Difference between revisions of "Geometry:TSAXS 3D"

From GISAXS
Jump to: navigation, search
(→‎cont)
(→‎Components (distances))
 
(23 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
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>
−
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:
 
:<math>
 
:<math>
 
\begin{alignat}{2}
 
\begin{alignat}{2}
Line 13: Line 16:
 
where <math>\scriptstyle d</math> is the sample-detector distance, <math>\scriptstyle \alpha_f ^{\prime} </math> is the out-of-plane component (angle w.r.t. to ''y''-axis, rotation about x-axis), and <math>\scriptstyle \theta_f </math> is the in-plane component (rotation about ''z''-axis). The alternate angle, <math>\scriptstyle \alpha_f </math>, is the elevation angle in the plane defined by <math>\scriptstyle \theta_f </math>.
 
where <math>\scriptstyle d</math> is the sample-detector distance, <math>\scriptstyle \alpha_f ^{\prime} </math> is the out-of-plane component (angle w.r.t. to ''y''-axis, rotation about x-axis), and <math>\scriptstyle \theta_f </math> is the in-plane component (rotation about ''z''-axis). The alternate angle, <math>\scriptstyle \alpha_f </math>, is the elevation angle in the plane defined by <math>\scriptstyle \theta_f </math>.
 
==Total scattering==
 
==Total scattering==
−
The full scattering angle is:
+
The full scattering angle is defined by a right-triangle with base ''d'' and height <math>\scriptstyle \sqrt{x^2 +d^2}</math>:
 
:<math>
 
:<math>
 
\begin{alignat}{2}
 
\begin{alignat}{2}
Line 43: Line 46:
 
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>
Line 51: Line 55:
 
If <math>\scriptstyle \alpha_f = 0 </math> (and <math>\scriptstyle \alpha_f ^{\prime} = 0</math>), then <math>\scriptstyle q_z = 0 </math>, <math>\scriptstyle 2 \theta_s = \theta_f </math>, and:
 
If <math>\scriptstyle \alpha_f = 0 </math> (and <math>\scriptstyle \alpha_f ^{\prime} = 0</math>), then <math>\scriptstyle q_z = 0 </math>, <math>\scriptstyle 2 \theta_s = \theta_f </math>, and:
 
:<math>
 
:<math>
−
q = k \sin \theta_f
+
\begin{alignat}{2}
 +
q & = 2 k \sin \left( \theta_f /2 \right) \\
 +
    & = 2 k \sqrt{ \frac{1- \cos(\theta_f)}{2} } \\
 +
    & = 2 k \sqrt{ \frac{1}{2} \left( 1 - \frac{1}{\sqrt{1+(x/d)^2}}  \right) } \\
 +
    & = 2 k \sqrt{ \frac{1}{2} \left( 1 - \frac{d}{\sqrt{d^2+x^2}} \right) }
 +
\end{alignat}
 +
</math>
 +
The other component can be thought of in terms of the sides of a right-triangle with angle <math>\scriptstyle \theta_f = 0 </math>:
 +
:<math>
 +
\begin{alignat}{2}
 +
q_x & = q \cos ( \theta_f /2 ) \\
 +
    & = 2k \sin(\theta_f /2 ) \cos ( \theta_f /2 ) \\
 +
    & = k \sin(\theta_f)\\
 +
q_y & = - q \sin ( \theta_f /2 ) \\
 +
    & = - 2k \sin(\theta_f /2 ) \sin ( \theta_f /2 ) \\
 +
    & = - k \left( 1 - \cos \theta_f \right) \\
 +
    & = k \left( \cos \theta_f - 1 \right) \\
 +
\end{alignat}
 +
</math>
 +
Summarizing:
 +
:<math>
 +
\mathbf{q} = \frac{2 \pi}{\lambda} \begin{bmatrix} \sin \theta_f \\ \cos \theta_f - 1 \\ 0 \end{bmatrix}
 +
</math>
 +
 
 +
==Out-of-plane only==
 +
If <math>\scriptstyle \theta_f = 0 </math>, then <math>\scriptstyle q_x = 0 </math>, <math>\scriptstyle \alpha_f^{\prime} = \alpha_f = 2 \theta_s </math>, and:
 +
:<math>
 +
\begin{alignat}{2}
 +
q & = 2 k \sin \left( \alpha_f /2 \right) \\
 +
    & = 2 k \sqrt{ \frac{1- \cos(\alpha_f)}{2} } \\
 +
    & = 2 k \sqrt{ \frac{1}{2} \left( 1 - \frac{d}{\sqrt{d^2+z^2}} \right) }
 +
\end{alignat}
 +
</math>
 +
The components are:
 +
:<math>
 +
\begin{alignat}{2}
 +
q_z & = q \cos ( \alpha_f /2 ) \\
 +
    & = 2k \sin(\theta_f /2 ) \cos ( \theta_f /2 ) \\
 +
    & = k \sin(\alpha_f)\\
 +
q_y & = - q \sin ( \alpha_f /2 ) \\
 +
    & = k \left( \cos \alpha_f - 1 \right) \\
 +
\end{alignat}
 +
</math>
 +
Summarizing:
 +
:<math>
 +
\mathbf{q} = \frac{2 \pi}{\lambda} \begin{bmatrix} 0 \\ \cos \alpha_f - 1 \\ \sin \alpha_f \end{bmatrix}
 
</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}
Line 62: Line 111:
 
q_z & = \frac{2 \pi}{\lambda} \sin \alpha_f  
 
q_z & = \frac{2 \pi}{\lambda} \sin \alpha_f  
 
\end{alignat}
 
\end{alignat}
 +
</math>
 +
In vector form:
 +
:<math>
 +
\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}
 
</math>
 
</math>
  
−
===Check===
+
===Total magnitude===
−
As a check of these results, consider:
+
Note that this provides a simple expression for ''q'' total:
 
:<math>
 
:<math>
 
\begin{alignat}{2}
 
\begin{alignat}{2}
 
q & = \sqrt{ q_x^2 + q_y^2 + q_z^2 } \\
 
q & = \sqrt{ q_x^2 + q_y^2 + q_z^2 } \\
 
     & = \frac{2 \pi}{\lambda} \sqrt{ \sin^2 \theta_f \cos^2 \alpha_f + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + \sin^2 \alpha_f } \\
 
     & = \frac{2 \pi}{\lambda} \sqrt{ \sin^2 \theta_f \cos^2 \alpha_f + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + \sin^2 \alpha_f } \\
−
\frac{q}{k}  
+
\left( \frac{q}{k} \right)^2
−
     & = \sqrt{ (\sin \theta_f)^2 (\cos \alpha_f)^2 + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + (\sin \alpha_f)^2 } \\
+
     & = \sin^2 \theta_f \cos^2 \alpha_f + \cos^2 \theta_f \cos^2 \alpha_f -2 \cos \theta_f \cos \alpha_f + 1 + \sin^2 \alpha_f \\
−
\frac{q^2}{k^2}
+
     & = \cos^2 \alpha_f (\sin^2 \theta_f + \cos^2 \theta_f)  +\sin^2 \alpha_f -2 \cos \theta_f \cos \alpha_f + 1 \\
−
    & = \left(\frac{x/d}{\sqrt{1+(x/d)^2}} \right)^2 \left(\cos \alpha_f \right)^2 + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + \left( \frac{z \cos \theta_f /d }{\sqrt{1+(z \cos \theta_f /d)^2}} \right)^2 \\
+
     & = \cos^2 \alpha_f (1)  +\sin^2 \alpha_f -2 \cos \theta_f \cos \alpha_f + 1 \\
−
     & = \left(\frac{x}{\sqrt{d^2+x^2}} \right)^2 \left(\cos \alpha_f \right)^2 + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + \left( \frac{z \cos \theta_f }{\sqrt{d^2+z^2 \cos^2 \theta_f }} \right)^2 \\
+
     & = 2 -2 \cos \theta_f \cos \alpha_f \\
−
     & = \frac{x^2}{d^2+x^2}  \left(\cos \alpha_f \right)^2 + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + \frac{z^2 \cos^2 \theta_f }{d^2+z^2 \cos^2 \theta_f }  \\
+
q & = \sqrt{2}k \sqrt{ 1 - \cos \theta_f \cos \alpha_f } \\
−
     & = \frac{x^2}{d^2+x^2}  \frac{d^4}{d^2+z^2 \cos^2 \theta_f} + \left ( \cos \theta_f \frac{d^2}{\sqrt{d^2+z^2 \cos^2 \theta_f}} - 1 \right )^2 + \frac{z^2 \cos^2 \theta_f }{d^2+z^2 \cos^2 \theta_f }
 
 
\end{alignat}
 
\end{alignat}
 
</math>
 
</math>
−
Where we used:
+
 
−
::<math>
+
====Check====
 +
As a check of these results, consider:
 +
:<math>
 +
\begin{alignat}{2}
 +
q & = \frac{4 \pi}{\lambda} \sin \left( \theta_s \right) \\
 +
    & = \frac{4 \pi}{\lambda} \sqrt{ \frac{1-\cos 2\theta_s }{2} } \\
 +
\left( \frac{q}{k} \right)^2
 +
    & = \frac{4}{2} \left( 1-\cos 2\theta_s \right)  \\
 +
    & = 2 \left( 1-\frac{1}{\sqrt{1+\left( \sqrt{\tan^2 \theta_f + \frac{ \tan^2 \alpha_f }{ \cos^2 \theta_f } } \right) ^2}} \right)  \\
 +
    & = 2 \left( 1-\frac{1}{\sqrt{1+\tan^2 \theta_f + \frac{ \tan^2 \alpha_f }{ \cos^2 \theta_f } }} \right)  \\
 +
    & = 2-\frac{2}{\sqrt{1+\frac{\sin^2 \theta_f}{\cos^2 \theta_f} + \frac{ \sin^2 \alpha_f }{ \cos^2 \alpha_f \cos^2 \theta_f } }}  \\
 +
\end{alignat}
 +
</math>
 +
And:
 +
:<math>
 
\begin{alignat}{2}
 
\begin{alignat}{2}
−
\sin( \arctan[u]) & = \frac{u}{\sqrt{1+u^2}} \\
+
    & \left( 1+\frac{\sin^2 \theta_f}{\cos^2 \theta_f} + \frac{ \sin^2 \alpha_f }{ \cos^2 \alpha_f \cos^2 \theta_f } \right) ^{-1/2}  \\
−
\sin \theta_f & = \sin( \arctan [x/d] ) \\
+
    = & \left( \frac{\cos^2 \alpha_f \cos^2 \theta_f}{\cos^2 \alpha_f \cos^2 \theta_f}+\frac{\cos^2 \alpha_f \sin^2 \theta_f}{\cos^2 \alpha_f \cos^2 \theta_f} + \frac{ \sin^2 \alpha_f }{ \cos^2 \alpha_f \cos^2 \theta_f } \right) ^{-1/2}   \\
−
& = \frac{x/d}{\sqrt{1 + (x/d)^2}} \\
+
    = & \left( \frac{\cos^2 \alpha_f \cos^2 \theta_f + \cos^2 \alpha_f \sin^2 \theta_f + \sin^2 \alpha_f}{\cos^2 \alpha_f \cos^2 \theta_f} \right) ^{-1/2}  \\
−
     & = \frac{x}{\sqrt{d^2+x^2}}
+
    = & \left( \frac{\cos^2 \theta_f \cos^2 \alpha_f }{\cos^2 \alpha_f \cos^2 \theta_f + \cos^2 \alpha_f \sin^2 \theta_f + \sin^2 \alpha_f} \right) ^{+1/2}   \\
 +
     = & \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
 
\end{alignat}
 
\end{alignat}
 
</math>
 
</math>
  
−
And, we further note that:
+
==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>
 
::<math>
 
\begin{alignat}{2}
 
\begin{alignat}{2}
−
\cos( \arctan[u]) & = \frac{1}{\sqrt{1+u^2}} \\
+
\frac{1}{\sqrt{d^2+z^2 \cos^2 \theta_f }}
−
\cos \theta_f & = \frac{1}{\sqrt{1 + (x/d)^2}} \\
+
    & = \frac{1}{\sqrt{d^2+z^2 \left( d^2/(d^2+x^2) \right) }} \\
−
    & = \frac{d^2}{\sqrt{d^2+x^2}}
+
    & = \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}
 
\end{alignat}
 
</math>
 
</math>
  
−
Continuing:
+
===Total magnitude===
 
:<math>
 
:<math>
 
\begin{alignat}{2}
 
\begin{alignat}{2}
 
\left( \frac{q}{k} \right)^2
 
\left( \frac{q}{k} \right)^2
−
     & = \frac{x^2}{d^2+x^2}  \frac{d^4}{d^2+z^2 \cos^2 \theta_f} + \left ( \frac{d^2}{\sqrt{d^2+x^2}} \frac{d^2}{\sqrt{d^2+z^2 \cos^2 \theta_f}} - 1 \right )^2 + \frac{z^2 }{d^2+z^2 \cos^2 \theta_f } \frac{d^4}{d^2+x^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 \\
−
     & = d^4\frac{x^2+z^2}{(d^2+x^2)(d^2+z^2 \cos^2 \theta_f)}  + \left ( \frac{d^4}{\sqrt{(d^2+x^2)(d^2+z^2 \cos^2 \theta_f)}} - 1 \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{d^4x^2+d^4z^2}{d^4+d^2x^2+d^4z^2}  + \left ( \frac{d^4}{\sqrt{d^4+d^2x^2+d^4z^2}} - 1 \right )^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{d^2x^2+d^2z^2}{d^2+x^2+d^2z^2}  + \left ( \frac{d^8}{d^4+d^2x^2+d^4z^2} -2 \frac{d^4}{\sqrt{d^4+d^2x^2+d^4z^2}} + 1 \right ) \\
+
     & = \frac{2 x^2 + 2 d^2 + 2 z^2 - 2d \sqrt{x^2 + d^2 + z^2 } }{x^2 + d^2 + z^2} \\
−
     & = \frac{d^2x^2+d^2z^2}{d^2+x^2+d^2z^2}  + \frac{d^6}{d^2+x^2+d^2z^2} -2 \frac{d^3}{\sqrt{d^2+x^2+d^2z^2}} + 1 \\
+
     & = 2 \frac{( x^2 + d^2 + z^2 ) - d \sqrt{x^2 + d^2 + z^2 } }{x^2 + d^2 + z^2} \\
−
     & = \frac{d^2x^2+d^2z^2 + d^6 -2d^3\sqrt{d^2+x^2+d^2z^2} + d^2+x^2+d^2z^2}{d^2+x^2+d^2z^2} \\
+
     & = 2 \left( 1  - \frac{d}{\sqrt{x^2 + d^2 + z^2}} \right) \\
−
     & = \frac{d^6 + d^2 + d^2x^2 + x^2 + 2d^2z^2 -2d^3\sqrt{d^2+x^2+d^2z^2}}{d^2+x^2+d^2z^2} \\
+
q & = \sqrt{2}k \sqrt{1  - \frac{d}{\sqrt{x^2 + d^2 + z^2}} }
−
    & = ? \\
 
 
\end{alignat}
 
\end{alignat}
 
</math>
 
</math>

Latest revision as of 13:11, 13 January 2016

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:

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} & = \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} }

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 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) } . The scattering angles are then:

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} \theta_f & = \arctan\left[ \frac{x}{d} \right] \\ \alpha_f ^\prime & = \arctan\left[ \frac{z}{d} \right] \\ \alpha_f & = \arctan \left[ \frac{z }{d / \cos \theta_f} \right] \end{alignat} }

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 \scriptstyle d} is the 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 \alpha_f ^{\prime} } is the out-of-plane component (angle w.r.t. to y-axis, rotation about x-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_f } is the in-plane component (rotation about z-axis). The alternate 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 \alpha_f } , is the elevation angle in the plane defined 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_f } .

Total scattering

The full scattering angle is defined by a right-triangle with base d and height 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 \sqrt{x^2 +d^2}} :

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} 2 \theta_s = \Theta & = \arctan\left[ \frac{ \sqrt{x^2 + z^2}}{d} \right] \\ & = \arctan\left[ \frac{ \sqrt{(d \tan \theta_f)^2 + (d \tan \alpha_f^\prime )^2}}{d} \right] \\ & = \arctan\left[ \sqrt{\tan^2 \theta_f + \tan^2 \alpha_f^\prime } \right] \\ & = \arctan\left[ \sqrt{\tan^2 \theta_f + \frac{ \tan^2 \alpha_f }{ \cos^2 \theta_f } } \right] \\ \end{alignat} }

The total momentum transfer 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 \begin{alignat}{2} q & = \frac{4 \pi}{\lambda} \sin \left( \theta_s \right) \\ & = \frac{4 \pi}{\lambda} \sin \left( \frac{1}{2} \arctan\left [ \frac{\sqrt{x^2 + z^2}}{d} \right ] \right) \end{alignat} }

Given 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 \begin{alignat}{2} \cos( \arctan[u]) & = \frac{1}{\sqrt{1+u^2}} \\ \cos( 2 \theta_s ) & = \frac{1}{\sqrt{1 + (\sqrt{x^2+z^2}/d)^2}} \\ & = \frac{d}{\sqrt{d^2+x^2+z^2}} \end{alignat} }

We can also write:

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 - \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} }

Where we take for granted that q must be positive.

In-plane only

If 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 \alpha_f = 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 \alpha_f ^{\prime} = 0} ), then 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 q_z = 0 } , 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 2 \theta_s = \theta_f } , 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 \begin{alignat}{2} q & = 2 k \sin \left( \theta_f /2 \right) \\ & = 2 k \sqrt{ \frac{1- \cos(\theta_f)}{2} } \\ & = 2 k \sqrt{ \frac{1}{2} \left( 1 - \frac{1}{\sqrt{1+(x/d)^2}} \right) } \\ & = 2 k \sqrt{ \frac{1}{2} \left( 1 - \frac{d}{\sqrt{d^2+x^2}} \right) } \end{alignat} }

The other component can be thought of in terms of the sides of a right-triangle with 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_f = 0 } :

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_x & = q \cos ( \theta_f /2 ) \\ & = 2k \sin(\theta_f /2 ) \cos ( \theta_f /2 ) \\ & = k \sin(\theta_f)\\ q_y & = - q \sin ( \theta_f /2 ) \\ & = - 2k \sin(\theta_f /2 ) \sin ( \theta_f /2 ) \\ & = - k \left( 1 - \cos \theta_f \right) \\ & = k \left( \cos \theta_f - 1 \right) \\ \end{alignat} }

Summarizing:

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} \sin \theta_f \\ \cos \theta_f - 1 \\ 0 \end{bmatrix} }

Out-of-plane only

If 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_f = 0 } , then 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 q_x = 0 } , 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 \alpha_f^{\prime} = \alpha_f = 2 \theta_s } , 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 \begin{alignat}{2} q & = 2 k \sin \left( \alpha_f /2 \right) \\ & = 2 k \sqrt{ \frac{1- \cos(\alpha_f)}{2} } \\ & = 2 k \sqrt{ \frac{1}{2} \left( 1 - \frac{d}{\sqrt{d^2+z^2}} \right) } \end{alignat} }

The components are:

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_z & = q \cos ( \alpha_f /2 ) \\ & = 2k \sin(\theta_f /2 ) \cos ( \theta_f /2 ) \\ & = k \sin(\alpha_f)\\ q_y & = - q \sin ( \alpha_f /2 ) \\ & = k \left( \cos \alpha_f - 1 \right) \\ \end{alignat} }

Summarizing:

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} 0 \\ \cos \alpha_f - 1 \\ \sin \alpha_f \end{bmatrix} }

Components (angular)

For arbitrary 3D scattering vectors, the momentum transfer components are:

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_x & = \frac{2 \pi}{\lambda} \sin \theta_f \cos \alpha_f \\ q_y & = \frac{2 \pi}{\lambda} \left ( \cos \theta_f \cos \alpha_f - 1 \right ) \\ q_z & = \frac{2 \pi}{\lambda} \sin \alpha_f \end{alignat} }

In vector form:

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} \sin \theta_f \cos \alpha_f \\ \cos \theta_f \cos \alpha_f - 1 \\ \sin \alpha_f \end{bmatrix} }

Total magnitude

Note that this provides a simple expression for q total:

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{ q_x^2 + q_y^2 + q_z^2 } \\ & = \frac{2 \pi}{\lambda} \sqrt{ \sin^2 \theta_f \cos^2 \alpha_f + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + \sin^2 \alpha_f } \\ \left( \frac{q}{k} \right)^2 & = \sin^2 \theta_f \cos^2 \alpha_f + \cos^2 \theta_f \cos^2 \alpha_f -2 \cos \theta_f \cos \alpha_f + 1 + \sin^2 \alpha_f \\ & = \cos^2 \alpha_f (\sin^2 \theta_f + \cos^2 \theta_f) +\sin^2 \alpha_f -2 \cos \theta_f \cos \alpha_f + 1 \\ & = \cos^2 \alpha_f (1) +\sin^2 \alpha_f -2 \cos \theta_f \cos \alpha_f + 1 \\ & = 2 -2 \cos \theta_f \cos \alpha_f \\ q & = \sqrt{2}k \sqrt{ 1 - \cos \theta_f \cos \alpha_f } \\ \end{alignat} }

Check

As a check of these results, consider:

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) \\ & = \frac{4 \pi}{\lambda} \sqrt{ \frac{1-\cos 2\theta_s }{2} } \\ \left( \frac{q}{k} \right)^2 & = \frac{4}{2} \left( 1-\cos 2\theta_s \right) \\ & = 2 \left( 1-\frac{1}{\sqrt{1+\left( \sqrt{\tan^2 \theta_f + \frac{ \tan^2 \alpha_f }{ \cos^2 \theta_f } } \right) ^2}} \right) \\ & = 2 \left( 1-\frac{1}{\sqrt{1+\tan^2 \theta_f + \frac{ \tan^2 \alpha_f }{ \cos^2 \theta_f } }} \right) \\ & = 2-\frac{2}{\sqrt{1+\frac{\sin^2 \theta_f}{\cos^2 \theta_f} + \frac{ \sin^2 \alpha_f }{ \cos^2 \alpha_f \cos^2 \theta_f } }} \\ \end{alignat} }

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 \begin{alignat}{2} & \left( 1+\frac{\sin^2 \theta_f}{\cos^2 \theta_f} + \frac{ \sin^2 \alpha_f }{ \cos^2 \alpha_f \cos^2 \theta_f } \right) ^{-1/2} \\ = & \left( \frac{\cos^2 \alpha_f \cos^2 \theta_f}{\cos^2 \alpha_f \cos^2 \theta_f}+\frac{\cos^2 \alpha_f \sin^2 \theta_f}{\cos^2 \alpha_f \cos^2 \theta_f} + \frac{ \sin^2 \alpha_f }{ \cos^2 \alpha_f \cos^2 \theta_f } \right) ^{-1/2} \\ = & \left( \frac{\cos^2 \alpha_f \cos^2 \theta_f + \cos^2 \alpha_f \sin^2 \theta_f + \sin^2 \alpha_f}{\cos^2 \alpha_f \cos^2 \theta_f} \right) ^{-1/2} \\ = & \left( \frac{\cos^2 \theta_f \cos^2 \alpha_f }{\cos^2 \alpha_f \cos^2 \theta_f + \cos^2 \alpha_f \sin^2 \theta_f + \sin^2 \alpha_f} \right) ^{+1/2} \\ = & \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 \end{alignat} }

Components (distances)

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} & = \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} }

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 \cos \theta_f = d/\sqrt{d^2+x^2}} , 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 \cos^2 \theta_f = d^2/(d^2+x^2)} so:

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} \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} }

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 \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} }

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} \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} }
Retrieved from "http://gisaxs.com/index.php?title=Geometry:TSAXS_3D&oldid=5428"