17.2 Fermi-Dirac distribution and applications

The Fermi-Dirac distribution is crucial for understanding how fermions behave in various systems. It describes the probability of these particles occupying energy states at different temperatures, accounting for the Pauli exclusion principle.

This distribution has wide-ranging applications, from explaining electrical properties of metals and semiconductors to determining the fate of stars. It's key to grasping how particle behavior influences larger-scale phenomena in physics and engineering.

Fermi-Dirac Distribution

Fermi-Dirac distribution function

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• Describes the probability of a fermion occupying a state with energy $E$ at a given temperature $T$
• Expressed as $f(E) = \frac{1}{e^{(E-\mu)/kT} + 1}$, where $\mu$ is the chemical potential (Fermi energy at $T = 0$), $k$ is the Boltzmann constant
• Applies to particles with half-integer spin (electrons, protons, neutrons, quarks) known as fermions
• Accounts for the Pauli exclusion principle prevents any two identical fermions from occupying the same quantum state simultaneously
• At absolute zero temperature ($T = 0$), the distribution becomes a step function all states below the Fermi energy are filled, while all states above are empty
• As temperature increases, the distribution "smears out" around the Fermi energy, allowing some fermions to occupy higher energy states

Fermi energy calculations

• Fermi energy $E_F$ represents the highest occupied energy state at absolute zero temperature ($T = 0$)
• Calculated using the particle density $n$ and the formula $E_F = \frac{\hbar^2}{2m}\left(\frac{3\pi^2n}{g}\right)^{2/3}$, where $\hbar$ is the reduced Planck constant, $m$ is the particle mass, and $g$ is the degeneracy factor
• At low temperatures ($T \ll T_F$), $E_F$ remains nearly constant, as most fermions remain in their ground states
• At high temperatures ($T \gg T_F$), $E_F$ decreases as temperature increases, as more fermions occupy higher energy states
• $E_F$ increases with increasing particle density higher density leads to more occupied states and a higher Fermi energy

Applications of Fermi-Dirac distribution

• In metals, the high electron density leads to a high Fermi energy, and electrons near the Fermi energy contribute to electrical conductivity (copper, aluminum)
• The temperature dependence of resistivity in metals can be explained using the Fermi-Dirac distribution
• In semiconductors (silicon, germanium), the Fermi energy lies within the bandgap, and electron and hole concentrations depend on the position of the Fermi level relative to the conduction and valence bands
• Doping shifts the Fermi level and changes the electrical properties of semiconductors
• In white dwarf stars, electron degeneracy pressure supports the star against gravitational collapse, and Fermi-Dirac statistics determine the relationship between the star's mass and radius

Degeneracy pressure in stars

• Degeneracy pressure arises from the Pauli exclusion principle, which states that fermions cannot be compressed indefinitely
• As density increases, particles are forced into higher energy states, resulting in a pressure that resists further compression
• In white dwarf stars, electron degeneracy pressure balances the gravitational force, preventing the star from collapsing under its own gravity
• White dwarf stars remain stable as long as their mass is below the Chandrasekhar limit ($\approx 1.4$ solar masses)
• If the Chandrasekhar limit is exceeded:
  1. Electron degeneracy pressure can no longer support the star
  2. Further collapse leads to the formation of a neutron star or a black hole, depending on the initial mass of the star