As far as I understand from what I'm studying, only when you have more than one kind of atom in the lattice structure, will the phonon have an optical branch, or else it should not have an optical branch, because as was written in the text, "two atoms per unit cell and more". Am I interpreting it correctly or am I being mislead? If I am correct, then why does Silicon have an optical phonon branch?
[Physics] When do optical phonon branches arise
lattice-modelphononssemiconductor-physicssolid-state-physics
Related Solutions
Since you're talking about quarters of atoms, I assume you consider the two-dimensional case. To answer your question, you have to go the 3D-case.
As you can see in the picture below, the FCC unit cell in three dimensions contains
$$\left( 8 \cdot \frac{1}{8} + 6 \cdot \frac{1}{2} \right) \text{Atoms} = 4 \,\text{Atoms}. \\[1cm]$$
(from https://wps.prenhall.com/wps/media/objects/3082/3156196/blb1107/bl11fg33.jpg)
Each of the eight atoms at the corners of the unit cell is shared with seven other unit cells. So in total these account only for one atom. Together with the atom in the middle you get two atoms per unit cell, as mentioned in the comments.
Best Answer
Just to be clear, the two or more atoms do not have to be of different type. Optical phonons are related to the relative vibrations of atoms within the unit cell, while acoustic phonons describe the relative vibration of different unit cells. Optical phonons arise whenever the unit cell has at least one such degree of freedom, meaning at least two atoms in the unit cell. This includes silicon.