Einstein solid

Crystal model of independent identical quantum oscillators; yields an explicit heat capacity curve with the Dulong–Petit high-temperature limit.

Einstein solid

Model

The Einstein solid models a crystal of $N$ atoms as $3N$ independent quantum harmonic oscillators, all with the same frequency $\omega_E$ (three vibrational modes per atom). Thermal equilibrium is taken in the canonical ensemble at temperature $T$ .

This is essentially the quantum harmonic oscillator replicated $3N$ times.

Partition function

Let $Z_{\mathrm{osc}}(\beta)$ be the single-oscillator partition function:

Z_{\mathrm{osc}}(\beta)=\frac{e^{-\beta\hbar\omega_E/2}}{1-e^{-\beta\hbar\omega_E}}.

Independence gives

Z_N(\beta)=\bigl(Z_{\mathrm{osc}}(\beta)\bigr)^{3N}.

Free energy, internal energy, and entropy

The Helmholtz free energy is

F(\beta)=-\beta^{-1}\ln Z_N(\beta) =3N\left(\frac{\hbar\omega_E}{2}+\beta^{-1}\ln\!\bigl(1-e^{-\beta\hbar\omega_E}\bigr)\right).

The internal energy is

U(\beta)=-\frac{\partial}{\partial\beta}\ln Z_N(\beta) =3N\left(\frac{\hbar\omega_E}{2}+\frac{\hbar\omega_E}{e^{\beta\hbar\omega_E}-1}\right).

Let $\bar n = (e^{\beta\hbar\omega_E}-1)^{-1}$ . Then the entropy can be written as

S(\beta)=3N k_B\Bigl[(\bar n+1)\ln(\bar n+1)-\bar n\ln\bar n\Bigr],

consistent with thermodynamic entropy in the canonical setting.

Heat capacity

The heat capacity at constant volume (see heat capacity at constant volume ) is

C_V =3N k_B(\beta\hbar\omega_E)^2\,\frac{e^{\beta\hbar\omega_E}}{(e^{\beta\hbar\omega_E}-1)^2}.

Defining the Einstein temperature $\Theta_E=\hbar\omega_E/k_B$ , this becomes

C_V = 3N k_B\left(\frac{\Theta_E}{T}\right)^2 \frac{e^{\Theta_E/T}}{(e^{\Theta_E/T}-1)^2}.

Limiting behavior and interpretation

High temperature $T\gg \Theta_E$ :
$C_V \to 3N k_B,$
reproducing the Dulong–Petit law.
Low temperature $T\ll \Theta_E$ :
$C_V \sim 3N k_B\left(\frac{\Theta_E}{T}\right)^2 e^{-\Theta_E/T},$
which decays exponentially and therefore misses the observed $T^3$ law in many crystalline solids.

The Debye model modifies the mode spectrum to recover the correct low-temperature behavior.