Simple harmonic crystal in one dimension

As a simple example we consider a monatomic one-dimensional solid with identical atoms, one per unit cell, characterized by the Hamiltonian 

It is convenient to define a wave-vector k in the direction of the particle s motion, whose magnitude is (p/u, where a is the lattice spacing 

Mfl is the characteristic position of an atom in the chain. Thus, for each value of k we got an independent equation of motion 

These are the normal modes of this harmonic system. A motion of this type is a collective motion of all atoms according to (from (4.17)) 

Each such oscillation constitutes a wave of wavelength A, = 27r/ A and a corresponding frequency  

These modes of motion with wavelengths and frequencies determined by k are called phonons. A relationship such as (4.25) between k and co is called a dispersion 

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