−2pi/a^0 −pi/a (^0) k pi/a 2pi/a
ω
Figure 16: Dispersion relation of a 1-dimensional chain of two different types of atoms with similar
masses
6.2 Simple Cubic
Now we take a look at the behavior of phonons in a simple cubic crystal (the indicesp,qandrdefine
the lattice site of the atom in thex-,y- andz-direction).
Figure 17: Simple cubic arrangement of atomsWe assume, that each atom is coupled to its neighbors through linear forces. The spring constant
C 1 shall be assigned to the force between the atom and the 6 nearest neighbors (fig. 17),C 2 shall be
the spring constant for the 12 second-nearest neighbors and so forth. At first we will only consider
the nearest neighbors and because the atoms in different directions are independent from each other,
Newton’s laws looks the same as for the 1-dimensional chain with identical atoms:
m
∂^2 xp,q,r
∂t^2=C 1 (xp− 1 ,q,r− 2 xp,q,r+xp+1,q,r)m
∂^2 yp,q,r
∂t^2
=C 1 (yp,q− 1 ,r− 2 yp,q,r+yp,q+1,r)