Motion of Atoms in a Diatomic Chain

Next consider a chain of two different atomic species in which the two species alternate. We can write the equations of motion for each species, which are similar to Equation 16.4. 

Now we have a coupled set of second-order differential equations that must be solved simultaneously. Notice that coupling comes about through the odd nearest neighbors of the even atoms and vice versa. The solutions can be written as 

Putting these back into the differential equations, we get  

If the inverse of the above matrix exists, the only solution possible is for and 17 = 0, which is the trivial solution. The only other way to satisfy this equation is for the inverse matrix not to exist, which requires its determinant to vanish. Thus the solution to the simultaneous differential equations requires that 

Dispersion relationship for chain of alternating atoms with different masses. For this plot, M = 1.2 m. There is an energy gap at ir/2fl which closes as m M. The frequencies in this gap cannot be propagated by the chain. If M = m, the gap would close and the lower branch would 

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