Fermi-Dirac distribution and electron gas

Intuition [Beginner]

Electrons, protons, and neutrons are fermions: particles with half-integer spin that obey the Pauli exclusion principle. No two identical fermions can occupy the same quantum state. Picture electrons in a metal filling up energy levels like seats in a theatre — each seat holds at most one person.

At absolute zero, electrons pack into the lowest available states. The energy of the highest occupied state is the Fermi energy $E_F$. Every state below $E_F$ is full; every state above is empty. At finite temperature the occupation becomes a smooth step: the Fermi-Dirac distribution gives the probability that a state at energy $E$ is occupied:

$$n(E)=\frac{1}{e^{(E-E_F)/kT}+1}.$$

At $T=0$ this is a sharp step: $n=1$ for $E<E_F$ and $n=0$ for $E>E_F$. At room temperature only electrons within about $kT$ of $E_F$ can change states — the rest are "frozen" below.

This explains why metals have a small electronic specific heat. A classical gas of $N$ electrons would carry specific heat $\frac{3}{2}N{k}_B$, but in a metal only the fraction $kT/E_F$ of electrons near the Fermi surface can absorb thermal energy, giving $C_v\sim (kT/E_F)N{k}_B$ — much smaller.

Copper has $E_F\approx 7$ eV. At 300 K, $kT\approx 0.026$ eV, so the thermal "fuzzy region" near $E_F$ is only about 0.4% of the full energy range.

Visual [Beginner]

Picture a tall building with many floors. Each floor is an energy level, and each floor has exactly two apartments (spin-up and spin-down). The Pauli principle says at most one electron per apartment. At $T=0$, tenants fill from the ground floor up — the building is full up to floor $E_F$ and empty above. The Fermi surface is the boundary between occupied and unoccupied floors.

Turn on a small amount of heat. Only the tenants on the top few floors near $E_F$ have enough energy to move to higher floors. Everyone else is stuck — there are no empty apartments nearby. This is why the electron gas in a metal contributes so little to heat capacity: almost all electrons are locked in place by the exclusion principle.

Worked example [Beginner]

Copper: how many electrons can participate in thermal physics?

Copper has one conduction electron per atom. The Fermi energy is $E_F\approx 7.0$ eV. At room temperature $T=300$ K, the thermal energy is $kT\approx 0.026$ eV.

The fraction of electrons within $kT$ of the Fermi surface is roughly $kT/E_F=0.026/7.0\approx 0.004$, or about 0.4%. Only these electrons can change their state when the metal absorbs heat.

The classical prediction for the electronic specific heat would be $C_{\text{classical}}=\frac{3}{2}N{k}_B$. The actual result from Fermi-Dirac statistics is smaller by the factor $kT/E_F$:

$$C_v\approx \frac{{\pi }^2}{2}\frac{kT}{E_F}N{k}_B\approx 0.006N{k}_B.$$

This is roughly 250 times smaller than the classical prediction — matching experimental measurements on metals.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Consider a system of non-interacting fermions that can occupy single-particle states labelled by $k$ with energies ${ϵ}_k$. The grand canonical ensemble 11.04.01 pending for fermions differs from the bosonic case in one critical respect: each state $k$ can be occupied by at most one particle ($n_k\in \{0,1\}$), enforced by the Pauli exclusion principle 12.01.02 pending.

For a single state $k$, the grand partition function is

$${\mathcal{Z}}_k=\sum _{n_k=0}^1{e}^{-\beta ({ϵ}_k-\mu )n_k}=1+e^{-\beta ({ϵ}_k-\mu )},$$

where $\beta =1/(k_{B}T)$ and $\mu$ is the chemical potential. Since the states are independent, the full grand partition function factorises:

$${\mathcal{Z}}_{GC}=\prod _k(1+e^{-\beta ({ϵ}_k-\mu )}).$$

The mean occupation number of state $k$ is

$$⟨n_k⟩=\frac{1}{{\mathcal{Z}}_k}\sum _{n_k=0}^1{n}_k{e}^{-\beta ({ϵ}_k-\mu )n_k}=\frac{e^{-\beta ({ϵ}_k-\mu )}}{1+e^{-\beta ({ϵ}_k-\mu )}}.$$

This gives the Fermi-Dirac distribution:

$$\boxed{f(ϵ)=\frac{1}{e^{\beta (ϵ-\mu )}+1}.}$$

At $T=0$ the chemical potential equals the Fermi energy: $\mu (T=0)=E_F$. The distribution becomes a step function: $f=1$ for $ϵ<E_F$, $f=0$ for $ϵ>E_F$. At low temperatures ($kT\ll E_F$), $\mu \approx E_F$ and the distribution changes from 1 to 0 in a narrow band of width $\sim kT$ around $E_F$.

The Fermi temperature is defined as $T_F=E_F/k_B$. For typical metals, $T_F\sim 10^4$ K, so room temperature is a small perturbation.

The free electron gas in a box

For $N$ free electrons in a box of volume $V$, the single-particle energies are ${ϵ}_{\mathbf{k}}={\hslash }^2{k}^2/(2m)$. The density of states per unit energy (counting spin degeneracy) is

$$g(ϵ)=\frac{V}{2{\pi }^2}{\left(\frac{2m}{{\hslash }^2}\right)}^{3/2}\sqrt{ϵ}.$$

At $T=0$, the total number of electrons fixes $E_F$:

$$N={\int }_0^{E_F}g(ϵ)dϵ=\frac{V}{3{\pi }^2}{\left(\frac{2m{E}_F}{{\hslash }^2}\right)}^{3/2},$$

giving

$$E_F=\frac{{\hslash }^2}{2m}{\left(3{\pi }^{2}n\right)}^{2/3},$$

where $n=N/V$ is the electron number density. The total energy at $T=0$ is

$$U_0={\int }_0^{E_F}ϵg(ϵ)dϵ=\frac{3}{5}N{E}_F,$$

and the pressure at $T=0$ is $P_0=2U_0/(3V)=2n{E}_F/5$ — the degeneracy pressure of the electron gas, which does not vanish even at zero temperature.

Counterexamples to common slips

• The chemical potential $\mu$ is not in general equal to $E_F$. The Fermi energy is defined as $\mu (T=0)$. At finite temperature $\mu (T)$ decreases slightly below $E_F$ for a 3D gas. The distinction matters at the level of the Sommerfeld expansion.
• The FD distribution $f(ϵ)$ is the mean occupation of a single state. When states are degenerate (e.g., spin), the total occupation of the energy level includes the degeneracy factor.
• The free electron gas model neglects electron-electron interactions and the periodic lattice potential. In real metals, band structure modifies the density of states substantially, but the Fermi-surface concept and the FD distribution remain valid — the Fermi surface becomes a surface in $\mathbf{k}$-space determined by the band dispersion rather than the free-particle parabola.
• "Degenerate" in this context means $T\ll T_F$, not related to energy-level degeneracy. A highly degenerate Fermi gas is one whose properties are dominated by the exclusion principle rather than thermal effects.

Key theorem with proof [Intermediate+]

Theorem (Sommerfeld electronic specific heat). For a free electron gas in three dimensions with $N$ electrons and Fermi energy $E_F$, the electronic specific heat at constant volume at low temperature ($kT\ll E_F$) is

$$\boxed{C_v=\frac{{\pi }^2}{2}\frac{k_{B}T}{E_F}N{k}_B.}$$

Proof. The total energy at temperature $T$ is

$$U(T)={\int }_0^{\infty }ϵg(ϵ)f(ϵ)dϵ,$$

where $g(ϵ)=C\sqrt{ϵ}$ with $C=(V/2{\pi }^2)(2m/{\hslash }^2{)}^{3/2}$ and $f(ϵ)=1/(e^{\beta (ϵ-\mu )}+1)$. Define $h(ϵ)=ϵg(ϵ)=C{ϵ}^{3/2}$. We need

$$U(T)={\int }_0^{\infty }h(ϵ)f(ϵ)dϵ.$$

The Sommerfeld expansion evaluates integrals of the form ${\int }_0^{\infty }\varphi (ϵ)f(ϵ)dϵ$ when $kT\ll E_F$. Integrating by parts:

$${\int }_0^{\infty }\varphi (ϵ)f(ϵ)dϵ=-{\int }_0^{\infty }\Phi (ϵ)f^{\prime }(ϵ)dϵ,$$

where $\Phi (ϵ)={\int }_0^{ϵ}\varphi ({ϵ}^{\prime })d{ϵ}^{\prime }$. The derivative $-f^{\prime }(ϵ)$ is sharply peaked around $ϵ=\mu$ with width $\sim k_{B}T$ and integrates to 1. Taylor-expanding $\Phi (ϵ)$ around $\mu$:

$$\Phi (ϵ)=\Phi (\mu )+{\Phi }^{\prime }(\mu )(ϵ-\mu )+\frac{1}{2}{\Phi }^{\prime \prime }(\mu )(ϵ-\mu {)}^2+\cdots$$

The zeroth term gives $\Phi (\mu )={\int }_0^{\mu }\varphi (ϵ)dϵ$. The first-order term vanishes by symmetry of $f^{\prime }$. The second-order term gives

$${\int }_0^{\infty }\varphi (ϵ)f(ϵ)dϵ={\int }_0^{\mu }\varphi (ϵ)dϵ+\frac{{\pi }^2}{6}(k_{B}T{)}^2{\varphi }^{\prime }(\mu )+O(T^4).$$

Applying this to $U(T)$ with $\varphi (ϵ)=h(ϵ)=C{ϵ}^{3/2}$:

$$U(T)={\int }_0^{\mu }C{ϵ}^{3/2}dϵ+\frac{{\pi }^2}{6}(k_{B}T{)}^2\cdot \frac{3}{2}C{\mu }^{1/2}+O(T^4).$$

The zeroth-order piece is $\frac{2}{5}C{\mu }^{5/2}=\frac{3}{5}N\mu$ (using $N=\frac{2}{3}C{\mu }^{3/2}$ at leading order). Similarly, applying the expansion to the particle number $N=\int g(ϵ)f(ϵ)dϵ$ gives $\mu =E_F[1-O(T^2/E_F^2)]$, so at leading order in $T$ we may replace $\mu$ by $E_F$ in the correction term. Then:

$$U(T)=\frac{3}{5}N{E}_F+\frac{{\pi }^2}{4}N\frac{(k_{B}T{)}^2}{E_F}+O(T^4).$$

Differentiating with respect to $T$:

$$C_v=\frac{\partial U}{\partial T}=\frac{{\pi }^2}{2}\frac{k_{B}T}{E_F}N{k}_B+O(T^3).\square$$

Corollary. The ratio $C_v/C_v^{\text{classical}}=({\pi }^2/3)(kT/E_F)\ll 1$ at room temperature. For copper ($E_F\approx 7$ eV, $T=300$ K): $C_v/C_v^{\text{classical}}\approx 0.006$. This resolves the century-old puzzle of why the Dulong-Petit law overestimates the specific heat of metals by counting all electrons as classical particles.

Worked example: Pauli paramagnetism

A free electron gas in a uniform magnetic field $B$ develops a net magnetisation because spin-up and spin-down electrons shift their Fermi energies by $\pm {\mu }_{B}B$ (where ${\mu }_B=e\hslash /(2m)$ is the Bohr magneton). The excess of spin-up over spin-down electrons near $E_F$ is $\delta N\sim g(E_F){\mu }_{B}B$. The magnetisation is $M={\mu }_B\delta N/V={\mu }_B^{2}g(E_F)B$, giving the Pauli susceptibility

$${\chi }_P={\mu }_B^{2}g(E_F)=\frac{3}{2}\frac{N{\mu }_B^2}{V{E}_F}.$$

This is temperature-independent at low $T$ (unlike the Curie law $1/T$ for local moments), because only electrons near $E_F$ can flip their spin, and their number does not change with $T$ at leading order.

Exercises [Intermediate+]

The Fermi surface and Landau's Fermi liquid theory [Master]

The Fermi surface is the surface in momentum space $\mathbf{k}$ defined by $ϵ(\mathbf{k})=E_F$. For a free electron gas this is a sphere of radius $k_F=(3{\pi }^{2}n{)}^{1/3}$. In a real metal, the periodic lattice potential breaks the spherical symmetry and the Fermi surface develops complex topography — sheets, necks, and open orbits — that encode the band structure. Experimental techniques such as de Haas-van Alphen oscillations measure the extremal cross-sectional areas of the Fermi surface directly.

Landau's Fermi liquid theory (1956) provides the framework for understanding why the free-electron picture works as well as it does, despite strong electron-electron interactions in metals. Landau postulated that the low-energy excitations of the interacting system are quasiparticles — long-lived excitations that are in one-to-one correspondence with the free-particle states, carrying the same quantum numbers but with renormalised parameters (effective mass $m^{\ast }$, effective $g$-factor, etc.).

The key insight is adiabatic continuity: if the interaction is turned on slowly, each free-particle state evolves into a quasiparticle state without crossing other states. The Pauli principle ensures stability: a quasiparticle near the Fermi surface can only decay by scattering into other states near $E_F$, and the phase space for such processes scales as $(ϵ-E_F{)}^2$, giving quasiparticle lifetimes $\tau \sim 1/(ϵ-E_F{)}^2$. The Fermi surface itself is protected by the Luttinger theorem: the volume enclosed by the Fermi surface is proportional to the total electron density, independent of interaction strength.

Landau introduced interaction parameters $F_l^s$ (symmetric) and $F_l^a$ (antisymmetric) characterising how quasiparticles interact in angular-momentum channel $l$. The specific heat, compressibility, and susceptibility are renormalised:

$$\frac{C_v}{C_v^0}=\frac{m^{\ast }}{m}=1+\frac{F_1^s}{3},\frac{\chi }{{\chi }_0}=\frac{m^{\ast }/m}{1+F_0^a},\frac{\kappa }{{\kappa }_0}=\frac{m/m^{\ast }}{1+F_0^s}.$$

Fermi liquid theory breaks down in one dimension (where the correct description is the Luttinger liquid with spin-charge separation) and in systems near a quantum critical point.

Density of states in real metals [Master]

The free-electron density of states $g(ϵ)\propto \sqrt{ϵ}$ is modified by band structure. In the nearly-free-electron model, Bragg planes open energy gaps and the density of states develops van Hove singularities — logarithmic divergences in 2D and square-root cusps in 3D — at the band edges. The Fermi energy may fall near such a singularity, dramatically enhancing the electronic specific heat and the superconducting transition temperature.

The measured Sommerfeld coefficient $\gamma =C_v/T=({\pi }^2/3)k_B^{2}g(E_F)$ is conventionally written as $\gamma ={\gamma }_0(m^{\ast }/m)$ where ${\gamma }_0$ is the free-electron value. Heavy-fermion materials (e.g., CeAl${}_3$, UPt${}_3$) have $m^{\ast }/m\sim 10^2$--$10^3$, reflecting enormous quasiparticle masses from strong correlations. These systems lie at the boundary of Fermi liquid theory and often exhibit unconventional superconductivity.

White dwarf stars as degenerate electron gas [Master]

A white dwarf is the remnant core of a low-to-intermediate mass star ($M\lesssim 8M_{\odot }$), supported against gravitational collapse by electron degeneracy pressure. The typical mass is $\sim 0.6M_{\odot }$ and the radius is $\sim R_{\oplus }$, giving electron densities $n_e\sim 10^{36}{\text{m}}^{-3}$ and $E_F\sim 0.3$ MeV — well into the relativistic regime.

The equation of state has two limits. Non-relativistic ($p_F\ll m_{e}c$): $P\propto {\rho }^{5/3}$, supporting any mass at some radius. Ultra-relativistic ($p_F\gg m_{e}c$): $P\propto {\rho }^{4/3}$, and the pressure scales the same way as gravitational force with density. Equating the two pressures yields a unique mass — the Chandrasekhar limit $M_{\mathrm{Ch}}\approx 1.4M_{\odot }$ (Exercise 9). Above this mass, no stable white dwarf exists; the star collapses further to a neutron star or black hole.

Connections [Master]

The Fermi-Dirac distribution connects to nearly every area of condensed matter and many-body physics:

• Superconductivity (BCS theory): The Cooper instability arises because electrons near the Fermi surface in the presence of an attractive interaction form bound pairs. The BCS ground state is a coherent superposition of occupied and unoccupied states near $E_F$, and the energy gap $\Delta$ opens at the Fermi surface.
• Quantum Hall effect: In two-dimensional electron gases, Landau quantisation discretises the density of states into delta-function-like levels. The integer and fractional quantum Hall effects are Fermi-surface phenomena in the presence of a strong magnetic field.
• Nuclear physics: Nucleons in a nucleus obey Fermi-Dirac statistics with $E_F\sim 38$ MeV. The nuclear Fermi gas model treats the nucleus as two degenerate Fermi gases (protons and neutrons) and explains the asymmetry term in the semi-empirical mass formula.
• Neutron stars: At densities above nuclear saturation, electrons inverse-beta-decay into neutrons. The resulting neutron degeneracy pressure ($E_F\sim 100$ MeV) supports the star — a white dwarf supported by electron degeneracy becomes a neutron star supported by neutron degeneracy.
• Cross-reference to chemistry 14.01.01: The aufbau principle for atomic electron configurations is the single-atom limit of the FD distribution. Each orbital in an atom is a "state" that holds at most two electrons (spin up/down), and they fill from lowest energy upward — the molecular-orbital and crystal-field treatments in inorganic chemistry 16.03.02 pending inherit this structure.

Historical and philosophical context [Master]

The Fermi-Dirac distribution was discovered independently by Enrico Fermi (1926) and Paul Dirac (1926), both building on the new quantum statistics of Bose (1924) and Einstein (1924--25). Fermi's approach started from the Pauli exclusion principle and constructed the statistical mechanics of particles that cannot share a quantum state. Dirac's approach identified the connection between particle statistics and the symmetry of the quantum-mechanical wave function: fermions have antisymmetric wave functions ($\psi (1,2)=-\psi (2,1)$), bosons have symmetric ones.

Arnold Sommerfeld (1928) applied the FD distribution to electrons in metals, resolving two major puzzles: the small electronic specific heat (the Drude model predicted a value 100 times too large) and the long mean free path of conduction electrons. Sommerfeld's free-electron treatment — the "Sommerfeld model" — remains the starting point for solid-state physics textbooks.

Lev Landau (1956) generalized the theory to interacting fermions with his Fermi liquid theory, providing the conceptual framework for understanding why the non-interacting picture works so well. Landau's theory predicted the existence of zero sound in liquid ${}^3$He (confirmed experimentally in 1966) and established the renormalisation-group language later adopted for quantum critical phenomena.

Subrahmanyan Chandrasekhar (1931) showed that relativistic electron degeneracy cannot support stars above $\sim 1.4M_{\odot }$, predicting the existence of neutron stars and black holes decades before their observational confirmation. This result was controversial — Eddington publicly opposed it — and Chandrasekhar did not receive the Nobel Prize for this work until 1983.

Bibliography [Master]

• Fermi, E. (1926). "Sulla quantizzazione del gas perfetto monoatomico." Rend. Lincei 3, 145--149. Originator paper for Fermi-Dirac statistics.
• Dirac, P. A. M. (1926). "On the theory of quantum mechanics." Proc. Roy. Soc. A 112, 661--677. Independent discovery of FD statistics; connection to antisymmetric wave functions.
• Sommerfeld, A. (1928). "Zur Elektronentheorie der Metalle." Naturwissenschaften 16, 374--410. The Sommerfeld free-electron model: electronic specific heat, Wiedemann-Franz law.
• Landau, L. D. (1956). "The theory of a Fermi liquid." Sov. Phys. JETP 3, 920. Fermi liquid theory.
• Chandrasekhar, S. (1931). "The maximum mass of ideal white dwarfs." Astrophys. J. 74, 81. The Chandrasekhar mass limit.
• Ashcroft, N. W. & Mermin, N. D. (1976). Solid State Physics. Saunders. Ch. 2: Sommerfeld theory of metals.
• Landau, L. D. & Lifshitz, E. M. (1980). Statistical Physics, Part 1, 3rd ed. Pergamon. §56--58: Fermi distribution, degenerate Fermi gas.
• Schroeder, D. V. (2000). An Introduction to Thermal Physics. Addison-Wesley. Ch. 7: Quantum statistics.
• Tong, D. "Statistical Physics." §4: Quantum statistics — Fermi-Dirac distribution. ^[tong]