Editing Carbon nanotube (section)

=== Basic details ===
[[File:Nanotube strip +03 +01.pdf|thumb|A "sliced and unrolled" representation of a carbon nanotube as a strip of a graphene molecule, overlaid on a diagram of the full molecule (faint background). The arrow shows the gap ''A2'' where the atom ''A1'' on one edge of the strip would fit in the opposite edge, as the strip is rolled up]]
[[File:Nanotube strip master.pdf|thumb|left|The basis vectors '''''u''''' and '''''v''''' of the relevant sub-lattice, the (n,m) pairs that define non-isomorphic carbon nanotube structures (red dots), and the pairs that define the enantiomers of the chiral ones (blue dots)]]

The structure of an ideal (infinitely long) single-walled carbon nanotube is that of a regular hexagonal lattice drawn on an infinite [[cylinder|cylindrical]] surface, whose vertices are the positions of the carbon atoms. Since the length of the carbon-carbon bonds is fairly fixed, there are constraints on the diameter of the cylinder and the arrangement of the atoms on it.<ref name="sinnott">{{cite journal | vauthors = Sinnott SB, Andrews R |title=Carbon Nanotubes: Synthesis, Properties, and Applications |journal=Critical Reviews in Solid State and Materials Sciences |date=July 2001 |volume=26 |issue=3 |pages=145–249 |doi=10.1080/20014091104189 |bibcode=2001CRSSM..26..145S |s2cid=95444574 }}</ref>

In the study of nanotubes, one defines a zigzag path on a graphene-like lattice as a [[path (graph theory)|path]] that turns 60 degrees, alternating left and right, after stepping through each bond. It is also conventional to define an armchair path as one that makes two left turns of 60 degrees followed by two right turns every four steps. On some carbon nanotubes, there is a closed zigzag path that goes around the tube. One says that the tube is of the '''zigzag type''' or configuration, or simply is a '''zigzag nanotube'''. If the tube is instead encircled by a closed armchair path, it is said to be of the '''armchair type''', or an '''armchair nanotube'''. An infinite nanotube that is of one type consists entirely of closed paths of that type, connected to each other.

The zigzag and armchair configurations are not the only structures that a single-walled nanotube can have. To describe the structure of a general infinitely long tube, one should imagine it being sliced open by a cut parallel to its axis, that goes through some atom ''A'', and then unrolled flat on the plane, so that its atoms and bonds coincide with those of an imaginary graphene sheet—more precisely, with an infinitely long strip of that sheet. The two halves of the atom ''A'' will end up on opposite edges of the strip, over two atoms ''A1'' and ''A2'' of the graphene. The line from ''A1'' to ''A2'' will correspond to the circumference of the cylinder that went through the atom ''A'', and will be perpendicular to the edges of the strip. In the graphene lattice, the atoms can be split into two classes, depending on the directions of their three bonds. Half the atoms have their three bonds directed the same way, and half have their three bonds rotated 180 degrees relative to the first half. The atoms ''A1'' and ''A2'', which correspond to the same atom ''A'' on the cylinder, must be in the same class. It follows that the circumference of the tube and the angle of the strip are not arbitrary, because they are constrained to the lengths and directions of the lines that connect pairs of graphene atoms in the same class.

Let '''''u''''' and '''''v''''' be two [[linear dependency|linearly independent]] vectors that connect the graphene atom ''A1'' to two of its nearest atoms with the same bond directions. That is, if one numbers consecutive carbons around a graphene cell with C1 to C6, then '''''u''''' can be the vector from C1 to C3, and '''''v''''' be the vector from C1 to C5. Then, for any other atom ''A2'' with same class as ''A1'', the vector from ''A1'' to ''A2'' can be written as a [[linear combination]] ''n'' '''''u''''' + ''m'' '''''v''''', where ''n'' and ''m'' are integers. And, conversely, each pair of integers (''n'',''m'') defines a possible position for ''A2''.<ref name="sinnott" /> Given ''n'' and ''m'', one can reverse this theoretical operation by drawing the vector '''''w''''' on the graphene lattice, cutting a strip of the latter along lines perpendicular to '''''w''''' through its endpoints ''A1'' and ''A2'', and rolling the strip into a cylinder so as to bring those two points together. If this construction is applied to a pair (''k'',0), the result is a zigzag nanotube, with closed zigzag paths of 2''k'' atoms. If it is applied to a pair (''k'',''k''), one obtains an armchair tube, with closed armchair paths of 4''k'' atoms.