The Bohr model correctly predicts the main energy levels not only for atomic hydrogen but also for other “oneelectron”
atoms where all but one of the atomic electrons has been removed, such as in He+ (one electron removed) or Li++ (two
electrons removed).
(a) Predict the energy levels in eV for a system consisting of a nucleus containing Z protons and just one electron. You
need not recapitulate the entire derivation for the Bohr model, but do explain the changes you have to make to take
into account the factor Z.
(b) The negative muon (μ−) behaves like a heavy electron, with the same charge as the electron but with a mass 207
times as large as the electron mass. As a moving μ-comes to rest in matter, it tends to knock electrons out of atoms
and settle down onto a nucleus to form a “onemuon” atom. For a system consisting of a lead nucleus (Pb208 has 82
protons and 126 neutrons) and just one negative muon, predict the energy in eV of a photon emitted in a transition
from the first excited state to the ground state. The high-energy photons emitted by transitions between energy
levels in such “muonic atoms” are easily observed in experiments with muons.
(c) Calculate the radius of the smallest Bohr orbit for a μ-
bound to a lead nucleus (Pb208 has 82 protons and 126
neutrons). Compare with the approximate radius of the lead nucleus (remember that the radius of a proton or
neutron is about 1 × 10−15 m, and the nucleons are packed closely together in the nucleus).
Comments: This analysis in terms of the simple Bohr model hints at the result of a full quantum-mechanical analysis, which
shows that in the ground state of the lead–muon system there is a rather high probability for finding the muon inside the
lead nucleus. Nothing in quantum mechanics forbids this penetration, especially since the muon does not participate in the
strong interaction. Electrons in an atom can also be found inside the nucleus, but the probability is very low, because on
average the electrons are very far from the nucleus, unlike the muon.
The eventual fate of the μ− in a muonic atom is that it either decays into an electron, neutrino, and antineutrino, or it reacts
through the weak interaction with a proton in the nucleus to produce a neutron and a neutrino. This “muon capture” reaction
is more likely if the probability is high for the muon to be found inside the nucleus, as is the case with heavy nuclei such as
lead.