Intrinsic Local Modes

Consider a chain of particles of mass m where the nearest-neighbors are connected by the anharmonic springs. The anharmonic interparticle potential has the following form
V = K2 x2 / 2 + K4 x4 / 4 ,
where K2>0 and K4>0 are the harmonic and quartic anharmonic terms, respectively, and x is the deviation of the spring's length from its equilibrium value.

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

where is the amplitude of the mode, and characterizes its ac displacement pattern. Substitution of the above ansatz into the classical equations of motion

allows one to find the mode eigenvector. The ILM's eigenvector is a wave package which transfers to a lattice envelope soliton in a limit of a weak anharmonicicty.

A similar ansatz can give the eigenvector of a moving ILM.

For a more complete description of intrinsic local modes look under "Some References" or in the review article: S. A. Kiselev, S. R. Bickham, and A. J. Sievers, "Properties of Intrinsic Localized Modes in One-Dimensional Lattices", Comments Cond. Mat. Phys, 17, 135-173 (1995).

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: m=1, K2=1, K4=10.

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, .