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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.

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The acoustic and optical branches come about from the fact that per spacial dimension, there is one mode where the atoms oscillate "mostly in phase", and all the others have at least one atom per unit cell "out of phase".

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.

Even though this is 1D, I drew the amplitudes vertically, so it's easier to see what's going on.

enter image description here

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.

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