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