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 2 of 3: Reduced mass from the spectrum

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\frac{1}{\lambda} = 109700\ \mathrm{cm}^{-1}\times\frac{\mu}{m_e}\left(\frac{1}{2^2}-\frac{1}{4^2}\right)\ \Rightarrow\ \mu = 185.9\,m_e = 1.693\times10^{-28}\ \text{kg}

Analysis

The transition from the third excited state ( $n = 4$ ) to the first excited state ( $n = 2$ ) obeys $1/\lambda = R_H (\mu/m_e)(1/4 - 1/16)$ . With $1/\lambda = 3.824\times10^{6}$ cm $^{-1}$ , $\mu/m_e = 185.9$ , giving $\mu = 1.693\times10^{-28}$ kg.

Common pitfall. “Third excited state” means

n = 4

, not

n = 3

: the ground state is

n = 1

and the first excited state is

n = 2