The metal that is yellow because it is heavy

Copper, silver and gold stand in one column of the periodic table, and they are there because their outermost electrons are arranged identically: a full shell of ten d electrons, and one s electron outside it. Chemistry follows the arrangement of outer electrons, and for most purposes the three behave as a family. Their colours do not. Silver is white, gold is yellow, and copper is a reddish orange, and a calculation of their electronic structure done with Schrödinger’s equation — the equation of every introductory course in quantum chemistry — gets copper roughly right, gets silver right, and makes gold white. The calculation is not wrong about anything it contains. What it leaves out is that in an atom with seventy-nine protons, the electrons nearest the nucleus move at more than half the speed of light.

How fast an inner electron moves #

The size of an atom is the outcome of a competition: confining an electron costs kinetic energy, and the nucleus pays for confinement with attraction. For an electron bound to a nucleus of charge $Z$, the balance puts the innermost electron at about $a_0/Z$ from the nucleus, and its typical speed at $Z$ times the speed of the electron in hydrogen — which is $\alpha c$, the fine-structure constant times the speed of light, about 2,200 kilometres per second. So the innermost electron of an element with nuclear charge $Z$ moves at roughly $Z\alpha$ times the speed of light.

For hydrogen that is 0.7 per cent of $c$ and relativity is a small correction, visible only as the fine structure of spectral lines. For iron it is a fifth of $c$. For gold, $Z\alpha = 79/137 = 0.58$: the innermost electrons move at more than half the speed of light, their Lorentz factor is about 1.22, and relativity is not a correction but a first-order fact about them.

The effect on the innermost orbital can be calculated exactly, because Dirac’s relativistic equation for one electron round a point nucleus has an exact solution. The ground state has a radial density proportional to $r^{2\gamma}e^{-2Zr/a_0}$, where Schrödinger’s has $r^2 e^{-2Zr/a_0}$ — the only change is the exponent of the power, $2\gamma$ in place of 2 — and its mean radius is $(1 + 2\gamma)a_0/2Z$ in place of $3a_0/2Z$. At $Z = 79$ that is 12 per cent smaller. The physical reading is the one special relativity gives any fast body: a moving electron’s inertia is its energy, so a fast electron behaves as a heavier particle, and the radius that the confinement-versus-attraction balance picks goes inversely as the mass. A heavier electron is held closer and more tightly.

The figure also shows where the exact solution runs out. As $Z\alpha$ approaches one, $\gamma$ approaches zero, and for a point nucleus of charge above 137 the ground state no longer exists — the quantum version of the classical orbit that falls into the centre when the attraction exceeds angular momentum times $c$. Real nuclei have finite size, which moves the threshold to around 170, but the table’s heaviest elements live close enough to the edge that relativity is the largest single fact about their inner electrons.

The same ten per cent in an X-ray tube #

The binding-energy curve in the second figure is not only a calculation. It has been measured, a century ago, by people who were not looking for relativity.

When an electron is knocked out of the innermost shell of a heavy atom, an electron from the next shell falls into the vacancy and emits an X-ray. Henry Moseley measured those X-rays across the periodic table in 1913 and found that their energies follow a simple rule: the energy of the strongest line, called K-alpha, goes as the square of the nuclear charge less one, the one being the screening of the nucleus by the other electron left in the innermost shell. It is the Bohr atom’s energy formula with $Z - 1$ in place of $Z$, and it was the measurement that settled that atomic number, not atomic weight, orders the table.

The rule is a non-relativistic one, and at the heavy end of the table it drifts. For copper it predicts a K-alpha line of 8.0 kiloelectronvolts, and the measured line is at 8.05. For gold it predicts about 62 kiloelectronvolts, and the measured line is at 68.8 — about 10 per cent higher. That is the same 10 per cent the Dirac binding ratio gives for the innermost electron of gold: the inner electrons are bound more tightly than Schrödinger’s equation allows, the vacancy they leave is deeper, and the X-ray that fills it is correspondingly more energetic. Every X-ray fluorescence instrument that identifies elements by their characteristic lines is, at the heavy end of its range, reading relativistic binding energies off a detector.

The same shift reaches the chemistry through a different route, and it is worth separating the two. The X-ray line measures the innermost electrons directly. The colour of gold measures the outermost ones, which are affected only through their penetration towards the nucleus and their orthogonality to the inner shells. That the same factor — electrons near the nucleus moving at an appreciable fraction of $c$ — shows up in both, a hundred thousand times apart in energy, is the strongest evidence that it is one effect and not a coincidence of two.

From the core to the colour #

The contraction of the innermost orbital would be of no chemical interest if it stayed there. It does not stay there, because the outer s orbitals are connected to the inner ones.

An s orbital of any shell has some probability of being found very close to the nucleus — it is the piece of the s orbital that penetrates inside the inner shells, and it is why 4s fills before 3d in potassium. Near the nucleus the outer s electron feels nearly the full nuclear charge and moves fast, just as the inner electrons do, and it too becomes effectively heavier. And an outer s orbital must be orthogonal to the inner s orbitals, which means its shape is tied to theirs: when the inner ones shrink, so must it. The net effect in gold is that the outermost orbital, 6s, is pulled in by about 17 per cent and its energy drops, bound more tightly than the non-relativistic calculation says.

The d electrons respond in the opposite direction. A d orbital has almost no probability near the nucleus, so it is not directly affected, and the contracted s shells inside it screen the nuclear charge more effectively than before. The 5d orbitals of gold therefore expand slightly and rise in energy. The s level falls and the d level rises, and the gap between them narrows.

In the metal, the gap is between the top of the filled band of d states and the Fermi level in the half-filled band of s states, and it sets the lowest photon energy that can lift a d electron into an empty state — the onset of strong absorption, an edge in the absorption spectrum like the ones X-rays show at much higher energy. For silver the edge is at 3.9 eV, in the ultraviolet. For gold without relativity, calculations put it at about 3.7 eV, also in the ultraviolet: non-relativistic gold would be a white metal, a heavier silver. For gold as it is, the edge is at about 2.4 eV, well inside the visible range.

The gap that matters is a gap in a metal, not in an isolated atom, and the distinction adds one step. When gold atoms come together in a crystal, their levels spread into bands: the ten 5d electrons of each atom form a narrow, full band, and the single 6s electron forms a broad band that is half full, with the Fermi level in the middle of it. Because each s state holds at most two electrons and there is one s electron per atom, the s band is exactly half full, and the empty states just above the Fermi level are the ones a d electron can be lifted into. The absorption edge is the energy from the top of the d band to those empty states. Relativity lowers the whole s band and raises the d band, so it closes that gap in the metal as it does in the atom.

Why an absorption edge makes a colour #

A metal’s colour is the colour it reflects, and a metal reflects everything its electrons can respond to without absorbing — which, below an absorption edge, is nearly everything.

Silver’s edge at 318 nanometres lies entirely below the visible range, so every visible colour is reflected almost equally and the metal looks white. Gold’s edge at about 517 nanometres cuts through the middle of the visible range. Light redder than the edge — yellow, orange, red — is reflected strongly; light bluer than it — blue and violet — is absorbed partly and reflected less. The reflected mixture, subtracting a share of the blue from white, is yellow. Copper’s edge, at about 590 nanometres, is further into the orange, so it removes more of the spectrum and looks redder still.

Copper’s colour is the non-relativistic one, and the contrast makes the point. Copper is light enough, at $Z = 29$, that relativity changes its levels by about a per cent, and its low edge comes from the ordinary chemistry of its 3d electrons, which are compact and high in energy. Silver’s 4d are lower, and its edge moves out into the ultraviolet. Gold, going by the pattern, should be further out still. It comes back into the visible because relativity closes its gap by about a third, and the colour that results is a direct, everyday observation of the fact that some electrons in a gold ring move at more than half the speed of light.

Mercury, and the pair that will not share #

The element next to gold is mercury, with one more proton and a second 6s electron to fill the shell. The same relativistic contraction that pulls gold’s single 6s electron in now pulls in a pair, and a filled, contracted, tightly bound pair behaves almost like the closed shell of a noble gas.

Metallic bonding holds a metal together by sharing outer electrons among many atoms, and it is only as strong as the outer electrons’ willingness to be shared. Zinc and cadmium, above mercury in the same column, have the same two outer s electrons and melt at 419 and 321 degrees Celsius. Mercury melts at minus 39. Its 6s pair is held so tightly by the contraction that it barely participates in bonding between atoms, and the attraction between mercury atoms is weak enough to be overcome at room temperature. A simulation of liquid and solid mercury published in 2013, repeating the calculation with relativity switched on and off, found the melting point about 105 degrees higher without it — a solid on a warm day.

The tightened s electrons also show up in two numbers that can be looked up for every element. Mercury’s first ionisation energy, the cost of removing one of its 6s electrons, is 10.4 electronvolts — higher than zinc’s 9.4 and cadmium’s 9.0, reversing the usual trend down a column in which outer electrons get further from the nucleus and easier to remove. And gold’s electron affinity, the energy released when a neutral atom captures an extra electron into its half-empty 6s orbital, is 2.3 electronvolts against silver’s 1.3, larger than that of any other metal and approaching the halogens’. In both cases the heavier element holds its s electrons more tightly than the lighter one above it, which is the wrong way round for everything except relativity.

The same contraction explains a string of other anomalies at the bottom of the table. Gold forms compounds with caesium in which gold is a negative ion, like a halogen, because its contracted 6s orbital binds an extra electron unusually well. Lead’s preference for the charge +2 over +4, the voltage of a lead–acid car battery — about four-fifths of which calculations attribute to relativity — and the unexpectedly low reactivity of mercury vapour all trace to the same tightening of s electrons in heavy atoms. None of them requires anything moving fast on a human scale. They require only a heavy nucleus and the electrons nearest it.

Where the one-electron picture stops #

The exact solution is for one electron. Dirac’s closed form describes a single electron round a point nucleus. In a real gold atom there are seventy-nine electrons, each screening the nucleus from the others, and the relativistic contraction of the outer orbitals comes from many-electron calculations — Dirac–Hartree–Fock and its successors — whose results, not closed forms, underlie the 17 per cent contraction of 6s and the gap of the third figure. The first two figures are exact for one electron; the rest is computation, and the essay says which is which.

Spin–orbit coupling is part of the story. Relativity also couples each electron’s spin to its orbital motion, splitting levels with orbital angular momentum — the same coupling that splits sodium’s yellow line, here large enough to split gold’s 5d band by more than an electronvolt. Its effect on the colour is folded into the measured edge and not drawn separately.

The simulated melting points are simulations. The 2013 figures come from a particular set of computational methods and approximations, and their agreement with the measured value, about 15 degrees too high with relativity included, is a measure of how good they are. The difference between the relativistic and non-relativistic runs is the robust part of the result.

The reflectance curve is a sketch of a shape. The positions of the edges are measured; the smooth step drawn at each is a model, and real reflectance spectra have structure above the edge from other transitions.

One electron among seventy-nine #

The radial densities are densities of a single electron in a single state, and in a real atom that electron is one of two in the 1s shell, surrounded by seventy-seven more. The figure shows where one electron is likely to be found, and nothing of the correlations between electrons that the true many-electron state contains.

Nor does a density show the speed that causes the whole effect. An electron in a stationary state has no trajectory, and “moves at 0.58 of the speed of light” is shorthand for a statement about the spread of its momentum — the root-mean-square momentum of the innermost electron of gold corresponds to that speed. The figure shows the consequence, a contracted distribution, and not the momentum that produces it.

Still open: how far relativity reshapes the heaviest elements #

For the superheavy elements beyond uranium, made a few atoms at a time in accelerators and lasting seconds or less, relativistic effects are expected to be large enough to break the periodic table’s pattern outright. Calculations predict that copernicium, below mercury, may be a volatile liquid or even a gas at room temperature; that flerovium, below lead, may behave almost like a noble gas; and that oganesson, nominally a noble gas, may be a semiconducting solid. A handful of chemistry experiments on single atoms of copernicium and flerovium have been done, by measuring how the atoms stick to gold surfaces as they pass along a cooled column, and their results are consistent with some predictions and ambiguous about others. Whether the heaviest elements’ chemistry follows their columns or follows relativity is being decided a few atoms at a time.

The next question the contraction raises is about bonding itself — what happens when two heavy atoms share their contracted s electrons in a molecule, and whether the tightened orbitals make stronger or weaker bonds. The habit worth carrying from here is to estimate how fast the fastest electrons are moving before trusting a non-relativistic calculation. The inner electrons of an element move at about Z/137 of the speed of light, and by the sixth row of the table that fraction is large enough to change the colour of a metal and the state of matter of another.