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Title: Lattice vibrations in the Frenkel-Kontorova model. I. Phonon dispersion, number density, and energy

We studied the lattice vibrations of two inter-penetrating atomic sublattices via the Frenkel-Kontorova (FK) model of a linear chain of harmonically interacting atoms subjected to an on-site potential, using the technique of thermodynamic Green's functions based on quantum field-theoretical methods. General expressions were deduced for the phonon frequency-wave-vector dispersion relations, number density, and energy of the FK model system. In addition, as the application of the theory, we investigated in detail cases of linear chains with various periods of the on-site potential of the FK model. Some unusual but interesting features for different amplitudes of the on-site potential of the FK model are discussed. In the commensurate structure, the phonon spectrum always starts at a finite frequency, and the gaps of the spectrum are true ones with a zero density of modes. In the incommensurate structure, the phonon spectrum starts from zero frequency, but at a non-zero wave vector; there are some modes inside these gap regions, but their density is very low. In our approximation, the energy of a higher-order commensurate state of the one-dimensional system at a finite temperature may become indefinitely close to the energy of an incommensurate state. This finding implies that the higher-order incommensurate-commensurate transitionsmore » are continuous ones and that the phase transition may exhibit a “devil's staircase” behavior at a finite temperature.« less
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  1. Brookhaven National Lab. (BNL), Upton, NY (United States)
Publication Date:
OSTI Identifier:
Report Number(s):
Journal ID: ISSN 1098-0121; PRBMDO; R&D Project: MA015MACA; KC0201010
Grant/Contract Number:
Accepted Manuscript
Journal Name:
Physical Review. B, Condensed Matter and Materials Physics
Additional Journal Information:
Journal Volume: 91; Journal Issue: 22; Journal ID: ISSN 1098-0121
American Physical Society (APS)
Research Org:
Brookhaven National Laboratory (BNL), Upton, NY (United States)
Sponsoring Org:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)
Country of Publication:
United States