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| − | Using the notation <math>T_i = T(\alpha_i)</math> for compactness, the DWBA equation can be written as: | + | ==DWBA Equation in thin film== |
| | + | Using the notation <math>T_i = T(\alpha_i)</math> for compactness, the DWBA equation inside a thin film can be written: |
| | + | <math> |
| | + | \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} |
| | + | </math> |
| | | | |
| | + | ==Expansion== |
| | + | If one expands the <math>|\ellipsis}^2</math> of the DWBA, one obtains 16 terms: |
| | + | <math> |
| | + | </math> |
| | + | The equation can thus be written as: |
| | <math> | | <math> |
| | \begin{align} | | \begin{align} |
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| | | | |
| | \end{align} | | \end{align} |
| − | </math>
| |
| − | The DWBA equation is:
| |
| − | <math>
| |
| − | 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
| |
| | </math> | | </math> |
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| − | Taking advantage of the fact that <math>|F(Q)|^2 = I(Q)</math> and <math>I(+Q)=I(-Q)</math>, we can rearrange to:
| + | We can rearrange to: |
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Revision as of 17:06, 6 March 2018
DWBA Equation in thin film
Using the notation 
for compactness, the DWBA equation inside a thin film can be written:
Expansion
If one expands the Failed to parse (unknown function "\ellipsis"): {\displaystyle |\ellipsis}^2}
of the DWBA, one obtains 16 terms:
The equation can thus be written as:
We can rearrange to:
![{\displaystyle {\begin{aligned}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}^{2}T_{f}R_{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}[2F(+Q_{z1})F(-Q_{z1})+F(+Q_{z1})F(+Q_{z2})+2F(+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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc7ce8eb06c1820f96123b1e63533c795cdc783f)
