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Linear Chain - Classical treatment[edit]

In order to simplify the analysis of a 3-dimensional lattice of atoms it is convenient to model a 1-dimensional lattice or linear chain. The forces between the atoms are assumed to be linear and nearest-neighbour, and they are represented by an elastic spring. Each atom is assumed to be a point particle and the nucleus and electrons move in step.(adiabatic approximation)

    n-1   n   n+1    d  


→→  →→→

Where labels the n'th atom, is the distance between atoms when the chain is in equilibrium and the displacement of the n'th atom from it's equilibrium position.
If C is the elastic constant of the spring and m the mass of the atom then the equation of motion of the n'th atom is :

This is a set of coupled equations and since we expect the solutions to be oscillatory, new coordinates can be defined by a discrete Fourier transform, in order to de-couple them.[1]


Here replaces the usual continuous variable . The are known as the normal coordinates. Substitution into the equation of motion produces the following decoupled equations.(This requires a significant amount of algebra using the orthonormality and completeness relations of the discrete fourier transform [2])

These are the equations for harmonic oscillators which have the solution:

Each normal coordinate represents an independent vibrational mode of the lattice with wavenumber which is known as a normal mode. The second equation for is known as the dispersion relation between the angular frequency and the wavenumber.[3]

  1. ^ Mattuck R. A guide to Feynman Diagrams in the many-body problem
  2. ^ Greiner & Reinhardt. Field Quantisation
  3. ^ Donovan B. & Angress J.; Lattice Vibrations