---
title: "Harmonic Systems: The Einstein Solid and Vibrational Heat Capacity"
module: The Canonical Ensemble
moduleNumber: 4
lessonNumber: 4
order: 404
summary: >
  A quantum harmonic oscillator has a geometric partition function summed in
  closed form, giving a mean energy $\hbar\omega(\tfrac12+\langle n\rangle)$ with
  the Bose occupation factor. Modeling a solid as $3N$ independent oscillators
  yields a heat capacity that rises from zero and saturates at the Dulong–Petit
  value $3Nk_B$. The Einstein temperature sets the crossover, and the model's
  exponential low-temperature falloff, too steep against the observed $T^3$,
  motivates the Debye theory.
topics: [The Canonical Ensemble]
sources:
  - book: Schroeder
    ref: "Ch. 2 §2.2, Ch. 3 §3.3, and Ch. 7 §7.5 (Debye comparison)"
  - book: Reif
    ref: "Ch. 7 — Simple Applications of Statistical Mechanics; §7.6–7.7"
  - book: Pathria & Beale
    ref: "Ch. 3 — The Canonical Ensemble; §3.8 The Statistics of the Harmonic Oscillator"
  - book: Tipler & Llewellyn
    ref: "Ch. 8 — Statistical Physics; §8-1 (Dulong–Petit and its failure)"
draft: false
---

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

$$
E_n = \hbar\omega\Big(n + \tfrac{1}{2}\Big), \qquad n = 0, 1, 2, \dots,
$$

with the ground state at $\tfrac12\hbar\omega$, the zero-point energy. The levels
are equally spaced by $\hbar\omega$, so the partition function is a geometric
series in the ratio $x = e^{-\beta\hbar\omega}$:

$$
z = \sum_{n=0}^{\infty} e^{-\beta\hbar\omega(n+1/2)}
= e^{-\beta\hbar\omega/2}\sum_{n=0}^{\infty} x^{n}
= \frac{e^{-\beta\hbar\omega/2}}{1 - e^{-\beta\hbar\omega}}
= \frac{1}{2\sinh(\beta\hbar\omega/2)}.
$$

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

$$
% caption: The oscillator has equally spaced levels $E_n=\hbar\omega(n+\tfrac12)$; the Boltzmann weights $e^{-\beta E_n}$, shown as bar lengths, decrease geometrically by the fixed ratio $e^{-\beta\hbar\omega}$ from one level to the next.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,-0.1)--(0,4.7) node[above,black]{$E$};
\foreach \n/\y/\w in {0/0.5/3.4, 1/1.35/2.0, 2/2.2/1.2, 3/3.05/0.72, 4/3.9/0.43}{
  \draw[black,very thick] (-0.15,\y)--(0.15,\y);
  \node[black!70,left,font=\scriptsize] at (-0.2,\y) {$E_\n$};
  \draw[acc,very thick,fill=acc!14] (0.5,\y-0.11) rectangle (0.5+\w,\y+0.11);
}
\node[acc,font=\scriptsize] at (4.6,0.5) {largest weight};
\draw[<->,black] (4.9,0.5)--(4.9,1.35) node[midway,right,font=\scriptsize,black!70]{even spacing};
\node[black!70,font=\scriptsize] at (2.2,4.4) {geometric decrease upward};
\end{tikzpicture}
$$

## Mean energy and the Bose occupation factor

The mean energy follows from $\langle E\rangle = -\partial\ln z/\partial\beta$.
With $\ln z = -\tfrac12\beta\hbar\omega - \ln(1 - e^{-\beta\hbar\omega})$,

$$
\langle E\rangle = -\frac{\partial\ln z}{\partial\beta}
= \frac{\hbar\omega}{2}
+ \frac{\hbar\omega\, e^{-\beta\hbar\omega}}{1 - e^{-\beta\hbar\omega}}
= \frac{\hbar\omega}{2} + \frac{\hbar\omega}{e^{\beta\hbar\omega} - 1}.
$$

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

$$
\langle n\rangle = \frac{1}{e^{\beta\hbar\omega} - 1},
$$

the mean number of quanta excited above the ground state, so that
$\langle E\rangle = \hbar\omega\big(\tfrac12 + \langle n\rangle\big)$. 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.[^reif-osc]

- **High temperature**, $\beta\hbar\omega \ll 1$. Expand
  $e^{\beta\hbar\omega} - 1 \approx \beta\hbar\omega$, so
  $\langle n\rangle \approx k_B T/\hbar\omega$ and
  $\langle E\rangle \approx k_B T$. 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**, $\beta\hbar\omega \gg 1$. The occupation is exponentially
  small, $\langle n\rangle \approx e^{-\beta\hbar\omega}\to 0$, and
  $\langle E\rangle \to \tfrac12\hbar\omega$: the oscillator is frozen into its
  ground state, holding only the zero-point energy. Thermal energy $k_B T$ cannot
  bridge the gap $\hbar\omega$ to the first excited state.

$$
% caption: The mean oscillator energy crosses over from the frozen zero-point value $\tfrac12\hbar\omega$ at low temperature to the classical line $k_BT$ at high temperature, where the level spacing is negligible against $k_BT$.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(7.2,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,4.4) node[above,black]{$U$};
\draw[black,dashed] (0,0.6)--(7.0,0.6);
\node[black!70,font=\scriptsize,right] at (5.1,0.35) {zero-point energy};
\draw[black,very thick,dashed] (0,0)--(6.6,3.4);
\node[black] at (5.7,3.25) {$k_BT$};
\draw[acc,very thick] plot[domain=0.28:6.6,samples=110]
  (\x,{0.6+0.721/(exp(1.4/\x)-1)});
\node[acc] at (3.4,2.35) {mean energy};
\end{tikzpicture}
$$

## 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 $\omega$.[^schroeder-einstein] 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:

$$
U = 3N\hbar\omega\left(\frac{1}{2} + \frac{1}{e^{\beta\hbar\omega} - 1}\right).
$$

The zero-point term $\tfrac32 N\hbar\omega$ is a constant, and only the second
term carries temperature dependence. Define the **Einstein temperature**
$\Theta_E = \hbar\omega/k_B$, 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
$u = \Theta_E/T = \beta\hbar\omega$,

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

This is the Einstein heat-capacity function. Its two limits are the content of the
model.[^tl-dulong]

- **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 $\hbar\omega$ 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.

$$
% caption: The Einstein heat capacity rises from zero at low temperature to the Dulong–Petit plateau $3Nk_B$; the Einstein temperature $\Theta_E$ marks the crossover, near which the curve turns over.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(7.4,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,4.2) node[above,black]{$C_V$};
\draw[black,very thick,dashed] (0,3.4)--(7.2,3.4);
\node[black,right,font=\scriptsize] at (5.3,3.68) {Dulong-Petit $3Nk_B$};
\draw[acc,very thick] plot[domain=0.25:7.0,samples=110]
  (\x,{3.4*(2.2/\x)^2*exp(-2.2/\x)/(1-exp(-2.2/\x))^2});
\node[acc] at (5.0,2.5) {Einstein curve};
\draw[black,dashed] (2.2,0)--(2.2,3.1);
\node[black!70,font=\scriptsize,below] at (2.2,0) {Einstein temperature};
\end{tikzpicture}
$$

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.

> **Worked example.** Evaluate the fraction $C_V/3Nk_B$ at $T = \Theta_E$ and at
> $T = \tfrac12\Theta_E$. At $T = \Theta_E$ the argument is $u = \Theta_E/T = 1$,
> so
> $$
> \frac{C_V}{3Nk_B} = \frac{u^2 e^{u}}{(e^{u}-1)^2}
> = \frac{e}{(e-1)^2} = \frac{2.718}{(1.718)^2} = 0.921.
> $$
> The solid already carries $92\%$ of its classical heat capacity at the Einstein
> temperature. At $T = \tfrac12\Theta_E$ the argument is $u = 2$, and
> $$
> \frac{C_V}{3Nk_B} = \frac{4\,e^{2}}{(e^{2}-1)^2}
> = \frac{4(7.389)}{(6.389)^2} = 0.724.
> $$
> Halving the temperature from $\Theta_E$ drops the heat capacity from $92\%$ to
> $72\%$ of Dulong–Petit; the decline steepens further below, reaching the
> exponential freeze-out. For diamond, with $\Theta_E$ of order $10^3\ \mathrm{K}$,
> room temperature is well below $\Theta_E$ and the measured heat capacity is far
> under $3R$, the anomaly Einstein's model was built to explain.

## 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.[^schroeder-debye]

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 $\hbar\omega \ll k_B T$ 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.

$$
% caption: At low temperature the Einstein heat capacity falls exponentially, well below the observed $T^3$ law; the single-frequency model freezes out too abruptly because it omits the low-frequency acoustic modes.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.2,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,4.2) node[above,black]{$C_V$};
\draw[black,very thick,dashed] plot[domain=0.2:2.05,samples=90] (2.7*\x,{0.45*\x*\x*\x});
\node[black] at (4.7,2.7) {observed $T^3$};
\draw[acc,very thick] plot[domain=0.2:2.05,samples=110]
  (2.7*\x,{3.4*(6/\x)^2*exp(-6/\x)/(1-exp(-6/\x))^2});
\node[acc] at (5.0,1.0) {Einstein};
\node[black!70,font=\scriptsize] at (1.9,2.4) {measured lies above};
\end{tikzpicture}
$$

## Summary

- The quantum harmonic oscillator has a geometric partition function
  $z = [2\sinh(\beta\hbar\omega/2)]^{-1}$, and its mean energy is
  $\langle E\rangle = \hbar\omega(\tfrac12 + \langle n\rangle)$ with the
  Bose occupation $\langle n\rangle = (e^{\beta\hbar\omega}-1)^{-1}$.
- At high temperature $\langle E\rangle\to k_B T$ (equipartition); at low
  temperature it freezes to the zero-point energy $\tfrac12\hbar\omega$.
- The **Einstein solid** of $3N$ identical oscillators has heat capacity
  $C_V = 3Nk_B(\Theta_E/T)^2 e^{\Theta_E/T}/(e^{\Theta_E/T}-1)^2$, rising from
  zero to the **Dulong–Petit** value $3Nk_B$ as $T$ passes the Einstein
  temperature $\Theta_E = \hbar\omega/k_B$.
- 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.

[^pathria-osc]: **Pathria & Beale**, _Statistical Mechanics_ (4th ed.), §3.8 — the partition function of the quantum harmonic oscillator, its closed geometric form, and the resulting thermodynamics.
[^reif-osc]: **Reif**, _Fundamentals of Statistical and Thermal Physics_, §7.6 — the mean energy of an oscillator, the Bose occupation factor, and the high- and low-temperature limits.
[^schroeder-einstein]: **Schroeder**, _An Introduction to Thermal Physics_, §2.2 and §3.3 — the Einstein model of a solid as $3N$ identical oscillators, its multiplicity, and its entropy and temperature. Companion material at <https://physics.weber.edu/schroeder/thermal/>.
[^tl-dulong]: **Tipler & Llewellyn**, _Modern Physics_, §8-1 — the Dulong–Petit law, its classical equipartition derivation, and the observed failure of the constant heat capacity at low temperature.
[^schroeder-debye]: **Schroeder**, _An Introduction to Thermal Physics_, §7.5 — the Debye theory of solids, the acoustic mode spectrum, and the low-temperature $T^3$ heat capacity that the Einstein model misses. **Reif**, §7.7, gives the parallel treatment.