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Final Lab Frame Moller Electron 4-momentum components in XY Plane
Figure 4: Definition of Moller electron variables in the Lab Frame in the x-y plane.
Similarly, [math]\displaystyle{ \phi '_2=\arccos \left( \frac{p^'_{2(x) Lab}}{p^'_{2(xy)}} \right) }[/math]
where [math]\displaystyle{ p_{2(xy)}^'=\sqrt{(p_{2(x)}^')^2+(p^'_{2(y)})^2} }[/math]
[math]\displaystyle{ (p^'_{2(xy)})^2=(p^'_{2(x)})^2+(p^'_{2(y)})^2 }[/math]
and using [math]\displaystyle{ p^2=p_{(x)}^2+p_{(y)}^2+p_{(z)}^2 }[/math]
this gives [math]\displaystyle{ (p^'_{2})^2=(p^'_{2(xy)})^2+(p^'_{2(z)})^2 }[/math]
[math]\displaystyle{ \Longrightarrow (p'_{2})^2-(p'_{2(z)})^2 = (p'_{2(xy)})^2 }[/math]
[math]\displaystyle{ \Longrightarrow p_{2(xy)}^'=\sqrt{(p^'_{2})^2-(p^'_{2(z)})^2} }[/math]
which gives[math]\displaystyle{ \phi '_2 = \arccos \left( \frac{p_{2(x)}'}{\sqrt{p_{2}^{'\ 2}-p_{2(z)}^{'\ 2}}}\right) }[/math]
| [math]\displaystyle{ \Longrightarrow p_{2(x)}'=\sqrt{p_{2}^{'\ 2}-p_{2(z)}^{'\ 2}} \cos(\phi) }[/math]
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Similarly, using [math]\displaystyle{ p_{2}^2=p_{2(x)}^2+p_{2(y)}^2+p_{2(z)}^2 }[/math]
[math]\displaystyle{ \Longrightarrow p_{2}^{'\ 2}-p_{2(x)}^{'\ 2}-p_{2(z)}^{'\ 2}=p_{2(y)}^{'\ 2} }[/math]
| [math]\displaystyle{ p_{2(y)}'=\sqrt{p_{2}^{'\ 2}-p_{2(x)}^{'\ 2}-p_{2(z)}^{'\ 2}} }[/math]
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