The Drude model successfully predicts the Wiedemann-Franz law (κ/σT = L₀) relating thermal and electrical conductivity. Why does this work despite the model's incorrect treatment of electron statistics?
AThe Wiedemann-Franz law is independent of electron statistics
BBoth thermal and electrical conductivity depend on the same relaxation time τ and carrier density n. When you form the ratio κ/σT, the incorrectly estimated velocity and specific heat per electron cancel, leaving a ratio that depends only on fundamental constants — the errors compensate in the ratio
CDrude used quantum mechanics for the thermal conductivity calculation
DThe Wiedemann-Franz law only works at high temperatures where classical statistics are valid
In the Drude model, σ = ne²τ/m and κ = (1/3)nv²τc_v, where the classical values of v² = 3k_BT/m and c_v = 3k_B/2 give κ/σT = 3(k_B/e)² / 2. The Sommerfeld model replaces v² → v_F² and c_v → (π²/2)(k_BT/E_F)k_B, but these changes cancel in the ratio, giving κ/σT = π²k_B²/(3e²) = L₀, the Lorenz number. The numerical prefactor changes (Sommerfeld gets π²/3 instead of 3/2) but is closer to experiment. The cancellation of errors is a beautiful accident.
Question 2 Multiple Choice
Why does the Drude model overestimate the electronic specific heat of metals by a factor of ~100?
AThe Drude model ignores electron-electron interactions
BClassically, all n electrons each contribute 3k_B/2 to the heat capacity. Quantum mechanically (Sommerfeld), only electrons within ~k_BT of the Fermi level can absorb thermal energy — a fraction ~k_BT/E_F of the total — reducing the specific heat by a factor of ~T/T_F ≈ 1/100 at room temperature
CThe Drude model uses the wrong value of the electron mass
DPhonon contributions mask the electronic specific heat in the Drude model
For typical metals, E_F ~ 5-10 eV and room temperature k_BT ~ 0.025 eV, so k_BT/E_F ~ 1/200-1/400. The Pauli exclusion principle 'freezes out' most electrons: those deep in the Fermi sea have no empty states nearby to be excited into. Only the thin shell of electrons within ~k_BT of E_F can participate in thermal processes. This was one of the great triumphs of applying quantum statistics to metals — it resolved the long-standing puzzle of why the electronic contribution to specific heat was far smaller than the classically expected value.
Question 3 True / False
The Drude model predicts a Hall coefficient R_H = -1/ne for a metal with n free electrons per unit volume. Some real metals (e.g., aluminum, beryllium) have positive Hall coefficients. Does this falsify the free-electron picture?
TTrue
FFalse
Answer: False
A positive Hall coefficient does not falsify free-electron physics per se — it reveals the limitations of the single-band free-electron model. In metals with multiple partially filled bands, the Hall coefficient depends on the contributions from both electron-like and hole-like carriers. If hole carriers dominate the Hall response (which can happen due to band structure effects and anisotropic scattering), R_H becomes positive. Beryllium and aluminum are classic examples: their band structures create hole pockets that dominate the Hall effect. A multi-band Drude model or full Boltzmann transport calculation correctly accounts for these cases.
Question 4 Short Answer
Explain the key physical improvement Sommerfeld made over Drude, and why it matters for understanding metals.
Think about your answer, then reveal below.
Model answer: Sommerfeld replaced the classical Maxwell-Boltzmann distribution with the quantum Fermi-Dirac distribution for the conduction electrons, while keeping the free-electron (no lattice potential) and relaxation-time approximations. This single change resolves the specific heat anomaly (C_el ∝ T instead of constant), correctly predicts the Pauli paramagnetic susceptibility (temperature-independent, proportional to g(E_F)), and fixes the magnitude of the thermopower. The essential physics is that Fermi-Dirac statistics restrict thermal excitations to the narrow energy window ~k_BT around E_F, rather than allowing all electrons to participate equally.
Historically, the specific heat problem was devastating for the Drude model — equipartition demanded a huge electronic contribution that experiments simply did not show. Sommerfeld's fix (1928) showed that quantum statistics, not some failure of the free-electron picture, was the missing ingredient.