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

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(Working results 1)
(Working results 1)
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\end{alignat}
 
\end{alignat}
 
</math>
 
</math>
Note that <math>\cos \theta_f = d^2/\sqrt{d^2+x^2}</math>, and <math>\cos^2 \theta_f = d^4/(d^2+x^2)</math> so:
+
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}
 
\frac{1}{\sqrt{d^2+z^2 \cos^2 \theta_f }}  
 
\frac{1}{\sqrt{d^2+z^2 \cos^2 \theta_f }}  
     & = \frac{1}{\sqrt{d^2+z^2 \left( d^4/(d^2+x^2) \right) }} \\
+
     & = \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)/(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 d^2  }} \\
+
     & = \frac{\sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2  }} \\
  
 
\end{alignat}
 
\end{alignat}
Line 32: Line 32:
 
\mathbf{q}  
 
\mathbf{q}  
 
& = \frac{2 \pi}{\lambda} \begin{bmatrix}  
 
& = \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 d^2  }}  \\  
+
\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 d^2  }} - 1 \\  
+
\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^2/\sqrt{d^2+x^2} \right) \sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2 d^2  }} \end{bmatrix} \\
+
\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{2 \pi}{\lambda} \begin{bmatrix}  
\frac{x}{ \sqrt{x^2 + d^2 + z^2 d^2  }}  \\  
+
\frac{x}{ \sqrt{x^2 + d^2 + z^2   }}  \\  
\frac{d }{\sqrt{x^2 + d^2 + z^2 d^2  }} - 1 \\  
+
\frac{d }{\sqrt{x^2 + d^2 + z^2   }} - 1 \\  
\frac{z d }{\sqrt{x^2 + d^2 + z^2 d^2  }} \end{bmatrix} \\
+
\frac{z }{\sqrt{x^2 + d^2 + z^2   }} \end{bmatrix} \\
  
  

Revision as of 12:47, 13 January 2016

Working results 1

Note that , and so:

And:

As a check:

Working results 2 (contains errors)

As a check of these results, consider:

Where we used:

And, we further note that:

Continuing: