The constant that depends on how fast it is asked

Two numbers are printed for water in the same reference book. Its relative permittivity is 80.1. Its refractive index is 1.333.

Maxwell’s equations say those are the same number: a wave in a non-magnetic medium travels at $c/\sqrt{\varepsilon_r}$, so $n = \sqrt{\varepsilon_r}$, and $\sqrt{80.1}$ is 8.95. That is not 1.333, and it is not near it.

What the static number measures #

The 80.1 is measured with a capacitor and a slowly changing voltage. It is large because a water molecule is a permanent electric dipole that can turn: put a field across the liquid and the molecules rotate, on average, until the field they produce inside partly cancels the field applied.

What the static number measures is a ratio of two capacitances — with the slab in, and without — and the phrase “with the slab in” is already hiding the question this essay is about, because it does not say how fast the voltage is being changed. The force pulling a dielectric into a capacitor is a useful check on that: the stored energy moves in opposite directions depending on whether the plates are held at fixed charge or fixed voltage, and the force comes out inward in both cases. It is a fact about the fringing field, which the parallel-plate idealisation deletes.

That capacitance is the same quantity a shape holds before anything is charged, multiplied by whatever the material between the plates contributes — and the multiplier is what the rest of this essay is about.

Turning a molecule takes time. A water molecule in liquid water reorients in about eight picoseconds, because it has to break and remake hydrogen bonds to do it. Below about twenty gigahertz the field changes slowly enough that the molecules keep up and the full 80 is available. Above it they cannot, they fall behind, and their contribution drops out — carrying away almost the whole of the permittivity, since the other mechanisms account for less than five of the eighty.

That is the first plateau ending. The visible light of the second measurement oscillates at $5 \times 10^{14}$ hertz, twenty-five thousand times faster than the molecules can turn, so a light wave in water sees a substance in which nothing is turning at all — only electrons being displaced within their own molecules, which is a much smaller effect and a much faster one.

The shape between the plateaus #

Every mechanism that responds to a field has a natural frequency and a rate at which it loses energy, and the response of any such thing to a driving field is one of the most reused curves in the subject.

The shape between the plateaus is a driven oscillator’s, and the two parts that matter are on either side of its resonance. Below it the charge follows the drive; above it, the charge moves against the drive. The quarter-cycle nobody mentions is the phase lag between them, and that lag is the whole of the difference between the two parts of a permittivity: the component of the response in phase with the field is the dispersion, and the component a quarter cycle behind it is the absorption. They are not two properties. They are one response resolved onto two axes.

Model an electron in a molecule as such an oscillator, with a restoring force, a damping and a charge, and the permittivity follows at once:

$$\varepsilon(\omega) = 1 + \frac{\omega_p^2}{\omega_0^2 - \omega^2 - i\gamma\omega}.$$

Water needs four such terms and a fifth of a different kind. The relaxation of the molecular dipoles is not a resonance at all — a molecule in a liquid is too heavily damped to ring, so its response is a Debye relaxation, a resonance so overdamped that its frequency has dropped out of the answer. Above it come the librational band near $2 \times 10^{13}$ hertz, the bending mode of the molecule, its two stretching modes, and finally the electronic transitions in the ultraviolet, which are the only ones left contributing by the time the light is visible.

The index, and the region where it behaves backwards #

Expressed as a refractive index and an absorption coefficient, the same function says something that reads as impossible until it is looked at.

The index is also what decides how much of a wave gets across a boundary at all: an impedance mismatch reflects, and for light the impedance is the index. So a frequency-dependent index is a frequency-dependent reflectance, which is why a window looks different in the infrared.

An index below one is entirely ordinary, and X-rays have it in every material — which is why a grazing-incidence mirror works at all, and why the total external reflection of X-rays is possible. What is not ordinary is the temptation to read a phase velocity as a signal velocity.

The extreme case has been arranged deliberately, and it is worth knowing about because it sounds impossible. Two gain lines placed either side of a carrier frequency give a group velocity that is negative, so a pulse’s peak leaves the far side of a cell before it entered. No information arrives early: the front of a signal always travels at exactly c, and the peak is a feature of a shape whose leading edge has already determined it.

The band where $n$ misbehaves is the band where the material is opaque, and that is not a coincidence either. It is the second and stronger version of the same statement.

The two halves are one function #

Suppose only the absorption of a substance has been measured — how much light of each colour it swallows, and nothing else. It turns out that fixes the refraction at every frequency, including frequencies at which the substance is perfectly clear.

The reason is causality, and the argument is worth stating because it has nothing to do with electrons. If a material cannot respond before it is driven, its response function has no poles in the upper half of the complex frequency plane, and Cauchy’s theorem then relates the real part on the real axis to an integral of the imaginary part over the whole axis. Nothing about the mechanism enters. Any linear response of anything to anything obeys the same relation — a circuit’s impedance, a solid’s compliance, a nucleus’s scattering amplitude — and it is the same relation each time.

Two consequences of the relation are worth having in hand. The first is that a substance which absorbs nothing at any frequency has an index of exactly one everywhere: there is nothing for the integral to integrate. Perfect transparency and perfect optical inertness are the same statement, and a material that bends light must swallow some of it somewhere. The second is a sum rule — integrate the absorption over all frequencies and the answer is fixed by the number of electrons per unit volume and nothing else. A material cannot absorb more in total than it has electrons to do it with, and moving a band to a new frequency means taking its strength from somewhere.

That relation is what makes a single number for an index defensible in a transparent window. Far from every absorption band, the integral is dominated by distant contributions and $\varepsilon'$ is slowly varying, so a material that absorbs nowhere near the visible has an index that is nearly constant across it. The residual variation is the dispersion a prism uses and a camera lens has to cancel.

The consequence in an instrument is chromatic aberration. Because the index depends on frequency, a single lens focuses blue nearer than red, and the fix is not a better glass but two glasses whose derivatives cancel. The quantity being cancelled is the slope of the dispersion curve, evaluated a long way from any resonance — which is why an achromat works over a band and not at a point.

What a single number is good for #

Once the frequency dependence is admitted, several familiar numbers have to be re-read.

A single number is good for more than it looks. The relation between one molecule’s polarisability and the bulk permittivity carries a factor of three, because the field a molecule sits in is not the applied field but the applied field plus whatever its neighbours produce. Written for optical frequencies the same equation is the Lorentz–Lorenz relation with $n^2$ in place of $\varepsilon_r$ — which only makes sense because those are the same quantity asked at the same rate.

An antireflection coating names a wavelength twice. It works when its index is the geometric mean of the two it sits between and its thickness is a quarter of a wavelength, so it is exact at one colour and approximate either side. The residual purple of a coated lens is three materials’ dispersion failing to stay in step — the coating designed at 550 nm, doing progressively less at both ends of the visible.

Why the sky is blue is a statement about the same function. Rayleigh scattering goes as the fourth power of frequency times the square of the polarisability — and the polarisability’s own rise towards the molecule’s ultraviolet resonances steepens the exponent slightly past four. That correction is measurable and is usually left out, which is a small example of the habit this essay is about.

And the wave itself carries the same ambiguity. Its speed is $1/\sqrt{\varepsilon\mu}$, which for vacuum is $c$ and for a medium is $c/n$. The permittivity in that expression is the one at the wave’s own frequency — not the static one, not the one in a data table — and writing it without saying so is where the whole confusion begins.

The instrument the shoulder makes #

The frequency dependence is not only a hazard to be navigated. It is a measurement.

Sweeping a sample’s permittivity across frequency and watching where it falls is called dielectric spectroscopy, and what it returns is a list of relaxation times — how long each mechanism in the substance takes to respond. A polymer above its glass transition has a relaxation of milliseconds and below it one of hours, so the fall moves through eight decades of frequency as the sample is cooled through a few tens of kelvin, and the temperature at which it crosses a chosen frequency is a definition of the transition. A protein in solution has a relaxation set by how long it takes to tumble, so the fall locates its hydrodynamic size. In each case the quantity read off is a time, taken from the frequency at which a curve bends.

The most familiar instance of the same physics is not a laboratory instrument at all. A microwave oven drives water at 2.45 gigahertz, and the reason is not that the molecule resonates there. It has no resonance within two orders of magnitude of it. The oven is working on the low-frequency shoulder of the Debye relaxation — the region where the molecules are just beginning to fall behind the field, so that each cycle leaves a little energy behind as heat.

Working at the loss peak instead, near nineteen gigahertz, would be worse rather than better. The absorption length there is a few millimetres: the surface of the food would receive everything and the interior nothing. On the shoulder the loss per cycle is a third of its maximum and the penetration depth is a few centimetres, which is the size of the thing being cooked. The frequency is a compromise between how strongly water absorbs and how far the field gets, and the answer to why 2.45 is a graph of the imaginary part of a permittivity with a domestic constraint drawn across it.

The same reasoning, run the other way, is why a radar altimeter uses a band water is nearly transparent in and a weather radar uses one it is not.

The same expression with the spring removed #

The oscillator model has three ingredients — a charge, a damping and a restoring force — and it is worth asking what happens if the third is deleted. An electron in a metal is not bound to any particular ion; it is free to wander. Setting $\omega_0 = 0$ in the expression above and letting the damping go with it leaves

$$\varepsilon(\omega) = 1 - \frac{\omega_p^2}{\omega^2},$$

which is a permittivity that is negative below $\omega_p$ and positive above it. A negative permittivity means an imaginary index, which means a wave that does not propagate but decays over a fixed distance and is thrown back — so the model says, without any further assumption, that a metal is a mirror below its plasma frequency and a window above it.

That is what metals do. For the alkali metals the plasma frequency falls in the near ultraviolet, and sodium becomes transparent below about 210 nanometres — a result that reads like a mistake and was confirmed by Robert Wood in 1933, who looked through a film of it. For copper and gold the same edge sits in the visible, which is why they are coloured rather than white: the reflectance falls away in the blue while the red is still on the mirror side of it.

The upper atmosphere does the same thing to radio. A layer of free electrons is a plasma with a plasma frequency set by their density, and a wave below it is reflected. That frequency is a few megahertz at night and roughly twice that in the afternoon, which is why shortwave broadcasts reach across an ocean by bouncing and why FM at a hundred megahertz simply leaves — and why a satellite can be spoken to at all. The same three-line model that explains the colour of gold decides which radio bands stay on the planet.

A conductor is a dielectric that is losing #

The other thing the free-electron case exposes is that the distinction between a conductor and an insulator is a statement about frequency rather than about a substance.

A current in a medium enters Maxwell’s equations beside the displacement current, and the two can be combined: a conductivity $\sigma$ is indistinguishable from an addition of $\sigma/\varepsilon_0\omega$ to the imaginary part of the relative permittivity. So there is one complex function, and calling a material a conductor means only that this term is the larger one at the frequency being used.

Seawater is the clean example, because it is both. Its conductivity is about four siemens per metre and its relative permittivity about eighty, and the two contributions are equal near nine hundred megahertz. Above that, seawater is a lossy dielectric and light behaves in it much as it does in fresh water. Below it, seawater is a conductor, and a wave entering it dies over a skin depth of $\sqrt{2/\mu_0\sigma\omega}$ — about five metres at three kilohertz, and about thirty at seventy-six hertz.

Those two numbers are why a submarine communication system operates at frequencies that sound absurd. At seventy-six hertz an antenna must be kilometres long, the usable data rate is a few characters a minute, and the transmitter is a national installation — and it is the only band that reaches a vessel that does not wish to come up. The choice is read straight off the imaginary part of a permittivity, exactly as the microwave oven’s was, and in the opposite direction.

A brief history of getting it in the wrong order #

The pieces arrived out of sequence, which is why the subject is taught confusingly.

Maxwell’s identity $n = \sqrt{\varepsilon_r}$ was published in 1865, and it was almost immediately embarrassing: the static permittivities available then gave indices far too large for every polar liquid, and for water it was out by a factor of seven. The relation was suspected for thirty years on the strength of a discrepancy that came entirely from comparing a measurement made with a battery against one made with a lamp.

Drude’s model of an electron in a restoring force, and Lorentz’s more careful version of it, supplied the shape between the plateaus in the 1890s and 1900s. Debye supplied the overdamped case, and the relaxation that carries water’s eighty, in 1913. And the relation that ties the two halves together came last, in 1926 and 1927, from Ralph Kronig and Hendrik Kramers working independently on X-ray dispersion — which is the piece with the least physics in it and the most consequence, because it holds for anything at all that responds to being driven and cannot respond first.

Where the model stops #

One oscillator is never enough. The single-resonance figures are a teaching object. A real substance has a forest of transitions, and the sum rule that binds them — the total oscillator strength is the number of electrons — is what makes the index approach one from below at high frequency in every material without exception.

The wave has been infinite. Every expression above is for a single frequency extending for ever. A pulse contains a band of them, travels at a speed that is not the speed of its own crests, and spreads — and all three are consequences of the slope and curvature of the curve in the first figure.

Linearity is assumed throughout. Every expression above is the first term of an expansion in the field. At the fields a pulsed laser reaches, the second and third terms produce frequency doubling and an intensity-dependent index, and the Kramers–Kronig relation as written no longer applies.

The Debye term is an average over an environment. A relaxation time of eight picoseconds is a fitted number describing an enormous population of molecules in different local surroundings, and the true relaxation is a distribution rather than a single exponential. Fitting one number to it is excellent for the shape of the fall and poor for its edges.

Locality is assumed. The polarisation at a point has been taken to depend on the field at that point. In a metal at short wavelengths, and in any medium near a sharp resonance, it depends on the field nearby as well, and the permittivity acquires a dependence on wavevector as well as frequency.

The medium is treated as homogeneous. A permittivity is an average over a volume containing many molecules, and it stops meaning anything at all on a scale where the molecules can be counted — which for water is a few nanometres. The field a single ion sits in, a molecular diameter from its neighbours, is not the field this framework computes, and correcting for that is what the factor of three in the Clausius–Mossotti relation is doing.

Nothing here is about the boundary. The permittivity decides the speed inside a medium; what happens at the surface where it changes is a separate calculation with its own conditions, and the index in it is the one at the frequency arriving.

And the model has nothing to say about the static value itself. The 80.1 is a fact about hydrogen bonding in a liquid, and no model of an oscillator produces it. What the framework here delivers is the shape between measured plateaus, not the plateaus.

What the pictures cannot show #

The hero figure is drawn with logarithmic axes over six decades, which makes the three plateaus legible and makes the transitions between them look sharper than they are. Each fall occupies about a decade of frequency, which on this scale is a corner and in the substance is a broad and gradual thing.

Nor can any of these figures show the phase. Both parts of the permittivity are drawn as magnitudes against frequency, and what physically happens is that the polarisation lags the field by an angle that sweeps through 180 degrees as the resonance is crossed. The lag is the quantity; the two curves are its cosine and its sine, and a figure of them cannot make the single rotating thing they came from visible.

Where this ladder goes next #

Two rungs stand on dielectrics. The first found that the field inside a piece of matter depends on the shape of the piece before it depends on what the piece is made of. This one finds that even the material constant in that answer is not a constant.

The habit worth carrying away is a question to ask of any material property. A quoted constant is a measurement made under conditions, and the first question is what was held fixed while it was taken. Permittivity has a frequency; a thermal conductivity has a temperature; a Young’s modulus has a strain rate, and a substance quoted with one number for it is a substance nobody has yet asked twice.

What is left on this ladder is the nonlinear response, where the polarisation stops being proportional to the field. Every effect that changes a beam’s colour lives there, and the first thing to go is the superposition that all of the above quietly assumes.