f9977d5a804c845)
Arbitrary Point
TBD
See Also