Fermi Energy Calculator

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:

Those figures come from the formula above, using the electron densities in the second column, so the calculator reproduces every row exactly. Textbook tables will differ by a percent or two, because they start from slightly different measured densities and quote slightly different values of the constants. If a table you are checking against says 4.74 eV for lithium where this one says 4.75, that gap is the density, not an error.

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

Worked example: copper

Copper contributes one free electron per atom, giving n = 8.49 × 10²⁸ per cubic meter.

(ħ²/2mₑ) = (1.0546 × 10⁻³⁴)² / (2 × 9.109 × 10⁻³¹) = 6.105 × 10⁻³⁹
(3π²n)^(2/3) = (29.609 × 8.49 × 10²⁸)^(2/3) = 1.849 × 10²⁰
E_F = 6.105 × 10⁻³⁹ × 1.849 × 10²⁰ = 1.129 × 10⁻¹⁸ J = 7.05 eV
T_F = E_F / k_B = 81,700 K

Pick Copper from the dropdown, or type 8.49 into the custom box, and you get that back.

Why the Fermi temperature is so absurd

T_F for copper is 81,700 K. The metal melts at 1,358 K and boils at 2,835 K. There is no temperature at which solid copper is anything but deeply degenerate, and that is the point: room temperature is well under half a percent of T_F, so to a very good approximation the electrons in every metal you have ever touched are sitting in their ground-state configuration.

That is also why the electronic heat capacity of a metal is so small. Only the sliver of electrons within roughly k_B T of E_F can absorb energy at all, which is a fraction of about T/T_F, and 293/81,700 is 0.36%. Classical physics predicted metals should have far more heat capacity than they do, and this is the resolution.

The Bose-Einstein distribution is the other half of the story: swap the +1 in the Fermi-Dirac denominator for a -1 and you get particles that pile into one state instead of stacking up to E_F.