Limit of Scattering Angle Theta in Lab Frame
The quantity is known as the
In the CM Frame
where and is the angle between the before and after momentum in the CM frame
Using the relativistic relationthis reduces to
There is no scattering, or no momentum transfer at 90 degrees since the incident momentum direction is the same as the scattered momentum direction. However, at a certain angle enough momentum must be transferred to provide the ionization energy to create a Moller electron.
For, by symmetry this implies
This can be rewritten again using the relativistic energy relation
In the Lab Frame
Minimum Moller Scattering Angle Theta in Lab Frame
Since u is invariant between frames
As found earlier, the Moller electron has a maximum energy possible of:
Using the relativistic energy relation,
Rewriting the expression relating the terms
Solving for the angle theta
The Moller electron is traveling near, but not on the beamline.