Example phonons in 2D periodic chain

We next apply the force-constant model to a simple example that includes all the essential features needed to demonstrate the behavior of phonons. Our example is based on a system that is periodic in one dimension (with lattice vector ai = (a/V5)[x -I- y]), but exists in a two-dimensional space, and has two atoms per unit cellatpositionsti = a/2 )k, ti = —(a/2V2)x. This type of atomic arrangement 

We notice that, since we have adopted the approximation of nearest neighbor interactions, for the diagonal elements (/ = 1,2 a = jc, y) we must only use 

The solution of this matrix gives for the frequency eigenvalues 

A plot of the two different eigenvalues, denoted in the following by cu from the sign in front of the square root, is given in Fig. 6.3. We have used values of the force constants such that Kr Kg since the cost of stretching bonds is always greater than the cost of bending bonds. 

It is instructive to analyze the behavior of the eigenvalues and the corresponding eigenvectors near the center (k = 0) and near the edge of the BZ (k = (7t/a)x). This gives the following results  

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