What it does
In a system at temperature T, the average number of particles in a quantum state of energy E depends on what kind of particle it is:
- Maxwell-Boltzmann (classical particles): f = e^(−(E−μ)/kT).
- Fermi-Dirac (fermions: electrons, protons, ³He): f = 1/(e^((E−μ)/kT) + 1). Never more than one particle per state (the Pauli principle), giving a sharp step at μ when T is low.
- Bose-Einstein (bosons: photons, ⁴He, ⁸⁷Rb): f = 1/(e^((E−μ)/kT) − 1). Bosons crowd into the lowest states; f grows without limit as E approaches μ.
The chemical potential μ is the energy at which a state is half full for fermions. The top half of the page plots the three curves for any T and μ and reads them off at one energy. The bottom half solves an ideal quantum gas: from the particles’ mass, spin degeneracy, number density and temperature it finds μ, the thermal wavelength, the degeneracy nλ³/g (above about 1 quantum effects dominate), the mean energy, the pressure, and the Fermi energy for fermions or the Bose-Einstein condensation temperature and condensate fraction for bosons.
How to use it
- Set the temperature, μ and the energy range (in eV or meV), and an energy E to read off. Untick any distribution to hide it.
- For the gas, pick particles (electrons, helium-3, helium-4, rubidium-87, or any mass in u), the statistics, the number density per m³ and the temperature.
- Download the occupancies as CSV, or the report as an image, PDF or Word file.
Worked example
At 300 K, kT = 0.02585 eV (25.85 meV). With μ = 0, at E = 0.05 eV (1.934 kT above μ), the occupation is 0.1446 for Maxwell-Boltzmann, 0.1263 for Fermi-Dirac and 0.1690 for Bose-Einstein: fermions sit below and bosons above the classical value.
For the conduction electrons of copper (8.5 × 10²⁸ per m³, rounded) at 300 K, the thermal wavelength is 4.303 nm, far longer than the 0.2274 nm spacing, so the degeneracy is 3387 and the gas is strongly quantum. The calculator finds μ = 7.049 eV (272.7 kT), the Fermi energy 7.049 eV and the Fermi temperature 81,804 K. The mean energy per electron is 4.230 eV, not the classical 0.039 eV, and the electron pressure is 3.84 × 10¹⁰ Pa.
Switch to rubidium-87 at 10²⁰ per m³ and 2e-7 K (200 nK): the condensation temperature is 398.3 nK and 64.4 % of the atoms are in the condensate.
Tips
- When E − μ is several kT, all three distributions agree: that is the classical limit.
- For a Fermi gas well below TF, μ ≈ EF and the mean energy approaches 3/5 EF.
- A Bose gas cannot have μ above its lowest energy (0 here); when the density passes ζ(3/2) = 2.612 per λ³ the extra atoms condense.
FAQ
Why can a Bose-Einstein value exceed 1? Any number of bosons may share a state.
Are interactions included? No. Real metals have band structure and real atom clouds have interactions and traps; this is the ideal gas, for study and planning.
What does g mean? The number of spin states per momentum state: 2 for spin-½ particles, 1 for spin 0.
Where the numbers come from
- Fermi-Dirac and Bose-Einstein statistics, the ideal Fermi and Bose gases, the Fermi energy and the condensation temperature: standard results, for example Pathria, R. K. and Beale, P. D., Statistical Mechanics.
- Constants: CODATA 2018 exact values of h, kB and e; electron mass and unified atomic mass unit from CODATA; atomic masses of ³He, ⁴He and ⁸⁷Rb rounded.
- The conduction-electron density of copper (about 8.5 × 10²⁸ per m³) is a rounded general value.
- Checked against mpmath at 30 digits: complete Fermi-Dirac integrals as −Li(−e^η), Bose-Einstein integrals as polylogarithms, and the chemical potential by root finding.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.