Available energy (particle collision) - Alchetron, the free social encyclopedia

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 = 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

This derivation will use the fact that:

( m c 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 + 2 E 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 + 2 E t E k

Square rooting both sides and we get

E a = ( m t c 2 ) 2 + ( m k c 2 ) 2 + 2 E t E k

References

Available energy (particle collision) Wikipedia

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