The static field a metal may or may not reflect

The force that forgets Planck’s constant followed the Casimir pressure between two plates out from the quantum regime, where it falls as the inverse fourth power of the gap, to the thermal one, where it falls as the inverse cube and contains Planck’s constant not at all. Far out, only the static fluctuations of the field survive, and the pressure is a purely classical $\zeta(3)kT/4\pi a^3$ — if both polarisations of those static fluctuations are reflected. For plates of real metal, that essay found, one of them may not be, and the force is then half as large. It drew the two possibilities with perfect mirrors and left the question of which is right.

This essay puts real gold in the gap. Gold’s response to light is measured, from the infrared to the ultraviolet, to a few per cent, and it is the input to the calculation Evgeny Lifshitz devised in 1955 for the force between any two materials. With real gold the question of the static term does not go away. It becomes sharper, and it turns out to be a question about one point on a curve that nobody can measure.

The force adds up a list of frequencies #

The Lifshitz formula at a temperature $T$ is a sum. Each term belongs to one of the Matsubara frequencies, $\xi_l = 2\pi l kT/\hbar$, which are the frequencies at which a thermal field at temperature $T$ can be decomposed, and each term depends on how strongly the plates reflect fluctuations of that frequency. At room temperature the first nonzero frequency, $\xi_1$, is $2.5 \times 10^{14}$ per second, an infrared frequency; the others are its multiples. The term with $l = 0$ is the static field, and it is weighted by one half.

The frequencies are imaginary: the formula needs the permittivity $\varepsilon(i\xi)$ on the imaginary axis, which is obtained from the measured real-frequency absorption by the dispersion relation that the answer that cannot come first derived from causality. That step is standard and uncontroversial. The difficulty is that the measurements stop at low frequency — optical data for gold run down to a few tenths of an electronvolt — and the dispersion relation needs an extrapolation below that. For a metal the extrapolation is the free-electron response, and there are two forms of it.

The plasma description treats gold’s conduction electrons as a gas that accelerates freely in a field: $\varepsilon(i\xi) = 1 + \omega_p^2/\xi^2$, with the plasma frequency $\hbar\omega_p = 9.0$ eV fixed by the density of electrons. The Drude description adds the collisions that give a real metal its resistance: $\varepsilon(i\xi) = 1 + \omega_p^2/\xi(\xi+\gamma)$, with a collision rate $\gamma$ of about $5 \times 10^{13}$ per second for gold at room temperature, the number that sets its electrical resistivity. The frequency below which nothing gets in used the plasma form for the ionosphere, where collisions are rare; a metal at room temperature is a plasma whose electrons collide every twenty femtoseconds.

The first figure plots the combination that enters the reflection of a transverse-electric fluctuation, $(\varepsilon - 1)\xi^2$, scaled by $\omega_p^2$. For the plasma description it is exactly one at every frequency. For the Drude description it follows the plasma line at high frequency and falls away below the collision rate. The Matsubara frequencies are marked, and they all lie above the fall: at the first the two descriptions differ by 18 per cent, at the second by 10, and at higher ones by less. Then there is the point at zero, which a logarithmic axis cannot show and the Lifshitz sum cannot omit. There the plasma description gives one and the Drude description gives zero.

A static field and a copper sheet #

The physical content of the zero-frequency point is easier to see than the formula. A transverse-electric fluctuation at zero frequency is a static magnetic field, varying in strength across the surface of the plate and unchanging in time. The second figure shows how strongly each description reflects it. The plasma description reflects it almost perfectly: for a field varying over the micrometre scale of the gap, the reflection coefficient is −0.993. The Drude description reflects it not at all.

Here the Drude description has the whole of classical electromagnetism behind it. A static magnetic field passes through a sheet of copper or gold as if the sheet were not there. When the field was switched on, it did induce currents in the metal that opposed it — the circuit that fights its own change is the rule — but those currents met resistance and died away, and with them the opposition. The magnet that falls slowly down a copper tube is slowed by exactly those currents, and a magnet held still inside the tube feels nothing. Only a superconductor, whose currents never decay, keeps a static field out, which is what the field that is pushed out made the defining property of superconductivity. The plasma description, with no collisions, describes a metal whose electrons are dissipationless and so gives an ordinary metal a superconductor’s response to static magnetic fields.

So the Drude description looks correct and the plasma description looks like an error. At the frequencies that matter for the force at short distances that error is harmless — the plasma line and the Drude line agree there — and the plasma form is often used as a convenient approximation. The trouble starts at the zero-frequency term, where the error, if it is one, is total.

Forty per cent at three micrometres #

The third figure computes the full Lifshitz pressure both ways, with gold’s measured plasma frequency and collision rate, and divides one by the other. At a tenth of a micrometre the two agree to two per cent. The gap is small compared with the thermal wavelength $\hbar c/kT$, so many Matsubara frequencies contribute, and the static term is a small share of the total. As the gap widens, fewer frequencies matter and the static term grows in relative importance. At a micrometre the Drude pressure is sixteen per cent lower; at three, forty-one per cent; at ten, half. Far out, the pressure is the static term alone, and in the Drude description the static term has lost its TE half.

The dashed curve is the perfect-mirror model of the earlier essay, with only that one term removed, and it lies almost on top of the gold curve. The comparison makes the point precisely. Everything about real gold — its finite plasma frequency, its collisions at the nonzero Matsubara frequencies, the penetration of fields into its surface — changes the ratio by a few per cent at the shortest gaps and by nothing further out. The whole of the forty per cent at three micrometres is the static TE term.

The fourth figure takes the three-micrometre pressure apart. The static TM term is the response to a static electric field, and both descriptions make gold a perfect mirror for that: the electrons rearrange until the field inside is zero, whatever their collisions, as the inside of a conductor showed. The sum over all the nonzero frequencies differs between the descriptions by less than one per cent. The static TE term is seven micropascals in the plasma description and nothing in the Drude. The disagreement between two calculations that share every measured input is one term, of one sign, at one frequency.

Why that term grows in importance with the gap has a simple reason. A fluctuation at Matsubara frequency $\xi_l$ contributes to the force only if it can reach across the gap, and its contribution falls as $e^{-2\xi_l a/c}$. At a tenth of a micrometre, $2\xi_1 a/c$ is 0.16 and dozens of frequencies contribute nearly equally, so the static term is one of many. At three micrometres the same exponent is 4.9: the first nonzero frequency contributes less than a hundredth of what it would at zero gap, and the higher ones nothing. The force has become a sum of one term and a correction, and whatever is uncertain about that one term is uncertain about the force. The crossover is set by the thermal wavelength $\hbar c/kT$, 7.6 micrometres at room temperature: at gaps well below it the force is quantum and the dispute is a small correction; at gaps above it the force is thermal and the dispute is the whole answer.

The same fluctuations do other work, and there the dissipative picture is not in doubt. The heat that crosses a gap too narrow for light found that two surfaces closer than a thermal wavelength exchange heat far faster than blackbody radiation allows, through evanescent fields that tunnel across the gap. For two metal surfaces, calculations find that this heat is carried largely by the transverse-electric fluctuations — slowly varying magnetic fields that induce eddy currents in the far surface and are absorbed by its resistance. Those fluctuations exist only because the electrons collide; a collisionless plasma neither emits nor absorbs them. The metal that must dissipate to carry heat across a gap is, in the force measurements at short range, behaving as though its static magnetic response did not dissipate at all.

The measurements that disagree with each other #

The question would be academic if it could not be tested, and it has been tested repeatedly, with results that do not agree. The last figure places the experiments on the curve of the difference.

At the short end, Ricardo Decca and colleagues measured the force between a gold-coated sphere and a gold-coated plate oscillating on a micromachined torsional oscillator, from 0.16 to 0.75 micrometres, repeatedly between 2003 and 2016 with improving precision. In that range the two descriptions differ by three to twelve per cent, and the measurements, precise to a fraction of a per cent, sit on the plasma prediction and exclude the Drude one by many standard deviations. Other measurements since — with an atomic-force microscope and surfaces of nickel, a magnetic metal for which the two descriptions differ in a different way, and with a differential arrangement designed so that the two predictions for the measured quantity differ enormously — have found the plasma description again.

At the long end, Alex Sushkov, Woo-Joong Kim, Diego Dalvit and Steve Lamoreaux measured the force between a gold-coated lens and plate on a torsion pendulum, from 0.7 to 7 micrometres, and published in 2011 a result that agreed with the Drude description. Here the difference between the descriptions is large, tens of per cent, but so is a competing force. Real metal surfaces are not equipotentials: grains of different crystal orientation have slightly different work functions, and the patches of different potential produce an electrostatic attraction that, at a few micrometres, is comparable with the Casimir force itself. The experiment measured a total force many times larger than the predicted thermal Casimir force and extracted it by fitting and subtracting the patch contribution. Critics have argued that the subtraction carries more uncertainty than was assigned; the authors that the short-range experiments neglect the patches, which matter less there but not nothing.

The standoff is the reverse of what one would expect. The experiments that see a small difference cleanly favour the description that appears physically wrong. The experiment that sees a large difference, but beside a large background, favours the description that appears physically right.

The entropy at absolute zero #

There is a theoretical argument against the Drude description as well, and it is about thermodynamics rather than electromagnetism. The Casimir energy between two plates is a free energy, and its derivative with respect to temperature is an entropy. For the Drude description with a collision rate that vanishes as the temperature falls — as it does in an ideal crystal with no impurities, where the only scattering is by lattice vibrations that freeze out — the Casimir entropy at absolute zero comes out not zero but a finite negative number that depends on the separation. That would violate the third law of thermodynamics, which the two laws absolute zero will not allow took apart, and it was pointed out by Vladimir Mostepanenko and colleagues in 2002 as evidence that the Drude description is being misapplied.

The answer offered by others is that real metals always have impurities, so the collision rate does not fall to zero but levels off at a residual value set by the defects, and with a residual rate the Drude entropy does go to zero, though only at temperatures so low that no experiment reaches them. Whether an argument about a perfect crystal at absolute zero can rule out a description of an imperfect crystal at room temperature is itself disputed. What the argument shows is that the static term is delicate at both ends: what a metal does with a fluctuation at zero frequency depends on processes slower than anything an optical measurement sees.

What the dispute is really about #

The Lifshitz formula treats the plates as having a local permittivity, a number at each frequency that relates the current at a point to the field at that point. At zero frequency and over micrometre distances that description may simply be wrong. A static magnetic field varying over a micrometre across a gold surface is not a uniform field, and the electrons that respond to it travel tens of nanometres between collisions at room temperature. Whether their response is local — the assumption behind both the Drude and plasma forms — or whether the spatial variation matters, as it does in the anomalous skin effect at low temperature, has been examined with nonlocal permittivities that make the two pictures blend, and the conclusions depend on details of the surface that are hard to know.

There is also the question of what the electrons’ collisions are. In the Drude description a collision randomises an electron’s momentum and dissipates energy, and a field that varies more slowly than the collision rate drives a current limited by resistance. A static fluctuation of the thermal field is a fluctuation, not a field applied by an outside agent, and some theorists have argued that dissipative and dissipationless responses enter the fluctuation–dissipation relation differently at zero frequency, so that the static term should be computed in a way neither standard description does. Half a kT in a piece of wire found the fluctuation–dissipation relation fixing the noise in a resistor from its resistance alone; whether the same reasoning applies without modification to the zero-frequency fluctuations of the field between two resistors is the point at issue.

What a reflection coefficient leaves out #

The figures show the pressure between two infinite, flat, smooth plates of gold in thermal equilibrium at 300 K, and every experiment differs from that in every respect. The surfaces are a sphere or a lens against a plate, handled by the proximity-force approximation, which is accurate to a fraction of a per cent at the gaps involved but is still an approximation. The surfaces are rough at the scale of tens of nanometres, the gold films are a few hundred nanometres thick and have their own grain structure, and their optical properties differ from tabulated bulk values by a few per cent. Each of these corrections is computed and applied, and the experimental groups have argued over each. None of them is drawn here, and none of them affects the main point the figures make: that the two descriptions share everything except the static TE term, so any measurement that separates them is a measurement of that one term.

Still open: how a superconductor would vote #

The most direct test proposed so far would change the static TE term while holding everything else fixed. A plate of a superconducting metal, cooled through its transition temperature, changes from a metal that lets a static magnetic field through to one that expels it. In the Drude description the static TE reflection would jump from zero to nearly one at the transition, adding a measurable step to the force between a plate and a sphere at a few micrometres; in the plasma description, which already treats the metal as reflecting static fields, nothing should change. Platforms that measure the force between superconducting surfaces at a few kelvin have been built, and the step has been looked for, but none has yet reached the precision needed to see it or to rule it out.

The habit worth carrying away is to locate a disagreement exactly. Two calculations of the Casimir force between gold plates that share every measured property of gold differ by 16 per cent at a micrometre and 41 per cent at three — and the whole difference is the reflection of a static magnetic fluctuation, about −1 if the electrons never collide and exactly 0 if they do, at a frequency no optical measurement reaches. When a prediction rests on an extrapolation, the part of it that depends on the extrapolation is the part a measurement is testing.