3. The symmetry of crystals
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Symmetry
by repetition of patterns in a wall drawing or in flowers. The wall
drawing shows repetition by translation. The flowers show repetitions
by rotations. The flower on the left shows repetition around an axis of
rotation of order 8 (8 identical petals around the rotation axis).
The flower in the middle shows an axis of rotation of order 3 (two
different families of petals that are distributed around
the rotation axis). In addition, each petal in both flowers
shows a plane of
symmetry which divides it into two identical parts (approximately).,
the same as it occurs with the butterfly shown on the right. If the
reader is surprised by the fact that we say that the two parts
separated by a plane of symmetry (mirror) are only "approximately"
identical, is because they are really not identical; they cannot be
superimposed, but this is an issue that will be explained in another
section.




Symmetry by repeating events: Night  Day  Night ... 
Symmetry in music. A fragment from "Six unisono melodies" by Bartók. (The diagram at the bottom represents the symmetrization of the one shown above) 
The word "Symmetry," carefully written with somewhat distorted letters, shows a twofold axis (a rotation of 180 degrees) perpendicular to the screen. 


Two objects related by a center of symmetry 
Polyhedron showing a twofold rotation axis passing through the centers of the top and bottom edges 
Polyhedron showing a symmetry plane that relates (as a mirror does) the top to the bottom 
The symmetry elements of types center or plane relate objects in a peculiar way; the same way that our two hands are related one to the other: they are not superimposable. Objects which in themselves do not contain any of these symmetry elements (center or plane) are called chiral and their repetition through these elements (center or plane) produce objects that are called enantiomers with respect to the original ones. The mirror image of one of our hands is the enantiomer of the one we put in front of the mirror.

The
mirror image of either of our hands is the enantiomer of the other hand.
They are objects not superimposable and as they do not contain
(in themselves) symmetry centers or symmetry planes, are called
chiral objects. Chiral molecules have different properties than their enantiomers and so it is important that we are able to differentiate them. The correct determination of the absolute configuration or absolute structure of a molecule (differentiation between enantiomers) can be done in a secure manner through Xray diffraction only, but this will be explained in another chapter 
Polyhedron showing a center of symmetry in its center 
Axis of improper rotation, shown vertically, in a crystal of urea. The numerical information shown in the figure will be discussed in another chapter. 
Stacking of unit cells forming an octahedral crystal 
Parameters which characterize the shape and size of an elementary cell (or unit cell) 
point group . crystal translational periodicity = 32 crystal classes 
Graphic representation of the 32 crystal classes 
The object represented by a single brick, which can also be represented by a reticular point, shows the point group symmetry 2mm 
Links below illustrate the 32 crystal classes using some crystal morphologies:  
Triclinic  1  1  
Monoclinic  2  m  2/m  
Orthorhombic  222  mm2  mmm  
Tetragonal  4  4  4/m  422  4mm  42m  4/mmm 
Cubic  23  m3  432  43m  m3m  
Trigonal  3  3  32  3m  3m  
Hexagonal  6  6  6/m  622  6mm  6m2  6/mmm 
Links below show animated displays of the symmetry elements in each of the 32 crystal classes: (taken from Marc De Graef) 

Triclinic  1  1  
Monoclinic  2  m  2/m  
Orthorhombic  222  mm2  mmm  
Tetragonal  4  4  4/m  422  4mm  42m  4/mmm 
Cubic  23  m3  432  43m  m3m  
Trigonal  3  3  32  3m  3m  
Hexagonal  6  6  6/m  622  6mm  6m2  6/mmm 





The adequacy of a lattice to the structure is illustrated in the twodimensional examples shown below. In all three cases two different lattices are shown, one oblique and primitive and one rectangular and centered. In the first two cases, the rectangular lattices are the most appropriate ones. However, the deformation of the structure in the third example leads to metric relationships that make that the most appropriate lattice, the oblique primitive, hexagonal in this case.
Adequacy of the lattice type to the structure. The blue lattice is the best one in each case. 
32 crystal classes + 14 Bravais lattices = 230 Space groups 
A
wall of bricks showing the most appropriate lattice which best
represents both the brick and its symmetry. Note that in this case the
point symmetry of the brick and the point symmetry of the reticular
point are coincident. The space group, considering the thickness of the brick, is Cmm2.

The 32 crystal classes, the 14 Bravais lattices and the 230 space groups can be classified, according to their hosted minimum symmetry, into 7 crystal systems. The minimum symmetry produces some restrictions in the metric values (distances and angles) which describe the shape and size of the lattice.
32 classes, 14 lattices, 230 space groups / crystal symmetry = 7 crystal systems 
All this is summarized in the following table:
(Laue with *) 
and their symmetry 
space groups 




1 





2/m 





mmm 




4mm 4222 42m 4/mmm* 
4/mmm 




432 43m m3m* 
m3m 




6mm 622 62m 6/mmm* 
6/mmm 


α=β=90 γ=120 

3m 32 3m* 
(R) 6/mmm 








The 230 crystallographic space groups are listed and described in the International Tables for Xray Crystallography, where they are classified according to point groups and crystal systems. A composition of part of the information contained in these tables is shown below, corresponding to the space group Cmm2, where C means that the structure is described in terms of a lattice centered on the faces separated by the c axis. The first m represents a mirror plane perpendicular to the a axis. The second m means another mirror plane (in this case perpendicular to the second main crystallographic direction), the b axis. The number 2 refers to the twofold axis parallel to the third crystallographic direction, the c axis.
Summary of the information shown in the International Tables for Xray Crystallography for the space group Cmm2. And this is another example for the space group P21/c, centrosymmetric and based on a primitive monoclinic lattice, as it appears in the International
Tables for Xray Crystallography
Summary of the information shown in the International Tables for Xray Crystallography for the space group P21/c As an example of the amount of information recorded in these tables, the reader can obtain some sample pages by clicking here. 
The advanced reader can also consult:
Some examples of structures made with bricks 
In any case, it doesn't end here! There are many more things to talk about. Go on...
To the next suggested chapter: Direct and reciprocal lattices 
