Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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1550002     Kinetic energy of particles resulting from the decay of a neutron.     1550002
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Question

 

A free neutron (mass ) is highly unstable, meaning that it won't exist for very long before it decays. When it decays, it decays into a proton (mass , an electron (mass , and an antineutrino.

  1. If a neutron at rest decays, what will be the total kinetic energy of the proton, electron, and antineutrino when they are very far apart? An antineutrino has no rest energy (or actually it has negligible rest energy).
  2. Why is it impossible for all three particles to travel in the same direction after the decay?

 



Solution

 

(a) Sketch a picture showing the system before and after the decay.


Figure: The final momentum of a proton.

Define the system. Before the decay, the system is a neutron at rest. After the decay, the system is a proton, an electron, and an antineutrino. There are no energy inputs or outputs from the system; it is a closed system. Assume that the particles after the decay are very far apart from each other. In other words, treat them as "free" non-interacting particles after the decay that only have particle energy ( ).

The initial energy of the system is

After the decay, the particles are very far apart. Therefore, there is no interaction energy and the final energy of the system is the sum of the energies of the particles. Note that an antineutrino has negligible rest energy.

Now, apply the Energy Principle and solve for the total kinetic energy of the system. Since there are no energy inputs or outputs

Substitute the rest energies of the particles. The units of for mass are quite convenient because multiplying by c-squared gives the rest energy of the particle.

The kinetic energy of the system is positive, as it should be since kinetic energy cannot be negative. Basically, the loss of rest energy in the system resulted in a gain in kinetic energy. All of the figures shown are significant because of the precision with which the rest energies are known.

(b) The Energy Principle does not tell us about direction. Though we can calculate the kinetic energy of the system after the decay, we do not know anything about the directions of the particles' momenta. To answer this question about their directions, we must employ the Momentum Principle. There are no external forces acting on the system. Thus, the net force is zero, and the momentum of the system is constant, as shown below.

In this situation, the neutron is at rest before the decay. Thus, the momentum of the system is zero. This means that the momentum of the system after the collision must also be zero. We don't know the direction in which the particles necessarily travel; however, we do know what at least one particle has to go in the opposite direction as the other two so that the sum of the momenta of two of the particles will be equal in magnitude, opposite in direction to the momentum of the third particle. An example is shown below. Note that the sum of the momenta of the electron and antineutrino add up to be the same in magnitude as the momentum of the proton. This way, the sum of the momenta of all three particles is zero, which is equal to the momentum of the neutron before the decay.


Figure: One possible outcome showing the final momenta of the three particles after the decay.