Hong Life CalculatorsUnderstanding the Fermi energy calculator — concept and principle
This calculator estimates the Fermi energy of a metal in the free-electron model, from either its density, molar mass and valence or a known electron density. It also gives the Fermi temperature, velocity and wavevector and the average electron energy at absolute zero. It is a standard tool in solid-state physics courses.
How it is calculated
Electron density n = (ρ ÷ M) × N_A × valence
k_F = (3π² n)^(1/3)
E_F = ħ² k_F² ÷ (2mₑ) = (h² ÷ 8mₑ)(3n/π)^(2/3)
T_F = E_F ÷ k_B; v_F = ħ k_F ÷ mₑ; average energy at 0 K = (3/5) E_F
- ρ is in g/cm³ (× 10⁶ to g/m³), M in g/mol, valence is the number of free electrons per atom.
The principle
In the free-electron model the conduction electrons move freely inside the metal like a gas. They are fermions, so the Pauli principle allows only one electron per quantum state (two with spin). At absolute zero they fill the lowest states one by one, up to a highest occupied energy, the Fermi energy. The more electrons per volume, the higher they must stack, which is why E_F grows as n^(2/3). Because E_F is several electronvolts, the Fermi temperature is tens of thousands of kelvin, so at room temperature only a thin layer of electrons near E_F can change state.
Worked example
Default inputs: copper, ρ = 8.96 g/cm³, M = 63.546 g/mol, 1 free electron per atom.
- n = 8.96 × 10⁶ ÷ 63.546 × 6.02214 × 10²³ = 8.491 × 10²⁸ m⁻³.
- k_F = (3π² × 8.491 × 10²⁸)^(1/3) = 13.598 per nm.
- E_F = ħ² k_F² ÷ 2mₑ = 7.045 eV.
- T_F = E_F ÷ k_B = 81,748 K; v_F = 1,574 km/s.
- Average energy at 0 K = 0.6 × 7.045 = 4.227 eV.
| Molar mass (g/mol) | Fermi energy E_F (eV) |
|---|---|
| 15.9 | 17.751 |
| 31.8 | 11.182 |
| 47.7 | 8.534 |
| 63.5 | 7.045 |
| 79.4 | 6.071 |
| 95.3 | 5.376 |
| 127 | 4.438 |
How to read the result
Simple metals have free-electron Fermi energies of a few to about 10 eV. Results near the measured values suggest the free-electron picture works well, as for copper, silver and the alkali metals. Large differences point to band-structure effects, which the model ignores.
Common mistakes
- Using the number of valence electrons instead of the number of conduction electrons.
- Forgetting to convert g/cm³ to g/m³.
- Reading T_F as a real temperature of the metal. It is only a scale for the electron energies.
Frequently asked questions
Why is the Fermi temperature so high?
It is just E_F divided by k_B; a few eV corresponds to tens of thousands of kelvin.
Is the Fermi level the same as the Fermi energy?
In a metal at 0 K they coincide; at higher temperatures the chemical potential shifts very slightly.
Why do only a few electrons carry heat?
Only electrons within roughly k_BT of E_F have empty states to move into.
Sources: OpenStax, University Physics Volume 3 (2016), section 9.4 "Free Electron Model of Metals"