Periodic Boundary Conditions and Dispersion Relations

We begin with the linear chain with two atoms of masses m and m per unit cell of dimension a (Fig.2.1). 

The chain considered contains N unit cells numbered by an index i = 1.N. In each unit cell the masses are numbered by an index k = 1, 2 and their equilibrium positions in each cell are x(k). We could of course choose the origin 

The Hamiltonian of the system is the sum of the kinetic energy T and the potential energy 0 

The equations of motion are obtained by calculating the force -9 /9u( ) acting on the atom ( )  

If q 0, (2.7a) with (2.8) represents a wave travelling to the right and describes the undamped propagation of a disturbance through the chain. 

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