Difference between revisions of "Talk:Polarization correction"

From GISAXS
Jump to: navigation, search
(Created page with "{| class="wikitable" |- ! Angle ! Adjacent ! Opposite ! Hypotenuse ! Sine = O/H ! Cosine = A/H ! Tangent = O/A |- | <math>\chi</math> | <math>z</math> | <math>x</math> | <math...")
 
 
(5 intermediate revisions by the same user not shown)
Line 2: Line 2:
 
|-
 
|-
 
! Angle
 
! Angle
 +
|
 
! Adjacent
 
! Adjacent
 
! Opposite
 
! Opposite
Line 10: Line 11:
 
|-
 
|-
 
| <math>\chi</math>
 
| <math>\chi</math>
 +
| Azimuth on detector <br/> (relative to <math>q_z</math> axis)
 
| <math>z</math>
 
| <math>z</math>
 
| <math>x</math>
 
| <math>x</math>
Line 18: Line 20:
 
|-
 
|-
 
| <math>2\theta</math>
 
| <math>2\theta</math>
 +
| Full scattering angle <br/> (between incident beam and scattering)
 
| <math>y</math>
 
| <math>y</math>
 
| <math>r = \sqrt{x^2 + z^2}</math>
 
| <math>r = \sqrt{x^2 + z^2}</math>
Line 26: Line 29:
 
|-
 
|-
 
| <math>\gamma</math>
 
| <math>\gamma</math>
 +
| In-plane angle
 
| <math>y</math>
 
| <math>y</math>
 
| <math>x</math>
 
| <math>x</math>
Line 32: Line 36:
 
| <math>\cos \gamma = \frac{y}{h}</math>
 
| <math>\cos \gamma = \frac{y}{h}</math>
 
| <math>\tan \gamma = \frac{x}{y}</math>
 
| <math>\tan \gamma = \frac{x}{y}</math>
 +
|-
 +
| <math>\delta</math>
 +
| Elevation angle
 +
| <math>h = \sqrt{x^2 + y^2}</math>
 +
| <math>z</math>
 +
| <math>R = \sqrt{x^2 + y^2 + z^2}</math>
 +
| <math>\sin \delta = \frac{z}{R}</math>
 +
| <math>\cos \delta = \frac{h}{R}</math>
 +
| <math>\tan \delta = \frac{z}{h}</math>
 
|}
 
|}
 +
 +
 +
So:
 +
:<math>
 +
\begin{alignat}{2}
 +
\tan \gamma & = \frac{x}{y} \\
 +
    & = \frac{r \sin \chi}{r / \tan 2 \theta} \\
 +
    & = \tan 2 \theta \sin \chi \\
 +
\gamma & = \tan^{-1} \left [ \tan 2 \theta \sin \chi \right ]
 +
 +
\end{alignat}
 +
</math>
 +
and:
 +
:<math>
 +
\begin{alignat}{2}
 +
\sin \delta & = \frac{z}{R} \\
 +
    & = \frac{r \cos \chi}{r / \sin 2 \theta} \\
 +
    & = \sin 2 \theta \cos \chi \\
 +
\delta & = \sin^{-1} \left [ \sin 2 \theta \cos \chi \right ]
 +
 +
\end{alignat}
 +
</math>
 +
 +
and so:
 +
:<math>
 +
\begin{alignat}{2}
 +
P_h & = 1 - \cos^2 \delta \sin^2 \gamma \\
 +
    & = 1 - \left( 1 - \left[ \sin 2  \theta \cos \chi \right]^2 \right ) \frac{ \left [ \tan 2 \theta \sin \chi \right ]^2 }{ \left [ \tan 2 \theta \sin \chi \right ]^2 + 1 } \\
 +
    & = 1 - \sin^2(2 \theta) \sin^2(\chi)
 +
\end{alignat}
 +
</math>

Latest revision as of 21:58, 22 November 2019

Angle Adjacent Opposite Hypotenuse Sine = O/H Cosine = A/H Tangent = O/A
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 \chi} Azimuth on detector
(relative 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 q_z} axis) 		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 z} 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 x} 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 r = \sqrt{x^2 + z^2}} 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 \sin \chi = \frac{x}{r}} 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 \cos \chi = \frac{z}{r}} 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 \tan \chi = \frac{x}{z}}
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\theta} Full scattering angle
(between incident beam and scattering) 		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 y} 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 r = \sqrt{x^2 + z^2}} 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 R = \sqrt{x^2 + y^2 + z^2}} 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 \sin 2\theta = \frac{r}{R}} 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 \cos 2\theta = \frac{y}{R}} 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 \tan 2\theta = \frac{r}{y}}
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 \gamma} In-plane angle 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 y} 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 x} 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 h = \sqrt{x^2 + y^2}} 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 \sin \gamma = \frac{x}{h}} 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 \cos \gamma = \frac{y}{h}} 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 \tan \gamma = \frac{x}{y}}
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 \delta} Elevation angle 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 h = \sqrt{x^2 + y^2}} 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 z} 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 R = \sqrt{x^2 + y^2 + z^2}} 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 \sin \delta = \frac{z}{R}} 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 \cos \delta = \frac{h}{R}} 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 \tan \delta = \frac{z}{h}}


So:

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{alignat}{2} \tan \gamma & = \frac{x}{y} \\ & = \frac{r \sin \chi}{r / \tan 2 \theta} \\ & = \tan 2 \theta \sin \chi \\ \gamma & = \tan^{-1} \left [ \tan 2 \theta \sin \chi \right ] \end{alignat} }

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{alignat}{2} \sin \delta & = \frac{z}{R} \\ & = \frac{r \cos \chi}{r / \sin 2 \theta} \\ & = \sin 2 \theta \cos \chi \\ \delta & = \sin^{-1} \left [ \sin 2 \theta \cos \chi \right ] \end{alignat} }

and so:

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{alignat}{2} P_h & = 1 - \cos^2 \delta \sin^2 \gamma \\ & = 1 - \left( 1 - \left[ \sin 2 \theta \cos \chi \right]^2 \right ) \frac{ \left [ \tan 2 \theta \sin \chi \right ]^2 }{ \left [ \tan 2 \theta \sin \chi \right ]^2 + 1 } \\ & = 1 - \sin^2(2 \theta) \sin^2(\chi) \end{alignat} }
Retrieved from "http://gisaxs.com/index.php?title=Talk:Polarization_correction&oldid=6008"