Available energy (particle collision)

For the concept in classical thermodynamics, see Exergy.

In particle physics, the available energy is the energy in a particle collision available to produce new matter from the kinetic energy of the colliding particles. Since the conservation of momentum must be held, a system of two particles with a net momentum may not convert all their kinetic energy into mass - and thus the available energy is always less than or equal to the kinetic energy of the colliding particles. The available energy for a system of one stationary particle and one moving particle is defined as:

$E_{a}={\sqrt {2E_{t}E_{k}+(m_{t}c^{2})^{2}+(m_{k}c^{2})^{2}}}$

where

$E_{t}$ is the total energy of the target particle,

$E_{k}$ is the total energy of the moving particle,

$m_{t}$ is the mass of the stationary target particle,

$m_{k}$ is the mass of the moving particle, and

$c$ is the speed of light.

Derivation

This derivation will use the fact that:

$(mc^{2})^{2}=E^{2}-P^{2}c^{2}$

From the principle of the conservation of linear momentum:

$P_{a}=P_{k}$

Where $P_{a}$ and $P_{k}$ are the momentums of the created and the initially moving particle respectively. From the conservation of energy:

$E_{T}=E_{t}+E_{k}$

Where $E_{T}$ is the total energy of the created particle. We know that after the collision:

$(E_{a})^{2}=(E_{T})^{2}-(P_{a})^{2}c^{2}$

$(E_{a})^{2}=(E_{t}+E_{k})^{2}-(P_{k})^{2}c^{2}$

$(E_{a})^{2}=(E_{t})^{2}+(E_{k})^{2}+2E_{t}E_{k}-(P_{k})^{2}c^{2}$

Donating this last equation (1). But

$(m_{k})^{2}c^{4}=(E_{k})^{2}-(P_{k})^{2}c^{2}$

and since the stationary particle has no momentum

$(m_{t})^{2}c^{4}=(E_{t})^{2}$

Therefore from (1) we have

$(E_{a})^{2}=(m_{k})^{2}c^{4}+(m_{t})^{2}c^{4}+2E_{t}E_{k}$

Square rooting both sides and we get

$E_{a}={\sqrt {(m_{t}c^{2})^{2}+(m_{k}c^{2})^{2}+2E_{t}E_{k}}}$

See also

• Threshold energy
• Matter creation