Electron Gas

Quantum model of free electrons in fermionic equilibrium. Theory, formulas, worked examples and interactive graphs.

Conceptual context

An electron gas is an idealized picture of many free electrons spread over quantum states in a metal, semiconductor, or dense plasma. They are not classical grains that can pile into the same state without restriction. Because electrons are fermions, Pauli exclusion fills low-energy states first and forces additional electrons into higher-momentum states. Thus $n_e$ is not merely a count per $V$: it sets the energy height of the filled quantum sea, determines $E_F$, and shapes $f$ near the $\mu$.

🟢 Essential level

The main idea is that increasing $n_e$ packs more free electrons into the same $V$. Since electrons are fermions, they cannot all occupy the same quantum state. Low $E$ states fill first, and added electrons must move into higher momentum and energy states. The upper edge of this filling is $E_F$. At very low $T$, states below that edge are almost occupied and states above it are almost empty.

Raising $T$ softens the edge: some states just above it gain $f$, while some just below it lose occupation. Therefore $n_e$ fixes the energy scale, while temperature controls how sharp the occupation transition appears.

This simple model explains why metals conduct so well: only the electrons near the filling edge can respond to an electric field, while the vast majority remain locked in place by Pauli exclusion, unable to absorb the tiny energy a field or a photon could offer them, since every nearby state is already occupied.

🔵 Formal level

Basic formulas

In the nonrelativistic three-dimensional model, $E_F$ is determined by $n_e$, $m_e$, and $\hbar$. The relation states that filling more momentum states raises the Fermi boundary:

Fermi energy of a 3D electron gas

To compare this energy scale with thermal agitation, one defines $T_F$ using $k_B$:

Fermi temperature of the electron gas

The $f$ of a state with $E$ follows the Fermi-Dirac form, centered near $\mu$ and broadened by $T$:

Fermi-Dirac electron occupation

Real student questions

Is $E_F$ the energy of every electron? No. It is the upper occupation edge in the ideal low $T$ limit.

Why use $n_e$ instead of only $N$? Because the competition for states depends on how many electrons are packed into a given $V$.

Can $f$ be greater than one? For a single fermionic state, no: it is a mean probability. For grouped states, one must distinguish a single state from a degenerate set.

Does increasing $T$ remove Pauli exclusion? No. It only smooths occupation around $\mu$.

Final synthesis

The electron gas shows density as a quantum key: increasing $n_e$ raises $E_F$ through Pauli filling and reshapes $f$. A correct graph turns that forced filling into an increasing energy boundary. Reading the model physically means tracking two coupled effects together: the growth of the Fermi boundary as more electrons are packed into the same volume, and the thermal smoothing of that boundary as temperature rises. Both originate from the same fermionic counting rule, and neither can be separated without losing the physical picture. The graph of $E_F$ against $n_e$ captures the first effect; the Fermi-Dirac curve captures the second. Together they give a complete description of how the electron gas organizes itself in equilibrium.