Optical phonons in fcc-lattice

[johannes919]

Aluminium has an fcc structure, which is a simple cubic lattice with four Al-atoms in the basis.

On the other hand, diamond has a diamond structure, which is a simple cubic lattice with 8 atoms in basis.

Now, diamond has optical modes in addition to acoustical, while Aluminium does not. What is the reason for this?

 

[DrDu]

Because diamond is a simple cubic lattice with two atoms per cell while Al is a fcc lattice with one atom per cell.
Note that the fcc lattice is a lattice on its own different from the primitive cubic lattice and that a primitive cell is smaller than the cubic cell you have in mind.

 

[johannes919]

Ah, ok. So the reason is that the primitive unit cell of the fcc lattice only consists of one atom, while the primitive unit cell of diamond consists of two?

 

[nasu]

Because diamond is a simple cubic lattice with two atoms per cell while Al is a fcc lattice with one atom per cell.
Note that the fcc lattice is a lattice on its own different from the primitive cubic lattice and that a primitive cell is smaller than the cubic cell you have in mind.

Don't you mean FCC lattice with a base of two atoms?

 

[DrDu]

You are certainly right, thank's for correcting this. The main point is that optical phonons will only occur if there are more than one atoms in the primitive cell as the optical mode is a motion where these atoms move relative to each other, while in the accoustical mode all atoms in the basis move in unison (well, this is exactly true only at the Gamma point, i.e. the center of the Brillouin zone).

 

[Wminus]

Why do you always have 3 modes for each atom in the basis?

 

[johannes919]

So, I get that there are two atoms in the primitive cell of the Diamond lattice, while there's only one in the fcc lattice. But do we get optical modes if the two atoms are identical?

Referring to your answer in Wminus' post, it would seem to me that as the atoms in the primitive cell are equal, we shouldn't get optical modes, but we would see an "extra" acoustical mode in the extended zone scheme?

 

[DrDu]

To your first point: As nasu pointed out, diamond, like aluminium, also crystallizes in a fcc lattice. There is no diamond lattice, only a diamond crystal structure. Diamond and aluminium both have a fcc lattice, but the former one has a basis with two atoms while the latter one a basis comprising only one atom. In Wminus's post, you can still call the upper mode an optical one if you like.
However in this example, the two atoms are not only equal but there positions are additionally spaced by a lattice vector. Therefore the two modes are degenerate at the zone boundary, i.e. there is no band gap between them.
In the case of diamond, the two atoms in the elementary cell cannot be made coincident via a shift by a lattice vector. This has the effect that the bands are not degenerate on the zone boundary (maybe with the exception of isolated points).

 

[johannes919]

Thanks!