# Available energy (particle collision)

For a discussion about the meaning of the term 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:

$E_a = \sqrt{2 E_t E_k + (m_t c^2)^2 + (m_k c^2)^2}$

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

Suppose the following:

$E_a = E_k + E_t$

We know that since

$E = m c^2$

must hold, then

$E_k = m_k c^2$

and

$E_t = m_t c^2$

should both hold. From here, we can see that

$E_a = m_k c^2 + m_t c^2$

Squaring both sides, we get

$(E_a)^2 = (m_k c^2 + m_t c^2)^2$

Expanding, we can see that

$(E_a)^2 = ((m_k c^2)^2 + (m_t c^2)^2 + 2 (m_t c^2) (m_k c^2))$

Taking the square root of both sides,

$E_a = \sqrt{(m_k c^2)^2 + (m_t c^2)^2 + 2 (m_t c^2) (m_k c^2)}$

Evaluating the inside of the square root, we see that

$E_a = \sqrt{(m_k c^2)^2 + (m_t c^2)^2 + 2 E_k E_t}$

Which can be rearranged to form the original equation,

$E_a = \sqrt{2 E_t E_k + (m_t c^2)^2 + (m_k c^2)^2}$