Why Gold Is Yellow: Einstein on Your Finger

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Silver and gold sit one beneath the other in the periodic table and their electron arrangements are almost identical; one shines white, the other yellow. Chemistry is not enough to explain that difference — you need the equations Einstein wrote in nineteen oh five. Because gold's innermost electrons are circling at one hundred and sixty thousand kilometres a second, and at that speed they grow heavier.

Picture two rings in a jeweller's window, lying side by side on the same velvet. Both were hammered into the same shape, polished on the same wheel, and sit under the same light. One is white, one is yellow. We take that difference so utterly for granted from childhood that we never think to ask about it. And yet chemistry expects these two metals to be near-twins.

On the periodic table, silver and gold sit in the same column. Copper is with them too, at the top of that same column. All three have the same arrangement of outer electrons: a completely filled d shell, and above it, standing entirely alone, a single s electron. The whole power of the periodic table is hidden in exactly that promise. Same column means similar behaviour. Drop sodium into water, then potassium, and both froth with the same fury. Fluorine and chlorine snatch electrons from anything that comes near them. The logic of the column works.

Gold and silver keep that promise, for the most part. Both are extraordinarily soft, spreading under a hammer without tearing. Both conduct electricity superbly; silver is the best conductor known, and gold comes in just behind copper. Both settle into the same crystal arrangement in the solid state, the face-centred cubic structure. Their atomic radii are even almost exactly the same, roughly 144 picometres. That fact alone is strange, because gold sits a full row below silver on the table; it has one extra shell, and it ought to be noticeably fatter. It is not. And because far more mass is packed into the same volume, gold is nearly twice as dense as silver: 19.3 grams per cubic centimetre against 10.5.

But one is white and one is yellow.

Let us first look at what the colour of a metal even means. Metals are lustrous because inside them there is a sea of electrons that belongs to no single atom, roaming freely through the whole mass. That sea does not care what frequency the light falling on it happens to have; the electrons oscillate along with the incoming wave and immediately radiate the energy back out. Every colour of visible light is sent back in equal measure. The result is a colourless brightness. This is how a mirror works, this is why steel looks grey, this is why silver is white. For a metal to acquire a colour, that equality has to break — the metal has to selectively swallow part of the visible spectrum.

A little way beneath the sea of free electrons lies a completely filled d band. An electron down there can leap up and join the sea, but doing so requires a certain threshold of energy. In silver that threshold is about 4 electronvolts. Convert it to a wavelength and you get 310 nanometres, which is in the ultraviolet. In other words, the light silver is capable of absorbing lies in a region our eyes cannot see. The whole of visible light comes back untouched, and silver shines white.

In gold, the threshold for that same leap is about 2.4 electronvolts. That means 520 nanometres, and that falls squarely inside visible light, on the border between blue and green. Gold swallows the violet and blue portion of the light falling on it and sends the rest back — part of the green, the yellow, the orange, the red. Our eye reads that mixture as a warm yellow. The yellow of gold is really the absence of the blue that gold cannot give back.

So the question sharpens. If the outer electron arrangement of the two metals is the same, if their crystal structures are the same, if even their atomic sizes are the same, then why does that threshold sit in the ultraviolet for one of them and right in the middle of our vision for the other? The gap between them is more than 1.5 electronvolts; on the atomic scale that is not a deviation to be shrugged off, it is a different world entirely.

Let us set copper aside, because the answer to copper's redness is the easier one. In copper the d band already sits very close to the conduction sea, the threshold naturally falls into the visible region, and no unusual explanation is needed for it. Copper is the easy member of the trio. The anomaly is gold.

And gold's anomaly does not come alone. Silver slowly tarnishes in air, a dark layer of sulphide forming on its surface. Gold does not tarnish. When a gold mask that has waited 3,000 years under the earth is lifted out, it comes out with the surface it had on the day it was buried. That one of two metals sitting in the same column should be so chemically reluctant is also something that demands explaining.

The answer to these three oddities — the colour, the size, and the reluctance — is not in the chemistry textbooks. The answer is in a paper written in 1905 on the electrodynamics of moving bodies. You do not need to look at black holes or satellite orbits to see Einstein. Looking at the ring on your finger is enough.

In the nucleus of a gold atom there are 79 protons. That is the number which determines how fierce the pull exerted on the electrons is. The innermost electron is held right at the bottom of that nucleus, in an extraordinarily narrow region, and a rough calculation can tell us how fast that electron is moving.

In the simple model of the atom, the speed of the innermost electron is found by dividing the number of protons in the nucleus by 137. That number 137 is not arbitrary; it is the inverse of the fine structure constant, one of nature's fundamental constants, and it sets the strength of the electromagnetic interaction. In hydrogen, that is to say in a nucleus with a single proton, the electron's speed is one 137th of the speed of light. Negligible. But in gold we have 79 protons. 79 divided by 137 comes to roughly 0.58.

Which is to say the innermost electron of a gold atom moves at 58 percent of the speed of light, roughly 175,000 kilometres per second.

A margin of honesty has to be left here: the electron does not really orbit like a planet, it is spread out as a cloud of probability, and assigning it a classical speed is a crude thing to do. But the order of magnitude of the momentum is right, and full calculations that include relativity from the outset confirm this rough estimate. The inner electrons of gold have long since entered the region where relativity is no longer a correction but the main effect.

What happens at 58 percent of the speed of light? This is where special relativity enters. The inertia of a moving body increases; the electron behaves as though its mass were 23 percent greater. And since the diameter of the region the electron occupies is inversely proportional to its mass, that region shrinks. The innermost shell is roughly one fifth smaller than it would be in a universe where relativity were switched off. It collapses onto the nucleus.

Who cares about the innermost electron, you might say. It is the outer electron that does the chemistry. And here is precisely the crux of the story: the contraction does not stay inside.

A strict rule of quantum mechanics requires electron regions of the same type to remain distinct from one another, never blending together. So when the innermost s region contracts, all the s regions outside it are forced to contract along with it. In gold, the chain reaches all the way out to the outermost electron. And that outermost electron is exactly what makes gold gold. The contraction that relativity sets off at the bottom of the nucleus is carried directly to the valence electron. That electron is drawn inward, bound more tightly to the nucleus, and its energy drops.

At the same time a second effect works in the opposite direction. The s and p electrons that have collapsed inward screen the positive charge of the nucleus more effectively. The d electrons out beyond them never venture down to the bottom of the nucleus, so they are left behind that screen and feel the nucleus more weakly than before. They loosen, they expand, their energy rises.

Now put the picture together. In gold, the energy of the lone outer electron goes down, and the energy of the filled d band just beneath it goes up. The gap between them closes from both sides at once, like a pair of pincers. Calculations that leave relativity out find that threshold in the ultraviolet for gold, that is, they give a result resembling silver. Calculations that include relativity bring the threshold down to 2.4 electronvolts and paint gold yellow.

This is not a claim that stayed on paper. A relativistic band calculation carried out in 1971 reproduced gold's measured reflectance curve properly for the first time; non-relativistic calculations could not manage it. Switch relativity off in the computer and gold turns the colour of silver. Switch it on and it goes yellow.

So why does it not happen in silver? Silver has 47 protons in its nucleus. 47 divided by 137 comes to roughly 0.34. A third of the speed of light — still an unthinkable speed — but relativistic effects do not depend on speed linearly, they depend on it far more sharply. In silver the mass increase is not 23 percent but around 6. The contraction is real, but it is not strong enough to drag the threshold out of the ultraviolet and into visible light. Silver stays white.

What is more, gold is particularly fortunate in this respect. Along the sixth row of the periodic table, the relativistic contraction of the outer electron does not simply grow steadily; it peaks at exactly 79 protons. Chemists call that peak the gold maximum. Gold sits at the very point where relativity is most visible in chemistry.

And here lies the irony of the story. Dirac, who wrote down the relativistic equation of quantum mechanics, stated in 1929 that in situations where high speeds do not come into play — that is, in atomic and molecular structure and in ordinary chemical reactions — relativity would have no appreciable effect. That view was generally accepted in chemistry for forty years; taking relativity seriously was the business of a minority who concerned themselves with the fine details of heavy nuclei. One of the works that really pulled the subject together was published in 1979, and it laid out how relativity runs through the entire periodic table. The counter-example, meanwhile, had been sitting on people's fingers for thousands of years.

The same pincer explains gold's other oddities. Because the outer electron is bound more tightly, tearing an electron off gold is markedly harder than tearing one off silver: 890 kilojoules per mole against 731. A metal that reluctant to give up its electron does not readily enter into reactions. Gold's refusal to tarnish, that stubborn cleanliness which made it money and sacred object throughout history, comes straight from here.

More surprising still, gold is willing to take an electron rather than give one. When it accepts an electron it gains 2.3 electronvolts; that value is far from anything typical of a metal and close instead to iodine's 3.1 electronvolts. Which is to say gold can behave chemically almost like a halogen. Combine it with caesium and what emerges, caesium auride, is not a metal alloy but plainly a salt: a semiconductor, close to transparent, and yellowish. Inside it, gold stands as a negatively charged ion. The only thing that makes this possible is the relativistically lowered energy of that electron.

And one question becomes unavoidable: if 79 protons do this, what do 80 protons do?

Immediately to the right of gold on the periodic table sits mercury. Its nucleus holds 80 protons, and unlike gold its outer shell is not half filled but completely filled; there is not a single electron there but a pair of them.

Relativistic contraction pulls that pair toward the nucleus and renders it unusually stable. The result is a kind of chemical stinginess. Mercury atoms hold their own electrons tightly and will not share them with their neighbours. What makes metals hard and durable is that their atoms contribute generously to a common pool of electrons; in mercury that pool stays weak. The atoms cling to one another with a slackness better suited to the noble gases than to a metal.

This is why mercury melts at minus 38.8 degrees Celsius and is the only metal that is liquid at room temperature. A computer simulation published in 2013 tested this directly: when relativistic effects were removed from the calculation, mercury's melting temperature shot up to 82 degrees Celsius. In other words, relativity pulls mercury's melting point down by about 105 degrees, and it does so passing precisely across room temperature on the way. Without relativity, mercury would be a grey metal you could pick up in your hand, just like lead. The barometer, the old thermometers, and the whole slippery, ungraspable image mercury summons in the human mind are all owed to those 105 degrees.

Let us go one row further: 82 protons, lead. Lead's most familiar use is the battery discovered by Gaston Planté in 1859, which despite all the time that has passed since is still sitting under the bonnet of your car. A lead acid cell produces about 2.1 volts; you wire 6 cells in series and out comes the standard 12 volt battery. A cheap, reliable arrangement capable of meeting the enormous current demand of the starter motor.

In 2011, researchers in Sweden and Finland calculated the voltage of that cell twice: once with equations that included relativity, and once without. The result was striking. Of the 2.1 volts per cell, roughly 1.7 volts — that is, about 80 percent — came directly from relativistic effects. Without relativity, the lead acid battery would never produce enough voltage to turn a starter motor. When you turn the key on a cold winter morning and hear the engine begin to spin, part of what is doing the work is electrons moving faster than half the speed of light at the bottom of a lead nucleus.

This trail, which began with the yellow of gold, gave birth to an entire field in chemistry. Relativistic quantum chemistry is a mature discipline today; no serious calculation involving heavy elements is done without accounting for relativity. If you want to get the bond length of a molecule right, or why a catalyst works, or how the platinum inside a drug behaves, you have to put Einstein into the equation.

At the very bottom of the table, among the artificial elements carrying more than 104 protons in their nuclei, the effect is expected to become still more dominant. The prediction is that in these elements the relativistic shifts grow so large that the habit of guessing an element's behaviour from the column it sits in may break down. This needs underlining: it is a prediction, not a settled result. The elements in question survive in the laboratory for fractions of a second, and their chemistry is tested one atom at a time, a few atoms per attempt.

But the real point, I think, has to do with scale. The examples we are used to when relativity is explained are all far away. Atomic clocks on satellites drifting by a few microseconds a day. Starlight bending during a solar eclipse. Black holes, the universe, billions of years. All of it true, all of it measured, but none of it tangible; we accept every bit of it on the word of an instrument.

The yellow of gold is not like that. It requires no instrument, no calculation, no intermediary. It is a colour looked at with the naked eye in broad daylight, and one that would not be there without relativity. Human beings have been gazing at it for more than 6,000 years; the oldest worked gold ever found dates from an age before writing itself existed. The two properties that make gold precious, that warm yellow and that stubborn refusal to tarnish, are, we now know, two faces of a single cause. Both are the work of electrons exceeding half the speed of light at the bottom of a nucleus.

For thousands of years humanity placed gold in temples, in graves, and in crowns without knowing why it was the way it was, simply because it was. Then, in the twentieth century, a physicist thinking about moving bodies and the speed of light wrote down the equations, and it turned out that those equations, even in their most abstract form, had long been sitting on people's fingers.

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