Einstein Solid Heat Capacity

In 1907, Albert Einstein proposed a model to explain the low-temperature deviation of the heat capacity of solids from the classical Dulong-Petit law. The Einstein solid models a lattice of N atoms as 3N independent quantum harmonic oscillators, all vibrating at the same frequency ω.

Worked example: Einstein Heat Capacity

Let us consider a solid modeled with 3N oscillators at a reduced temperature τ = T / \theta_E = 1, where \theta_E = ℏω / k_B is the Einstein temperature. The thermal energy is given by the mean occupation of the oscillators.

The mean mode occupation is 1 / (e^(1/τ) - 1). For τ = 1, this is 1 / (e^1 - 1) ≈ 0.5819767.
The classical heat capacity is 3Nk_B. The ratio of the quantum heat capacity to the classical limit is C_V / (3Nk_B) = (1/τ)² e^(1/τ) / (e^(1/τ) - 1)².
For τ = 1, the heat capacity ratio is (1)² e^1 / (e^1 - 1)² ≈ 0.9206736.
The quantum capacity deficit is the difference from the classical limit: 1 - C_V / (3Nk_B) ≈ 0.0793264.

Native experiment

Open Einstein Solid Heat Capacity to explore how the heat capacity responds to different reduced temperatures.

Set Reduced temperature to 5. Notice that the Heat capacity ratio is very close to 1 (about 0.9967), approaching the classical Dulong-Petit limit.
Set Reduced temperature to 0.5. The Heat capacity ratio drops significantly to roughly 0.724, showing the beginning of quantum suppression.
Set Reduced temperature to 0.1. The Heat capacity ratio plummets to about 0.0045. The modes are "frozen out" because the thermal energy k_B T is much less than the level spacing ℏω.
Set Reduced temperature to 0. At absolute zero, the Heat capacity ratio is exactly 0 and the quantum capacity deficit is 1.

Model limitations

The Einstein model assumes that all 3N oscillators vibrate at the exact same frequency (a single-mode assumption). While this correctly predicts the exponential suppression of heat capacity at very low temperatures, it fails to match the precise T³ dependence observed in real low-temperature crystals. Peter Debye later improved this model by treating the solid as a continuous elastic medium with a spectrum of frequencies, which correctly reproduces the T³ law at low temperatures.

References

A detailed derivation of the Einstein model of a solid from statistical mechanics, including the mean occupation number and heat capacity calculations, is available in Thermodynamics and Statistical Mechanics by Richard Fitzpatrick.

Einstein Solid Heat Capacity