Difference between revisions of "Talk:Geometry:WAXS 3D"

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(Check of Total Magnitude #2: Doesn't work)
(Check of Total Magnitude #2: Doesn't work)
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     & = 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 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) \\
 
     & = 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>
 
 
====Check of Total Magnitude #2: Doesn't work====
 
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
 
\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)
 
     & = 2 \left( 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{d^{\prime} } \right)
 
\end{alignat}
 
\end{alignat}
 
</math>
 
</math>

Revision as of 17:21, 13 January 2016

Check of Total Magnitude #1: Doesn't work