Fermi Energy Calculator
Calculate the Fermi energy of a metal from its electron number density.
Shows Fermi energy in eV and Fermi temperature.
Includes presets for common metals.
Fermi Energy
The Fermi energy is the highest occupied electron energy level in a metal at absolute zero temperature:
E_F = (ħ²/2m_e)(3π²n)^(2/3)
Fermi temperature: T_F = E_F / k_B
Where:
- E_F = Fermi energy (joules)
- ħ = Reduced Planck constant = 1.0546 × 10⁻³⁴ J·s
- m_e = Electron mass = 9.109 × 10⁻³¹ kg
- n = Free electron number density (electrons/m³)
- k_B = Boltzmann constant = 1.381 × 10⁻²³ J/K
Fermi energy of common metals:
| Metal | n (×10²⁸ /m³) | E_F (eV) | T_F (K) |
|---|---|---|---|
| Lithium | 4.70 | 4.74 | 55,000 |
| Sodium | 2.65 | 3.24 | 37,600 |
| Aluminum | 18.1 | 11.7 | 135,800 |
| Copper | 8.49 | 7.04 | 81,700 |
| Gold | 5.90 | 5.53 | 64,200 |
| Silver | 5.86 | 5.49 | 63,700 |
Why Fermi energy matters:
The Fermi energy is critical for understanding:
- Electrical conductivity: Only electrons near E_F can be excited into higher states by an applied voltage. Metals conduct because E_F lies in the middle of an energy band.
- Thermoelectric effects: The Seebeck coefficient depends on the density of states near E_F
- X-ray emission: When electrons fall back to fill inner shell vacancies, they emit X-rays with energies related to E_F
- White dwarf stars: The Fermi energy of degenerate electrons supports the star against gravitational collapse
The Fermi temperature T_F is typically ~50,000–100,000 K — far above room temperature. This is why electrons in metals are highly degenerate (quantum effects dominate) even at room temperature.