I'm a programmer trying to simulate the movement of the particles involved in $\beta^-$ decay, or at least an approximation of it, for fun, in a 2D universe. I would like to keep the simulation realistic in some ways though, so I would like to obey conservation of energy and conservation of momentum.

So far I have calculated the total energy and momentum of the original neutron (and checked to ensure there is enough energy for $\beta^-$ decay to occur). I have made the proton continue along the original trajectory of the neutron at the same velocity, and created a neutrino traveling at near light speed (trajectory as yet undecided). Then I subtracted the momentum of the proton from the original momentum (per axis), leaving me with the available momentum to distribute between the electron and neutrino (per axis). Then I took the total energy and subtracted from it the total energies of the proton and neutrino and the mass-energy of the electron, leaving me with the kinetic energy of the electron, which I then used to find its velocity (sans trajectory).

This leaves me with a system of equations, $$ |v_n|^2 = \mathbf{v_{xn}}^2 + \mathbf{v_{yn}}^2 $$ $$ |v_e|^2 = \mathbf{v_{xe}}^2 + \mathbf{v_{ye}}^2 $$ $$ p_x = m_e \mathbf{v_{xe}} + m_n \mathbf{v_{xn}} $$ $$ p_y = m_e \mathbf{v_{ye}} + m_n \mathbf{v_{yn}} $$ where $_e$ and $_n$ stand for electron and neutrino, and $T$ stands for kinetic energy. All unknowns are bolded (4 total).

  1. Are there infinite solutions?
  2. Given that, how can I pick a solution at random?

1 Answer 1


Yes there are an infinite number of solutions.

Indeed, this is how the neutrino was discovered. If a neutron simply decayed to a proton and an electron then the electron KE would always be the same fixed value. However when we measure the electron energy in beta decay we find there is a spread of energies. Wikipedia shows a typical spectrum of electron energies in beta decay:

Beta decay

This can only happen if there is some third body involved, and it was for that reason that Pauli proposed a neutrino was also produced in beta decay.

  • $\begingroup$ Interesting, thanks! I changed my question slightly and added equations. Do you have an answer for question #2? $\endgroup$
    – Ken
    Sep 5, 2018 at 20:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.