Posted by Steve Simon on May 27, 2014, 4:49 pm, in reply to "Periodicity phonon dispersion relation"
This is done the same way it is done in one dimension. The wave motion for the atoms in a crystal can be written as U_a(R) = A_a e^{i k dot R  i omega t} where k is the 3d wavevector, R is the 3d position of the lattice vector of a given unit cell U_a is the displacement vector of atom a within the unit cell associated with lattice vector R. Given this, if we displace k to k+G, using e^{i G R} = 1 for G a reciprocal lattice vector, then we get back exactly the same physical displacements. 
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