Debye Model

Debye Model of Heat Capacity

The Debye model treats lattice vibrations (phonons) as sound waves with a cutoff frequency ω_D. It correctly predicts the T³ dependence of heat capacity at low temperatures.

Debye Frequency and Temperature

The Debye cutoff frequency is set by the condition that there are exactly 3N phonon modes per N atoms:

$\omega_D = v_s \left(6\pi^2 n\right)^{1/3}$

The corresponding Debye temperature:

$\theta_D = \frac{\hbar \omega_D}{k_B}$

Where v_s is the (Debye-averaged) sound speed and n is the atom number density.

Heat Capacity: Low Temperature Limit (T ≪ θ_D)

$C_V \approx \frac{12\pi^4}{5} N k_B \left(\frac{T}{\theta_D}\right)^3$

This Debye T³ law matches experimental data well below about θ_D/10.

Heat Capacity: High Temperature Limit (T ≫ θ_D)

The Dulong-Petit law — classical equipartition:

$C_V \rightarrow 3 N k_B$

For one mole, this gives ~24.9 J/K regardless of the material.

Physical Constants (inside your functions)

• ħ = 1.055 × 10⁻³⁴ J·s
• k_B = 1.381 × 10⁻²³ J/K