# Reducing a conventional cell to a primitive cell

Suppose I only have a conventional cell with atomic positions obtained from a structural database, and I need to know the primitive cell for that lattice. Is there some general way to reduce the conventional cell to the primitive cell?

That is, to determine the correct primitive vectors and the correct basis with atomic positions referenced from the lattice point contained in the primitive cell? Any help or examples would be much appreciated.

If the unit cell is not primitive, there must be at least two equivalent basis atoms.

This suggests the following:

1. Find all translation vectors connecting any two basis atoms.
2. Eliminate translation vectors that are not a symmetry of the crystal.
3. For each of the remaining translations T
3.1 Find cycle length n of T, i.e. how many applications of T before a basis atom maps back to itself.
3.2 Reduce size of unit cell by factor n

Should work. Maybe you could do better.

The problem is that there is not just one unique "correct" primitive unit cell. There are many possible choices.

In crystallography, you only have a few cases of non-primitive conventional unit cells. Remember that only primitive translations play a role. Of the 14 Bravais lattices, 7 are primitive anyways. Possible choices for the remaining 7 are:

• Face centered cubic (FCC): A primitive cell is obtained by taking the vectors from the corner to the adjacent face centers. (Also works for face-centered tetragonal and orthorhombic).
• Body centered cubic (BCC): pick two edges of the conventional basis vectors (edges of the cube) and from the origin to the body center (also works for body-centered orthorhombic)
• End centered: pick a and c and replace b by the vector from the origin to the face center. Or pick b and c and replace a by face center (monoclinic and orthorhombic)

I guess the systematic way is as Sam_bell pointed out:

(1) Select the origin. There is not necessarily an atom at the origin.
(2) Find all primitive translation vectors of the lattice. (Screw axes, glide planes, mirror operations etc. are not primitive translations!)
(3) From these, retain only the nearest neighbors.
(4) Pick any 3 provided that they are not all in the same plane.
(5) check the volume of your unit cell to make sure. Should be 1/4 of the FCC cubic volume or 1/2 of the conventional volume for the other non-primitives.

http://www.cryst.ehu.es/cryst/celltran.html

A unique primitive unit cell is the Wigner-Seitz cell. However, that often has a complicated shape (rather than just a parallelepiped you get with primitive unit cells spanned by 3 vectors.

http://en.wikipedia.org/wiki/Wigner–Seitz_cell

Thanks for the replies! I'm still very new to all of this crystallography stuff, so any bit helps. Would either of you happen to know of an example of such a procedure I could look at?

Pick one of the structures in your database and try to use what me or M Quack said to construct a primitive cell. You can post your attempt on here. If there is a mistake, we'll point it out. As M Quack mentioned, there are not many possibilities you will encounter. You should definitely be familiar with primitive cells for FCC, BCC, and HCP lattices. The descriptions of those are at the beginning of every solid state physics text (e.g. Ashcroft & Mermin).

Yup I've been doing tons of reading, and I feel comfortable with those lattices, but it's always hard to put something into practice the first time you do it, no matter how much you read. I will work on it tomorrow and hopefully I can share my efforts!

Hi, sorry I'm having some difficulty.

I am not sure how to find translation vectors that point to any basis atom, because the only vectors I know are those that define the unit cell, and any (integer) linear combination of those vectors can't point to a basis atom because the basis atoms are within the cell it's self...

Also, what do you mean by "reduce the size of the unit cell by factor n"? Do you mean divide a,b,c by n?

Here's an example of a cell I am working with and unsure of how to reduce (the link will open a download to a .amc file which can be opened in VESTA, a program I'm sure either of you are familar with).

Now supposedly this is a BCC cell, could you possibly give me some indication on where to start?

Edit: It's not so much finding a primitive cell for a given unit cell, that procedure is fairly straight forward I believe, my problem is discovering the correct basis for a primitive cell, and what atoms needs to go where.

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The space group is Im3 (should probably be Im-3 or Im 3bar). That tells you immideately that this is cubic (the 3 in that position is the clue), and that it is body-centered (I).

The file then lists positions for the Ir and Sb atoms. The easiest way to figure out all symmetry-equivalent positions is to look up the space group in the International Tables for Crystallography, vol A

http://it.iucr.org/A/

then select space group 204. The IT will list all relevant information, in particular the site symmetry, and the number and positions of all equivalent positions within the conventional unit cell.

There is no trivial way of generating these positions with just a pencil and a piece of paper, although experienced people will be able to list all symmetry operations from just looking at the space group symbol - and from that they will be able to generate the positions.

Alternatively, the file lists a reference for the structure determination. Looking up that paper is never a bad idea if you want to understand the crystal structure.