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| In transmission-[[SAXS]] ([[TSAXS]]), the x-ray beam hits the sample at normal incidence, and passes directly through without [[refraction]]. TSAXS is normally considered in terms of the one-dimensional [[momentum transfer]] (''q''); however the full 3D form of the ''q''-vector is necessary when considering [[scattering]] from anisotropic materials. The ''q''-vector in fact has three components: | | In transmission-[[SAXS]] ([[TSAXS]]), the x-ray beam hits the sample at normal incidence, and passes directly through without [[refraction]]. TSAXS is normally considered in terms of the one-dimensional [[momentum transfer]] (''q''); however the full 3D form of the ''q''-vector is necessary when considering [[scattering]] from anisotropic materials. The ''q''-vector in fact has three components: |
| :<math> | | :<math> |
− | \mathbf{q} = \begin{bmatrix} q_x \\ q_y \\ q_z \end{bmatrix} | + | \begin{alignat}{2} |
| + | \mathbf{q} & = \begin{bmatrix} q_x \\ q_y \\ q_z \end{bmatrix} \\ |
| + | & = \frac{2 \pi}{\lambda} \begin{bmatrix} \sin \theta_f \cos \alpha_f \\ \cos \theta_f \cos \alpha_f - 1 \\ \sin \alpha_f \end{bmatrix} |
| + | \end{alignat} |
| </math> | | </math> |
| This vector is always on the surface of the [[Ewald sphere]]. Consider that the [[x-ray]] beam points along +''y'', so that on the [[detector]], the horizontal is ''x'', and the vertical is ''z''. We assume that the x-ray beam hits the flat 2D area detector at 90° at detector (pixel) position <math>\scriptstyle (x,z) </math>. The scattering angles are then: | | This vector is always on the surface of the [[Ewald sphere]]. Consider that the [[x-ray]] beam points along +''y'', so that on the [[detector]], the horizontal is ''x'', and the vertical is ''z''. We assume that the x-ray beam hits the flat 2D area detector at 90° at detector (pixel) position <math>\scriptstyle (x,z) </math>. The scattering angles are then: |
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| q & = \frac{4 \pi}{\lambda} \sin \left( \theta_s \right) \\ | | q & = \frac{4 \pi}{\lambda} \sin \left( \theta_s \right) \\ |
| & = \pm \frac{4 \pi}{\lambda} \sqrt{ \frac{1-\cos 2\theta_s }{2} } \\ | | & = \pm \frac{4 \pi}{\lambda} \sqrt{ \frac{1-\cos 2\theta_s }{2} } \\ |
− | & = \frac{4 \pi}{\lambda} \sqrt{ \frac{1}{2}\left(1 - \frac{d}{\sqrt{d^2+x^2+z^2}} \right) } | + | & = \frac{4 \pi}{\lambda} \sqrt{ \frac{1}{2}\left(1 - \frac{d}{\sqrt{d^2+x^2+z^2}} \right) } \\ |
| + | & = \sqrt{2} \frac{2 \pi}{\lambda} \sqrt{ 1 - \frac{d}{\sqrt{x^2+d^2+z^2}} } |
| \end{alignat} | | \end{alignat} |
| </math> | | </math> |
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| </math> | | </math> |
| | | |
− | ==Components== | + | ==Components (angular)== |
− | The [[momentum transfer]] components are:
| + | For arbitrary 3D scattering vectors, the [[momentum transfer]] components are: |
| :<math> | | :<math> |
| \begin{alignat}{2} | | \begin{alignat}{2} |
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| = & \frac{\cos \theta_f \cos \alpha_f }{ \sqrt{ \cos^2 \alpha_f (\cos^2 \theta_f + \sin^2 \theta_f) + \sin^2 \alpha_f }} \\ | | = & \frac{\cos \theta_f \cos \alpha_f }{ \sqrt{ \cos^2 \alpha_f (\cos^2 \theta_f + \sin^2 \theta_f) + \sin^2 \alpha_f }} \\ |
| = & \cos \theta_f \cos \alpha_f | | = & \cos \theta_f \cos \alpha_f |
| + | \end{alignat} |
| + | </math> |
| + | |
| + | ==Components (distances)== |
| + | :<math> |
| + | \begin{alignat}{2} |
| + | \mathbf{q} & = \frac{2 \pi}{\lambda} \begin{bmatrix} \sin \theta_f \cos \alpha_f \\ \cos \theta_f \cos \alpha_f - 1 \\ \sin \alpha_f \end{bmatrix} \\ |
| + | & = \frac{2 \pi}{\lambda} \begin{bmatrix} \sin \left( \arctan\left[ \frac{x}{d} \right] \right) \cos \left( \arctan \left[ \frac{z }{d / \cos \theta_f} \right] \right) \\ \cos \left( \arctan\left[ \frac{x}{d} \right] \right) \cos \left( \arctan \left[ \frac{z }{d / \cos \theta_f} \right] \right) - 1 \\ \sin \left( \arctan \left[ \frac{z }{d / \cos \theta_f} \right] \right) \end{bmatrix} \\ |
| + | |
| + | & = \frac{2 \pi}{\lambda} \begin{bmatrix} |
| + | \frac{x/d}{\sqrt{1+\left(x/d \right)^2}} \frac{d}{\sqrt{d^2+z^2\cos^2 \theta_f}} \\ |
| + | \frac{1}{\sqrt{1+\left(x/d \right)^2}} \frac{d}{\sqrt{d^2+z^2\cos^2 \theta_f}} - 1 \\ |
| + | \frac{z \cos \theta_f}{\sqrt{d^2+z^2 \cos^2 \theta_f }} \end{bmatrix} \\ |
| + | |
| + | & = \frac{2 \pi}{\lambda} \begin{bmatrix} |
| + | \frac{x d}{\sqrt{d^2+x^2 }} \frac{1}{\sqrt{d^2+z^2\cos^2 \theta_f}} \\ |
| + | \frac{d}{\sqrt{d^2+x^2}} \frac{d}{\sqrt{d^2+z^2\cos^2 \theta_f}} - 1 \\ |
| + | \frac{z \cos \theta_f}{\sqrt{d^2+z^2 \cos^2 \theta_f }} \end{bmatrix} \\ |
| + | |
| + | \end{alignat} |
| + | </math> |
| + | Note that <math>\cos \theta_f = d/\sqrt{d^2+x^2}</math>, and <math>\cos^2 \theta_f = d^2/(d^2+x^2)</math> so: |
| + | ::<math> |
| + | \begin{alignat}{2} |
| + | \frac{1}{\sqrt{d^2+z^2 \cos^2 \theta_f }} |
| + | & = \frac{1}{\sqrt{d^2+z^2 \left( d^2/(d^2+x^2) \right) }} \\ |
| + | & = \frac{1}{\sqrt{d^2} \sqrt{((d^2+x^2)+z^2)/(d^2+x^2) }} \\ |
| + | & = \frac{\sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2 }} \\ |
| + | |
| + | \end{alignat} |
| + | </math> |
| + | And: |
| + | :<math> |
| + | \begin{alignat}{2} |
| + | \mathbf{q} |
| + | & = \frac{2 \pi}{\lambda} \begin{bmatrix} |
| + | \frac{x d}{\sqrt{d^2+x^2 }} \frac{\sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2 }} \\ |
| + | \frac{d}{\sqrt{d^2+x^2}} \frac{d \sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2 }} - 1 \\ |
| + | \frac{z \left( d/\sqrt{d^2+x^2} \right) \sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2 }} \end{bmatrix} \\ |
| + | |
| + | & = \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} \\ |
| + | |
| + | |
| + | \end{alignat} |
| + | </math> |
| + | |
| + | ===Total magnitude=== |
| + | :<math> |
| + | \begin{alignat}{2} |
| + | \left( \frac{q}{k} \right)^2 |
| + | & = \left( \frac{x}{ \sqrt{x^2 + d^2 + z^2 }} \right)^2 + \left( \frac{d - \sqrt{x^2 + d^2 + z^2 } }{\sqrt{x^2 + d^2 + z^2 }} \right)^2 + \left( \frac{z }{\sqrt{x^2 + d^2 + z^2 }} \right)^2 \\ |
| + | & = \frac{x^2 + \left( d - \sqrt{x^2 + d^2 + z^2 }\right)^2 + z^2 }{x^2 + d^2 + z^2} \\ |
| + | & = \frac{x^2 + \left( d^2 - 2d \sqrt{x^2 + d^2 + z^2 } + x^2 + d^2 + z^2 \right) + z^2 }{x^2 + d^2 + z^2} \\ |
| + | & = \frac{2 x^2 + 2 d^2 + 2 z^2 - 2d \sqrt{x^2 + d^2 + z^2 } }{x^2 + d^2 + z^2} \\ |
| + | & = 2 \frac{( x^2 + d^2 + z^2 ) - d \sqrt{x^2 + d^2 + z^2 } }{x^2 + d^2 + z^2} \\ |
| + | & = 2 \left( 1 - \frac{d}{\sqrt{x^2 + d^2 + z^2}} \right) \\ |
| + | q & = \sqrt{2}k \sqrt{1 - \frac{d}{\sqrt{x^2 + d^2 + z^2}} } |
| \end{alignat} | | \end{alignat} |
| </math> | | </math> |
In transmission-SAXS (TSAXS), the x-ray beam hits the sample at normal incidence, and passes directly through without refraction. TSAXS is normally considered in terms of the one-dimensional momentum transfer (q); however the full 3D form of the q-vector is necessary when considering scattering from anisotropic materials. The q-vector in fact has three components:
This vector is always on the surface of the Ewald sphere. Consider that the x-ray beam points along +y, so that on the detector, the horizontal is x, and the vertical is z. We assume that the x-ray beam hits the flat 2D area detector at 90° at detector (pixel) position . The scattering angles are then:
where is the sample-detector distance, is the out-of-plane component (angle w.r.t. to y-axis, rotation about x-axis), and is the in-plane component (rotation about z-axis). The alternate angle, , is the elevation angle in the plane defined by .
Total scattering
The full scattering angle is defined by a right-triangle with base d and height :
The total momentum transfer is:
Given that:
We can also write:
Where we take for granted that q must be positive.
In-plane only
If (and ), then , , and:
The other component can be thought of in terms of the sides of a right-triangle with angle :
Summarizing:
Out-of-plane only
If , then , , and:
The components are:
Summarizing:
Components (angular)
For arbitrary 3D scattering vectors, the momentum transfer components are:
In vector form:
Total magnitude
Note that this provides a simple expression for q total:
Check
As a check of these results, consider:
And:
Components (distances)
Note that , and so:
And:
Total magnitude