Phonons in a Diatomic Chain

Diatomic Chain

Teach me

Speed

Displacement u(x) and ω(k)

atom A (4 u) atom B (variable) drive line marker (line meets branch)

Guided Inquiry

1 · A wave down a chain

Let’s start gently. The shaker drives the left atom at 2 THz. Watch the long ripple run down the chain — neighbours swaying almost together. Then find the white marker where the drive line meets the lower branch on the ω(k) plot, and read the marker’s wavelength λ in the Drive & marker panel.

2 · Climbing the dial

Now let’s climb higher. Suppose you raise the drive into the frequency region between the two branches. What do you predict the chain will do there?

3 · Into the gap

Let’s find out. Drag the drive up into the region between the two branches — around 8 THz — and watch the chain closely. What happens to the motion?

4 · Closing the gap

So that shaded band on the plot is the forbidden band — you can read its edges in the band panel. Now, atom A is fixed at 4 u. Which mass for atom B do you think makes the band vanish completely?

5 · Masses matched

Let’s test it. With the drive still at 8 THz, slide atom B down toward 4 u and watch what happens. The band narrows and the once-dead 8 THz springs onto the lower branch; keep going and the two branches touch exactly at m_B = 4 u — the width readout hits 0.00 and the band is gone.

6 · Why "optical"?

One last idea to explore. Drag the drive up onto the upper branch and watch neighbouring atoms. In NaCl the two ions carry opposite charge — so what does that antiphase motion create, and what kind of light could grab it?

Listen

Drive frequency f2.00 THz

0.2 THz14 THz

Mass of atom B12.0 u

1 u16 u

Atom A is fixed at 4 u — the slider crosses equality when B = 4 u.

Spring constant K15.0 N/m

5 N/m30 N/m

Drive f 2.00 THz

Meets lower branch

k 0.241 π/a

λ 8.31 cells

U_B / U_A +1.038

neighbours move together (in phase)

Two branches over the first Brillouin zone: ω₋ (acoustic) and ω₊ (optical).

Band edges at the zone edge k = π/a.

Optical maximum at k = 0; the acoustic branch starts at exactly zero.

Inside the gap k is complex — the displacement decays as e^(−κx): evanescent, not travelling.

The Physics

Phonons in a diatomic chain · what this sim shows, the physics behind it, and every control

At a glance

Shake one end of a chain of alternating light and heavy atoms and ask what travels down it. This sim shows how the answer depends on the drive frequency and on the two masses, in the simplest model of a vibrating crystal.

What you'll learn

How the dispersion relation ω(k) links the drive frequency to the wave on the chain.
How neighbouring atoms move relative to each other, and how that depends on frequency.
How the masses and the spring constant set the frequencies at which the chain's behaviour changes.
How to read k, λ and U_B/U_A off the Drive & marker panel.

Atoms A and B alternate and every spring has the same constant K. The left end is shaken.

What's on screen

Atom A (blue) is always 4 u.
Atom B (orange) has the mass set by its slider.
Displacement trace u(x): how far each atom sits from rest, right now. The shaded right end absorbs the wave.
Lower curve (teal) of the ω(k) plot: f in THz against k across the zone −π/a to +π/a.
Upper curve (amber): the second branch.
Drive line at the shaker's frequency, and a marker where it meets a curve. The marker's k gives the wavelength.

See itDrive at the opening 2 THz. Watch for: the marker on the lower curve and a long ripple along the chain.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Drive frequency f	The frequency at which the left end is shaken. It moves the dashed drive line.	0.2–14 THz opens at 2 THz
Mass of atom B	The mass of every B atom. Atom A stays at 4 u. Both curves reshape.	1–16 u opens at 12 u
Spring constant K	The stiffness of every spring.	5–30 N/m opens at 15 N/m
Readouts	Drive & marker: f, which curve the drive meets, k, λ and U_B/U_A. A second group, Band readout, appears once the drive first lands between the two curves.	panel
▶ Play / ⏸ Pause	Runs the chain, or freezes it so you can read the trace.	top bar
Speed	Plays the motion faster or slower on screen. The physics is unchanged.	0.25×–4×
↻ Reset	Restarts the chain from rest. The sliders keep their values.	top bar

Fixed: atom A = 4 u; cell length a = 0.5 nm; springs join nearest neighbours only; the right end of the chain absorbs.

Two branches, not one

A unit cell holds two atoms, so each wavevector k allows two ways for the chain to vibrate. They are the two roots of the dispersion relation, one for each sign:

m_A and m_B are the two masses and S = 1/m_A + 1/m_B.

The lower (acoustic) branch goes to ω = 0 at k = 0. Over a long wavelength the whole cell moves together, which is sound.
The upper (optical) branch never reaches zero. At k = 0 it sits at ω = √(2K/μ), where μ = m_Am_B/(m_A+m_B) is the reduced mass.
The plot covers one zone, −π/a to +π/a. A wavevector outside it describes atom motion already drawn inside.

The sim's own plot at its opening values: 4 u and 12 u atoms on 15 N/m springs.

See itDrive at 2 THz, then at 11.5 THz, on the opening chain. Watch for: the marker jumping from the lower curve to the upper curve.

What the atoms do on each branch

The Drive & marker panel prints the ratio of the two atoms' displacements. Its sign tells the branches apart:

U_B/U_A from the equations of motion, for a wave at wavevector k and frequency ω.

Lower branch: the ratio is positive, so the two atoms of a cell move the same way.
Upper branch: the ratio is negative, so they move against each other. At k = 0 it equals −m_A/m_B, which means m_AU_A + m_BU_B = 0: the centre of mass stays still and the lighter atom swings more.
That k = 0 mode is a two-body oscillator with stiffness 2K and reduced mass μ, which is smaller than either mass. That is why the upper branch sits high.

The two patterns at the opening values: the lower branch at the 2 THz marker, the upper branch at k = 0.

See itDrive onto the upper branch. Watch for: U_B/U_A turning negative, and neighbouring atoms swinging against each other on the chain.

Frequencies with no wave

Some drive frequencies meet neither curve: those between the two branches, and those above the top of the upper one. No real k solves the dispersion relation there, so no travelling wave exists. The solution has a complex k:

Between the branches k = π/a + iκ. Above the upper branch k = iκ.

The motion shrinks by e^−κa from each cell to the next. At 8 THz on the opening chain that is ×0.35 per cell, so it is under 5% of the drive by the third cell.
The equations are linear, so the decay per cell does not depend on the drive strength. Shaking harder does not make a wave travel.
Above the upper branch the motion also dies out, here by ×0.23 per cell at 13 THz.

See itDrive at 8 THz, between the branches. Watch for: no marker on the plot, and a trace that dies within a few cells of the shaker.

See itDrive at 13 THz, above the upper branch. Watch for: the trace dying out again, with the drive line above both curves.

Where the branches end

Both band edges sit at the zone edge k = π/a. There one sublattice stands still, and each moving atom hangs between two motionless neighbours on two springs, so ω = √(2K/m).

The lower branch ends at the heavy atom's value, the upper branch starts at the light atom's.

On the lower branch's last point only the heavy atoms move. On the upper branch's first point only the light atoms move.
At the opening values the edges are 6.18 and 10.70 THz. The upper branch peaks at 12.35 THz, at k = 0.

See itDrive just under the lower edge (6.17 THz), then just over the upper one (10.70 THz). Watch for: U_B/U_A near +19.5, where B swings and A barely moves, then near −0.02, where A swings and B barely moves.

Mass contrast sets the width

The edge that belongs to atom A, √(2K/m_A), does not move when you change m_B. The other edge travels.
The ratio of the two edges is √(m_heavy/m_light). It is 1 only when the masses are equal, and then the band closes: the chain is a single chain of identical atoms, spacing a/2, and the upper branch is the lower one folded back at k = π/a.
K multiplies every frequency by √K. Doubling it widens the band by √2 but leaves the edge ratio alone: stiffness shifts the band, mass contrast sizes it.

m_B K

These move the sim's own sliders. Atom A stays at 4 u.

See itKeep the drive at 8 THz and slide m_B from 12 u to 6 u, then to 4 u. Watch for: the Width readout shrinking from 4.52 THz to 0.00 THz.

Where it matters

The lower branch at long wavelength is a straight line, ω = v_sk.

Sound in solids. The slope of the lower branch at k = 0 is the speed of sound. At the opening values v_s = 8.4 km/s, the right order of magnitude for a solid.
Infrared absorption in ionic crystals. In a crystal like NaCl the two ions carry opposite charge, so the upper branch's antiphase motion is an oscillating dipole that infrared light can drive. A photon's k is tiny on this scale (about 4×10⁻⁵ of π/a at 12.35 THz in vacuum), so light couples only near k = 0.
Heat. Quantise these same vibrations and each energy step is a phonon. Phonons carry heat in insulators.

Go deeper: what this model leaves out

It is one-dimensional, with nearest-neighbour springs only. Real crystals have three dimensions and longer-range forces.
There is no long-range Coulomb force, so it does not show the LO–TO splitting of a real ionic crystal.
The motion is classical. The quantum statement is only that each mode's energy comes in steps of ħω.

Summary

In one line. A two-atom cell gives two branches, with no travelling wave between them or above them. The edges are √(2K/m) for each atom, and equal masses close the band.

Key points

The lower branch goes to 0 at k = 0; the upper starts at √(2K/μ).
Cell partners move together on the lower branch and against each other on the upper.
A frequency that meets no branch gives complex k, and a decay of e^−κa per cell at any drive strength.
K scales every frequency by √K; mass contrast sets the edge ratio.

Common mix-ups

"Frequencies in the band get through, only weaker." No real k exists there. The motion decays within a few cells at any amplitude.
"More mass contrast closes the band." It widens it. Equal masses close it.
"Above the upper branch the wave travels again." There is no mode there either, so k is imaginary.

Things to try on your own

Set K = 30 N/m (m_B = 12 u). Divide each band edge by its opening value, 6.18 and 10.70 THz, and compare with √2.
Set m_B = 4 u and sweep f from 0.2 to 14 THz. Note where the marker moves from the lower curve to the upper one.
Set m_B = 1 u and drive at 14 THz. Use the Band readout to explain the trace.

Predict before reveal	Students commit to what the chain does in the gap, and which mass closes it, before dragging.
Discovery, not diagram	The band is unshaded and unnamed until a student has driven a frequency inside it and watched motion die.
What is invariant	One band edge stays fixed while the other moves as the masses change — watch which, and what happens when they meet.
Emergent, not painted	Decay and phase come from integrating the coupled chain, matching the dispersion and evanescence oracles.
Operational numbers	Every readout — k, λ, U_B/U_A, band edges — is read off or compared by a step; no standing meters.