S. Scandolo, ICTP

This lesson continues the study of the **Bravais lattice **in three dimensions were in more details and starts with the **Body-centered ****cubic **lattice this is an example that can be described by both a **Bravais lattice **or a lattice with a base. Even if the **Bravais lattice **description is simpler from the mathematical point of view but it breaks the symmetry while in the lattice with a base description the primitive vectors are symple to visualize and conserve the symmetry of the problem. Then it was introduced the **Face****-centered ****cubic **(fcc) lattice and it was shown that together with the **hexagonal close-packed **(hcp) are an examples of efficient **close-packing **of equal spheres, that is a dense arrangement of congruent spheres in an infinite, regular arrangement.

In the second part of the lesson it was introduced the concept of **unit cell **a region of the space such that when translated by all the primitive vectors of the lattice it covers all the space with no overlaps. It was emphasized that the choice of the unit cell is not unique and is in general arbitrary. It was shown that any arbitrary point in the space can be written as a linear combination of a **Bravais lattice **vector and a vector inside the unit cell. Then it was introduced the **Wigner–Seitz unit cell**, the set of points for witch **R**=0 is the closest **Bravais lattice **point. Some examples were considered to illustrate how to construct the **Wigner–Seitz unit cell **the square lattice and the triangular lattice in this particular lattice the **Wigner–Seitz unit cell **is a hexagon. The **Wigner–Seitz unit cell **has the properties that it not depends on the choice of the primitive vectors and it preserves the symmetry of the **Bravais lattice **cell.