Einstein Model

Einstein Model of Heat Capacity

The Einstein model (1907) was the first quantum mechanical treatment of lattice heat capacity. It assumes all N atoms oscillate independently at the same frequency ω_E.

Einstein Temperature

$\theta_E = \frac{\hbar \omega_E}{k_B}$

This characteristic temperature separates the quantum (T ≪ θ_E) from the classical (T ≫ θ_E) regimes.

Heat Capacity

$C_V = 3Nk_B \left(\frac{\theta_E}{T}\right)^2 \frac{e^{\theta_E/T}}{\left(e^{\theta_E/T} - 1\right)^2}$

Limits:

• High T (T ≫ θ_E): C_V → 3Nk_B (Dulong-Petit law)
• Low T (T ≪ θ_E): C_V → 3Nk_B (θ_E/T)² exp(-θ_E/T) (exponential suppression)

The Einstein model correctly captures the quantum freeze-out of vibrations at low T, though it falls off too fast compared to experiment (the Debye T³ law is better at very low T).

Mean Energy per Oscillator

Including the zero-point energy (½ħω per mode, 3 modes):

$\langle E \rangle = \frac{3k_B\theta_E}{e^{\theta_E/T} - 1} + \frac{3}{2}k_B\theta_E$

Physical Constants (inside your functions)

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