The frequency a lattice cannot carry

Pluck a string and it carries whatever it is given: a low note, a high note, and in principle a note of any frequency at all. The wave equation has a solution at every frequency, with the wavelength adjusting to keep the speed the same, and there is no upper limit anywhere in it.

That is a property of a continuum, and matter is not one. A row of masses joined by springs — which is what a crystal is, at the scale that matters — has a highest frequency, and the reason is easy to see once the question is asked. The fastest thing the chain can do is have every mass move opposite to both its neighbours, and that is one particular motion at one particular frequency. There is nothing faster available.

The dashed straight line is the continuum the chain is often taken to be. The two agree well where the wavelength is long compared with the spacing, and the departure is second order in the wavenumber — which is exactly why the discreteness of matter is invisible to sound and unavoidable to anything at the atomic scale.

What happens above the top #

The behaviour above the cutoff is worth being careful about, because the word for it — evanescent — is usually met somewhere else and carries the wrong associations.

Nothing is absorbing. The model has no damping in it at all: the springs are perfect, the masses are frictionless, and energy is conserved exactly. What has happened is that the equation has no travelling solution at that frequency, and the only solution that stays bounded is one that decays with distance. The energy the drive delivers is returned to it; the chain reflects everything.

The decay rate is not a new piece of physics either. It is the dispersion relation continued past the point where it has a real answer: the wavenumber acquires an imaginary part, and the imaginary part is the decay per site. That is the same continuation that makes light beyond the critical angle decay into the far medium instead of travelling in it, and the same one that makes a quantum particle’s wavefunction decay inside a barrier rather than oscillate. Three subjects, one analytic continuation.

The alternating sign is the tell that the cutoff has been passed at the top rather than at the bottom. A waveguide below its cutoff also has an evanescent solution, and there the neighbouring elements move together; here they move in opposition, which is what a wavenumber at the zone edge means.

Where the frequency limit is a real number #

The chain is a model and its cutoff is a real quantity. For a crystal, the highest lattice frequency is a few times $10^{13}$ hertz — a few tens of terahertz, or a wavelength in the infrared — and it is set by the same arithmetic: the square root of an interatomic spring constant over an atomic mass.

That number decides several things at once.

It is why the heat capacity of a solid falls at low temperature. There is a highest frequency, so there is a highest quantum of vibrational energy, and below a temperature corresponding to it the modes near the top cannot be excited at all. The temperature scale that appears in the Debye account of heat capacity is the cutoff frequency written as a temperature.

It is why sound in a solid is dispersionless. Audible sound has a wavelength of centimetres and the spacing is a fraction of a nanometre, so the wavenumber is smaller than the zone edge by eight orders of magnitude and the curve is indistinguishable from its tangent. Every measurable property of sound in a solid is a property of the straight-line part.

And it sets the frequency above which a solid stops transmitting vibration at all. Ultrasound at gigahertz frequencies is still far below it; the terahertz region is where the discreteness begins to matter, and measuring the whole dispersion curve rather than its slope is what neutron scattering is for.

Two things in the cell #

Making the repeating unit contain two masses instead of one changes the picture qualitatively, and the change is the point.

There are now two solutions at each wavenumber rather than one. The lower branch starts at zero and behaves like sound; the upper one does not go to zero at all, and at long wavelength it is a motion in which the two masses in each cell move against each other while the cell as a whole stays still. Between them is a band of frequencies at which the chain has no mode whatever.

The gap is opened by the repeat rather than by the springs, and the figure’s parameter is a mass ratio with the springs untouched throughout. As the masses become equal the gap closes — and it must, because a chain of equal masses described as though its cell held two of them is the same chain, with the upper branch being the lower one folded back at half the wavenumber.

That is the general statement about periodic media, and it is the same statement as the one about a repeat opening a gap in any wave problem. A periodicity of some length forbids a band of frequencies near where the wavelength is twice that length, because there the forward and backward waves are strongly coupled and the two combinations they form have different energies. The mechanism is identical for a diatomic chain, a stack of dielectric layers, an electron in a crystal and a corrugated waveguide.

The upper branch’s name — the optical branch — comes from the ionic case, where the two masses carry opposite charges and their opposed motion is a dipole that couples to light. That is why an ionic crystal has a strong infrared absorption at a frequency with nothing to do with any electronic transition, and why the same crystal is highly reflective in a band just above it.

Comparing the two mass ratios makes the argument’s shape clear. Nothing about the springs differs between the figures; what differs is how unlike the two masses in a cell are, and the gap grows in proportion to that difference. A modulation of any kind — mass, spring, spacing — opens a gap in the same way and to a size set by how strong the modulation is.

That has a design consequence that is exploited constantly. A weak periodic modulation opens a narrow gap; a strong one opens a wide gap. So a filter, a mirror or an insulating band can be made to any width by choosing how much of a contrast to build in, and the frequency at which the gap sits is set separately, by the period. Two knobs, doing two independent jobs, which is why periodic structures are such useful things to design with.

Seeing the modes themselves is worth it, because it settles what the zone edge means. A chain of ten masses has ten modes and no more — there is nowhere for an eleventh to come from — and the tenth is the alternating one. Adding masses adds modes, filling in the curve between; it does not add any above the top, because the top mode is the fastest motion the springs and masses allow and lengthening the chain does not change either.

That is a different kind of counting from a string’s. A string has infinitely many modes because it has infinitely many degrees of freedom; a chain has as many as it has masses. The number of modes is the number of things that can move, which is a statement that is trivially true and constantly forgotten, and it is the reason a solid’s vibrational spectrum stops.

Where the same picture is doing other work #

The chain is worth keeping in mind well outside mechanics, because a great many systems are one.

A row of coupled pendulums, a ladder of inductors and capacitors, a line of coupled optical cavities and a row of quantum wells brought close enough to interact all have the same dispersion relation with different names for the quantities. In each the same three facts hold: a linear region at long wavelength, a cutoff where neighbours move in opposition, and evanescent decay above it.

The electron case is the one with the most riding on it. An electron in a periodic potential has bands and gaps for the same reason, and the width of a band is set by how strongly neighbouring sites are coupled — which is the tight-binding picture and is the chain with a different quantity travelling along it. That a metal conducts and an insulator does not is decided by where the electrons fall relative to a gap that exists for the reason drawn here.

What travels along the chain differs; that a periodic array has a top frequency and a forbidden band does not.

Reading the curve as a speed #

The dispersion curve carries two speeds and they are not the same, which is the second thing the flattening at the edge is about.

The phase velocity is the frequency over the wavenumber — the speed of a crest — and it is the slope of the line from the origin to the point on the curve. It falls steadily from the sound speed at long wavelength to about two thirds of it at the zone edge.

The group velocity is the slope of the curve itself, and it is the speed a disturbance and its energy actually travel at. It equals the sound speed at long wavelength and falls to zero at the zone edge, where the curve is flat.

A mode at the zone edge therefore does not go anywhere at all. Its crests move — the phase velocity is not zero — and nothing is transported, because the mode is a standing wave. That is a clean instance of the distinction between a phase velocity and the speed of anything real, and here the two differ by a factor of infinity rather than by a correction.

The consequence for a real solid is a large one. The density of modes is inversely proportional to the group velocity, so a flat region of the dispersion curve is a region with a great many modes crowded into a narrow band of frequencies. Every sharp feature in a solid’s vibrational spectrum sits where the dispersion curve is flat, and the flattest place is the zone edge.

Where the model stops #

The chain is one-dimensional and a crystal is not. In three dimensions there are three branches for each atom in the cell — one longitudinal and two transverse — and their dispersion depends on the direction of travel through the lattice. The cutoff becomes a surface rather than a number, and the density of modes near the top is what actually enters a heat capacity.

The springs are linear. Real interatomic forces are not, and the anharmonic term does the work the harmonic chain cannot: it lets modes exchange energy, which is what gives a solid a thermal conductivity at all — a perfectly harmonic crystal has infinite conductivity, because its modes never scatter.

Nearest neighbours only. Including further neighbours changes the shape of the dispersion curve without changing its structure: there is still a top frequency and still a zone edge, and the curve between them acquires more detail. Fitting that detail to measured dispersion curves is how interatomic force constants are extracted.

And the chain is infinite. A finite chain of $N$ masses has $N$ modes at discrete wavenumbers rather than a continuous curve, which is the same discreteness a finite string has — and the two effects, discreteness of the medium and finiteness of the sample, are often confused. The first gives a highest frequency; the second gives a spacing between frequencies.

Two ways of failing to be a continuum #

It is worth separating two effects that both make a discrete system unlike a continuous one, because they are often run together and they scale differently.

The first is the one this essay is about: the medium has a spacing, so there is a shortest meaningful wavelength and therefore a highest frequency. It is a property of the medium, it does not change with the size of the sample, and it appears as a ceiling.

The second is that a sample has ends, so the allowed wavenumbers are discrete rather than continuous — only some notes fit. It is a property of the boundary, its spacing shrinks as the sample is made longer, and it appears as a comb.

A chain of ten masses a metre long shows both: ten modes, spaced by a tenth of the zone width, with the highest at the cutoff. Lengthen it to a thousand masses and the comb becomes fine while the ceiling stays exactly where it was. Every question about a discrete medium is helped by asking which of the two is doing the work, and the test is whether the answer depends on the sample’s length.

The reason the distinction matters practically is that the second effect is usually a nuisance to be removed — by making the sample large, or by absorbing at the ends — and the first is a real property to be measured.

What the pictures cannot show #

The dispersion figure draws frequency against wavenumber and cannot show what the modes look like. At the zone edge the mode is a standing wave with alternate masses stationary, carrying nothing — which is why the group velocity is zero there — and the picture of a flat curve is a picture of that motion drawn in the wrong variables.

The evanescent figure draws a steady state and hides how it is reached. Switch the drive on and there is a transient in which the disturbance does travel, briefly, before settling into the decaying profile; the steady state is the end of a process whose beginning looks quite different, and a measurement short enough to catch the beginning would report a propagating wave above the cutoff.

How the curve is actually measured #

The dispersion relation is not an abstraction: it is measured, routinely, and the method is worth a paragraph because it explains why the subject is stated in wavenumbers at all.

A neutron entering a crystal can create or absorb one lattice vibration, and when it does, its energy and its momentum both change — by the vibration’s energy and by its wavevector. Measure the neutron’s energy and direction before and after, and the difference gives one point on the dispersion curve. Repeat over directions and energies and the whole curve comes out.

Neutrons are used rather than light because of a mismatch of scales. A vibration at the zone edge has a wavelength of a few tenths of a nanometre and a frequency of tens of terahertz; a photon of that frequency has a wavelength of tens of micrometres, five orders of magnitude too long to carry the required momentum. A neutron of the right energy has a wavelength of exactly the right order, because its energy and its wavelength are related by a mass rather than by the speed of light.

That mismatch is worth carrying, because it recurs. Probing a wavevector requires a probe of comparable wavevector, and whether a given probe can reach a given part of a dispersion curve is decided by its own dispersion relation rather than by how energetic it is. Light can reach the vibrations near the centre of the zone and nothing else, which is why infrared spectroscopy of a crystal sees the optical branch at zero wavenumber and is blind to everything the neutrons see.

Where the ladder goes next #

The periodic-media ladder began with the gap a repeat opens and the mode that lives in the mistake. This rung asks what discreteness itself does. The rungs after it: the density of modes, which is the quantity a heat capacity or a spectrum actually contains and which has a sharp feature wherever the dispersion curve flattens; anharmonicity, where the modes stop being independent and a solid acquires a thermal conductivity; and the two-dimensional and three-dimensional cases, where the gap may exist in one direction and not another and a full gap becomes something to be engineered rather than assumed.

The habit worth carrying away is to ask what a continuum description has thrown away. Every continuum is an approximation to something with a spacing in it, and the approximation fails at a wavelength comparable with that spacing — bringing a top frequency, a zero group velocity and an evanescent regime that the continuum has no room for at all.