Dispersion relation diagrams, phonons

Optical phonons have non-zero components at the center of the Brillouin zone.

 

The reason this is possible (as Dr T says, the optical branch has non-zero energy at k=0) is because k is not really a wavenumber.

p = [hbar]k is the 'crystal momentum', which is not a real momentum.

 

Also, you could easily be reading a reduced zone scheme, in which the band from the next zone is folded back into the first zone.

Zz.

 

Optical phonons occur in crystals which have more than one atom per unit cell. If you have a phonon with k=0 that means the displacement of atoms is the same in every cell. When you have only one atom per cell, then a k=0 displacement is just a shift of the whole crystal, so there can't be a restoring force (hence, [tex]\omega=0[/tex]). But if you have more than one atom per unit cell then the atoms could displace relative to one another (eg. like a bond-stretching mode). Then you can have a k=0 wave, where the displacement is the same in each cell, but the atoms in the cell move relative to one another. Then you will have a restoring force, and have [tex]\omega > 0[/tex] for this type of phonon.