Phonons as harmonic oscillators

We next draw the analogy between the phonon hamiltonian and that of a collection of independent harmonic oscillators. We begin with the most general expression for the displacement of ions, Eq. (6.22). In that equation, we combine the amplitude c with the time-dependent part of the exponential exp(—i t) into a new variable Q t), and express the most general displacement of ions as  

These quantities must be real, since they describe ionic displacements in real space. Therefore, for the time dependence we should only consider the real part, that is cos( j 0- Moreover, the coefficients of factors exp(ik R ) and exp(i(—k) R ) that appear in the expansion must be complex conjugate, leading to the relation 

The kinetic energy of the system of ions will be given in terms of the displacements Snj(t) as 

To obtain the last expression we have also used Eqs. (6.10) and (6.11) to relate the displacements to the force-constant matrix and the frequency eigenvalues Combining the expressions for the kinetic and potential energies, we obtain the total energy of a system of phonons  

In practice, if the atomic displacements are not too large, the harmonic approximation to phonon excitations is reasonable. For large displacements anharmonic terms become increasingly important, and a more elaborate description is necessary which takes into account phonon-phonon interactions arising from the anharmonic terms. Evidently, this places a limit on the number of phonons that can be excited before the harmonic approximation breaks down. 

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