Fermi Energy of Free Electron Gas

\[E_F=\frac{\hbar^2}{2m}\left(3\pi^2 n\right)^{2/3}\]

Variables

Ef[E_F]Fermi energy (J)

hbar[ℏ]reduced Planck constant (J·s)

melectron mass (kg)

nelectron concentration (1/m³)

Description

What is this formula?

The Fermi energy formula calculates the highest occupied electron energy level in a free electron gas at absolute zero temperature. It is a fundamental concept in quantum mechanics and solid state physics.

When to use it

This equation is used when analyzing metals, electron gases, semiconductors, and quantum systems where electron statistics follow Fermi-Dirac behavior.

Example

A metal has an electron concentration of:

n = 8.5×10^28 1/m³

Using:

E_F = (ℏ²/(2m))(3π²n)^(2/3)

Where:

ℏ = 1.055×10^-34 J·s

m = 9.11×10^-31 kg

Substitution:

E_F = ((1.055×10^-34)²/(2×9.11×10^-31)) × (3×π²×8.5×10^28)^(2/3)

Result:

E_F ≈ 8.8×10^-19 J

Applications

- Free electron gas theory

- Quantum statistics

- Metal conductivity analysis

- Semiconductor physics

- Fermi surface calculations

- Electronic material analysis