Suppose you want to make a muon-antimuon pair (μ) by colliding and annihilating a positron and an electron.

(1) |
---|

The (muon) μ particles are each about 200 times the mass of an electron, so most of the kinetic energy of the electron and positron must be used to create them.

The positron e^{+} could be fired
at a stationary electron (in practice the electron would be bound in an
atom with a kinetic energy of a few eV, which is negligible compared with
the high energies involved in Particle Physics).

The incident positron (e^{+}) has
momentum p_{e} and total energy
E_{e} and collides with a stationary
electron (e^{−}) thereby annihilating
to produce a 'virtual photon'. The virtual photon materialises into
a new e^{+} and
e^{−} pair or any other heavier particle-antiparticle pair,
such as μ^{+} and
μ^{−}, provided sufficient energy
is available. The μ^{+} and
μ^{−} pair must have total momentum
equal to p_{e} in order to conserve
momentum in the collision.

Combining the relativistic energy and the momentum conservation equations
yields the equation below:

(2) |
---|

This is the total energy of the positron at which particles are produced with the least possible momentum, so it is, therefore, the threshold energy.

**Put the masses into this equation and calculate (use Excel etc. or
a calculator) the threshold total energy for pair production by positrons
hitting a stationary target:**

Particle |
Mass in MeV/c^{2} |
Colour Key |

electron (e^{+}/e^{−}) |
0.511 | Blue/Red |

muon (μ^{+}/μ^{−}) |
105.7 | Green/Dark Green |

pion (π^{+}/π^{−}) |
139.6 | Pink/Purple |

kaon (K^{+}/K^{−}) |
493.7 | Yellow/Orange |

proton (p^{+}/p^{−}) |
938.3 | Grey/Black |

The actual type of particle-antiparticle pair created is random once you are above the threshold energy. Thus you might simply create a new electron-positron pair even though you have enough energy to make muons. You need to repeat the experiment a few times before increasing the beam energy.

**Now run the animation and test your prediction for the threshold energy.**