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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 Centering Vector(s)
Lattice Points per Unit Cell
Face Centered F �a + �b�a + �c�b + �c
4
Body Centered I �a + �b + �c
2
Base Centered C �a + �b 2
B �a + �c
A �b + �c
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
Bravais Lattice Unit Cell Dimensions
Required Symmetry Element
Triclinic Primitive (P) a � b � c��
None
Monoclinic Primitive (P)Base-Centered (C)
a � b � c= =90� �
Either a mirror (glide) plane or a 2-fold axis
Orthorhombic Primitive (P)Base-Centered (C) Body-Centered (I)Face-Centered (F)
a � b � c= ==90�
Any combination of three mutually 2-fold axes or mirror (glide)
planes
Rhombohedral
Primitive (P) a = b = c= =�90�
One 3-fold axis
Trigonal R-Centered (R) a = b � c= =90�, =120�
Hexagonal Primitive (P) a = b � c= =90� =120�
One 6-fold axis
Tetragonal Primitive (P)Body-Centered (I)
a = b � c= ==90�
One 4-fold axis
Cubic Primitive (P)Body-Centered (I)Face-Centered (F)
a = b = c= ==90�
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
Axis to the Glide Plane
Displacement Vector
a b or c a/2
b a or c b/2
c a or b c/2
n* abc
b/2+c/2a/2+c/2a/2+b/2
d** abc
b/4+c/4a/4+c/4a/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
Number of Point
Groups
Herman-Mauguin Point
Group*
Schoenflies
Point Group
Triclinic 2 1,1 C1, Ci
Monoclinic 3 2, m, 2/m C2, Cs, C2h
Orthorhombic 3 222, mm2, mmm D2, C2v, D2h
Trigonal 5 3,3, 32, 3m,3m
C3, S6, D3, C3v, D3d
Hexagonal 7 6,6, 6/m, 622, C6, C3h, C6h, D6,
6mm,62m, 6/mmm
C6v, D3h, D6h
Tetragonal 7 4,4, 4/m, 422, 4mm,42m,
4/mmm
C4, S4, C4h, D4,C4v, D2d, D4h
Cubic 5 23, m3, 432,432, m3m
T, Th, O,Td, Oh
*Unfortunately html coding does not properly write symbols for improper rotations. The bar should be written directly above the number (not to the left of the number).
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. 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 2-fold 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
Bravais Lattice
Space Group(Long Symbol)
Space Group(Short Symbol)
2 (C2) Primitive P121 P2
2 (C2) Primitive P1211 P21
2 (C2) C-centered
C121 C2
m (Cs) Primitive P1m1 Pm
m (Cs) Primitive P1c1 Pc
m (Cs) C-centered
C1m1 Cm
m (Cs) C-centered
C1c1 Cc
2/m (C2h) Primitive P 1 2/m 1 P2/m
2/m (C2h) Primitive P 1 21/m 1 P21/m
2/m (C2h) C-centered
C 1 2/m 1 C2/m
2/m (C2h) Primitive P 1 2/c 1 P2/c
2/m (C2h) Primitive P 1 21/c 1 P21/c
2/m (C2h) C-centered
C 1 2/c 1 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.