5  Lecture notes for GEOL3010
Until this point, we have been concerned with the external form of perfect crystals, using the basic symmetry operations of reflection, inversion, rotation and rotoinversion or other combinations of operations.
By now you should be getting familiar with the HermannMauguin symbols that describe symmetry elements in a crystal form without translation. For example,
It can be shown through trial and error (or it can be proven mathematically) that there are only 32 possible combinations of symmetry operations to completely describe all possible external space filling symmetry properties (recall, this is without translations, glide planes or screw axes). These are the 32 point groups and they make up the 32 crystal classes in the table below.
32 Combinations of symmetry elements 

Rotations 
1 
2 
3 
4 
6 
Rotoinversons 
bar1 
bar2 
bar3 
bar4 
bar6 
Multiple rotation axes 
222 
32 
422 
622 

Rotation axis with ^ mirror 
2/m 
3/m 
4/m 
6/m 

Rotation axis with // mirrors 
2mm 
3m 
4mm 
6mm 

Rotoinversion with rotation axis and mirror  bar3 2/m  bar4 2m  bar6 2m  
3 rotation axes and ^ mirrors  2/m 2/m 2/m  4/m 2/m 2/m  6/m 2/m 2/m  
Combinations in isometric system 
23 
432 
4/m bar3 2 /m  
2/m bar3 
bar4 3m 
These 32 classes have been grouped into six crystal systems with the each group basis being similarities in the degree of symmetry elements.
The 32 crystal classes grouped by symmetry. 

Crystal system 
Center of symmetry 
No center of symmetry 
Triclinic 
i 
1 
Monoclinic 
2 / m 
2, m 
Orthorhombic 
2/m 2/m 2/m 
222, mm2 
Tetragonal 
4/m, 4/m 2/m 2/m 
4, bar4, 422, 4mm, bar42 m 
Hexagonal 
bar3, bar3 2/m, 6/m, 6/m 2/m 2/m 
3, 32, 3m, 6, bar6, 622, 6mm, bar62m 
Isometric 
2/m bar3, 4/m bar3 2 /m 
23, 432, bar43m 
The physical properties of minerals can sometimes be related back to whether they have a center of symmetry or whether they have enantiomorphism (left or right handed) or polar properties (i.e., no center of symmetry).
One example involves the property of piezoelectricity. If there is no center of symmetry, then sometimes the opposite ends of the crystal structure have different forms. If pressure is exerted at one of the polar ends, then a flow of electrons is started, with negative and positive electrical charges building at the respective ends. The 432 class is the only noncentered class that is nonpolar.
The best known example of the piezoelectric effect is the use of quartz to control the frequency of a digital clock.
We now want to make the transition from generating the external form of a crystal (i.e., its habit) to generating the internal arrangement of atoms within the unit cell of a crystal.
Each of the six crystal systems have a set of unique crystallographic axes that are defined by their unit length and the angles at which the axes intersect.
Crystal System  # independent variables 
Low
s y m m e t r y
High 
Triclinic 
6  a, b, c, alpha,
beta, gamma 

Monoclinic 
4  a, b, c, beta  
Orthorhombic 
3  a, b, c  
Tetragonal 
2  a, b,  
Hexagonal 
2  a, c  
Isometric 
1  a 
Now instead of using symmetry operations to define the position of crystal faces in space, we want to be able to describe the positions of atoms in space.
In most cases it is convenient to describe the unit cell of in terms of the lattice points located at the corners of parallelepipeds. The shape outlined by the lengths and angles of the axes constitute the unit cell. In these cases, we term the unit cell as primitive (P) with their edges lying along the crystallographic axes.
The dimensions along the axes are given as the unit cell lengths (a, b, c). A common unit of length is the angstrom (Å), where 1 Å = 10^{10 } m or 10^{8} cm. Also, 1 nm = 10^{9 } m = 10 Å.
1. The edges of the unit cell should coincide with the symmetry axes of the lattice.
2. Chose the smallest possible primitive cell volume (unless a higher symmetry element can be located with a nonprimitive cell).
Sometimes it is desirable to choose a unit cell that is not primitive. It turns out that there are only 14 space lattice types that are compatible with the 32 crystal classes or 32 point groups. These 14 space lattices are made by stacking the five planar lattices in various ways.
Nonprimitive unit cells may choose the convention of centering on opposite faces. If only one set of faces are centered, then the lattice type is designated by the axis that is perpendicular to the faces (A, B or Ccentered).
If all the faces are centered then the lattice is designated as F or all facecentered. In some cases the unit cell can be designated as bodycentered (I) (innenzentrierte  Gr. innen = inside, + zentri = center, translated  to produce from the inside center).
A hexagonal unit cell may also be represented by a primitive unit cell or a centered unit cell. A special case of the hexagonal symmetry is the rhombohedral (R) unit cell, outlined by a rhombohedron edges which has equal units lengths (a=b=c) and equal angles (a=b=g ≠ 90°).
Crystal System  Lattice Type (click on symbol for zoom view) or click on movie file for animation  
Triclinic 

Monoclinic 
C  
Orthorhombic 

Tetragonal 

Hexagonal 

Isometric 
F  
Total 
14 
The 14 lattices shown above are known as the 14 Bravais Lattice Types. Note that only Ccentered nonprimitive cells are indicated. If one chooses an axis other than c as the principle axis (a or b) then the lattice convention is changed and designated as either Acentered or Bcentered. The choice of Ccentering is a matter of convention, as using A, B or C would still result in the same symmetry elements being present in a given nonprimitive lattice.
When the 32 crystal classes are combined with the 14 Bravais lattice types and symmetry operations that include translation (e.g., the elements of a screw axis or glide plane) we form the 230 space groups.
Space groups symbols describe the symmetry elements which relate equivalent points (i.e., atomic sites) in the crystal structure. The capital letter indicates whether the lattice is primitive (P), centered (C), bodycentered (I), facecentered (F) or rhombohedral (R).
The remaining part refers to the same elements as the 32 point group symbols.
However, if a translation element is present then it is included. Translations are denoted by glide planes where:
a, b, c when the translation is half the unit cell (i.e., a/2, b/2, c/2)
n if the translation is along half the diagonal (i.e., (a+b)/2, (b+c)/2, (a+c)/2 )
d if a diamond glide plane (i.e., the diagonal divided by (a+b)/4, (b+c)/4, (a+c)/4 ).
Translation in the screw axis will contain a subscript to the rotation axis with its magnitude being some unit fraction of the number of rotations. There are (n1) possible screw axes for an nfold rotation axis. The translational component is 1/n.
Facecentered
Sphalerite (ZnS) Isometric Space group F bar43m
Body centered
Chalcopyrite  CuFeS_{2 }Tetragonal  Space group I bar4 2d