Phonon spectrum and vibrational characteristics of linear nanostructures in solid matrices

Abstract

Introduction:The relative simplicity of one-dimensional models often helps to solve the problem in an analytical form from the beginning to the end. For many years, one-dimensional models have retained attractive for a qualitative description of quasi-particle spectra and consequently physical characteristics of solids (see, e.g., [1]. Nondegeneracy of the energy eigenvalues and the absence of singularities on the spectral densities within the continuous spectrum band significantly simplify the problem of determining spectra perturbations induced by various defects [1,2]. At once it should be noted that the formation of localized states in lowdimensional and three-dimensional systems are quite different (see, e. g., [3]). On the one hand, the use of one-dimensional models for the description of vibrations localized on defects in threedimensional crystals reduces the value of this description. On the other hand, the zero-threshold forming of local discrete levels outside the quasi-continuous spectrum band and the strong dependence of energy levels on the defect parameters are of great general importance including applied purposes.The in-stability of the crystalline phase of one-dimensional structures is the main disadvantage for the description of phonon spectra and vibrational characteristics of real systems.There is no long-range order in one-dimensional systems because even at its mean-square displacements diverge [4]. However, if the phonon spectrum of a linear chain starts with a frequency different from zero, this divergence disappears and thus the linear chains can exist in reality and can be studied experimentally. They can exist as the linear chains of atoms in the bulk or on the surface of some solid matrices, i.e., quasi-chains of carbon atoms between graphene monolayers on silicon substrates [5] and chains of gold atoms deposited on the silicon surface [6-8]. The bundles of closed at the ends nanotubes enriched with gases containing quasi-one-0 T =

Uniaxial Heat Balance Model of the Solar Collector

“…As discussed earlier, if we do not take into account the heat transfer at х  0 and temperature Tp above the tubes is constant [16], the following boundary conditions can be specified:…”

Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles

“…It was shown in [10] that a chain of atoms adsorbed in a groove between nanotubes with a sufficiently high degree of accuracy can be considered as a chain in an external field that determines the initial frequency of its quasicontinuous spectrum. Basing on this result, we will consider the chain of atoms adsorbed in the grooves between nanotubes as a chain of atoms in an external periodic field.…”

“…It was shown in [10] that starting with a certain frequency 0 ω , the vibrations of the linear chain deposited on the crystal surface or in the bulk, actually do not extend through the crystal matrix and are completely localized on the chain. The frequency 0 ω is determined by the contribution of the interaction of an atom in the chain with the atoms of the crystal-matrix in the self-interaction matrix of the atom in the chain, Thus, аt 0 ω > ω , neither the structure nor the phonon spectrum of the crystal-matrix can considerably change the spectral characteristics of atoms in the chain.…”

“…At 0 ω > ω , the vibrations of the atoms of the chain are either quasi-localized or their propagation has onedimensional character and the spectral densities of these atoms are well described by simple analytical expressions obtained for one-dimensional models [11]. The bandwidth of the quasi-continuous spectrum of the chain is determined by the interaction between the atoms of the chain [10,11], which depends on the distance r between them. Naturally, this distance is significantly affected by the interaction of adsorbed atoms of the chain with the carbon atoms of nanobundles.…”

“…At 0 r r < , the parameter of noncentral interaction, which determines the width of the spectrum of transverse vibrations, is negative. It will result in a shift of the minimum frequency of the quasi-continuous spectrum to the lowfrequency region [11] and, consequently, to the displacement of the linear part of the heat capacity [10] to the region of lower temperatures. [11].…”