The Harmonic Crystal and Phonon Dispersion

A crystal is not a rigid array of fixed points. Each atom sits in a potential well formed by its neighbors and oscillates about its equilibrium site, and because the wells are coupled the oscillations propagate as waves. Those waves carry the crystal's sound, most of its heat capacity, and the scattering that limits electrical conduction. This lesson treats them classically: expand the interatomic potential to second order, reduce the equations of motion to an eigenvalue problem, and solve the one-dimensional chains that already contain every qualitative feature of the three-dimensional result.

The harmonic approximation

Label the equilibrium positions of the ions by the Bravais lattice vectors $R$ and let $u (R)$ be the displacement of the ion nominally at $R$ . The potential energy of the crystal is some function $U$ of all the instantaneous positions $R + u (R)$ . Expand it about the equilibrium configuration in powers of the displacements:

U = U_{eq} + R, i \sum \frac{\partial U}{\partial u _{i} ( R )} u_{i} (R) + \frac{1}{2} R, i R^{'}, j \sum \frac{\partial ^{2} U}{\partial u _{i} ( R ) \partial u _{j} ( R ^{'} )} u_{i} (R) u_{j} (R^{'}) + \dots

The constant $U_{eq}$ is the cohesive energy and sets the zero. The linear term vanishes: equilibrium is defined by every first derivative of $U$ being zero, so no net force acts when all displacements vanish. The first surviving term is quadratic. Truncating there is the harmonic approximation, and it defines the force constants

Φ_{ij} (R, R^{'}) = \frac{\partial ^{2} U}{\partial u _{i} ( R ) \partial u _{j} ( R ^{'} )}_{eq},

a matrix of second derivatives evaluated at equilibrium. Because the crystal is periodic, $Φ_{ij} (R, R^{'})$ depends only on the separation $R - R^{'}$ , not on $R$ and $R^{'}$ separately.

The force on the ion at $R$ is $- \partial U / \partial u_{i} (R)$ , giving Newton's equations

M \overset{u}{¨}_{i} (R) = - R^{'}, j \sum Φ_{ij} (R - R^{'}) u_{j} (R^{'}) .

These are coupled linear differential equations, one for each Cartesian component of each ion. Their solution is the program of lattice dynamics.

The monatomic linear chain

Strip the problem to its essentials: a line of identical ions of mass $M$ , spacing $a$ , connected by springs of stiffness $K$ that couple only nearest neighbors. Let $u_{n}$ be the longitudinal displacement of the $n$ -th ion from its site $na$ .

A one-dimensional monatomic chain: identical masses M at spacing a, joined by nearest-neighbor springs of stiffness K. The displacement of ion n is u_n, measured from its equilibrium site.

The stretch of the spring to the right of ion $n$ is $u_{n + 1} - u_{n}$ ; the stretch of the spring to the left is $u_{n} - u_{n - 1}$ . The net force is the difference, so the equation of motion is

M \overset{u}{¨}_{n} = K (u_{n + 1} - u_{n}) - K (u_{n} - u_{n - 1}) = K (u_{n + 1} + u_{n - 1} - 2 u_{n}) .

Look for travelling-wave solutions $u_{n} = A e^{i (k na - ω t)}$ : a plane wave sampled at the lattice sites, with wavevector $k$ and angular frequency $ω$ . Substituting and cancelling the common factor $A e^{i (k na - ω t)}$ ,

- M ω^{2} = K (e^{ik a} + e^{- ik a} - 2) = 2 K (cos k a - 1) = - 4 K sin^{2} \frac{k a}{2} .

Solving for $ω$ gives the dispersion relation of the monatomic chain:

ω (k) = 2 \frac{K}{M} sin \frac{k a}{2} .

Frequency is not proportional to wavevector except near $k = 0$ . The chain is a dispersive medium: waves of different wavelength travel at different phase speeds.

Dispersion of the monatomic chain over the first Brillouin zone. The curve rises from a linear (sound-like) region near k = 0 to a maximum of 2 sqrt(K/M) at the zone boundary k = pi/a, where the group velocity vanishes.

Two limits read off the curve. Near $k = 0$ , $sin (k a /2) \approx k a /2$ and

ω \approx a \frac{K}{M} ∣ k ∣ = v_{s} ∣ k ∣, v_{s} = a \frac{K}{M},

a linear (nondispersive) branch: long-wavelength vibrations are ordinary sound, travelling at the speed $v_{s}$ . At the zone boundary $k = π / a$ the frequency reaches its maximum $ω_{max} = 2 K / M$ , and the group velocity $d ω / d k = a K / M cos (k a /2)$ falls to zero: the boundary mode is a standing wave in which alternate ions move in opposite directions and no energy propagates.

Brillouin-zone periodicity

The dispersion depends on $k$ only through $sin (k a /2)$ , which is periodic in $k$ with period $2 π / a$ . Shifting $k \to k + 2 π / a$ leaves $ω$ unchanged. More than that, it leaves the physical motion unchanged. The displacement pattern is determined by the phase $k na$ evaluated at integer $n$ , and

e^{i (k + 2 π / a) na} = e^{ik na} e^{i 2 π n} = e^{ik na} .

A wavevector outside the first Brillouin zone $(- π / a, π / a]$ describes exactly the same lattice motion as its image inside the zone. There is no physical content in $k$ beyond the zone: sampling a wave only at discrete sites cannot resolve wavelengths shorter than $2 a$ . This aliasing is the lattice analogue of the Nyquist limit.

Aliasing on the lattice. A short-wavelength wave (light curve) and a long-wavelength wave (accent curve) pass through identical displacements at every lattice site, so they represent the same motion; only the wave with wavevector inside the first zone is retained.

Counting confirms that one zone holds every mode. Impose Born–von Kármán periodic boundary conditions on a ring of $N$ ions: $u_{n + N} = u_{n}$ forces $e^{ik N a} = 1$ , so $k = 2 π m / (N a)$ for integer $m$ . Exactly $N$ of these values fall in the first zone, one allowed $k$ per ion. A chain of $N$ ions has $N$ degrees of freedom and $N$ normal modes, and the first Brillouin zone accounts for all of them.

The diatomic chain

Real crystals often have more than one atom per primitive cell, and the extra atoms produce qualitatively new modes. Take the simplest case: a chain with two different masses $M_{1}$ and $M_{2}$ alternating along the line, joined by identical springs $K$ , with lattice constant $a$ (so the two-atom cell repeats every $a$ ). Let $u_{n}$ be the displacement of the mass- $M_{1}$ atom in cell $n$ and $v_{n}$ that of the mass- $M_{2}$ atom.

A diatomic chain: two masses M1 and M2 alternate along the line, joined by identical springs K. The primitive cell of length a holds one of each; u_n and v_n are the two displacements in cell n.

Each $M_{1}$ atom is pulled by the $M_{2}$ atoms on either side, and vice versa:

M_{1} \overset{u}{¨}_{n} M_{2} \overset{v}{¨}_{n} = K (v_{n} + v_{n - 1} - 2 u_{n}), = K (u_{n + 1} + u_{n} - 2 v_{n}) .

Try coupled travelling waves $u_{n} = A e^{i (k na - ω t)}$ and $v_{n} = B e^{i (k na - ω t)}$ with different amplitudes $A$ and $B$ . Substituting gives a $2 \times 2$ homogeneous linear system for $A$ and $B$ :

(2 K - M_{1} ω^{2}) A - K (1 + e^{- ik a}) B - K (1 + e^{ik a}) A + (2 K - M_{2} ω^{2}) B = 0, = 0.

A nontrivial solution requires the determinant to vanish, giving the characteristic equation

M_{1} M_{2} ω^{4} - 2 K (M_{1} + M_{2}) ω^{2} + 2 K^{2} (1 - cos k a) = 0.

This is quadratic in $ω^{2}$ , so at each $k$ there are two frequencies:

ω_{\pm}^{2} (k) = K (\frac{1}{M _{1}} + \frac{1}{M _{2}}) \pm K (\frac{1}{M _{1}} + \frac{1}{M _{2}})^{2} - \frac{4 sin ^{2} ( k a /2 )}{M _{1} M _{2}} .

The two roots are the two branches of the dispersion.

Acoustic and optical branches

The two roots separate cleanly in the long-wavelength limit. For small $k$ , $sin^{2} (k a /2) \approx (k a /2)^{2}$ , and expanding the square root gives

ω_{-}^{2} \approx \frac{K /2}{M _{1} + M _{2}} (k a)^{2}, ω_{+}^{2} \approx 2 K (\frac{1}{M _{1}} + \frac{1}{M _{2}}) .

The lower root $ω_{-}$ is the acoustic branch: it starts from zero with slope $v_{s} = a K /2 (M_{1} + M_{2})$ , and the two atoms move together as a rigid unit — the amplitude ratio $B / A \to 1$ . The upper root $ω_{+}$ is the optical branch: it approaches a finite frequency at $k = 0$ where the two atoms move exactly out of phase against each other, keeping the center of mass fixed ( $M_{1} A + M_{2} B = 0$ ). At the zone boundary $k = π / a$ the two branches reach

ω_{-} (π / a) = \frac{2 K}{M _{heavy}}, ω_{+} (π / a) = \frac{2 K}{M _{light}},

leaving a frequency gap between $2 K / M_{heavy}$ and $2 K / M_{light}$ in which no travelling wave exists. Waves at those frequencies are evanescent and decay into the crystal; the gap is the mechanical analogue of the electronic band gap.

The two branches of the diatomic chain over the first Brillouin zone. The acoustic branch rises from zero; the optical branch runs across the top. Between the zone-boundary values a frequency gap opens, forbidden to travelling waves. Insets show the in-phase and out-of-phase cell motions.

Cell motions at k near zero. In the acoustic mode the two atoms of a cell move in the same direction (in phase); in the optical mode they move in opposite directions (out of phase), the lighter atom swinging farther.

The optical name comes from ionic crystals such as NaCl. There the two atoms carry opposite charges, so the out-of-phase optical motion at $k \approx 0$ produces an oscillating electric dipole. That dipole couples strongly to infrared light at the optical frequency, giving the reststrahlen (residual-ray) absorption band that reflects a narrow range of infrared wavelengths almost perfectly.

The three-dimensional generalization

A real crystal replaces the scalar displacement $u_{n}$ with a vector $u (R)$ and the scalar force constant with the tensor $Φ_{ij} (R - R^{'})$ . The same plane-wave substitution $u (R) = ϵ e^{i (k \cdot R - ω t)}$ reduces the equations of motion to an eigenvalue problem for the dynamical matrix

D_{ij} (k) = \frac{1}{M} R \sum Φ_{ij} (R) e^{- i k \cdot R}, ω^{2} (k) ϵ = D (k) ϵ .

$D (k)$ is a $3 p \times 3 p$ Hermitian matrix for a crystal with $p$ atoms per cell. Its eigenvalues $ω_{s}^{2} (k)$ are the squared frequencies of the $3 p$ branches; its eigenvectors $ϵ_{s}$ are the polarization vectors giving the direction each atom moves. The branch structure generalizes directly:

Three acoustic branches, one for each polarization (one longitudinal, two transverse), all with $ω \to 0$ linearly as $k \to 0$ . These are the three sound waves of the elastic continuum.
$3 (p - 1)$ optical branches, finite frequency at $k = 0$ , in which atoms within a cell oscillate against one another.

A crystal with $p = 1$ (a monatomic Bravais lattice such as copper) has only the three acoustic branches and no optical modes. A crystal with $p = 2$ (silicon, NaCl, diamond) has three acoustic and three optical branches. The dynamical matrix, its symmetry, and the eigenvectors organize the full three-dimensional problem, but the counting and the acoustic-versus-optical distinction are already visible in the one-dimensional chains.

So far the treatment is entirely classical: the modes are continuous oscillations of arbitrary amplitude. The quantization of these normal modes turns each into a ladder of energy levels and gives the phonon, the quantized unit of lattice vibration that carries crystal momentum and sets the thermal properties of the solid.