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 (incorrect)

WARNING: This incorrectly ignores the complex components.

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 expanded as:

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

Simplification

We can rearrange to:

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

We can rewrite in a more compact form using the notation $T_{i}=T(\alpha _{i})$ and $F_{+1}=F(+Q_{z1})$:

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:

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

Breaking into components

The experimental data $I_{d}(q_{z})$ 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)} :

${\begin{aligned}I_{d}(q_{z})&=[|T_{i}T_{f}|^{2}+|R_{i}R_{f}|^{2}]I_{Tc}(q_{z})+[|T_{i}R_{f}|^{2}+|R_{i}T_{f}|^{2}]I_{Rc}(q_{z})\\&=|Tc|^{2}I_{Tc}(q_{z})+|Rc|^{2}I_{Rc}(q_{z})\\\end{aligned}}$

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} }