Since only the z direction is considered to be the relativistic direction of motion, this implies that the x and y components are not effected by a Lorentz transformation and remain the same in the CM and Lab frame. Holding the angle Phi constant at an initial value of 10 degrees, allows us to find the x and y components.
Figure 4: Definition of Moller electron variables in the Lab Frame in the x-y plane.
Similarly, [math]\phi '_2=\arccos \left( \frac{p^'_{2(x) Lab}}{p^'_{2(xy)}} \right)[/math]
where [math]p_{2(xy)}^'=\sqrt{(p_{2(x)}^')^2+(p^'_{2(y)})^2}[/math]
[math](p^'_{2(xy)})^2=(p^'_{2(x)})^2+(p^'_{2(y)})^2[/math]
and using [math]p^2=p_{(x)}^2+p_{(y)}^2+p_{(z)}^2[/math]
this gives [math](p^'_{2})^2=(p^'_{2(xy)})^2+(p^'_{2(z)})^2[/math]
[math]\Longrightarrow (p'_{2})^2-(p'_{2(z)})^2 = (p'_{2(xy)})^2[/math]
[math]\Longrightarrow p_{2(xy)}^'=\sqrt{(p^'_{2})^2-(p^'_{2(z)})^2}[/math]
which gives[math]\phi '_2 = \arccos \left( \frac{p_{2(x)}'}{\sqrt{p_{2}^{'\ 2}-p_{2(z)}^{'\ 2}}}\right)[/math]
| [math]\Longrightarrow p_{2(x)}'=\sqrt{p_{2}^{'\ 2}-p_{2(z)}^{'\ 2}} \cos(\phi)[/math]
|
Similarly, using [math]p_{2}^2=p_{2(x)}^2+p_{2(y)}^2+p_{2(z)}^2[/math]
[math]\Longrightarrow p_{2}^{'\ 2}-p_{2(x)}^{'\ 2}-p_{2(z)}^{'\ 2}=p_{2(y)}^{'\ 2}[/math]
| [math]p_{2(y)}'=\sqrt{p_{2}^{'\ 2}-p_{2(x)}^{'\ 2}-p_{2(z)}^{'\ 2}}[/math]
|
Checking on the sign from the cosine results for [math]\phi '_2[/math]
We have the limiting range that [math]\phi[/math] must fall within:
| [math]-\pi \le \phi '_2 \le \pi\ Radians[/math]
|
Examining the signs of the components which make up the angle [math]\phi[/math] in the 4 quadrants which make up the xy plane:
| [math]For\ 0 \ge \phi '_2 \ge \frac{-\pi}{2}\ Radians[/math]
|
| px=POSITIVE
|
| py=NEGATIVE
|
| [math]For\ 0 \le \phi '_2 \le \frac{\pi}{2}\ Radians[/math]
|
| px=POSITIVE
|
| py=POSITIVE
|
| [math]For\ \frac{-\pi}{2} \ge \phi '_2 \ge -\pi\ Radians[/math]
|
| px=NEGATIVE
|
| py=NEGATIVE
|
| [math]For\ \frac{\pi}{2} \le \phi '_2 \le \pi\ Radians[/math]
|
| px=NEGATIVE
|
| py=POSITIVE
|
