Assignment of the chiralities of double-walled carbon nanotubes using two radial breathing modes

The chiralities of double-walled carbon nanotubes (DWNTs) are assigned using two radial breathing modes (RBMs) of the DWNTs unambiguously, which are calculated theoretically in this paper by use of the modified continuum model. According to experimental data, we assign the specific chiralities to the inner and outer layers by use of two RBM frequencies of the DWNTs, and then calculate the diameters of the two layers and even the interlayer distance. The interlayer spacing between two layers b is found to be not a constant, but is in the range 0.321 to 0.402 nm, which agrees well with experimental data [W. Ren et al., Chem. Phys. Lett. 359, 196 (2002); R. Pfeiffer et al., Phys. Rev. Lett. 90, 225501 (2003)] and calculated result [R. Saito et al., Chem. Phys. Lett. 348, 187 (2001)]. The values of both the upshift and downshift of DWNTs' RBM frequencies are not constant, but vary with the interlayer spacings b and their chiralities. By calculating and comparing the case of the chiralities being taken into account with that of the chiralities being neglected in the van der Waals forces, we find that the effect of chiralities should not be neglected in order to assign the chiralities of DWNTs though the difference between the two RBM frequencies caused by the different chiralities of inner and outer tubes is small. In addition, it is found that there can be several values of ω_B1 (ωB₂) corresponding to the same value of ω_B2 (ω_B1), arising because one inner (outer) tube can be paired with several outer (inner) tubes, which can be explained by two RBM frequencies of the DWNTs.