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Does a diatomic crystal posses both optical and acoustic branch simultaneously; Or, whether the vibration is either optical or acoustic?

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It's not clear to me what exactly you're asking, but for a crystal with $N$ atoms in the unit cell, you will get $3N$ phonon branches. The first 3 will be acoustic, and the remaining will be optical. So yes, for a diatomic crystal, we have 3 acoustic modes and 3 optical modes.

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    $\begingroup$ If diatomic crystal means that the crystalline structure is described with a Bravais lattice with a basis of two atoms, there are 3 acoustic and 3 optic branches, also in according with the total $3N$ you report. $\endgroup$ – GiorgioP Apr 4 '19 at 6:56

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