How are the direction of primitive vectors determined in BCC crystal structures?

[potatowhisperer]

hello
i have browsed this forum for topics about this, and i found them very enlightening and helped a lot in terms of finding the length of the primitive vectors , the problem that i have is with the direction of said primitive vectors , while in fcc they are more or less easy to visualize in BCC they are not .
so my question is , how is the direction in BCC of the primitive vectors found ? i mean what are the steps and conditions followed to get those directions ?
thank you in advance.

 

[M Quack]

There are several possible solutions.
In general any unit cell that contains exactly one basis is a valid primitive unit cell.

For the basic BCC structure the basis is one atom.
For more complicated crystal structures there may be more than one atom. E.g. for rock salt, NaCl, the primitive unit cell must contain one Na and one Cl atom.
For the diamond structure the primitive unit cell must contain two atoms, as they are not related by a simple translation.

One possible solution is to take the vectors from the origin to two corners and to the body center. You see immediately that there are 3 ways to pick 2 out of 3 corners...
This is due to the cubic symmetry.
In fact, the reason the BCC cubic unit cell is chosen is because it nicely represents the symmetry. For any of the possible primitive unit cells the symmetry is much less obvious. The same is true for FCC.