[web] [lect]

The number of lattices that can fill two- or three-dimensional space with periodically repeating units
without leaving gaps or causing overlaps is limited. Therefore, there is a finite number of different
crystal structures, and different crystalline solids may crystallise according to the same pattern.
The
*metrics*

of the lattice may be different, but the
*symmetry*

is the same in such
cases. Lattices which fill space without gaps are called
*Bravais lattices*.

There are five
of them in two dimensions and 14 in three dimensions.

The most general and least symmetric Bravais lattice in two dimensions is the
*oblique*

lattice.
If the angle between the two lattice vectors is 90°, the higher symmetry of the cell gives rise to
a distinct Bravais lattice, either rectangular or
*square*

depending on whether the unit
cell vectors have different length or not. In the case of a rectangular lattice, we can distinguish
between a
*primitive rectangular*

lattice and a
*centred rectangular*

lattice, which has
an extra lattice point (atom) at the centre. The centred rectangular lattice could be set up as a
primitive lattice with lower symmetry (unit cell shown in green), but convention prefers the more
symmetric description. Finally, if the lattice vectors are the same length and
the angle is 120°, we have another special case with higher symmetry, the
*hexagonal*

lattice.

Clearly, the difference between the Bravais lattices boils down to symmetry. If the lattice vectors are
at right angles, the unit cell can be folded over along lines intersecting either lattice vector at its
centre: there are
*mirror lines*

crossing each pair of edges of the unit cell in the middle, and the two
mirror lines intersect at the centre of the cell. This explains the special position of the central point
in the centred rectangular lattice - it is located at the centre of symmetry of the cell and always maps
onto itself under any symmetry operation.

**Exercise.**

What symmetry elements can you identify in the rose window in Lugo Cathedral (Galicia)? Ignore the colours of the individual pieces of glass.
How about the rugby pitch?

Click images for solution.

In two dimensions, the effect of a mirror line and a rotation by 180° is the same, so the rectangular
lattices both have a two-fold
*rotation axis*

at their centre. In the case of the square lattice, it is even
a four-fold axis - turning the cell by 90° maps it onto itself because both lattice vectors are the same
length. This is indicated by a green diamond shape in the diagram. There is also a four-fold axis in each
corner of the cell: a 90° rotation around any of the lattice points again maps the lattice (though not
the specific unit cell) onto itself. The hexagonal lattice has a three-fold symmetry by rotating in 120°
steps around a lattice point, indicated by a triangular marker.

Wikimedia Commons images © Daniel Mayer/Stannered under Creative Commons Licence.

The same symmetry principles apply in three dimensions. The concept of a centred lattice expands into
three distinct cases, depending on whether the additional point is at the centre of the unit cell
(*body-centred*),

of one face and, because of the translational periodicity, its opposite
(*side-centred*)

or on all its faces
(*face-centred*).

By convention, the lattice vectors are named a, b and
c and the angles are given the Greek letter corresponding to the lattice vector
that is *not* spanning the angle, *i.e.* the angle between a and
c is *β*.

The equivalent of the two-dimensional oblique lattice in three dimensions is the triclinic Bravais lattice. All angles are irregular and the three lattice vectors have different lengths. More symmetric lattices arise when some or all angles are 90° or 120° or when two or all three lattice vectors have the same length.

Among the lattices with exclusively right angles are the
*orthorhombic*,

*tetragonal*

and
*cubic*

lattices depending on whether there are three, two or just one distinct lattice vectors in terms of
their length. If only two angles are 90°, the cell is monoclinic, resulting in four rectangular and
two skewed faces to the unit cell. If none of the angles is a right angle, the cell is
*trigonal*

if all lattice vectors are the same length but
*triclinic*

if they are different. If only one angle is skewed, the resulting cell is called
*monoclinic*.

Finally, the
*hexagonal*

lattice has one angle at 120° and two at 90°.

The
*gaps in the table*
arise because the missing lattices can be expressed in terms of one of the others
by choosing a different unit cell. For example, a side-centred cubic lattice is the same as a
primitive tetragonal one with a smaller unit cell based on four of the corner atoms and four side-centred
atoms of the cubic cell.

Next (in the *Concepts* lecture), we'll discuss some common crystal
structure types
found in materials
and minerals, and how they are structurally related to one another.

For a more in-depth view of crystal structures (in the *Structure Determination* lecture) we'll
see how to deal with crystals containing more than one type of atom by introducing the wider concept
of symmetry groups.