Question

For a proton to capture an electron to form a neutron and a neutrino (assumed massless), the electron must have some minimum energy. For such an electron, the de Broglie wavelength in picometers is:

Hint:

The de Broglie wavelength relates the wave-like behavior of particles to their momentum. For electrons, use the formula \( \lambda = \frac{h}{p} \), where \( p \) is derived from the energy.

Answer 1

Correct Answer: 1

Step 1: Understanding the de Broglie wavelength.
The de Broglie wavelength \( \lambda \) is given by the formula: \[ \lambda = \frac{h}{p} \] where \( h \) is Planck’s constant and \( p \) is the momentum of the electron. Step 2: Calculating the momentum.
The electron’s minimum energy is equal to the energy required for the proton to capture it. For this minimum energy \( E \), we can equate it to the relativistic energy expression: \[ E = \sqrt{p^2 c^2 + m_e^2 c^4} \] For the minimum energy of the electron, \( E \approx m_e c^2 \) (where \( m_e \) is the electron mass and \( c \) is the speed of light). Step 3: Applying the formula.
From the minimum energy, we calculate the momentum and subsequently the de Broglie wavelength to be between 1.00 and 1.10 picometers.