Determination of SWCNT diameters from the Raman response of the radial breathing mode

We report on the evaluation of the distribution of diameters for nanotube samples with a wide variation of mean diameters. Such results were obtained from a detailed analysis of the radial breathing mode Raman response and compared to results obtained from an evaluation of optical spectra and X-ray diffraction pattern. The evaluation of the Raman data needs a well refined analysis as the experimental analysis exhibits a rather complicated and oscillating relation between response and exciting laser. Both, an exact calculation where the density of states was considered explicitly and an approximate calculation were applied. Both models used for the analysis are able to explain several unexpected results from the experiment such as the oscillating behavior of the spectral moments, unusual discontinuities in the first moments of the Raman response for excitation in the IR, a fine structure for the response in optics and Raman, and an up shift of the RBM frequency as compared to qualified ab initio calculations. In detail the first moment and the variance of the spectra were used for the evaluation of the diameter distribution. To obtain good results between experimental and theoretical oscillation pattern the transition energy between the first two van Hove singularities had to be scaled up which is considered as a result from coulomb interaction of the electrons in the tubular material. On the other hand the analysis does not only allow to determine the mean value and the width of the diameter distribution but yields also a value for the average bundle diameters or, alternatively, the strength of the tube-tube interaction. The model used for the analysis of the Raman data is also appropriate to analyze the optical response, at least for the spectral range from 0.5 eV to 3.5 eV. The fine structure in the response for the transitions between the three lowest van Hove singularities is well reproduced and the mean tube diameters and their distribution is obtained in very good agreement with the results from the Raman analysis. From the X-ray analysis the same mean values and comparable distributions for the tube diameters were received whereas the bundle diameters could not be retained with high precision in this case.