Whereas crystal lattices can only build up using translation of the unit cell, the description of the crystal lattices often additionally constitutes a proper characterization of the internal SYMMETRY. Symmetry elements passing through a point of a finite object, define the total symmetry of the object, which is known as the point group symmetry. The use of symmetry becomes immediately clear, when you try to solve a crystal structure. Here simply by using symmetry you can determine the whole structure from a very small subset of known positions.
There are various symmetry operations, that can be applied on the crystal in order to describe all its features.
A Symmetry operation is an operation which results in no change in the appearance of the object.
This is a very weird definition, so let me say in other words. Imagine looking at a crystal unit cell, then close your eyes while performing the symmetry operation. If you open your eyes and the crystal unit cell looks exactly like it did before the operation, then it is a valid operation.
Is it still weird? Don’t worry, you will get it once we discussed some of them. Actally symmetry operation are not really performed on the crystal, they are just a way of describing the arrangement of the atoms and features inside the crystal. The symmetry of a crystal is an internal characteristic.
Besides the simple translation, there are three basic symmetry operations: Rotation, Reflection and Inversion
Rotation
Symmetry operation: Rotation
Symmetry element: Rotation axis -> two coordinates change
Space group denotion: number (e.g. 2 for 2-fold rotation symmetry)
In crystals we find only 1-fold, 2 -fold, 3-fold, 4-fold and 6-fold rotation symmetry, with a 360°, 180°, 120°, 90° and 60° rotation around an axis. Everything in the three dimensional world has a 1-fold rotation axis, therefore space groups that don’t have any symmetry elements in a particular direction, just have a 1 standing at the that position.
Other rotational symmetry operation do of course exist, but cannot be applied to crystals. However quasi-crystals (solid matter composed of multiple different crystal forms, that are not periodic in a sense that one crystal class can fill the whole 3-dimensional space alone) can show a 5-fold symmetry.
Reflection
Symmetry operation: Reflection
Symmetry element: Mirror plane –> one coordinate changes
Space group denotion: m
Inversion
Symmetry operation: Inversion
Symmetry element: Inversion point –> all coordinates change
Space group denotion: i
From the inversion point, the distance of each lattice point to the point is mirrored directly in the other direction.
Combined symmetry elements:
Each of these three basic symmetry elements can be combined to give another couple symmetry element
Rotation + Inversion = RoToinversion
Symmetry operation: Rotoinversion
Symmetry element: Rotoinversion axis
Space group denotion: bar Number (e.g. 3-bar for 3-fold rotoinversion axis)
There are also 2-fold, 3-fold, 4-fold and 6-fold rotoinversion, where you rotation around the rotoinversion axis 180°, 120°, 90° and 60° and than invert through an inversion point on the axis. Note, that 2-fold rotoinversion is equivalent with a reflection operation, which is perpenticular the rotoinversion axis and therefore not used.
Reflection + Translation = Glide Reflection
Symmetry operation: Glide Reflection
Symmetry element: Glide Plane
Space group denotion: letters (a ,b ,c , n, d or e)
Glide reflection combine mirroring and translation by 0.5 or 0.25 of the unit cell.
There are six types of glide reflections, each indicated in the space group denotion with another letter.
a, b, c indicate glide reflection along the a, b or c direction respectively.
n and d indicates a reflection along a diagonal glide plane and a translation for 0.5 or 0.25 of the unit cell, respectively
e shows a reflection along two glide planes
Explanation by Frank Hoffmann in unit 4.1 and 4.2: https://www.youtube.com/watch?v=5XwZj0m8zEQ&list=PL6C90-24AMSPjrP_hrxSkXjazWiqKwxIG
Rotation + Translation = Screw Rotation
Symmetry operation: Screw rotation
Symmetry element: Screw axis
Space group denotion: two number containing the angle of rotation and the fractional coordinates of the translation
Note, that all screw axis apply to right-handed coordinate systems, how that look like is shown below.
All combinations of screw axis:
| Label | Rotation angle | Translation part | enantiomorphous |
| 21 | 180° | 1/2 | |
| 31 | 120° | 1/3 | 32 |
| 41 | 90° | 1/4 | 43 |
| 42 | 90° | 2/4 = 1/2 | |
| 61 | 60° | 1/6 | 65 |
| 62 | 60° | 2/6 = 1/3 | 64 |
| 63 | 60° | 3/6 = 1/2 |
In the International Table of Crystallography each symmetry operation has its own symbol:
These and all other symbols can be found and are further explained here: http://img.chem.ucl.ac.uk/sgp/misc/symbols.htm and here: https://www.xtal.iqfr.csic.es/Cristalografia/parte_03_1-en.html
Each of these symbols are not only used in the International tables of Crystallography to symbolize the symmetry operations, but also in short forms as numbers and letters (see below Nomenclature).
The application on these symmetry operations: translation, rotation, mirroring and inversion to the outer shape of crystals, one can describe 32 crystal classes ( which are also called crystallographic point groups). That is to say, that there are 32 different kinds of outer shapes that crystals can possess.
Each of these crystal classes, in addition to their outer symmetry, can have multiple internal symmetries, that are then described by the 230 space groups.
All of these space groups are listed here: http://img.chem.ucl.ac.uk/sgp/large/sgp.htm
Before moving on, lets recap:
There are 7 crystal classes, with primitive unit cells. Once we assemble these unit cells together to form lattices, we see, that there are 14 Bravais lattice types. Using the symmetry operation translation, rotation, mirroring and inversion we can describe 32 outer shapes of crystals (crystal classes). By including not only the external but also the internal symmetries inside these crystals, we get 230 space groups, for different arrangements of the atoms inside these crystals.
Symmetry nomenclature
Let’s drive right inside the notation of the symmetries in a space group.
The nomenclature after Hermann and Mauguin is done according to the following rules:
- First letter is the Bravais symbol describing translational symmetry of the unit cell (P ,A ,B ,C ,R ,I ,F ) followed by a blank (space).
- P – primitive: all lattice points sum up to one
- A –
- B
- C – Side – centered
- R
- I (“Internal”) – Body – centered (bcc – body centered cubic)
- F Face – centered (fcc – face centered cubic)
- The symmetry operator symbols generating the equipoints in the unit cell follow. They are all separated by a blank – except if a slash combines them (denoting that the symmetry plane is perpendicular to the corresponding symmetry axis). Example : I 4/m m m (not case sensitive)
- A screw axis (usually denoted by a subscript following the axis) is entered as two adjacent numbers :
P 63/M M C, P 21 21 21 - An inversion (shown by the bar above the axis) is entered as a minus sign. Example : R -3 C. The exception is the triade in the cubic systems, which is automatically recognized as an inversion in the centrosymmetric groups and thus can be omitted: F D 3 M is ok, and so is F d -3 m.
- Settings with origin at center must be used.
- Use monoclinic setting with b as unique axis (alpha, gamma = 90.0°).
- Use obverse hexagonal setting for rhombohedral space groups.
According to the Cambridge Structural Database (CSD) two space groups make up over 50% of all crystals: the P2(1)/c and the P -1
Every crystal system has they own “viewing direction”, this means in the most cases, in what direction do I have to look, then I want to see additional symmetries.
Asymmetric unit: minimal number of atoms, that with the proper symmetry operations can reproduce the whole unit cell and in turn the whole crystal lattice
Symmetry in the NaCl (sodium chlorid salt) crystal
Everybody knows salt from cocking and maybe even from science classes in school, because the salt crystals are very regular, as shown here.
Here each chlorid ion is coordinated by 6 sodium, which form a tetrahedra and also each sodium is coordinated by 6 chlorid ions froming a tetrahedra as shown above.
This structure was actually one of the first solved crystal structure by the Bragg’s (father and son) in 1912.
NaCl crystalizes in the cubic crystal system, with a = b = c = 5.625 Å in the unit cell (shown in red), and all angles being 90°. The unit cell was choosen by focusing on the Na (sodium). The unit cell could also be drawn from connecting each Cl (chlorid). As a rule of thumb, crystallographers look first of all for the easiest unit cell and second for the unit cell, that best represents the symmetry of the crystal. For this latter reason, NaCl is not a primitive crystal. That is to say, that the unit cell does not just contain the outer Na ions (making up the corners of the cube), but additional Na ions inside the unit cell. In fact, in addition to each Na ion at the corner of the cube (totalling 1 whole lattice point, because each ion participates in 8 cells, by 1/8th each. 8 x 1/8 = 1), each of the sides (called faces) also feature a Na ion, indistingushable from the other Na ions at the corner.
Due to that fact, NaCl belongs to the face centered (Bravais) crystal lattice class, F. Each Na ion at the faces is “shared” so to say between two neighboring cells. Since a cube has 6 sides, the face-centered ions therefore contribute 3 lattice points (6 x 1/2 = 3), totalling with the corner lattice points to 4 lattice points per unit cell.
This is just gargon, but the thing to remember is all 7 primitive unit cells only contain 1 lattice point per unit cell, that is to say, only have atoms/ions at the corners of the cube, whereas the additional unit cells (non-primitive unit cells) have more than 1 lattice point per unit cell. Non-primitive unit cells are chosen be able to fully describe the symmetry of the crystal.
According to the International Table of Crystallography, NaCl crystalizes in the space group F m -3 m. Let’s take a look at that:
| Space group (short) | F m -3 m |
| Space group (Full symbol) | F 4/m -3 2/m |
| Crystal class | m -3 m |
| Crystal system | cubic |
| Entry in the International Table | #225 |
Nomenclature for the description of NaCl
Each crystal system as stated above, has it’s own viewing direction, for the description of the crystals symmetry.
For the cubic crystal systems, we first looks along the a direction, then the [111] direction and finally the [110].
So, after the initial F, for all-side face-centered unit cell, the other three parameters, describe the symmetry that can be seen in each of the three direction.
| F | 4/m | -3 | 2/m |
| general | [100] | [111] | [110] |
| all-side face-centered unit cell | 4-fold rotational axis + perpendicular mirror plane | 3-fold rotational axis | 2-fold rotational symmetry + mirror plane |
Let’s start the with a direction [100]:
Here, we have a 4-fold axis of rotation and perpendicular to that axis is a mirror plane, therefore 4/m.
Next along the body-diagonal, [111]:
Here, we have a 3-fold axis of rotation right in the center.
Last, along the ab-plane [110]:
Here, we have a 2-fold axis of rotation in the center through which a mirror plane can be placed as well.
Frank Hoffmann video on the NaCl crystal: https://www.youtube.com/watch?v=wCjtD75nJG8
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