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}

Denoting 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}}}

References

External links

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Available energy (particle collision)

Derivation

See also

References

External links