Schottky Anomaly and the Limits of Equipartition
A solid contains N = 1.0×10^22 independent two-level systems with energy gap Δ = 0.050 eV. A calorimetry student measures the heat capacity between 10 K and 600 K. At 300 K the student uses classical equipartition, assigns k_B/2 per two-level system, and predicts a constant C ≈ 69 J/K. The measured value is about 58 J/K, and the measured curve has a pronounced maximum near 240 K before falling at higher temperature. The student is confused because the two-level system has a fixed gap, not a continuum. You are asked to derive the heat capacity from the canonical partition function, locate the peak temperature, and explain why equipartition fails even though k_B T is comparable to Δ.
1. Derive the partition function, internal energy, and heat capacity for one two-level system.
Read the worked answer
Take the lower level as zero energy and the upper level as Δ. The single-system partition function is
Z = 1 + e^(-βΔ), β = 1/(k_B T).
The mean energy is
U = -(∂ ln Z/∂β) = Δ e^(-βΔ)/(1 + e^(-βΔ)) = Δ/(e^(βΔ) + 1).
The heat capacity follows from C = dU/dT. With x = βΔ,
C/Nk_B = x² e^x/(1 + e^x)².
This is the Schottky heat capacity. It is not a constant: it vanishes both as T → 0 and as T → ∞, because the population difference saturates at low T and both levels become equally populated at high T.
2. Find the temperature of the heat-capacity maximum and evaluate C at 300 K.
Read the worked answer
Using the identity e^x/(1+e^x)² = 1/[4 cosh²(x/2)],
C/Nk_B = x²/[4 cosh²(x/2)].
Differentiating ln(C/Nk_B) gives the maximum condition
2/x = tanh(x/2),
whose nonzero solution is x ≈ 2.399. Therefore
T_max = Δ/(k_B x) = 0.050 eV/(8.617×10^-5 eV/K × 2.399) ≈ 242 K,
consistent with the observed peak near 240 K. At 300 K, x = 0.050/(8.617×10^-5 × 300) ≈ 1.93. Then C/Nk_B ≈ 0.413, so C ≈ 0.413 × 1.0×10^22 × 1.381×10^-23 ≈ 57 J/K, close to the measured 58 J/K.
3. Explain why equipartition fails and describe the low- and high-temperature limits.
Read the worked answer
Equipartition assigns k_B/2 per quadratic classical degree of freedom, but a two-level system is not a harmonic coordinate with a continuous energy. Its energy is bounded between 0 and Δ, so its heat capacity must vanish when thermal fluctuations are either too weak to excite the upper level or so strong that both levels are already equally populated. At low T, with k_B T << Δ, C/Nk_B ≈ x² e^(-x), exponentially small. At high T, with k_B T >> Δ, C/Nk_B ≈ x²/4, which falls as 1/T². The peak near 242 K is the Schottky anomaly. The student's constant 69 J/K is not a valid equipartition limit; it happens to be close to the 300 K value only because that point lies on the shoulder of the peak, not on a plateau.