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

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 $F_{+1}=F(+Q_{z1})$:

${\begin{aligned}I_{d}(q_{z})=\,\,&T_{i}^{2}T_{f}^{2}|F_{+1}|^{2}+T_{i}^{2}R_{f}^{2}|F_{-2}|^{2}+R_{i}^{2}T_{f}^{2}|F_{+2}|^{2}+R_{i}^{2}R_{f}^{2}|F_{-1}|^{2}\\&+2\times T_{i}^{2}T_{f}R_{f}F_{+1}F_{-2}+2\times T_{i}R_{i}T_{f}^{2}F_{+1}F_{+2}\\&+2\times T_{i}R_{i}T_{f}R_{f}[F_{+1}F_{-1}+F_{+2}F_{-2}]\\&+2\times T_{i}R_{i}R_{f}^{2}F_{-1}F_{-2}+2\times R_{i}^{2}T_{f}R_{f}F_{-1}F_{+2}\end{aligned}}$

Breaking into components

The experimental data $I_{d}(q_{z})$ can be broken into contributions from the transmitted channel $I_{Tc}(qz)$ and reflected channel $I_{Rc}(qz)$:

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:

${\begin{aligned}w&={\frac {I_{Tc}(q_{z})}{I_{Tc}(q_{z})+I_{Rc}(q_{z})}}\end{aligned}}$

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:

${\begin{aligned}I_{Rc}(q_{z})&={\frac {I_{d}(q_{z})-|Tc|^{2}I_{Tc}(q_{z})}{|Rc|^{2}}}\end{aligned}}$