FIGURE 1. Relation between the hexagonal carbon lattice and the chirality of carbon nanotubes.
From the following article:
Electronic structure of atomically resolved carbon nanotubes
Jeroen W. G. Wilder, Liesbeth C. Venema, Andrew G. Rinzler, Richard E. Smalley and Cees Dekker
Nature 391, 59-62(1 January 1998)
doi:10.1038/34139
A, the construction of a carbon nanotube from a single graphene sheet. By rolling up the sheet along the wrapping vector C, that is, such that the origin (0,0) coincides with point C, a nanotube indicated by indices (11,7) is formed. Wrapping vectors along the dotted lines lead to tubes that are zigzag or armchair. All other wrapping angles lead to chiral tubes whose wrapping angle is specified relative to either the zigzag direction (
) or to the armchair direction (
= 30° - ). Dashed lines are perpendicular to C and run in the direction of the tube axis indicated by vector T. The solid vector H is perpendicular to the armchair direction and specifies the direction of nearest-neighbour hexagon rows indicated by the black dots. The angle between T and H is the chiral angle . B, Atomically resolved STM images of individual single-walled carbon nanotubes. The lattice on the surface of the cylinders allows a clear identification of the tube chirality. Dashed arrows represent the tube axis T and the solid arrows indicate the direction of nearest-neighbour hexagon rows H. Tubes no. 10, 11 and 1 are chiral, whereas tubes no. 7 and 8 have a zigzag and armchair structure, respectively. Tube no. 10 has a chiral angle = 7° and a diameter d = 1.3 nm, which corresponds to the (11,7) type of panel A. A hexagonal lattice is plotted on top of image no. 8 to clarify the non-chiral armchair structure. Carbon nanotubes were synthesized as described in ref. 14. TEM studies14 have shown that the material consists mainly of 
1.4-nm-thick single-walled nanotubes. These were deposited from a dispersion in 1,2 dichloroethane on single-crystalline Au(111) facets. Topographic images were obtained by recording the tip height at constant tunnel current in a home-built STM25 operated at 4 K. The Pt/Ir tips were cut in ambient air by scissors. Typical bias parameters are those of image no. 10, that is, a tunnel current I = 60 pA, and a bias voltage Vbias = 500 mV.
