Robust Bessel-function-based method for determination of the(n,m)indices of single-walled carbon nanotubes by electron diffraction

Abstract

We report a calibration-free method for the determination of chiral indices ͑n , m͒ of single-walled carbon nanotubes from their electron diffraction patterns based on Bessel function analysis of the diffracted layer lines. An approach has been developed for confident identification of the orders of the Bessel functions from the intensity modulations of the diffraction layer lines, to which ͑n , m͒ are correlated. In particular, we critically evaluate the effect of nanotube inclination on the validity of the method and show that the layer lines governed by high-order Bessel functions tolerate higher tilt angles than those of low-order Bessel functions and thus are favored for ͑n , m͒ evaluation. The method is of particular significance in that it considerably enhances the precision of chiral indexing and makes possible the analysis of high-order Bessel functions, especially when EDPs are of relatively low pixel resolution. The technique can be extended to structural analysis of doublewalled carbon nanotubes.

Accurate determination of the chiral indices of individual carbon nanotubes by combining electron diffraction and Resonant Raman spectroscopy

“…4. Second method, introduced by Liu and Qin et al [51,54], analyses the orders of Bessel functions of the main oscillations in the EDP and from them determines the indices n or m of each layer.…”

Determination of helicities in unidirectional assemblies of graphitic or graphiticlike tubular structures

“…Efforts have been undertaken to analyze the structures of such assemblies in CNTs [11][12][13][14][15][16] and in BNNTs [17][18][19] from their electron diffraction patterns ͑EDPs͒, usually with uncertainties due to unmeasured tilt effects of the nanotube with respect to the electron beam and the streaking effects of reflections forming the so-called diffraction "layer lines." 8 Errors are especially large when layer lines overlap, if two or more helicities are close to each other. In general, current methods for electron diffraction analysis are inadequate and there is an urgent need for a reliable and universal method to characterize structural helicity properties in nanotube assemblies.…”

“…Moreover, in combination with high-resolution electron microscopy, the morphology of nanotubes, i.e., whether the tubes are individual, bundled, or multiwalled, can simultaneously be established. In particular, electron diffraction analysis of individual SWCNTs [5][6][7][8][9] or BNNTs 10 enables explicit determination of their atomic structures specified by chiral indices ͑n , m͒.…”

Transport measurements on carbon nanotubes structurally characterized by electron diffraction

“…[25][26][27] Measurements of the axial distance of each layer line from the equatorial line allow both the chiral angle 21 θ and the ratio of the chiral indices 22,23 n/m to be calculated. For the diffraction pattern shown in Fig.…”