Geometry:WAXS 3D

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 $\scriptstyle \phi _{g}$ be the in-plane angle of the goniometer arm (rotation about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle z } -axis), and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \theta_g } be the elevation angle (rotation away from $\scriptstyle xy$ plane and towards Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle z } axis).

The final scattering vector depends on:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle x } : Pixel position on detector (horizontal).
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle z } : Pixel position on detector (vertical).
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d } : Sample-detector distance.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \theta_g } : Elevation angle of detector.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \phi_g } : In-plane angle of detector.

Note that $\scriptstyle x$ and $\scriptstyle z$ are defined relative to the direct-beam. That is, for $\scriptstyle \theta _{g}=0$ and $\scriptstyle \phi _{g}=0$, the direct beam is at position $\scriptstyle (x,z)=(0,0)$ on the area detector.

Central Point

The point $\scriptstyle (x,z)=(0,0)$ can be thought of in terms of a vector that points from the source-of-scattering (center of goniometer rotation) to the detector:

$\mathbf {v} _{i}={\begin{bmatrix}0\\d\\0\end{bmatrix}}$

This vector is then rotated about the $\scriptstyle x$-axis by $\scriptstyle \theta _{g}$:

${\begin{alignedat}{2}\mathbf {v} _{1}&=R_{x}(\theta _{g})\mathbf {v} _{i}\\&={\begin{bmatrix}1&0&0\\0&\cos \theta _{g}&-\sin \theta _{g}\\0&\sin \theta _{g}&\cos \theta _{g}\\\end{bmatrix}}{\begin{bmatrix}0\\d\\0\end{bmatrix}}\\&={\begin{bmatrix}0\\d\cos \theta _{g}\\d\sin \theta _{g}\end{bmatrix}}\end{alignedat}}$

And then rotated about the $\scriptstyle z$-axis by $\scriptstyle \phi _{g}$:

${\begin{alignedat}{2}\mathbf {v} _{f}&=R_{z}(\phi _{g})\mathbf {v} _{1}\\&={\begin{bmatrix}\cos \phi _{g}&-\sin \phi _{g}&0\\\sin \phi _{g}&\cos \phi _{g}&0\\0&0&1\\\end{bmatrix}}{\begin{bmatrix}0\\d\cos \theta _{g}\\d\sin \theta _{g}\end{bmatrix}}\\&=d{\begin{bmatrix}-\sin \phi _{g}\cos \theta _{g}\\\cos \phi _{g}\cos \theta _{g}\\\sin \theta _{g}\end{bmatrix}}\end{alignedat}}$

The point $\scriptstyle (x,z)=(0,0)$ on the detector probes the total scattering angle $\scriptstyle \Theta =2\theta _{s}$, which is simply the angle between $\scriptstyle \mathbf {v} _{i}$ and $\scriptstyle \mathbf {v} _{f}$:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{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{alignat} }

Thus:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{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{alignat} }

See Also

• Geometry:TSAXS 3D