What is Fermi energy?

The Fermi energy is the highest occupied electron energy level in a free electron gas at absolute zero (0 K). It marks the boundary between fully occupied and empty states in the quantum mechanical treatment of metals.

What formula does this calculator use?

EF = (ℏ²/2mₑ)(3π²n)^(2/3), where ℏ is the reduced Planck constant (1.0546×10⁻³⁴ J·s), mₑ is the electron mass (9.1094×10⁻³¹ kg), and n is the free-electron number density in m⁻³.

What is a typical electron number density?

For copper the free-electron density is about 8.49×10²⁸ m⁻³, giving EF ≈ 7.04 eV. Aluminum is about 1.81×10²⁹ m⁻³ (EF ≈ 11.7 eV). These are order-of-magnitude estimates from the free-electron model.

Why does the result differ slightly from published values?

This tool uses the ideal free-electron model. Real metals have band-structure corrections, and published values sometimes include zero-point energy shifts. For most educational purposes the difference is under 1%.

Can I enter density in electrons per cm³?

Yes — multiply the cm⁻³ value by 10⁶ to convert to m⁻³, then paste it into the input. The preset buttons already use m⁻³ values.

Fermi Energy Calculator

Enter electron number density to compute Fermi energy using EF = (ℏ²/2m)(3π²n)^(2/3).

Electron Number Density

Preset metals

Or enter manually

m⁻³

Free-electron number density (m⁻³). Use scientific notation: 8.49e28

Enter electron density or select a preset metal.

Copied to clipboard

Summary

This tool computes the Fermi energy EF of a free electron gas given the free-electron number density n. It applies the standard quantum-mechanical result EF = (ℏ²/2mₑ)(3π²n)^(2/3) and reports the answer in both joules and electron-volts, along with intermediate values useful for coursework verification.

How it works

The free-electron model treats conduction electrons as non-interacting particles in a box.
Filling energy states from the bottom up (Pauli exclusion principle) at T = 0 K defines the Fermi level.
Evaluating the Fermi wavevector kF = (3π²n)^(1/3) gives EF = ℏ²kF²/2mₑ.

Use cases

Verify textbook Fermi energy values for common metals like copper or aluminum.
Compute Fermi energy for custom electron densities in condensed matter research.
Check intermediate steps when learning solid-state physics derivations.
Compare Fermi energies across different materials by entering their free-electron densities.

Frequently Asked Questions

Last updated: 2026-06-19 · Reviewed by Nham Vu