Dynamics of one and two dimensional solid $^4$He adsorbed on nanotubes

In a previous experiment[1], we showed that one dimensional (1D) solid helium can be created on the surface of nanotube bundles. Specifically, when $^4He$ is first adsorbed on nanotubes, it forms a 1D linear solid along the grooves between two nanotubes on the bundle surface with lattice parameter, a$_1$ = 3.40 $\pm$ 0.02 \AA. When more helium is added, 2D solid helium covers the whole bundle surface. We have now determined the vibrational dynamics of these 1D and 2D solids, the dynamic structure factor, $S(Q,\omega)$. From the inelastic intensity integrated over all $\omega$ we obtain the MS amplitude of vibration along 1D chain $\langle u^2\rangle = 0.28$ \AA$^2$or Lindemann ratio $\gamma = (\langle u^2\rangle)^{1/2}/a_1 = 0.15 $ which is less than the bulk solid value near melting. The vibrational density states (DOS) of the 2D solid shows a gap at $\omega\simeq$ 0.75 meV indicating a commensurate solid as found for $^3He$ and $^4He$ on graphite surfaces. In contrast the 1D DOS shows little or no gap and the DOS goes uniformly to zero as $\omega \rightarrow 0$. [1] Pearce et al. Phys. Rev. Lett. 95, 185302 (2005).