Electrons in a metal fill energy levels up to the Fermi energy – several electronvolts, equivalent to tens of thousands of kelvins. Only electrons near the Fermi level take part in conduction and heat capacity.
How to use the Fermi Level Calculator
- Metal
- Free-electron density (m⁻³)
- Energy E (for the occupation)
- Temperature T
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose a metal (or enter the electron density), and optionally an energy and temperature for the Fermi–Dirac occupation.
Inputs explained
| Field | What to enter |
|---|---|
| Metal | Choose one: Lithium, Sodium, Potassium, Copper … |
| Free-electron density (m⁻³) | A number |
| Energy E (for the occupation) | Units: eV, keV, MeV, J, zJ |
| Temperature T | Units: K, °C, °F |
What each result means
- Fermi wave vector k_F = (3×pi^2×[n])^(1÷3)
- Fermi energy E_F = 1.054571817e-34^2×[Fermi wave vector k_F]^2÷(2×9.1093837015e-31)
- Fermi temperature T_F = [Fermi energy E_F]÷1.380649e-23
- Fermi velocity v_F = 1.054571817e-34×[Fermi wave vector k_F]÷9.1093837015e-31
- Occupation probability f(E) = 1÷(exp(([Energy E (for the occupation)] – [Fermi energy E_F])÷(1.380649e-23×[Temperature T])) + 1)
Example values
How the main result changes when Free-electron density changes and everything else stays at the example values above:
| Free-electron density (m⁻³) | Fermi energy E_F (eV) |
|---|---|
| 4.25e28 | 7.0328 |
| 6.38e28 | 7.0328 |
| 8.50e28 | 7.0328 |
| 1.06e29 | 7.0328 |
| 1.28e29 | 7.0328 |
| 1.70e29 | 7.0328 |
Formula
k_F = (3π²n)^{1/3}, E_F = ħ²k_F²/(2mₑ), T_F = E_F/k_B, v_F = ħk_F/mₑ, f(E) = 1/(e^{(E − E_F)/(k_BT)} + 1)
Worked example
Inputs — Metal: Copper; Energy E (for the occupation): 7 eV; Temperature T: 300 K.
Results — Fermi wave vector k_F = 13,586,308,449.70909 m⁻¹; Fermi energy E_F = 7.03276 eV; Fermi temperature T_F = 81,611.80589 K; Fermi velocity v_F = 1,572.85481 km/s; Occupation probability f(E) = 0.780274.
Key terms
- Fermi level
- Energy at which occupation probability is ½.
Frequently asked questions
What is the Fermi energy of copper?
About 7.0 eV.
Why is the Fermi temperature so high?
E_F is far larger than thermal energy at room temperature.
What is the occupation at E = E_F?
Exactly ½ at any temperature.
Is the free-electron model accurate?
Good for simple metals like Na, K, Cu, Ag.