Latest revision as of 08:06, 10 August 2017

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 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 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 xy } plane and towards $\scriptstyle z$ axis).

The final scattering vector depends on:

$x$ : Pixel position on detector (horizontal).
$z$ : 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 d } : Sample-detector distance.
$\theta _{g}$ : 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 \phi_g } : In-plane angle of detector.

Note that 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 } 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 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}_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{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}_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} }

Total scattering

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) } \\ & = \sqrt{2} k \sqrt{ 1 - \cos \phi_g \cos \theta_g } \end{alignat} }

Components

The momentum transfer vector is (for elastic scattering):

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{q} & = \mathbf{k}_f - \mathbf{k}_i \\ & = \frac{2 \pi}{\lambda} \begin{bmatrix} -\sin \phi_g \cos \theta_g \\ \cos \phi_g \cos \theta_g \\ \sin \theta_g\end{bmatrix} - \frac{2 \pi}{\lambda} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \\ & = \frac{2 \pi}{\lambda} \begin{bmatrix} -\sin \phi_g \cos \theta_g \\ \cos \phi_g \cos \theta_g - 1 \\ \sin \theta_g\end{bmatrix} \end{alignat} }

This vector is of course the surface of the Ewald sphere.

Arbitrary Point

For other points on the detector face, we can combine the above result with the known results for the Geometry of TSAXS. The incident beam is:

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} }

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{k}_i = \frac{2 \pi}{\lambda} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} }

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{q}_i = \frac{2 \pi}{\lambda} \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} }

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 can compute the vector onto the detector face:

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} x \\ d \\ z \end{bmatrix} }

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{k}_1 = \frac{2 \pi}{\lambda} \begin{bmatrix} \frac{x}{ \sqrt{x^2 + d^2 + z^2 }} \\ \frac{d }{\sqrt{x^2 + d^2 + z^2 }} \\ \frac{z }{\sqrt{x^2 + d^2 + z^2 }} \end{bmatrix} }

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{q}_1 = \frac{2 \pi}{\lambda} \begin{bmatrix} \frac{x}{ \sqrt{x^2 + d^2 + z^2 }} \\ \frac{d }{\sqrt{x^2 + d^2 + z^2 }} - 1 \\ \frac{z }{\sqrt{x^2 + d^2 + z^2 }} \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}_2 & = R_x(\theta_g) \mathbf{v}_1 \\ & = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta_g & -\sin \theta_g \\ 0 & \sin \theta_g & \cos \theta_g \\ \end{bmatrix} \begin{bmatrix} x \\ d \\ z \end{bmatrix} \\ & = \begin{bmatrix} x \\ d \cos \theta_g - z \sin \theta_g \\ d \sin \theta_g + z \cos \theta_g \end{bmatrix} \end{alignat} }

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{k}_2 = \frac{2 \pi}{\lambda} \frac{1}{\sqrt{x^2 + d^2 + z^2} } \begin{bmatrix} x \\ d \cos \theta_g - z \sin \theta_g \\ d \sin \theta_g + z \cos \theta_g \end{bmatrix} }

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{q}_2 = \frac{2 \pi}{\lambda} \begin{bmatrix} \frac{x}{\sqrt{x^2 + d^2 + z^2}} \\ \frac{ d \cos \theta_g - z \sin \theta_g}{\sqrt{x^2 + d^2 + z^2}} - 1 \\ \frac{d \sin \theta_g + z \cos \theta_g}{\sqrt{x^2 + d^2 + z^2}} \end{bmatrix} }

The 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 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}_2 \\ & = \begin{bmatrix} \cos \phi_g & -\sin \phi_g & 0 \\ \sin \phi_g & \cos \phi_g & 0\\ 0 & 0 & 1\\ \end{bmatrix} \begin{bmatrix} x \\ d \cos \theta_g - z \sin \theta_g \\ d \sin \theta_g + z \cos \theta_g \end{bmatrix} \\ & = \begin{bmatrix} x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ d \sin \theta_g + z \cos \theta_g \end{bmatrix} \end{alignat} }

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{k}_f = \frac{2 \pi}{\lambda} \frac{1}{\sqrt{x^2 + d^2 + z^2} } \begin{bmatrix} x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ d \sin \theta_g + z \cos \theta_g \end{bmatrix} }

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{q} = \frac{2 \pi}{\lambda} \begin{bmatrix} \frac{x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g )}{\sqrt{x^2 + d^2 + z^2}} \\ \frac{ x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{\sqrt{x^2 + d^2 + z^2}} - 1 \\ \frac{d \sin \theta_g + z \cos \theta_g}{\sqrt{x^2 + d^2 + z^2}} \end{bmatrix} }

Components

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{q} = \frac{2 \pi}{\lambda} \frac{1}{d^{\prime}} \begin{bmatrix} x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) - d^{\prime} \\ d \sin \theta_g + z \cos \theta_g \end{bmatrix} }

Where:

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 d^{\prime} = \sqrt{x^2 + d^2 + z^2} }

Total magnitude

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{k}_i \cdot \mathbf{k}_f }{ \left\| \mathbf{k}_i \right\| \left\| \mathbf{k}_f \right\|} \\ & = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \cdot \frac{1}{\sqrt{x^2 + d^2 + z^2} } \begin{bmatrix} x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ d \sin \theta_g + z \cos \theta_g \end{bmatrix} \\ & = \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{\sqrt{x^2 + d^2 + z^2} } \\ \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 & = \sqrt{2} k \sqrt{ 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{\sqrt{x^2 + d^2 + z^2} } } \end{alignat} }

Check

We define:

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} d^{\prime} & = \sqrt{x^2 + d^2 + z^2} = \| \mathbf{v}_1 \| \\ ( v_{2y} ) & = ( d \cos \theta_g - z \sin \theta_g ) \\ ( v_{2y} )^2 & = ( d \cos \theta_g - z \sin \theta_g )^2 \\ & = d^2 \cos^2 \theta_g - 2dz \cos \theta_g \sin\theta_g + z^2 \sin^2 \theta_g \end{alignat} }

And calculate:

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^2 & = [ (q_x)^2 + (q_y)^2 + (q_z)^2 ] \\ \left ( \frac{q}{k} \right )^2 d^{\prime 2} & = \begin{alignat}{2} [ & \left( x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right)^2 \\ & + \left( x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) - d^{\prime} \right)^2 \\ & + \left( d \sin \theta_g + z \cos \theta_g \right)^2 ] \end{alignat} \\ & = \begin{alignat}{2} [ & \left( x \cos \phi_g -\sin \phi_g ( v_{2y} ) \right)^2 \\ & + \left( x \sin \phi_g + \cos \phi_g ( v_{2y} ) - d^{\prime} \right)^2 \\ & + \left( d \sin \theta_g + z \cos \theta_g \right)^2 ] \end{alignat} \\ & = \begin{alignat}{2} [ & x^2 \cos^2 \phi_g - 2 x \cos \phi_g \sin \phi_g ( v_{2y} ) + \sin^2 \phi_g ( v_{2y} )^2 \\ & + x^2 \sin^2 \phi_g + x \sin \phi_g \cos \phi_g ( v_{2y} ) - d^{\prime} x \sin \phi_g \\ & + x \sin \phi_g \cos \phi_g ( v_{2y} ) + \cos^2 \phi_g ( v_{2y} )^2 - d^{\prime} \cos \phi_g ( v_{2y} ) \\ & - d^{\prime} x \sin \phi_g - d^{\prime} \cos \phi_g ( v_{2y} ) + d^{\prime 2} \\ & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g ] \end{alignat} \\ \end{alignat} }

Grouping and rearranging:

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} \left ( \frac{q}{k} \right )^2 d^{\prime 2} & = \begin{alignat}{2} [ & x^2 + ( v_{2y} )^2 \\ & - 2 d^{\prime} x \sin \phi_g \\ & - 2 d^{\prime} \cos \phi_g ( v_{2y} ) \\ & + d^{\prime 2} \\ & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g ] \end{alignat} \\ & = \begin{alignat}{2} [ & d^{\prime 2} + x^2 + ( d^2 \cos^2 \theta_g - 2dz \cos \theta_g \sin\theta_g + z^2 \sin^2 \theta_g ) \\ & - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} ) \\ & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g ] \end{alignat} \\ & = \begin{alignat}{2} [ & d^{\prime 2} + x^2 + d^2 + z^2 \\ & - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} ) ] \end{alignat} \\ & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} )\\ & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ & = 2 d^{\prime} \left( d^{\prime} - x \sin \phi_g - \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right) \\ \left( \frac{q}{k} \right)^2 & = 2 \left( 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{d^{\prime} } \right) \end{alignat} }

Area Detector on Goniometer Arm, with offsets

In the most general case, the sample may not sit at the exact center of the goniometer rotation. In such a case, corrections must be applied.

TBD

Difference between revisions of "Geometry:WAXS 3D"

Latest revision as of 08:06, 10 August 2017

Contents

Area Detector on Goniometer Arm

Central Point

Total scattering

Components

Arbitrary Point

Components

Total magnitude

Check

Area Detector on Goniometer Arm, with offsets

See Also

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@@ Line 5: / Line 5: @@
 The final scattering vector depends on:
-* <math>\scriptstyle x </math>: Pixel position on detector (horizontal).
+* <math> x </math>: Pixel position on detector (horizontal).
-* <math>\scriptstyle z </math>: Pixel position on detector (vertical).
+* <math> z </math>: Pixel position on detector (vertical).
-* <math>\scriptstyle d </math>: Sample-detector distance.
+* <math> d </math>: Sample-detector distance.
-* <math>\scriptstyle \theta_g </math>: Elevation angle of detector.
+* <math> \theta_g </math>: Elevation angle of detector.
-* <math>\scriptstyle \phi_g </math>: In-plane angle of detector.
+* <math> \phi_g </math>: In-plane angle of detector.
 Note that <math>\scriptstyle x </math> and <math>\scriptstyle z </math> are defined relative to the direct-beam. That is, for <math>\scriptstyle \theta_g = 0 </math> and <math>\scriptstyle \phi_g =0 </math>, the direct beam is at position <math>\scriptstyle (x,z)=(0,0) </math> on the area detector.
@@ Line 76: / Line 76: @@
 :<math>
 \mathbf{v}_i = \begin{bmatrix} 0 \\ d \\ 0 \end{bmatrix}
+</math>
+:<math>
+\mathbf{k}_i = \frac{2 \pi}{\lambda} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}
 </math>
 :<math>
@@ Line 84: / Line 87: @@
 :<math>
 \mathbf{v}_1 = \begin{bmatrix} x \\ d \\ z \end{bmatrix}
+</math>
+:<math>
+\mathbf{k}_1
+ = \frac{2 \pi}{\lambda} \begin{bmatrix}
+\frac{x}{ \sqrt{x^2 + d^2 + z^2   }}  \\
+\frac{d }{\sqrt{x^2 + d^2 + z^2   }}  \\
+\frac{z }{\sqrt{x^2 + d^2 + z^2   }} \end{bmatrix}
 </math>
@@ Line 105: / Line 116: @@
 \end{alignat}
 </math>
+:<math>
+\mathbf{k}_2
+ = \frac{2 \pi}{\lambda} \frac{1}{\sqrt{x^2 + d^2 + z^2} } \begin{bmatrix} x \\ d \cos \theta_g - z \sin \theta_g \\ d \sin \theta_g + z \cos \theta_g \end{bmatrix}
+</math>
+:<math>
+\mathbf{q}_2
+ = \frac{2 \pi}{\lambda} \begin{bmatrix} \frac{x}{\sqrt{x^2 + d^2 + z^2}} \\ \frac{ d \cos \theta_g - z \sin \theta_g}{\sqrt{x^2 + d^2 + z^2}} - 1 \\ \frac{d \sin \theta_g + z \cos \theta_g}{\sqrt{x^2 + d^2 + z^2}} \end{bmatrix}
+</math>
+The vector is then rotated about the <math>\scriptstyle z</math>-axis by <math>\scriptstyle \phi_g</math>:
+:<math>
+\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 & 1\\
+\end{bmatrix} \begin{bmatrix} x \\ d \cos \theta_g - z \sin \theta_g \\ d \sin \theta_g + z \cos \theta_g \end{bmatrix} \\
+    & = \begin{bmatrix}
+      x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\
+      x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\
+      d \sin \theta_g + z \cos \theta_g \end{bmatrix}
+\end{alignat}
+</math>
+:<math>
+\mathbf{k}_f
+ = \frac{2 \pi}{\lambda} \frac{1}{\sqrt{x^2 + d^2 + z^2} } \begin{bmatrix}
+      x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\
+      x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\
+      d \sin \theta_g + z \cos \theta_g \end{bmatrix}
+</math>
+:<math>
+\mathbf{q}
+ = \frac{2 \pi}{\lambda} \begin{bmatrix} \frac{x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g )}{\sqrt{x^2 + d^2 + z^2}} \\ \frac{ x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{\sqrt{x^2 + d^2 + z^2}} - 1 \\ \frac{d \sin \theta_g + z \cos \theta_g}{\sqrt{x^2 + d^2 + z^2}} \end{bmatrix}
+</math>
+===Components===
+:<math>
+\mathbf{q}
+ = \frac{2 \pi}{\lambda} \frac{1}{d^{\prime}} \begin{bmatrix} x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) - d^{\prime} \\ d \sin \theta_g + z \cos \theta_g \end{bmatrix}
+</math>
+Where:
+::<math>
+d^{\prime} = \sqrt{x^2 + d^2 + z^2}
+</math>
+===Total magnitude===
+:<math>
+\begin{alignat}{2}
+\cos \Theta & = \frac{ \mathbf{k}_i \cdot \mathbf{k}_f }{ \left\| \mathbf{k}_i \right\| \left\| \mathbf{k}_f \right\|} \\
+    & = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \cdot \frac{1}{\sqrt{x^2 + d^2 + z^2} } \begin{bmatrix}
+      x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\
+      x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\
+      d \sin \theta_g + z \cos \theta_g \end{bmatrix} \\
+    & = \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{\sqrt{x^2 + d^2 + z^2} } \\
+\end{alignat}
+</math>
+Thus:
+:<math>
+\begin{alignat}{2}
+q
+    & = \sqrt{2} k \sqrt{ 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{\sqrt{x^2 + d^2 + z^2} } }
+\end{alignat}
+</math>
+====Check====
+We define:
+::<math>
+\begin{alignat}{2}
+d^{\prime} & = \sqrt{x^2 + d^2 + z^2} = \| \mathbf{v}_1 \| \\
+( v_{2y} ) & = ( d \cos \theta_g - z \sin \theta_g ) \\
+( v_{2y} )^2 & = ( d \cos \theta_g - z \sin \theta_g )^2 \\
+    & = d^2 \cos^2 \theta_g - 2dz \cos \theta_g \sin\theta_g + z^2 \sin^2 \theta_g
+\end{alignat}
+</math>
+And calculate:
+:<math>
+\begin{alignat}{2}
+q^2 & = [ (q_x)^2 + (q_y)^2 + (q_z)^2 ] \\
+\left ( \frac{q}{k} \right )^2 d^{\prime 2}
+    & = \begin{alignat}{2} [ & \left( x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right)^2 \\ & + \left( x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) - d^{\prime} \right)^2 \\ & + \left( d \sin \theta_g + z \cos \theta_g \right)^2  ] \end{alignat}  \\
+    & = \begin{alignat}{2} [
+      & \left( x \cos \phi_g -\sin \phi_g ( v_{2y} ) \right)^2 \\
+      & + \left( x \sin \phi_g + \cos \phi_g ( v_{2y} ) - d^{\prime} \right)^2 \\
+      & + \left( d \sin \theta_g + z \cos \theta_g \right)^2  ] \end{alignat}  \\
+    & = \begin{alignat}{2} [
+      & x^2 \cos^2 \phi_g - 2 x \cos \phi_g \sin \phi_g ( v_{2y} ) + \sin^2 \phi_g ( v_{2y} )^2  \\
+      & + x^2 \sin^2 \phi_g + x \sin \phi_g \cos \phi_g ( v_{2y} ) - d^{\prime} x \sin \phi_g \\
+      & + x \sin \phi_g \cos \phi_g ( v_{2y} ) + \cos^2 \phi_g ( v_{2y} )^2 - d^{\prime} \cos \phi_g ( v_{2y} ) \\
+      & - d^{\prime} x \sin \phi_g - d^{\prime} \cos \phi_g ( v_{2y} ) + d^{\prime 2} \\
+      & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g  ] \end{alignat}  \\
+\end{alignat}
+</math>
+Grouping and rearranging:
+:<math>
+\begin{alignat}{2}
+\left ( \frac{q}{k} \right )^2 d^{\prime 2}
+    & = \begin{alignat}{2} [
+      & x^2 + ( v_{2y} )^2  \\
+      & - 2 d^{\prime} x \sin \phi_g \\
+      & - 2 d^{\prime} \cos \phi_g ( v_{2y} ) \\
+      & + d^{\prime 2} \\
+      & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g  ] \end{alignat}  \\
+    & = \begin{alignat}{2} [
+      & d^{\prime 2} + x^2 + ( d^2 \cos^2 \theta_g - 2dz \cos \theta_g \sin\theta_g + z^2 \sin^2 \theta_g )  \\
+      & - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} ) \\
+      & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g  ] \end{alignat}  \\
+    & = \begin{alignat}{2} [
+      & d^{\prime 2} + x^2 + d^2 + z^2   \\
+      & - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} ) ] \end{alignat}  \\
+    & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} )\\
+    & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )  \\
+    & = 2 d^{\prime} \left( d^{\prime} - x \sin \phi_g - \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right) \\
+\left( \frac{q}{k} \right)^2
+    & = 2 \left( 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{d^{\prime} } \right)
+\end{alignat}
+</math>
+=Area Detector on Goniometer Arm, with offsets=
+In the most general case, the sample may not sit at the exact center of the goniometer rotation. In such a case, corrections must be applied.
+TBD
 =See Also=
 * [[Geometry:TSAXS 3D]]