QUEEN MARY, UNIVERSITY OF LONDON
SCHOOL OF PHYSICS AND ASTRONOMY
Condensed Matter A
Week 2: Crystal structure (II)
References for crystal structure: Dove chapters 2–3; Sidebottom chapter 1.
Last week we learnt about some common lattice types, including simple
cubic, body-centred cubic, and face-centred cubic. This week we will list all
the possible types of lattice in three dimensions. In order to do this, though,
we first need to consider in more detail the possible symmetries of a crystal
structure.
Recall that a symmetry of some object is a geometric transformation that
leaves the object – for instance, a crystal structure – unchanged. We have
already seen that crystals have translational symmetry: specifically, that
translation by any lattice vector leaves the crystal structure unchanged. But
you know from everyday usage that there are many other types of symmetry.
In particular, we will now discuss point symmetry: symmetry operations
that, unlike translation, leave at least one point in space fixed.
One point symmetry operation is rotation about some axis. Note the
words we use, which are conventional: rotation is a symmetry operation; the
axis about which we rotate is a symmetry element. If repeating the rotation n
times will give a full revolution – or in other words, if the rotation is through
360◦ /n – we call this an n-fold rotation axis.
Another familiar operation is reflection in some mirror plane. Here
reflection is the operation; the mirror plane is the corresponding element.
An operation with which you may be less familiar is inversion (the
operation) through a point called a centre of symmetry (the element). If the
centre of symmetry is at the origin, inversion is the transformation that takes
a point at (x, y, z) to (−x, −y, −z).1 This is equivalent to reflection through
three perpendicular mirror planes.
Finally, we have the operation of improper rotation, consisting of rotation
about some axis followed by inversion.2 In fact, reflection is equivalent to
twofold improper rotation – rotation through 180◦ followed by inversion
– about an axis perpendicular to the mirror plane. Similarly, inversion is
trivially equivalent to onefold improper rotation, rotation through 360◦
which of course does nothing, followed by inversion. Thus we can describe
all point symmetry operations as proper or improper rotation.
As an example of some of these points, consider a set of atoms forming
a tetrahedron, as seen, for example, in the diamond structure:
1 More
generally, inversion through a centre of symmetry at (p, q, r) takes (x, y, z) to (2p −
x, 2q − y, 2r − z).
2 We can call rotation proper rotation if we want to explicitly distinguish it from improper
rotation.
1
Some of the point symmetry elements of this collection of atoms are illustrated in the figure. There are four threefold rotation axes, of which one is
shown in blue, and three mirror planes of which one is shown in red. There
is no proper fourfold rotation axis, but there are three improper fourfold
rotation axes, of which one is shown in green. There is no centre of inversion.
We will continue occasionally to note symmetry in the structures we study
this week.
Symmetry is important in many different parts of condensed matter
physics, some of which are well beyond the scope of this module. For our
purposes there are two main consequences of the symmetry of a structure.
The first is that point symmetry operations are closely related to the lattice
underlying a given crystal structure.
Consider the graphene structure once more. By inspection we concluded
that the lattice parameters were related by a = b and γ = 120◦ . Looking for
symmetry elements, though, we see that there are clear sixfold rotation axes,
shown by hexagons in the diagram below:
A little consideration of possible lattice vectors a and b will show that only
these lattice parameters are compatible with this symmetry.3
More generally, and also in three dimensions, the particular symmetries
of a crystal structure limit the possibilities for the lattice parameters. Taking
these possibilities into account, there are only seven different lattice systems
containing fourteen Bravais lattices, which are listed in the table below:
simple
base-centred
body-centred
face-centred
Triclinic
Monoclinic
α = γ = 90◦
one twofold axis
Orthorhombic
α = β = γ = 90◦
three twofold axes
Rhombohedral
a = b = c, α = β = γ
one threefold axis
Tetragonal
a = b, α = β = γ = 90◦
one fourfold axis
Hexagonal
a = b, α = β = 90◦ , γ = 120◦
one threefold or sixfold axis
Cubic
a = b = c, α = β = γ = 90◦
four threefold axes
In particular the three different cubic lattices and the hexagonal lattice are
3 There
are also twofold and threefold rotation axes in the graphene structure; putting these
in the correct spots is left as an exercise!
2
familiar from the crystal structures we have already studied. This diagram
shows a further reason why it is useful to consider centred cells: as well as
making calculations easier, the conventional cells also make the symmetry
requirements of the lattice clearer. It is much easier, for instance, to see
the symmetry of the face-centred cubic conventional unit cell than the
corresponding primitive unit cell with α = β = γ = 60◦ .
The second reason why symmetry is so important is that it has a direct
impact on the physical properties of a material. For instance, in many
materials electric polarisation is important. In ferroelectrics, the polarisation
can be switched back and forth by applying an external electric field, much
like the magnetisation in ferromagnets; this property finds applications, for
instance, in computer memory (“FeRAM”). However, certain symmetry
elements destroy this property. For instance, a centre of symmetry will mean
that any dipole moment is exactly cancelled out by an identical one pointing
in the opposite direction; thus centrosymmetric materials – those with a
centre of symmetry – can never have a permanent electric dipole moment.
Deeper analysis of a material’s symmetry can similarly rule out or constrain
many of its physical properties.
Having established the different lattice systems, we will continue our
study of common crystal structures. Last week we considered elemental
structures; this week we will look at materials containing two or three
elements. All of the common structures we will learn about are based
on cubic lattices. We have already seen materials in which the dominant
interactions were covalent or metallic. Diatomic structures will also include
materials in which the major bonding type is ionic. Thus we are building up
a repertoire of different types of material to discuss the energies associated
with crystal packing next week. Similarly, we have examples of metals,
insulators, and semiconductors for when we discuss electronic properties
later in the module.
The simplest diatomic crystal structure is the sodium chloride structure,
also called halite or rock salt (both of which are names for sodium chloride
as a naturally occurring mineral). This is very common in simple ionic
materials, and simply consists of a checkerboard pattern of the two different
ions:
1
2
0
1
2
1
2
0
1
2
1
2
0
1
2
0
0
1
2
0
1
2
0
1
2
0
0
It is important to note that, although the arrangement of atoms looks similar
to that in a primitive cubic elemental structure, in fact closer inspection will
show that this is based on a face-centred cubic lattice with a motif consisting
of one type of atom at (0, 0, 0) and the other at ( 12 , 0, 0). The coordination
number of both cations (positively charged ions) and anions (negatively
charged ions) is 6.
3
Note also that, although the primitive cubic elemental structure is unstable with respect to shearing, the sodium chloride structure is not, because in
this case shearing would bring pairs of ions with the same charge together,
which is unfavourable by Coulomb’s law:
Simple cubic
Rock salt
Another common diatomic structure is the caesium chloride structure,
common in ionic materials with bigger cations:
0
0
1
2
0
0
Again, although this looks similar to the body-centred cubic structure, it is
based on a simple cubic lattice with a motif of one type of atom at (0, 0, 0)
and the other at ( 12 , 12 , 21 ). This time the coordination number for both cations
and anions is 8.
Just as we saw for the elements, the coordination number of diatomic
materials tends to be lower where the bonding is more covalent than when
the bonding is more ionic. An example of such a structure is the cubic
zinc sulfide structure, also known as the zinc blende or sphalerite structure.
(Again, both of these are names for zinc sulfide as a mineral. The “cubic” is
important because there is also a form of zinc sulfide that has a hexagonal
lattice.)
1
2
0
3
4
1
2
1
4
1
2
0
1
4
0
0
3
4
1
2
0
This is also based on the face-centred cubic lattice, but with a motif of one
type of atom at (0, 0, 0) and the other at ( 14 , 41 , 41 ). In fact, this is identical
to the diamond structure, except that the two motif atoms are of different
types. Like the atoms in the diamond structure, both cations and anions have
a coordination number of four.
Closely related to this is the calcium fluoride or fluorite structure, which
simply has twice the number of anions, so that while the anions remain
4
tetrahedrally coordinated, the cations now have a coordination number of
eight:
1
2
0
0
1 3
4, 4
1
2
1 3
4, 4
1
2
0
1 3
4, 4
0
1 3
4, 4
1
2
0
Again, this has a face-centred cubic lattice, with a motif of a cation at (0, 0, 0)
and two anions at ± 14 , ± 14 , ± 41 . One important difference between the fluorite and zinc sulfide structures is that the fluorite structure is centrosymmetric
whereas the zinc sulfide structure is not.
Finally, the perovskite structure is the only triatomic structure that we
will consider as part of this module. It has general formula ABX3 , where
A and B are usually metal ions and X oxide. However, part of its interest
comes from the fact that there is substantial compositional flexibility, and
materials of this general structure can also be made with fluoride or other
linker anions, and with many different mono- and polyatomic cations on the
A and B sites.
The B site, shown in black below, is octahedrally coordinated to six
X atoms (the coordination octahedron is itself shown for clarity in the 3D
picture below); the A site is surrounded by 12 X atoms:
1
2
0
0
1
2
1
2
1
2
0
1
2
0
0
An alternative unit cell in which the B atoms are at the corners, with the A
atom at the centre, is also a common representation of this structure.
If, as previously, we model the ions as rigid spheres, in a perfectly
fitting structure the B and X ions would just touch along an edge, in the
[100] direction, so that 2rB + 2rX = a. On the other hand, the A and X
ions would just touch along a face diagonal, in the [110] direction, so that
√
2rA +2rX = 2a. In general, given arbitrary A, B, and X, it is not possible to
find an a that solves both of this equations. Instead, the perovskite structure
is able to distort to accommodate mismatch in the size of the component
ions. If the A ion is too small, the BX6 octahedra will rotate so that the
cavity in which the A ion sits is smaller:
5
This, for instance, happens in strontium titanate, SrTiO3 .
On the other hand, if the B ion is too small, it will tend to be displaced
from the centre of the BX6 octahedra:
(Note that in both of these figures the scale of the distortion has been
exaggerated for clarity.)
The perovskite structure is centrosymmetric.4 However, displacement
of the B ion as shown above breaks the symmetry so that there are no longer
any centres of symmetry in the material, allowing a net polarisation. Lead
titanate, PbTiO3 , is a material with this structure used, as mentioned above,
in ferroelectric RAM.
Other well-known perovskites include lithium niobate, used in the
telecommunications and photonics industries; magnesium silicate, found in
the Earth’s lower mantle, and methylammonium lead iodide, a promising
solar cell material.
Next week: What forces between atoms give rise to these different
crystal structure types? How can we calculate the energy associated
with crystal packing?
4 Can
you find some centres of symmetry? There are several.
6