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 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 } be the in-plane angle of the goniometer arm (rotation about $\scriptstyle z$-axis), and $\scriptstyle \theta _{g}$ be the elevation angle (rotation away from $\scriptstyle xy$ plane and towards $\scriptstyle z$ axis).

The final scattering vector depends on:

• $\scriptstyle x$: Pixel position on detector (horizontal).
• $\scriptstyle z$: Pixel position on detector (vertical).
• $\scriptstyle d$: Sample-detector distance.
• $\scriptstyle \theta _{g}$: Elevation angle of detector.
• $\scriptstyle \phi _{g}$: 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 $\scriptstyle \theta _{g}$:

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

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

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} \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{alignat} }

The point 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,z)=(0,0) } on the detector probes the total scattering angle 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 = 2 \theta_s} , which is simply the angle between 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 \mathbf{v}_i} 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 \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} }

Arbitrary Point

For other points on the detector face, we can combine the above result with the known results for the Geometry of TSAXS. For 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 = 0} 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 = 0} , we have:

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 \mathbf{v}_1 = \begin{bmatrix} 0 \\ d \\ 0 \end{bmatrix} }

See Also

• Geometry:TSAXS 3D