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I am trying to understand the dispersion curve (as shown below) of a 1D lattice with diatomic basis.
Here are my questions:
Can both optical and acoustic branch of phonon simultaneously exist in crystal?
Why is there a band gap between the optical and acoustic phonon branch ?
In answer to your first question: yes, as that figure shows.
As for your second question, consider this: what would happen if you keep the diatomic basis but set the masses of the two atoms to be equal. The answer is that it'd look like the above figure but without a bandgap; it's the monoatomic case with the ends of the dispersion relation folded over to fit in the smaller Brillouin zone of the diatomic case.
If you increase the difference in the masses, the band gap grows. This is because you've broken a symmetry, and as is usual in quantum mechanics, breaking some symmetry can drive degenerate (or nearly degenerate) energy levels apart.
