The Einstein solid is a simple model of a crystalline solid, introduced by Einstein in 1907 to explain why the heat capacity of solids decreases at low temperatures—a phenomenon classical physics could not explain. It was one of the early triumphs of quantum theory applied to condensed matter.
The Model
Einstein made two key simplifying assumptions:
- Each atom in the solid vibrates independently about its equilibrium lattice position
- All atoms vibrate at the same frequency \(\omega\)
A solid of \(N\) atoms thus becomes \(3N\) independent quantum harmonic oscillators (three spatial directions per atom), each with energy levels
\[ E_n = \hbar\omega\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \ldots \]
The model ignores the fact that atomic vibrations are actually coupled (a simplification the Debye model later addressed), but it captures the essential quantum behaviour.
Partition Function
For a single oscillator, the partition function is
\[ Z_1 = \sum_{n=0}^{\infty} e^{-\beta \hbar\omega(n + 1/2)} = e^{-\beta\hbar\omega/2} \sum_{n=0}^{\infty} \left(e^{-\beta\hbar\omega}\right)^n \]
The geometric series sums to give
\[ Z_1 = \frac{e^{-\beta\hbar\omega/2}}{1 - e^{-\beta\hbar\omega}} = \frac{1}{2\sinh(\beta\hbar\omega/2)} \]
Since the oscillators are independent, the total partition function for \(3N\) oscillators is
\[ Z = Z_1^{3N} = \left(\frac{1}{2\sinh(\beta\hbar\omega/2)}\right)^{3N} \]
Average Energy
The average energy follows from
\[ \langle E \rangle = -\frac{\partial \ln Z}{\partial \beta} = 3N\hbar\omega\left(\frac{1}{2} + \frac{1}{e^{\beta\hbar\omega} - 1}\right) \]
This can be written more transparently by introducing the Einstein temperature
\[ \Theta_E = \frac{\hbar\omega}{k_B} \]
so that
\[ \langle E \rangle = 3Nk_B\Theta_E\left(\frac{1}{2} + \frac{1}{e^{\Theta_E/T} - 1}\right) \]
The first term is the zero-point energy; the second is the thermal excitation energy, with the Bose-Einstein occupation factor for each mode.
Heat Capacity
The heat capacity at constant volume is
\[ C_V = \frac{\partial \langle E \rangle}{\partial T} = 3Nk_B \left(\frac{\Theta_E}{T}\right)^2 \frac{e^{\Theta_E/T}}{(e^{\Theta_E/T} - 1)^2} \]
This is often written in terms of the Einstein function \(f_E(x)\) where \(x = \Theta_E/T\):
\[ C_V = 3Nk_B \, f_E\left(\frac{\Theta_E}{T}\right), \quad f_E(x) = \frac{x^2 e^x}{(e^x - 1)^2} \]
Limiting Behaviour
High temperature (\(T \gg \Theta_E\)):
When \(x = \Theta_E/T \ll 1\), we can expand \(e^x \approx 1 + x\) to get \(f_E(x) \to 1\), so
\[ C_V \to 3Nk_B \]
This is the Dulong-Petit law: each quadratic degree of freedom contributes \(\frac{1}{2}k_B T\) to the energy (and \(\frac{1}{2}k_B\) to the heat capacity), giving \(k_B T\) per oscillator, or \(3k_B\) per atom. Classical equipartition is recovered.
Low temperature (\(T \ll \Theta_E\)):
When \(x \gg 1\), we have \(e^x \gg 1\), so
\[ f_E(x) \approx x^2 e^{-x} \]
and the heat capacity vanishes exponentially:
\[ C_V \approx 3Nk_B \left(\frac{\Theta_E}{T}\right)^2 e^{-\Theta_E/T} \]
This was Einstein’s key result: quantum mechanics naturally explains why heat capacities decrease at low temperatures. When \(k_B T \ll \hbar\omega\), there isn’t enough thermal energy to excite oscillators out of the ground state, so they “freeze out” and stop contributing to the heat capacity.
Historical Significance
Before Einstein’s model, the Dulong-Petit law (\(C_V = 3Nk_B\)) was a well-established empirical rule, but it was known to fail at low temperatures—diamond, for instance, has anomalously low heat capacity at room temperature. Classical physics predicted a temperature-independent heat capacity, in clear contradiction with experiment.
Einstein’s 1907 paper showed that quantising the oscillator energies resolves the puzzle. The model gives qualitatively correct behaviour: \(C_V\) interpolates smoothly from zero at \(T = 0\) to the classical value at high \(T\).
Limitations and the Debye Model
The Einstein model predicts an exponential decay of \(C_V\) at low temperatures, but experiments show a power law: \(C_V \propto T^3\). The discrepancy arises because real solids have a spectrum of vibrational frequencies (phonons), not a single frequency.
The Debye model improves on this by treating the solid as an elastic continuum with a distribution of phonon modes up to a cutoff frequency. It correctly predicts the \(T^3\) behaviour at low temperatures while still recovering Dulong-Petit at high temperatures.
Despite its simplicity, the Einstein model remains valuable pedagogically: it’s exactly solvable, demonstrates the essential role of quantisation, and gives the right qualitative physics with minimal mathematical machinery.