Consider a monatomic 1D lattice of N particles of mass
m, where the nearest-neighbors are connected by springs having
a positive even order anharmonicity. The potential could consist of a
harmonic and quartic terms in the form
V(x) = K2/2 x2 + K4/4 x4
where K2, K4
and x is the deviation of
the spring's length from its equilibrium value.
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.
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
(Next >> 1 ) 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
(K4 < 0).
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.
 A.J.Sievers and S.Takeno, Phys. Rev. Lett. 61, 970 (1988).
Last modified: August 12, 1997