**Coexistence of Intrinsic Local Mode**

and Plane Waves

Perhaps these vibrations can be treated as localized normal modes,
i.e. localized degrees of freedom, of the perfect anharmonic lattice,
and if so, are these degrees of freedom distinguishable from those
associated with extended plane waves? If so, then MD
simulations should clearly show that the extended degree of freedom
converts to the local type in the presence of anharmonicity. To test
this proposal, consider a monatomic chain of 21 particles with a
nearest neighbor anharmonic potential and cyclic boundary
conditions. As initial conditions for the first of the two MD
simulation tests, all particles are randomly displaced from their
equilibrium positions with small amplitudes. The Fourier spectrum of
the particles motion shown in the upper panel of the Figure consists of a
set of peaks corresponding to all of the allowed normal mode
frequencies. Because of the cyclic boundary conditions, there are two
modes at each frequency so that ten peaks in the power spectrum with
nonzero frequencies are expected and observed. Those together with the
translational mode give the 21 expected degrees of freedom.
The situation changes dramatically when the vibrational amplitude at
some site increases enough to make the effective anharmonicity
comparable to the harmonic forces. When the eigenvector of the
odd-parity anharmonic local mode is excited on top of a background of
small amplitude random displacements, the resultant power spectrum is
shown in the lower panel of the Figure. It contains a high frequency peak
corresponding to the anharmonic local excitation, as well as the set
of small peaks at the expected plane wave frequencies. The local mode
peak is clearly separated from the plane wave spectrum, and the number
of plane wave peaks is decreased from 10 to 9.
Because of the translational symmetry requirement, this
localized excitation evolves from two plane wave modes.
The number of degrees of
freedom of this anharmonic system is therefore conserved
(18+2+1).
Note that the frequencies of the plane waves shown in the lower panel
have shifted slightly to higher frequencies, but are still confined to
the phonon band of the harmonic spectrum as expected from Rayleigh
theorem[1].

[1] A. A. Maradudin, et al., *Theory of Lattice Dynamics in the
Harmonic Approximation.* (Academic Press, New York, 1971), vol. 3.

Last modified: August 12, 1997