Question

A muonic atom consists of a muon (m = 106 MeV/c2) in place of an electron. For the muon in a hydrogen atom, what are the following?

(a) the smallest radius in the ground state
(b) the binding energy of the muon in the ground state
(c) the series limit of the wavelength for the first three series

Answer 1

A muonic atom's properties, including the smallest ground-state radius, binding energy of the muon, and series limit of the wavelength, are calculated using modified physics formulas from the Bohr model, taking into account the muon's mass. Comparing the muonic orbit radius to that of a uranium nucleus allows for nuclear studies via photon emission analysis.

Step-by-step explanation:

The properties of a muonic atom, specifically a muonic hydrogen atom, where a muon replaces an electron, can be determined using physics formulas adapted from the Bohr model to account for the greater mass of the muon. When addressing such properties for a muon in a hydrogen-like atom, three aspects are necessary:

1. The smallest ground-state radius, determined by the modified Bohr radius formula for a muonic atom.
2. The binding energy of the muon in the ground state, calculated using the energy level formulas for hydrogen-like atoms.
3. The series limit of the wavelength for various spectral series, which can be found using the Rydberg formula modified for the mass of the muon.

Comparing the ground state radius of a muonic atom to the radius of atomic nuclei such as a uranium nucleus can yield insights into nuclear properties, as the muon's orbital radius may overlap with the dimensions of the nucleus. Measurement of photon emission as a muon transitions to a lower orbit can thus provide valuable information regarding the nucleus itself.