Dynamic polarization effects on the angular distributions of protons channeled through carbon nanotubes in dielectric media

The best level of ordering and straightening of carbon nanotube arrays is often achieved when they are grown in a dielectric matrix, so such structures present the most suitable candidates for future channeling experiments with carbon nanotubes. Consequently, we investigate here how the dynamic polarization of carbon valence electrons in the presence of various surrounding dielectric media affects the angular distributions of protons channeled through (11,9) single-wall carbon nanotubes. Proton speeds between 3 and 10 a.u., corresponding to energies of 0.223 and 2.49 MeV, are chosen with the nanotube’s length varied between 0.1 and $1\mu \text{m}$. We describe the repulsive interaction between a proton and the nanotube’s atoms in a continuum-potential approximation based on the Doyle-Turner potential, whereas the attractive image force on a proton is calculated using a two-dimensional hydrodynamic model for the dynamic response of the nanotube valence electrons, while assigning to the surrounding medium an appropriate (frequency dependent) dielectric function. The angular distributions of channeled protons are generated using a computer simulation method which solves the proton equations of motion in the transverse plane numerically. Our analysis shows that the presence of a dielectric medium can strongly affect both the appearance and positions of maxima in the angular distributions of channeled protons.

Figure 1

The image force of a proton traveling paraxially at three distances from the nanotube axis: $r=3$ (lower), 7 (middle), and 10 a.u. (upper), as a function of proton speed $v$ in a.u., due to an (11,9) SWNT in vacuum (dashed curves) and encapsulated by an (a) ${\text{SiO}}_2$ channel, (b) ${\text{Al}}_2{\text{O}}_3$ channel, and (c) Ni channel (solid curves). The radius of the nanotube is $a=13.01\text{a.u.}$ and the radius of the dielectric channel is $b=16.22\text{a.u.}$ In panel (c) we also show the image force for the same three distances inside Ni channels without encapsulated nanotube for two channel radii: $b=13.01\text{a.u.}$ (thin solid lines) and $b=16.22\text{a.u.}$ (dotted lines).

Figure 2

The image force of a proton channeled in an (11,9) SWNT in vacuum (dashed curve) and encapsulated by ${\text{SiO}}_2$, ${\text{Al}}_2{\text{O}}_3$, and Ni channels (thin, medium, and thick solid curves) as functions of the proton position $x$ in a.u. across the nanotube radius, for three proton speeds: (a) $v=3\text{a.u.}$, (b) $v=5\text{a.u.}$, and (c) $v=8\text{a.u.}$ The dotted curve gives the repulsive force $F_{\text{rep}}$, while the lower branches of solid and dashed curves give the attractive image force alone, $F_{\text{image}}$, and the upper branches give the total force $F$, due to an (11,9) SWNT in vacuum (dashed curve) and encapsulated by the three channels (solid curves). The radius of the nanotube is $a=13.01\text{a.u.}$ and the radius of the dielectric channel is $b=16.22\text{a.u.}$

Figure 3

The (a) deflection functions and (b) corresponding angular distributions of protons channeled in an (11,9) SWNT in vacuum (dashed curve) and encapsulated by a ${\text{SiO}}_2$ channel (solid curve). The proton speed is $v=5\text{a.u.}$, the nanotube length is $L=0.1\mu \text{m}$, the nanotube radius is $a=13.01\text{a.u.}$, and the dielectric channel radius is $b=16.22\text{a.u.}$ The angular distribution’s bin size is 0.0213 mrad.

Figure 4

The (a) deflection functions and (b) corresponding angular distributions of protons channeled in an (11,9) SWNT in vacuum (dashed curve) and encapsulated by a ${\text{SiO}}_2$ channel (solid curve). The proton speed is $v=5\text{a.u.}$, the nanotube length is $L=0.3\mu \text{m}$, the nanotube radius is $a=13.01\text{a.u.}$, and the dielectric channel radius is $b=16.22\text{a.u.}$ The angular distribution’s bin size is 0.0213 mrad.

Figure 5

The (a) deflection functions and (b) corresponding angular distributions of protons channeled in an (11,9) SWNT in vacuum (dashed curve) and encapsulated by a ${\text{SiO}}_2$ channel (solid curve). The proton speed is $v=5\text{a.u.}$, the nanotube length is $L=0.5\mu \text{m}$, the nanotube radius is $a=13.01\text{a.u.}$, and the dielectric channel radius is $b=16.22\text{a.u.}$ The angular distribution’s bin size is 0.0213 mrad.

Figure 6

The (a) deflection functions and (b) corresponding angular distributions of protons channeled in an (11,9) SWNT in vacuum (dashed curve) and encapsulated by a ${\text{SiO}}_2$ channel (solid curve). The proton speed is $v=8\text{a.u.}$, the nanotube length is $L=0.8\mu \text{m}$, the nanotube radius is $a=13.01\text{a.u.}$, and the dielectric channel radius is $b=16.22\text{a.u.}$ The angular distribution’s bin size is 0.0213 mrad.

Figure 7

The (a) deflection functions and (b) corresponding angular distributions of protons channeled in an (11,9) SWNT in vacuum (dashed curve) and encapsulated by an ${\text{Al}}_2{\text{O}}_3$ channel (solid curve). The proton speed is $v=8\text{a.u.}$, the nanotube length is $L=0.8\mu \text{m}$, the nanotube radius is $a=13.01\text{a.u.}$, and the dielectric channel radius is $b=16.22\text{a.u.}$ The angular distribution’s bin size is 0.0213 mrad.

Figure 8

The (a) deflection functions and (b) corresponding angular distributions of protons channeled in an (11,9) SWNT in vacuum (dashed curve) and encapsulated by a Ni channel (solid curve). The proton speed is $v=8\text{a.u.}$, the nanotube length is $L=0.8\mu \text{m}$, the nanotube radius is $a=13.01\text{a.u.}$, and the dielectric channel radius is $b=16.22\text{a.u.}$ The angular distribution’s bin size is 0.0213 mrad.

Figure 9

Dependence of the ZDF versus dwell time $L/v$ (in units of $\mu \text{m}/v_b=0.4571\text{ps}$) for protons channeled in an (11,9) SWNT in vacuum (dashed curve) and encapsulated by a ${\text{SiO}}_2$ channel (solid curve). The longitudinal proton speeds are (a) $v=3\text{a.u.}$, (b) $v=5\text{a.u.}$, and (c) $v=8\text{a.u.}$, while the proton yield at ZDF is obtained by collecting all transverse proton speeds such that $v_t<{10}^{-5}\text{a.u.}$

Figure 10

Dependence of the proton deflection angle ${\Theta }_x$ in mrad multiplied by the proton speed $v$ in a.u. on the dwell time $L/v$ (in units of $\mu \text{m}/v_b=0.4571\text{ps}$) for a proton impact parameter of $x_0=12\text{a.u.}$ and the three protons speeds (a) $v=3\text{a.u.}$, (b) $v=5\text{a.u.}$, and (c) $v=8\text{a.u.}$, due to an (11,9) SWNT in vacuum (dashed curve) and encapsulated by ${\text{SiO}}_2$ channels (solid curve).

Figure 11

Dependence of the proton deflection angle ${\Theta }_x$ in mrad multiplied by the proton speed $v$ in a.u. on the dwell time $L/v$ (in units of $\mu \text{m}/v_b=0.4571\text{ps}$) for proton impact parameters of $x_0=3\text{a.u.}$ and $x_0=8\text{a.u.}$ and three proton speeds (a) $v=3\text{a.u.}$, (b) $v=5\text{a.u.}$, and (c) $v=8\text{a.u.}$, due to an (11,9) SWNT in vacuum (dashed curve) and encapsulated by ${\text{SiO}}_2$ channels (solid curve).