# Limit of Energy in Lab Frame

The t quantity is known as the square of the 4-momentum transfer

# In the CM Frame

where and is the angle between the before and after momentum in the CM frame

Using the relativistic relation this reduces to

There is no scattering, or no momentum transfer at 0 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.

The maximum momentum is transfered at 90 degrees, i.e.

This can be rewritten again using the relativistic energy relation

The maximum momentum is transfered at 180 degrees, i.e.

This can be rewritten again using the relativistic energy relation

with

and

# Maximum Moller Energy in Lab Frame

Since t is invariant between frames

with for

The Moller electron has a maximum energy possible of:

# Minimum Moller Energy in Lab Frame

Since t is invariant between frames

This implies that the Moller electron has a non-zero momentum, hence it's total energy is more than it's rest mass energy. The momentum that the Moller electron would have would have to be transfered from the incident electron to the "stationary" electron bound to the detector. The binding energy of an electron bound to a hydrogen atom is 13.6 eV

At