The orange-stacking problem
Most metals are made of one kind of atom, and to a very good approximation those atoms behave like identical hard spheres — like a crate of oranges. So nature faces a grocer's everyday puzzle: what is the tidiest, tightest way to stack a pile of identical balls? Pack them loosely and you waste room; pack them well and you fit more in the same space. The arrangements that squeeze the spheres together as tightly as geometry allows are called close-packed structures, and they are the single most important idea in this whole guide.
Start with a single flat layer of oranges. The tightest way to lay them out is the obvious one: each orange nestled into the dimple between its neighbours, so every ball touches six others around it, like a honeycomb. There is no denser way to cover a floor with circles — your grocer knew this without being told. The interesting choices all come on the *next* layer.
Two ways to stack, two famous crystals
Lay down your second layer and you have no real choice but to drop each new orange into a dimple of the first layer. But when you reach the third layer, a fork appears. You can place the third layer's oranges directly above the first layer's, so the stack alternates A, B, A, B, A, B — or you can shift them sideways into a fresh set of dimples, giving A, B, C, A, B, C. Both stack equally tightly. Both waste exactly the same amount of space. Yet they are different crystals, and almost every common metal picks one or the other.
The A-B-A-B stack — third layer back over the first — is the hexagonal close-packed structure, hcp for short. Zinc, magnesium, titanium, and cobalt all live here. The A-B-C-A-B-C stack — third layer shifted to a new position — is, surprisingly, a cube viewed from a corner; it is the face-centered cubic structure, fcc, and it is home to copper, aluminium, gold, silver, nickel, and lead. That the cosy-looking A-B-C orange pile is secretly a cube is one of the genuinely delightful facts of crystallography; tilt an fcc cube onto one corner and the close-packed layers spring into view.
Counting your neighbours
There is a wonderfully simple number that captures how snugly an atom is packed: how many other atoms are touching it. This count is the coordination number, and it is one of the first things a crystallographer asks about any structure. A high coordination number means each atom is hugged by many neighbours, which usually means a dense, tightly bonded, sturdy material.
In both close-packed structures, fcc and hcp, every atom touches twelve others — six in its own layer, three in the layer above, three below. Twelve is the maximum any arrangement of equal spheres can achieve, which is just another way of saying these are the densest packings. Not every metal goes for maximum density, though. A great many — iron at room temperature, chromium, tungsten, the alkali metals — settle instead into the body-centered cubic structure, bcc: a plain cube of atoms with one extra atom sitting at its very centre. In bcc each atom touches only eight neighbours, and the spheres fill about 68% of space rather than 74%. It is slightly looser, slightly less hugged — but for these elements it happens to be the lower-energy choice.
structure stacking neighbours space filled --------- -------- ---------- ------------ fcc A-B-C-A-B-C 12 ~74% hcp A-B-A-B-A-B 12 ~74% bcc cube + centre atom 8 ~68% diamond open, 4 stiff bonds 4 ~34%
When the bonds have opinions: diamond
The orange-stacking picture works beautifully when atoms behave like featureless balls that just want to huddle. But some atoms are pickier. A carbon atom does not merely want to touch as many neighbours as possible — it insists on bonding to exactly four others, and it insists those four point outward in a particular splayed arrangement, like the four corners of a tripod plus the leg you stand it on. When carbon gets its way, the result is the diamond structure.
Diamond is a strikingly *open* structure — its atoms fill only about a third of space, far less than the close-packed metals. Each carbon clings to just four neighbours, not twelve. By the logic of pure packing, this is a terrible, wasteful arrangement. And yet diamond is the hardest natural material known. The lesson is important and a little humbling: density is not the same as strength. What makes diamond unbreakable is not how tightly its atoms are crammed but how rigid and directional its bonds are. Every carbon is locked to its four partners by stiff, fussy bonds that refuse to bend, and that scaffolding of obstinate bonds, not close packing, is the source of its hardness.
When two kinds of atom share a crystal: salt
So far our crystals have been made of a single element. Most interesting materials are not. Table salt is sodium chloride — two different atoms, sodium and chlorine — and they have to share one crystal. In salt, sodium has given an electron to chlorine, so each is a charged ion: sodium positive, chlorine negative. Opposite charges attract, like charges repel, and the crystal's job is to surround every positive ion with negatives and every negative with positives. The arrangement that does this best is the rock-salt structure: two interpenetrating cubic grids, one of each ion, slotted together like a three-dimensional chessboard where the colours are the two elements.
In rock salt each ion is surrounded by six of the opposite kind — a coordination number of six. This is a lovely example of why the basis idea from earlier matters so much. The lattice of salt is just face-centered cubic, the same grid as copper. But the basis is not one atom; it is a pair, one sodium and one chlorine. Same grid, different contents, completely different material. It shows how a small handful of lattices, dressed with different bases, can generate the whole sprawling variety of the mineral world.