S. Scandolo, ICTP

This lesson starts by recalling the concept of **crystals **introduced in the previous lesson, then it is introduced the concept of **Bravais lattice **an infinite array of discrete points generated by a set of discrete translation operations. Then it was given some examples in two dimensions like the **square lattice****, **the **rectangular lattice **and the **triangular lattice. **As an example of an structure that **is not **a **Bravais lattice **it is introduced the **Honeycomb lattice **because if a Bravais lattice vector belong to the lattice then any integer multiple of that vector must belong to the lattice and this property does not hold on the **Honeycomb lattice**. Thus a more general definition of **lattice **is needed to describe the **Honeycomb lattice **for this porpoise is introduced the concept of crystal structure or "lattice with a basis" is a **Bravais lattice **to witch are adding more points for each **Bravais lattice **point. This new points are given as a basis set on *n *vectors. Then it was shown how the **Honeycomb lattice **is describe by using a lattice with a basis. Finally some examples of the **Bravais lattice **in three dimensions were given like the **cubic**, the **tetragonal, **the **orthorhombic**, the **hexagonal **and the **triclinic **lattices