Free-electron gas and the Fermi level

Equipment

• 3D Fermi sphere in k-space with occupied states
• “Temperature” slider (5–180 simulation units)
• “Density / frequency” and “Number of states” sliders
• Occupation-distribution plot versus ε/E_F and the “Pause” button

Procedure

1. Change the density and gauge the Fermi radius

   Raise “Density / frequency”: the Fermi sphere inflates in k-space as more states get occupied. Compare its radius with $k_F = (3\pi^2 n)^{1/3}$ and the edge energy $E_F = \hbar^2k_F^2/2m \propto n^{2/3}$ — press “Pause” to read the occupied-state count.

2. Sweep the temperature and watch the Fermi edge blur

   At low T the occupation is nearly a step: all states ε < E_F full, ε > E_F empty. Raise “Temperature”: only a band $\sim k_BT$ wide around $E_F$ blurs — electrons deep inside the sphere cannot be excited since nearby states are occupied (Pauli principle).

3. Change the state count and draw consequences

   Sweep “Number of states” to see the density of states change, then combine with n to predict the electronic heat capacity: only the small fraction $\sim T/T_F$ of electrons near the Fermi surface participates, so $C_V \propto T$ — far from the classical $3k_B/2$ per particle.

Simulation

Experiment history

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