Normal Modes in a Linear Chain of Atoms

Thus far, we have introduced no quantum effects into the discussion. However, we also considered the linear chain to be infinite in extent thus we avoided applying any boundary conditions at the end points. However, now we must consider finite systems and apply appropriate boundary conditions that will allow waves to traverse the solid. Let the length of the chain be L and assume it contains N +1 atoms with spacing a = L/N. Recall from the treatment of electrons in a box, forcing the wave function to be zero at the boundaries of the crystal allowed only standing waves. To permit traveling waves, we chose periodic boundary conditions which, for the case of elastic waves, can be written as u(x + L,t) = u(x,t). Thus the solution must be in the form exp(ikna) = exp [ik(na + L)], which requires k to be given by 

The only values of k that produce unique frequencies lie between n/fl and ir/ i. If the allowed values of k are spaced 2tt/L apart, the total number of states is 

Thus we arrive at the important conclusion that a finite chain having N +1 particles has N normal modes of longitudinal vibration. The same arguments used above can be generalized to three dimensions and it can be shown that two transverse modes of vibration are also possible. Therefore, a chain of N +1 atoms can have 3N discrete modes of vibration. Thus the fc-vector, which was continuous for an infinite chain, is now quantized into 3N discrete states. The energy of each of these states is assigned to be hoj. Because of the close resemblance of these quantized vibrational states to photons in a cavity, these quantized waves are called phonons. 

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