Crystals: difference between basis and unit cell

[fisico30]

Hello Forum,

a lattice is a set of points. We can place a basis at each set of points.
The basis can be one atom or a group of atoms.

I thought that a translation of the basis would produce the whole crystal...
How is a basis different from the unit cell? Are they the same thing?

There is the primitive cell then...how does it compare to the basis?

thanks
fisico30

 

[M Quack]

If I remember correctly from a long time ago. the basis is the set of atoms (one or more) to be placed on each point of the lattice, but without defining the actual lattice.

You can see that if the basis is a single atom, for example in a simple metal, then you can construct a whole load of different lattices from this basis. Indeed, there are FCC, BCC, HCP, and DHCP metals to name just a few.

The unit cell has both types of information. The translation information to build the lattice, and the basis, which atom to place where.

For high symmetry systems, there can be two types of unit cells. The primitive unit cell is the smallest unit cell from which you can construct the whole lattice. The conventional unit cell is larger, but shows the symmetry better.

For FCC, for example, the conventional unit cell is a cube that contains four primitive unit cells, as the face-centers are related by simple translation to the corner.

 

[fisico30]

Hi M Quack,

thanks for the reply. I am not clear on your last example:

"For FCC, for example, the conventional unit cell is a cube that contains four primitive unit cells, as the face-centers are related by simple translation to the corner."

I get what a lattice and a basis is. In a 2D crystal there is an infinite number of independent primitive vectors. The weighted sum of these 2 vectors locate any point in the lattice...

So, in 2D, is the primitive cell the area described by these two primitive vectors?

For a FCC structure there is then an infinite number of possible primitive cells, one for each possible set of primitive vectors...is that correct?

 

[M Quack]

The primitive unit cell is the smallest unit cell from which the lattice can be built. The choice is not unique, and I guess in principle it is infinite. In practice, you try to respect the symmetry of the crystal as best as you can. The volume of the primitive unit cell (or area in 2D) does not depend on this choice and is always the same.

The vectors that span the primitive unit cell are usually chosen to go from (0,0,0) to the nearest equivalent point on the lattice. For FCC that are the 3 face centers at (1/2,1/2,0), (0.,1/2,1/2) and (1/2,0,1/2).

In reciprocal space one chooses a different construction, the Brillouin zone. The definition is unique, there are no choices to make, and the BZ perfectly represents the symmetry of the (reciprocal) lattice. It often has a non-trivial shape, though.

If you are interested, look at one of the standard textbooks on solid state physics. Ashcroft and Mermin is one of the better ones. if you are interested in the exact mathematical definitions, then the International Tables for Crystallography are very good, but that's not where you will get an overview of what is going on.