II. Crystal Structure A.Lattice, Basis, and the Unit Cell B.Common Crystal Structures C.The Reciprocal Lattice

A. Lattice, Basis, and Unit Cell An ideal crystalline solid is an infinite repetition of identical structural units in space. The repeated unit may be a single atom or a group of atoms. An important concept: crystal structure = lattice + basis =+

The translational symmetry of a lattice is given by the base vectors or lattice vectors. Usually these vectors are chosen either: 1. to be the shortest possible vectors, or 2. to correspond to a high symmetry unit cell lattice: a periodic array of points in space. The environment surrounding each lattice point is identical. basis: the atom or group of atoms attached to each lattice point in order generate the crystal structure.

Example: a 2-D lattice These two choices of lattice vectors illustrate two types of unit cells: Conventional (crystallographic) unit cell: larger than primitive cell; chosen to display high symmetry unit cell Primitive unit cell: has minimum volume and contains only one lattice point

B. Common Crystal Structures 2-Donly 5 distinct point lattices that can fill all space 3-Donly 14 distinct point lattices (Bravais lattices) The 14 Bravais lattices can be subdivided into 7 different crystal classes, based on our choice of conventional unit cells (see text, handout). Attaching a basis of atoms to each lattice point introduces new types of symmetry (reflection, rotation, inversion, etc.) based on the arrangement of the basis atoms. When each of these point groups is combined with the 14 possible Bravais lattices, there are a total of 230 different possible space groups in 3-D. We will focus on the few that are common for metals, semiconductors, and simple compounds.

Analysis of Common Crystal Structures 1. NaCl structure (many ionic solids) 2. CsCl structure (some ionic solids and intermetallic alloys) lattice: face-centered cubic (fcc) basis: Na at 000, Cl at ½½½ lattice: simple cubic (sc) basis: Cs at 000, Cl at ½½½

Common Crystal Structures, contd 3. hexagonal-close-packed (divalent metals) 4. diamond structure (C, Si, Ge) lattice: hexagonal basis: 000, 2/3 1/3 1/2 lattice: face-centered-cubic (fcc) basis: 000, ¼¼¼ (see text for an alternate choice of lattice and basis) 5. zincblende structure (ZnS, GaAs, InP, compound semiconds) lattice: face-centered-cubic (fcc) basis: Zn at 000, S at ¼¼¼