# Why are there $3N - 3$ optical branches for a system in 3-Dimensions with $N$ different atoms per unit cell?

I am specifically referring to when we can approximate the atoms to be in a chain and they interact via harmonic potentials, for instance, if $N = 2$, we have a diatomic cubical lattice. For the acoustic modes, I can see we have 3 acoustic branches (one for every dimension) but I don't understand why we have $3N - 3$ optical branches.

The relation also holds for onedimensional systems, where it is easier to visualize. In that case, we have only 1 acoustic, and $$s-1$$ optical branches, $$s$$ being the number of atoms per unit cell.
Excuse the crude drawing, but I think it will get the point across. The upper sketch is the acoustic mode at some $$k$$-value, the lower is the optical mode at the same $$k$$. The lower mode will have a higher energy, since the "springs" connecting neighboring atoms are stretched much further, resulting in stronger forces and faster oscillations. For 3 atoms in the basis, things get a little more complicated, but the acoustic mode looks pretty much the same. However, when there are 3 kinds of atoms, there are 2 different modes like the lower one.
A more mathematical take on this would be to look at the dimesion of the dynamical matrix. In 1D we get an $$s \times s$$ matrix that we need to diagonalize, giving $$s$$ eigenvalues for the frequencies. In 3D we have $$3s \times 3s$$ giving $$3s$$ eigenvalues, resulting in $$3s$$ branches in total. This does, however, not explain their distribution to 3 acoustic and $$3s-3$$ optical branches.