Number of modes for a 2d crystal

[kuecken]

Hi, I have problems with the following:

Number of modes for a 2d crystal with one atom per primittive unit cell.

1. Number of discrete vibrational modes?
Should there be 2N of them? That is what I come up with if I think of one transverse and one longitudinal mode for each k. But I am unsure.

2. So am I able to obtain two branches from this crystal?
Both of them are acoustical?

Thank you :D

 

[DrDu]

Yes, both conclusions seem to be correct to me.