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

We report a calibration-free method for the determination of chiral indices $(n,m)$ 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 $(n,m)$ 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 $(n,m)$ 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 double-walled carbon nanotubes.

Figure 1

(Color online) A simulated electron diffraction pattern from a (12,7) chiral SWCNT in normal incidence. The innermost two visible diffraction hexagons from one set of patterns are schematically illustrated, with vertices of hexagons indexed based on the honeycomb lattice structure of a graphene layer.

Figure 2

(Color online) (a) Two Bessel factors with neighboring orders $J_{\mid \nu \mid }^2\left(x\right)$ and $J_{\mid \nu \mid +1}^2\left(x\right)$. A higher-order Bessel factor has a wider intensity gap than a lower order Bessel factor. (b) The distance between the first pair of roots $W_{\mathrm{MZ}1}$, and the pair of highest peaks $W_{\mathrm{TP}1}$, and the interval between the first two positive roots ${\Delta }_{12}$ are shown as functions of the Bessel order.

Figure 3

(Color online) Planar cuts of the reciprocal counterpart of a SWCNT by the Ewald sphere which is approximated to be a plane for high-energy electrons: (a) in normal incidence; (b) when the tube is tilted by an angle $\tau$ with respect to the incident electron beam.

Figure 4

(Color online) (a) A simulated diffraction pattern of a chiral (40,5) SWCNT. (b) The diffraction disk for the corresponding (0,1) reflection. (The diffraction disk for the (0,1) reflection of the (40,5) SWCNT shown in (b) is duplicated from Ref. 21.) (c)–(e) The intensity profiles along the secant lines for tilts at $\tau =0°$, 6.5°, and 10.5°, respectively.

Figure 5

(Color online) Simulated electron diffraction patterns of a (29,2) SWCNT, with tilt angles of (a) 0°, (b) 3°, and (c) 6°, respectively. (d)–(i) Intensity profiles along layer lines $L_1$ and $L_2$ in the diffraction patterns of respective tilt angles.

Figure 6

(Color online) (a) A high-resolution TEM image of an individual SWCNT. (b) The corresponding experimental electron diffraction pattern. (c)–(d) Intensity profiles along the $L_1$ and $L_2$ layer lines indicated in (b).