Such lattice supports intrinsic local modes (ILMs) with their
frequencies above the phonon band characterized by the maximal
harmonic plane waves frequency
The eigenvector of the intrinsic local mode can be found within the rotating-wave apporximation (RWA) where the displacement of the nth particle from its equilibrium position un is described by the following ansatz
A similar ansatz can give the eigenvector of a moving ILM.
For a more complete description of intrinsic local modes look under
The above applet allows you to watch vibrating ILMs in a nonlinear lattice
of 15 particles with periodic boundaries. The vibrational evolution of the
chain is calculated by the molecular-dynamics technique. The parameters of
the lattice are the following:
You can launch either an Odd-Parity ILM (where a central particle has the largest amplitude) or an Even-Parity ILM (where two central particles have the largest but opposite amplitudes). You can also launch a Moving ILM. To see how a harmonic lattice behaves under the same initial conditions turn off the anharmonicity.
The time is shown in units of the smallest period (highest frequency) of the small amplitude plane wave vibrations.
The kinetic energyd of the particle and the potential energy of the bond are shown as the red and the yellow bars, respectively. They are given in arbitrary units and the sum should be a constant, which it is to a very good approximation.
If you wait for 20, 40, or 80 periods you will see the resulting frequency spectrum of the particles with increased resolution. It will appear in the lower panel. The frequency axis is given in terms of the maximal plane wave frequency, .