Difference between revisions of "Talk:DWBA"
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===Equation=== | ===Equation=== | ||
| − | The equation can thus be | + | The equation can thus be expanded as: |
<math> | <math> | ||
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& \begin{align} | & \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^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 | + | & && + 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^2 R_f^2 | F(-Q_{z2}) |^2 && + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\ | ||
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\end{align} | \end{align} | ||
</math> | </math> | ||
| + | |||
===Simplification=== | ===Simplification=== | ||
We can rearrange to: | We can rearrange to: | ||
Revision as of 18:27, 6 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
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 R_f 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^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\ & + T_i R_i T_f R_f [ 2 F(+Q_{z1})F(-Q_{z1}) + F(+Q_{z1})F(+Q_{z2}) + 2 F(+Q_{z2})F(-Q_{z2}) ] \\ & + 2 \times 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}) \\ & + 2 \times R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ \end{align} }
