From long-range order to one repeating tile
Guide 1 drew the line between a crystalline solid, where atoms sit in a pattern that repeats without end, and an amorphous one, where the same atoms are frozen in a jumble. That leaves an awkward practical problem: a fingernail-sized crystal holds something like 10^22 atoms. Nobody can draw them, and nobody needs to — because in a crystal they all say the same thing over and over. So we do what a tiler does with a floor: instead of describing every tile, we describe one tile and the rule for repeating it. Find the smallest chunk of the pattern that, stamped edge to edge in all three directions, rebuilds the entire crystal, and you have captured the whole solid in one small box.
Two ideas make this precise. The space lattice is an imaginary infinite grid of points spread through the crystal so that every point has exactly the same surroundings as every other — it is the pure skeleton of the repetition, with no atoms yet attached. The unit cell is the smallest box, cut from that grid, that tiles the whole lattice when you stack copies of it. Its shape is fixed by six numbers: the three edge lengths a, b, c (the lattice parameters, usually a fraction of a nanometre) and the three angles between those edges. Learn to read those six numbers and you can name any crystal's geometry.
Seven shapes, fourteen lattices
The box need not be a cube. Let the three edge lengths and three angles vary, and you find there are exactly seven distinct box shapes a repeating pattern can have — the seven crystal systems. They run from the perfectly symmetric cubic (all edges equal, all angles 90 degrees) through tetragonal, orthorhombic, hexagonal, and on down to the lopsided triclinic, where no edges match and no angle is a right angle. The cube is the friendliest and, happily, the one most common metals choose, so it will do most of the work below.
Now add the second freedom: within each shape, the repeating points can sit at the corners only, or also at the body centre, or at every face centre, or at just one opposing pair of faces. Auguste Bravais proved in 1848 that once you throw out the redundant combinations, exactly fourteen genuinely different arrangements survive — the fourteen Bravais lattices. That is the whole vocabulary of crystalline order: every crystal ever measured, from table salt to a jet-engine blade, is one of these fourteen lattices decorated with atoms. Metals use only a small corner of that vocabulary, which is why we can now zoom straight to the three arrangements that carry almost the entire metals world.
The big three: FCC, BCC, HCP
Why do metals pick such simple structures? Recall from the bonding rung that the metallic bond is non-directional — the shared electron sea does not care which way a neighbour lies, only how close it is. So metal atoms behave almost exactly like hard, equal spheres that want to pack as tightly and symmetrically as they can, the way oranges settle into a crate. Out of all the possibilities, three winners dominate: face-centred cubic, body-centred cubic, and hexagonal close-packed.
In face-centred cubic (FCC) the atoms sit at the eight corners of a cube plus the centre of each of the six faces; they touch along the face diagonal, and counting the shared corner and face atoms leaves 4 whole atoms per cell. Copper, aluminium, nickel, silver, and gold are all FCC. In body-centred cubic (BCC) there are atoms at the eight corners plus one lone atom in the very centre of the cube; they touch along the body diagonal, giving 2 atoms per cell — this is room-temperature iron, tungsten, chromium, and molybdenum. Hexagonal close-packed (HCP) is not a cube at all: it stacks close-packed layers in a two-beat ABAB rhythm, and the conventional cell holds 6 atoms. Zinc, magnesium, titanium, and cobalt are HCP.
FCC (face-centred cubic) BCC (body-centred cubic)
o-----------o o-----------o
/| o /| /| /|
o-----------o | o-----------o |
| | o | | | | o | | <- one atom
| o------o--|-o | o---------|-o dead centre
|/ o |/ |/ |/
o-----------o o-----------o
corners + 6 face centres corners + 1 body centre
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Structure Atoms/cell Coordination APF a-R relation Example
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FCC 4 12 0.74 a = 2R x sqrt2 Cu, Al
BCC 2 8 0.68 a = 4R / sqrt3 a-Fe, W
HCP 6 12 0.74 c/a ~ 1.633 Zn, Ti
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R = atomic radius a = cube edge (lattice parameter)
FCC and HCP both reach 0.74 (close-packed); BCC is looser at 0.68Counting the inside: coordination, packing, density
Two numbers turn these pictures into physics. The coordination number is simply how many nearest neighbours each atom touches. In FCC and HCP that number is 12 — the most that equal spheres can manage — which is exactly why both are called close-packed. BCC gets only 8 nearest neighbours, so it is dense but not maximally so. The atomic packing factor (APF) then asks what fraction of the cell's volume the spheres actually fill: FCC and HCP reach 0.74 (74 percent solid, 26 percent empty space, the tightest equal spheres can ever be), while BCC settles at 0.68. Guide 3 derives these fractions from scratch; here just hold the ranking, because it explains real behaviour — the extra open space in BCC is one reason iron can absorb carbon and be hardened into steel.
The reward for all this counting is that you can predict a metal's density from geometry alone — its theoretical density. The recipe just weighs one unit cell and divides by its volume: take the atoms inside the cell, multiply by the mass of one atom, and divide by the cell's volume. Run it for copper and watch how close it lands to the value you would measure on a scale.
- Count the atoms per unit cell. Copper is FCC, so n = 4.
- Get the mass ingredients: copper's atomic weight is A = 63.5 g/mol, and Avogadro's number is N_A = 6.022 x 10^23 atoms per mole.
- Find the cell volume from the lattice parameter: copper has a = 0.3615 nm = 3.615 x 10^-8 cm, so V_c = a^3 = 4.72 x 10^-23 cm^3.
- Combine: density = n times A divided by (V_c times N_A) = (4 x 63.5) / (4.72 x 10^-23 x 6.022 x 10^23) = about 8.9 g/cm^3.
- Sanity-check against reality: measured copper is 8.96 g/cm^3. A near-perfect match — real evidence that the hard-sphere FCC picture is not a cartoon but genuinely how the atoms sit.
One metal, two structures: polymorphism and allotropy
A structure is not stamped on an element forever. The same atoms can crystallise in more than one arrangement depending on temperature and pressure — call it polymorphism in general, or allotropy when the substance is a pure element. Iron is the star example, and it is no exaggeration to say the whole steel industry hangs on it: below 912 degrees C iron is BCC (called alpha-iron, or ferrite), but heat it past 912 degrees C and its atoms rearrange into FCC (gamma-iron, or austenite). That single switch, which you will meet again in the iron-carbon rung, is the hinge every steel heat treatment turns on.
Here is a fact that catches everyone off guard, and it falls straight out of the packing numbers: FCC (0.74) is denser than BCC (0.68), so when iron transforms from BCC to FCC on heating through 912 degrees C, it actually shrinks — a piece of metal that contracts as you heat it through that point. Carbon makes the same lesson vivid at the extreme: as diamond every atom is locked in a rigid covalent cage, giving the hardest natural material, transparent and insulating; as graphite the same atoms lie in slippery sheets that make a soft, grey, electrically conducting solid you write with. Same element, opposite personalities — structure, not composition, is doing the talking.
How we know — and what the tidy cell hides
How can anyone be sure atoms really sit in these boxes, when they are far too small to see with light? We fire X-rays, whose wavelength happens to match the spacing between atomic planes. The orderly planes reflect the beam, and the reflections interfere to produce sharp spots at very specific angles, governed by Bragg's law (n times lambda = 2d sin theta). Read off the angles and you can back out the exact spacing and structure — this is how copper was pinned as FCC and iron as BCC. An amorphous solid, having no repeating planes, gives only a broad fuzzy halo instead of sharp spots — the diffraction fingerprint of order versus disorder. Guide 5 tells this story in full.