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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?
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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.

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  • $\begingroup$ Interesting, thanks! I changed my question slightly and added equations. Do you have an answer for question #2? $\endgroup$ – Ken Sep 5 '18 at 20:09

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