Engineering & Science
Sommerfeld Electronic Heat Capacity (γT) & Phonon Crossover Temperature
Calculate the free-electron Sommerfeld coefficient γ = (π^2 / 2) * Z * R / T_F (mJ/(mol·K^2)), linear electronic heat capacity C_el = γT, and cryogenic electron-phonon crossover temperature T_cross = SQRT(γ / A_ph).
How Fermi-Dirac quantum statistics resolved the classical heat-capacity catastrophe!
Classical equipartition predicted every conduction electron in copper should contribute (3/2)k_B to heat capacity—an extra 12.47 J/(mol·K) that experiments never saw! Arnold Sommerfeld (1928) applied Pauli exclusion to the degenerate Fermi sea and proved that only electrons within ~k_B T of the Fermi temperature T_F can thermally excite, suppressing molar electronic heat capacity by T / T_F into the linear law C_el = γ * T, where γ = (π^2 / 2) * Z * R / T_F!
At room temperature (300 K), copper’s Fermi temperature T_F = 81,600 K (valence Z = 1) yields γ = 0.5028 mJ/(mol·K^2) and C_el = 150.85 mJ/(mol·K)—less than 1% of lattice Dulong-Petit heat capacity! Yet below liquid-helium temperatures, Debye lattice heat capacity A_ph * T^3 collapses much faster than γ * T: for copper (A_ph = 0.0478 mJ/(mol·K^4)), electrons overtake phonons below T_cross = SQRT(γ / A_ph) = 3.24 K!
| Metal Cryogenic Regime (Cu: TF = 81,600 K, Z = 1) | Sommerfeld γ (mJ/mol·K^2) | Electronic C_el = γT | Dominant Thermal Carrier |
|---|---|---|---|
| Liquid Helium Bath (T = 4.0 K) | 0.5028 | 2.011 mJ/(mol·K) | Comparable to Debye A·T^3 (3.059) |
| Crossover Point (T_cross = 3.24 K) | 0.5028 | 1.631 mJ/(mol·K) | Exact C_el = C_phonon Equality! |
| Room Temperature (T = 300 K) | 0.5028 | 150.85 mJ/(mol·K) | Lattice Phonons Dominate (>99%) |
Sommerfeld Electronic Molar Heat Capacity C_el = γ * T (mJ/(mol·K))
Evaluates γ = 1000 * (PI()^2 / 2) * Z * 8.3144626 / T_F in mJ/(mol·K^2) and multiplies by temperature T (B4).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0), "Invalid Fermi Inputs", ROUND(1000 * (PI() ^ 2 / 2) * B2 * 8.3144626 / B3 * B4, 3))Cryogenic Electron-Phonon Crossover Temperature T_cross = SQRT(γ / A_ph) (K)
Finds the exact temperature below which linear electronic heat capacity exceeds cubic Debye phonon heat capacity A_ph * T^3.
=IF(OR(B2 <= 0, B3 <= 0, B5 <= 0), "Invalid Fermi Inputs", ROUND(SQRT((1000 * (PI() ^ 2 / 2) * B2 * 8.3144626 / B3) / B5), 2))Try the example
Use the original values or change them to test your understanding.