the Weiss zone law
/ Weiss: VYSS /
The Weiss zone law is a single, checkable equation that answers a common question: does this direction lie flat inside this plane? Given a direction [uvw] and a plane (hkl), the law says the direction lies in the plane exactly when hu + kv + lw = 0. It is a wonderfully quick test — multiply the matching indices together, add up the three products, and if you get zero the answer is yes.
The reason it works is that lying in a plane means being at right angles to the plane's normal, and in the language of the reciprocal lattice the numbers (hkl) point along that normal. So hu + kv + lw is a dot product, and a zero dot product means perpendicular, which means the direction lies in the plane. A quick check: does [1 -1 0] lie in (111)? Compute 1(1) + 1(-1) + 1(0) = 0 — yes. Does [100] lie in (111)? Compute 1 + 0 + 0 = 1, not zero — no.
You use the law in three everyday ways: to test whether a direction lies in a plane, to find all the planes belonging to a given zone, and (via the cross product of two planes) to find their zone axis. Its most remarkable feature is that it holds in every crystal system, even triclinic ones with no right angles anywhere. It is a pure incidence relation, blind to the cell's shape — unlike the spacing and angle formulas, which must be rewritten for each system.
Which planes contain the [001] axis? The law hu + kv + lw = 0 becomes 0h + 0k + 1l = 0, so l = 0. Every plane of the form (hk0) — like (100), (010), (110), (120) — belongs to the [001] zone, and no others do.
Multiply matching indices, add, and check for zero — a one-line test valid in every crystal system.
The law hu + kv + lw = 0 is a metric-free incidence relation, so it holds in all seven crystal systems. That is unusual — the d-spacing and interplanar-angle formulas both depend on the cell shape and must be rewritten per system.