Consider a chain of particles of mass

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
*n*th particle from its equilibrium position
** u_{n}**
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"***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:
**
**.

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, .

Last modified: August 14, 1997