= Einstein solid
{c}
{title2=$C_V/(3Nk_B)=x^2e^x/(e^x-1)^2$}
{wiki}
= Einstein model of a solid
{c}
{synonym}
The <Einstein model of a solid> represents the vibrations of $N$ <chemical atoms> by $3N$ independent one-dimensional <quantum harmonic oscillators>, all with the same positive <angular frequency> $\omega$. Equivalently, each <chemical atom> has three independent oscillator directions. The common <angular frequency> is a simplifying model assumption, not a claim about every vibration of a real <solid>. Hold the <volume (thermodynamics)>, $N$ and $\omega$ fixed. Define the characteristic <temperature> $\theta_E=\hbar\omega/k_B$ and the dimensionless ratio $x=\theta_E/T$. The vibrational <internal energy> is
$$
U=3N\hbar\omega\left[\frac12+\frac{1}{e^x-1}\right],\qquad
x=\frac{\hbar\omega}{k_BT}=\frac{\theta_E}{T}.
$$
The first term is the <zero-point energy> $3N\hbar\omega/2$. It is constant at fixed $\omega$, so it contributes no <heat capacity at constant volume>. The <temperature>-dependent term gives
$$
\frac{C_V}{3Nk_B}=\frac{x^2e^x}{(e^x-1)^2}
=\frac{x^2e^{-x}}{(1-e^{-x})^2}.
$$
This is the vibrational <heat capacity at constant volume> of the model, with no expansion or other contributions included. https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/d3fa883afa8df7cf7aa0432fcaadde8f_lec22t.pdf[MIT 3.012, Lecture 22, pages 9–13: Einstein-solid partition function, energy and heat capacity]{ref}
For a worked comparison, choose the model parameter $\theta_E=300\,\mathrm K$. The normalized <heat capacity at constant volume> depends only on $T/\theta_E$:
|| $T/\theta_E$
|| <Temperature> $T$ in $\mathrm K$
|| $C_V/(3Nk_B)$
| $0.1$
| $30$
| $0.00454$
| $0.5$
| $150$
| $0.72406$
| $1$
| $300$
| $0.92067$
| $2$
| $600$
| $0.97942$
At $T=\theta_E$, one <mole> of <chemical atoms> therefore has vibrational <heat capacity at constant volume> $3R(0.92067)\simeq22.965\,\mathrm{J\,mol^{-1}\,K^{-1}}$, using the unrounded model value. Even at twice the characteristic <temperature>, the model remains about $2.06\%$ below the <Dulong-Petit law> limit. These are calculations for the ideal <Einstein solid>, not material measurements.
\Image[/einstein-solid-heat-capacity.png]
{title=<Einstein solid> vibrational <heat capacity> approaches the <Dulong-Petit law> limit}
{description=The model ratio $C_V/(3Nk_B)$ over $0.02\leq T/\theta_E\leq3$, with the four computed cases marked. The dashed classical line is $1$; quantum freeze-out suppresses the low-<temperature> response. All <quantum harmonic oscillators> have the same stipulated <angular frequency>.}
A short derivation uses the <thermal partition function of a quantum harmonic oscillator>. The <energy eigenvalues> $E_n=\hbar\omega(n+1/2)$ and their <Boltzmann factors> form a <geometric series>, giving the one-oscillator <canonical partition function> $z=e^{-x/2}/(1-e^{-x})$. The mean occupation is $1/(e^x-1)$, hence the displayed $U$ after summing over $3N$ independent <quantum harmonic oscillators>. Taking its <derivative> with $dx/dT=-x/T$ gives the displayed <heat capacity at constant volume>. https://davidtong.org/pdfs/teaching/statistical-physics/omg1.pdf[David Tong, Statistical Physics, Example Sheet 1 (January 2012), problem 5: oscillator energy, the Einstein model and its limits]{ref}
The two limits explain the shape. At high <temperature>, a <Taylor series> in small $x$ gives
$$
\frac{C_V}{3Nk_B}=1-\frac{x^2}{12}+O(x^4)\longrightarrow1
\qquad(T\gg\theta_E).
$$
At low <temperature>, $1-e^{-x}\to1$, so
$$
\frac{C_V}{3Nk_B}\sim x^2e^{-x}\longrightarrow0
\qquad(T\ll\theta_E).
$$
The <zero-point energy> remains even when the thermal response is tiny: freeze-out means that changing <temperature> scarcely changes the occupations, not that the <chemical atoms> have zero <energy>. This is the useful lesson of the <Einstein model of a solid>: the classical counting rule reappears when many levels are accessible, while discrete excitation spacings explain its failure upon cooling.
The exponential low-<temperature> decay is also the model's limitation. Real three-dimensional crystalline <solids> have <acoustic phonon branches> with arbitrarily small <angular frequencies> in the bulk limit. The <Debye model> includes these collective <normal modes> and predicts a low-<temperature> vibrational contribution $C_V\propto T^3$, which the single-frequency <Einstein solid> cannot reproduce. This qualification concerns the vibrational contribution; additional excitations can change a material's total <heat capacity>. https://davidtong.org/teaching/statistical-physics/statmechhtml/S3#S3.SS3[David Tong, Statistical Physics, section 3.3: acoustic phonons and the Debye low-temperature law]{ref}
Einstein introduced the quantum treatment of <solid> vibrations in his 1907 paper \i[Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme]. Its historical importance is that quantum excitation energies explained why the classical <Dulong-Petit law> could fail; the modern <quantum harmonic oscillator> formulation above includes the <zero-point energy> explicitly. https://onlinelibrary.wiley.com/doi/10.1002/andp.19063270110[A. Einstein, Annalen der Physik 327, 180–190 (1907), original-paper publication metadata]{ref}
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