Phi Dependent Components

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]\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]

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]

Checking on the sign from the cosine results for [math]\displaystyle{ \phi '_2 }[/math]

We have the limiting range that [math]\displaystyle{ \phi }[/math] must fall within:

[math]\displaystyle{ -\pi \le \phi '_2 \le \pi\ Radians }[/math]

Examining the signs of the components which make up the angle [math]\displaystyle{ \phi }[/math] in the 4 quadrants which make up the xy plane:

[math]\displaystyle{ For\ 0 \ge \phi '_2 \ge \frac{-\pi}{2}\ Radians }[/math]

px=POSITIVE

py=NEGATIVE

[math]\displaystyle{ For\ 0 \le \phi '_2 \le \frac{\pi}{2}\ Radians }[/math]

px=POSITIVE

py=POSITIVE

[math]\displaystyle{ For\ \frac{-\pi}{2} \ge \phi '_2 \ge -\pi\ Radians }[/math]

px=NEGATIVE

py=NEGATIVE

[math]\displaystyle{ For\ \frac{\pi}{2} \le \phi '_2 \le \pi\ Radians }[/math]

px=NEGATIVE

py=POSITIVE