Difference between revisions of "Talk:DWBA"

Latest revision as of 09:21, 5 April 2018

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

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

Expansion

Terms

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

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

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

Equation

We take advantage of a more compact form using the notation $T_{i}=T(\alpha _{i})$ and $F_{+1}=F(+Q_{z1})$ . The DWBA equation can thus be expanded as:

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

Simplification

We can rearrange to:

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

We define $I_{+1}=|F_{+1}|^{2}$ , and note that for any complex number $c$ , it is true that $c+c^{*}=2\mathrm {Re} [c]$ . Thus:

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

Breaking into components

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

${\begin{aligned}I_{d}(q_{z})&=[|T_{i}T_{f}|^{2}+|R_{i}R_{f}|^{2}]I_{R}(+Q_{z1})+[|T_{i}R_{f}|^{2}+|R_{i}T_{f}|^{2}]I_{R}(+Q_{z2})\\&=[|T_{i}T_{f}|^{2}+|R_{i}R_{f}|^{2}]I_{R}(q_{z}-\Delta q_{z,\mathrm {Tc} })+[|T_{i}R_{f}|^{2}+|R_{i}T_{f}|^{2}]I_{R}(q_{z}-\Delta q_{z,\mathrm {Rc} })\\&=[|T_{i}T_{f}|^{2}+|R_{i}R_{f}|^{2}]I_{d,\mathrm {Tc} }(q_{z})+[|T_{i}R_{f}|^{2}+|R_{i}T_{f}|^{2}]I_{d,\mathrm {Rc} }(q_{z})\\&=|Tc|^{2}I_{d,\mathrm {Tc} }(q_{z})+|Rc|^{2}I_{d,\mathrm {Rc} }(q_{z})\\\end{aligned}}$

We define the ratio between the channels to be:

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

Such that one can compute the two components from:

${\begin{aligned}I_{d}(q_{z})&=|Tc|^{2}(I_{d,\mathrm {Tc} }(q_{z}))+|Rc|^{2}(I_{d,\mathrm {Rc} }(q_{z}))\\I_{d}(q_{z})&=|Tc|^{2}(I_{d,\mathrm {Tc} }(q_{z}))+|Rc|^{2}\left({\frac {I_{d,\mathrm {Tc} }(q_{z})-wI_{d,\mathrm {Tc} }(q_{z})}{w}}\right)\\I_{d}(q_{z})&=I_{d,\mathrm {Tc} }(q_{z})\times \left(|Tc|^{2}+|Rc|^{2}{\frac {1}{w}}-|Rc|^{2}{\frac {w}{w}}\right)\\I_{d,\mathrm {Tc} }(q_{z})&={\frac {I_{d}(q_{z})}{|Tc|^{2}+{\frac {|Rc|^{2}}{w}}-|Rc|^{2}}}\\\end{aligned}}$

and:

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

or:

${\begin{aligned}I_{d}(q_{z})&=|Tc|^{2}(I_{d,\mathrm {Tc} }(q_{z}))+|Rc|^{2}(I_{d,\mathrm {Rc} }(q_{z}))\\&=|Tc|^{2}\left({\frac {w}{1-w}}I_{d,\mathrm {Rc} }(q_{z})\right)+|Rc|^{2}(I_{d,\mathrm {Rc} }(q_{z}))\\I_{d,\mathrm {Rc} }(q_{z})&={\frac {I_{d}(q_{z})}{|Tc|^{2}{\frac {w}{1-w}}+|Rc|^{2}}}\end{aligned}}$

@@ Line 12: / Line 12: @@
 </math>
-==Expansion==
+==Expansion (incorrect)==
+'''WARNING: This incorrectly ignores the complex components.'''
 ===Terms===
 If one expands the <math>|...|^2</math> of the DWBA, one obtains 16 terms:
@@ Line 109: / Line 110: @@
          & + 2 \times T_iR_iR_f^2 F_{-1}F_{-2}
            + 2 \times R_i^2T_fR_f F_{-1}F_{+2}
+\end{align}
+</math>
+==Expansion==
+===Terms===
+If one expands the <math>|...|^2</math> of the DWBA, one obtains 16 terms:
+<math>
+\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 T_i^* 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}
+</math>
+<math>
+\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_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 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 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 R_f |^2    \\
+\end{matrix}
+</math>
+===Equation===
+We take advantage of a more compact form using the notation <math>T_i = T(\alpha_i)</math> and <math>F_{+1} = F(+Q_{z1})</math>. The DWBA equation can thus be expanded as:
+<math>
+\begin{align}
+I_d(q_{z}) & =  |
+      T_i T_f F_{+1}
+    + T_i R_f F_{-2}
+    + R_i T_f F_{+2}
+    + R_i R_f F_{-1}  | ^{2} \\
+    & \begin{align}
+        = \,\, &  |T_i T_f|^2 | F_{+1} |^2 && + |T_i|^2 T_f R_f^* F_{+1}F_{-2}^* \\
+          & && + T_i R_i^* |T_f|^2 F_{+1}F_{+2}^* + T_i R_i^* T_f R_f^* F_{+1} F_{-1}^*  \\
+          & + |T_i R_f|^2 | F_{-2} |^2 && + |T_i|^2T_f^*R_f F_{+1}^* F_{-2} \\
+          & && + T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} + T_i R_i^* |R_f|^2 F_{-1}^* F_{-2}  \\
+          & + |R_i T_f|^2 | F_{+2} |^2 && + T_i^* R_i |T_f|^2 F_{+1}^* F_{+2} \\
+          & && + T_i^* R_i T_f R_f^* F_{+2}^*F_{-2} + |R_i|^2 T_f R_f^* F_{-1}^* F_{+2}  \\
+          & + |R_i R_f|^2 | F_{-1} |^2 && + T_i^* R_i T_f^* R_f F_{+1}^* F_{-1} \\
+          & && + T_i^* R_i |R_f|^2 F_{-1}F_{-2}^* + |R_i|^2 T_f^* R_f F_{-1} F_{+2}^*  \\
+        \end{align} \\
+\end{align}
+</math>
+===Simplification===
+We can rearrange to:
+<math>
+\begin{align}
+I_d(q_{z}) = \, \, &  |T_i T_f|^2 | F_{+1} |^2 + |T_i R_f|^2 | F_{-2} |^2 + |R_i T_f|^2 | F_{+2} |^2 + |R_i R_f|^2 | F_{-1} |^2 \\
+    & + |T_i|^2 T_f R_f^* F_{+1}F_{-2}^* + |T_i|^2T_f^*R_f F_{+1}^* F_{-2} \\
+    & + T_i R_i^* |T_f|^2 F_{+1}F_{+2}^* + T_i^* R_i |T_f|^2 F_{+1}^* F_{+2} \\
+    & + |R_i|^2 T_f R_f^* F_{-1}^* F_{+2} + |R_i|^2 T_f^* R_f F_{-1} F_{+2}^* \\
+    & + T_i R_i^* |R_f|^2 F_{-1}^* F_{-2} + T_i^* R_i |R_f|^2 F_{-1}F_{-2}^*\\
+    & + T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* \\
+    & + T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} \\
+    & +  T_i^* R_i T_f R_f^* F_{+2}^*F_{-2} \\
+    & +  T_i^* R_i T_f^* R_f F_{+1}^* F_{-1} \\
+= \, \, &  |T_i T_f|^2 | F_{+1} |^2 + |T_i R_f|^2 | F_{-2} |^2 + |R_i T_f|^2 | F_{+2} |^2 + |R_i R_f|^2 | F_{-1} |^2 \\
+    & + |T_i|^2 [ T_f R_f^* F_{+1}F_{-2}^* + T_f^*R_f F_{+1}^* F_{-2} ] \\
+    & + |T_f|^2 [ T_i R_i^*  F_{+1}F_{+2}^* + T_i^* R_i F_{+1}^* F_{+2} ] \\
+    & + |R_i|^2 [ T_f R_f^* F_{-1}^* F_{+2} + T_f^* R_f F_{-1} F_{+2}^* ] \\
+    & + |R_f|^2 [ T_i R_i^*  F_{-1}^* F_{-2} + T_i^* R_i F_{-1}F_{-2}^* ]\\
+    & + [ T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* + T_i^* R_i T_f^* R_f F_{+1}^* F_{-1} ] \\
+    & + [ T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} + T_i^* R_i T_f R_f^* F_{+2}^*F_{-2} ] \\
+\end{align}
+</math>
+We define <math>I_{+1}=|F_{+1}|^2</math>, and note that for any complex number <math>c</math>, it is true that <math>c+c^*=2 \mathrm{Re}[c]</math>. Thus:
+<math>
+\begin{align}
+I_d(q_{z})
+= \, \, &  |T_i T_f|^2 I_{+1} + |T_i R_f|^2 I_{-2} + |R_i T_f|^2 | I_{+2} + |R_i R_f|^2 I_{-1} \\
+    & + 2 |T_i|^2 \mathrm{Re}[ T_f R_f^* F_{+1}F_{-2}^* ] \\
+    & + 2 |T_f|^2 \mathrm{Re}[ T_i R_i^*  F_{+1}F_{+2}^* ] \\
+    & + 2 |R_i|^2 \mathrm{Re}[ T_f R_f^* F_{-1}^* F_{+2} ] \\
+    & + 2 |R_f|^2 \mathrm{Re}[ T_i R_i^*  F_{-1}^* F_{-2} ]\\
+    & + 2 \mathrm{Re}[ T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* ] \\
+    & + 2 \mathrm{Re}[ T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} ] \\
+= \, \, &  |T_i T_f|^2 I_{+1} + |T_i R_f|^2 I_{-2} + |R_i T_f|^2 | I_{+2} + |R_i R_f|^2 I_{-1} \\
+    & + 2 |T_i|^2 \mathrm{Re}[ T_f R_f^* F_{+1}F_{-2}^* ] + 2 |T_f|^2 \mathrm{Re}[ T_i R_i^*  F_{+1}F_{+2}^* ] \\
+    & + 2 |R_i|^2 \mathrm{Re}[ T_f R_f^* F_{-1}^* F_{+2} ] + 2 |R_f|^2 \mathrm{Re}[ T_i R_i^*  F_{-1}^* F_{-2} ]\\
+    & + 2 \mathrm{Re}[ T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* ] + 2 \mathrm{Re}[ T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} ] \\
 \end{align}
 </math>
 ==Breaking into components==
-The experimental data <math>I_d(q_z)</math> can be broken into contributions from the transmitted channel <math>I_{Tc}(qz)</math> and reflected channel <math>I_{Rc}(qz)</math>:
+The experimental data <math>I_d(q_z)</math> can be broken into contributions from the transmitted channel <math>I_{d,\mathrm{Tc}}(qz)</math> and reflected channel <math>I_{d,\mathrm{Rc}}(qz)</math>:
 <math>
 \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) \\
+   & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(+Q_{z1}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(+Q_{z2}) \\
-   & = |Tc|^2 I_{Tc}(q_z) + |Rc|^2 I_{Rc}(q_z) \\
+  & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(q_z-\Delta q_{z,\mathrm{Tc}}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(q_z-\Delta q_{z,\mathrm{Rc}}) \\
+  & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{d,\mathrm{Tc}}(q_z) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{d,\mathrm{Rc}}(q_z) \\
+   & = |Tc|^2 I_{d,\mathrm{Tc}}(q_z) + |Rc|^2 I_{d,\mathrm{Rc}}(q_z) \\
 \end{align}
 </math>
@@ Line 127: / Line 256: @@
 <math>
 \begin{align}
-w
+w (q_z)
-   & = \frac{ I_{Tc}(q_z) }{ I_{Tc}(q_z) + I_{Rc}(q_z) }
+   & = \frac{ I_{d,\mathrm{Tc}}(q_z) }{ I_{d,\mathrm{Tc}}(q_z) + I_{d,\mathrm{Rc}}(q_z) }
 \end{align}
 </math>
@@ Line 136: / Line 265: @@
 <math>
 \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_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 ( I_{d,\mathrm{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}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 \left ( \frac{ I_{d,\mathrm{Tc}}(q_z) - w I_{d,\mathrm{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_d(q_{z}) & = I_{d,\mathrm{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  } \\
+I_{d,\mathrm{Tc}}(q_z)  & = \frac{ I_d(q_{z}) }{  |Tc|^2  +  \frac{ |Rc|^2 }{w}  - |Rc|^2  } \\
 \end{align}
 </math>
@@ Line 148: / Line 277: @@
 <math>
 \begin{align}
-I_{Rc}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{Tc}(q_z) }{|Rc|^2}
+I_{d,\mathrm{Rc}}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{d,\mathrm{Tc}}(q_z) }{|Rc|^2} \\
+    & = \frac{ I_d(q_{z}) }{|Rc|^2} - \frac{|Tc|^2}{|Rc|^2} I_{d,\mathrm{Tc}}(q_z)
+\end{align}
+</math>
+or:
+<math>
+\begin{align}
+I_d(q_{z}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\
+    & = |Tc|^2 \left( \frac{w}{1-w}  I_{d,\mathrm{Rc}}(q_z) \right) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\
+I_{d,\mathrm{Rc}}(q_z) & = \frac{ I_d(q_{z}) }{|Tc|^2 \frac{w}{1-w} + |Rc|^2}
 \end{align}
 </math>

Difference between revisions of "Talk:DWBA"

Latest revision as of 09:21, 5 April 2018

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DWBA Equation in thin film

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