# Conventional unit cell and point group symmetries?

A definition of a conventional unit cell of a lattice is one that contains the same point group symmetries as the overall lattice and is the smallest such cell.

I can understand how a (infinite) lattice can have a point group symmetry about any lattice point such as rotational symmetry, mirror symmetry etc.

But I cannot see the same for a unit cell. Please can someone explain how we go about comparing the point group symmetries of a unit cell to that of an overall lattice? (e.g. what points do we use, for a cell what exactly is meant by a symmetry when most transformations move it from its original position etc...)

Edit

Consider the following diagram of a simple 2d cubic lattice:

In this diagram their is a unit cell in green. This cell clearly shares a the symmetry of reflection through the line A with the lattice. However the lattice is also symmetric by reflection through the line B but the unit cell is not even though for the lattice it is a point group symmetry of one of the lattice points within the unit cell. I would therefore say that this unit cell and the lattice do not share the same symmetry and therefore this unit cell is not a conventional unit cell. I however know (/am pretty confident) that this is indeed a conventional unit cell, given the above definition I, however cannot cell how this holds and where my reasoning is wrong.

• 'Conventional unit cell' is not a precise thing, in that there are many 'conventional' unit cells so named by different authors. Now, a Wigner-Seitz unit cell, that perhaps you can understand given Wigner's broad use of group theory. Aug 9 '16 at 0:09
• @JonCuster Although the term 'Conventional Unit Cell' may not be precise I think that my definition given above is, and for a given lattice specifies a unique cell (I have changed it slightly since you posted your comment). Aug 9 '16 at 9:30
• Concerning your final phrase, recall that point groups operations do not translate the object. The transformations do not move the cell. Aug 11 '16 at 11:12
• @garyp consider a cube, unless you rotate it about its center and by very specific amounts, the cube will occupy a different space (i.e. its corners before and after rotation will not line up). Aug 11 '16 at 12:58
• Ok, but that's not a group operation. So I guess I don't understand why you bring up this possibility. I'm worried that I do not understand your question. Aug 11 '16 at 13:46

A definition of a conventional unit cell of a lattice is one that contains the same point group symmetries as the overall lattice and is the smallest such cell.

I don't think this is the definition of a "conventional unit cell".

The "smallest cell" that fully describes any structure is the primitive cell, which is the smallest cell that contains only one lattice point.

https://en.wikipedia.org/wiki/Primitive_cell

The important symmetries for the primitive cell are the translational symmetries, which are part of the space group symmetry of the lattice, not the point group symmetry. Because the definition of the primitive cell does not specify the position of the cell origin with respect to the contained lattice point, the point group symmetry of a primitive cell is not uniquely defined, and depends on the choice of cell origin. You may, or may not, be able to find point group symmetry operations within a given cell that you would expect from looking at the full crystal.

• I'd like to add that there is a procedure to find the primitive cell called the Wigner-Seitz procedure. See this webpage, for instance. Dec 4 '20 at 20:50

The unit cell is a 3D figure which possess certain symetry (e.g. cube, tetragon etc.). The unit cell is selected after you have found out what is the symmetry of the crystal, and it is selected in a way that it has the symmetry of the crystal (can be rather complicated, like here). You cannot build unit cell if you don't know what your crystal look like (just from the number of atoms, etc.). With a given unit cell you can reproduce your crystal.

• Hi, thanks for your answer. You said "it is selected in a way that it has the symmetry of the crystal", this is the crux of my question. I cannot see how we compare the symmetry of the unit cell with the infinite crystal. I would appreciate it if you could please go into more depth surrounding this statement. Also see the edits to the question above (I have tried to clarify more my problems). Aug 13 '16 at 19:14
• After I've read your edit, I guess I can see what is missing. The key thing is that we are considering POINT groups, i.e. you should care that all the elements of the group are related to the same point. For the unit cell this is typically a "center" of the unit cell, i.e. its' most symmetric point (center of the square in your case). Now if you will draw the horizontal line through the center of the cell both cell, and lattice will be symmetric with respect to reflection. Aug 14 '16 at 21:40
• In other words when you are looking for a symmetry element in your lattice or unit cell, you should ask the question "Can I find a point/line with respect to which my unit cell/lattice will possess this symmetry?". Crude example of how your previous way of thinking will not work on lattice: In your drawing - another symmetry element is rotation by Pi/4. If I will randomly select any point (let say 1/5 of the lattice constant away from point A ) my lattice will not be symmetric to rotation by Pi/4. But if I will look for a point for which it does, I will find some. Aug 14 '16 at 21:46
• Finally, in general case it can be quite non-obvious to draw a unit cell, or to find a point in the lattice for which all the symmetries are respected. I failed my first attempt on group theory exam, because I could not find one:) Aug 14 '16 at 21:52
• An extra comment: there is a certain freedom of choice of the unit cell as you might know, which does not limit the generality. For instance you may place center of the square in the B point. In this case you will still have horizontal and diagonal reflection planes, though the diagonal now will be required to pass through B. Aug 14 '16 at 23:01

Part of your difficulty is that you are not choosing a point about which to define your symmetry operations (after all, they are called point symmetries). In the case of your specific square, the symmetry operations are defined w/r to the center of the square.

This makes it clear that the reflections - indicated in blue or red - are about planes going through the symmetry point. In particular, a reflection about the horizontal axis interchange $$(12)\leftrightarrow (43)$$.

If you choose an atom in the square as symmetry point, then you will need to use (discrete) translations to bring the transformed square back to its original position. These translations are also included in the symmetry group of the lattice so no real harm is done since any two cells are equivalent.

There are several good sources on this but one I like is

A.W. Joshi, Elements of group theory for physicists.

• Isn't reflection a symmetry which is defined through a line? Dec 6 '20 at 16:41
• yes but this line goes through the symmetry point. Dec 6 '20 at 17:24