After this lecture, you should be able to:
- distinguish heat capacity, molar heat capacity, and specific heat capacity;
- explain why the classical Dulong-Petit law works at high temperature but fails at low temperature;
- derive the quantum-mechanical lattice internal energy from phonon occupation numbers;
- compare the Einstein and Debye approximations for the phonon spectrum;
- derive the Debye \(T^3\) law for the low-temperature lattice heat capacity;
- interpret the Debye temperature as an energy scale for lattice vibrations.
8.1 Heat Capacity and Specific Heat
The heat capacity of a body is the amount of heat required to raise its temperature by one kelvin:
\[ C \equiv \frac{\Delta Q}{\Delta T}. \]
It is an extensive quantity. A larger sample has a larger heat capacity even if the material is the same. To compare materials, one usually normalizes the heat capacity. The molar heat capacity is
\[ c_m \equiv \frac{C}{\nu}, \]
where \(\nu\) is the amount of substance in moles. One may also define heat capacity per mass or per volume,
\[ c_{\mathrm{mass}} = \frac{C}{m_{\mathrm s}}, \qquad c_{\mathrm{vol}} = \frac{C}{V}. \]
For a solid, the useful thermodynamic distinction is between constant volume and constant pressure. From the first law,
\[ dQ=dU+p dV, \]
because the work done on the system is \(dW=-p,dV\). If the volume is fixed, \(dV=0\), and therefore
\[ C_V \equiv \left(\frac{\partial Q}{\partial T}\right)_V = \left(\frac{\partial U}{\partial T}\right)_V. \]
Thus \(C_V\) is directly determined by how the internal energy changes with temperature.
Most measurements on solids are closer to constant pressure conditions. The measured quantity is then
\[ C_p \equiv \left(\frac{\partial Q}{\partial T}\right)_p. \]
Thermodynamics gives
\[ C_p-C_V = T V B \alpha_V^2, \]
where \(\alpha_V\) is the volume expansion coefficient and \(B\) is the bulk modulus. Since \(B>0\) and \(\alpha_V^2\ge 0\), one has
\[ C_p \ge C_V. \]
The difference is usually small for crystalline solids at ordinary temperatures, so we will mostly discuss \(C_V\) and interpret it as the lattice specific heat.
At constant pressure, part of the supplied heat is used to expand the solid. At constant volume, no expansion work is done. This is why \(C_p\) is slightly larger than \(C_V\).
8.2 Classical Lattice Heat Capacity: The Dulong-Petit Law
Consider a crystal with \(N\) unit cells and \(r'\) atoms per unit cell. There are \(3r'N\) normal modes of vibration. In a classical harmonic solid, each mode contributes
\[ \frac{1}{2}k_B T \]
as mean kinetic energy and
\[ \frac{1}{2}k_B T \]
as mean potential energy. Therefore each normal mode contributes \(k_B T\) to the internal energy.
The classical internal energy is
\[ U = U_{\mathrm{eq}} + 3r'N k_B T. \]
The static lattice energy \(U_{\mathrm{eq}}\) is temperature independent, so
\[ C_V = \left(\frac{\partial U}{\partial T}\right)_V = 3r'N k_B. \]
This is the Dulong-Petit law.
For one mole of unit cells, \(N=N_A\), and the molar heat capacity is
\[ c_{m,V} = 3r' N_A k_B = 3r'R. \]
For a monatomic crystal, \(r'=1\), so
\[ c_{m,V} = 3R \simeq 24.9\ \mathrm{J,mol^{-1}K^{-1}}. \]
8.2.1 Why the Classical Result Cannot Be Complete
The Dulong-Petit law works well at sufficiently high temperatures, but experiments show two key deviations:
- at low temperature, the heat capacity becomes much smaller than \(3r'Nk_B\);
- in many insulating crystals, the low-temperature behavior is approximately proportional to \(T^3\).
The classical model fails because it assumes that every vibrational mode can absorb arbitrarily small amounts of energy. Quantum mechanics changes this: a mode of frequency \(\omega\) can only be excited in energy steps of size \(\hbar\omega\).
8.3 Quantum Oscillators and Phonon Occupation
For a quantum harmonic oscillator, the allowed energies are
\[ E_n = \left(n+\frac{1}{2}\right)\hbar\omega, \qquad n=0,1,2,\ldots \]
The term \(\hbar\omega/2\) is the zero-point energy. The oscillator cannot have less energy than this even at \(T=0\).
The essential comparison is between the thermal energy \(k_B T\) and the excitation energy \(\hbar\omega\):
- if \(k_B T\gg \hbar\omega\), the oscillator can easily exchange energy with the heat bath and behaves almost classically;
- if \(k_B T\ll \hbar\omega\), the first excited state is too costly and the oscillator remains mostly in its ground state.
A useful analogy is an entrance fee. If a mode requires an energy ticket \(\hbar\omega\) and the heat bath only has a typical budget \(k_B T\), then modes with \(\hbar\omega\gg k_B T\) are effectively inaccessible.
8.3.1 Mean Occupation Number
For a harmonic oscillator in thermal equilibrium, the mean occupation number is
\[ \langle n\rangle = \frac{1}{e^{\hbar\omega/(k_B T)}-1}. \]
This is the Bose-Einstein distribution with zero chemical potential. Phonon number is not conserved, so no chemical potential appears.
The mean oscillator energy is
\[ \langle E\rangle = \hbar\omega \left( \frac{1}{2} + \langle n\rangle \right) = \hbar\omega \left[ \frac{1}{2} + \frac{1}{e^{\hbar\omega/(k_B T)}-1} \right]. \]
At high temperature, \(\hbar\omega\ll k_B T\), and
\[ \langle n\rangle \simeq \frac{k_B T}{\hbar\omega}. \]
Thus
\[ \langle E\rangle \simeq k_B T + \frac{1}{2}\hbar\omega. \]
The temperature derivative of the zero-point term vanishes, so the heat capacity approaches the classical value.
8.4 Internal Energy of the Phonon Gas
A real crystal has many normal modes. We label them by a wave vector \(\mathbf q\) and a branch index \(r\). The angular frequency of such a mode is \(\omega_{\mathbf q r}\).
For a crystal with \(N\) unit cells and \(r'\) atoms per unit cell, the total number of vibrational modes is
\[ 3r'N. \]
The quantum-mechanical mean internal energy is
\[ \langle U\rangle = U_{\mathrm{eq}} + \sum_{\mathbf q,r} \frac{1}{2}\hbar\omega_{\mathbf q r} + \sum_{\mathbf q,r} \frac{\hbar\omega_{\mathbf q r}} {e^{\hbar\omega_{\mathbf q r}/(k_B T)}-1}. \]
The three contributions are:
- \(U_{\mathrm{eq}}\): static equilibrium lattice energy;
- \(\sum_{\mathbf q,r}\hbar\omega_{\mathbf q r}/2\): zero-point vibration energy;
- the final sum: thermal energy stored in excited phonons.
Since \(U_{\mathrm{eq}}\) and the zero-point term are temperature independent in the harmonic approximation, only the thermal phonon term contributes to \(C_V\):
\[ C_V = \left(\frac{\partial \langle U\rangle}{\partial T}\right)_V = \sum_{\mathbf q,r} \frac{\partial}{\partial T} \left[ \frac{\hbar\omega_{\mathbf q r}} {e^{\hbar\omega_{\mathbf q r}/(k_B T)}-1} \right]. \]
Equivalently, defining
\[ x_{\mathbf q r} = \frac{\hbar\omega_{\mathbf q r}}{k_BT}, \]
one obtains the mode-resolved heat capacity
\[ C_{V,\mathbf q r} = k_B \frac{x_{\mathbf q r}^2 e^{x_{\mathbf q r}}} {(e^{x_{\mathbf q r}}-1)^2}. \]
This expression has two important limits:
\[ C_{V,\mathbf q r} \rightarrow k_B \qquad (x_{\mathbf q r}\ll 1), \]
and
\[ C_{V,\mathbf q r} \rightarrow 0 \qquad (x_{\mathbf q r}\gg 1). \]
Each mode either contributes approximately \(k_B\) if thermally active or almost nothing if frozen out.
8.4.1 Density-of-States Form
For a large crystal, the allowed \(\mathbf q\) values are very dense. It is then convenient to replace sums over modes by integrals using the phonon density of states \(D(\omega)\):
\[ \langle U\rangle = U_{\mathrm{eq}} + \int_0^{\omega_D} \frac{\hbar\omega}{2}D(\omega) d\omega + \int_0^{\omega_D} \hbar\omega D(\omega) \langle n(\omega,T)\rangle d\omega. \]
Here
\[ \langle n(\omega,T)\rangle = \frac{1}{e^{\hbar\omega/(k_B T)}-1}. \]
The heat capacity is obtained by differentiating the thermal part with respect to \(T\).
8.5 Einstein Approximation
The Einstein approximation assumes that all \(3N\) normal modes of a monatomic crystal have the same angular frequency \(\omega_E\):
\[ D(\omega) = 3N\delta(\omega-\omega_E). \]
This is a drastic simplification. It ignores the dispersion of acoustic modes but keeps the quantum nature of the oscillators.
With \(U_{\mathrm{eq}}\) omitted for simplicity, the mean internal energy becomes
\[ \langle U\rangle = 3N\hbar\omega_E \left[ \frac{1}{2} + \frac{1}{e^{\hbar\omega_E/(k_B T)}-1} \right]. \]
Introduce the Einstein temperature
\[ \Theta_E = \frac{\hbar\omega_E}{k_B}. \]
Then the Einstein heat capacity is
\[ C_V^{E} = 3Nk_B \left(\frac{\Theta_E}{T}\right)^2 \frac{e^{\Theta_E/T}} {\left(e^{\Theta_E/T}-1\right)^2}. \]
8.5.1 Limiting Cases
At high temperature, \(T\gg \Theta_E\), we have \(\Theta_E/T\ll 1\). Expanding the exponential gives
\[ C_V^{E} \rightarrow 3Nk_B. \]
Thus the Einstein model recovers the Dulong-Petit law.
At low temperature, \(T\ll \Theta_E\), one obtains
\[ C_V^{E} \simeq 3Nk_B \left(\frac{\Theta_E}{T}\right)^2 e^{-\Theta_E/T}. \]
The heat capacity is exponentially small.
The Einstein model therefore explains why the heat capacity decreases at low temperature, but it does not reproduce the experimentally observed \(T^3\) law. The reason is that the model assigns a finite energy gap \(\hbar\omega_E\) to every vibrational mode. Real crystals have acoustic phonons with arbitrarily small frequency as \(q\rightarrow 0\).
8.6 Debye Approximation
The Debye approximation improves the low-temperature physics by replacing the real phonon spectrum with three acoustic branches of linear dispersion:
\[ \omega_i(q) = v_i q, \qquad i=1,2,3. \]
Here \(v_i\) are the sound velocities of the three acoustic branches. At low temperature, this approximation is natural because optical phonons are high-energy modes and are barely occupied.
8.6.1 From Mode Counting to the Debye Cutoff
The linear acoustic approximation cannot be valid to arbitrarily large \(q\). Debye therefore introduces a cutoff wave vector \(q_D\). It is chosen so that the number of allowed \(\mathbf q\) states per acoustic branch is exactly \(N\).
In \(q\)-space, one state occupies the volume
\[ \left(\frac{2\pi}{L}\right)^3, \]
where \(L^3=V\). The Debye sphere must therefore satisfy
\[ N \left(\frac{2\pi}{L}\right)^3 = \frac{4\pi}{3}q_D^3. \]
Solving for \(q_D\) gives
\[ q_D = \left( 6\pi^2\frac{N}{V} \right)^{1/3}. \]
For branch \(i\), the corresponding Debye frequency is
\[ \omega_{D,i} = v_i q_D. \]
For an averaged sound velocity \(v_s\), the Debye temperature is defined by
\[ \Theta_D \equiv \frac{\hbar\omega_D}{k_B} = \frac{\hbar v_s q_D}{k_B} = \frac{\hbar v_s}{k_B} \left( 6\pi^2\frac{N}{V} \right)^{1/3}. \]
The averaged sound velocity is defined through
\[ \frac{1}{v_s^3} = \frac{1}{3} \sum_{i=1}^{3} \int \frac{d\Omega}{4\pi} \frac{1}{v_i^3}. \]
This average is weighted by \(1/v_i^3\) because slower acoustic modes produce a larger density of states.
The Debye sphere is a bookkeeping device. It replaces the real first Brillouin zone by a sphere with the same number of allowed vibrational states.
8.6.2 Debye Heat Capacity
Using the Debye approximation, the heat capacity of a monatomic three-dimensional crystal can be written as
\[ C_V^D = \frac{V}{(2\pi)^3} \frac{\partial}{\partial T} \sum_{i=1}^{3} \int_0^{q_D} 4\pi q^2 dq \frac{\hbar v_iq} {e^{\hbar v_iq/(k_B T)}-1}. \]
Using the averaged sound velocity \(v_s\) and introducing
\[ x = \frac{\hbar v_s q}{k_B T}, \]
this becomes
\[ C_V^D = 9Nk_B \left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D/T} \frac{x^4e^x}{(e^x-1)^2} dx. \]
This is the Debye heat-capacity formula.
8.6.3 High-Temperature Limit
For \(T\gg\Theta_D\), all modes are thermally occupied. The Debye expression reduces to
\[ C_V^D \rightarrow 3Nk_B. \]
Thus the Debye model also recovers the Dulong-Petit law.
8.6.4 Low-Temperature Limit and the \(T^3\) Law
For \(T\ll\Theta_D\), the upper integration limit \(\Theta_D/T\) is very large. We may replace it by infinity:
\[ \int_0^{\Theta_D/T} \frac{x^4e^x}{(e^x-1)^2} dx \simeq \int_0^\infty \frac{x^4e^x}{(e^x-1)^2} dx. \]
The integral has the value
\[ \int_0^\infty \frac{x^4e^x}{(e^x-1)^2} dx = \frac{4\pi^4}{15}. \]
Therefore
\[ C_V^D = \frac{12\pi^4}{5} Nk_B \left(\frac{T}{\Theta_D}\right)^3, \qquad T\ll\Theta_D. \]
This is the Debye \(T^3\) law.
A more direct low-temperature form is obtained by extending the acoustic integral to infinity:
\[ C_V = \frac{2\pi^2}{5} V k_B \left( \frac{k_B T}{\hbar v_s} \right)^3. \]
This expression shows explicitly that the low-temperature heat capacity is larger when the sound velocity is smaller. Softer materials have more low-frequency vibrational states.
8.7 Meaning of the Debye Temperature
The Debye temperature is not a phase-transition temperature. It is a characteristic temperature scale for lattice vibrations:
\[ \Theta_D = \frac{\hbar\omega_D}{k_B}. \]
It separates two limiting regimes:
\[ T<\Theta_D: \quad \text{quantum regime; modes progressively freeze out}, \]
\[ T>\Theta_D: \quad \text{classical regime; most modes are thermally active}. \]
Large \(\Theta_D\) usually indicates high characteristic phonon frequencies. This often correlates with stiff bonding and high sound velocity. Diamond, for example, has a very high Debye temperature because its bonds are stiff and its atoms are light.
8.8 Phonon Number and Zero-Point Energy
The number of thermally excited phonons is
\[ N_{\mathrm{ph}} = \int_0^{\omega_D} D(\omega)\langle n(\omega,T)\rangle d\omega. \]
In the Debye model for a three-dimensional crystal,
\[ D(\omega) = \frac{3V}{2\pi^2} \frac{\omega^2}{v_s^3}. \]
Substituting the Bose-Einstein occupation gives
\[ N_{\mathrm{ph}} = \frac{3V}{2\pi^2v_s^3} \left(\frac{k_B T}{\hbar}\right)^3 \int_0^{\Theta_D/T} \frac{x^2}{e^x-1} dx. \]
From this expression one obtains
\[ N_{\mathrm{ph}} \propto \begin{cases} T^3, & T\ll\Theta_D,\\ T, & T\gg\Theta_D. \end{cases} \]
The zero-point energy in the Debye model is
\[ U_0 = \int_0^{\omega_D} \frac{\hbar\omega}{2} D(\omega) d\omega. \]
For a three-dimensional Debye solid this gives
\[ U_0 = \frac{9}{8}Nk_B\Theta_D. \]
The zero-point energy contributes to the internal energy, but in the harmonic approximation it does not contribute to \(C_V\), because it is independent of temperature.
The heat capacity does not measure the absolute internal energy. It measures how the internal energy changes with temperature. This is why the large static lattice energy and the zero-point energy drop out of \(C_V\) in the harmonic approximation.
8.9 Connection to the Next Lecture: Anharmonicity and Thermal Expansion
In this lecture we treated the lattice as a set of harmonic normal modes. This explains the central features of lattice heat capacity:
- the Dulong-Petit law at high temperature;
- the freezing out of high-frequency modes at low temperature;
- the Debye \(T^3\) law for three-dimensional acoustic phonons.
However, a purely harmonic crystal has important limitations. It predicts no thermal expansion, no phonon-phonon interactions, no finite phonon lifetime, and \(C_p=C_V\). Real solids do expand when heated, and phonons scatter from one another. These effects require anharmonic terms in the interatomic potential.
The next lecture therefore moves from harmonic lattice thermodynamics to anharmonic lattice effects: thermal expansion, the relation between \(C_p\) and \(C_V\), and ultimately lattice thermal conductivity.
- Heat capacity measures the temperature derivative of internal energy, not the absolute internal energy.
- The Dulong-Petit law is the classical high-temperature limit of lattice vibrations.
- Classical equipartition fails at low temperature because phonon excitations are quantized.
- The Bose-Einstein occupation factor determines which phonon modes are thermally active.
- The Einstein model captures quantum freezing but misses the acoustic low-frequency spectrum.
- The Debye model succeeds at low temperature because it keeps the acoustic modes and counts them correctly.
- The Debye \(T^3\) law is a direct consequence of three-dimensional acoustic phonon phase space.
- The Debye temperature is a characteristic lattice-vibration scale, not a sharp transition temperature.
Problem Set
Definitions and Thermodynamic Meaning. A crystalline solid has heat capacity \(C\), molar heat capacity \(c_m\), and heat capacities \(C_V\) and \(C_p\).
A. Define \(C\), \(c_m\), \(C_V\), and \(C_p\).
B. Explain why \(C_V\) is directly related to the internal energy.
C. Explain physically why \(C_p\) is usually slightly larger than \(C_V\).
Dulong-Petit Law. Consider a crystal with \(N\) unit cells and \(r'\) atoms per unit cell.
A. Use classical equipartition to derive the lattice internal energy \[ U = U_{\mathrm{eq}} + 3r'Nk_BT. \]
B. Derive the Dulong-Petit law for \(C_V\).
C. Give the molar heat capacity for one mole of unit cells.
Einstein Approximation. In the Einstein approximation, all \(3N\) normal modes of a monatomic crystal have the same frequency \(\omega_E\).
A. Write the Einstein density of states \(D(\omega)\).
B. Derive the Einstein heat capacity \[ C_V^{E} = 3Nk_B \left(\frac{\Theta_E}{T}\right)^2 \frac{e^{\Theta_E/T}} {(e^{\Theta_E/T}-1)^2}. \]
C. Derive the high-temperature limit.
D. Derive the low-temperature limit and explain why it is not the observed Debye \(T^3\) law.
Debye Wave Vector and Debye Temperature. For a monatomic crystal of volume \(V\) with \(N\) unit cells, the Debye approximation replaces the first Brillouin zone by a sphere of radius \(q_D\).
A. Derive \[ q_D = \left( 6\pi^2\frac{N}{V} \right)^{1/3}. \]
B. If the averaged sound velocity is \(v_s\), derive \[ \Theta_D = \frac{\hbar v_s}{k_B} \left( 6\pi^2\frac{N}{V} \right)^{1/3}. \]
C. Explain in words what physical information is contained in \(\Theta_D\).
Debye \(T^3\) Law. Starting from \[ C_V^D = 9Nk_B \left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D/T} \frac{x^4e^x}{(e^x-1)^2} dx, \] show that for \(T\ll\Theta_D\), \[ C_V^D = \frac{12\pi^4}{5} Nk_B \left(\frac{T}{\Theta_D}\right)^3. \] Explain why this result is expected for a three-dimensional crystal with acoustic phonons.