An address system for arrows in the lattice
In the last rung you built the unit cell and measured it with the lattice parameters a, b, and c, and you learned to give an atom a home with fractional coordinates. Now we do the opposite job: not locating a point, but pointing in a heading. A crystallographic direction is nothing more than an arrow — a straight line drawn through the lattice — and we want a compact, unambiguous name for it. Think of it like calling out a compass bearing, except the three reference axes ('north, east, up') are the cell edges a, b, and c themselves.
The key trick is this: measure in lattice units. We record how far the arrow runs along each axis as a multiple of the cell edge, not in nanometres. Because of that, the name comes out dimensionless and portable — the same [uvw] labels that heading in a 3-angstrom cell and in a 30-angstrom cell, and it keeps working even in a skewed cell whose axes are not at right angles. Bear that last point in mind: those three axes need not be equal in length, nor perpendicular to one another.
Reading off [uvw]: the recipe
To name any direction, slide the arrow (keeping it parallel to itself) until its tail sits at the origin, then read where its head lands. Here is the whole drill in four moves.
- Slide the arrow parallel to itself until its tail sits at the cell origin — a direction has no fixed position, so this move is always allowed.
- Read the coordinates of the head as multiples of the cell edges a, b, and c, giving three numbers u', v', w'.
- Clear any fractions, then divide out any common factor, reducing to the smallest set of three integers u, v, w.
- Enclose them in square brackets with no commas: [uvw]. A negative index is written with a bar over it, as in [11bar0].
head (units of a,b,c) reduce [uvw] what it is --------------------- -------------- ---------- -------------------- (1, 0, 0) already ok [100] a cube edge (0, 1, 0) already ok [010] another cube edge (1, 1, 0) already ok [110] a face diagonal (1, 1, 1) already ok [111] the body diagonal (1/2, 1, 0) x2 -> (1, 2, 0) [120] an in-face arrow (1, -1, 0) already ok [1 1bar 0] the other face diag note: "1bar" means minus one, drawn with a bar over the digit.
Walk the fifth row slowly. An arrow from the origin to the point (1/2, 1, 0) carries a fraction, so multiply through by 2 to get (1, 2, 0): the direction is [120]. The body diagonal runs corner-to-opposite-corner, (0,0,0) to (1,1,1), giving [111]; a face diagonal is [110]. And remember [111] and [222] would be the identical arrow — a direction carries no length, so we always keep the smallest integers.
Minus signs, and a warning about skewed cells
When the arrow runs backwards along an axis, that index is negative, shown by an overbar rather than a minus sign. So [11bar0] heads along +a and -b — the face diagonal that crosses [110] at a right angle inside a cube. Flip every sign and [1bar 1bar 1bar] is the exact reverse of [111]: it lies on the same line but points the opposite way, and that antiparallel arrow counts as a genuinely different direction.
One more thing to file away for the next guide. Beginners love the rule 'the direction [hkl] is perpendicular to the plane (hkl).' It is a genuinely useful fact — but only in cubic crystals. In the general case a plane's normal is not the like-numbered direction, and untangling the two calls for the reciprocal lattice from a rung further up. For now, treat directions [uvw] and the Miller indices (hkl) for planes as two separate languages that happen to coincide only in the cubic special case.
Families: the <uvw> brackets
In a highly symmetric crystal, many directions are physically indistinguishable. In a cube the edge [100] is no different from [010] or [001] — give the cube a 90-degree turn and one becomes another, so no experiment could tell them apart. Crystallographers gather all such symmetry-equivalent arrows into a family of directions, written in angle brackets: <100>. Where [100] is one specific arrow, <100> is the whole set the crystal's point-group symmetry treats as the same.
In the cubic system a family is easy to list, because you just write down every arrangement of the numbers with every choice of sign. <100> collects the six cube-edge directions ([100], [010], [001] and their three negatives); <110> collects the twelve face-diagonal directions; <111> collects the eight body-diagonal directions — the four body diagonals and their reverses. Those counts, 6, 12, and 8, are worth committing to memory.
Why we bother: packing, slip, and angles
Directions are far more than bookkeeping, because atoms are spaced differently along different headings. Count how many atoms per unit length a line threads and you have its linear density; the headings packed tightest, with atoms touching along the line, are the close-packed directions. In a face-centered-cubic metal these are the <110> face diagonals, where the atoms sit shoulder to shoulder — the densest lines in the whole structure.
Those close-packed directions are exactly where a crystal prefers to shear. A dislocation glides most easily along a close-packed direction, so the slip system — the plane-and-direction pair on which a metal yields — is built from them: FCC metals slip along <110>, BCC metals along <111>. This is the straight line from an index we just learned to write, all the way to why one metal is soft and another hard — and, recalling the bonding rung, to why real crystals are ten to a hundred times weaker than a flawless lattice, because dislocations let slip run one row of atoms at a time instead of tearing the whole plane loose at once.
Two loose ends this rung will tie off. You will often want the angle between two directions; in a cubic crystal it is a plain dot product, cos theta = (u1 u2 + v1 v2 + w1 w2) / (sqrt(u1^2 + v1^2 + w1^2) times sqrt(u2^2 + v2^2 + w2^2)) — for instance [100] and [110] give cos theta = 1/sqrt(2), so 45 degrees — though, as warned, this tidy form is cubic-only, and guide 5 handles the general case. And any direction can be plotted as a single dot, a pole, on the stereographic projection from the symmetry rung, which is how a crystal's orientation gets recorded. Next up: we turn the very same indexing trick loose on planes.