Special frequencies and Lifshitz singularities in binary random harmonic chains |
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Authors: | Th M Nieuwenhuizen J M Luck J Canisius J L van Hemmen and W J Ventevogel |
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Institution: | (1) Institute for Theoretical Physics, R.U.U., 3508 TA Utrecht, The Netherlands;(2) Service de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette, France;(3) Sonderforschungsbereich 123, Universität Heidelberg, 6900 Heidelberg 1, Germany;(4) Present address: Institut für Theoretische Physik A, RWTH Aachen, 5100 Aachen, Germany |
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Abstract: | We consider a one-dimensional chain of coupled harmonic oscillators; the mass of each atom is a random variable taking only two values (M or 1). We investigate the integrated density of statesH(2) near special frequencies: a given frequency
s
with rational wavelength becomes special if the mass ratioM exceeds a certain critical valueM
c
. We show thatH has essential singularities of the typesH
sg
exp(–C
1 ¦2–
s
2
¦–1/2) or exp(–C
2¦2–
s
2
¦–1), according to the value ofM and the sign of (2–
s
2
). The Lifshitz singularity at the band edge is analyzed in the same way. In each case, the constantC
1 orC
2 is evaluated explicitly and compared with a vast amount of numerical work. All these exponential singularities are modulated by periodic amplitudes. The properties of the eigenfunctions with frequencies close to the special values are also discussed, and are illustrated by numerical data. |
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Keywords: | Density of states random harmonic chains one-dimensional systems special frequencies Lifshitz singularities |
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