Law of Rational Indices

In crystallography, crystal faces exist at specific orientations that follow simple geometric relationships.

To describe these relationships, imagine three crystallographic axes—OX, OY, and OZ —intersecting at a common origin. These axes act as reference directions in the crystal. A crystal face may cut these axes at certain distances from the origin, called intercepts.

Consider a plane PQR that intersects the axes at OP, OQ, and OR, which represent the unit intercepts a, b, and c.

Now observe another plane, KLM, intersecting the same axes. Here, the intercepts appear as simple whole-number multiples of the unit intercepts.

For example, the plane may intersect the axes at a, 2b, 3c, but not at arbitrary distances.

This pattern represents the Law of Rational Indices, which states that a crystal face intersects the crystallographic axes at distances equal to simple whole-number multiples of the unit intercepts.

This law helps relate crystal structure to measurable properties such as diffraction patterns, surface energies, and catalytic behavior.