Radial breathing mode frequencies of carbon nanotubes for determination of their diameters

In this paper, exact formulas are obtained for the radial breathing mode (RBM) frequencies of triple-walled carbon nanotubes (TWCNTs) using symbolic package in MAPLE software. For this purpose, TWCNT is considered as triple concentric elastic thin cylindrical shells, which are coupled through van der Waals (vdW) forces between two adjacent tubes. Lennard–Jones potential is used to calculate the vdW forces between adjacent tubes. Then, explicit formulas for RBM frequencies of single-walled (SW), and double-walled (DW) CNTs have been deduced from TWCNT formulas that show an excellent agreement with the available experimental results and the other theoretical model results. The advantage of this analytical approach is that the elastic shell model considers all degrees of freedom in the vibrational analysis of CNTs. To demonstrate the accuracy of this work, the RBM frequencies of different multi-walled carbon nanotubes (MWCNTs) are compared with the available experimental or atomistic results with relative errors of less than 1.5%. To illustrate the application of this approach, the diameters of DWCNTs are obtained from their RBM frequencies which show an excellent agreement with the available experimental results. Also, this approach can be used to determine the diameters of the TWCNTs and MWCNTs. The influence of changing the geometrical and mechanical parameters of a TWCNT on its RBM frequencies has been investigated, too.