Harmonic Systems: The Einstein Solid and Vibrational Heat Capacity

The harmonic oscillator is the one interacting system whose quantum partition function sums exactly, and the sum is a geometric series. That single result carries a large fraction of statistical mechanics: a solid is a collection of atoms vibrating about their lattice sites, and to a first approximation each atom is three independent oscillators. Einstein's 1907 model builds a solid from $3N$ such oscillators sharing one frequency and, with nothing but the canonical partition function, explains why the heat capacity of a solid falls to zero at low temperature — the fact classical equipartition cannot produce. The model's quantitative failure at the lowest temperatures is as instructive as its success, and it points directly to the Debye refinement.

The harmonic oscillator partition function

A one-dimensional quantum harmonic oscillator of angular frequency $\omega$ has the energy spectrum

with the ground state at $\frac{1}{2}\hslash \omega$, the zero-point energy. The levels are equally spaced by $\hslash \omega$, so the partition function is a geometric series in the ratio $x=e^{-\beta \hslash \omega }$:

The series converges for any $\beta >0$ because $x<1$. The closed form is the single ingredient the rest of the lesson differentiates.¹

Mean energy and the Bose occupation factor

The mean energy follows from $⟨E⟩=-\partial \ln z/\partial \beta$. With $\ln z=-\frac{1}{2}\beta \hslash \omega -\ln (1-e^{-\beta \hslash \omega })$,

The second term is $\hslash \omega$ times the Bose–Einstein occupation factor

the mean number of quanta excited above the ground state, so that $⟨E⟩=\hslash \omega (\frac{1}{2}+⟨n⟩)$. The same factor governs photons and phonons in later modules; here it counts the thermal excitation of one vibrational mode. Two limits fix the physics.²

• High temperature, $\beta \hslash \omega \ll 1$. Expand $e^{\beta \hslash \omega }-1\approx \beta \hslash \omega$, so $⟨n⟩\approx k_{B}T/\hslash \omega$ and $⟨E⟩\approx k_{B}T$. The oscillator holds $k_{B}T$, the equipartition value for two quadratic degrees of freedom (kinetic and potential), and the discreteness of the levels is invisible.
• Low temperature, $\beta \hslash \omega \gg 1$. The occupation is exponentially small, $⟨n⟩\approx e^{-\beta \hslash \omega }\to 0$, and $⟨E⟩\to \frac{1}{2}\hslash \omega$: the oscillator is frozen into its ground state, holding only the zero-point energy. Thermal energy $k_{B}T$ cannot bridge the gap $\hslash \omega$ to the first excited state.

The Einstein model of a solid

Model a monatomic solid of $N$ atoms as a set of independent oscillators. Each atom vibrates in three directions about its lattice site, and Einstein's approximation takes every one of the $3N$ resulting oscillators to have the same frequency $\omega$.³ The oscillators are distinguishable by lattice site, so the partition function factorizes as $Z=z^{3N}$ and the free energy and internal energy are $3N$ times the single-oscillator results:

The zero-point term $\frac{3}{2}N\hslash \omega$ is a constant, and only the second term carries temperature dependence. Define the Einstein temperature${\Theta }_E=\hslash \omega /k_B$, the temperature at which $k_{B}T$ matches the level spacing. It is the single material parameter of the model, set by the stiffness of the interatomic bonds: stiff, light lattices (diamond) have ${\Theta }_E$ of order $10^3\mathrm{K}$, soft, heavy ones (lead) a few tens of kelvin.

The heat capacity

Differentiating $U$ with respect to $T$ gives the heat capacity. Writing $u={\Theta }_E/T=\beta \hslash \omega$,

This is the Einstein heat-capacity function. Its two limits are the content of the model.⁴

• High temperature, $T\gg {\Theta }_E$. With $u\to 0$, $e^u/(e^u-1{)}^2\approx 1/u^2$, so $C_V\to 3N{k}_B$. This is the Dulong–Petit law: the molar heat capacity of a solid approaches $3R$, independent of temperature and of the material. Classical equipartition, giving $k_{B}T$ per oscillator and $C_V=3N{k}_B$ at all temperatures, reproduces only this limit.
• Low temperature, $T\ll {\Theta }_E$. The exponential dominates, $C_V\approx 3N{k}_B({\Theta }_E/T{)}^2{e}^{-{\Theta }_E/T}$, which vanishes as $T\to 0$. The heat capacity is quenched because the energy gap $\hslash \omega$ far exceeds $k_{B}T$ and the oscillators cannot be excited. This freeze-out is the qualitative success the classical theory could not deliver: the third law requires $C_V\to 0$, and equipartition violates it.

The crossover is fast. The heat capacity reaches most of its Dulong–Petit value within a factor of two of the Einstein temperature, and freezes out quickly below it.

The low-temperature discrepancy

The Einstein model gets the low-temperature trend right — $C_V\to 0$ — but the rate wrong. Measured heat capacities of insulating crystals fall as $T^3$ at low temperature, a power law, whereas the Einstein prediction $C_V\sim ({\Theta }_E/T{)}^2{e}^{-{\Theta }_E/T}$ falls exponentially, far faster. Experiment lies above the Einstein curve at low $T$, so the model underestimates the heat capacity there.⁵

The source of the discrepancy is the assumption of a single frequency. A real lattice supports a spectrum of vibrational modes, from long-wavelength acoustic waves of arbitrarily low frequency up to a maximum set by the interatomic spacing. The low-frequency modes have $\hslash \omega \ll k_{B}T$ even at low temperature and remain thermally active after the high-frequency modes have frozen out; their number grows as ${\omega }^2$, and integrating the oscillator heat capacity over that spectrum produces the $T^3$ law. The Debye model, which replaces the single Einstein frequency by exactly this acoustic spectrum with a cutoff, is the subject of a later module; the Einstein model is its single-frequency caricature, correct wherever one optical frequency dominates and qualitatively right everywhere.

Summary

• The quantum harmonic oscillator has a geometric partition function $z=[2\sinh (\beta \hslash \omega /2){]}^{-1}$, and its mean energy is $⟨E⟩=\hslash \omega (\frac{1}{2}+⟨n⟩)$ with the Bose occupation $⟨n⟩=(e^{\beta \hslash \omega }-1{)}^{-1}$.
• At high temperature $⟨E⟩\to k_{B}T$ (equipartition); at low temperature it freezes to the zero-point energy $\frac{1}{2}\hslash \omega$.
• The Einstein solid of $3N$ identical oscillators has heat capacity $C_V=3N{k}_B({\Theta }_E/T{)}^2{e}^{{\Theta }_E/T}/(e^{{\Theta }_E/T}-1{)}^2$, rising from zero to the Dulong–Petit value $3N{k}_B$ as $T$ passes the Einstein temperature ${\Theta }_E=\hslash \omega /k_B$.
• The model correctly gives $C_V\to 0$ at low $T$ but with an exponential falloff, too steep against the measured $T^3$; the missing low-frequency acoustic modes are supplied by the Debye model.