How to simulate possible trajectories of particles after $\beta$ decay?
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I'm a programmer trying to simulate the movement of the particles involved in β−β− 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 β−β− 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,
|vn|2=vxn2+vyn2|vn|2=vxn2+vyn2
|ve|2=vxe2+vye2|ve|2=vxe2+vye2
px=mevxe+mnvxnpx=mevxe+mnvxn
py=mevye+mnvynpy=mevye+mnvyn
where ee and nn stand for electron and neutrino, and TT stands for kinetic energy. All unknowns are bolded (4 total).
Are there infinite solutions?
Given that, how can I pick a solution at random?
So far I have calculated the total energy and momentum of the original neutron (and checked to ensure there is enough energy for β−β− 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,
|vn|2=vxn2+vyn2|vn|2=vxn2+vyn2
|ve|2=vxe2+vye2|ve|2=vxe2+vye2
px=mevxe+mnvxnpx=mevxe+mnvxn
py=mevye+mnvynpy=mevye+mnvyn
where ee and nn stand for electron and neutrino, and TT stands for kinetic energy. All unknowns are bolded (4 total).
Are there infinite solutions?
Given that, how can I pick a solution at random?
Answered by
0
I'm a programmer trying to simulate the movement of the particles involved in 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 β- 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).
So far I have calculated the total energy and momentum of the original neutron (and checked to ensure there is enough energy for β- 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).
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