Density of states in random lattices with translational invariance |
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Authors: | Y M Beltukov D A Parshin |
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Institution: | 1.St. Petersburg State Polytechnical University,St. Petersburg,Russia |
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Abstract: | We propose a random matrix approach to describe vibrations in disordered systems. The dynamical matrix M is taken in the form M = AA
T
, where A is a real random matrix. It guaranties that M is a positive definite matrix. This is necessary for mechanical stability of the system. We built matrix A on a simple cubic lattice with translational invariance and interaction between nearest neighbors. It was found that for
a certain type of disorder acoustical phonons cannot propagate through the lattice and the density of states g(ω) is not zero at ω = 0. The reason is a breakdown of affine assumptions and inapplicability of the macroscopic elasticity
theory. Young modulus goes to zero in the thermodynamic limit. It reminds of some properties of a granular matter at the jamming
transition point. Most of the vibrations are delocalized and similar to diffusons introduced by Allen, Feldman, et al., Phil.
Mag. B 79, 1715 (1999). We show how one can gradually return rigidity and phonons back to the system increasing the width of the so-called
phonon gap (the region where g(ω) ∝ ω2). Above the gap the reduced density of states g(ω)/ω2 shows a well-defined Boson peak which is a typical feature of glasses. Phonons cease to exist above the Boson peak and diffusons
are dominating. It is in excellent agreement with recent theoretical and experimental data. |
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