Calculating energy released from beta decay

[tyrant91101]

I have been learning particle physics lately but it's been mostly from a theoretical perspective and not a mathematical one so I have yet to come across any such math but my curiosity is peaked.

From what I understand it, this is the process:

$n \rightarrow p + W^{-}$

Followed by:

$W^{-} \rightarrow e^{-} + \bar{v}_{e}$

Since $m_{e} ~= .511 MeV/c^2$ and $m_{v_{e}} << m_{e}$, there is about $79.9995 GeV/c^{2}$ missing. I am unclear where this energy goes. Does it go into the momentum of the electron (since it is ultrarelitivistic during the decay)?

I've tried to solve the equation $E = m^{2}c^{4} + p^{2}c^{2}$ but I get a weird mass for the electron ($5.68 x 10^{-12} kg$) and I am all around confused by the equation.

If I have the wrong equation, which other one do I use for this type of math? If I am using the write equation, what is the best way of dealing with the units?

 

[Astronuc]

The kinetic energy of the proton (or nucleus), electron and anti-neutrino comes from the difference in mass between the products $p + e^{-} + \bar{v}_{e}$ and the neutron. The partition of the energy is governed by conservation of momentum. If the neutron is in a nucleus, the nucleus acquires very little kinetic energy because it's some massive. Even the case of the bare neutron, the vast majority of kinetic energy is manifest in the electron and anti-neutrino, and the partition of energy is largely determined by the angle between the electron and anti-neutrino trajectories.

http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/beta.html

http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/beta2.html#c1

 

[sir-pinski]

First of all, it's great that you're curious and interested in learning about particle physics! Calculating the energy released from beta decay involves using the equation E = mc^2, where m is the mass difference between the initial and final particles. In this case, the initial particle is a neutron (mass of 939.565 MeV/c^2) and the final particles are a proton (mass of 938.272 MeV/c^2), an electron (mass of 0.511 MeV/c^2), and an electron antineutrino (mass is negligible). Plugging these values into the equation, we get an energy release of 0.782 MeV. This energy is shared between the proton, electron, and antineutrino.

The missing energy that you mentioned is due to the fact that the electron and antineutrino are not the only particles involved in the decay. There are also other particles, such as neutrinos and photons, that are produced but are not directly detected. These particles carry away some of the energy, resulting in the missing energy you calculated. This is a common occurrence in particle physics and is accounted for in the calculations.

As for the units, it's important to keep track of them and make sure they are consistent throughout the calculation. In this case, all the masses are in units of MeV/c^2, so the resulting energy will also be in MeV.

I hope this helps clarify the process of calculating energy released from beta decay. Keep exploring and learning, and don't be afraid to ask questions!