The inner tube volume within the cavity is ≈ 6.6 mm^{3}, of which ≈ 0.40 mm^{3} is nanotube material (i.e. a packing fraction of about 6% by volume, based on the sample mass of 1.3 mg). From this, we conclude that the magnetic field is effectively screened within this low density powder, both from within the nanotubes themselves and from a much larger volume (×3) in the space outside. For this to happen, we also conclude that the nanotube sheet resistance must be extremely small, i.e. an upper limit of *R*_{sq} ≈ 10 μΩ based on a mean nanotube radius of *a* ≈ 0.7 nm. Since a trace of metallic Fe is present in the sample, it is prudent to quantify how much this contributes to the microwave screening. The 0.8% atomic ratio of Fe atoms corresponds to a volume of about 0.006 mm^{3}. This is likely to be dispersed as very small particles of radii less than 1 μm. Given that the skin depth of Fe at 3 GHz at room temperature is about 2.8 μm, we would not expect such small particles to screen the microwave magnetic field. Even if the Fe formed a single large particle, which gives rise to the greatest screened volume, the contribution to increase in resonant frequency will be only 0.6% of that observed for the whole CNT sample. Hence, we conclude that the screening effect of the Fe is negligible.

Does such a small value of *R*_{sq} contradict the observation of the large microwave losses of Figure 4? The answer can be found in the huge surface area of carbon nanotube powders, e.g. our 1.3-mg sample has a total surface area *S* ≈ 1.1 m^{3}. Sheet resistance can be extracted from Equation 1b in the limit of strong screening *ωτ* > > 1, resulting in *R*_{sq} ≈ 2πμ_{0}*V*_{eff} Δ*f*_{B} / *S* ≈ 30 μΩ. Therefore, these two independent measurements (one of microwave screening, the other of microwave loss) both point to very small values of sheet resistance due to ballistic transport. In experimental studies of carbon nanotube films, the sheet resistance is found to be of the order of 1 kΩ [7] with similar results for graphene films [8]. In these studies, the intrinsic sheet resistance of the carbon nanotubes can be difficult to extract from the measurements as charge transport is limited by contacts between the nanotubes. Other types of measurement such as the optical conductivities reported in [8] do not circumvent this problem as they characterise relatively large areas of the films. In theoretical studies of devices based on discrete carbon nanotubes and of carbon nanotube films, charge transport is often assumed to be ballistic, and therefore, the sheet resistance is zero. Ballistic transport has been observed in ohmically contacted metallic single wall carbon nanotubes having lengths less than approximately 300 nm [9].

Not all of the nanotube powders studied to date show such striking behaviour, so future experiments will concentrate on systematic studies of a wider range of materials (semiconducting and metallic), over a wider range of frequencies (particularly from kilohertz to megahertz), also trying to identify the nature of the defects giving rise to the finite sheet resistance. Indeed, isolating metallic samples is, itself, an important problem, and the method we have proposed here may serve as a means of quantifying the volume fraction of metallic nanotubes within a given powder.