Difference between revisions of "Talk:DWBA"

Revision as of 09:09, 13 March 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:

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

Breaking into components

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

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

We define the ratio between the channels to be:

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

Such that one can compute the two components from:

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

and:

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

@@ Line 176: / Line 176: @@
 ==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,Tc}(qz)</math> and reflected channel <math>I_{d,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,Tc}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(q_z-\Delta q_{z,Rc}) \\
+  & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{d,Tc}(q_z) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{d,Rc}(q_z) \\
+   & = |Tc|^2 I_{d,Tc}(q_z) + |Rc|^2 I_{d,Rc}(q_z) \\
 \end{align}
 </math>
@@ Line 191: / Line 193: @@
 \begin{align}
 w
-   & = \frac{ I_{Tc}(q_z) }{ I_{Tc}(q_z) + I_{Rc}(q_z) }
+   & = \frac{ I_{d,Tc}(q_z) }{ I_{d,Tc}(q_z) + I_{d,Rc}(q_z) }
 \end{align}
 </math>
@@ Line 199: / Line 201: @@
 <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,Tc}(q_z) ) + |Rc|^2 ( I_{d,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,Tc}(q_z) ) + |Rc|^2 \left ( \frac{ I_{d,Tc}(q_z) - w I_{d,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,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,Tc}(q_z)  & = \frac{ I_d(q_{z}) }{  |Tc|^2  +  \frac{ |Rc|^2 }{w}  - |Rc|^2  } \\
 \end{align}
 </math>
@@ Line 211: / Line 213: @@
 <math>
 \begin{align}
-I_{Rc}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{Tc}(q_z) }{|Rc|^2}
+I_{d,Rc}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{d,Tc}(q_z) }{|Rc|^2}
 \end{align}
 </math>

Difference between revisions of "Talk:DWBA"

Revision as of 09:09, 13 March 2018

Contents

DWBA Equation in thin film

Expansion (incorrect)

Terms

Equation

Simplification

Expansion

Terms

Equation

Breaking into components

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