Zones, Planar Density, and Angles

Where the rung leaves off

Four guides in, you can now name any line and any plane inside a crystal. Guide 1 turned a direction into a set of whole numbers [uvw], and gathered symmetry-equivalent directions into a family <uvw>. Guide 2 read a plane off its intercepts and reciprocals to give the Miller indices (hkl), with families {hkl}. Guide 3 added the fourth hexagonal index so the three equal axes of a hexagon get treated fairly. Guide 4 measured the perpendicular gap between neighbouring planes, the interplanar spacing d_hkl, and tied it to the cell edges — the very quantity Bragg's law will need.

This last guide makes those indices earn their keep. Four jobs remain, and each is a small piece of geometry you can do by hand. First, which planes belong together because they share a common edge — that is a zone. Second, how tightly atoms sit along a chosen line or across a chosen plane — the linear and planar density, which quietly decide how a metal bends and how a mineral splits. Third, the exact angle between two directions or two planes. And fourth, how to fold a crystal's full three-dimensional orientation flat onto a single disc you can read at a glance.

Zones: planes that share an edge

Open a book and look at the spine. Every page is a different flat plane, yet they all pivot on the same line — the spine. Crystals do the same trick: a whole set of planes can all be parallel to one common direction, and that shared direction is the zone axis [uvw]. The planes that share it form a zone, and the zone axis is the crease they all bend around. A single plane belongs to many zones, and a single zone gathers many planes, so zones are the connective tissue that links the (hkl) world of planes to the [uvw] world of directions.

There is a one-line test for membership. A plane (hkl) belongs to the zone [uvw] exactly when h u + k v + l w = 0. This is the Weiss zone law, and it is nothing more than the statement that the plane is parallel to the axis (the plane's normal is perpendicular to the axis, so their dot product vanishes). Try it: does the plane (1 -1 0) lie in the zone [111]? Compute (1)(1) + (-1)(1) + (0)(1) = 1 - 1 + 0 = 0. Yes — so the direction [111] runs inside the (1 -1 0) plane. Change the plane to (11-1) and you get 1 + 1 - 1 = 1, not zero, so that plane tilts out of the zone.

1. Write the two planes as index triples (h1 k1 l1) and (h2 k2 l2); their zone axis is the line where the two planes intersect.
2. Cross-multiply: u = k1 l2 - k2 l1, v = l1 h2 - l2 h1, w = h1 k2 - h2 k1.
3. Divide out any common factor to reach the smallest whole numbers [uvw].
4. Sanity-check with the Weiss zone law: h u + k v + l w must equal 0 for both starting planes. Example: (100) and (010) give the axis [001] — two vertical walls meet along the vertical line.

Angles: the cubic dot-product shortcut

Ask how steeply two directions lean apart and, in a cubic crystal, the answer is the plain vector dot product. For [u1 v1 w1] and [u2 v2 w2], the angle between them obeys cos(theta) = (u1 u2 + v1 v2 + w1 w2) / (sqrt(u1^2 + v1^2 + w1^2) times sqrt(u2^2 + v2^2 + w2^2)). Feed it [100] and [110]: cos(theta) = 1 / (1 times sqrt(2)) = 0.707, so theta = 45 degrees. Feed it a cube edge [100] and a body diagonal [111]: cos(theta) = 1 / (1 times sqrt(3)) = 0.577, giving the famous 54.74 degrees that a diamond bond makes with the cube axis.

Angles between planes ride the same formula, but only because of a cubic coincidence. In a cubic crystal the normal to the plane (hkl) happens to point exactly along the direction [hkl], so the angle between two planes equals the angle between [h1 k1 l1] and [h2 k2 l2] — just drop the indices into the same cosine. That coincidence is a cubic-only gift.

How densely atoms sit: linear and planar density

Not every direction is equally crowded, and neither is every plane. The linear density of a direction is simply the number of atom centres you meet per unit length as you walk along it; the planar density of a plane is the number of atom centres per unit area lying in it. In a face-centered cubic metal the [110] face diagonal is a close-packed direction: the atoms actually touch, sitting a full diameter 2R apart, so its linear density is a maximum, 1 atom per 2R. Any other direction leaves gaps and packs fewer atoms per length.

The densest plane in that same metal is the {111} family, its close-packed plane. Lay one such layer down and each atom is ringed by six touching neighbours; stack the layers in the offset ABCABC rhythm and you build the face-centered cubic structure — filling 74 percent of space (packing factor 0.74) with every atom touching twelve others (coordination number 12). No arrangement of equal spheres does better. The (111) plane therefore carries more atoms per unit area than any other plane in the cell.

FCC (111) -- the densest plane in the cell

   one layer:    o   o   o      each atom touches 6 in-plane
                o   o   o        neighbours (2D coordination 6)
   stacking:    A B C A B C...   close-packed direction <110>,
                                 atoms touch, spacing = 2R

  linear density along [110] = 1 atom / 2R        = 1/(2R)
  planar density of (111)    = 2 / (sqrt(3)*d^2),  d = 2R
                             = 1/(2*sqrt(3)*R^2)  ~= 0.29 / R^2
  packing factor (FCC) = 0.74     coordination number = 12

This bookkeeping is not idle. When a metal deforms, its planes slide over one another, and slip prefers the densest plane sliding along its densest direction — that pairing is a slip system, and for FCC it is {111}<110>, twelve of them. The reason is intuitive: widely spaced, smooth close-packed planes are the easiest to shear past each other, like decks of cards. The same geometry runs in reverse for fracture — many brittle crystals cleave apart along particular low-index planes. And here is the honest twist: a real metal yields at 10 to 100 times less stress than a perfect crystal would need, because slip does not shift a whole plane at once. A line defect, a dislocation, glides through one row at a time (guide territory from the defects rung), so the theoretical shear strength is almost never reached.

Orientation on a flat map: the stereographic projection

One last problem: a crystal sits in space at some arbitrary tilt, and we want to record and compare those tilts without drawing clumsy 3D boxes. The trick is to represent each plane by a single point. Imagine the crystal at the centre of a globe; the normal of a plane (hkl) pierces the sphere at one spot called its pole. A whole crystal orientation becomes a sprinkle of poles on the globe — and to read it, we flatten the globe. The stereographic projection does this the way a cartographer flattens Earth: draw a line from each surface pole to the far pole of the sphere and mark where it crosses the equatorial disc.

The projection has one precious property: it preserves angles. Two poles that lie 54.74 degrees apart on the globe still measure 54.74 degrees on the flat map (read off with a ruled overlay called a Wulff net), so every angle from the cubic dot-product formula can be checked straight off the picture. That is why the stereographic map is the working language for orientation — matching one crystal's directions to a neighbour's across a boundary, describing how a new phase grows on an old one, or plotting the preferred orientations of a rolled sheet as a pole figure, the raw shorthand of texture.