|
This should be easy, but I think I have a mind-block... For $\beta^-$-decay, what is the maximum possible momentum for the electron? The two equations I can use are conservation of energy and conservation of momentum, but I have three unknowns: Momentum of electron, nucleus and anti-neutrino, so what am I missing? Could I just set the kinetic energy of the nucleus to zero and work from there? |
|||||||||||
|
|
Answer of Zassounotsukushi is correct, I'd like to expand it consider center of mass frame, a resting neutron for simplicity. four-momentum is $\underline{p}_n=\binom{m_n}{\vec{0}}$. neutron decays into proton, electron, antineutrino: $n \rightarrow p^+ e^- \bar{\nu}$ $$ \binom{m_n}{\vec{0}}=\binom{E_p}{\vec{p}_p}+\binom{E_e}{\vec{p}_e}+\binom{E_\nu}{\vec{p}_\nu} $$ neutrino mass $(m_\nu < 0.2eV)$ is negligible $\Rightarrow E_\nu=|\vec{p}_\nu|$, to maximize electron energy neutrino has to be left without any $\Rightarrow E_\nu=0$ $$ \Rightarrow\binom{m_n}{\vec{0}}=\binom{E_p}{\vec{p}_p}+\binom{E_e}{\vec{p}_e} $$ for momentum conservation to be satisfied , $\vec{p}_p+\vec{p}_e=\vec0 $, proton momentum has to be $\vec{p}_p=-\vec{p}_e$ $$ \Rightarrow\binom{m_n}{\vec{0}}=\binom{E_p}{-\vec{p}_e}+\binom{E_e}{\vec{p}_e} $$ Energy-momentum relation can be used: $E_p=\sqrt{p_p^2+m_p^2}=\sqrt{p_e^2+m_p^2}$, $E_e=\sqrt{p_e^2+m_e^2}$ energy conservation: $$ \Rightarrow m_n = \sqrt{p_e^2 + m_p^2} + E_e $$ $$ \Rightarrow m_n = \sqrt{E_e^2 - m_e^2 + m_p^2} + E_e $$ $$ \Rightarrow (m_n - E_e)^2 = E_e^2 - m_e^2 + m_p^2 $$ $$ \Rightarrow E_{e,max} = \frac{m_n^2 - m_p^2 + m_e^2}{2m_n} $$ $$ \Rightarrow p_{e,max} = \left(\frac{m_n^2 - m_p^2 + m_e^2}{2m_n}\right)^2 - m_e^2 $$ the "mass defect" in transition $n\rightarrow p$ is in the term $m_n^2 - m_p^2$ included I assumed $c=\hbar=1$ |
|||||||||
|
|
The answer to your question is that the beta endpoint energy is equal to the total Q value of the reaction. Here is a graph of the spectrum from a beta emission, which is why we use internal conversion instead when looking for a monoenergetic source of beta particles. This graph is used in the German Wikipedia, the y-axis is counts per unit energy and the x-axis is energy (of the electron).
You asked for momentum instead, so use $E^2=(m_e c^2)^2 + (pc)^2$. Do not use the classical analog because it is extremely common that beta decay gives relativistic electrons. |
|||||||||||||
|
