Talk:DWBA

DWBA Equation in thin film

Using the notation $T_{i}=T(\alpha _{i})$ for compactness, the DWBA equation inside a thin film can be written:

${\begin{aligned}I_{d}(q_{z})&=|T_{i}T_{f}F(+Q_{z1})+T_{i}R_{f}F(-Q_{z2})+R_{i}T_{f}F(+Q_{z2})+R_{i}R_{f}F(-Q_{z1})|^{2}\\\end{aligned}}$

Expansion

Terms

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

${\begin{matrix}&(T_{i}T_{f})&(T_{i}R_{f})&(R_{i}T_{f})&(R_{i}R_{f})\\(T_{i}T_{f})&T_{i}^{2}T_{f}^{2}&T_{i}^{2}T_{f}R_{f}&T_{i}R_{i}T_{f}^{2}&T_{i}R_{i}T_{f}R_{f}\\(T_{i}R_{f})&T_{i}^{2}T_{f}R_{f}&T_{i}^{2}R_{f}^{2}&T_{i}R_{i}T_{f}R_{f}&T_{i}R_{i}R_{f}^{2}\\(R_{i}T_{f})&T_{i}R_{i}T_{f}^{2}&T_{i}R_{i}T_{f}R_{f}&R_{i}^{2}T_{f}^{2}&R_{i}^{2}T_{f}R_{f}\\(R_{i}R_{f})&T_{i}R_{i}T_{f}R_{f}&T_{i}R_{i}R_{f}^{2}&R_{i}^{2}T_{f}R_{f}&R_{i}^{2}R_{f}^{2}\\\end{matrix}}$

Equation

The equation can thus be written as:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}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{aligned}=\,\,&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}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}R_{f}^{2}|F(-Q_{z2})|^{2}&&+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})\\&+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{aligned}}\\\end{aligned}}}

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} }