Chapter 5: Lattice Planes

The foundation rests on Miller indices, a standardized notation that encodes plane orientation through crystallographic axes intercepts. The derivation process involves identifying where a plane intersects each axis, calculating reciprocals of these intercept values, and reducing the resulting fractions to the smallest whole numbers, thereby eliminating infinities and creating a compact symbolic representation. The notation distinguishes critically between individual planes and families of planes—symmetry-equivalent sets that possess identical spacing and diffraction properties. Hexagonal crystal systems present special complications because their inherent sixfold rotational symmetry cannot be fully captured by the conventional three-index system; the Miller-Bravais notation extends the description to four indices, introducing a dependent index that preserves mathematical consistency when rotating coordinate systems and enables meaningful comparison across equivalent planes. The chapter connects abstract lattice descriptions to observable crystal morphology by examining crystal forms, the distinct geometric shapes—including cubes, octahedrons, and more complex polyhedra—that arise when families of lattice planes bound the crystal's exterior. A systematic enumeration identifies forty-seven possible crystal forms across all crystal systems. The geometric principle of truncation explains how different families of planes intersect and combine to generate intricate polyhedral shapes. Historical development receives attention through the law of constancy of interfacial angles, an empirical principle demonstrating that the angles between crystal faces remain invariant despite variations in size, growth conditions, or physical damage, making this measurement method exceptionally reliable for material identification and classification before modern analytical techniques became available.