Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

I've been reading about nanotubes lately, and I keep seeing the $ (n,m) $ notation. How does this describe a nanotube's structure? How do I determine which is $n$ and which is $m$ ?

I'm familiar with matrix notation referring to rows and columns, but I couldn't connect it with the nanotube structure which is, albeit predictable, not quite like a row-column grid.

share|improve this question

2 Answers 2

up vote 4 down vote accepted

In the picure below, you can see how the row-column grid correspond to the graphene structure.

Graphene structure

To have a (n,m) nanotube, you "just" have to roll your graphene sheet so that the (0,0) hexagon coincides with the (n,m) hexagon. Of course, it is much easier said than done !

share|improve this answer
édéric: Does "coincide" mean that the northwest edge of $ (0,0) $ will merge with the southeast edge of $ (4,3) $, so that it will form a seamless nanotube? –  Kit Dec 3 '10 at 15:54
@Kit : No, the twot northwest edges will merge, as well as the two southwest edges. –  Frédéric Grosshans Dec 3 '10 at 16:43

A carbon nanotube can be seen as a sheet of graphene that is "rolled up".

Now, graphene is a two-dimensional lattice and hence has two lattice vectors, $\vec{a}_1$ and $\vec{a}_2$. (If you are unfamiliar with lattice vectors let me know and I will expand on this).

The numbers $(n,m)$ simply state that your tube is obtained from taking one atom of the sheet and rolling it onto that atom that is at located $n \vec{a}_1 + m \vec{a}_2$ away from your original atom.

EDIT: Graphene is a tridiagonal lattice with two atoms per unit cell. Left: Lattice structure of graphene, right: Brillouin zone (not of interest here)

share|improve this answer
+1 The way I understand it, for an ordinary row-column grid, its lattice vectors are $90\,^{\circ}$ apart. How does this generalize to a graphene lattice? –  Kit Dec 3 '10 at 4:50
@Kit : You can have oblique coordinates in the plane. See mathworld.wolfram.com/ObliqueCoordinates.html –  Frédéric Grosshans Dec 3 '10 at 13:59
I actually found the image you used in Figure 2 of Neto, et al. The electronic properties of graphene. 2009. I think it would be a courtesy to the authors to cite them, and also for me (and others) to help us know the relevant papers to read :) –  Kit Feb 1 '11 at 12:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.