# Available energy (particle collision)

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:

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

where

${\displaystyle E_{t}}$ is the total energy of the target particle,
${\displaystyle E_{k}}$ is the total energy of the moving particle,
${\displaystyle m_{t}}$ is the mass of the stationary target particle,
${\displaystyle m_{k}}$ is the mass of the moving particle, and
${\displaystyle c}$ is the speed of light.

## Derivation

This derivation will use the fact that:

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

From the principle of the conservation of linear momentum:

${\displaystyle P_{a}=P_{k}}$

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

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

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

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

Denoting this last equation (1). But

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

and since the stationary particle has no momentum

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

Therefore from (1) we have

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

Square rooting both sides and we get

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