Image potential and stopping force in the interaction of fast ions with carbon nanotubes: The extended two-fluid hydrodynamic model

Nuclear Instruments and Methods in Physics Research B 366 (2016) 83–89

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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb

Image potential and stopping force in the interaction of fast ions with carbon nanotubes: The extended two-fluid hydrodynamic model L. Karbunar a, D. Borka b,⇑, I. Radovic´ b a b

School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbia Vincˇa Institute of Nuclear Sciences, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 7 August 2015 Accepted 7 October 2015

We study the interaction of charged particles with a (6, 4) single-walled carbon nanotube (SWNT) under channeling conditions by means of the linearized, two dimensional (2D), two-fluid extended hydrodynamic model. We use the model to calculate analytically and numerically the image potential and the stopping force for a proton moving parallel to the axis of the SWNT, both inside and outside the nanotube at the speeds from 0.5 a.u. to 15 a.u. The effects of different angular modes on the velocity dependence of the image potential are compared for a proton moving in different types of SWNTs. We also compute the spatial and angular distributions of protons in the 2D two-fluid extended hydrodynamic model and compare them with the 2D two-fluid hydrodynamic model with zero damping. Ó 2015 Elsevier B.V. All rights reserved.

Keywords: Nanotubes Channeling Dynamic polarization

1. Introduction Carbon nanotubes were discovered in the beginning of 1990s [1]. They can be described as sheets of carbon atoms at the (twodimensional) hexagonal lattice sites rolled up into cylinders [2]. Soon after the discovery of carbon nanotubes, Klimov and Letokhov [3] realized that the effect of channeling of positively charged particles in nanotubes might occur. After that, a number of theoretical groups have studied ion channeling in nanotubes [4–12], with the main objective to explore the possibility of guiding chargedparticle beams with nanotubes. Because of technical problems that involve ordering, straightening and holding nanotubes, the experimental study of ion channeling through carbon nanotubes is still in the initial phase [13,14]. For ion channeling in carbon nanotubes at the low (<100 keV) and high (>10 MeV) ends of the energy range, the dynamic polarization effect of valence electrons in the nanotubes can be omitted. However, ions moving with the energies (100 keV–10 MeV) will induce strong dynamic polarization of valence electrons in the nanotubes which in turn will give rise to a sizeable image force on the ions, as well as a considerable energy loss due to the collective, or plasma, electron excitations [15–17]. The dynamic image force obtained from a one-fluid hydrodynamic model was shown to have big influence in the angular distributions of protons ⇑ Corresponding author. Tel.: +381 11 6455 451; fax: +381 11 6308 425. E-mail addresses: [email protected] (L. (D. Borka), [email protected] (I. Radovic´).

Karbunar),

http://dx.doi.org/10.1016/j.nimb.2015.10.033 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.

[email protected]

channeled through short (11, 9) single-wall carbon nanotubes [18] in vacuum. However, in Ref. [19] we treated the r and p electron orbitals in carbon nanostructures as separate but superimposed fluids with zero restoring frequencies and zero damping and calculated the image force and stopping force based on a 2D two-fluid hydrodynamic model. We found that those forces are affected, both at the low and high proton speeds, by a splitting of the collective electron excitation modes occurring due to different acoustic speeds in the r and p electron fluids that may be related to the so-called high-energy r + p plasmons and the low-energy p plasmons [17,19]. In this work, we use a more realistic approach for calculations of the image potential and stopping force based on a 2D two-fluid hydrodynamic model, which includes finite restoring frequencies and finite damping of the r and p electron that are determined from the EELS measurements [20]. Hence, we designate such version of the two-fluid hydrodynamic model as extended in contrast to the previously used two-fluid hydrodynamic model, which will be understood as having zero restoring frequencies and zero damping of the collective modes [19]. This paper is a continuation of the investigations in our previous paper [19] where we studied the image potential in the interaction of fast ions with carbon nanotubes and made comparisons between the one-fluid and two-fluid hydrodynamic models. In this work we present the analytically and numerically calculated image potential and the stopping force in the case of a 2D two-fluid extended hydrodynamic model [20]. We suppose that protons move parallel to the axis of the SWNTs at the speeds from 0.5 a. u. to 15 a.u. We investigate the cases when the particle position

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is inside and also outside the nanotube. We analyze the effects of different values of the angular mode on the velocity dependence of the reduced image potential for a proton moving in the SWNT, and we compare those effects for different types of nanotubes (SWNT (6, 4), SWNT (8, 6), SWNT (11, 9) and SWNT (15, 10)). We also compute the spatial and angular distributions of channeled protons using the two-fluid extended and the two-fluid hydrodynamic models [19,21,22]. First, we outline the basic theory used in modeling the image potential and stopping force of carbon nanotubes. After that, we discuss the obtained results for the image potential and stopping force, as well as the Monte Carlo simulations of spatial and angular distributions. At the end of the paper, we give our concluding remarks. Atomic units are used throughout unless explicitly stated otherwise. 2. Basic theory A SWNT is modeled as an infinitesimally thin cylindrical shell with the radius R and the length L. We assume that the valence electrons in the ground state may be considered as a freeelectron gas distributed uniformly over a cylindrical surface, with the number density per unit area n0 ¼ n0r þ n0p ¼ 0:428 where n0r ¼ 3n0 =4 ¼ 0:321 and n0p ¼ n0 =4 ¼ 0:107 are the unperturbed number densities corresponding to three r electrons and one p electron per carbon atom, respectively [17]. We use cylindrical coordinates ~ r ¼ ðq; u; zÞ and assume that a charged particle with the charge Q moves outside the SWNT, with its trajectory parallel to the nanotube axis z, at fixed distance q0 > R, such that the particle’s instantaneous position is given by ~ r0 ðtÞ ¼ ðq0 ; u0 ; v tÞ, where v is the particle’s speed. The Fourier–Bessel (FB) transform Aðq; m; k; xÞ of an arbitrary function Aðq; u; z; tÞ may be defined (assuming L ! 1) by [19]

Aðq; u; z; tÞ ¼

1

1 Z X

ð2pÞ3 m¼1

1

1

Z

1

1

the high-resolution reflection EEL spectra of a SL graphene supported by a metal substrate [24]. Note that the previously used two-fluid hydrodynamic model is obtained by simply setting xrr , xpr , cr and cp to zero [19]. By using Eqs. (1) and (2) one can obtain an expression for the induced potential as follows

Uind ðq; u; z; tÞ ¼ 4RQ

1 Z X

1

m¼1 1

eimuþikðzv tÞ K m ðjkjqÞK m ðjkjq0 Þ

 I2m ðjkjRÞvðm; k; kv Þdk

ð5Þ

The image potential, or the self energy, U im , for a point-charge ion Q on the trajectory ~ r 0 ðtÞ ¼ ðq0 ; u0 ; v tÞ is defined by

U im ¼

  Q Uind ðq; u; z; tÞ 2 q¼q0 ; u¼u0 ; z¼v t

ð6Þ

Substituting Eq. (5) into Eq. (6) and setting u0 ¼ 0, it is easy to obtain the following relation for the image potential

U im ¼ 4Q 2 R

1 Z X m¼1

0

1

I2m ðkRÞK 2m ðkq0 ÞRe½vðm; k; kv Þdk

ð7Þ

where we have used the symmetry properties of the real and imaginary parts of the density response function of a carbon nanotube from Eq. (3).

Aðq; m; k; xÞeimuþikzixt dkdx ð1Þ

Following Ref. [19], one can express the induced potential Uind due to dynamic polarization of the electron fluids as

Uind ðq; m; k; xÞ ¼ 2pRQg mk ðq; RÞg mk ðR; q0 Þvðm; k; xÞdðx  kv Þ ð2Þ where g mk ðq; RÞ ¼ 4pIm ðjkjRÞK m ðjkjqÞ and g mk ðR; q0 Þ ¼ 4pIm ðjkjRÞ K m ðjkjq0 Þ, while Im and K m are cylindrical Bessel functions of integer order m, and the density response function of a carbon nanotube is given by

vðm; k; xÞ ¼

v0 ðm; k; xÞ 1 þ 4pRIm ðjkjRÞK m ðjkjRÞv0 ðm; k; xÞ

ð3Þ

where the two-fluid extended non-interacting response function is given by v0 ðm; k; xÞ ¼ v0r ðm; k; xÞ þ v0p ðm; k; xÞ with [23] 0 i ðm; k;

v

  2 2 n0i k þ mR2  xÞ ¼  2 2 s2i k þ mR2 þ x2ir  x2  ici x

ð4Þ

Note that si , xir and ci are the acoustic speed of the density perturbations of valence electrons (which is given in Ref. [23] in the Thomas–Fermi–Dirac approximation for the 2D hydrodynamic model), restoring frequency and the damping rate in the ith fluid (where the index i takes values r and p), respectively. As regards the adjustable parameters in Eq. (4), we use xrr = 0.48, xpr = 0.15, cr = 0.1 and cp = 0.09. Those parameters are used in our previous publications devoted to modeling of the experimental electron energy loss (EEL) spectra of single-layer (SL) and multilayer graphene in a transmission electron microscope [20], and

Fig. 1. Effects of the proton position q0 on the velocity dependence of the image potential for a proton moving in the case of SWNT (6, 4) (a) inside and (b) outside the nanotube. The proton positions inside the nanotube are q0 = 2, 3, 4 and 4.5 a.u., and outside the nanotube are q0 = 2 R-2, 2 R-3, 2 R-4 and 2 R-4.5 a.u. Results are presented using two-fluid (solid curves) and two-fluid extended (dashed curves) hydrodynamic models. The nanotube radius of SWNT (6, 4) is R = 0.346 nm = 0.654 a.u.

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The stopping force on an ion of charge Q travelling paraxially to the SWNT with speed v at ~ r0 ðtÞ ¼ ðq0 ; u0 ; v tÞ is the force opposing the ion’s motion and its magnitude is given by

S¼Q

 @ Uind ðq; u; z; tÞ  @z q¼q0 ; u¼u0 ; z¼v t

ð8Þ

Substituting Eq. (5) into Eq. (8), setting u0 ¼ 0 and using the symmetry properties of the real and imaginary parts of the density response function of a carbon nanotube, we obtain for the stopping force 2

S ¼ 8Q R

1 Z X m¼1

0

1

q0 ÞIm½vðm; k; kv Þdk

2 kIm ðkRÞK 2m ðk

ð9Þ

A derivation of expressions for the image potential in the 2D two-fluid when the charged particle moves inside the SWNT, with its trajectory parallel to the nanotube axis z, at fixed distance q0 < R is given in our previous paper [19]. If we take those expressions for the image potential in the 2D two-fluid extended hydrodynamic model we can also compute spatial and angular distributions of protons channeled through a nanotube. The system under investigation is a proton moving through an (6, 4) SWNT positioned in vacuum. The z-axis coincides with the nanotube axis and the origin lies in its entrance plane. The initial proton velocity vector ~ v is taken to be parallel to the z-axis of the nanotube with the length L = 0.3 lm. We suppose that the chosen nanotube length

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is sufficiently short for the proton energy loss to be neglected, but is also long enough allowing us to omit the effect of nanotube edges on the image potential. We also assume that the repulsive interaction between the proton and nanotube atoms may be treated classically and we use the Doyle–Turner expression for the proton-nanotube atom interaction potential averaged axially and azimuthally [25–27]. The repulsive potential for proton channeling through the nanotube is of the form

U rep ðq0 Þ ¼

4 n h io 32pZ 1 Z 2 R X 2 2 2 aj bj I0 ð2bj Rq0 Þ exp bj q20 þ R2 pffiffiffi 2 3 3l j¼1

ð10Þ

where Z 1 ¼ 1 and Z 2 ¼ 6 are the atomic numbers of the proton and nanotube atom, respectively, R is the nanotube radius, l = 0.144 nm is the interatomic separation in the nanotube [2], q0 is the distance between the proton and nanotube axis, I0 designates the 0th order modified Bessel function of the first kind, and aj = (0.115, 0.188, 0.072, 0.020) and bj = (0.547, 0.989, 1.982, 5.656) are the fitting parameters [27]. The dynamic polarization interaction is treated by a 2D two-fluid extended hydrodynamic model of the nanotube valence electrons, based on a jellium-like description of the ion cores on the nanotube wall, which includes the axial and azimuthally averaging [28] consistent with the treatment of the repulsive interaction. The total interaction potential between the proton and the nanotube is

Fig. 2. Effects of the different values of the angular mode m on the velocity dependence of reduced image potential for a proton moving in the SWNT at distance q0 ¼ R  asc in the case of (a) SWNT (6, 4), (b) SWNT (8, 6), (c) SWNT (11, 9), (d) SWNT (15, 10). Results are obtained using a two-fluid extended hydrodynamic model. The values that are used in the panels (a), (b), (c) and (d) to reduce the image potential are 0.4782, 0.4754, 0.4675 and 0.4554, respectively. The nanotubes radii are R = 0.346, 0.483, 0.689 and 0.865 nm for (6, 4), (8, 6), (11, 9) and (15, 10) SWNTs, respectively.

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Uðq0 Þ ¼ U rep ðq0 Þ þ U im ðq0 Þ

ð11Þ

Using a computer simulation method (which includes the numerical solution of the proton equations of motion in the transverse plane) we generated the spatial and angular distributions of channeled protons in the exit transverse plane [18]. The initial proton position, x0 and y0 , is chosen randomly from a 2D uniform 1=2

distribution with the condition r0 ¼ ðx20 þ y20 Þ

< R  asc , where

1=3

asc ¼ ½9p =ð128Z 2 Þ a0 is the nanotube atom screening radius and a0 is the Bohr radius. The nanotube radius of SWNT (6, 4) is R = 0.346. 2

3. Results and discussion In this section we shall first discuss in Fig. 1 the effects of the proton position q0 on the velocity dependence of the image potential for a proton moving in the case of a SWNT (6, 4), both (a) inside and (b) outside the nanotube. The proton positions are q0 = 2, 3, 4 and 4.5 a.u. Results are presented in the two-fluid (xrr = 0, xpr = 0, cr = 0.001 and cp = 0.001) and in the two-fluid extended (xrr = 0.48, xpr = 0.15, cr = 0.1 and cp = 0.09) hydrodynamic models. In Fig. 1a we did calculations for SWNT (6, 4) in the cases of four different proton distances from the nanotube center assuming that particle is inside the nanotube. Ion velocity is in the range of 0–15 a.u. The results obtained lead us to conclude that the

magnitudes of the image potential are generally smaller in case of two-fluid extended model. In the expressions we used for calculations in the case of the two-fluid extended model there are terms designated as restoring frequencies that may be related to the interband transitions r ! r and p ! p giving xrr = 0.48 = 13.06 eV and xpr = 0.15 = 4.08 eV [20]. In the case of the two-fluid model xrr = 0 and xpr = 0. The damping factor c in the case of the two-fluid model has values c = cr = cp = 0.001, and in the case of the two-fluid extended model cr = 0.1 = 2.72 eV and cp = 0.09 = 2.45 eV [20]. We may conclude that finite values of the restoring frequencies and damping factors are the reason for the observed weakening of the image potential in the twofluid extended hydrodynamic model compared to the two-fluid hydrodynamic model. It is interesting to note that in the two fluid model for proton velocity above the 8.5 a.u. the values for the image potential become positive, with an absolute maximum for velocity v  11.5 a.u. In Fig. 1b we performed similar calculations as we did in Fig. 1a. Here we show how the image potential depends on the incoming proton velocity when proton is outside the nanotube. The proton positions are q0 = 2 R-2, 2 R-3, 2 R-4 and 2 R-4.5 a.u. We analyze the SWNT (6, 4) nanotube. Velocity range of incoming proton is the same as in the previous case. Proton distances from the nanotube wall are chosen to be the same as in the case of internal trajectories, but this time we assume that the proton is outside the nanotube area. The results we obtained here

Fig. 3. The image potential on a proton channeled in an SWNT (6, 4) in vacuum as a function of proton position q0 in a.u. across the nanotube radius q0 < R  asc , for four proton speeds: (a) v = 0.5 a.u., (b) v = 0.71 a.u., (c) v = 3 a.u. and (d) v = 15 a.u. The case of SWNT (6, 4) is presented using two-fluid (solid curves) and two-fluid extended (dashed curves) hydrodynamic models. Vertical bars on the horizontal axes indicate the radii of the four types of carbon nanotubes.

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are very similar for proton velocities in both models, but are not the same because the cases inside and outside nanotubes are not fully symmetric. The image potential values are again lower in the case of the two-fluid extended model. For the values of proton velocities >9 a.u. in the case of the two-fluid model there is a slightly larger gradient of the image potential outside the nanotube than inside the nanotube for the same values of the proton distances from the wall. In the all presented cases in Fig. 1 we can conclude that the magnitude of the image potential is very sensitive on the presence of the damping factors and restoring frequencies. In Fig. 2 we analyze the influence of different values of the angular mode m on the reduced image potential U im =U 0 (the image potential divided with the maximum absolute value of the total image potential in the velocity range from 0.1 to 1.2 a.u.). The values used for (a) SWNT (6, 4), (b) SWNT (8, 6), (c) SWNT (11, 9), (d) SWNT (15, 10) to reduce the image potential are 0.4782, 0.4754, 0.4675 and 0.4554, respectively. We perform our analysis in the cases of four different types of SWNT: (6, 4), (8, 6), (11, 9) and (15, 10), where we take the proton distance of q0 ¼ R  asc (R is nanotube radius and depends on nanotube type and asc is the nanotube atom screening radius) from the center of each nanotube. In that way we compare the values of the angular mode at the same distance asc from the nanotube wall for four different types of SWNT. We calculated the individual terms for the image potential modes for each value of m for 3 6 m 6 3. We have chosen m in that range as higher m values contribute very little to the shape of the reduced image potential curve in the chosen speed range. For each SWNT we can notice that the angular mode of the reduced image potential has a small jump around v  0.55 a.u. and that this jump is approximately at the same positions in all four cases. For the bigger radius of the nanotube the contribution of terms in the range 3 6 m 6 3 to the total reduced image potential is smaller. In the case of the two-fluid hydrodynamic model we had a jump around v  0:71 a.u., but in the case of the two-fluid extended hydrodynamic model we have a significantly lower jump, and its position is moved to around v  0:55 a.u. It seems that this effect is caused by the influence higher damping factors and restoring frequencies [20]. Fig. 3 shows how image potential depends on proton distance from the center of a SWNT (6, 4) nanotube (i.e. across the nanotube radius q0 < R  asc ) for four different proton velocities: (a) v = 0.5 a.u., (b) v = 0.71 a.u., (c) v = 3 a.u. and (d) v = 15 a.u. An analysis is performed for the two-fluid and the two-fluid extended hydrodynamic models. Absolute values for the image potential are lower in the case of the two-fluid extended model regardless of the proton position. This agrees with what we have previously concluded about the relation between the images potential values in the two adopted models. In the case of proton velocities v = 0.5 a. u. and v = 0.71 a.u. the difference between the potentials in both models decreases when proton is closer to the nanotube wall. For higher proton velocities the difference between the potentials in both models remains almost the same along the nanotube radius. In Fig. 3d we can see that the curve for the image potential is almost flat in both models, which means that the image potential does not depend on the proton position at higher speeds and is very close to zero. In Fig. 4 we analyze how the stopping force that resists proton motion depends on proton speed. We perform calculations for the SWNT (6, 4) in two different situations. In the first we assume that the proton is inside the nanotube and we consider four proton distances. The proton positions are q0 = 2, 3, 4 and 4.5 a.u. In the second we assume proton is outside the nanotube and again we take the same proton distances from nanotube wall, i.e., the proton positions are q0 = 2 R-2, 2 R-3, 2 R-4 and 2 R-4.5 a.u. The results

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we obtained for the stopping force in both cases are similar. Stopping force is significantly lower in the case of the two-fluid extended model for the particle speed in the range of 0 6 v 6 9 a:u:. This behavior could be explained in similar manner as with the image potential in terms of different parameters in the two-fluid (xrr = 0, xpr = 0, cr = 0.001 and cp = 0.001) and in the two-fluid extended (xrr = 0.48, xpr = 0.15, cr = 0.1 and cp = 0.09) hydrodynamic models. Their values strongly influence those of the stopping force. The maximum value of the stopping force is between 3 a.u. and 5 a.u. and when proton position is closer to the nanotube wall the maximum occurs at a lower proton speed. Fig. 5 shows how stopping force depends on the position q0 of a proton channeled in an SWNT (6, 4) (i.e. across the nanotube radius q0 < R  asc ) in vacuum. We assume that the proton is inside the nanotube area. We calculate the stopping force in the case of the two-fluid and the two-fluid extended models and make comparison. We assume that the proton distance from the nanotube center is in the range of 0 6 q0 6 R  asc . Calculations are done for four proton speeds: (a) v = 0.5 a.u., (b) v = 0.71 a.u., (c) v = 3 a.u. and (d) v = 15 a.u., where we assume the speed is constant. Absolute values for the stopping force are lower in the case of the twofluid model regardless of the proton position. The reason is in the damping factors which in the case of the two-fluid model have

Fig. 4. Effects of the proton position q0 on the velocity dependence of the stopping force for a proton moving in the case of SWNT (6, 4) (a) inside and (b) outside the nanotube. Results are presented in two-fluid (solid curves) and in two-fluid extended (dashed curves) hydrodynamic models. The nanotube radius of SWNT (6, 4) is R = 0.346.

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Fig. 5. The stopping force on a proton channeled in an SWNT (6, 4) in vacuum as a function of proton position q0 in a.u. across the nanotube radius q0 < R  asc , for four proton speeds: (a) v = 0.5 a.u., (b) v = 0.71 a.u., (c) v = 3 a.u. and (d) v = 15 a.u. The case of SWNT (6, 4) is presented using two-fluid (solid curves) and two-fluid extended (dashed curves) hydrodynamic models. Vertical bars on the horizontal axes indicate the radii of the four types of carbon nanotubes.

vanishingly small values, cr ¼ cp ¼ c = 0.001, while in the case of the two-fluid extended model their values are much bigger, cr = 0.1 = 2.72 eV and cp = 0.09 = 2.45 eV. For a lower proton velocity, differences between the stopping force in these two models are very large, and when the proton velocity increases those differences decrease. When the proton position becomes closer to the nanotube wall, values of the stopping force increase in both models, as expected. In Fig. 6(a) we present spatial distributions of protons channeled in an SWNT (6, 4) in vacuum for proton speed v = 3 a.u. The results are obtained using the two-fluid and two-fluid extended hydrodynamic models. Fig. 6(b) gives the angular distributions of protons channeled in an SWNT (6, 4) in vacuum for proton speed v = 3 a.u. The results are also obtained using the two-fluid and two-fluid extended hydrodynamic models. We see that the spatial and angular distributions obtained using the two-fluid extended model are different from those obtained using the two-fluid model for the same proton speed. Yields of the central maxima are larger in the case of the two-fluid extended model for spatial distributions, but the opposite holds true for angular distributions. The positions of the lateral maxima in the spatial and angular distributions are slightly moved compared to the two-fluid model, and the widths of those maxima are slightly different. The difference is bigger between the two-fluid and two-fluid extended models than between two-fluid and one-fluid models [19]. It seems that this effect is caused by the influence of different damping factors and restoring frequencies, i.e., the

values of these parameters have bigger influence on the spatial and angular distributions than the choice between the one or two fluid models.

4. Concluding remarks We have studied the dynamic polarization effect (image potential and stopping) of charged particles with carbon nanotube under channeling condition. We performed calculations for SWNT (6, 4) by means of linearized, 2D, two-fluid and two-fluid extended hydrodynamic models and compared those models. This study is a continuation of our previous work [19]. We calculated the image potential and stopping force for a proton moving parallel to the axis of SWNTs, both inside and outside the SWNTs, and we analyzed the influence of the nanotube radius, and the proton position on its image potential and stopping force. We investigated the effects of different values of the angular mode on the velocity dependence of the reduced image potential for a proton moving in different types of SWNTs. We also computed the spatial and angular distributions using these two models and compared them. We may conclude that the interaction between the channeled proton and electrons on the nanotube wall is the strongest for the proton velocity between 3 and 5 a.u. Also, this interaction becomes stronger as proton gets closer to the nanotube’s wall, as well as within nanotubes of smaller radius. The interactions between the channeled proton and electrons on the nanotube wall

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the largest contribution. We also showed that interactions between the channeled proton and electrons on the nanotube wall are different in the two-fluid extended and the two-fluid models in the whole range of the investigated proton speeds (0–15 a.u.). We showed that the spatial and angular distributions obtained using the two-fluid extended and two-fluid hydrodynamic models differ. The difference is bigger between the two-fluid extended and the two-fluid models than between two-fluid and one-fluid models. The reason is that the image potential (and consequently spatial and angular distributions) is very sensitive to the finite values of the damping factors and restoring frequencies of the extended model. We can conclude that it is important to take into account dynamic polarization effects in future simulations and experiments in ion channeling through carbon nanotubes at the (100 keV–10 MeV) energy range. Also, it is very important to adopt an accurate model that describes the dynamic polarization effects, because the spatial and angular distributions are strongly influenced by the parameter values of the model. Acknowledgements This work is supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (Project No. 45005). Authors would also like to thank Professor Z. L. Miškovic´ for many useful discussions. References

Fig. 6. (a) The spatial distributions of protons channeled in an SWNT (6, 4) in vacuum for proton speed v = 3 a.u. The case presented is obtained using two-fluid (thin blue solid curve) and two-fluid extended (thick red solid curve) hydrodynamic models. (b) The angular distributions of protons channeled in SWNT (6, 4) in vacuum for proton speed v = 3 a.u. The case presented is obtained using two-fluid (thin blue solid curve) and two-fluid extended (thick red solid curve) hydrodynamic models. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

when proton is moving inside and outside the nanotube (but at the same distance from the nanotube wall) are very similar. In the two fluid model for proton velocity above 8.5 a.u. values for the image potential become positive, and an absolute maximum occurs for the proton velocity v  11:5 a.u. We notice that around the proton velocity v = 0.71 a.u. (when the particle speed matches the phase velocity of the quasiacoustic p plasmon) both the image potential and stopping force exhibit a jump, but only in the two-fluid model. In the two-fluid extended model this jump is smaller and is moved to around v  0.55 a.u. The reason for this is that in the case of the two-fluid extended model the damping factors and restoring frequencies have finite values, which exert strong influence. We conclude that values of the angular mode m in the range (3, 3) have the largest contribution to the magnitude of the image potential in all four types of the considered nanotubes. If the radius of the nanotube is smaller, the values of m in the range (3, 3) have

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