Difference between revisions of "Talk:DWBA"

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(Expansion)
(Breaking into components)
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==Breaking into components==
 
==Breaking into components==
The experimental data <math>I_d(q_z)</math> can be broken into contributions from the transmitted channel <math>I_{Tc}(qz)</math> and reflected channel <math>I_{Rc}(qz)</math>:
+
The experimental data <math>I_d(q_z)</math> can be broken into contributions from the transmitted channel <math>I_{d,Tc}(qz)</math> and reflected channel <math>I_{d,Rc}(qz)</math>:
  
 
<math>
 
<math>
 
\begin{align}
 
\begin{align}
 
I_d(q_{z})  
 
I_d(q_{z})  
   & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{Tc}(q_z) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{Rc}(q_z) \\
+
   & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(+Q_{z1}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(+Q_{z2}) \\
   & = |Tc|^2 I_{Tc}(q_z) + |Rc|^2 I_{Rc}(q_z) \\
+
  & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(q_z-\Delta q_{z,Tc}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(q_z-\Delta q_{z,Rc}) \\
 +
  & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{d,Tc}(q_z) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{d,Rc}(q_z) \\
 +
   & = |Tc|^2 I_{d,Tc}(q_z) + |Rc|^2 I_{d,Rc}(q_z) \\
 
\end{align}
 
\end{align}
 
</math>
 
</math>
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\begin{align}
 
\begin{align}
 
w
 
w
   & = \frac{ I_{Tc}(q_z) }{ I_{Tc}(q_z) + I_{Rc}(q_z) }
+
   & = \frac{ I_{d,Tc}(q_z) }{ I_{d,Tc}(q_z) + I_{d,Rc}(q_z) }
 
\end{align}
 
\end{align}
 
</math>
 
</math>
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<math>
 
<math>
 
\begin{align}
 
\begin{align}
I_d(q_{z}) & = |Tc|^2 ( I_{Tc}(q_z) ) + |Rc|^2 ( I_{Rc}(q_z) ) \\
+
I_d(q_{z}) & = |Tc|^2 ( I_{d,Tc}(q_z) ) + |Rc|^2 ( I_{d,Rc}(q_z) ) \\
I_d(q_{z}) & = |Tc|^2 ( I_{Tc}(q_z) ) + |Rc|^2 \left ( \frac{ I_{Tc}(q_z) - w I_{Tc}(q_z) }{w}  \right ) \\
+
I_d(q_{z}) & = |Tc|^2 ( I_{d,Tc}(q_z) ) + |Rc|^2 \left ( \frac{ I_{d,Tc}(q_z) - w I_{d,Tc}(q_z) }{w}  \right ) \\
I_d(q_{z}) & = I_{Tc}(q_z) \times \left ( |Tc|^2  +  |Rc|^2 \frac{ 1}{w}  - |Rc|^2 \frac{w }{w}  \right ) \\
+
I_d(q_{z}) & = I_{d,Tc}(q_z) \times \left ( |Tc|^2  +  |Rc|^2 \frac{ 1}{w}  - |Rc|^2 \frac{w }{w}  \right ) \\
I_{Tc}(q_z)  & = \frac{ I_d(q_{z}) }{  |Tc|^2  +  \frac{ |Rc|^2 }{w}  - |Rc|^2  } \\
+
I_{d,Tc}(q_z)  & = \frac{ I_d(q_{z}) }{  |Tc|^2  +  \frac{ |Rc|^2 }{w}  - |Rc|^2  } \\
 
\end{align}
 
\end{align}
 
</math>
 
</math>
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<math>
 
<math>
 
\begin{align}
 
\begin{align}
I_{Rc}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{Tc}(q_z) }{|Rc|^2}
+
I_{d,Rc}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{d,Tc}(q_z) }{|Rc|^2}
 
\end{align}
 
\end{align}
 
</math>
 
</math>

Revision as of 09:09, 13 March 2018

DWBA Equation in thin film

Using the notation for compactness, the DWBA equation inside a thin film can be written:

Expansion (incorrect)

WARNING: This incorrectly ignores the complex components.

Terms

If one expands the of the DWBA, one obtains 16 terms:

Equation

The equation can thus be expanded as:

Simplification

We can rearrange to:


We can rewrite in a more compact form using the notation and :

Expansion

Terms

If one expands the of the DWBA, one obtains 16 terms:




Equation

We take advantage of a more compact form using the notation and . The DWBA equation can thus be expanded as:

Breaking into components

The experimental data can be broken into contributions from the transmitted channel and reflected channel :

We define the ratio between the channels to be:

Such that one can compute the two components from:

and: