# What Are the Threshold Kinetic Energies in Particle Decay Reactions?

• nv125
In summary, the conversation discusses the decay of a particle and finding the threshold kinetic energies of the incident particles in two different reactions. The teacher suggests using equations such as ƩPiu=0 and PuPu=m2c2= E2/C2-P2 to solve the problem. The student has attempted to solve for part i, but is unsure how to continue, as having all three final state particles at rest is impossible due to conservation of momentum.
nv125

## Homework Statement

Consider the decay of a particle.

(a) Find the threshold kinetic energies of the incident particles in each of
the following reactions:

(i) ∏+ + p →∏++ ∏° + p
(ii) p + n →p + n + ∏° + ∏°

The incident particle is the first particle listed in the reactions; the target particle is at rest. The masses of the ∏+ and ∏° ions are 139.567MeV/c2 and 134.963 MeV/c2 respectively while the masses of the neutron and proton are respectively 939.566 MeV/c2 and 938.272 MeV/c2.

## Homework Equations

The teacher suggested that I will need to use the following equations to get the solution
(Here P is the momentum and Px is the momentum of the incident particle ∏+in the X direction, assuming it hits the target particle p(proton) horizontally, so no P(momentum) in y or z direction)

ƩPiu=0

PuPu=m2c2= E2/C2-P2

Also said that using this notations will be useful to solve the problem

For ∏+, the incident particle:
Pu=(E/C, Px, 0, 0)
Pu=(E/C, -Px, 0, 0) where Px is the momentum of the incident particle, and no need to worry about y and z direction so just put zeros.

## The Attempt at a Solution

This is what I have done so far to solve for part i, but I have no clue how to continue.
Since we are solving for the threshold kinetic energy, so assuming that ∏+, ∏° and p are at rest after the reaction.

(E/C, Px, 0, 0) +(mpC2, 0, 0, 0) - (m+C2,0, 0, 0 ) -(m∏°C2,0, 0, 0 ) - (mpC2,0, 0, 0 ) = 0

mp, m+, m∏° are just the rest mass.

nv125 said:
Since we are solving for the threshold kinetic energy, so assuming that ∏+, ∏° and p are at rest after the reaction.

(E/C, Px, 0, 0) +(mpC2, 0, 0, 0) - (m+C2,0, 0, 0 ) -(m∏°C2,0, 0, 0 ) - (mpC2,0, 0, 0 ) = 0

As you see, trying to have all 3 final state particles at rest is impossible due to conservation of momentum. You need to allow for some final momenta and then think about what being at threshold implies.

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Electrodynamics is the study of the interactions between electric and magnetic fields, while relativity is the study of how space and time are affected by the presence of massive objects. While both fields deal with fundamental physical phenomena, they have different focuses and mathematical frameworks.

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