Difference between revisions of "Talk:DWBA"

From GISAXS
Jump to: navigation, search
(→‎Expansion)
(→‎TBD)
Line 143: Line 143:
 
</math>
 
</math>
  
−
===TBD===
+
 
−
We take advantage of a more compact form using the notation <math>T_i = T(\alpha_i)</math> and <math>F_{+1} = F(+Q_{z1})</math>:
+
 
 +
 
 +
===Equation===
 +
We take advantage of a more compact form using the notation <math>T_i = T(\alpha_i)</math> and <math>F_{+1} = F(+Q_{z1})</math>. The DWBA equation can thus be expanded as:
 +
 
 +
<math>
 +
\begin{align}
 +
I_d(q_{z}) & =  |
 +
      T_i T_f F_{+1}
 +
    + T_i R_f F_{-2}
 +
    + R_i T_f F_{+2}
 +
    + R_i R_f F_{-1}  | ^{2} \\
 +
 
 +
    & \begin{align}
 +
        = \,\, &  T_i^2 T_f^2 | F(+Q_{z1}) |^2 && + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\
 +
          & && + T_i R_i T_f^2 F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1})  \\
 +
 
 +
          & + T_i^2 R_f^2 | F(-Q_{z2}) |^2 && + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\
 +
          & && + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2})  \\
 +
 
 +
          & + R_i^2 T_f^2 | F(+Q_{z2}) |^2 && + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\
 +
          & && + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2})  \\
 +
 
 +
          & + R_i^2 R_f^2 | F(-Q_{z1}) |^2 && + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\
 +
          & && + T_i R_i R_f^2 F(-Q_{z1})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2})  \\
 +
 
 +
        \end{align} \\
 +
 
 +
\end{align}
 +
</math>
  
 
==Breaking into components==
 
==Breaking into components==

Revision as of 08:56, 13 March 2018

DWBA Equation in thin film

Using the notation 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 T_i = T(\alpha_i)} for compactness, the DWBA equation inside a thin film can be written:

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{align} I_d(q_{z}) & = | T_i T_f F(+Q_{z1}) + T_i R_f F(-Q_{z2}) + R_i T_f F(+Q_{z2}) + R_i R_f F(-Q_{z1}) | ^{2} \\ \end{align} }

Expansion (incorrect)

WARNING: This incorrectly ignores the complex components.

Terms

If one expands the 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 |...|^2} of the DWBA, one obtains 16 terms:

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{matrix} & (T_i T_f) & (T_i R_f) & (R_i T_f) & (R_i R_f) \\ (T_i T_f) & T_i^2T_f^2 & T_i^2 T_f R_f & T_iR_iT_f^2 & T_iR_iT_fR_f \\ (T_i R_f) & T_i^2T_fR_f & T_i^2R_f^2 & T_iR_iT_fR_f & T_iR_iR_f^2 \\ (R_i T_f) & T_iR_iT_f^2 & T_iR_iT_fR_f & R_i^2T_f^2 & R_i^2T_fR_f \\ (R_i R_f) & T_iR_iT_fR_f & T_iR_iR_f^2 & R_i^2T_fR_f & R_i^2R_f^2 \\ \end{matrix} }

Equation

The equation can thus be expanded as:

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{align} I_d(q_{z}) & = | T_i T_f F(+Q_{z1}) + T_i R_f F(-Q_{z2}) + R_i T_f F(+Q_{z2}) + R_i R_f F(-Q_{z1}) | ^{2} \\ & \begin{align} = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 && + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\ & && + T_i R_i T_f^2 F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & + T_i^2 R_f^2 | F(-Q_{z2}) |^2 && + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\ & && + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2}) \\ & + R_i^2 T_f^2 | F(+Q_{z2}) |^2 && + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\ & && + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ & + R_i^2 R_f^2 | F(-Q_{z1}) |^2 && + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & && + T_i R_i R_f^2 F(-Q_{z1})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ \end{align} \\ \end{align} }

Simplification

We can rearrange to:


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{align} I_d(q_{z}) = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\ & + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\ & + T_i R_i T_f^2 F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\ & + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2}) \\ & + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\ & + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ & + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & + T_i R_i R_f^2 F(-Q_{z1})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\ & + 2 \times T_i^2T_fR_f F(+Q_{z1})F(-Q_{z2}) \\ & + 2 \times T_iR_iT_f^2 F(+Q_{z1})F(+Q_{z2}) \\ & + 2 \times T_i R_i T_f R_f [ F(+Q_{z1})F(-Q_{z1}) + F(+Q_{z2})F(-Q_{z2}) ] \\ & + 2 \times T_iR_iR_f^2 F(-Q_{z1})F(-Q_{z2}) \\ & + 2 \times R_i^2T_fR_fx F(-Q_{z1})F(+Q_{z2}) \\ = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\ & + 2 \times T_i^2T_fR_f F(+Q_{z1})F(-Q_{z2}) + 2 \times T_iR_iT_f^2 F(+Q_{z1})F(+Q_{z2}) \\ & + 2 \times T_i R_i T_f R_f [ F(+Q_{z1})F(-Q_{z1}) + F(+Q_{z2})F(-Q_{z2}) ] \\ & + 2 \times T_iR_iR_f^2 F(-Q_{z1})F(-Q_{z2}) + 2 \times R_i^2T_fR_fx F(-Q_{z1})F(+Q_{z2}) \\ \end{align} }

We can rewrite in a more compact form using the notation 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 T_i = T(\alpha_i)} 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 F_{+1} = F(+Q_{z1})} :

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{align} I_d(q_{z}) = \,\, & T_i^2 T_f^2 | F_{+1} |^2 + T_i^2 R_f^2 | F_{-2} |^2 + R_i^2 T_f^2 | F_{+2} |^2 + R_i^2 R_f^2 | F_{-1} |^2 \\ & + 2 \times T_i^2T_fR_f F_{+1}F_{-2} + 2 \times T_iR_iT_f^2 F_{+1}F_{+2} \\ & + 2 \times T_i R_i T_f R_f [ F_{+1}F_{-1} + F_{+2}F_{-2} ] \\ & + 2 \times T_iR_iR_f^2 F_{-1}F_{-2} + 2 \times R_i^2T_fR_f F_{-1}F_{+2} \end{align} }

Expansion

Terms

If one expands the 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 |...|^2} of the DWBA, one obtains 16 terms:

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{matrix} & (T_i^* T_f^*) & (T_i^* R_f^*) & (R_i^* T_f^*) & (R_i^* R_f^*) \\ (T_i T_f) & T_i T_i^* T_f T_f^* & T_i T_i^* T_f R_f^* & T_i R_i^* T_f T_f^* & T_i R_i^* T_f R_f^* \\ (T_i R_f) & T_i T_i^* T_f^* R_f & T_i T_i^* R_f R_f^* & T_i R_i^* T_f^* R_f & T_i R_i^* R_f R_f^* \\ (R_i T_f) & T_i^* R_i T_f T_f^* & T_i^* R_i T_f R_f^* & R_i R_i^* T_f T_f^* & R_i R_i^* T_f R_f^* \\ (R_i R_f) & T_i^* R_i T_f^* R_f & T_i^* R_i R_f R_f^* & R_i R_i^* T_f^* R_f & R_i R_i^* R_f R_f^* \\ \end{matrix} }


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{matrix} & (T_i^* T_f^*) & (T_i^* R_f^*) & (R_i^* T_f^*) & (R_i^* R_f^*) \\ (T_i T_f) & |T_i T_f|^2 & |T_i|^2 T_f R_f^* & T_i R_i^* |T_f|^2 & T_i R_i^* T_f R_f^* \\ (T_i R_f) & |T_i|^2 T_f^* R_f & |T_i R_f|^2 & T_i R_i^* T_f^* R_f & T_i R_i^* |R_f|^2 \\ (R_i T_f) & T_i^* R_i |T_f|^2 & T_i^* R_i T_f R_f^* & |R_i T_f|^2 & |R_i|^2 T_f R_f^* \\ (R_i R_f) & T_i^* R_i T_f^* R_f & T_i^* R_i |R_f|^2 & |R_i|^2 T_f^* R_f & | R_i R_f |^2 \\ \end{matrix} }



Equation

We take advantage of a more compact form using the notation 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 T_i = T(\alpha_i)} 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 F_{+1} = F(+Q_{z1})} . The DWBA equation can thus be expanded as:

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{align} I_d(q_{z}) & = | T_i T_f F_{+1} + T_i R_f F_{-2} + R_i T_f F_{+2} + R_i R_f F_{-1} | ^{2} \\ & \begin{align} = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 && + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\ & && + T_i R_i T_f^2 F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & + T_i^2 R_f^2 | F(-Q_{z2}) |^2 && + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\ & && + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2}) \\ & + R_i^2 T_f^2 | F(+Q_{z2}) |^2 && + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\ & && + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ & + R_i^2 R_f^2 | F(-Q_{z1}) |^2 && + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & && + T_i R_i R_f^2 F(-Q_{z1})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ \end{align} \\ \end{align} }

Breaking into components

The experimental data 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 I_d(q_z)} can be broken into contributions from the transmitted channel 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 I_{Tc}(qz)} and reflected channel 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 I_{Rc}(qz)} :

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{align} 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) \\ & = |Tc|^2 I_{Tc}(q_z) + |Rc|^2 I_{Rc}(q_z) \\ \end{align} }

We define the ratio between the channels to be:

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{align} w & = \frac{ I_{Tc}(q_z) }{ I_{Tc}(q_z) + I_{Rc}(q_z) } \end{align} }

Such that one can compute the two components from:

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{align} I_d(q_{z}) & = |Tc|^2 ( I_{Tc}(q_z) ) + |Rc|^2 ( I_{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}) & = I_{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 } \\ \end{align} }

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{align} I_{Rc}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{Tc}(q_z) }{|Rc|^2} \end{align} }

Retrieved from "http://gisaxs.com/index.php?title=Talk:DWBA&oldid=5871"