- Explain why a strictly harmonic crystal has no thermal expansion.
- Introduce the quasi-harmonic picture and define the mode and average Grüneisen parameters.
- Derive the Grüneisen relation for the thermal expansion coefficient \(\alpha(T)\).
- Interpret the difference between \(C_P\) and \(C_V\) in a solid and connect it to thermal expansion.
- Explain qualitatively how anharmonicity gives phonons finite lifetimes and thereby produces thermal resistance.
9.1 Roadmap
- Start from the harmonic crystal and show why it cannot expand with temperature.
- Introduce anharmonic terms in the lattice potential and motivate the quasi-harmonic approximation.
- Derive the Grüneisen relation and discuss the physical meaning of the Grüneisen parameter.
- Relate thermal expansion to the difference \(C_P-C_V\).
- Outlook: anharmonicity also causes phonon decay and thermal resistance.
| Symbol | Meaning |
|---|---|
| \(V\) | crystal volume |
| \(p\) | pressure |
| \(K\) | bulk modulus, \(K=-V(\partial p/\partial V)_T\) |
| \(\alpha\) | linear thermal expansion coefficient, \(\alpha=(1/3V)(\partial V/\partial T)_p\) |
| \(F(T,V)\) | Helmholtz free energy |
| \(\omega_\lambda(q)\) | phonon frequency of branch \(\lambda\) and wave vector \(q\) |
| \(n_\lambda(q)\) | Bose occupation number of phonon mode \((q,\lambda)\) |
| \(\gamma_\lambda(q)\) | mode Grüneisen parameter |
| \(\gamma_G\) | heat-capacity-weighted Grüneisen parameter |
| \(C_V,\ C_P\) | constant-volume and constant-pressure heat capacities |
| \(c_V,\ c_P\) | corresponding heat capacities per unit volume, with \(c_V=C_V/V\) and \(c_P=C_P/V\) |
| \(\beta\) | inverse temperature, \(\beta=1/(k_B T)\) |
| \(G\) | reciprocal-lattice vector in momentum-conservation relations |
9.2 Why a Harmonic Crystal Does Not Expand
We now leave the strictly harmonic crystal of Lecture 8 and ask a new question: why do real solids expand when heated, whereas the harmonic model does not?
For a harmonic crystal, the Helmholtz free energy can be written as \[ F(T,V) = E_0(V) + k_B T \sum_{q,\lambda} \ln \left[ 2\sinh \left(\frac{\beta \hbar \omega_\lambda(q)}{2}\right) \right]. \]
Here \(E_0(V)\) is the zero-temperature reference energy, \(\omega_\lambda(q)\) is the phonon frequency of mode \((q,\lambda)\), and \(\beta=1/(k_B T)\) is the inverse temperature.
The key statement of the strict harmonic approximation is that the phonon frequencies do not change when the equilibrium volume changes: \[ \left(\frac{\partial \omega_\lambda(q)}{\partial V}\right)=0. \]
This means that the temperature-dependent phonon part of \(F(T,V)\) carries no explicit volume dependence. Therefore \[ p =-\left(\frac{\partial F}{\partial V}\right)_T -\frac{\partial E_0}{\partial V}, \] so the pressure is independent of temperature at fixed volume: \[ \left(\frac{\partial p}{\partial T}\right)_V=0. \]
Here \(p\) is the pressure and the derivative is taken at constant volume.
Now define the linear thermal expansion coefficient and bulk modulus as \[ \alpha = \frac{1}{3V}\left(\frac{\partial V}{\partial T}\right)_p, \qquad K=-V\left(\frac{\partial p}{\partial V}\right)_T. \]
Here \(\alpha\) measures the relative change of a linear dimension with temperature in an isotropic solid, and \(K\) measures the resistance to compression.
Using the cyclic thermodynamic identity, one obtains \[ \alpha = \frac{1}{3K}\left(\frac{\partial p}{\partial T}\right)_V. \]
Therefore, in the harmonic approximation,
This is the formal reason why a harmonic crystal does not expand.
A simple sanity check comes from a single harmonic oscillator. In a purely harmonic potential \[ U(u)=\frac{1}{2} f u^2, \] the potential is symmetric about its minimum. Here \(u\) is the displacement and \(f\) is the spring constant. A thermal distribution in such a symmetric well has \(\langle u\rangle=0\) at every temperature, so the average position does not shift. Thermal expansion therefore requires asymmetry of the potential, not just larger vibration amplitudes.
9.3 From Anharmonicity to the Quasi-Harmonic Idea
To go beyond the harmonic model, the lattice potential energy must be expanded beyond second order around the equilibrium structure: \[ \begin{aligned} U(\{u\}) &= U_0 + \frac{1}{2!} \sum_{l\kappa\alpha} \sum_{l'\kappa'\beta} \Phi^{(2)}_{\alpha\beta}(l\kappa,l'\kappa') \,u_{\alpha}(l,\kappa)\,u_{\beta}(l',\kappa') \\[4pt] &\quad+ \frac{1}{3!} \sum_{l\kappa\alpha} \sum_{l'\kappa'\beta} \sum_{l''\kappa''\gamma} \Phi^{(3)}_{\alpha\beta\gamma}(l\kappa,l'\kappa',l''\kappa'') \,u_{\alpha}(l,\kappa)\,u_{\beta}(l',\kappa')\,u_{\gamma}(l'',\kappa'') \\[4pt] &\quad+ \frac{1}{4!} \sum_{l\kappa\alpha} \sum_{l'\kappa'\beta} \sum_{l''\kappa''\gamma} \sum_{l'''\kappa'''\delta} \Phi^{(4)}_{\alpha\beta\gamma\delta}(l\kappa,l'\kappa',l''\kappa'',l'''\kappa''') \,u_{\alpha}(l,\kappa)\,u_{\beta}(l',\kappa')\,u_{\gamma}(l'',\kappa'')\,u_{\delta}(l''',\kappa''')\\[4pt] &\quad+\cdots \end{aligned} \]
Here \(U(\{u\})\) is the crystal potential energy as a function of the full set of ionic displacements \(\{u\}\), and \(U_0\) is the potential energy of the equilibrium crystal, that is, the value at \(\{u\}=0\). The quantity \(u_{\alpha}(l,\kappa)\) denotes the displacement of atom \(\kappa\) in primitive cell \(l\) along Cartesian direction \(\alpha\in\{x,y,z\}\) from its equilibrium position. The indices \(l,l',l'',\dots\) run over primitive cells, \(\kappa,\kappa',\kappa'',\dots\) run over the atoms in the primitive basis, and \(\alpha,\beta,\gamma,\delta\) denote Cartesian components.
The expansion coefficients are the interatomic force constants evaluated at equilibrium: \[ \Phi^{(2)}_{\alpha\beta}(l\kappa,l'\kappa') = \left. \frac{\partial^2 U}{\partial u_{\alpha}(l,\kappa)\,\partial u_{\beta}(l',\kappa')} \right|_{\{u\}=0}, \]
\[ \Phi^{(3)}_{\alpha\beta\gamma}(l\kappa,l'\kappa',l''\kappa'') = \left. \frac{\partial^3 U}{\partial u_{\alpha}(l,\kappa)\,\partial u_{\beta}(l',\kappa')\,\partial u_{\gamma}(l'',\kappa'')} \right|_{\{u\}=0}, \]
\[ \Phi^{(4)}_{\alpha\beta\gamma\delta}(l\kappa,l'\kappa',l''\kappa'',l'''\kappa''') = \left. \frac{\partial^4 U}{\partial u_{\alpha}(l,\kappa)\,\partial u_{\beta}(l',\kappa')\,\partial u_{\gamma}(l'',\kappa'')\,\partial u_{\delta}(l''',\kappa''')} \right|_{\{u\}=0}. \]
The factorial prefactors \(1/2!\), \(1/3!\), and \(1/4!\) are the usual Taylor-series coefficients. The second-order term is the harmonic contribution, while the third- and fourth-order terms are the leading anharmonic corrections. The cubic term introduces asymmetry of the potential and enables three-phonon processes; the quartic term is the next correction and helps stabilize the potential for larger displacements.
For a monatomic basis, the basis index \(\kappa\) may be dropped, and one simply writes \(u_{\alpha}(l)\).
Interpretation:
- the cubic term introduces asymmetry of the potential;
- the quartic term stabilizes the energy for large displacements;
- once these terms matter, the exact decoupling into independent normal modes is lost.
That last point is crucial: in a truly anharmonic crystal, phonons are no longer exact eigenstates of the full Hamiltonian.
A very useful intermediate step is the quasi-harmonic approximation. In this picture, the crystal is still treated as a gas of noninteracting phonons at any fixed volume, but the phonon frequencies are allowed to depend on the equilibrium volume: \[ \omega_\lambda(q) \rightarrow \omega_\lambda(q,V). \]
The free energy then becomes \[ F(T,V) = U_0(V) + k_B T \sum_{q,\lambda} \ln \left[ 2\sinh \left(\frac{\beta\hbar\omega_\lambda(q,V)}{2}\right) \right]. \]
Here the phonons remain harmonic at fixed \(V\), but the spectrum shifts when \(V\) changes. This is the simplest controlled way to encode thermal expansion in phonon thermodynamics.
The natural quantity that measures this volume sensitivity is the mode Grüneisen parameter: \[ \gamma_\lambda(q) = - \frac{\partial \ln \omega_\lambda(q)}{\partial \ln V} = -\frac{V}{\omega_\lambda(q)} \left(\frac{\partial \omega_\lambda(q)}{\partial V}\right). \]
Here \(\gamma_\lambda(q)\) is dimensionless. It tells us how strongly a given phonon mode softens or stiffens when the crystal is expanded.
Physical meaning:
- if \(\gamma_\lambda(q)>0\), the mode frequency decreases upon expansion, so heating tends to favor larger volume;
- if \(\gamma_\lambda(q)<0\), the mode frequency increases upon expansion, so that mode tends to oppose expansion;
- the observed expansion is therefore a weighted average over all thermally populated modes.
9.4 Grüneisen Parameter and the Thermal Expansion Coefficient
The Bose occupation number of mode \((q,\lambda)\) is \[ n_\lambda(q) = \frac{1}{e^{\beta\hbar\omega_\lambda(q)}-1}. \]
Here \(n_\lambda(q)\) is the thermal mean occupation number of the phonon mode.
Using the quasi-harmonic free energy, one finds that the temperature derivative of the pressure at fixed volume is \[ \left(\frac{\partial p}{\partial T}\right)_V = \frac{1}{V} \sum_{q,\lambda} \gamma_\lambda(q) \hbar\omega_\lambda(q) \frac{\partial n_\lambda(q)}{\partial T}. \]
The factor \(\hbar\omega_\lambda(q)\partial n_\lambda(q)/\partial T\) is the heat-capacity weight carried by that mode.
Define the modal contribution to the heat capacity per unit volume as \[ c_{V,\lambda}(q) = \frac{1}{V} \hbar\omega_\lambda(q) \frac{\partial n_\lambda(q)}{\partial T}, \] and the total volume heat capacity as \[ c_V = \sum_{q,\lambda} c_{V,\lambda}(q). \]
Then \[ \left(\frac{\partial p}{\partial T}\right)_V = \sum_{q,\lambda}\gamma_\lambda(q)c_{V,\lambda}(q). \]
This motivates the heat-capacity-weighted Grüneisen parameter \[ \gamma_G = \frac{\sum_{q,\lambda}\gamma_\lambda(q)c_{V,\lambda}(q)} {\sum_{q,\lambda}c_{V,\lambda}(q)}. \]
By construction, \(\gamma_G\) is the average Grüneisen parameter of the modes that actually contribute thermally at temperature \(T\).
Substituting this into expression for the coefficient of thermal expansion \[ \alpha = \frac{1}{3K}\left(\frac{\partial p}{\partial T}\right)_V \] gives the Grüneisen relation
Here \(\alpha(T)\) is the linear thermal expansion coefficient, \(c_V(T)\) is the constant-volume heat capacity per unit volume, \(K(T)\) is the bulk modulus, and \(\gamma_G(T)\) is the weighted Grüneisen parameter.
This formula is the central thermodynamic result of the lecture.
Interpretation:
- if \(K\) and \(\gamma_G\) vary only weakly with temperature, then \(\alpha(T)\) follows the same temperature dependence as \(c_V(T)\);
- at low temperature, where acoustic phonons dominate and \(c_V\propto T^3\), one expects \(\alpha\propto T^3\);
- at high temperature, once the phonon heat capacity is nearly constant, \(\alpha\) is also approximately constant;
- negative thermal expansion is possible if modes with negative \(\gamma_\lambda(q)\) dominate the weighted average over some temperature range.
9.5 Why \(C_P\) and \(C_V\) Differ in a Real Crystal
In Lecture 8 we could treat the measured heat capacity of a harmonic solid as essentially \(C_V\). Anharmonicity changes that.
A useful thermodynamic identity is \[ C_P = C_V - T \frac{\left[(\partial p/\partial T)_V\right]^2} {(\partial p/\partial V)_T}. \]
Here \(C_P\) is the heat capacity at constant pressure and \(C_V\) is the heat capacity at constant volume.
Now use \[ K=-V\left(\frac{\partial p}{\partial V}\right)_T, \qquad \alpha=\frac{1}{3K}\left(\frac{\partial p}{\partial T}\right)_V. \]
Eliminating the derivatives gives
Equivalently, per unit volume, \[ c_P-c_V = 9\alpha^2 K T. \]
Here \(V\) is the crystal volume, \(\alpha\) is the linear thermal expansion coefficient, and \(K\) is the bulk modulus.
Interpretation:
- at constant volume, all supplied heat goes into raising the internal energy;
- at constant pressure, part of the supplied heat also goes into the mechanical work associated with expansion;
- therefore \(C_P>C_V\) whenever \(\alpha\neq 0\).
Sanity checks:
- in the harmonic limit, \(\alpha=0\), so \(C_P=C_V\);
- the difference is usually modest in solids because \(\alpha\) is small;
- the difference grows with temperature and with the ease of expansion.
This is the clean thermodynamic signature of anharmonicity in calorimetry.
9.6 Outlook on the Next Lecture: Anharmonicity, Phonon Lifetimes, and Thermal Resistance
Thermal expansion is not the only consequence of anharmonicity. The same cubic and quartic terms that make \(\omega_\lambda(q)\) volume dependent also make phonons interact with one another.
In the cubic term, the relevant three-phonon processes obey crystal-momentum conservation modulo a reciprocal-lattice vector: \[ q_1+q_2=q_3+G, \] together with energy conservation \[ \hbar\omega_1+\hbar\omega_2=\hbar\omega_3. \]
Here \(G=0\) corresponds to a normal process, while \(G\neq 0\) corresponds to an umklapp process.
Why this matters:
- in a harmonic crystal, phonons have infinite lifetime;
- in an anharmonic crystal, one phonon can decay into others or merge with them;
- phonons therefore acquire a finite lifetime \(\tau\) and a finite mean free path \(\Lambda\).
That is the microscopic origin of a finite lattice thermal conductivity.
A kinetic picture then gives \[ \kappa_{ph} \sim \frac{1}{3} c_V v_g \Lambda, \qquad \Lambda=v_g\tau, \] where \(\kappa_{ph}\) is the phonon thermal conductivity, \(v_g\) is a characteristic group velocity, and \(\Lambda\) is the phonon mean free path.
The forward link is therefore direct:
- thermal expansion comes from the volume dependence of phonon frequencies;
- thermal resistance comes from phonon-phonon scattering and finite phonon lifetime.
The next lecture will build on this second point and focus on phonon transport, especially the role of normal and umklapp processes.
- A strictly harmonic crystal does not thermally expand.
- Thermal expansion requires anharmonicity, because only an asymmetric potential can shift the thermal average position.
- The quasi-harmonic approximation keeps phonons as useful degrees of freedom while allowing their frequencies to depend on volume.
- The Grüneisen parameter measures how strongly a phonon mode responds to expansion or compression.
- The thermal expansion coefficient is controlled by the heat-capacity-weighted Grüneisen parameter, the heat capacity, and the bulk modulus.
- The difference between constant-pressure and constant-volume heat capacities is another direct consequence of thermal expansion.
- The same anharmonicity that causes expansion also gives phonons finite lifetimes and finite thermal conductivity.
9.7 Problem Set
Harmonic Limit and Thermal Expansion. Starting from \[ \alpha=\frac{1}{3K}\left(\frac{\partial p}{\partial T}\right)_V, \] show that a strictly harmonic crystal has \(\alpha=0\) if its phonon frequencies are volume independent.
Mode Grüneisen Parameter. Suppose one phonon mode obeys \[ \omega(V)=\omega_0\left(\frac{V}{V_0}\right)^{-2}. \] Compute the mode Grüneisen parameter \(\gamma_\lambda(q)\) for this mode and state whether this mode favors positive or negative thermal expansion.
Low-Temperature Behavior of \(\alpha(T)\). Assume that at low temperature \[ c_V(T)=A T^3, \] while \(\gamma_G\) and \(K\) may be treated as constants. Use the Grüneisen relation to find the low-temperature scaling of \(\alpha(T)\).
Deriving \(C_P-C_V\). Starting from \[ C_P = C_V - T \frac{\left[(\partial p/\partial T)_V\right]^2}{(\partial p/\partial V)_T}, \] together with \[ K=-V\left(\frac{\partial p}{\partial V}\right)_T, \qquad \alpha=\frac{1}{3K}\left(\frac{\partial p}{\partial T}\right)_V, \] derive the relation \[ C_P-C_V=9\alpha^2KVT. \]
Normal Versus Umklapp Processes. Explain the difference between normal and umklapp three-phonon processes. Why are umklapp processes especially important for thermal resistance?