Problem 31 - Physics for Scientists and Engineers with Modern Physics

Question

Calculate the maximum kinetic energy of the electron when a muon decays from rest via \(\mu^{-} \rightarrow \mathrm{e}^{-}+\bar{\nu}_{\mathrm{e}}+\nu_{\mu}\). [Hint: In what direction do the two neutrinos move relative to the electron in order to give the electron the maximum kinetic energy? Both energy and momentum are conserved; use relativistic formulas.

Step-by-Step Solution

Answer

The electron's maximum kinetic energy is 105.19 MeV.

Step 1: Identify the Particles and Their Masses

In the decay \( \mu^{-} \rightarrow \mathrm{e}^{-} + \bar{u}_{\mathrm{e}} + u_{\mu} \), the initial particle is a muon \( \mu^{-} \) and the decay products are an electron \( \mathrm{e}^{-} \), an electron antineutrino \( \bar{u}_{\mathrm{e}} \), and a muon neutrino \( u_{\mu} \). The masses are: \( m_{\mu} = 105.7 \ \text{MeV}/c^2 \) and \( m_{e} = 0.511 \ \text{MeV}/c^2 \). Neutrinos are considered to have negligible mass.

Step 2: Analyze the Decay Using Conservation Laws

Since the muon is initially at rest, its initial energy is its rest mass energy \( E_i = m_{\mu}c^2 \). The final energy \( E_f \) consists of the kinetic energies and rest mass energies of the decay products. Energy conservation gives: \( E_i = E_{e} + E_{\bar{u}_{e}} + E_{u_{\mu}} \). Momentum conservation requires that the total momentum is zero since the muon was initially at rest.

Step 3: Consider Direction for Maximum Kinetic Energy

To maximize the electron's kinetic energy, the two neutrinos should move in the opposite direction to the electron. Thus, the electron takes most of the energy to balance the momentum with neutrinos, which contribute minimal momentum due to their negligible mass.

Step 4: Apply Relativistic Energy Conservation

The total energy available \( E_i = m_{\mu}c^2 \) is converted to the kinetic and rest mass energy of the electron \( E_e = \sqrt{(p_e c)^2 + (m_e c^2)^2} \) and kinetic energies of neutrinos. For the maximum electron kinetic energy, let the neutrinos take negligible energy (ideal case), so \( E_e \approx m_{\mu}c^2 \).

Step 5: Calculate Maximum Electron Kinetic Energy

The maximum kinetic energy of the electron \( K_{ ext{max}} \) is when it utilizes the mass energy difference between the muon and the electron: \( K_{ ext{max}} = (m_{\mu} - m_e)c^2 = 105.7 \ \text{MeV} - 0.511 \ \text{MeV} = 105.19 \ \text{MeV} \).

Key Concepts

Kinetic Energy Calculation

Kinetic energy is a measure of the motion energy of a particle. To calculate it in the context of muon decay, we particularly look at how energy distributes among decay products. In the decay process \( \mu^{-} \rightarrow \mathrm{e}^{-} + \bar{u}_{\mathrm{e}} + u_{\mu} \), the energy initially present is stored in the rest mass energy of the muon \( E_i = m_{\mu} c^2 \). After decay, this energy is redistributed among the electron, electron antineutrino, and muon neutrino. For the maximum kinetic energy of the electron, we consider the case where it absorbs most of this energy, thereby maximizing \( K_{\text{max}} \), the kinetic energy. To achieve this, both neutrinos should move in the opposite direction of the electron, minimizing their kinetic contributions. Consequently, the kinetic energy relies heavily on the rest mass energy difference:

• Initial rest mass energy: \( E_i = m_{\mu} c^2 \)
• Final electron energy: \( E_{e} = \sqrt{(p_{e} c)^2 + (m_{e} c^2)^2} \)

For maximum \( K_{\text{max}} \), assuming minimal neutrino energy, the difference translates to kinetic energy for the electron: \[ K_{\text{max}} = (m_{\mu} - m_e) c^2 = 105.19 \, \text{MeV} \]

Conservation Laws in Relativistic Physics

The laws of conservation play a critical role in understanding muon decay. These laws ensure that both energy and momentum are balanced before and after decay. Initially, the muon is stationary, meaning its momentum is zero, and all energy is in its rest mass. After the decay, energy conservation requires that the total energy \( E_i \) divides into energy portions of each decay particle: \[ E_i = E_e + E_{\bar{u}_{e}} + E_{u_{\mu}} \]Momentum conservation is particularly crucial due to the relativistic nature of particles involved. The total momentum before the decay is zero, so the sum of the electron's momentum and the neutrinos' momentum must equal zero. This setup is achieved by having the neutrinos move in a direction opposite to that of the electron, ensuring that the vector sum is zero. This strategic alignment enables maximal kinetic energy for the electron, respecting both energy and momentum conservation laws in a relativistic framework.

Particle Masses

Understanding the significance of particle masses is crucial in muon decay analysis. Each particle involved in decay carries a distinct mass that affects the energy distribution:

• The muon \( (\mu^-) \) has a significant rest mass: \( 105.7 \, \text{MeV}/c^2 \)
• The electron \( (\mathrm{e}^-) \) is much lighter: \( 0.511 \, \text{MeV}/c^2 \)
• Neutrinos are effectively massless, contributing negligibly to the total mass

This mass difference explains why, during decay, most of the muon's rest energy can be transferred as kinetic energy to the electron. Because neutrinos contribute minimal mass, their effect on the energy tally is minor, allowing the electron to capture the majority of the excess energy. Hence, knowledge of these particle masses helps in predicting energy and momentum tendencies in a relativistic particle decay scenario.