Editing Carbon nanotube (section)

== Types ==
The structure of the nanotube is not changed if the strip is rotated by 60 degrees clockwise around ''A1'' before applying the hypothetical reconstruction above. Such a rotation changes the corresponding pair (''n'',''m'') to the pair (−2''m'',''n''+''m''). It follows that many possible positions of ''A2'' relative to ''A1'' — that is, many pairs (''n'',''m'') — correspond to the same arrangement of atoms on the nanotube. That is the case, for example, of the six pairs (1,2), (−2,3), (−3,1), (−1,−2), (2,−3), and (3,−1). In particular, the pairs (''k'',0) and (0,''k'') describe the same nanotube geometry. These redundancies can be avoided by considering only pairs (''n'',''m'') such that ''n'' > 0 and ''m'' ≥ 0; that is, where the direction of the vector '''''w''''' lies between those of '''''u''''' (inclusive) and '''''v''''' (exclusive). It can be verified that every nanotube has exactly one pair (''n'',''m'') that satisfies those conditions, which is called the tube's '''type'''. Conversely, for every type there is a hypothetical nanotube. In fact, two nanotubes have the same type if and only if one can be conceptually rotated and translated so as to match the other exactly. Instead of the type (''n'',''m''), the structure of a carbon nanotube can be specified by giving the length of the vector '''''w''''' (that is, the circumference of the nanotube), and the angle ''α'' between the directions of '''''u''''' and '''''w''''',
may range from 0 (inclusive) to 60 degrees clockwise (exclusive). If the diagram is drawn with '''''u''''' horizontal, the latter is the tilt of the strip away from the vertical.
{{multiple image
| caption           = Unrolled nanotube diagrams
| image1            = nanotube strip +03 +01.pdf
| caption1          = Chiral nanotube of the (3,1) type
| image2            = nanotube strip +01 +03.pdf
| caption2          = Chiral nanotube of the (1,3) type, mirror image of the (3,1) type
| image3            = nanotube strip +02 +02.pdf
| caption3          = Nanotube of the (2,2) type, the narrowest "armchair" one
| image4            = nanotube strip +03 000.pdf
| caption4          = Nanotube of the (3,0) type, the narrowest "zigzag" one
| align             = center
}}

=== Chirality and mirror symmetry ===

A nanotube is [[chirality|chiral]] if it has type (''n'',''m''), with ''m'' > 0 and ''m'' ≠ ''n''; then its [[enantiomer]] (mirror image) has type (''m'',''n''), which is different from (''n'',''m''). This operation corresponds to mirroring the unrolled strip about the line ''L'' through ''A1'' that makes an angle of 30 degrees clockwise from the direction of the '''''u''''' vector (that is, with the direction of the vector '''''u'''''+'''''v'''''). The only types of nanotubes that are [[achiral]] are the (''k'',0) "zigzag" tubes and the (''k'',''k'') "armchair" tubes. If two enantiomers are to be considered the same structure, then one may consider only types (''n'',''m'') with 0 ≤ ''m'' ≤ ''n'' and ''n'' > 0. Then the angle ''α'' between '''''u''''' and '''''w''''', which may range from 0 to 30 degrees (inclusive both), is called the "chiral angle" of the nanotube.

=== Circumference and diameter ===

From ''n'' and ''m'' one can also compute the circumference ''c'', which is the length of the vector '''''w''''', which turns out to be:

: <math> c = \left|\boldsymbol{u}\right| \sqrt{(n^2 + n m + m^2)} \approx 246 \sqrt{((n+m)^2-n m)} </math>

in [[picometre]]s. The diameter <math>d</math> of the tube is then <math>c/\pi</math>, that is

: <math>d \approx 78.3 \sqrt{((n+m)^2-n m)}</math>

also in picometres. (These formulas are only approximate, especially for small ''n'' and ''m'' where the bonds are strained; and they do not take into account the thickness of the wall.)

The tilt angle ''α'' between '''''u''''' and '''''w''''' and the circumference ''c'' are related to the type indices ''n'' and ''m'' by:

: <math> \alpha \;=\; \arg(n + m/2,\, m \sqrt{3}/2) \;=\; \mathop{\mathrm{arc}}\cos\frac{n + m/2}{c} </math>

where arg(''x'',''y'') is the clockwise angle between the ''X''-axis and the vector (''x'',''y''); a function that is available in many programming languages as <code>atan2</code>(''y'',''x''). Conversely, given ''c'' and ''α'', one can get the type (''n'',''m'') by the formulas:

: <math>m = \frac{2 c}{\sqrt{3}}\sin \alpha \quad\quad n = c \cos\alpha - \frac{m}{2}</math>
which must evaluate to integers.