Interactions of fast ions with carbon nanotubes: Two-fluid model

We propose a simple, parameter-free two-fluid model for the collective electron response of a single-walled carbon nanotube, which treats the $\sigma$ and $\pi$ electrons as separate two-dimensional fluids constrained to the same cylindrical surface. The electrostatic interaction between the fluids gives rise to splitting of the plasmon frequencies into two groups closely following the experimentally determined longitudinal dispersions of the $\sigma +\pi$ and $\pi$ plasmons. The model is used to calculate the induced electron density on the nanotube, as well as the stopping power for a charged particle moving parallel to the axis of the nanotube. It is found that these quantities exhibit novel features when the particle speed matches the phase velocity of the quasiacoustic $\pi$ plasmon.

Figure 1

Dispersion curves (in eV) ${\omega }_{+}(m,k)$ and ${\omega }_{-}(m,k)$ from Eq. (9), corresponding to the $\sigma +\pi$ and $\pi$ plasmons in a nanotube with radius $a=7\mathrm{\AA }$, shown versus longitudinal momentum $k$ (in ${\mathrm{\AA }}^{-1}$) for several angular modes $m$. Experimental points are from Refs. 1, 2.

Figure 2

Induced electron densities $n_{\text{one}}$ and $n_{\text{two}}$ (in a.u.) from the one-fluid and two-fluid models, versus distance $z$ (in a.u.) in the moving frame of reference, for a proton moving along the axis of a nanotube with radius $a=7\mathrm{\AA }$, at several speeds: $v=2$, 2.5, and 3 (a.u.).

Figure 3

Induced electron densities $n_{\text{one}}$ and $n_{\text{two}}$ (in a.u.) from the one-fluid and two-fluid models, versus distance $z$ (in a.u.) in the moving frame of reference, for a proton moving along the axis of a nanotube with radius $a=7\mathrm{\AA }$, at several speeds: $v=0.70$, 0.72, and 0.74 (a.u.).

Figure 4

Stopping power (in a.u.) versus speed $v$ (in a.u.) for a proton moving along the axis of a nanotube with radius $a=7\mathrm{\AA }$. (a) Stopping powers $S_{\text{one}}$ and $S_{\text{two}}$ (in a.u.) from the one-fluid and two-fluid models, with the friction constant $\gamma ={10}^{-3}{\mathrm{\Omega }}_p$. (b) Stopping power $S_{\text{two}}$ for two friction constants, $\gamma ={10}^{-3}{\mathrm{\Omega }}_p$ and $\gamma ={10}^{-2}{\mathrm{\Omega }}_p$.

Figure 5

Stopping power (in a.u.) versus speed $v$ (in a.u.) for a proton moving at distance $r_0=a∕2$ from the axis of a nanotube with radius $a=7\mathrm{\AA }$. (a) Stopping powers $S_{\text{one}}$ and $S_{\text{two}}$ (in a.u.) from the one-fluid and two-fluid models, with the friction constant $\gamma ={10}^{-3}{\mathrm{\Omega }}_p$. (b) Stopping power $S_{\text{two}}$ for two friction constants, $\gamma ={10}^{-3}{\mathrm{\Omega }}_p$ and $\gamma ={10}^{-2}{\mathrm{\Omega }}_p$.