I Love Gold

Start with why a metal is shiny at all.

A metal’s outermost electrons are not tied to any particular atom. They form a shared sea that runs through the whole block, and a sea of loose charge is very good at refusing light: an arriving wave pushes the electrons, the electrons slosh, and the sloshing sends the wave straight back out. That is what a mirror is. It is also why almost every metal is the same colour, which is to say no colour at all – if everything comes back, what you see is whatever was falling on it, and in daylight that is white.

So a coloured metal is a metal that has failed to send something back. Gold is not adding yellow. It is subtracting blue, and yellow is what daylight looks like with the blue taken out.

The question is what does the subtracting. Somewhere inside gold there is a process that will accept a blue photon and not a red one, and the price of admission is about two and a half electron-volts. Find out what sets that price and you have found out why gold is yellow.

Before going further it is worth stopping on a phrase that sounds like it means something simpler than it does.

The innermost electron of a gold atom is not a small ball whizzing round a track at 58% of light speed. It is not going anywhere. Its cloud sits still, and if you ask which way it is travelling on average, the answer is exactly nowhere: there is as much of it heading left as right, always, and the two cancel perfectly.

What is large is not the electron’s travel but its spread. Think of a guitar string held down hard against a fret. The string as a whole goes nowhere – the pattern just sits there humming – and yet every point on it is whipping up and down very fast indeed. Press the string shorter and the motion gets faster, even though the string still goes nowhere.

That is the honest picture. Squeezing the electron into a small space does not give it a direction. It gives it speed without direction: a bigger range of momentum, in every direction at once, cancelling in the average and not cancelling in size. “58% of light speed” is the size of that spread. It is the same wiggle you already paid for when you asked why the electron does not fall into the nucleus – just quoted in miles per hour instead of in energy.

Now the consequence.

Something moving that fast is harder to push around than something slow. Relativity’s way of saying this is that the electron responds as though it had gained weight – at 58% of light speed, about 22% more of it.

A heavier electron is held on a shorter leash. Orbits shrink in proportion to how heavy the orbiting thing is, so gold’s innermost shell pulls in by about 18%. That much is unsurprising and, on its own, would not matter to anybody: the innermost electrons of a heavy atom are buried far below anything chemistry can reach.

The surprise is which other shell comes with it. The electrons at the very outside of a gold atom are not all the same shape. One family dives right through the middle of the atom, passing through the crowded region near the nucleus on every pass, and so it feels the contraction too and is dragged in with the core. Another family is shaped so that it never goes near the middle at all. That one is left behind – and worse, it now finds itself screened by the family that moved inward, so it drifts outward.

Two outer families, squeezed together from opposite sides. The gap between them narrows. And the size of that gap is the price of admission we were looking for in the last section.

Here is the test that settles it.

Build a model of a metal with two ingredients: the electron sea that makes it a mirror, and one absorption threshold below which nothing is absorbed at all. Tune it, once, so it reproduces gold’s measured reflectance – the fraction of each colour that a real gold surface actually sends back, which has been in tables for a century.

Then change exactly one number. Leave the electron sea alone, leave the shape of the absorption alone, and move the threshold from where relativity puts it, 2.4 electron-volts, to where it would sit without relativity, 3.7 electron-volts. Feed what comes back through a model of the human eye and see what colour it is.

At 2.4, only 40% of the blue returns while 93% of the red does, and the result is unmistakably gold. At 3.7, the threshold has moved out past the end of the visible band entirely, nothing is absorbed, everything comes back, and the swatch turns white – landing on silver so closely that you would need a colour meter to tell them apart.

That is the whole claim, and you can drag the slider yourself.

Gold is the famous case, but it is not the only one, and the next two are stranger for being so ordinary.

Mercury sits one place along from gold, at 80 protons instead of 79, so its contraction is very slightly stronger. Its outermost pair of electrons is pulled in so tightly that the atom becomes reluctant to share them, and sharing electrons is what holds a metal together. Mercury’s atoms therefore barely bond to each other, and a metal that barely bonds is a metal that melts easily. It melts at −39 °C. Calculations that leave relativity out get a solid that would not melt until around +82 °C. Relativity is the reason the only liquid metal at room temperature is liquid.

Then there is the car outside. A lead-acid battery gives about 2.1 volts per cell, and it has done so since 1859. Work out how much of that voltage survives if you switch relativity off, and the answer is about 0.4 volts. Roughly four-fifths of your car battery is a relativistic effect. Turn the key on a cold morning and you are being started by the same physics that makes the ring on your finger yellow.