I would like to know why $\mu^+$ muons can easily penetrate a solid metal such as Fe with negligible interactions while $\pi^+$ mesons lose their energy a lot faster when traveling through the metal. Is it because the interaction between electrons and muons is forbidden or should I somehow see it from the crossection?
$\begingroup$
$\endgroup$
asked Jan 25, 2014 at 16:19
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
2
It actually depends on the energy of the particles.
As a charged particle passes through a metal, it is basically pushing/pulling sea electrons near its path, due to Coulomb repulsion/attraction. That is, the particle is doing work on the electrons in the material, and so it loses energy. The rate of energy loss is given (approximately) by the Bethe formula:
$$ -\frac{dE}{dx}=\frac{4\pi}{m_e c^2}\frac{n z^2}{\beta^2}\left(\frac{e^2}{4\pi\epsilon_0}\right)^2\left[ln\left(\frac{2m_e c^2 \beta^2}{I(1-\beta^2)}\right) - \beta^2 \right] $$
where $\beta$ and $z$ are the velocity and charge of the particle, respectively. The other quantities are properties of the material. Note that $m_e$ and $n$, the electron mass and number density, repsectively, appear in the equation.
So, to answer your question specifically, we find that for a given fixed velocity, the muon and pion would loose energy at the same rate in matter. But, at a fixed energy, they will loose energy at different rates, since the two particles have different masses.
-
1$\begingroup$ I would note that as the pion (273 me) and muon (207me) do not have very different masses, hence the 'a lot faster` is likely due to an energy difference. $\endgroup$– KeithJan 30, 2014 at 5:32
-
$\begingroup$ @Keith, agreed. I should have said they will loose energy at slightly different rates. But in any case, I'll stick to my original one-liner and say it depends on energy ;) $\endgroup$– chaseJan 30, 2014 at 5:35
$\begingroup$
$\endgroup$
Pions in the GeV range also interact hadronically via the strong force. Therefore in iron blocks with a thickness of the order of the hadronic interaction length they will loose energy a lot faster than muons. The dE/dx piece is almost the same but due to the hadronic interactions, e.g. 6 GeV pions in iron of few 10 of cm will loose about twice the energy.
