Question

The Fermi-Dirac distribution function \( [n(\epsilon)] \) is \[ n(\epsilon) = \frac{1}{e^{(\epsilon - \epsilon_F) / k_B T} + 1} \] where \( k_B \) is the Boltzmann constant, \( T \) is the temperature and \( \epsilon_F \) is the Fermi energy.

• A. \( n(\epsilon) = \frac{1}{e^{(\epsilon - \epsilon_F)} / k_B T - 1} \)
• B. \( n(\epsilon) = \frac{1}{e^{(\epsilon - \epsilon_F)} / k_B T + 1} \)
• C. \( n(\epsilon) = \frac{1}{e^{(\epsilon - \epsilon_F)} / k_B T + 1} \)
• D. \( n(\epsilon) = \frac{1}{e^{(\epsilon - \epsilon_F) / k_B T + 1} }\)

Hint:

The Fermi-Dirac distribution is crucial in describing the statistical behavior of fermions, especially in solids and at low temperatures.

Answer 1

The Correct Option is C

Step 1: Understanding the Fermi-Dirac distribution.
The Fermi-Dirac distribution gives the probability that a state with energy \( \epsilon \) is occupied by a fermion, like an electron. The formula is given by: \[ n(\epsilon) = \frac{1}{e^{(\epsilon - \epsilon_F)/k_B T} + 1} \] where: - \( \epsilon \) is the energy of the state, - \( \epsilon_F \) is the Fermi energy, - \( T \) is the temperature, - \( k_B \) is the Boltzmann constant.
Step 2: Comparing the options.
Option (C) correctly follows the Fermi-Dirac formula as shown above. The denominator contains \( e^{(\epsilon - \epsilon_F)/k_B T} + 1 \), which matches the given formula.