Difference between revisions of "Differential Cross-Section"

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<center><math>\frac{-2s}{t}=\frac{8E^{*2}}{2p^{*2}\left(1-\cos{\theta}\right)}=\frac{4E^{*2}}{p^{*2}\left(1-\cos{\theta}\right)}</math></center>
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<center><math>\frac{-2s}{t}=\frac{8E^{*2}}{2p^{*2}\left(1-\cos{\theta}\right)}=\frac{4E^{*2}}{p^{*2}\left(1-\cos{\theta}\right)}=\frac{4E^{*2}\left(1+\cos{\theta}\right)}{p^{*2}\left(1-\cos{\theta}\right)\left(1+\cos{\theta}\right)}</math></center>
  
  
  
<center><math>\frac{-2s}{u}=\frac{8E^{*2}}{2p^{*2}\left(1+\cos{\theta}\right)}=\frac{4E^{*2}}{p^{*2}\left(1+\cos{\theta}\right)}</math></center>
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<center><math>\frac{-2s}{u}=\frac{8E^{*2}}{2p^{*2}\left(1+\cos{\theta}\right)}=\frac{4E^{*2}}{p^{*2}\left(1+\cos{\theta}\right)}=\frac{4E^{*2}\left(1-\cos{\theta}\right)}{p^{*2}\left(1+\cos{\theta}\right)\left(1-\cos{\theta}\right)}</math></center>
  
  

Revision as of 22:10, 31 December 2018

[math]\underline{\textbf{Navigation}}[/math]

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Differential Cross-Section

[math]\frac{d\sigma}{d\Omega}=\frac{1}{64\pi ^2 s}\frac{\mathbf p_{final}}{\mathbf p_{initial}} |\mathfrak{M} |^2[/math]


Working in the center of mass frame

[math]\mathbf p_{final}=\mathbf p_{initial}[/math]


Determining the scattering amplitude in the center of mass frame


[math]\mathfrak{M}=e^2 \left ( \frac{u-s}{t}+\frac{t-s}{u} \right )[/math]


[math]\mathfrak{M}^2=e^4 \left ( \frac{u-s}{t}+\frac{t-s}{u} \right )\left ( \frac{u-s}{t}+\frac{t-s}{u} \right )[/math]


[math]\mathfrak{M}^2=e^4 \left ( \frac{(u-s)^2}{t^2}+\frac{(t-s)^2}{u^2} +2\frac{(u-s)}{t}\frac{(t-s)}{u}\right )[/math]


[math]\mathfrak{M}^2=e^4 \left ( \frac{(u^2-2us+s^2)}{t^2}+\frac{(t^2-2ts+s^2)}{u^2} +2\frac{(ut-st+s^2-us)}{tu}\right )[/math]


[math]\mathfrak{M}^2=e^4 \left ( \frac{(t^2+s^2)}{u^2}+\frac{2s^2}{tu}+2-\frac{2us}{t^2}-\frac{2s}{t}-\frac{2ts}{u^2}-\frac{2s}{u}+\frac{(u^2+s^2)}{t^2}\right )[/math]


Using the fine structure constant ([math]with\ c=\hbar=\epsilon_0=1[/math])

[math]\alpha \equiv \frac{e^2}{4\pi}[/math]


[math]\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{2s}\left ( \frac{(t^2+s^2)}{u^2}+\frac{2s^2}{tu}+2-\frac{2us}{t^2}-\frac{2s}{t}-\frac{2ts}{u^2}-\frac{2s}{u}+\frac{(u^2+s^2)}{t^2}\right )[/math]


In the center of mass frame the Mandelstam variables are given by:

[math]s \equiv 4E^{*2}[/math]


[math]t \equiv -2p^{*2}(1-\cos{\theta})[/math]



[math]u \equiv -2p^{*2}(1+\cos{\theta})[/math]


Calculating the parts:

[math]\frac{t^2}{u^2}=\frac{4p^{*2}\left(1-\cos{\theta}\right)^2}{4p^{*2}\left(1+\cos{\theta}\right)^2}=\tan^4{\frac{\theta}{2}}[/math]


[math]\frac{s^2}{u^2}=\frac{16E^{*4}}{4p^{*4}\left(1+\cos{\theta}\right)^2}=\frac{E^{*4}\sec^4{\frac{\theta}{2}}}{p^{*4}}[/math]


[math]\frac{u^2}{t^2}=\frac{4p^{*2}\left(1+\cos{\theta}\right)^2}{4p^{*2}\left(1-\cos{\theta}\right)^2}=\cot^4{\frac{\theta}{2}}[/math]


[math]\frac{s^2}{t^2}=\frac{16E^{*4}}{4p^{*4}\left(1-\cos{\theta}\right)^2}=\frac{E^{*4}\csc^4{\frac{\theta}{2}}}{p^{*4}}[/math]


[math]\frac{2s^2}{tu}=\frac{32E^{*4}}{4p^{*4}\left(1+\cos{\theta}\right)\left(1-\cos{\theta}\right)}=\frac{8E^{*4}}{p^{*4}\sin^2{\theta}}[/math]


[math]\frac{-2s}{t}=\frac{8E^{*2}}{2p^{*2}\left(1-\cos{\theta}\right)}=\frac{4E^{*2}}{p^{*2}\left(1-\cos{\theta}\right)}=\frac{4E^{*2}\left(1+\cos{\theta}\right)}{p^{*2}\left(1-\cos{\theta}\right)\left(1+\cos{\theta}\right)}[/math]


[math]\frac{-2s}{u}=\frac{8E^{*2}}{2p^{*2}\left(1+\cos{\theta}\right)}=\frac{4E^{*2}}{p^{*2}\left(1+\cos{\theta}\right)}=\frac{4E^{*2}\left(1-\cos{\theta}\right)}{p^{*2}\left(1+\cos{\theta}\right)\left(1-\cos{\theta}\right)}[/math]


[math]\frac{-2ts}{u^2}=\frac{4p^{*2}\left(1-\cos{\theta}\right)4E^{*2}}{4p^{*2}\left(1+\cos{\theta}\right)^2}=\frac{E^{*2}\left(1-\cos{\theta}\right)\sec^4{\frac{\theta}{2}}}{p^{*2}}[/math]


[math]\frac{-2us}{t^2}=\frac{4p^{*2}\left(1+\cos{\theta}\right)4E^{*2}}{4p^{*2}\left(1-\cos{\theta}\right)^2}=\frac{E^{*2}\left(1+\cos{\theta}\right)\csc^4{\frac{\theta}{2}}}{p^{*2}}[/math]

[math]\underline{\textbf{Navigation}}[/math]

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