One effect of such a positive anharmonicity is to increase slightly the frequency of each mode in the plane wave spectrum, but the eigenvectors will still extend over the entire lattice as long as the frequencies are within the phonon band.

Another effect is the localization of a vibrational mode in this
perfect anharmonic lattice if the localized amplitude at a site is
sufficiently large with respect to that of a plane wave[1].
The rms amplitude of the vibration of a particle is
.
For the case of an extended mode, the number of particles
vibrating within such a mode is large
(** N** ) so the rms amplitude at each
site will be small, as shown in Fig.1.
Since the highest frequency mode is not bounded from above by
another mode it can split off from the top of the plane wave
spectrum as shown in Fig.2 and become localized.

Another example of the anharmonic localization of lattice
vibrations can be given with a 1-D diatomic chain with a negative
anharmonicity in the interaction between nearest-neighbors
(** K**).
Figure 2(b) illustrates the anharmonic localization of the
mode that originally came from the bottom of the optic branch. In this
case the localization still produces a larger amplitude at a few atoms
which again increases the effective anharmonicity of the potential,
but now a lower frequency results.
Strictly speaking, such anharmonically
driven localization of lattice vibrations is only possible when there
is a frequency gap in the plane wave spectrum.

[1] A.J.Sievers and S.Takeno, Phys. Rev. Lett.

Last modified: August 12, 1997