Debye Solid Heat Capacity

Worked Example: Low-Temperature Heat Capacity

Let's trace the calculation for the low-temperature limit of the Debye heat capacity when the reduced temperature is τ = 0.1. In this regime, the approximation C / (3Nk_B) ≈ (4π^4)/5 τ³ holds very well.

1. Identify the given parameters. The reduced temperature is τ = 0.1.
2. Cube the reduced temperature: τ³ = (0.1)³ = 0.001.
3. Calculate the prefactor: (4π^4)/5 ≈ (4 × (3.14159)^4)/5 ≈ (4 × 97.409)/5 ≈ 77.927.
4. Multiply the prefactor by τ³: C / (3Nk_B) ≈ 77.927 × 0.001 = 0.077927.

This value closely matches the numerical integration of the exact Debye formula, which yields approximately 0.0758. At higher temperatures, this cubic approximation breaks down, and the heat capacity limits to the classical value C / (3Nk_B) → 1.

Experiment

You can interact with the Debye Solid Heat Capacity module to explore these behaviors interactively.

1. Set the Reduced temperature to 0.1. Notice how the Debye model output gives a significantly higher heat capacity than the Einstein model, reflecting the contribution of low-frequency acoustic phonons.
2. Change the Reduced temperature to 2. Observe that both models approach the classical limit C / (3Nk_B) → 1, where all phonon modes are fully thermally excited.
3. Modify the Debye temperature K. This shifts the Actual temperature K proportionally without changing the normalized heat capacity C/(3Nk_B) because the physics depends only on the ratio τ = T / \theta_D.

Limitations

The Debye model assumes the solid is an isotropic continuous elastic medium up to a maximum cutoff frequency (the Debye frequency). This approximation accurately captures the low-frequency acoustic modes responsible for the T³ behavior at low temperatures. However, it fails to capture the detailed van Hove singularities present in the true phonon density of states of real crystals. Furthermore, this is purely a lattice heat capacity model; it excludes electronic heat capacity (which is linear in T for metals) and anharmonic effects that become significant near the melting point.

References

This module is based on standard solid state physics resources.

• Tong, D. "Lectures on Statistical Physics." University of Cambridge. Section 3 covers the Debye model of a solid. Read here.