Problem 25

Muon. The muon (

\mu

) is a subatomic particle of the lepton family with the same charge and magnetic behaviour as the electron, but a different mass; it is unstable and disintegrates within microseconds. Determine the mass of the muon by two different approaches. (a) The most common spontaneous disintegration is

\ce{\mu- -> e- + \bar{\nu}_e + \nu_{\mu}}

. In an experiment with a stationary muon, the two neutrinos carried away a total energy of

2.000\times10^{-12}

J while the electron had a kinetic energy of

1.4846\times10^{-11}

J. Determine the mass of the muon. (

m_e = 9.109\times10^{-31}

kg,

c = 2.998\times10^{8}

m s

^{-1}

.) (b) Exotic atoms in which a muon replaces the electron are formed in various excited states. For an atom consisting of a

\ce{^1H}

nucleus with an attached muon, the transition from the third excited state to the first excited state is observed at a wavelength of 2.615 nm. Determine the mass of the muon using the Bohr model with reduced-mass correction. (

E_n = -109700\ \mathrm{cm}^{-1}\times hc\times\mu/m_e \times 1/n^2

with

\mu = m_\mu m_H/(m_\mu + m_H)

;

m_H = 1.673\times10^{-27}

kg.)

Step 3 of 3: Solve for the muon mass

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m_\mu = \frac{\mu\, m_H}{m_H - \mu} = \frac{1.693\times10^{-28}\times1.673\times10^{-27}}{1.673\times10^{-27} - 1.693\times10^{-28}} = 1.884\times10^{-28}\ \text{kg}

Analysis

Inverting the reduced-mass relation $\mu = m_\mu m_H/(m_\mu + m_H)$ gives $m_\mu = \mu m_H/(m_H - \mu)$ ; inserting $\mu = 1.693\times10^{-28}$ kg and $m_H = 1.673\times10^{-27}$ kg yields $m_\mu = 1.884\times10^{-28}$ kg — in excellent agreement with the decay-energy value.