Lattice Vibrations, Phonons & Thermal Properties

1. Equation of Motion for a Linear Monatomic Chain

Consider an infinite one-dimensional lattice of identical atoms, each of mass $M$, separated at equilibrium by lattice spacing $a$. Let $u_n(t)$ denote the displacement of the $n$-th atom from its equilibrium position $x_n = n a$. Under the harmonic approximation, the restoring force on atom $n$ arises from Hooke's law interactions with its nearest neighbors $n-1$ and $n+1$ with spring constant $C$:

2. Derivation of the Phonon Dispersion Relation $\omega(k)$

Seeking traveling plane-wave normal-mode solutions of the form $u_n(t) = A e^{i(k n a - \omega t)}$, where $k$ is the wavevector and $\omega$ is the angular frequency:

Dividing through by $A e^{i(k n a - \omega t)}$ and recalling Euler's identity $e^{i k a} + e^{-i k a} = 2\cos(k a)$:

Taking the positive square root yields the acoustic dispersion relation for the monatomic chain:

3. Long-Wavelength Limit and Brillouin Zone Boundary

• Long-Wavelength (Continuum / Acoustic) Limit ($k a \ll 1$):

  As $k \to 0$, $\sin(k a / 2) \approx k a / 2$, which gives a linear dispersion relation:

  $$\omega \approx 2 \sqrt{\frac{C}{M}} \left(\frac{k a}{2}\right) = a \sqrt{\frac{C}{M}} k = v_s k$$

  where $v_s = a \sqrt{C/M}$ is the macroscopic sound velocity in the crystal. In this limit, the group velocity equals the phase velocity: $v_g = d\omega/dk = v_s = v_p = \omega/k$.

• First Brillouin Zone Boundary ($k = \pm \pi / a$):

  At the edge of the first Brillouin zone, $\sin(\pi / 2) = 1$, yielding the maximum cut-off angular frequency:

  $$\omega_{\text{max}} = 2 \sqrt{\frac{C}{M}}$$

  The group velocity vanishes at the zone boundary: $\left. v_g \right|_{k = \pi/a} = \left. \frac{d\omega}{dk} \right|_{\pi/a} = a \sqrt{\frac{C}{M}} \cos\left(\frac{\pi}{2}\right) = 0$. The wave becomes a standing wave with adjacent atoms oscillating in exact antiphase: $u_{n+1}/u_n = e^{i\pi} = -1$.

1. Coupled Equations of Motion for a Diatomic Chain

Consider a 1D lattice with a two-atom basis: alternating masses $M_1$ and $M_2$ ($M_1 > M_2$) connected by identical nearest-neighbor springs of force constant $C$ with unit cell dimension $a$ (interatomic separation $a/2$). Let $u_n(t)$ denote the displacement of mass $M_1$ in unit cell $n$, and $v_n(t)$ denote the displacement of mass $M_2$ in unit cell $n$:

2. Secular Determinant & Two Frequency Branches

Substituting harmonic normal-mode trial solutions $u_n = A e^{i(k n a - \omega t)}$ and $v_n = B e^{i(k n a - \omega t)}$ yields the linear algebraic system:

For non-trivial solutions, the secular determinant must vanish:

Solving the quadratic equation in $\omega^2$ yields two distinct branches:

3. Physical Significance of Acoustic and Optical Branches

• Acoustic Branch (Minus Sign):

  As $k \to 0$, $\omega_{\text{ac}} \to 0$. In this limit, $A/B \to +1$: both masses in each unit cell oscillate in phase with identical amplitudes and directions, representing long-wavelength sound waves.

• Optical Branch (Plus Sign):

  At $k = 0$, $\omega_{\text{opt}}(0) = \sqrt{2C \left(\frac{1}{M_1} + \frac{1}{M_2}\right)} = \sqrt{\frac{2C}{\mu}}$, where $\mu$ is the reduced mass. In this limit, $M_1 A + M_2 B = 0 \implies A/B = - M_2 / M_1$: the center of mass of the unit cell remains stationary while adjacent opposite ions oscillate in antiphase against each other. In ionic crystals, this creates an oscillating electric dipole moment that couples strongly to electromagnetic light waves (infrared absorption), hence the name optical branch.

• Forbidden Band Gap at Brillouin Zone Boundary ($k = \pm \pi / a$):

  At $k = \pi / a$, $\sin^2(k a / 2) = 1$, yielding two discrete frequencies:

  $$\omega_{\text{opt}}\left(\frac{\pi}{a}\right) = \sqrt{\frac{2C}{M_2}}, \quad \omega_{\text{ac}}\left(\frac{\pi}{a}\right) = \sqrt{\frac{2C}{M_1}}$$

  Between $\sqrt{2C/M_1}$ and $\sqrt{2C/M_2}$, no real solution for $k$ exists. This frequency interval is a forbidden phononic band gap where lattice waves cannot propagate and are exponentially attenuated.

1. Quantum Mechanics of Lattice Vibrations: Phonons

When the classical harmonic Hamiltonian of $N$ coupled lattice oscillators is transformed into normal coordinates $q_{\vec{k}, s}$ and canonically quantized, it decouples into a sum of $3N$ independent quantum harmonic oscillators:

where $\hat{a}_{\vec{k},s}^\dagger$ and $\hat{a}_{\vec{k},s}$ are creation and annihilation operators. A quantum of lattice vibrational energy is called a phonon, carrying energy $\hbar \omega_s(\vec{k})$.

Because phonons are indistinguishable bosons with zero chemical potential ($\mu = 0$), their thermal equilibrium occupation number at temperature $T$ is governed strictly by the Planck distribution:

2. Phonon Crystal Momentum $\hbar \vec{q}$

A phonon carries wavevector $\vec{q}$ and a quantity $\hbar \vec{q}$ known as crystal momentum. Unlike physical momentum (the center of mass of the whole crystal does not move during internal vibrations), crystal momentum is conserved only modulo a reciprocal lattice vector $\vec{G}$:

3. Inelastic Neutron Scattering

The experimental dispersion relation $\omega(\vec{q})$ across the entire Brillouin zone is determined by Inelastic Neutron Scattering (INS). Thermal neutrons with mass $M_n \approx 1.675 \times 10^{-27}\text{ kg}$ have de Broglie wavelengths comparable to lattice spacings ($\lambda \sim 1 - 2\text{ \AA}$) and kinetic energies comparable to phonon energies ($E \sim 10 - 100\text{ meV}$):

• Energy Conservation: $E' - E = \frac{\hbar^2 k'^2}{2M_n} - \frac{\hbar^2 k^2}{2M_n} = \pm \hbar \omega(\vec{q})$ ($+$ for phonon emission, $-$ for phonon absorption).
• Crystal Momentum Conservation: $\vec{k}' - \vec{k} = \mp \vec{q} + \vec{G}$.

By measuring the scattered neutron angle and energy, both $\omega$ and $\vec{q}$ are mapped simultaneously.

1. Classical Dulong-Petit Law & Quantum Failure

By the classical equipartition theorem, each of the $3N$ vibrational degrees of freedom in a 3D crystal of $N$ atoms possesses an average thermal energy of $k_B T$, yielding total internal energy $U = 3 N k_B T$. The molar lattice heat capacity at constant volume is the constant Dulong-Petit value:

Experimentally, as $T \to 0\text{ K}$, $C_V$ vanishes rapidly, violating classical theory.

2. The Einstein Specific Heat Model

Einstein (1907) assumed that all $3N$ normal modes oscillate independently with the same identical frequency $\omega_E$ (characteristic Einstein temperature $\Theta_E = \hbar\omega_E / k_B$):

• High-Temperature Limit ($T \gg \Theta_E$): $C_V \to 3 N k_B$ (recovers Dulong-Petit).
• Low-Temperature Limit ($T \ll \Theta_E$): $C_V \approx 3 N k_B (\Theta_E / T)^2 e^{-\Theta_E / T}$. While this vanishes at $T = 0$, the exponential drop is much steeper than experimental data, which follows a power law.

3. The Debye Model & The $T^3$ Law

Debye (1912) treated the crystal as an elastic isotropic continuum with linear acoustic dispersion $\omega = v_s k$ across all three acoustic polarizations ($1$ longitudinal, $2$ transverse). The phonon density of states in 3D is:

Debye imposed a high-frequency cut-off, the Debye cut-off frequency $\omega_D$, to ensure the total number of modes equals $3N$:

The total thermal vibrational internal energy is:

Differentiating with respect to $T$ in the low-temperature limit ($T \ll \Theta_D$), the upper limit goes to infinity: $\int_0^\infty \frac{x^3}{e^x - 1} dx = \frac{\pi^4}{15}$. This yields the celebrated Debye $T^3$ Law:

This matches experimental specific heat measurements for all non-magnetic insulating solids.

1. Anharmonic Effects & Thermal Expansion

In a purely harmonic potential $V(x) = \frac{1}{2} c x^2$, the potential well is symmetric about $x = 0$. The thermal average displacement vanishes at all temperatures: $\langle x \rangle = \frac{\int x e^{-V(x)/k_BT} dx}{\int e^{-V(x)/k_BT} dx} = 0$, implying zero thermal expansion. Thermal expansion is an intrinsically anharmonic phenomenon.

Expanding the interatomic potential including cubic ($g x^3$) and quartic ($f x^4$) anharmonic perturbations:

Calculating the thermal average position using the Boltzmann distribution yields:

The linear thermal expansion coefficient $\alpha_{\text{th}} = \frac{1}{a} \frac{d\langle x \rangle}{dT} = \frac{3g k_B}{4c^2 a}$ is directly proportional to the cubic anharmonic coefficient $g$.

2. Lattice Thermal Conductivity $\kappa$

Heat conduction by phonons is described by the kinetic theory formula:

where $C_V$ is the phonon heat capacity per unit volume, $v_s$ is the mean sound velocity, and $\ell_{\text{ph}} = v_s \tau_{\text{ph}}$ is the phonon mean free path.

3. Normal ($N$) vs. Umklapp ($U$) Phonon Scattering Processes

In three-phonon scattering collisions ($\vec{q}_1 + \vec{q}_2 = \vec{q}_3 + \vec{G}$):

• Normal ($N$) Processes ($\vec{G} = 0$): $\vec{q}_1 + \vec{q}_2 = \vec{q}_3$. Total phonon crystal momentum is conserved. $N$-processes redistribute energy among phonon modes but do not produce thermal resistance (they cannot decay a net heat current).
• Umklapp ($U$) Processes ($\vec{G} \neq 0$): $\vec{q}_1 + \vec{q}_2 = \vec{q}_3 + \vec{G}$. When two high-energy phonons collide such that $\vec{q}_1 + \vec{q}_2$ falls outside the First Brillouin Zone, it is Bragg-reflected back into the zone by subtracting a reciprocal lattice vector $\vec{G}$. This reverses the direction of the resultant energy flow, destroying net crystal momentum and creating intrinsic lattice thermal resistivity.

At high temperatures ($T \gg \Theta_D$), the number of excited high-energy phonons scales as $n_{\text{ph}} \propto T$, so $\ell_{\text{ph}} \propto 1/T$, yielding $\kappa \propto 1/T$. At very low temperatures, $U$-processes freeze out as $e^{-\Theta_D / 2T}$, and $\ell_{\text{ph}}$ is limited only by sample boundary scattering ($\ell_{\text{ph}} \approx \text{const}$), so $\kappa \propto C_V \propto T^3$.