Determination of the inner diameter of a double-walled carbon nanotube from its Raman spectra

Abstract

In this paper, an exact formula is obtained for the inner diameter of double-walled carbon nanotube (DWCNT) as a function of its higher radial breathing mode (RBM) frequency, using the symbolic package of maple software. Its outer diameter is obtained from the inner diameter formula by considering the constant interlayer spacing between two tubes of DWCNT. For this purpose, DWCNT is considered as double concentric elastic thin cylindrical shells, which are coupled through the van der Waals (vdW) forces between two tubes. Lennard-Jones potential is used to calculate the vdW forces between tubes. The advantage of this analytical approach is that in the double concentric elastic shell model all degrees of freedom in the vibrational analysis of DWCNTs are considered. To demonstrate the accuracy of this work, the relationship between RBM frequency of a single-walled carbon nanotube and its radius is deduced from the DWCNT formula that is well consistent with other publications. To illustrate the application of this approach, the diameters of DWCNTs are obtained from their known RBM frequencies which show an excellent agreement with the available experimental results. Also, the influence of changing the geometrical and mechanical parameters of a DWCNT on its RBM frequencies has been investigated.