22
Symmetry of Bravais Lattices
From the definitions given earlier it is clear that Bravais lattices
are symmetric under all translations through their lattice
vectors. However, these transformations are only a subset of
the set of rigid operations that take the lattice into itself.
The full set of these operations is called the symmetry group
or space group of a Bravais lattice. The space group often
includes rotations, reflections and inversions. Notice below
how after each operation the infinite lattice appears the same
as before.
21
34
32
41
2
3
1
4
Rotation Reflection
21
34
Inversion
Any symmetry operation of a Bravais lattice can be broken down
into a translation through a lattice vector combined with a rigid
operation that leaves at least one lattice point fixed (Proof on
next slide).
Take, for example, the rotation of the
lattice through 90o.
This can be broken down into a
translation through the lattice vector
and a rotation about point 1.
21
34
32
41
21
34
The rotation about the point 1 above is a member of a subset
of the space group, called the point group. The point group of
a lattice is the set of symmetry operations that hold one point
of the lattice in place while moving each remaining point to the
position of another point in the lattice.
a
Symmetry of Bravais Lattices
“Any symmetry operation of a Bravais lattice can be broken down
into a translation through a lattice vector combined with a rigid
operation that leaves at least one lattice point fixed.”
Proof
Consider a symmetry operation S that leaves no lattice point fixed.
Suppose this operation takes the origin, O, into another point P by a
vector R.
R must be one of the Bravais vectors because of the very notion of
symmetric operations: After S is applied the lattice falls into the same
lattice.
Now consider an operation, which consists of first applying S and then
applying a translation through –R called T-R
.
Combining the two operators provides a new symmetry operation, ST-R
which holds the origin in place.
Combining ST-R
with TRgives simply the operation S.
So the operation S can be broken down into ST-R
and TR. ST
-Rbeing a
point group operation and TR
being a translation.
Symmetry of Bravais Lattices
The full symmetry group of a Bravais lattice contains only
operations of the form;
1. Translations through Bravais lattice
vectors;
2. Operations that leave a particular point
of the lattice fixed (operations of this
type are in the point group subset);
3. Operations that can be constructed by
successive applications of the
operations of types (1) and (2);
Symmetry of Bravais Lattices
Point Group Subset
RECALL:
The point group of a lattice is the set of symmetry
operations that hold at least one point of the lattice in place
while moving the remaining points to the positions of other
points in the lattice.
There are seven distinct point groups that a Bravais lattice
can have. Point groups are considered identical if they
contain exactly the same symmetry operations.
The Cubic and Octahedral
structures have the same point
group since they both contain
the same symmetries
Seven Crystal Systems
Crystal structures can be categorised into one of the seven
crystal systems based on the symmetries of its underlying
Bravais lattice. Each crystal system corresponds to a different
point group.
Cubic Tetragonal Orthorhombic Monoclinic
Triclinic
Rombohedral
(Trigonal)
Hexagonal
a
a
a
a
a
c
ab
c
a
bc
α = 90°
Table 1
The Seven Crystal
systems, based on
the point group of
the underlying
Bravais lattice.ab
c
a
aa
aa
a
c
Point Group Operations
1. Rotations through Integral Multiples of 2π/n about some Axis
(n- fold rotation axis).
2. Rotations-Reflections A rotation through 2π/n that is not a
symmetry element can sometimes be made symmetric by
following the rotation with a reflection in a plane perpendicular to
the axis. (n- fold rotation-reflection axis).
3. Rotation-Inversions A rotation through 2π/n that is not a
symmetry element can sometimes be made symmetric by
following the rotation with an inversion in a point lying on the
rotation axis. (n- fold rotation-inversion axis).
4. Reflections Takes every point into its mirror image in a plane
(mirror plane).
5. Inversions A single point remains fixed. If that point is taken as
the origin, then every other point r is taken into –r.
a
a
c
Point Operation Examples
Tetragonal
• 4-fold rotation axis (vertical).
• 2-fold rotation axis perpendicular
to the 4-fold axis.
• Three mirror planes perpendicular
to the axes.
High-symmetry case
Point Operation Examples
ab
c
Triclinic
Low-symmetry case
• 2-fold rotation-reflection axis
The Triclinic crystal system has
the lowest symmetry of all seven
systems
Cubic Tetragonal
Monoclinic Triclinic
Rombohedral
(Trigonal)
Hexagonal
Bravais Space Groups
When considering the full symmetry groups, not just the
point groups, of a Bravais lattice it is discovered that there
are fourteen distinct space groups a lattice can have, and
hence fourteen types of Bravais lattice.
Orthorhombic
Crystal Structure = Bravais Lattice + Basis
Usually, in nature, a basis will contain atoms of several
kinds
Example 1:
Sodium Chloride (NaCl)
NaCl structure consists of a
FCC lattice with a two point
basis.
zyxa
atClatNa ˆˆˆ2
,0
Example 2:
Cesium Chloride (CsCl)
The structure of CsCl consists
of two interlocking simple
cubic lattices of both
elements.
So it is an SC lattice with a
two point basis.
zyxa
atClatCs ˆˆˆ2
,0
Crystal Structure = Bravais Lattice + Basis
A multi-atom basis does not have to have atoms of different
kinds. They could be of the same type.
Example:
Diamond Structure
The diamond structure is
made up of a face centred
cubic lattice along with a two
point basis.
The basis consists of two
atoms of the same type.
a
zyxa
ˆˆˆ4
xy
z zyxa
atCandatC ˆˆˆ4
0
Crystal Structure = Bravais Lattice + Basis
Some Bravais lattices themselves can be broken down
further into a basis and a simpler Bravais lattice.
Example:
Face Centred Cubic
The FCC lattice can be
considered a simple cubic
lattice with a four point
basis.
Therefore, the Diamond
structure can be considered
a simple cubic lattice with
an eight point basis.
a1
a2
a3
Crystal Structure = Bravais Lattice + Basis
Diamonds VS Graphite (Coal)
Diamond and graphite are two allotropes of carbon: pure forms
of the same element that differ in structure. Their difference
structures lead to vastly different properties.
Diamond Graphite
Diamond Graphite
Diamonds VS Graphite (Coal)
Electrical insulator
Colour: Transparent to
opaque
Hardest naturally
occurring material
Electrical conductor
Colour: Steel black, to gray
One of the softest known
materials.
Hexagonal Lattices
It can be viewed as two interpenetrating
simple hexagonal Bravais lattices
displaced vertically by a distance c/2
and also horizontally so that the points
of one lie directly above the centres of
the triangles formed by the points of the
other.
Ideal ratio of c/a: 3
8
a
c
Hexagonal Close Packed Crystal Structure (Non-Bravais)
Simple Hexagonal Lattice (Bravais Lattice)
Two-dimensional triangular nets are
stacked directly above each other
Primitive Vectors:
za
yxa
xa
c
a
a
3
2
1
)2/3(2/
x
y
z
a1
= a
a2
= b
a3
= c
Stacking
Consider the problem of stacking
cannonballs. The first layer is fine and
consists of a hexagonal pattern of spheres.
However, when you reach the third layer there is a choice to make.
You can:
1. Place the spheres directly above those in the first layer. (ABA)
With the second layer the spheres are placed
above alternative interstices in the first.
2. Place the sphere above the interstices in the first that were not
covered by spheres in the second. (ABC)
a1a2
a3
The two options for packing spheres result in the hexagonal close
packed lattice or the face-centred cubic lattice respectively.
Hexagonal Close Packed
(ABABABAB…)
Face Centred Cubic
(ABCABCABC…)
Stacking
Exercise
FCC lattice is the most dense of the three Bravais cubic lattices
with Simple cubic being the least dense.
One measure of this is the coordination number.
Diamond structure has a coordination number of 4 and so is
even less dense than SC.
Suppose the lattice points are identical solid spheres of unit
density, show that the density (or “packing fraction”) of the four
structures are:
FCC: 0.74 =
BCC: 0.68 =
SC: 0.52 =
Diamond: 0.34 =
6
2
8
3
6
16
3
Problems/Questions?
What are the seven crystal systems?
What are the possible Point group operations.
Can you visualise them?
I would urge you to know the answers to these questions before
next time.
Good resources
Solid State Physics ~ Ashcroft, Ch. 7
The Physics and Chemistry of Solids ~ Elliott, Ch. 2
