Energy distribution plot of neutrinos in beta decay

[Phys pilot]

TL;DR Summary

what's the shape and how I get the spectrum of the kinetic energy of the neutrinos?
Thanks

Hello,
When you have a beta decay you get the typical continuos spectrum representing counts against the kinetic energy of the electron. But what's the shape and how I get the spectrum of the kinetic energy of the neutrinos?
Thanks

 

[Astronuc]

It's a 3 body problem, since there is a nucleus, an electron (beta particle) and an anti-neutrino. Given that the nucleus is massive, A * ~931.49 MeV, the beta and anti-neutrino will get most of the energy.

One can approximate the shape of the anti-neutrino spectrum by assuming the total energy of the beta decay, e.g., E_β, and subtracting the beta energy spectrum. The anti-neutrino energy is approximately the total energy minus the beta energy. The most probably energy of a beta particle in the decay is about E_β/3, and corresponding anti-neutrino energy would be ~2/3 E_β.

 

[Phys pilot]

It's a 3 body problem, since there is a nucleus, an electron (beta particle) and an anti-neutrino. Given that the nucleus is massive, A * ~931.49 MeV, the beta and anti-neutrino will get most of the energy.

One can approximate the shape of the anti-neutrino spectrum by assuming the total energy of the beta decay, e.g., E_β, and subtracting the beta energy spectrum. The anti-neutrino energy is approximately the total energy minus the beta energy. The most probably energy of a beta particle in the decay is about E_β/3, and corresponding anti-neutrino energy would be ~2/3 E_β.

I see, so the spectrum would be also continuos and a similar shape?
Thanks

 

[Astronuc]

I see, so the spectrum would be also continuos and a similar shape?
Thanks

No, the shape for the neutrino would be continuous but complementary to the spectrum of the beta decay, so if the proportion of energy for the beta particle was given by f(E), then the function describing the anti-neutrino energy would be 1-f(E).

See this spectrum for beta particles - http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/beta2.html

Draw a horizontal line across the top. The difference between the horizontal line and the curve for the beta particle intensity is approximately the anit-neutrino intensity. Plot the difference vs energy, and that is the anti-neutrino spectrum.

 

[mfb]

Draw a horizontal line across the top. The difference between the horizontal line and the curve for the beta particle intensity is approximately the anit-neutrino intensity. Plot the difference vs energy, and that is the anti-neutrino spectrum.

Huh? The difference between intensity at a given energy and the peak intensity isn't relevant.
If we neglect the recoil of the nucleus then the sum of electron and neutrino energy is constant, so you need to flip the graph across the vertical axis. The largest electron energy corresponds to the smallest neutrino energy and vice versa.