Set of supplementary notes.

42 CHAPTER 3. FROM ATOMS TO SOLIDS
3.3 The description of periodic solids
An ideal crystal is constructed from the infinite repetitition of identical structural units in
space. The repeating structure is called the lattice, and the group of atoms which is repeated is
called the basis. The basis may be as simple as a single atom, or as complicated as a polymer
or protein molecule. This section discusses briefly some important definitions and concepts.
For a more complete description with examples, see any of the textbooks recommended in the
introduction.
Lattice. The lattice is defined by three fundamental (called primitive ) translation vectors
a i , i = 1, 2, 3, which define primitive lattice translation operations. An arbitrary lattice
translation operation can be written as
T = ∑ i
n i a i (3.8)
The atomic arrangement looks the same from equivalent points in the unit cell:
r ′ = r + ∑ i
n i a i ∀ integer n i . (3.9)
The lattice so formed is called a Bravais lattice.
Primitive unit cell. A unit cell is a part of the crystal, which – when translated repeatedly
by the primitive translation vectors – can fill the entire volume of the crystal. However,
there may be overlapping regions. For example, a face centred cubic lattice (fcc) can have a
simple cubic unit cell. Because this is the simplest one to draw and to work with, it is called
the conventional unit cell. It contains more than one lattice site (four, in fact). When this
cell is translated by one of the primitive translation vectors (e.g. a(1/2, 1/2, 0)), there will
be overlaps. A primitive unit cell, however, would exactly fill the volume of the crystal,
when repeatedly translated. It contains exactly one lattice point. One way of constructing a
primitive unit cell, is by forming a parallelepiped from a set of primitive translation vectors a i .
Wigner-Seitz cell. A convenient alternative primitive unit cell to use is the Wigner-Seitz
cell. This is the region in space around a lattice point, which is closer to this lattice point than
to any other lattice point. It can be constructed as follows: Draw lines to connect a given lattice
point to all of its near neighbours. Then draw planes normal to each of these lines from the
midpoints of the lines. The smallest volume enclosed in this way is the Wigner-Seitz primitive
unit cell. You can show that the Wigner-Seitz cell has the same symmetry properties as the
lattice itself, which is not true of all the choices for primitive unit cell. The first Brillouin zone
is the Wigner-Seitz cell in reciprocal space.
Point group. The are other symmetry operations than just translations that can be
performed on a lattice, for example rotations and reflections. The collection of symmetry
operations which, when applied about a lattice point, map the lattice onto itself is the lattice
point group. This includes reflections and rotations; for example a 2D square lattice is invariant
under reflections about the x and y axes, as well as through axes at an angle of π/4 to the x
and y axes, and rotations through any multiple of π/2. Remember that adding a basis to a
primitive lattice may destroy some of the point group symmetry operations.
Space group. The translational symmetry and the point group symmetries are subgroups
of the full symmetry of the lattice which is the space group. Every operation in the space group