3. The symmetry of crystals
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We don't often notice it, but we continuously live with symmetry...

Symmetry is the consistency, the repetition of something in space and/or in time, as is shown in the examples below: a wall drawing, the petals of flowers, the two sides of a butterfly, the succession of night and day, a piece of music, etc.

 
A wall drawing with translational repetition
Flower wit a rotation axis of order 8
Flower wit a rotation axis of order 3
Mirror symmetry in a butterfly
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 two-fold axis (a rotation of 180 degrees) perpendicular to the screen.


This phrase also serves to illustrate the concept of symmetry:

A MAN, A PLAN, A CANAL: PANAMA

where, if we forget the commas and the colon, it becomes:

AMANAPLANACANALPANAMA

which can be read from right to left with exactly the same meaning as above. It is a case similar to the "palindromic" numbers (232 or 679976).

There are many links in which the reader can find information on the concept of symmetry and we have selected some of them: symmetry and shape of space, some others in the context of crystallographic concepts, some with decorative patterns, or in the context of minerals. There is even an international society for the study of symmetry.



The essential knowledge on crystal morphology, symmetry elements and their combination to generate repetitive objects in space, were well established between the 17th and 19th Centuries, as stated elsewhere in these pages...



Specifically, in finite objects, there are a number of operations (elements of symmetry) describing repetitions. In the wall-drawing (shown above) we find translational operations (the motif is repeated by translation). The repetition of the petals in the flowers show us rotational operations (the motif is repeated by rotation) around a symmetry axis and (although not exactly) the symmetry shown in the phrase or in the music fragment (shown above), lead us to consider other symmetry operations known as symmetry planes (or mirror planes). The same operation occurs when you look into a mirror. Similarly, for example, if we look at the relationship between the three-dimensional objects in the pictures shown below, we will discover a new element of symmetry called center of symmetry, which is an imaginary point between objects shown below.


 
Polyhedron showing a two-fold 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



Hands and molecular models related by a twofold axis perpendicular to the drawing plane


Hands and molecular models related through a mirror plane perpendicular to the drawing plane




Hands (left and right) related through a center of symmetry








Two objects related by a center of symmetry



The association of elements of rotation with centers or planes of symmetry generates new elements of symmetry called improper rotations.

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.




Combining the rotation axes and the mirror planes with the characteristic translations of the crystals (which we'll see below), new symmetry elements appear, with some "sliding" components: screw axes (or helicoidal axes) and glide planes.



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



The symmetry elements of types center or mirror 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 X-ray diffraction only, but this will be explained in another chapter 


Thus, any finite object (such as a quartz crystal, a chair or a flower) shows that certain parts of it are repeated by symmetry operations that go through a point of the object. This set of symmetry operations form what we called a symmetry point group. The advanced reader has also the opportunity to visit the nice work on point group symmetry elements offered through these links:


But to keep talking about the symmetry in crystals, it is necessary to recall a fundamental concept related to the repetition by translation...

Periodic repetition, which is a characteristic of the internal structure of crystals, is represented by a set of translations in the three directions of space, so that crystals can be seen as the stacking of the same block in three dimensions. Each block, of a certain shape and size (but all of them being identical), is called a unit cell or elementary cell. Its size is determined by the length of its three edges (a, b, c) and the angles between them (alpha, beta, gamma: αβγ).


Stacking of unit cells forming an octahedral crystal






Parameters which characterize the shape and size of an elementary cell (or unit cell)



In crystals, the axes of symmetry (rotation axes) can only be two-fold (2), three-fold (3), four-fold (4) or six-fold (6), depending on the number of repetitions of the motif which can occur (order of rotation). Thus, an axis of order 3 (3-fold) produces 3 repetitions of the motif, one every 360 / 3 = 120 degrees of rotation. If the reader wonders why only symmetry axes of order 2, 3, 4 and can occur in crystals, and not 5-, 7-fold, etc., we recommend the explanations given in another section.

Improper rotations (rotations followed by reflection through a plane perpendicular to the rotational axis) are designated by the order of rotation, with a bar above the number.

The screw axes(or helicoidal axes, ie, symmetry axes involving rotation followed by a translation along the axis) are represented by the order of rotation, with an added subindex that quantifies the translation along the axis. Thus, a screw axis of type 
62 means that in each of the 6 rotations an associated translation occurs of 2/6 of the axis of the elementary cell in that direction.

The mirror planes are represented by the letter m.

The glide planes (mirror planes involving reflexion and a translation parallel to the plane) are represented by the letters a, b, c, n or d, depending if the translation associated with the reflection is parallel to the reticular translations (a, b, c), parallel to the diagonal of a reticular plane (n), or parallel to a diagonal of the unit cell (d).

The letters and numbers that are used to represent the symmetry elements also have an equivalence with some graphic symbols. These symbols and the operations of the most common symmetry elements can be seen through this link.



As mentioned above, all symmetry elements of a finite object, passing by a point, define the total symmetry of the object, which is known as the point group symmetry of the object.

There are many symmetry point groups, but in crystals they must be consistent with the crystalline periodicity (repetition by translation). On the other hand, for instance, the symmetry axes of order 5 (5-fold axes) are not possible in crystals and therefore only 32 point groups are allowed in the crystalline state of matter. These 32 point groups are also known in Crystallography as the 32 crystal classes.

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
 
In addition, we recommend to use the Java applet offered by Nicolas Schoeni y Gervais Chapuis of the Ecole Polytechnique Fédéral de Lausanne (Switzerland) as an initiation to discover the point group symmetry of platonic and other polyhedra. In case of problems using this applet, please follow the indications shown in this link.



Of the 32 crystal classes, only 11 contain the operator center of symmetry, and these 11 centro-symmetric crystal classes are known as Laue groups.

crystal class . center of symmetry = 11 Laue groups
Graphic representation of the 11 Laue groups  (centro-symmetric crystal classes)

In addition, the repetition modes by translation in crystals must be compatible with the possible point groups (the 32 crystal classes), and this is why we find only 14 types of translational lattices which are compatible with the crystal classes. These types of lattices (translational repetiton modes) are known as the Bravais lattices (you can see them here). The translational symmetry of an ordered distribution of 3-dimensional objects can be described by many types of lattices, but there is always one of them more suited to the object, ie: the one that best describes the symmetry of the object. As the lattices themselves have their own distribution of symmetry elements, we must fit them to the symmetry elements of the structure.

crystal translational periodicity . 32 crystal classes  =  14 Bravais lattices
Graphic representation of the 14 Bravais lattices

 
A brick wall can be structured with many different types of lattices, of different origins, and defining reticular points representing the brick. But there is a lattice that is more appropriate to the symmetry of the brick and to the way the bricks build the wall. 

The adequacy of a lattice to the structure is illustrated in the two-dimensional 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.


Finally, combining the 32 crystal classes (crystallographic point groups) with the 14 Bravais lattices, we find up to 230 different ways to replicate a finite object (motif) in 3-dimensional space. These 230 ways to repeat patterns in space, which are compatible with the 32 crystal classes and with the 14 Bravais lattices, are called space groups, and represent the 230 different ways to fit the Bravais lattices to the symmetry of the objects. The interested reader should also consult the excellent work on the symmetry elements present in the space groups, offered by Margaret Kastner,
Timathy Medlock and Kristy Brown  through this link of the Bucknell University.

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 with *)
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
α=γ=90
Monoclinic
222    2mm    mmm*
P   C  (A,B)  I   F
mmm
59
Three  2  or  2
α=β=γ=90
Orthorhombic
4      4     4/m* 
4mm    4222    42m   4/mmm*
P     I
4/mmm
68
One  4  or  4
a=b
α=β=γ=90
Tetragonal
23      m3* 
432     43m    m3m*
P     I     F
m3m
36
Four  3  or  3
a=b=c
α=β=γ=90
Cubic
6       6     6m* 
6mm   622     62m    6/mmm*
P
6/mmm
27
One  6  or  6
a=b
α=β=90  γ=120
Hexagonal
3       3
3m     32    3m*
P      3m
(R)   6/mmm
25
One  3  or  3
a=b=c
α=β=γ 
(or Hexagonal)
Trigonal
Total: 32,  11*
14 independent
230
   
7

The 230 crystallographic space groups are listed and described in the International Tables for X-ray 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 two-fold axis parallel to the third crystallographic direction, the c axis.



Summary of the information shown in the International Tables for X-ray 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 X-ray Crystallography

Space group P21/c

Summary of the information shown in the International Tables for X-ray 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:


Crystallographers never get bored! Try to enjoy the beauty, looking for the symmetry of the objects around you, and particularly in the objects shown below ...
 

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
Go to the introduction