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Since the flexural rigidity of a graphene sheet is isotropic, thermal expansion of a nanotube should depend only on its diameter and not on the chiral angle. The increased flexibility of nanotubes narrower than the (10, 10) tube is expected to augment the length contraction we find. Wider nanotubes, on the other hand, cannot bend easily, but show a tendency to collapse radially, augmenting the volumetric contraction.
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In achiral nanotubes aligned with the z axis, we define the ensemble of atoms with the same z coordinate as a "layer." In the (10, 10) nanotube, a layer consists of a 20 atom ring along the tube circumference.
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