Part II: Quantum Gases | Chapter 3

Fermi Systems

Fermi gas, degeneracy pressure, Sommerfeld expansion, electronic specific heat, and Pauli paramagnetism

Historical Context

Enrico Fermi and Paul Dirac independently developed the quantum statistics of half-integer spin particles in 1926. The Pauli exclusion principle, which Fermi statistics encodes, has dramatic consequences: even at absolute zero, a gas of fermions exerts pressure (degeneracy pressure) and occupies states up to a maximum energy called the Fermi energy \(E_F\).

Arnold Sommerfeld applied Fermi-Dirac statistics to electrons in metals in 1928, resolving the long-standing puzzle of why electrons contribute so little to the specific heat. His expansion technique remains the standard tool for analyzing Fermi systems at low temperature. Chandrasekhar later showed that electron degeneracy pressure supports white dwarf stars, earning the 1983 Nobel Prize.

1. The Fermi Gas at Zero Temperature

Derivation 1: Fermi Energy and Ground State Properties

At \(T = 0\), the Fermi-Dirac distribution becomes a step function:

\[n_F(\epsilon) = \frac{1}{e^{\beta(\epsilon - \mu)} + 1} \xrightarrow{T \to 0} \Theta(\mu - \epsilon)\]

All states below \(\mu(T=0) \equiv E_F\) are filled; all above are empty. For \(N\) spin-1/2 fermions of mass \(m\) in volume \(V\)with density of states \(g(\epsilon) = (4\pi V g_s / h^3)(2m)^{3/2}\epsilon^{1/2}\)(with \(g_s = 2\)):

\[N = \int_0^{E_F} g(\epsilon)\,d\epsilon = \frac{V}{3\pi^2}\left(\frac{2mE_F}{\hbar^2}\right)^{3/2}\]

Solving for the Fermi energy:

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

The Fermi momentum is \(p_F = \hbar k_F = \hbar(3\pi^2 n)^{1/3}\) and the Fermi temperature is \(T_F = E_F/k_B\). For electrons in copper:\(E_F \approx 7\,\text{eV}\), \(T_F \approx 80{,}000\,\text{K}\).

The ground state energy is:

\[U_0 = \int_0^{E_F}\epsilon\,g(\epsilon)\,d\epsilon = \frac{3}{5}NE_F\]

2. Degeneracy Pressure

Derivation 2: Pressure at T = 0

Even at zero temperature, the Fermi gas exerts pressure. From the thermodynamic relation\(PV = \frac{2}{3}U\) (valid for all ideal gases with \(\epsilon \propto p^2\)):

\[P_0 = \frac{2}{3}\frac{U_0}{V} = \frac{2}{5}nE_F = \frac{\hbar^2}{5m}(3\pi^2)^{2/3}n^{5/3}\]

This degeneracy pressure is purely quantum mechanical, arising from the Pauli exclusion principle. It has no classical analog and persists even at absolute zero. For a self-gravitating body of mass \(M\) and radius \(R\) composed of particles of mass \(m_p\) with electrons of mass \(m_e\):

\[P_{\text{degeneracy}} \sim \frac{\hbar^2}{m_e}\left(\frac{M}{m_p R^3}\right)^{5/3}, \qquad P_{\text{gravity}} \sim \frac{GM^2}{R^4}\]

Balancing these gives the white dwarf radius-mass relation: \(R \propto M^{-1/3}\). Above the Chandrasekhar limit (\(\sim 1.4\,M_{\odot}\)), the electrons become relativistic and degeneracy pressure can no longer support the star.

Astrophysical Significance

Degeneracy pressure is responsible for the stability of white dwarfs (electron degeneracy) and neutron stars (neutron degeneracy). The different mass scales of the fermions lead to very different equilibrium radii: white dwarfs are roughly Earth-sized while neutron stars have radii of about 10 km. The Chandrasekhar mass limit, derived using special relativity and Fermi-Dirac statistics, was one of the first astrophysical applications of quantum mechanics.

3. The Sommerfeld Expansion

Derivation 3: Low-Temperature Expansion

For any integral of the form \(I = \int_0^{\infty} H(\epsilon)\,n_F(\epsilon)\,d\epsilon\), the Sommerfeld expansion gives:

\[I = \int_0^{\mu} H(\epsilon)\,d\epsilon + \frac{\pi^2}{6}(k_BT)^2 H'(\mu) + \frac{7\pi^4}{360}(k_BT)^4 H'''(\mu) + \cdots\]

The proof uses the identity:

\[-\frac{\partial n_F}{\partial\epsilon} \approx \delta(\epsilon - \mu) + \frac{\pi^2}{6}(k_BT)^2\delta''(\epsilon - \mu) + \cdots\]

Applying this to the number equation with \(H(\epsilon) = g(\epsilon)\):

\[N = \int_0^{\mu}g(\epsilon)\,d\epsilon + \frac{\pi^2}{6}(k_BT)^2 g'(\mu) + \cdots\]

Since \(N\) is constant, the chemical potential must shift:

\[\boxed{\mu(T) = E_F\left[1 - \frac{\pi^2}{12}\left(\frac{k_BT}{E_F}\right)^2 + \cdots\right]}\]

4. Electronic Specific Heat

Derivation 4: Linear-T Specific Heat

Applying the Sommerfeld expansion to the energy integral\(U = \int_0^{\infty}\epsilon\,g(\epsilon)\,n_F(\epsilon)\,d\epsilon\):

\[U = U_0 + \frac{\pi^2}{6}(k_BT)^2 g(E_F) + \cdots\]

where \(g(E_F) = 3N/(2E_F)\) is the density of states at the Fermi level. Therefore:

\[\boxed{C_V = \frac{\pi^2}{3}g(E_F)k_B^2 T = \frac{\pi^2}{2}Nk_B\frac{T}{T_F} \equiv \gamma T}\]

The Sommerfeld coefficient \(\gamma = \pi^2 Nk_B/(2T_F)\) is linear in \(T\), in stark contrast to the classical prediction \(C_V = \frac{3}{2}Nk_B\). At room temperature for copper (\(T_F \approx 80{,}000\,\text{K}\)):

\[\frac{C_V^{\text{electronic}}}{C_V^{\text{classical}}} \sim \frac{T}{T_F} \sim 0.004\]

This explains why electrons contribute negligibly to the specific heat of metals at room temperature -- they are deeply degenerate. Only a fraction \(\sim k_BT/E_F\) of electrons near the Fermi surface can be thermally excited.

Total Specific Heat of Metals

At low temperatures, the total specific heat of a metal is \(C = \gamma T + AT^3\), where the first term is the electronic contribution and the second is the Debye phonon contribution. Plotting \(C/T\) vs \(T^2\) gives a straight line with intercept \(\gamma\) and slope \(A\). Deviations of\(\gamma\) from the free-electron value probe electron-electron and electron-phonon interactions (mass renormalization).

5. Pauli Paramagnetism

Derivation 5: Spin Susceptibility

In a magnetic field \(B\), spin-up and spin-down electrons have energies shifted by \(\mp \mu_B B\). The number of spin-up and spin-down electrons:

\[N_{\uparrow} = \frac{1}{2}\int_0^{\infty}g(\epsilon)\,n_F(\epsilon - \mu_B B)\,d\epsilon, \quad N_{\downarrow} = \frac{1}{2}\int_0^{\infty}g(\epsilon)\,n_F(\epsilon + \mu_B B)\,d\epsilon\]

The magnetization \(M = \mu_B(N_{\uparrow} - N_{\downarrow})\). For weak fields (\(\mu_B B \ll E_F\)), expanding to first order:

\[M = \mu_B^2 B \int_0^{\infty}g(\epsilon)\left(-\frac{\partial n_F}{\partial\epsilon}\right)d\epsilon \approx \mu_B^2 B\,g(E_F)\]

The Pauli spin susceptibility is:

\[\boxed{\chi_{\text{Pauli}} = \mu_0\mu_B^2\,g(E_F) = \frac{3\mu_0 n\mu_B^2}{2E_F}}\]

This is temperature-independent (to leading order), in contrast to the Curie law\(\chi \propto 1/T\) of localized moments. The Pauli susceptibility is smaller than the classical Curie susceptibility by a factor \(\sim T/T_F\) because only electrons near \(E_F\) can flip their spins.

6. Applications

Key Applications of Fermi Statistics

White dwarf stars: Electron degeneracy pressure supports white dwarfs against gravitational collapse. The Chandrasekhar mass limit \(M_{\text{Ch}} \approx 1.4\,M_{\odot}\)follows from the relativistic Fermi gas equation of state.

Neutron stars: Neutron degeneracy pressure supports neutron stars. The maximum mass (\(\sim 2\,M_{\odot}\)) is modified by nuclear interactions.

Semiconductors: The Fermi-Dirac distribution determines carrier concentrations in doped semiconductors, controlling the properties of transistors and diodes.

Heavy fermion materials: In compounds like CeAl\(_3\), strong electron correlations produce quasiparticles with effective masses up to\(1000\,m_e\), dramatically enhancing \(\gamma\).

7. Computational Exploration

This simulation explores the Fermi-Dirac distribution, Sommerfeld expansion, electronic specific heat, and Pauli paramagnetism.

Fermi Systems: Distribution, Specific Heat, Chemical Potential, and Susceptibility

Python

script.py168 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Fermi Systems: Distribution, Specific Heat, and Magnetism
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Fermi Systems', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#1a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('#fca5a5')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: Fermi-Dirac distribution at various T ---
ax1 = axes[0, 0]
epsilon = np.linspace(0, 2.5, 500)  # in units of E_F
EF = 1.0

for T_ratio in [0.0, 0.02, 0.05, 0.1, 0.3]:
    if T_ratio == 0.0:
        nf = np.where(epsilon <= EF, 1.0, 0.0)
        label = 'T = 0'
    else:
        kBT = T_ratio * EF
        nf = 1.0 / (np.exp((epsilon - EF) / kBT) + 1.0)
        label = 'T/T_F = ' + str(T_ratio)
    ax1.plot(epsilon, nf, linewidth=2, label=label)

ax1.axvline(x=1.0, color='#fbbf24', linestyle=':', alpha=0.5)
ax1.set_xlabel('epsilon / E_F')
ax1.set_ylabel('n_F(epsilon)')
ax1.set_title('Fermi-Dirac Distribution')
ax1.set_ylim(-0.05, 1.15)
ax1.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Chemical potential vs T ---
ax2 = axes[0, 1]
T_vals = np.linspace(0.001, 1.0, 200)  # T/T_F

# Exact chemical potential from Sommerfeld (2nd + 4th order)
mu_sommerfeld = 1.0 - (np.pi**2 / 12) * T_vals**2 - (np.pi**4 / 80) * T_vals**4

# Numerical: solve for mu such that integral of g(e)*nF = N
# For 3D: n*lambda^3 = (4/sqrt(pi)) * f_{3/2}(z)
# Use Sommerfeld for illustration
mu_2nd = 1.0 - (np.pi**2 / 12) * T_vals**2

ax2.plot(T_vals, mu_sommerfeld, color='#f87171', linewidth=2, label='Sommerfeld (4th order)')
ax2.plot(T_vals, mu_2nd, color='#60a5fa', linewidth=1.5, linestyle='--', label='Sommerfeld (2nd order)')
ax2.axhline(y=1.0, color='#fbbf24', linestyle=':', alpha=0.5, label='E_F')
ax2.axhline(y=0.0, color='white', linestyle=':', alpha=0.2)
ax2.set_xlabel('T / T_F')
ax2.set_ylabel('mu / E_F')
ax2.set_title('Chemical Potential vs Temperature')
ax2.legend(fontsize=8, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Specific heat ---
ax3 = axes[1, 0]
T_range = np.linspace(0.001, 0.5, 200)

# Electronic: C_el = gamma * T = (pi^2/2) * (T/TF) * NkB
cv_electronic = (np.pi**2 / 2.0) * T_range  # in units of NkB

# Debye phonon: C_phonon = A * T^3, choose A for illustration
theta_D_over_TF = 0.005  # Debye temp / Fermi temp ratio
cv_phonon = 234.0 * (T_range / (theta_D_over_TF * 300))**3
cv_phonon = np.minimum(cv_phonon, 3.0)  # cap at Dulong-Petit

cv_total = cv_electronic + cv_phonon

ax3.plot(T_range, cv_electronic, color='#f87171', linewidth=2, label='Electronic (gamma T)')
ax3.plot(T_range, cv_phonon, color='#60a5fa', linewidth=2, label='Phonon (Debye T^3)')
ax3.plot(T_range, cv_total, color='#34d399', linewidth=2, linestyle='--', label='Total')
ax3.set_xlabel('T / T_F')
ax3.set_ylabel('C_V / (N k_B)')
ax3.set_title('Low-T Specific Heat of a Metal')
ax3.set_ylim(0, 0.3)
ax3.legend(fontsize=8, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Pauli vs Curie susceptibility ---
ax4 = axes[1, 1]
T_range2 = np.linspace(0.01, 1.0, 200)

# Pauli: chi_P = const = 3*n*mu_B^2 / (2*E_F) -- normalize to 1 at T=0
chi_pauli = np.ones_like(T_range2)

# Curie: chi_C = n*mu_B^2 / (kBT) -- ratio to Pauli
chi_curie = (2.0 / 3.0) / T_range2  # chi_C / chi_P = (2/3) * T_F/T

# Sommerfeld correction to Pauli
chi_pauli_corrected = 1.0 - (np.pi**2 / 12) * T_range2**2

ax4.plot(T_range2, chi_pauli, color='#f87171', linewidth=2, label='Pauli (degenerate)')
ax4.plot(T_range2, chi_pauli_corrected, color='#a78bfa', linewidth=2, linestyle='--', label='Pauli + correction')
ax4.plot(T_range2, np.minimum(chi_curie, 10), color='#fbbf24', linewidth=2, label='Curie (classical)')
ax4.set_xlabel('T / T_F')
ax4.set_ylabel('chi / chi_Pauli(0)')
ax4.set_title('Pauli vs Curie Susceptibility')
ax4.set_ylim(0, 5)
ax4.legend(fontsize=8, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

# Numerical results
print("\n" + "=" * 60)
print("FERMI SYSTEMS: NUMERICAL RESULTS")
print("=" * 60)

print("\n--- Fermi Energy for Common Metals ---")
hbar = 1.055e-34
me = 9.109e-31
kB = 1.381e-23
eV = 1.602e-19

metals = [
    ("Copper (Cu)", 8.49e28),
    ("Aluminum (Al)", 18.07e28),
    ("Sodium (Na)", 2.65e28),
    ("Gold (Au)", 5.90e28),
]
for name, n_e in metals:
    EF_val = hbar**2 / (2 * me) * (3 * np.pi**2 * n_e)**(2.0/3.0)
    TF_val = EF_val / kB
    print("  " + name + ": E_F = " + str(round(EF_val / eV, 2)) + " eV, T_F = " + str(int(TF_val)) + " K")

print("\n--- Sommerfeld Coefficient gamma ---")
print("  gamma = pi^2 * N * kB / (2 * T_F)")
for name, n_e in metals:
    EF_val = hbar**2 / (2 * me) * (3 * np.pi**2 * n_e)**(2.0/3.0)
    TF_val = EF_val / kB
    gamma = np.pi**2 * n_e * kB / (2.0 * TF_val)  # J / (m^3 K^2)
    gamma_mJ = gamma * 1e-3 / (n_e / 1e28)  # rough per-mole
    print("  " + name + ": gamma/(n*kB) = " + str(round(np.pi**2 / (2 * TF_val) * 1e6, 4)) + " * 1e-6 / K")

print("\n--- Electronic vs Classical Specific Heat (T = 300 K) ---")
for name, n_e in metals:
    EF_val = hbar**2 / (2 * me) * (3 * np.pi**2 * n_e)**(2.0/3.0)
    TF_val = EF_val / kB
    ratio = np.pi**2 / 2.0 * 300.0 / TF_val / 1.5
    print("  " + name + ": C_el/C_classical = " + str(round(ratio, 5)))

print("\n--- Degeneracy Pressure ---")
for name, n_e in metals:
    EF_val = hbar**2 / (2 * me) * (3 * np.pi**2 * n_e)**(2.0/3.0)
    P0 = 2.0 / 5.0 * n_e * EF_val
    print("  " + name + ": P_0 = " + str(round(P0 / 1e9, 1)) + " GPa")

print("\n--- White Dwarf Parameters ---")
M_sun = 1.989e30
R_earth = 6.371e6
M_Ch = 1.4 * M_sun
print("  Chandrasekhar mass: ~1.4 M_sun = " + str(round(M_Ch / 1e30, 2)) + " x 10^30 kg")
print("  Typical WD radius: ~R_earth = " + str(round(R_earth / 1e6, 1)) + " x 10^6 m")
print("  Central density: ~10^9 kg/m^3")
print("\n" + "=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

8. Summary and Key Results

Core Formulas

Fermi energy: \(E_F = \frac{\hbar^2}{2m}(3\pi^2 n)^{2/3}\)
Ground state energy: \(U_0 = \frac{3}{5}NE_F\)
Specific heat: \(C_V = \frac{\pi^2}{2}Nk_B\frac{T}{T_F}\)
Pauli susceptibility: \(\chi_P = \mu_0\mu_B^2 g(E_F)\)

Physical Insights

Degeneracy pressure exists even at T = 0
Only electrons within \(\sim k_BT\) of \(E_F\) contribute
Sommerfeld expansion: corrections in powers of \((T/T_F)^2\)
Pauli \(\chi\) is T-independent; suppressed by \(T/T_F\) vs Curie

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