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 $\scriptstyle z$-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:

• 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.
• $\scriptstyle \theta _{g}$: Elevation angle of detector.
• $\scriptstyle \phi _{g}$: In-plane angle of detector.

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

Central Point

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) } can be thought of in terms of a vector that points from the source-of-scattering (center of goniometer rotation) to the 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 \mathbf{v}_i = \begin{bmatrix} 0 \\ d \\ 0 \end{bmatrix} }

This vector is then rotated about the 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} -axis by 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} :

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}_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{alignat} }

And then rotated about the 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 by 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} :

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}_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{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} }

See Also

• Geometry:TSAXS 3D