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Inelastic collisions

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I'll give two separate derivations. In case of the collision:

P + P ---> P + n + pi^{+}

we can exploit the symmetry w.r.t. interchanging the two protons in the center of mass frame.

In case of the collision

e + P ---> e + P + P + P^{-}

we have to derive a general formula.

In both cases I'll only derive the formulas, you'll have to look up the masses of the particles and plug that in the formulas to find the answers. Also, I'll put c = 1 everywhere.

In both cases we can reason as follows. To find the threshold kinetic energy we must determine how low the total energy of the particles can be. We can't just say that the total energy at threshold equals the sum of the rest masses of the particles on the right hand side of the reaction, because if that energy is to be delivered partially in the form of kinetic energy of one particle in the Lab frame, then the total momentum is not zero before the collision, so after the collision the total momentum must be nonzero as well, hence the particles must have nonzero kinetic energy after the collision.

To deal with this complication we first consider the collision in the center of mass frame. In the center of mass frame the total momentum is zero. The particles after the collision will each have some energy and some velocity, but the total momentum of all the particles added together is zero. How low can the total energy in the center of mass frame be? Clearly, the slower the particles move after the collision, the lower the energy is. Momentum must be conserved as well, but because the total momentum is zero, we can make the velocity of each particle after the collision zero. The total ...

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