- Use the phonon density of states from Lecture 9 to write the lattice internal energy and heat capacity.
- Derive the Einstein heat capacity and explain why it reproduces the Dulong–Petit limit but fails at low temperature.
- Derive the Debye heat capacity and obtain its low-temperature \(T^{3}\) law and high-temperature limit.
- Interpret the Debye temperature as a physically meaningful energy scale rather than a sharp transition temperature.
- Read low-temperature heat-capacity data and extract \(\beta_{\mathrm{LT}}\), \(\Theta_D\), and the metallic \(\gamma T\) term.
8.1 Roadmap
- Recall the general phonon formula for \(U(T)\) and \(C_V\).
- Use the Einstein model as the simplest quantum correction to classical specific heat.
- Replace the one-frequency picture by the Debye acoustic continuum.
- Derive the low- and high-temperature asymptotics.
- Learn the experimental logic behind \(C/T\) versus \(T^2\) plots.
- Bridge to the next lecture: why metals need an additional electronic term.
| Symbol | Meaning |
|---|---|
| \(r\) | number of atoms in the primitive cell |
| \(N\) | number of primitive cells |
| \(V\) | crystal volume |
| \(g(\omega)\) | phonon density of states per unit volume |
| \(\omega_E,\Theta_E\) | Einstein frequency and Einstein temperature, with \(k_B\Theta_E=\hbar\omega_E\) |
| \(\omega_D,\Theta_D\) | Debye frequency and Debye temperature, with \(k_B\Theta_D=\hbar\omega_D\) |
| \(U(T), C_V\) | lattice internal energy and constant-volume heat capacity |
| \(\beta\) | inverse temperature, \(\beta=1/(k_B T)\) |
| \(C_{V,m}\) | molar constant-volume heat capacity |
| \(\beta_{\mathrm{LT}}, \gamma\) | low-temperature phonon coefficient and linear electronic coefficient |
| \(R\) | gas constant, \(R=N_A k_B\) |
8.2 From the Phonon DOS to Heat Capacity
We take the end point of Lecture 7 as the starting point here: once the phonon density of states is known, lattice thermodynamics reduces to a one-dimensional frequency integral:
Here \(U(T)\) is the lattice internal energy, \(g(\omega)\) is the phonon density of states per unit volume, and \(\beta=1/(k_B T)\) is the inverse temperature. The term \(\hbar\omega/2\) is the zero-point contribution of each mode.
The heat capacity is defined as the derivative of the internal energy of the lattice with respect to temperature \[ C_V = \frac{\partial U(T)}{\partial T} \]
This gives
Since the zero-point term is temperature independent in the harmonic approximation, it drops out of the heat capacity.
The dimensionless variable \(x\) compares the phonon energy \(\hbar\omega\) with the thermal scale \(k_B T\). This formula shows immediately that the heat capacity is controlled by two inputs only: the Bose factor and the phonon DOS.
A useful sanity check:
- low-frequency modes have small \(x\) and are easy to excite;
- high-frequency modes have large \(x\) and are thermally frozen out;
- therefore the low-\(\omega\) part of \(g(\omega)\) controls low-temperature thermodynamics.
8.3 Einstein Model: Quantum Oscillators at One Frequency
The Einstein model is the simplest quantum model of lattice heat capacity: all vibrational modes are assigned the same frequency \(\omega_E\). It is therefore a model with the correct quantum statistics but an oversimplified spectrum.
8.3.1 Einstein DOS and Heat Capacity
In DOS language, the Einstein model is
\[ g_E(\omega) = \frac{3rN}{V}\ \delta(\omega-\omega_E). \]
The prefactor counts the \(3rN\) vibrational modes of the crystal, and the delta function means that all of them sit at the same angular frequency \(\omega_E\).
Substituting \(g_E(\omega)\) into the general formula for \(C_V\) gives
Here \(\Theta_E\) is the Einstein temperature, defined by \(k_B\Theta_E=\hbar\omega_E\). The factor \[ F_E(x)=\frac{x^2 e^x}{(e^x-1)^2} \] is the Einstein function.
Interpretation:
- one model parameter, \(\omega_E\) or equivalently \(\Theta_E\);
- all branches are treated as if they were equally hard to excite;
- no true acoustic low-energy continuum is present.
8.3.2 High-Temperature Limit: Dulong–Petit Recovered
For \(T \gg \Theta_E\), one has \(x_E \ll 1\), and therefore
\[ \frac{x_E^2 e^{x_E}}{(e^{x_E}-1)^2} = 1+\mathcal O(x_E^2). \]
Hence
This is the Dulong–Petit law. Per mole of primitive cells, the corresponding limit is \(3rR\). In this regime every vibrational mode contributes essentially \(k_B\) to the heat capacity.
The high-temperature result is therefore not a special property of the Einstein model. It follows much more generally from mode counting plus classical equipartition.
8.3.3 Why the Einstein Model Fails at Low Temperature
For \(T \ll \Theta_E\), one has \(x_E \gg 1\), so
\[ \frac{x_E^2 e^{x_E}}{(e^{x_E}-1)^2} \sim x_E^2 e^{-x_E}. \]
Therefore
The heat capacity falls off exponentially.
This is too fast. The physical reason is simple: the Einstein model has no modes with arbitrarily small \(\omega\). Real crystals do. Long-wavelength acoustic phonons cost very little energy and remain thermally accessible even when \(T\) is small. The Einstein model therefore captures quantization, but it suppresses the very modes that dominate low-temperature thermodynamics.
8.4 Debye Model: Acoustic Modes Dominate Low-Temperature Thermodynamics
The Debye model keeps the one ingredient the Einstein model misses: an acoustic continuum with linear dispersion at low frequency.
8.4.1 Debye DOS and the Debye Temperature
In Lecture 7 we obtained the Debye density of states
\[ g_D(\omega) = \begin{cases} \dfrac{9rN}{V}\dfrac{\omega^2}{\omega_D^3}, & 0\le \omega \le \omega_D,\\ 0, & \omega>\omega_D. \end{cases} \]
The cutoff \(\omega_D\) is chosen so that the total number of modes remains correct: \[ V\int_{0}^{\omega_D}\ g_D(\omega) d\omega = 3rN. \]
Thus the Debye model does not preserve the detailed branch structure of the real phonon spectrum, but it does preserve the total number of modes and the correct low-frequency behavior \(g(\omega)\propto \omega^2\).
The Debye temperature is defined by \[ k_B\Theta_D=\hbar\omega_D. \]
It packages the cutoff energy into a temperature scale. In the Debye model, once \(\Theta_D\) is fixed, the entire temperature dependence of the lattice heat capacity is fixed as well.
8.4.2 Debye Internal Energy and Heat Capacity
We insert \(g_D(\omega)\) into the thermal part of the internal energy, omit the zero-point term, and change variables to \[ t=\frac{\hbar\omega}{k_B T}. \] The symbol \(t\) is dimensionless; it measures phonon frequency in units of temperature.
We obtain
\[ U_T = 9rN k_B T \left(\frac{T}{\Theta_D}\right)^3 \int_{0}^{\Theta_D/T} dt\ \frac{t^3}{e^t-1} . \]
Differentiating with respect to \(T\) yields the Debye heat capacity
This is the central working formula of the Debye model.
Sanity checks:
- the prefactor is extensive in the number of modes, \(3rN\);
- the only material parameter is \(\Theta_D\);
- the entire nontrivial temperature dependence sits in a dimensionless integral.
8.4.3 Low-Temperature Limit: The \(T^{3}\) Law
For \(T \ll \Theta_D\), the upper integration limit \(\Theta_D/T\) is very large, so the integral may be extended to infinity:
\[ \int_{0}^{\Theta_D/T} dt\ \frac{t^4 e^t}{(e^t-1)^2} \approx \int_{0}^{\infty} dt\ \frac{t^4 e^t}{(e^t-1)^2} = \frac{4\pi^4}{15}. \]
Therefore
This is the Debye \(T^3\) law.
Why \(T^3\)?
- in three dimensions, the acoustic DOS starts as \(g(\omega)\propto\omega^2\);
- the relevant thermally occupied frequencies satisfy \(\hbar\omega \lesssim k_B T\);
- counting modes up to \(\omega\sim T\) therefore gives a phase-space factor \(\sim T^3\).
So the low-temperature heat capacity is not an arbitrary curve fit. It is a direct fingerprint of three-dimensional acoustic phonons.
8.4.4 High-Temperature Limit: Dulong–Petit Recovered Again
For \(T \gg \Theta_D\), the upper limit \(\Theta_D/T\) is small, and the Bose factor may be expanded as \[ e^t-1 \approx t. \]
Equivalently, each mode approaches the classical result \(\langle E\rangle \approx k_B T\). Since the total number of modes is \(3rN\), one recovers
So both Einstein and Debye recover Dulong–Petit at high temperature, but only Debye gives the correct low-temperature asymptote.
8.4.5 What the Debye Temperature Means
We can interpret the Debye temperature \(\Theta_D\) in the following way:
- \(\Theta_D\) is the temperature scale associated with the Debye cutoff energy \(\hbar\omega_D\);
- it increases with increasing characteristic sound velocity and with stiffer bonding;
- it marks the crossover between the low-temperature acoustic regime and the high-temperature Dulong–Petit regime;
- it is model dependent.
The last point matters in practice. Real solids do not have an exact Debye DOS over the whole spectrum, so a \(\Theta_D\) extracted from the low-temperature \(T^3\) coefficient need not coincide with a value inferred from fitting the full heat-capacity curve at higher temperature.
8.5 Data Literacy: How to Read Low-Temperature Heat-Capacity Data
The Debye law becomes especially useful when it is converted into a fitting form.
8.5.1 Insulators: Extracting the Phonon Coefficient
For a nonmetallic crystal at sufficiently low temperature, the leading lattice contribution is
\[ C_{V,m} \approx \beta_{\mathrm{LT}} T^3. \]
Here \(C_{V,m}\) is the molar heat capacity, and \(\beta_{\mathrm{LT}}\) is the fitted low-temperature phonon coefficient. For one mole of the chosen chemical unit, the coefficient is
\[ \beta_{\mathrm{LT}} = \frac{12\pi^4}{5} \frac{n_{\mathrm{at}} R}{\Theta_D^3}. \]
The integer \(n_{\mathrm{at}}\) is the number of atoms in the chemical unit used to report the molar heat capacity. For a monatomic elemental solid quoted per mole of atoms, \(n_{\mathrm{at}}=1\).
Thus \[ \Theta_D = \left( \frac{12\pi^4 n_{\mathrm{at}} R}{5\beta_{\mathrm{LT}}} \right)^{1/3}. \]
This is the standard route from a low-temperature fit to an effective Debye temperature.
8.5.2 Metals: Why Plot \(C/T\) Versus \(T^2\)?
In a metal, the low-temperature heat capacity contains both lattice and electronic pieces. Keeping only the leading terms,
Dividing by \(T\) gives
Now the logic is purely graphical:
- plot \(C_{V,m}/T\) on the vertical axis;
- plot \(T^2\) on the horizontal axis;
- the intercept is \(\gamma\);
- the slope is \(\beta_{\mathrm{LT}}\).
This is why \(C/T\) versus \(T^2\) is such a standard low-temperature analysis plot.
The linear term \(\gamma T\) is absent in ionic insulators but present in metals. It comes from the conduction electrons, not from the phonons. We only mention it here and will derive it in the upcoming lecture when we introduce the electron gas.
8.5.3 What a Straight Line Does and Does Not Mean
A straight line in \(C/T\) versus \(T^2\) means:
- the data are in a regime where the leading terms \(\gamma T\) and \(\beta_{\mathrm{LT}}T^3\) dominate;
- the phonon part is consistent with Debye’s low-temperature asymptote;
- one can extract \(\gamma\) and \(\beta_{\mathrm{LT}}\) with minimal modeling.
It does not mean:
- the Debye model is exact at all temperatures;
- the full phonon DOS is featureless;
- the same \(\Theta_D\) must fit the entire heat-capacity curve.
That last point is important whenever one compares low-temperature fits with room-temperature material tables.
8.5.4 Experimental Note
Strictly speaking, the derivations above concern \(C_V\). In many low-temperature solid-state measurements, the experimentally accessible quantity is closer to \(C_P\), but at low temperature the difference is usually very small, so the same fitting logic is routinely applied. Within the strictly harmonic approximation, the two are equal.
8.6 Bridge to the Next Lecture: Why the Pure Phonon Picture Is Not the Whole Story
Today’s main message is that the harmonic phonon gas already explains two cornerstone facts:
- the Dulong–Petit limit at high temperature;
- the \(T^3\) lattice law at low temperature.
But real solids often show additional structure:
- metals carry a linear electronic term \(\gamma T\) at low temperature;
- anharmonicity becomes important at higher temperature;
- then thermal expansion appears, phonons acquire finite lifetimes, and eventually \(C_P\) and \(C_V\) are no longer effectively identical.
So the next conceptual step is to consider the contribution from electrons and anharmonic effects.
- The heat capacity of a harmonic crystal is determined by the phonon DOS together with Bose statistics.
- The Einstein model fixes the classical overestimate of specific heat but misses the low-frequency acoustic modes.
- The Debye model succeeds at low temperature because it builds in an acoustic DOS that starts quadratically in frequency.
- The low-temperature \(T^3\) law is a phase-space result for three-dimensional acoustic phonons.
- The Dulong–Petit limit is recovered when all vibrational modes are thermally populated.
- The Debye temperature is a crossover scale and an effective stiffness measure, not a phase-transition temperature.
- In low-temperature experiments, the slope of \(C/T\) versus \(T^2\) isolates the phonon coefficient, while the intercept isolates the electronic one.
- A single fitted Debye temperature should always be interpreted together with the temperature range from which it was extracted.
8.7 Problem Set
Einstein asymptotics. Starting from \[ C_V^{(E)} = 3rN k_B \frac{x_E^2 e^{x_E}}{(e^{x_E}-1)^2}, \qquad x_E=\frac{\Theta_E}{T}, \] derive the high-temperature and low-temperature asymptotic forms. State clearly which physical feature of the model causes the low-temperature behavior.
Debye \(T^3\) law. Starting from \[ C_V^{(D)} = 9rN k_B \left(\frac{T}{\Theta_D}\right)^3 \int_{0}^{\Theta_D/T} dt\ \frac{t^4 e^t}{(e^t-1)^2}, \] show that for \(T\ll\Theta_D\), \[ C_V^{(D)} \approx \frac{12\pi^4}{5},rN k_B \left(\frac{T}{\Theta_D}\right)^3. \]
Estimating \(\Theta_D\) from a fit. A monatomic metal is fitted at low temperature by \[ \frac{C_{V,m}}{T} = 18.0 \mathrm{mJ\ mol^{-1}K^{-2}} + 0.20 \mathrm{mJ\ mol^{-1}K^{-4}} T^2. \] Identify \(\gamma\) and \(\beta_{\mathrm{LT}}\). Then estimate the Debye temperature, assuming the heat capacity is quoted per mole of atoms.
Why \(C/T\) versus \(T^2\)? Explain why an insulating crystal and a metal behave differently in a \(C/T\) versus \(T^2\) plot at low temperature. What information is contained in the slope and in the intercept?