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We develop a continuous theory of low-frequency dynamics for single-walled carbon nanotubes (SWCNTs) weakly interacting with the environment. In the frame of the approach proposed we obtain temperature dependence of SWCNTs specific heat in the low (T<40 K) and ultra-low (T<2.5 K) temperature ranges. We take into account the main term in the coupling between SWCNT and the environment that slightly increases the frequencies of those SWCNT modes, which possess predominantly radial polarization. The coupling drastically decreases the density of phonon states in the lowest frequencies region. The theoretically predicted fall of the specific heat in the interval T<2.5 K properly explains available experimental data in contrast to the preceding approaches. The theory proposed can be the basis for studies of low-temperature heat capacity and phonon dynamics of many other single-walled and multi-walled tubular structures (boron nitride, transition metal dichalcogenides) which have emerged in the past decade. PACS number(s): 78.67.Ch, 62.30.+d.

Most likely, both layers in double-walled carbon nanotubes have the same handedness.

A new group-theory method to relate a spectrum of carbon nanotubes with that of a graphene monolayer is elaborated. The spectrum reconstruction is performed using a virtual intermediate planar periodic structure. Selected irreducible representations of its space symmetry group and all those of the nanotube one are isomorphic and therefore span the correlated excitations. The method is applied to study the origin of the zone-center phonon modes of chiral and achiral carbon nanotubes. The structure of the G-band for particular types of nanotubes is determined. The results obtained could be useful for experimental indexing of the carbon nanotubes by means of Raman spectroscopy.Keywords: single-walled carbon nanotubes, graphene, Raman spectroscopy, zone-folding theory, factor-group analysis, phonon modes A graphene layer (GL) possesses a higher symmetry with respect to a single wall carbon nanotube (SWCNT) and therefore, has relatively simple phonon dispersion branches structure [1,2]. Comprehensive understanding of general correlations between phonons of GL and those of the SWCNT is obviously required. The Brillouin zone folding scheme, explaining formation of the one-dimensional reciprocal space of SWCNTs from the two-dimensional space of the GL was developed more than 10 years ago [3]. This fruitful theory explained electronic zone structure of the SWCNT and the criterion for classification of SWCNTs as metallic or semiconducting ones was elaborated [3]. Besides, the factor-group analysis of SWCNTs vibrational spectra was carried out as well, and the lists of Raman-and IR-active modes (in both chiral and achiral cases) were found [4,5]. Later, it was demonstrated that all SWCNTs irrespective to their chirality possess the higher symmetry than it had been considered before and the factor-group analysis of SWCNTs vibrational spectra was revised [6]. Since the higher symmetry imposed more rigorous selection rules the number of Raman-and IR-active modes reported in [6] was smaller in comparison with earlier results [4,5]. Although the authors of Refs. 3-5 admitted [7] the results of [6], the revision of the GL reciprocal space folding theory still has not been carried out (see, for example, the latest review [9]). In our opinion, an application of the initial theory [3] leads to the correct results in those cases only, for which the exact symmetry of SWCNT vibrational or electronic states is not very important. The earlier theory [3] describes correctly the origin, number and polarization of the Raman-active vibrational modes in the case of the lowest-symmetry chiral SWCNTs, but it is not so for higher-symmetry achiral tubes. However, an approach, considering the origin of all IR-and Raman-active modes in achiral SWCNTs and consisting with the correct symmetry [6,12] is not known till now.The aim of this Letter is to develop a general approach describing both the origin and symmetry of vibrational modes in the SWCNTs of all possible types. We pay a particular attention to the origin of Raman-active modes of achiral S...

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