Definitions

A crystal or crystalline solid is an ordered array of atoms, molecules, or ions whose pattern or lattice is repeated periodically. A “single crystal” is perfectly ordered. Most crystalline solids are polycrystalline, meaning that they are composed of many small single crystals with defective, bounding surfaces between them. Large single crystals occur in nature, but often it is necessary to prepare them by special crystal-growth methods in laboratories. Single crystals are the preferred form for studying the intrinsic properties of a crystalline material.

Theoretical analysis of single-crystal properties usually assumes the crystalline structure is infinite or imposes periodic boundary conditions requiring the wave-function to repeat itself after a sufficiently large number of chemical units. This is a reasonable approximation, since a cubic-centimeter crystal contains on the order of 1022 to 1024 repeated units.

To sharpen our description of a lattice some definitions are useful.

1. • Bravais lattice. A Bravais lattice is a space-filling array of points generated by three primitive lattice vectors, a, b, and c. The vectors of the infinite set of vectors {R(l, m, n)} terminate on the Bravais lattice points. These vectors are R(l, m, n) = la + mb + nc, where l, m, and n are positive or negative integers or zero. Instead of the vectors a, b, and c, a Bravais lattice can be specified by the magnitudes of the primitive vectors, a = |a|, b = |b|,and c =|c|, and the angles between them, α, β, and γ.