Fermi Energy Calculator

The Fermi energy EF is the highest occupied electron energy in a metal at absolute zero (0 K), where electrons fill all available states up to EF. It is determined by the electron number density n using the free electron model: EF = (hbar^2 / 2m) * (3 pi^2 n)^(2/3). For typical metals it ranges from 1.5 eV (cesium) to 11.7 eV (beryllium).

The Fermi energy determines the electronic properties of metals: thermal capacity, electrical conductivity, magnetic susceptibility, and optical properties. It sets the scale of electronic energies in the material. Only electrons within about kT (at room temperature, about 0.025 eV) of EF can be thermally excited and contribute to conduction.

The free electron model treats conduction electrons in a metal as a gas of non-interacting electrons in a box (the metal). Despite its simplicity, it correctly predicts many metallic properties including the Fermi energy, density of states, and electronic heat capacity. It is the starting point for more sophisticated band theory.

The Fermi temperature TF = EF / kB, where kB is Boltzmann's constant. For copper (EF = 7.0 eV), TF = 81,300 K. Since room temperature (300 K) is much less than TF, the electron gas is highly degenerate (nearly T = 0 behavior), justifying the use of the T = 0 Fermi energy formula even at room temperature.

Number density n = (valence electrons per atom) * (atoms per m^3) = Z_val * (rho * NA / M), where rho is mass density, NA is Avogadro's number, and M is molar mass. For copper: density 8,960 kg/m^3, molar mass 63.55 g/mol, Z_val = 1, giving n = 8.49 x 10^28 electrons/m^3.