The attraction that weakens when light is too slow

Everything written so far about retardation has been about a source someone is driving: a charge that is pushed, an antenna fed with current, an array steered by its feeds. The field that points where the charge is now found radiation where a charge’s motion changes; the distance where a field changes its mind found the radius, a wavelength over $2\pi$, beyond which a source stops storing energy and starts losing it; the solution that is thrown away asked why the retarded solution is the one kept at all.

This essay turns the argument on a force nobody drives. Two neutral atoms attract each other weakly, even though neither has a charge or a permanent dipole. The attraction that needs no charge treated the molecular version of that force with dipoles that sit still. Between atoms with no permanent dipole the attraction comes from fluctuations, and London’s classic calculation of it assumes that each atom feels the other’s field at the instant it is produced. Take retardation seriously and London’s law survives only close in. Beyond a certain distance the attraction changes its law and weakens faster, and the distance at which it changes is the same radian-sphere distance, $c/\omega$, that separated the near field from the far field of an antenna. The atoms are antennas, driven by nothing but their own fluctuations.

The force between two fluctuations #

A hydrogen atom in its ground state has no dipole moment on average: its electron cloud is spherical. But the electron is not a cloud; it is a quantum object whose position fluctuates, and at any moment the atom has an instantaneous dipole. Its average is zero and its average square is not. That fluctuating dipole makes a field at a neighbouring atom, which polarises it, and the induced dipole in the neighbour points so as to be attracted back. The two fluctuations fall into step, and averaged over time the energy is negative.

London computed it in 1930 with each atom modelled by one oscillator, the dominant transition, at a frequency $\omega_0$. The two oscillators coupled by their dipole field have zero-point energies slightly lower than two uncoupled oscillators’, the coupling mixing them into modes of which one is lower in frequency by more than the other is higher. The energy lowering, the whole attraction, is $-\tfrac{3}{4}\hbar\omega_0\alpha^2/r^6$, with $\alpha$ the polarisability. It is a quantum effect through and through — the motion that cannot be stopped is what keeps the dipoles fluctuating — and it holds liquid helium together, makes noble gases condense, and supplies the attractive half of every van der Waals interaction.

London’s derivation has one silent assumption. It uses the instantaneous dipole field between the atoms, the electrostatic $1/r^3$ field, as though a change in one atom’s dipole were felt by the other at the same moment. For atoms a few ångströms apart the error is negligible, because light crosses that distance in a small fraction of an oscillation. It is not negligible when the atoms are far enough apart that light takes a significant part of an oscillation to cross.

Where London’s law gives out #

Casimir and Polder, working in 1947 at the Philips laboratories on the stability of colloidal suspensions, did the calculation with the full, retarded electromagnetic field. Their colleague Overbeek had found that the measured forces between colloid particles fell off faster at long range than London’s law predicted, and had suggested that retardation might be the reason.

The figure plots their result for two identical atoms, each modelled with a single resonance, as the energy times the sixth power of the distance, so that London’s law is a horizontal line at one. Distance is measured in units of $c/\omega_0$, the reduced wavelength of the atoms’ dominant transition. Close in, the curve sits on London’s line. It begins to fall near one reduced wavelength, has fallen to a half by 3.8, and far out it falls as one over the distance. So the energy itself goes as the inverse seventh power:

$$U(r) \to -\frac{23\,\hbar c\,\alpha^2}{4\pi r^7}.$$

At ten reduced wavelengths the attraction is 23 per cent of what London’s law would give, and at a hundred, 2.4 per cent.

The far-field formula has a remarkable feature: it contains no transition frequency. It depends on the static polarisability, the speed of light and Planck’s constant, and nothing else about the atom. London’s formula contains $\omega_0$ explicitly. Somewhere between the two regimes the detailed spectrum of the atom has dropped out of the answer.

Which fluctuations still count #

The reason is visible in the Casimir–Polder integral itself, which adds up contributions from fluctuations at every frequency.

The integral is taken over imaginary frequency, which is a mathematical convenience with a physical reading: each value of $\xi$ labels fluctuations on a timescale of $1/\xi$. The figure plots how much each contributes at three distances. At a tenth of a reduced wavelength, every frequency up to about the resonance contributes, weighted by how strongly the atom responds at that frequency, which is London’s picture. At ten reduced wavelengths a factor $e^{-2\xi r/c}$ has cut off every fluctuation that changes faster than light can cross the gap and return, and only the slowest tenth of the spectrum survives.

That factor is retardation in its cleanest form. A fluctuation in atom A sends out a field that reaches B a time $r/c$ later, polarises B, and B’s induced field returns to A after another $r/c$. For the energy to be lowered, the returning field must find A’s dipole still pointing the way it was when the field left. If A’s dipole has changed sign in the meantime, the returning field pushes instead of pulling. Fluctuations slower than the round trip keep their phase and contribute; faster ones arrive out of step and average away.

At large distances only the slow fluctuations survive, and a slow fluctuation sees the atom’s response to a slowly varying field, which is its static polarisability. That is why the far-field law depends only on the static polarisability and not on the resonance. The extra power of distance is also accounted for. The range of surviving frequencies shrinks as $c/r$, and that one factor of $1/r$ turns London’s $1/r^6$ into $1/r^7$.

The same argument appears in when the source is not heard all at once, where the spread of retarded times across an antenna decides whether it behaves as a point. Here the spread is across the gap between two atoms, and it decides whether their fluctuations can cooperate.

One power, whatever the geometry #

Retardation adds one power of distance, and the rule is not special to two atoms.

The figure plots the local exponent of the energy — the power of the distance it falls as, at each distance — for two geometries. Between two atoms the exponent runs smoothly from London’s 6 to Casimir and Polder’s 7, passing halfway at 2.5 reduced wavelengths. An atom in front of a perfectly reflecting mirror is attracted to its own image, and the unretarded energy goes as the inverse cube of the distance to the mirror. Retarded, it goes as the inverse fourth power, and the crossover is closer in, halfway at 1.1 reduced wavelengths, because the round trip to the image is twice the distance to the mirror.

In both cases the change is exactly one power, because the only new ingredient at long range is the travel time of the field, which brings in one factor of distance over $c$. Two parallel metal plates, the geometry Casimir himself turned to next, follow the same rule: their attraction per unit area goes from the inverse cube of the gap for the unretarded van der Waals force between dielectric slabs to the inverse fourth power, $\pi^2\hbar c/240d^4$, for the Casimir force between perfect mirrors. The Casimir force is this essay’s force summed over two surfaces and taken to the long-range limit, where no property of the metal survives but its ability to reflect.

An atom pulled towards a mirror #

The atom–mirror force was the first version measured with individual atoms.

The figure plots the attraction for a sodium atom, whose strong yellow transition at 589 nanometres sets $c/\omega_0$ at 94 nanometres, against its distance from a perfect mirror. At 10 nanometres the attraction is equivalent to 0.069 kelvin and follows the inverse cube. By a micrometre it has fallen to $6.5\times10^{-9}$ kelvin — a few nanokelvin — and is 11 times weaker than the inverse cube would give.

In 1993 Sukenik, Hinds and their colleagues measured it directly. They sent a beam of sodium atoms through a narrow channel between two gold plates, with gaps from 0.7 to 1.2 micrometres, and counted how many atoms got through. Atoms passing near a wall were pulled onto it and lost, so the transmitted fraction depended on how strongly the walls attracted. The inverse-cube law predicted far too few transmitted atoms. The retarded law fitted the data, and the measurement confirmed Casimir and Polder’s calculation for individual atoms forty-five years after it was made. Later experiments bounced cold atoms off surfaces and watched Bose–Einstein condensates of rubidium oscillate near a surface, measuring the shift in their frequency; both confirmed the retarded law and reached distances where the thermal radiation of the surface begins to add its own contribution.

Where each atom changes its law #

The crossover distance is fixed by one number per atom, the wavelength of its dominant transition from the ground state.

The figure lists seven. Helium, whose ground state is held by an ultraviolet transition at 58 nanometres, changes its law within 9 nanometres. Hydrogen, with its Lyman-alpha line at 122 nanometres, within 19. The alkali atoms, whose outer electrons are loosely held and whose first transitions are in the red and near infrared, do not change until about a hundred nanometres: sodium at 94, caesium at 136. That is why the alkali atoms are the ones used to measure the retarded force. Their crossover is far enough out that the fully retarded regime can be reached at distances where the force is still measurable.

For helium the scale connects to another essay here. The molecule that lives outside its own well found the helium dimer’s two atoms about 52 ångströms apart on average, with much of their wavefunction beyond 100 ångströms. That is beyond a reduced wavelength for helium, so the tail of the potential holding the dimer together is partly retarded. Calculations of the dimer’s binding energy to the precision now measured must include the retardation correction, which shifts the binding by about a per cent.

The colloid chemists’ question #

The calculation began as a practical problem, and the problem shows why the change of law matters outside atomic physics. In the 1940s Verwey and Overbeek were building the theory of why particles suspended in water — clay, paint pigments, the droplets of an emulsion — stay dispersed or clump together. Two forces compete. The particles carry charge and are surrounded by clouds of counterions that repel one another over a distance set by the salt concentration, the screening that the like charges that pull together found could even reverse sign. Against that repulsion acts the van der Waals attraction, summed over every pair of atoms in the two particles. The balance, known as DLVO theory, decides whether a suspension is stable.

For two particles the atom–atom attraction integrates into a law that depends on their shapes: for two flat surfaces, the energy per unit area falls as the inverse square of the gap, and the pressure as the inverse cube. The screening repulsion falls exponentially. So at large separations the attraction always wins in London’s form, since a power law beats an exponential eventually, and the theory predicted a weak attractive minimum far out where particles could loosely gather. Overbeek’s measurements of real suspensions showed a weaker long-range attraction than that. Retardation, which makes every atom–atom term fall faster beyond the reduced wavelength, made the summed attraction fall faster too, and removed the discrepancy. Casimir and Polder’s paper of 1948 was, in origin, a correction to a theory of paint.

The same competition runs through the thin films that the film that goes black before it bursts describes. The disjoining pressure that decides where a soap film stops thinning has a van der Waals part, and in films more than a few tens of nanometres thick that part is retarded. It is weaker than the unretarded estimate and falls one power faster with thickness. The thickness at which black films form, and the stability of wetting films on surfaces, depend on which regime the film is in.

And the same force, at the shortest range and without retardation, is the attraction that the first correction to the gas law found lowering a real gas’s pressure below the ideal value. Molecules in a gas collide at separations of a few ångströms, far inside the reduced wavelength of any molecular transition, so the gas’s second virial coefficient is set entirely by London’s law and the repulsion at contact. Retardation touches the gas only through the rare pairs that are far apart, where the force is already negligible. The regime in which the change of law matters is the intermediate one: nanometres to micrometres, the scale of colloids, films, cold atoms near surfaces and the gaps in micromachines. Machines built at that scale have parts that stick together by exactly this force, and their designers need both laws.

What the one-oscillator atom leaves out #

The figures model each atom by a single resonance, and three simplifications follow.

Real atoms have many transitions. The polarisability at imaginary frequency is a sum over every transition, and the short-range $C_6$ coefficient depends on all of them. The crossover is spread out rather than sharp, and for atoms like helium with several comparable transitions the effective $\omega_0$ is an average. The far-field law is untouched, because it depends only on the static polarisability, which includes every transition.

Real mirrors are not perfect. A metal reflects well only below its plasma frequency, and a dielectric reflects partially at every frequency. Lifshitz’s theory of 1955 generalises the atom–wall and plate–plate forces to real materials through their dielectric functions, and the measured forces depend on the material at short range and approach the ideal-mirror result only for good conductors at long range.

Temperature. The calculation is for zero temperature, where the only fluctuations are quantum ones. At a finite temperature the surfaces and the field also fluctuate thermally, and beyond a distance of $\hbar c/k_BT$ — about 7.6 micrometres at room temperature — thermal fluctuations dominate. The force then goes back to an inverse-cube law, with a strength proportional to the temperature and no Planck constant in it.

Still open: the force at the temperature of the plates #

The thermal correction to the Casimir force between metal plates has been the subject of a long dispute. The answer depends on how the metal’s electrons are described at low frequencies — as a free-electron plasma or with the finite conductivity that the Drude model gives them — and the two descriptions predict thermal corrections that differ by a factor of two at separations of a few micrometres. Precision measurements between gold surfaces at separations below a micrometre have repeatedly favoured the plasma description, which ignores the electrons’ resistance at low frequency, while theory and other measurements suggest the Drude description, which includes it, should be right. The disagreement has persisted for two decades, and no resolution has been generally accepted: it may point to something missing in how fluctuating fields interact with conduction electrons, or to systematic errors in delicate measurements, and experiments at larger separations, where the thermal term is larger, are designed to decide.

The habit worth carrying away is to ask of any interaction how long its field takes to travel compared with how fast its source changes. A force carried by a field is instantaneous only for sources that change slowly compared with the field’s travel time; past the distance at which the two are equal, the fast parts of the source stop contributing, and the force loses one power of the distance. London’s force is the near-field limit, Casimir and Polder’s the far-field limit, and the distance between them is the same radian-sphere boundary that separates an antenna’s stored field from its radiation.