3
$\begingroup$

Once we start dealing with nuclei the size of gold onwards the inner 1s orbital of an electron approaches relativistic speed. Now it's well known that muonic helium (hydrogen 4.1) has a mean life-time about equivalent to the single muon orbiting the nucleus ($2.2 \mu s$).

So the natural question is, if the nuclei were much larger, would the muon orbital be faster and therefore perhaps would such an atom have a longer mean lifetime?

In many collider experiments relativistic time dilation is seen to increase the lifetime of muons so I would imagine similar principles should work on exotic atom orbitals.

$\endgroup$

1 Answer 1

5
$\begingroup$

Yes, the lab-frame muon free decay lifetime in a heavy muonic atom is probably extended by relativistic effects, but the muonic atom lifetime actually gets shorter for heavier nuclei.

The shorter muonic atom lifetime is because heavier nuclei have larger electric charges ($Z$) which make the orbital radius of the muon smaller so it spends more and more time inside the nucleus (which is also getting larger as $Z$ increases). Instead of decaying it becomes more likely to disappear by interacting with a nuclear proton via $$\mu^- + p \rightarrow n + \nu_\mu$$ As shown in Figure 2 of "Study of nuclear properties with muonic atoms", the lifetime of muonic atoms rapidly decreases as $Z$ increases. This muon capture processe dominates for atoms with $Z>10$, and the lifetime of the heaviest muonic atoms (e.g. lead, uranium) is about 80 ns. The lifetime stops decreasing once the muon orbital radius is small enough that it spends essentially all its time within the nucleus. The radius of a uranium nucleus ($Z=92$) is about 6 fm, compared to the naive muonic uranium Bohr radius ($a_\mu = 260/Z\,\textrm{fm}$) of about 3 fm.

The free muon decay lifetime was measured back in the 1960s by looking for the energetic electrons from free muon decays, and this lifetime agreed with relativistic theoretical calculations. For example, the theoretically expected time dilation for lead was 16-20%, and the experimental value was 14±4%. Not a very precise test, but enough to strongly suggest the atomic muons are subject to time dilation.

$\endgroup$
3
  • 2
    $\begingroup$ Is this argument valid only for muons on the lowest energy level (the 1s orbital)? Can a muon replace one of the outer electrons (say, 4d)? $\endgroup$
    – Martino
    Oct 4, 2022 at 17:44
  • 2
    $\begingroup$ Muons can initially be captured in any atomic level, but they cascade down to the 1S level in about 0.0001 nanoseconds. (See Section 3.2 of "The nuclear physics of muon capture".) This rapid cascade is because muons have about 200 times more binding energy than electrons, so there is plenty of energy to very quickly displace electrons. Since the cascade is so much faster than the decay, there is no way to study muon decays in higher orbitals. $\endgroup$ Oct 4, 2022 at 18:36
  • $\begingroup$ This was good food for thought. The way electron capture half lives can be tweaked by messing with the electronic environment something similar should be possible for muon capture half lives. I wonder between mixing up the size of the atom, the electronic environment and ratio of neutrons to protons, and adding energy how much we can push this. We might want to have something like a “Rydberg exotic atom” with a neutron rich nucleus to really prolong the half life. $\endgroup$ Oct 6, 2022 at 3:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.