While it is not surprising that the loss of periodicity in
defect crystals leads to localized vibrational phenomena, the standard
description of the dynamics of defect-free periodic lattices in terms
of plane wave phonons is so deeply ingrained that it was surprising to
many researchers when it was argued theoretically in the latter half
of the 1980's [1-3] that the presence of strong quartic anharmonicity
in perfect lattices can also lead to localized vibrational modes,
henceforth called "intrinsic localized modes" (ILMs). Sievers and
Takeno studied the classical vibrational dynamics of a simple
one-dimensional monatomic chain of particles interacting via
nearest-neighbor harmonic and quartic anharmonic springs, and they
used a "rotating wave approximation" (RWA), in which just one
frequency component was kept in the time dependence. For the case of
sufficiently strong positive quartic anharmonicity, it was found that
the lattice could sustain stationary localized vibrations having the
approximate mode pattern
In the years since the appearance of these papers, there has been a rapidly increasing number of theoretical studies published related to ILMs. The ILM papers published since 1988 roughly divide into two broad and overlapping categories. The first of these involves the relation of the new excitations to the general behavior of discrete nonlinear systems, with emphasis on connections to soliton-like behavior in lattices. The second category is more tightly focused on generalizations and extensions of the 1988 papers, in order to discover the properties of highly-localized large-amplitude ILMs and also to assess their likely importance for real solids. Computer studies of the dynamics of nonlinear lattices extend back to the pioneering work on one-dimensional chains by Pasta, Fermi and Ulam [5] , and they have been a vital adjunct to much of the subsequent work on solitons in anharmonic lattices. Discussions of both analytic and numerical aspects of soliton behavior in one-dimensional lattices with intersite cubic and quartic anharmonic interactions are found in Flytzanis, et al. [6] for the monatomic case and in Pnevmatikos, et al. [7] for the diatomic case. Of particular interest for ILMs is their relationship to lattice envelope solitons. Stationary lattice solitons were studied theoretically over 20 years ago by Kosevich and Kovalev [8] for anharmonic chains with cubic and quartic onsite and intersite anharmonicity. The emphasis was on onsite anharmonicity, and solutions were obtained for low-anharmonicity stationary envelope solitons whose spatial extent is broad compared with the lattice constant. Later MD simulations by Yoshimura and Watanabe [9] for linear chains with quadratic plus quartic interactions showed that the stationary envelope soliton solutions account well for ILMs whose spatial widths are sufficiently broad, e.g. 10 or more lattice constants. However, as was subsequently emphasized by Kosevich [10], the stationary envelope soliton solutions do not describe the large-anharmonicity highly localized ILMs which are stabilized by lattice discreteness.
This history is after Ref. [11] ...
(Under construction)
[1] | A.S. Dolgov, Sov. Phys. Solid State 28, 907 (1986). |
[2] | A.J. Sievers and S. Takeno, Phy. Rev. Lett. 61, 970 (1988). |
[3] | S. Takeno and A.J. Sievers, Sol. St. Comm. 67, 1023 (1988). |
[4] | A.J. Sievers and S. Takeno, Phys. Rev. B 39, 3374 (1989). |
[5] | E. Fermi, J.R. Pasta, and S.M. Ulam, in Collected Works of E. Fermi, edited by E. Segre (University of Chicago Press, Chicago, 1955), |
[6] | N. Flytzanis, S. Pnevmatikos, and M. Remoissenet, J. Phys. C. 18, 4603 (1985). |
[7] | S. Pnevmatikos, N. Flytzanis, and M. Remoissenet, Phys. Rev. B 33, 2308 (1986). |
[8] | A.M. Kosevich and A.S. Kovalev, Sov. Phys.-JETP 40, 891 (1974). |
[9] | K. Yoshimura and S. Watanabe, J. Phys. Soc. Japan 60, 82 (1991). |
[10] | Y.A. Kosevich, Phys. Lett. A 173, 257 (1993). |
[11] | A.J. Sievers and J.B. Page, in Dynamical Properties of Solids: Phonon Physics The Cutting Edge, edited by G.K. Norton and A.A. Maradudin (North Holland, Amsterdam, 1995), Vol. VII, p. 137. |