Crystal Structures

Summary

In this chapter first we have introduced basic geometrical concepts useful in describing periodic arrays of objects and crystal structures both in real and reciprocal spaces assuming that the atoms sit at lattice sites.

A lattice is an infinite array of points obtained from three primitive translation vectors a₁, a₂, a₃. Any point on the lattice is given by

$${\rm n} = n_1{\rm a}_1 + n_2{\rm a}_2 + n_3{\rm a}_3.$$

Any pair of lattice points can be connected by a vector of the form

$${\rm T}_{n_1 n_2 n_3 } =n_1{\rm a}_1 + n_2{\rm a}_2 + n_3{\rm a}_3.$$

Well defined crystal structure is an arrangement of atoms in a lattice such that the atomic arrangement looks absolutely identical when viewed from two different points that are separated by a lattice translation vector. Allowed types of Bravais lattices are discussed in terms of symmetry operations both in two and three dimensions. Because of the requirement of translational invariance under operations of the lattice translation, the rotations of 60°, 90°, 120°, 180°, and 360° are allowed.