An address system for the inside of a crystal
So far in this rung you have met the unit cell, learned the three metal structures FCC, BCC and HCP, and worked out their packing and density. That tells you how atoms sit. But a huge amount of a material's behavior depends not on how it is packed but on which way you look: a metal slips most easily along particular directions, is stiffest along others, and reflects X-rays off particular sets of planes. To talk about any of that, we first need a way to name directions and planes inside the cell.
The trick is exactly like putting graph paper over a city map. Lay three axes x, y, z along the three edges of the unit cell, but measure distance in a clever unit: each axis is counted in cell-edge lengths (a, b, c), not in nanometres. So a corner is at 0, the far corner is at 1, and the middle of the cell is at (1/2, 1/2, 1/2). Every crystallographer, working on any material, uses this same fractional address book — which is what lets a copper result and a silicon result be written in the same language.
Naming a direction: [uvw]
A direction is just an arrow. To turn it into three numbers — its crystallographic direction indices — you slide the arrow so it starts at a corner, read off where its head lands, and tidy the result up. The recipe is short and mechanical.
- Slide the arrow so its tail sits at the origin (a corner of the cell). Directions are free to move, so this is always allowed.
- Read the coordinates of the head in units of a, b, c — for example (1, 1, 0) for a face diagonal.
- Clear any fractions and reduce to the smallest set of whole numbers (so 1/2, 1/2, 1 becomes 1, 1, 2).
- Enclose the three integers in square brackets with no commas: [uvw]. A negative value gets a bar (written here as a minus).
A few land immediately. A cube edge is [100]. A face diagonal is [110]. The body diagonal from one corner to the opposite corner is [111]. That [110] face diagonal is not just any arrow — in an FCC metal it is the close-packed direction, the line along which atoms actually touch (recall from the packing guide that 4R equals a times the square root of 2 along a face diagonal). Because a cube is so symmetric, [100], [010], [001] and their negatives are all physically the same kind of direction; we bundle them as the family <100>.
One honest caveat about families: [100], [010] and [001] are equivalent because a cube treats its three axes identically. In a stretched or skewed cell — tetragonal, orthorhombic, hexagonal — the axes are no longer interchangeable, so directions with the same three numbers need not be equivalent at all. Symmetry, not the digits, decides who belongs to a family.
Naming a plane: the reciprocal trick (hkl)
Planes are harder, because a plane can be tilted every which way. The clever fix, invented by William Miller, is to describe a plane not by where it crosses the axes but by the reciprocals of those crossings. These are the Miller indices (hkl), and the recipe is again short.
- Find where the plane crosses the x, y and z axes, in units of a, b, c. If the plane runs through the origin, shift the origin to a neighbouring corner first.
- Take the reciprocal of each intercept. A plane that never crosses an axis (it is parallel to it) has intercept infinity, whose reciprocal is a tidy 0.
- Clear fractions to the smallest whole numbers.
- Enclose in round brackets: (hkl). No commas; negatives get a bar (a minus here).
plane intercepts (a, b, c) reciprocals Miller (hkl) ------ -------------------- ----------- ------------ (100) 1, inf, inf 1, 0, 0 (100) (110) 1, 1, inf 1, 1, 0 (110) (111) 1, 1, 1 1, 1, 1 (111) (200) 1/2, inf, inf 2, 0, 0 (200) inf = plane is parallel to that axis -> reciprocal is 0
Why go through reciprocals at all? Two reasons make it worth the bother. First, a plane parallel to an axis never crosses it, and infinity is an awkward label — its reciprocal, 0, is not. Second, the labels stay meaningful for parallel planes at different spacings: (100), (200), (300) are all parallel, but (200) sits at half the spacing and gets its own name, which matters enormously the moment X-rays enter. And, like directions, planes come in symmetry families: in a cube the braces {100} gather all six cube faces into one set.
Linear and planar density — the close-packed plane
Now the indices start to pay for themselves. Once you can name a line or a plane, you can count how crowded it is. Linear density is atoms per unit length along a direction; planar density is atoms per unit area on a plane. Both are just headcount divided by size, and both tell you where a crystal is most tightly packed.
Take the [110] direction in FCC. The face diagonal has length a times the square root of 2, and it threads two half-atoms at the ends plus one whole atom at the face centre — two atoms in all. Since 4R equals a times the square root of 2, the linear density works out to 2 divided by (a times root 2), which is 2 over 4R, i.e. 1 over 2R. That is the densest a line can be: the atoms are bumper-to-bumper. Lift this idea up one dimension and FCC's (111) plane turns out to be the single most crowded plane in the whole structure, and it is stitched together from exactly these close-packed <110> directions.
Be honest about the other two structures. BCC has no truly close-packed plane at all — its 0.68 packing falls short of FCC's 0.74 — so its densest plane {110} is merely the best on offer, and its close-packed direction is the body diagonal [111], where 4R equals a times the square root of 3. HCP does have a genuine close-packed plane, the flat basal plane, but hexagonal cells are awkward enough that they use four indices (hkil), with the extra one fixed by i = -(h + k). Keep the throughline in view: which planes are densest is precisely what governs where a metal will deform — the next rung on slip is built on exactly this.
Why indices earn their keep: stiffness, slip, and X-rays
First payoff: direction-dependent properties, or anisotropy. A single crystal is not the same in every direction. BCC iron is a vivid case: measured on one crystal, its Young's modulus is about 125 GPa pulled along [100] but roughly 275 GPa pulled along [111] — more than double the stiffness, same block of iron, only a different direction. The body diagonal simply packs its bonds more densely, so it resists stretching harder. Only because we can name [100] and [111] can we even state a fact like that.
Second payoff: slip. Metals bend not by stretching every bond at once but by planes gliding over planes, and the plane that glides is almost always the densest plane moving along a close-packed direction — a pairing we call a slip system. That is why FCC's (111)/<110> combination makes copper and aluminium so ductile: the crowded plane offers a smooth, low-hurdle path for slip. Naming planes and directions is what lets us predict, in advance, where a crystal will yield.
Third payoff: seeing the atoms. X-rays reflect off whole families of (hkl) planes, and each family has its own interplanar spacing d — for a cubic crystal, d equals a divided by the square root of (h^2 + k^2 + l^2). Feed a beam in, measure the angles where reflections flare up, and Bragg's law hands you those d-values, and from them the structure itself. That is the whole subject of the next guide — and it only works because every plane in the crystal already has a name.