3. The symmetry of crystals
If you do not see a menu on the left, please, use this link. 



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 5 (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 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. 


Polyhedron showing a twofold rotation axis (2) passing through the centers of the top and bottom edges 
Polyhedron showing a reflection plane (m) that relates (as a mirror does) the top to the bottom 
Hands and molecular models related by a twofold axis (2) perpendicular to the drawing plane 
Hands and molecular models related through a mirror plane (m) perpendicular to the drawing plane 
Hands (left and right) related through a center of symmetry 
Two objects related by a center of symmetry Polyhedron showing a center of symmetry in its center 
The association of elements of rotation with centers or planes of symmetry generates new elements of symmetry called improper rotations. 

A fourfold improper axis implies 90º rotations followed
by reflection through a mirror plane perpendicular to the axis.

Axis
of
improper rotation, shown vertically, in a crystal of urea.
The meaning of numerical triplets shown will be discussed in another chapter. 
Twofold screw axis. A screw axis consists of a rotation followed by a translation 
Glide plane. A glide plane consists of a reflection followed by a translation 
Twofold screw axis applied to a left hand. The hand rotates 180º and moves a half of the lattice translation in the direction of the screw axis, and so on. Note that the hand always remains as a left hand. (Animation taken from M. Kastner, T. Medlock & K. Brown, Univ. of Bucknell) 
Glide plane.applied to a left hand. The left hand reflects on the plane, generating a right hand that moves a half of the lattice translation in the direction of the glide operation. (Animation taken from M. Kastner, T. Medlock & K. Brown, Univ. of Bucknell) 

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 
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 motif, represented by a single brick, can also be represented by a lattice point. It shows the point symmetry 2mm 
Links below illustrate the 32
crystal classes using some crystal morphologies These interactive animated drawings need the Java environment and therefore will not run with all navigators 

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 illustrate the 32
crystal classes using some crystal morphologies These are noninteractive animated gifs obtained from the Java animations appearing in http://webmineral.com. They will run with all navigators 

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:
Crystal
classes (* Laue) 
Compatible
crystal lattices and their symmetry 
Number
of space groups 
Minimum symmetry  Metric restrictions  Crystal system 
1 1 *  P 1 
2  1 or 1  none  Triclinic 
2 m 2/m *  P
C (I) 2/m 
13  One 2 or 2 

Monoclinic 
222 2mm mmm *  P
C (A,B) I
F mmm 
59  Three 2 or 2 

Orthorhombic 
4
4 4/m * 422 4mm 42m 4/mmm * 
P
I 4/mmm 
68  One 4 or 4  a=b 
Tetragonal 
3
3
* 32 3m 3m * 
P
(R) 3m 6/mmm 
25  One 3 or 3  a=b=c 
Trigonal 
6
6
6m * 622 6mm 6m2 6/mmm * 
P 6/mmm 
27  One 6 or 6  a=b α=β=90 γ=120 
Hexagonal 
23
m3 * 432 43m m3m * 
P
I
F m3m 
36  Four 3 or 3  a=b=c 
Cubic 
Total: 32, 11 * 
14 independent  230  7 
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. Chiral compounds that are prepared as a single enantiomer (for instance, biological molecules) can crystallize in only a subset of 65 space groups, those that do not have mirror and/or inversion symmetry operations.
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 P2_{1}/c 
The advanced reader can also consult:
Look for possible unitcells and symmetry elements in these structures made with bricks (the solution clicking on the image) 
There is a question that surely the readers will have considered... In this chapter we have shown elements of symmetry that operate inside the crystals, but we have not yet said how we can find out the existence of such operations, when in fact, and in the best of cases, we could only visualize the external habit of the crystals if they are well formed! Although we will not answer this question here, we can anticipate that this response will be given by the behavior of the crystals when we illuminate them with that special light that we know as Xrays, but this will be the subject of another chapter.
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 

