Computational Physics

6.1 Introduction: definitions 123

x

y

z

a 1

a 1 a 1

a 2 a (^2) a 2
a 3
a 3 a 3
Figure 6.1. Lattice structure of the simple cubic (left), body-centred cubic (middle)
and face-centred cubic (left) lattices with basis vectors.
Before going into these methods, we start with a brief review of the theory of
electronic structure of solids. For further reading concerning the material in the first
three sections of this chapter, we refer to general books on solid state physics [ 1 , 2 ].
An excellent reference for computational band structures is the book by
R. M. Martin [3].

6.1 Introduction: definitions

6.1.1 Crystal lattices

We consider crystals in which the atomic nuclei are perfectly ordered in a periodic
lattice. Such a lattice, a so-calledBravaislattice, is defined by three basis vec-
tors. The lattice sitesRare given by the integer linear combinations of the basis
vectors:

R=

∑^3

i= 1

niai, niinteger. (6.1)

Each cell may contain one or more nuclei – in the latter case we speak of a lat-
tice with a basis. The periodicity implies that the arrangement of these nuclei
must be the same within each cell of the lattice. Of course, in reality solids will
only be approximately periodic: thermal vibrations and imperfections will destroy
perfect periodicity and moreover, periodicity is destroyed at the crystal surface.
Nevertheless, the infinite, perfectly periodic lattice is usually used for calculat-
ing electronic structure because periodicity facilitates calculations and a crystal
usually contains large regions in which the structure is periodic to an excellent
approximation.
Three common crystal structures, the simple cubic (sc), body-centred cubic (bcc)
and face-centred cubic (fcc) structures, are shown inFigure 6.1.