Determination of the chiral indices (n,m) of carbon nanotubes by electron diffraction

Abstract

The atomic structure of a carbon nanotube can be defined by the chiral indices, (n,m), that specify its perimeter vector (chiral vector), with which the diameter and helicity are also determined. The fine electron beam available in a modern Transmission Electron Microscope (TEM) offers a unique and powerful probe to reveal the atomic structure of individual nanotubes. This article covers two aspects related to the use of the electron probe in the TEM for the study of carbon nanotubes: (i) to express the electron diffraction intensity distribution in the electron diffraction patterns of carbon nanotubes and (ii) to obtain the chiral indices (n,m) of carbon nanotubes from their electron diffraction patterns. For a nanotube of given chiral indices (n,m), the electron scattering amplitude from the carbon nanotube can be expressed analytically in closed form using the helical diffraction theory, from which its electron diffraction pattern can be calculated and understood. The reverse problem, i.e., assignment of the chiral indices (n,m) of a carbon nanotube from its electron diffraction pattern, is approached from the relationship between the electron diffraction intensity distribution and the chiral indices (n,m). The first method is to obtain indiscriminately the chiral indices (n,m) by reading directly the intensity distribution on the three principal layer lines, l₁, l₂, and l₃, which have intensities proportional to the square of the Bessel functions of orders m, n, and n + m: I_l1 ∝ |J_m (πdR)|², I_l2 ∝ |J_n (πdR)|², and I_l3 ∝ |J_n+m (πdR)|². The second method is to obtain and use the ratio of the indices n/m = (2D₁ − D₂)/(2D₂ − D₁) in which D₁ and D₂ are the spacings of principal layer lines l₁ and l₂, respectively. Examples of using these methods are also illustrated in the determination of chiral indices of isolated individual single-walled carbon nanotubes, a bundle of single-walled carbon nanotubes, and multi-walled carbon nanotubes.