The shape that takes five numbers

The attraction that needs no charge follows a water molecule’s field out to where its net charge contributes nothing and finds the dipole moment doing all the work: three numbers, a strength and a direction, and a field falling as the cube of the distance. It ends by naming the next term in the tower, the quadrupole, and saying that shape at that order needs five numbers rather than three.

That is a claim worth making precise, because the obvious count gives nine, the next obvious count gives six, and the right answer is five. The count is not bookkeeping. It says what a distant observer can learn about a charge distribution’s shape from its field — which is exactly two numbers, plus which way the shape is pointing — and every property of the quadrupole field follows from it, including an angle that turns up in a completely different part of physics.

A term in the expansion that knows about shape #

The potential of any set of charges, far away compared with the set’s size, is a series in the inverse distance, as one number for every point sets up. The first term is the total charge over the distance. The second is the dipole moment, $\sum q\,\mathbf r$, dotted into the direction of the observer, over the distance squared. The third involves the second moment of the charges, $\sum q\,x_ix_j$, and falls as the distance cubed.

The dipole moment says how far the centre of positive charge sits from the centre of negative charge. The second moment says how the charge is spread: along a line, in a ring, in a square. It is the first term in which the arrangement has a shape rather than merely a position — the term that decides the pattern field lines are a choice draws when two equal charges are placed side by side and the lines between them have to turn away — and the shape decides the falloff is right that the exponent of the leading term belongs to the geometry: a distribution with no charge and no dipole moment has a field that falls as $1/r^4$ and a potential as $1/r^3$, whatever it is made of.

The simplest quadrupole is three charges in a line, $+q$, $-2q$, $+q$: no net charge, and no dipole moment because the arrangement is symmetric. Its potential is positive near each end and negative around the middle, and far away it has the angular pattern $3\cos^2\theta - 1$ — twice as strong along the axis as across it, with the opposite sign. Between the two regions it passes through zero on a cone.

The drawing finds that cone rather than assuming it, which is the point the field before the lines were drawn on it makes about any picture of a field: the contours here are computed from the three charges, and the cone is read off them. On a circle of radius $2d$ the exact potential changes sign at 50.89 degrees from the axis; at $4d$, 53.76; at $8d$, 54.49; at $16d$, 54.67. The sequence closes on $\arccos(1/\sqrt3) = 54.74°$, where $3\cos^2\theta - 1$ vanishes. Near the charges the zero is pulled in towards the waist, because higher terms of the series still matter; far away only the quadrupole is left, and its node is fixed by geometry alone.

Nine numbers, then six, then five #

The second moment $\sum q\,x_ix_j$ is a three-by-three table: nine numbers. Two of them, $\sum q\,xy$ and $\sum q\,yx$, are obviously the same number, and so on for each pair, so a symmetric table has six. The final reduction is the one that matters. The trace of the table, $\sum q\,(x^2+y^2+z^2) = \sum q\,r^2$, contributes a term to the potential proportional to $\sum q\,r^2/r^3$ with no dependence on direction — and a potential that falls as $1/r^3$ with no angular dependence does not satisfy Laplace’s equation in empty space. It cannot appear in the far field. Only the traceless part does, $Q_{ij} = \sum q\,(3x_ix_j - r^2\delta_{ij})$, and a symmetric traceless table has five independent numbers.

The general statement is that the terms of order $l$ in the far field are harmonic polynomials of degree $l$ — polynomials whose Laplacian vanishes — divided by the appropriate power of the distance. The figure counts them directly for each order, by writing down every monomial of degree $l$ in three variables, building the matrix of the Laplacian that maps them to monomials of degree $l-2$, and finding its rank. What survives is 1, 3, 5, 7, 9: always $2l+1$. The spherical harmonics of quantum mechanics, with their $2l+1$ values of the magnetic quantum number, are these same polynomials restricted to a sphere, and the count is the same count.

So a distant observer measuring a quadrupole field learns five numbers. A symmetric traceless table can always be rotated so that it is diagonal, and then three of the five are the angles that do the rotating — which way the distribution is pointing — and two are the diagonal entries, the third being fixed by the zero trace. A quadrupole’s shape is two numbers.

Stretched, squashed, or neither #

The two shape numbers are the eigenvalues, and they sort quadrupoles into three kinds. The linear arrangement has one large positive eigenvalue and two equal negative ones: it is prolate, stretched along its axis, like a rugby ball. The ring of charge round a central negative charge has the pattern reversed, one large negative eigenvalue and two equal positive ones: it is oblate, squashed along its axis, like a lens. The square has eigenvalues $+1$, $0$, $-1$ and no axis of symmetry at all. Every quadrupole in nature is one of these, or in between.

The same classification runs through physics under other names. A nucleus with a quadrupole moment is prolate or oblate — the deuteron, the simplest, is slightly prolate, and its quadrupole moment was the first evidence that the force between a proton and a neutron depends on the direction of their spins. A planet’s gravitational field has a quadrupole term that measures its equatorial bulge, which the field outside that cannot find the core uses and shows cannot say what is inside. And the square arrangement, which near its centre has a potential proportional to $x^2 - z^2$ — a saddle — is the field of every quadrupole mass filter and ion trap. Nothing can be held still by a static field explains why that saddle can never hold a charge by itself and how turning it slowly makes it one; the trap is a square quadrupole spun by a radio-frequency supply.

The first moment that survives is the only one that belongs to the object #

There is a qualification on “the object’s quadrupole” that is easy to miss.

Move the origin about which the moments are computed and every moment after the first non-zero one changes. A single charge placed away from the origin acquires a dipole moment and a quadrupole moment that grows as the square of the displacement — not because it has any shape, but because it has been described from somewhere else. A dipole acquires a quadrupole moment proportional to the displacement. Only the linear quadrupole, which has neither net charge nor dipole moment, has a quadrupole moment that stays put.

The leading multipole of a distribution is a property of the distribution; everything after it is partly a property of the origin. A molecule’s quadrupole moment is a meaningful number only for a molecule with no dipole, or once the origin has been fixed by convention at its centre of mass; for a charged ion it is not a property of the ion at all. The same rule governs the planetary case: a planet’s gravitational quadrupole is referred to its centre of mass, where the dipole term vanishes by definition, and it is only there that $J_2$ describes the bulge.

Molecules with nothing but shape #

Carbon dioxide and nitrogen are linear molecules with no charge and no dipole moment, and benzene is a flat ring with neither. Each has a quadrupole moment, measured by how the molecule orients in a field gradient, and each quadrupole acts on anything charged nearby.

The energy of a charge on the axis of a linear quadrupole falls as the cube of the distance and has the sign of the quadrupole moment times the charge. Carbon dioxide’s moment is about three times nitrogen’s, so at the same distance a cation holds a carbon dioxide molecule three times as strongly. That ratio is a large part of why porous materials lined with metal cations — zeolites — take up carbon dioxide from a gas that is mostly nitrogen, which is what makes them candidates for capturing it from flue gas. The absolute energies in the figure are from treating each molecule as a point quadrupole, which is poor at contact distances where the molecule’s size is comparable to the separation; the sign and the ratio survive that.

Benzene is the dramatic case. Its quadrupole is large and negative, with negative charge concentrated above and below the ring in the π electrons and positive charge round the rim, so a cation sitting over the face of the ring is strongly attracted. Chemists call it the cation–π interaction, and it holds potassium ions in the selectivity filters of some ion channels and positively charged side-chains against aromatic rings in proteins. Replace the hydrogens with fluorines and the quadrupole changes sign: hexafluorobenzene’s face repels a cation and attracts an anion. Nothing about the two molecules’ charge or dipole differs — both are zero — and their behaviour towards an ion is reversed by the sign of a single number.

A nucleus in a crystal’s field gradient #

The quadrupole works in the other direction too: a quadrupolar charge distribution placed in a non-uniform field has an energy that depends on how it is oriented, and the orientation is quantised. A nucleus with spin 1 or more usually has a quadrupole moment — the charge of its protons is spread in a prolate or oblate ellipsoid rather than a sphere — and inside a crystal it sits in the electric field gradient made by the surrounding electrons and ions. A uniform field would exert no torque on a quadrupole, just as a uniform field exerts no net force on a dipole; the gradient is what couples to it. The result is a set of energy levels split by the orientation of the nuclear spin relative to the gradient’s axes, and transitions between them absorb radio waves at a few megahertz.

Nuclear quadrupole resonance needs no applied magnetic field at all, which is its practical virtue. Nitrogen-14 has a quadrupole moment, and the field gradient at a nitrogen nucleus depends sensitively on its chemical surroundings, so the nitrogen in a particular crystalline compound absorbs at a frequency almost nobody else’s nitrogen shares. Instruments have been built on that to detect specific explosives and drugs inside sealed packages from their nitrogen’s quadrupole lines. The measured frequency is the product of the nucleus’s shape and the gradient of the field at the nucleus — two quadrupole quantities, one from the nucleus and one from the crystal — and each can be extracted when the other is known. That is how most nuclear quadrupole moments, including the deuteron’s, have been measured.

The same pairing sets the order of gravitational radiation. The field of a distant mass distribution is a monopole plus a quadrupole, because the equivalence principle removes the dipole, and a changing mass quadrupole is what radiates — the wave that stretches one way and squeezes the other draws the stretch and squeeze as exactly the pattern of a traceless two-index table, with its two polarisations at 45 degrees to each other. The radiation a difference in falling would make asks what would follow if the dipole were not removed. In both the electric and the gravitational case, the quadrupole is where a source’s shape first reaches anyone far away.

The angle a spinning sample chooses #

The zero cone of the first figure has a use a long way from electrostatics. In a solid, each nuclear spin sits in the magnetic field of its neighbours, and the energy of two magnetic dipoles depends on the angle between the line joining them and the applied field as $3\cos^2\theta - 1$ — the same second-order harmonic, because the interaction of two dipoles is itself a quadrupole-like, rank-two object. In a crystal or a powder the neighbours lie at every angle, their couplings spread every nuclear magnetic resonance line across tens of kilohertz, and the spectrum is a smear.

In 1958 Raymond Andrew, and independently Irving Lowe, spun the sample rapidly about an axis tilted at 54.74 degrees to the applied field. Averaged over the spinning, every coupling acquires a factor of $3\cos^2 54.74° - 1 = 0$, and the lines collapse to widths comparable with those of liquids. Magic-angle spinning is now how the structures of solids, catalysts and membrane proteins are studied by magnetic resonance, and the angle that makes it work is the node of a linear quadrupole’s potential. The magic angle is where $3\cos^2\theta - 1$ is zero, and every interaction with that angular shape — electric, magnetic or elastic — vanishes there on average.

Where the quadrupole picture stops #

Far compared with the size. The quadrupole describes the field only where the distance is large compared with the distribution. The zero cone approaches 54.74 degrees only beyond a few times the charge spacing, and the molecular energies in the last figure are at distances where the molecule’s own size is not small, so the higher moments contribute tens of per cent.

Fixed charges. A real molecule polarises in the field of a nearby ion, adding an induced attraction that falls as the fourth power of the distance and has no preferred sign; for benzene and a potassium ion it is comparable to the quadrupole term. The figures leave it out to isolate the part that depends on shape.

Classical point charges. The measured quadrupole moments come from the quantum-mechanical electron distributions and include the nuclei; the figures use the numbers, not a model of where the charge is. The three shapes in the eigenvalue figure are drawn from point charges and are illustrations of the classification, not models of any molecule.

The square’s field out of the plane #

The drawings are slices through a plane containing the axis. The linear and ring quadrupoles are symmetric about that axis, so a slice shows everything, but the square is not, and its field out of the plane is not shown; the five numbers of a general quadrupole describe a three-dimensional pattern that no single slice captures. The count figure shows how many numbers there are and nothing about what they look like: the five independent quadrupole patterns on a sphere, the $l = 2$ harmonics, are shapes with two, one or no axes of symmetry, and none is drawn. And the molecular figure places each ion on the molecule’s axis only; off the axis the quadrupole’s attraction weakens and at 54.74 degrees it vanishes — which for benzene is why a cation over the rim is not held at all.

Still open: how much of a binding is the quadrupole’s #

For the cation–π interaction and for carbon dioxide in a zeolite, the quadrupole is one of several contributions — electrostatics, induced polarisation, dispersion, and the partial transfer of charge at contact — and how the total divides among them depends on how the division is defined. Different schemes for splitting a computed binding energy into parts give different shares, and for benzene with a cation estimates of the electrostatic fraction have ranged from a half to nearly all of it. The total binding energy is computed reliably now; which part of it the shape is responsible for is a question about definitions that chemists have not settled.

The habit worth carrying away is to count before describing. The number of independent numbers an object takes is fixed by its symmetry before any physics is done, and it says exactly how much a measurement from outside can learn. Nine components, reduced to five by symmetry and by Laplace’s equation, is the statement that a distant observer can see a distribution’s elongation and its orientation and nothing else — and the angle at which one of those five patterns is zero is the same angle whether the thing being spun is a molecule, a spin or a planet.