Fermi-Dirac Distribution Calculator
Calculate occupancy for fermions.
Formula first
Overview
The Fermi-Dirac distribution is a probability function used in statistical mechanics to describe the occupancy of energy states by fermions in thermal equilibrium. It incorporates the Pauli Exclusion Principle, which prevents multiple fermions from occupying the same quantum state simultaneously.
Symbols
Variables
E = Energy State, \mu = Fermi Level, T = Temperature, f(E) = Occupancy
Apply it well
When To Use
When to use: Apply this equation to systems of identical particles with half-integer spin, such as electrons, protons, or neutrons. It is required when particle density is high or temperatures are low enough that quantum exclusion effects become dominant over classical Boltzmann behavior.
Why it matters: This distribution explains the electronic properties of metals and semiconductors, forming the basis for modern solid-state electronics. It is also crucial in astrophysics for describing the internal pressure of degenerate matter in white dwarf stars and neutron stars.
Avoid these traps
Common Mistakes
- Confusing with Bose (denominator -1).
- Using wrong units for k.
One free problem
Practice Problem
Calculate the occupancy probability (f) for an energy state at 0.10 eV in a system where the Fermi level (u) is 0.05 eV and the temperature is 300 K. Use Boltzmann constant kB = 8.617 × 10⁻⁵ eV/K.
Solve for:
Hint: First calculate the exponent (E-u)/(kB × T) and then evaluate the exponential function.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Statistical Mechanics by Donald A. McQuarrie
- Thermodynamics and an Introduction to Thermostatistics by Herbert B. Callen
- Physical Chemistry by Peter Atkins and Julio de Paula
- Wikipedia: Fermi-Dirac statistics
- NIST CODATA
- Atkins' Physical Chemistry
- Callen, Thermodynamics and an Introduction to Thermostatistics
- McQuarrie Statistical Mechanics