Lattice Vibrations and Phonons

Consider $N$ atoms of mass $m$ connected by springs of constant $K$ with equilibrium spacing $a$.

The equation of motion for the $n$-th atom:

$m\ddot{u}_n = K(u_{n+1} - u_n) + K(u_{n-1} - u_n) = K(u_{n+1} + u_{n-1} - 2u_n)$

Derivation of the dispersion relation. Assuming solutions $u_n = u_0\, e^{i(qna - \omega t)}$:

$-m\omega^2 = K(e^{iqa} + e^{-iqa} - 2) = 2K(\cos qa - 1) = -4K\sin^2\left(\frac{qa}{2}\right)$

$\omega(q) = 2\sqrt{\frac{K}{m}}\left|\sin\left(\frac{qa}{2}\right)\right|$

$\blacksquare$

Key features:

• The Brillouin zone is $-\pi/a \leq q \leq \pi/a$.
• Linear for small $q$: $\omega \approx v_s \lvert q\rvert$ where $v_s = a\sqrt{K/m}$ is the speed of sound.
• Group velocity: $v_g = d\omega/dq = a\sqrt{K/m}\cos(qa/2)$.
• Maximum frequency: $\omega_{\mathrm{max} = 2\sqrt{K/m}}$.
• Phase velocity: $v_p = \omega/q$Which exceeds $v_s$ and diverges as $q \to 0$.

For a chain with alternating masses $m_1$ and $m_2$ (e.g., NaCl):

$\omega^2 = K\left(\frac{1}{m_1} + \frac{1}{m_2}\right) \pm K\sqrt{\left(\frac{1}{m_1} + \frac{1}{m_2}\right)^2 - \frac{4\sin^2(qa/2)}{m_1 m_2}}$

This gives two branches:

• Acoustic branch ($-$ sign): $\omega \to 0$ as $q \to 0$. Atoms in the unit cell move in phase.
• Optical branch ($+$ sign): $\omega \neq 0$ at $q = 0$. Atoms in the unit cell move out of phase. Can interact with light (hence the name).

At $q = 0$The optical frequency is $\omega_0 = \sqrt{2K(1/m_1 + 1/m_2)}$ and the acoustic branch Has $\omega = v_s q$ with $v_s = a\sqrt{2K/(m_1 + m_2)}$.

Lattice vibrations are quantised. Each normal mode of wave vector $\mathbf{q}$ and branch $s$ has Energy:

$E_{\mathbf{q}s} = \left(n_{\mathbf{q}s} + \frac{1}{2}\right)\hbar\omega_{\mathbf{q}s}$

Where $n_{\mathbf{q}s}$ is the phonon occupation number. Phonons are bosons obeying Bose-Einstein Statistics:

$\langle n_{\mathbf{q}s} \rangle = \frac{1}{e^{\beta\hbar\omega_{\mathbf{q}s}} - 1}$

In three dimensions, there are 3 acoustic branches (1 longitudinal, 2 transverse) and $3p - 3$ Optical branches for a crystal with $p$ atoms per primitive cell.

The Debye model approximates the phonon spectrum as linear ($\omega = v_s q$) up to a cutoff frequency $\omega_D$ (the Debye frequency):

$\omega_D = v_s\left(\frac{6\pi^2 N}{V}\right)^{1/3}$

The Debye temperature: $\Theta_D = \hbar\omega_D / k_B$.

Derivation of the phonon density of states. The number of modes with wave vector $\lvert\mathbf{q}\rvert \leq q$ In 3D is $N(q) = 3 \cdot \frac{V}{(2\pi)^3} \cdot \frac{4\pi q^3}{3}$ (factor of 3 for polarisations). Differentiating: $g(q)\,dq = dN/dq\,dq = (Vq^2/\pi^2)\,dq$. Converting to frequency with $\omega = v_s q$:

$g(\omega)\,d\omega = \frac{Vq^2}{\pi^2}\frac{dq}{d\omega}\,d\omega = \frac{V\omega^2}{\pi^2 v_s^3}\,d\omega$

Since there are $3N$ total modes, the cutoff is determined by $\int_0^{\omega_D} g(\omega)\,d\omega = 3N$Giving $g(\omega) = \frac{3V\omega^2}{2\pi^2 v_s^3}$ For $0 \leq \omega \leq \omega_D$. $\blacksquare$

Lattice heat capacity:

$C_V = 9Nk_B\left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D/T} \frac{x^4 e^x}{(e^x - 1)^2}\,dx$

High-temperature limit ($T \gg \Theta_D$): $C_V = 3Nk_B$ (Dulong—Petit law).

Low-temperature limit ($T \ll \Theta_D$): $C_V = \frac{12\pi^4}{5}Nk_B\left(\frac{T}{\Theta_D}\right)^3$ (Debye $T^3$ law).

The Einstein model treats all atoms as independent quantum harmonic oscillators with the same frequency $\omega_E$:

$C_V = 3Nk_B\left(\frac{\Theta_E}{T}\right)^2 \frac{e^{\Theta_E/T}}{(e^{\Theta_E/T} - 1)^2}$

Where $\Theta_E = \hbar\omega_E/k_B$.

High-temperature limit ($T \gg \Theta_E$): expanding $e^{\Theta_E/T} \approx 1 + \Theta_E/T + \frac{1}{2}(\Theta_E/T)^2$ gives $C_V \to 3Nk_B$ (Dulong—Petit), matching Debye.

Low-temperature limit ($T \ll \Theta_E$): $C_V \approx 3Nk_B(\Theta_E/T)^2 e^{-\Theta_E/T}$ Which is exponentially suppressed. This disagrees with the Debye $T^3$ law (and experiment).

Phonons carry heat through the lattice. By the kinetic theory formula:

$\kappa_{\mathrm{ph} = \frac{1}{3}C_V v_s \ell_{\mathrm{ph}}}$

Where $\ell_{\mathrm{ph}}$ is the phonon mean free path.

Scattering mechanisms that limit $\ell_{\mathrm{ph}}$:

1. Phonon—phonon scattering: At high $T$, $\ell_{\mathrm{ph} \propto 1/T}$ (Umklapp processes dominate, where the total phonon momentum is not conserved). At low $T$Only normal processes ($N$-processes, conserving momentum) contribute, and $\ell_{\mathrm{ph}}$ grows exponentially.
2. Boundary scattering: At very low $T$, $\ell_{\mathrm{ph}}$ is limited by the sample size $L$.
3. Defect scattering: Point defects, dislocations, and grain boundaries scatter phonons, reducing $\kappa_{\mathrm{ph}}$.

Temperature dependence:

• Low $T$ ($T \ll \Theta_D$): $\kappa_{\mathrm{ph} \propto T^3}$ (from $C_V \propto T^3$With $\ell_{\mathrm{ph}}$ limited by boundaries).
• Intermediate $T$: $\kappa_{\mathrm{ph}}$ peaks.
• High $T$ ($T \gtrsim \Theta_D$): $\kappa_{\mathrm{ph} \propto 1/T}$ (from $\ell_{\mathrm{ph} \propto 1/T}$ and $C_V \approx \mathrm{const}$).

The Debye model captures the correct low-$T$ behaviour because long-wavelength acoustic phonons Dominate the specific heat at low temperatures. The Einstein model is more appropriate for Describing the optical branch contribution, which is nearly flat (dispersionless) and hence well Approximated by a single frequency.

For crystals with optical branches (e.g., NaCl, SiO$_2$), a combined model using Debye for Acoustic modes and Einstein for optical modes gives better agreement with experiment across all Temperatures.

:::caution Common Pitfall The Debye and Einstein models describe the lattice contribution to specific heat. At low Temperatures, the electronic specific heat $C_e = \gamma T$ also contributes and can dominate over The lattice $T^3$ term in metals. The total low-$T$ specific heat of a metal is $C_V = \gamma T + \beta T^3$Where $\beta$ is related to $\Theta_D$. A plot of $C_V/T$ versus $T^2$ yields $\gamma$ (intercept) and $\beta$ (slope).

Neutrons are an ideal probe of phonons because their de Broglie wavelength ($\sim 1$ Å) matches Lattice spacings, and their energy ($\sim 10$—$100$ meV) matches phonon energies. In an inelastic Neutron scattering experiment, the energy and momentum transfer are measured:

$\hbar\omega = E_i - E_f, \quad \mathbf{q} = \mathbf{k}_i - \mathbf{k}_f$

The scattering cross-section is proportional to the dynamical structure factor $S(\mathbf{q}, \omega)$ Which has peaks when $\hbar\omega = \hbar\omega_{\mathbf{q}s}$ (phonon creation) or $\hbar\omega = -\hbar\omega_{\mathbf{q}s}$ (phonon annihilation). This allows direct measurement of The full phonon dispersion relation $\omega(\mathbf{q})$.

Time-of-flight and triple-axis spectrometers are the primary instruments used. Neutron scattering Has provided definitive measurements of phonon dispersions in virtually all important crystals.

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