Threshold energy

In particle physics, the threshold energy for production of a particle is the minimum kinetic energy a pair of traveling particles must have when they collide. The threshold energy is always greater than or equal to the rest energy of the desired particle. In most cases, since momentum is also conserved, the threshold energy is significantly greater than the rest energy of the desired particle - and thus there will still be considerable kinetic energy in the final particles.

Example

Look at the reaction of a proton hitting a stationary proton, $p + p \to p + p + \pi^0$.

By going into the center of mass frame, and assuming the outgoing particles have no kinetic energy the conservation of energy equation is:

$E = 2\gamma m_pc^2 = 2 m_pc^2+ m_\pi c^2$

$\gamma = \frac{1}{\sqrt{1-\beta^2}} = \frac{2 m_pc^2+ m_\pi c^2}{2 m_pc^2}$

$\beta^2 = 1-(\frac{2 m_pc^2}{2 m_pc^2+ m_\pi c^2})^2 \approx 0.13$

Using relativistic velocity additions:

$v_{lab} = \frac{u_{cm} + v_{cm}}{1+u_{cm}v_{cm}/c^2} \approx 0.64 c$

So the energy of the proton must be $E = \gamma m_p c^2 = \frac{m_p c^2}{\sqrt{1-\beta^2}} = 1221 MeV$

See also

• Threshold displacement energy

References

• http://galileo.phys.virginia.edu/classes/252/particle_creation.html