If pair production of a muon-antimuon pair can occur for a photon of wavelength 5.85 fm or less, what is the mass of a muon? (muons and antimuons have the same mass.) A) 212 MeV/c² B) 106 MeV/c² C) 424 MeV/c² D) 0.511 MeV/c² E) 1.02 MeV/c²

Question

Quizplus · Accepted Answer

The threshold energy for pair production of a muon-antimuon pair for a single photon is given by:

$E_{	ext{threshold}}=2m_{\mu}c^2\left(1+\frac{\alpha^2}{2}ight)\frac{\lambda_{\gamma}}{\lambda_c}$

Where $m_{\mu}$ is the mass of a muon, $c$ is the speed of light, and $\alpha$ is the fine structure constant. $\lambda_{\gamma}$ is the wavelength of the photon and $\lambda_c$ is the reduced Compton wavelength given by $\lambda_c=\frac{h}{m_{\mu}c}$.

Plugging in the given values, we get:

$E_{	ext{threshold}}=2m_{\mu}c^2\left(1+\frac{\alpha^2}{2}ight)\frac{\lambda_{\gamma}}{\lambda_c}=2\cdot m_{\mu}\cdot (3\cdot 10^8\ 	ext{m/s})^2\cdot\left(1+\frac{1}{2}\alpha^2ight)\frac{5.85\ 	ext{fm}}{2.43\cdot10^{-12}	ext{ m}}\approx 105.7\ 	ext{MeV}$

The mass of a muon can be found by dividing this energy by $c^2$:

$m_{\mu}=\frac{105.7\ 	ext{MeV}}{(3\cdot 10^8\ 	ext{m/s})^2}\approx 0.1057\ 	ext{GeV/c}^2\approx 106\ 	ext{MeV/c}^2$

Therefore, the best choice is B.

If Pair Production of a Muon-Antimuon Pair Can Occur for a Photon