Crystal Systems and Bravais Lattices
Now that we have considered symmetry in 2D we can apply the same
concepts to 3D crystals. The concepts are the same, but the possible
combinations are greater and the visualization can also be more difficult.
Let’s begin by identifying the possible combinations of crystal systems
(primitive lattices) and Bravais lattices.
In 3D there are 7 crystal systems:
 Triclinic
 Monoclinic
 Orthorhombic
 Trigonal (Rhombohedral)
 Hexagonal
 Tetragonal
 Cubic
In 2D centering was very simple, since a parallelogram has only one face. In
3D we have more options, leading to six different centering arrangements.
Centering
Symbol
Face Centered
F
Body Centered
Base Centered
I
C
Centering
Vector(s)
½a + ½b
½a + ½c
½b + ½c
½a + ½b + ½c
½a + ½b
B
½a + ½c
A
½b + ½c
Lattice Points
per Unit Cell
4
2
2
Note that there are three different ways to center in only one set of
opposing faces of the unit cell (base centered), depending upon whether the
opposing faces lie in the ab plane (C-centered), the ac plane (B-centered) or
the bc plane (A-centered).
Combining the crystal systems (primitive translational symmetry) and the
centering translations, produces 14 unique Bravais Lattices.
Crystal System
Triclinic
Bravais Lattice
Primitive (P)
Monoclinic
Rhombohedral
Primitive (P)
Base-Centered (C)
Primitive (P)
Base-Centered (C)
Body-Centered (I)
Face-Centered (F)
Primitive (P)
Trigonal
R-Centered (R)
Hexagonal
Primitive (P)
Tetragonal
Primitive (P)
Body-Centered (I)
Primitive (P)
Body-Centered (I)
Face-Centered (F)
Orthorhombic
Cubic
Unit Cell Dimensions
abc
  
abc
= =90 
abc
= ==90
Required Symmetry Element
None
a=b=c
= =90
One 3-fold axis
a=bc
= =90, =120
a=bc
= =90 =120
a=bc
= ==90
a=b=c
= ==90
Either a mirror (glide) plane
or a 2-fold axis
Any combination of three
mutually  2-fold axes or
mirror (glide) planes
One 6-fold axis
One 4-fold axis
Four 3-fold axes
Glide Planes and Screw Axes
Before going any further we need to expand upon our earlier treatment of
glide planes and introduce screw axes. Both are composite symmetry
operations that involve first a point symmetry operation (reflection for a
glide plane, and rotation for a screw axis) followed by a translation. The
details of each are given below:
Glide Reflection (SE = Glide Plane)
As detailed in our earlier discussion of 2D symmetry a glide reflection
consists of reflection through a mirror plane followed by a displacement
parallel to the mirror plane. In 2D there is no ambiguity in the direction
which is parallel to the mirror plane. However, in 3D we have several choices
of translation vectors parallel to the glide plane. This leads to a greater
variety of glide planes, each of which has a unique symbol. The various glide
planes, their symbols and the displacement vector associated with each are
given in the table below.
Herman-Mauguin
Symbol
a
b
c
n*
d**
Axis  to the
Glide Plane
b or c
a or c
a or b
a
b
c
a
b
c
Displacement
Vector
a/2
b/2
c/2
b/2+c/2
a/2+c/2
a/2+b/2
b/4+c/4
a/4+c/4
a/4+b/4
*
In rare cases the diagonal glide (n-glide) can also have a displacement
vector of a/2+b/2+c/2.
**
Diamond glides (d-glide) can only occur in F and I centered lattices. As can
the diagonal glide they may also take the form a/4+b/4+c/4 in rare cases.
Screw Rotation (SE = Screw Axis)
A NM screw rotation operation consists of a 360/N rotation followed by a
displacement of M/N the unit cell dimension parallel to the axis.
For example a 31 screw axis parallel to the c-axis represents a 120 rotation
followed by a displacement of 1/3c.
Point Groups and Space Groups
Those of you who have a background in group theory and symmetry
determination of molecules, will be familiar with determining the point group
of a molecule (see Chapter 3 of Cotton’s book “Chemical Applications of
Group Theory” for a review). For example, water has C2v symmetry and NH3
has C3v symmetry. In both of these cases I’ve used the Schoenflies
symbolism for describing the point group of the molecule.
Point group symmetry is also important in crystals. The point group
symmetry describes the non-translational symmetry of the crystal. In other
words, the point group symmetry describes the symmetry of the primitive
unit cell. The rest of the crystal is then generated by translational
symmetry (unit cell translations + centering translations).
I’m not going to go into the details of deriving point groups from inspection
of molecules or crystals. This is done quite well in many other places. What
I will do is say a few words about Herman-Mauguin point groups, how they
are used in crystallography and how they relate to Schoenflies point groups.
First of all since we’ve shown that only 2, 3, 4 and 6-fold rotation axes are
compatible with translational symmetry, we don’t need to worry about point
groups containing other types of rotation axes (i.e. D5d). Furthermore, there
is no such thing as a linear 3D crystal, so we can throw out the linear point
groups as well. This leaves us with 32 crystallographic point groups. They
are shown in the table below:
Crystal System
Triclinic
Monoclinic
Orthorhombic
Trigonal
Number of Point
Groups
2
3
3
5
Hexagonal
7
Tetragonal
7
Cubic
5
Herman-Mauguin
Point Group
1,1
2, m, 2/m
222, mm2, mmm
3,3, 32,
3m,3m
6,6, 6/m, 622,
6mm,62m, 6mm
4,4, 4/m, 422,
4mm,42m, 4/mmm
23, m3, 432,
432, m3m
Schoenflies
Point Group
C1, Ci
C2, Cs, C2h
D2, C2v, D2h
C3, S6, D3,
C3v, D3d
C6, C3h, C4h, D6,
C6v, D3h, D6h
C4, S4, C4h, D4,
C4v, D2d, D4h
T, Th, O,
Td, Oh
Note that because each crystal class has certain symmetry elements
associated with it (i.e. the presence of a 6-fold axis defines the crystal
system as hexagonal), we can group the point groups according to their
crystal system. We will learn later about how the order of the HM symbols
relates to the symmetry elements present in the structure.
You may notice that of the symmetry elements discussed (mirror planes,
rotation axes, etc.) both glide planes and screw axes are absent from the
list of point group symbols. That because for the purposes of determining
the crystallographic point group glide planes are treated as mirror planes,
and screw axes as rotation axes. What I mean by this will be made more
clear in the examples below.
If we now combine the 32 crystallographic point groups (point symmetry)
with the 14 Bravais lattices, we obtain 230 space groups in 3D.
As an illustration of how this works I will show how the 13 monoclinic space
groups can be derived from the combination of point groups and Bravais
Lattices.
There are two types of monoclinic Bravais lattices, primitive monoclinic and
base-centered monoclinic, and the minimum symmetry element is either a 2fold axis (or a 21 screw axis) or a mirror plane (or a glide plane). If we
define our unit cell so that the 2-fold axis is parallel to the b-axis and/or
the mirror plane is perpendicular to the b-axis, then the base centered
lattice becomes a C-centered lattice. Lets see how these combine with the
three monoclinic point groups: 2, m and 2/m.
Point Group
2 (C2)
2 (C2)
2 (C2)
m (Cs)
m (Cs)
m (Cs)
m (Cs)
2/m (C2h)
2/m (C2h)
2/m (C2h)
2/m (C2h)
2/m (C2h)
2/m (C2h)
Bravais
Lattice
Primitive
Primitive
C-centered
Primitive
Primitive
C-centered
C-centered
Primitive
Primitive
C-centered
Primitive
Primitive
C-centered
Space Group
(Long Symbol)
P121
P1211
C121
P1m1
P1c1
C1m1
C1c1
P 1 2/m 1
P 1 21/m 1
C 1 2/m 1
P 1 2/c 1
P 1 21/c 1
C 1 2/c 1
Space Group
(Short Symbol)
P2
P21
C2
Pm
Pc
Cm
Cc
P2/m
P21/m
C2/m
P2/c
P21/c
C2/c
Both the long and the short Herman-Mauguin space group symbols are listed.
The long symbol describes the symmetry elements which are either parallel
(for axes) or perpendicular (for planes) to the a, b and c axes of the unit
cell. In a monoclinic system there are no symmetry elements in two of the
three directions (if there were it would be a higher symmetry such as
orthorhombic). So all of the necessary information is contained in the short
symbol (which only shows the symmetry elements parallel and perpendicular
to b).
At first it would be logical to suppose that combining 2 Bravais lattices (P &
C) with 3 point groups (2, m & 2/m) would give 2  3 = 6 space groups (P2, C2,
Pm, Cm, P2/m & C2/m). In fact all six of these space groups are present*,
but we also generate other space groups if we replace the 2-fold axis with a
21 screw axis, or replace the mirror plane with a c-glide plane. When we do
this the additional space groups are generated. This brings up the question
of how to treat glide planes and screw axes when referring to the point
group symmetry, since these symmetry elements contain both a point
symmetry operation and a translational symmetry operation.
Crystallography’s answer to this problem is to treat glide planes as mirror
planes and screw axes as rotation axes, when it comes to signifying the point
group.
You might also wonder why C21, C21/m and C21/c are missing from the above
list. The reason for this is that when you combine C-centering with a 2-fold
axis (parallel to b), you create 21 axes. If you look at the space group
diagram for C2 (see the Int. Tables for Crystallography) you will find both
2-fold rotation axes and 21 screw axes. For this reason C2 and C21 are not
distinct space groups. The name C2 is chosen by convention.
*Space groups that can be generated without using glide planes and/or screw
axes are called symmorphic space groups. There are 73 symmorphic space
groups.