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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