The muon is a subatomic particle with the same charge as an electron but with a mass that is 207 times greater: m_μ = 207m_e. Physicists think of muons as “heavy electrons.” However, the muon is not a stable particle; it decays with a half-life of 1.5 μs into an electron plus two neutrinos. Muons from cosmic rays are sometimes “captured” by the nuclei of the atoms in a solid. A captured muon orbits this nucleus, like an electron, until it decays. Because the muon is often captured into an excited orbit (n > 1), its presence can be detected by observing the photons emitted in transitions such as 2 → 1 and 3 → 1. Consider a muon captured by a carbon nucleus (Z = 6). Because of its large mass, the muon orbits well inside the electron cloud and is not affected by the electrons. Thus the muon “sees” the full nuclear charge Ze and acts like the electron in a hydrogenlike ion.
a. What are the orbital radius and speed of a muon in the n = 1 ground state? Note that the mass of a muon differs from the mass of an electron.
b. What is the wavelength of the 2 → 1 muon transition?
c. Is the photon emitted in the 2 → 1 transition infrared, visible, ultraviolet, or x ray?
d. How many orbits will the muon complete during 1.5 μs? Is this a sufficiently large number that the Bohr model “makes sense,” even though the muon is not stable?