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Influence of the dynamic polarization effect on the angular distributions of protons channeled in double-wall carbon nanotubes
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 256 (2007) 131–136 www.elsevier.com/locate/nimb
Influence of the dynamic polarization effect on the angular distributions of protons channeled in double-wall carbon nanotubes D. Borka a
a,*
, S. Petrovic´ a, N. Nesˇkovic´ a, D.J. Mowbray b, Z.L. Misˇkovic´
b
Laboratory of Physics (010), Vincˇa Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia b Department of Applied Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L3G1 Available online 16 January 2007
Abstract We investigate the influence of the effect of dynamic polarization of the carbon atom valence electrons on the angular distributions of protons channeled in (6, 4)@(11, 9) double-wall carbon nanotubes. The proton velocities are 3 and 5 a.u., corresponding to energies of 0.223 and 0.621 MeV, while the nanotube lengths are 0.1–0.2 lm. The interaction between a proton and the nanotube atoms is described by the Doyle–Turner potential whereas the image force acting on the proton is calculated by a two-dimensional hydrodynamic model of the dynamic response of the nanotube valence electrons. The angular distributions of channeled protons are generated by a computer simulation method using the numerical solution of the proton equations of motion in the transverse plane. Our analysis shows that the image force induces additional extrema in the proton deflection functions, giving rise to the rainbow maxima in the angular distributions that do not exist when the proton is subject to the atomic force only. These maxima could be used to probe the total interaction potential in proton channeling in nanotubes. Ó 2006 Elsevier B.V. All rights reserved. PACS: 61.85.+p; 41.75.Ht; 61.82.Rx Keywords: Nanotubes; Channeling; Dynamic polarization; Rainbows
1. Introduction It is well-known that meteorological rainbows appear as a consequence of photon scattering from water droplets [1,2]. Rainbows also occur in nucleus–nucleus collisions [3–5], atom or ion collisions with atoms or molecules [6], electron–molecule collisions [7], atom or electron scattering from crystal surfaces [8,9], and ion channeling in crystals [10,11]. We mention here recent studies of the rainbow effect in grazing atom scattering from metal surfaces under channeling conditions [12]. In those studies, precise measurements of the well-defined maxima in the angular distributions of scattered atoms, attributed to the rainbow effect, gave detailed information on the interaction potential of the *
Corresponding author. Tel./fax: +381 11 2455041. E-mail address: [email protected] (D. Borka).
0168-583X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.11.102
atoms with the metal surfaces. In addition, we mention here the theory of crystal rainbows [13], which was proven to be the proper theory of ion channeling in thin crystals. It has been applied successfully to ion channeling in short carbon nanotubes as well [14–17]. The reported simulations of ion channeling in carbon nanotubes [14–20] gave virtually no information on the effect of dynamic polarization of the nanotube valence electrons, which contribute to the ion energy loss and give rise to the image force acting on the ion [21]. This force attracts the ion to the nanotube walls. Unlike a crystal channel, the relatively large ‘‘empty’’ space inside a nanotube provides conditions similar to those in grazing atomsurface scattering, where the image force plays an important role [22]. This force was studied for cylindrical channels in solids [23], where it was shown to affect the angular distributions of highly charged ions transmitted through the channels [24]. The importance of the image force has been
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demonstrated recently in a simulation of ion channeling in a single-wall nanotube [25], where a significant redistribution of the ion flux after several oscillations of the ions between the nanotube walls has been predicted. In analogy to the grazing atom-surface scattering under channeling conditions [12], we expect that measurements of the positions of the rainbow maxima in the angular distributions of ions channeled in short carbon nanotubes can provide detailed information on the corresponding interaction potential. With that in mind, we have recently studied the rainbow effect with short (11, 9) single-wall carbon nanotubes and found that the image interaction gives rise to additional rainbow singularities in the angular distributions, which would be absent if ion channeling were governed only by the repulsive interaction with the nanotube wall [26]. The present work is an extension of our previous study of single-wall nanotubes [26] to the case of double-wall nanotubes. In particular, we are interested in the effects of the dynamic polarization interaction on ion trajectories in between the constituent nanotubes of a double-wall carbon nanotube. We investigate here the influence of the dynamic polarization effect on the angular distribution of protons channeled in (6, 4)@(11, 9) double-wall carbon nanotubes. The proton velocities are 3 and 5 a.u., corresponding to energies of 0.223 and 0.621 MeV, while the nanotube lengths are 0.1 and 0.2 lm. The image force acting on the ion has already been calculated for the doublewall [27] and multi-wall carbon nanotubes [28]. In those cases, the electrostatic coupling between the nanotube walls leads to a rich dependence of the image force on ion velocity [27,28]. 2. Theory The system we investigate is a proton moving along a (6, 4)@(11, 9) double-wall carbon nanotube, i.e. inside the (6, 4) constituent nanotube and in between the (6, 4) and (11, 9) constituent nanotubes. We assume that the axes of the constituent nanotubes coincide and are aligned along the z-axis of a Cartesian coordinate system with the origin placed in the entrance plane of the nanotube. The initial proton velocity vector, V~0 , is taken to be parallel to the z-axis. We also assume that the repulsive interaction between the proton and the nanotube atoms can be treated classically, and use the Doyle–Turner expression for the proton–nanotube atom interaction potential [29] averaged axially [30] and azimuthally [31]. The corresponding force repels the proton from the nanotube walls. The dynamic polarization interaction is treated by a two-dimensional (2D) 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 azimuthal averaging [27], consistent with the treatment of the repulsive interaction. We note that, while the axial averaging (the continuum approximation) is a standard procedure in ion channeling [30], the azimuthal averaging of the interaction is performed because the constituent
nanotubes are chiral. It has been shown that the 2D hydrodynamic model of the dynamic response of the electron gas provides a good approximation for both the ion energy loss due to the collective electron excitations, and the image force acting on the ion [21]. The angular distribution of transmitted protons is generated by the computer simulation method. The components of the proton impact parameter, x0 and y0, are chosen randomly within the uniform distribution in the region of the entrance plane defined by the outer constituent nanotube. The protons with impact parameters inside the rings defined by ½Rin a; Rin þ a and ½Rout a; Rout , where Rin and Rout are the radii of the inner and outer con1=3 stituent nanotubes, respectively, a ¼ ½9p2 =ð128Z 2 Þ a0 is the nanotube atom screening radius, and a0 is the Bohr radius, are treated as if they are backscattered, and are disregarded. The proton equations of motion in the transverse plane are solved numerically. Since the bond length of two nanotube atoms is 0.144 nm [32], we find Rin ¼ 0:346 nm and Rout ¼ 0:689 nm. The initial number of protons is 2 883 443. The components of the proton scattering angle, i.e. of the deflection function, Hx and Hy, are obtained via the expressions Hx ¼ V x =V 0 and Hy ¼ V y =V 0 , where Vx and Vy are the transverse components of the final proton velocity vector, V~. It has been demonstrated that proton channeling in nanotubes can be analyzed successfully via the corresponding mapping of the impact parameter plane, the x0y0 plane, to the scattering angle plane, the HxHy plane. However, in the case under consideration, the total interaction potential, which is the sum of the repulsive and attractive interaction potentials, is axially symmetric. This means that the analysis of the mapping is reduced to the analysis of 1=2 the scattering angle, H ¼ ðH2x þ H2y Þ , as a function of 1=2 the impact parameter, r0 ¼ ðx20 þ y 20 Þ . Therefore, we take that y ¼ 0, and analyze the deflection function Hx ðx0 Þ only. The extrema of this function are the rainbow extrema, and the corresponding singularities appearing in the angular distribution of channeled protons are the rainbow singularities [2]. 3. Results and discussion Fig. 1 gives the repulsive, attractive and total interaction potentials of a proton moving along the (6, 4)@(11, 9) double-wall carbon nanotube as a function of the distance of the proton from the nanotube axis, r, for the proton velocities V 0 ¼ 3 and 5 a.u. The vertical dotted lines separate the regions inside the inner constituent nanotube, r < Rin , in between the inner and outer constituent nanotubes, Rin < r < Rout , and outside the outer constituent nanotube, r > Rout . We show in Fig. 2(a) the deflection functions of protons channeled in the nanotube for the total interaction potential, i.e. with both the repulsive and attractive interaction potentials, and for the repulsive interaction potential only,
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Fig. 1. The interaction potentials of a proton channeled in a (6, 4)@(11, 9) double-wall carbon nanotube as a function of the distance of the proton from the nanotube axis for the proton velocities V 0 ¼ 3 and 5 a.u. The two vertical dotted lines represent the radii of the constituent nanotubes (Rin ¼ 6:54 a.u. and Rout ¼ 13:01 a.u.). The dotted curves give the repulsive interaction potential, the thick and thin dashed curves the attractive interaction potential for the proton velocities of 3 and 5 a.u., respectively, and the thick and thin solid curves the total interaction potential for the two proton velocities.
i.e. without the attractive interaction potential, for a proton velocity of 3 a.u. and nanotube length of 0.1 lm. There are 12 rainbow extrema. It is clear that the two extrema designated by 1 are due to the action of the attractive force, and that the other 10 extrema, i.e. the four extrema designated by 2, the two extrema designated by 3 and the four extrema designated by 4, are due to the joint action of the repulsive and attractive forces. Fig. 2(b) shows the corresponding angular distributions of channeled protons for the total interaction potential and the repulsive interaction potential only. There are four rainbow singularities. The singularity designated by 1 is attributed to the image force acting on the proton, and singularities 2, 3 and 4 to both the atomic and image forces. We give in Fig. 3(a) the deflection functions of protons channeled in the nanotube for the total interaction potential and the repulsive interaction potential only for a proton velocity of 3 a.u. and nanotube length of 0.2 lm. There are 24 rainbow extrema. It is clear that the two extrema designated by 1a, the two extrema designated by 1b and the two extrema designated by 2 are due to the action of the image force acting on the proton, and that the other 18 extrema, i.e. the two extrema designated by 3a, the four extrema designated by 3b, the four extrema designated by 4, the two extrema designated by 5, the four extrema designated by 6 and the two extrema designated by 7 are due to the joint action of the atomic and image forces. Fig. 3(b) gives the corresponding angular distributions of channeled protons for the total interaction potential and the repulsive interaction potential only. There are seven rainbow singularities. The singularities designated by 1
Fig. 2. (a) The deflection functions of protons channeled in a (6, 4)@(11, 9) double-wall carbon nanotube for the total interaction potential – solid curves, and without the attractive interaction potential – dashed curves. The vertical dotted lines represent the walls of the constituent nanotubes. (b) The corresponding angular distributions of channeled protons for the total interaction potential – solid curve, and without the attractive interaction potential – dashed curve. The proton velocity is V 0 ¼ 3 a.u. and the nanotube length is L ¼ 0:1 lm.
(1a and 1b) and 2 are attributed to the image force, and singularities 3 (3a and 3b), 4, 5, 6 and 7 to both the atomic and image forces. We calculated also the deflection functions of protons moving outside the (6, 4)@(11, 9) double-wall carbon nanotube for the total interaction potential and the repulsive interaction potential alone for a proton velocity of 3 a.u. and nanotube lengths of 0.1 and 0.2 lm. Only the extrema due to the image force acting on the proton were found. The results are very similar to the analogous results obtained earlier with the protons moving inside the (11, 9) single-wall carbon nanotubes [26]. This is explained by the facts that both the repulsive and attractive interaction potentials outside the (6, 4)@(11, 9) double-wall carbon nanotube are similar to those on either side of the (11, 9) single-wall carbon nanotube, and that for the shorter (11, 9) nanotubes only the extrema due to the image force appear.
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Fig. 3. (a) The deflection functions of protons channeled in a (6, 4)@(11, 9) double-wall carbon nanotube for the total interaction potential – solid curves, and without the attractive interaction potential – dashed curves. The vertical dotted lines represent the walls of the constituent nanotubes. (b) The corresponding angular distributions of channeled protons for the total interaction potential – solid curve, and without the attractive interaction potential – dashed curve. The proton velocity is V 0 ¼ 3 a.u. and the nanotube length is L ¼ 0:2 lm.
Fig. 4(a) and (b) show the proton trajectories corresponding to the extrema of the deflection functions for the total interaction potential shown in Figs. 2(a) and 3(a), respectively. We call them the rainbow proton trajectories [26]. Our analysis shows that these trajectories can be placed into five classes. The nth class, n ¼1–5, includes the rainbow proton trajectories with n proton deflections within the total interaction potential well. The corresponding rainbows are the rainbows of the nth order. One can see in Fig. 4(a) that the proton trajectories designated by 1 and 3, corresponding to rainbow extrema 1 and 3 in Fig. 2(a), belong to the first class, that trajectory 2, corresponding to extremum 2 in Fig. 2(a), belongs to the second class, and that trajectory 4, corresponding to extremum 4 in Fig. 2(a), belongs to the third class. Similarly, one can see in Fig. 4(b) that the rainbows corresponding to the proton trajectories designated by 2 and 3a are the primary rainbows, that the rainbows corresponding to trajectories
1a, 1b and 5 are the secondary rainbows, that the rainbows corresponding to trajectories 3b and 7 are the tertiary rainbows, that the rainbow corresponding to trajectory 4 is the rainbow of the fourth order, and that the rainbow corresponding to trajectory 6 is the rainbow of the fifth order. The analysis of the deflection functions and angular distributions of protons channeled in the nanotubes for the total interaction potential and the repulsive interaction potential only for a proton velocity of 5 a.u. and nanotube lengths of 0.1 and 0.2 lm showed that the influence of the dynamic polarization effect on the angular distributions was hardly visible. This is explained by the fact that the image force acting on the proton weakens considerably when one changes the proton velocity from 3 to 5 a.u. [26]. Also, the analysis of the deflection functions of protons channeled inside the (6, 4) constituent nanotube and in between the (6, 4) and (11, 9) constituent nanotubes showed that the influence of the dynamic polarization effect
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positions of the rainbow singularities in the angular distributions resulting from the repulsive forces alone. These effects are qualitatively similar to those found in our previous simulations of the rainbow effect in (11, 9) single-wall carbon nanotubes [26]. On the other hand, it is shown here that the influence of the dynamic polarization effect on the deflection functions of protons channeled between the constituent nanotubes of the (6, 4)@(11, 9) double-wall carbon nanotube is much weaker than its influence on the deflection functions of protons channeled inside the (6, 4) constituent nanotube. In particular, the inclusion of the dynamical polarization effect in our simulations produces no additional rainbow singularities in the angular distributions of protons channeled between the constituent nanotubes. This novel result can be rationalized by the fact that the space between the constituent nanotubes is too narrow to allow for a full development of the long-range effects in the image interaction. These observations may prove useful in designing future studies of the rainbow effect in multi-wall carbon nanotubes. Finally, we also show that the proton trajectories corresponding to the rainbow maxima in the angular distributions of channeled protons in question can be classified depending on the number of proton deflections within the total interaction potential well. The number of classes appearing for the proton velocity of 3 a.u. and the nanotube lengths of 0.1 and 0.2 lm was five, i.e. we found the primary, secondary and tertiary rainbows as well as rainbows of the fourth and fifth orders. References Fig. 4. (a) The proton trajectories corresponding to the extrema of the deflection function for the total interaction potential shown in Fig. 2(a). The proton velocity is V 0 ¼ 3 a.u. and the nanotube length is L ¼ 0:1 lm. (b) The proton trajectories corresponding to the extrema of the deflection function for the total interaction potential shown in Fig. 3(a). The proton velocity is V 0 ¼ 3 a.u. and the nanotube length is L ¼ 0:2 lm. The horizontal dotted lines represent the upper and lower walls of the inner constituent nanotube, and the upper wall of the outer constituent nanotube.
in between the constituent nanotubes is much weaker than inside the (6, 4) constituent nanotube. 4. Conclusions We report here on the influence of the dynamic polarization effect, or the image force, on the angular distributions of protons channeled in (6, 4)@(11, 9) double-wall carbon nanotubes. The proton velocities are 3 and 5 a.u. while the nanotube lengths are 0.1 and 0.2 lm. It is shown that the dynamic polarization effect introduces additional extrema in the deflection functions for protons channeled inside the (6, 4) constituent nanotube. As a result, the corresponding angular distributions of channeled protons contain additional rainbow singularities, which are not present when proton channeling is governed by the repulsive forces alone. Also, the image force acting on protons changes the
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[25] D.-P. Zhou, Y.-N. Wang, L. Wei, Z.L. Misˇkovic´, Phys. Rev. A 72 (2005) 023202. [26] D. Borka, S. Petrovic´, N. Nesˇkovic´, D.J. Mowbray, Z.L. Misˇkovic´, Phys. Rev. A 73 (2006) 062902. [27] D.J. Mowbray, S. Chung, Z.L. Misˇkovic´, F.O. Goodman, Y.-N. Wang, Nucl. Instr. and Meth. B 230 (2005) 142; G. Gumbs, A. Balassis, P. Fekete, Phys. Rev. B 73 (2006) 75411. [28] S. Chung, D.J. Mowbray, Z.L. Misˇkovic´, F.O. Goodman and Y.-N. Wang, Rad. Phys. Chem., in press. [29] P.A. Doyle, P.S. Turner, Acta Crystallogr. A 24 (1968) 390. [30] J. Lindhard, K. Dan. Vidensk. Selsk, Mat.-Fys. Medd. 34 (14) (1965) 1. [31] N.K. Zhevago, V.I. Glebov, Phys. Lett. A 250 (1998) 360; N.K. Zhevago, V.I. Glebov, J. Exp. Theor. Phys. 91 (2000) 504. [32] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 2001.
Nuclear Instruments and Methods in Physics Research B 256 (2007) 131–136 www.elsevier.com/locate/nimb
Influence of the dynamic polarization effect on the angular distributions of protons channeled in double-wall carbon nanotubes D. Borka a
a,*
, S. Petrovic´ a, N. Nesˇkovic´ a, D.J. Mowbray b, Z.L. Misˇkovic´
b
Laboratory of Physics (010), Vincˇa Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia b Department of Applied Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L3G1 Available online 16 January 2007
Abstract We investigate the influence of the effect of dynamic polarization of the carbon atom valence electrons on the angular distributions of protons channeled in (6, 4)@(11, 9) double-wall carbon nanotubes. The proton velocities are 3 and 5 a.u., corresponding to energies of 0.223 and 0.621 MeV, while the nanotube lengths are 0.1–0.2 lm. The interaction between a proton and the nanotube atoms is described by the Doyle–Turner potential whereas the image force acting on the proton is calculated by a two-dimensional hydrodynamic model of the dynamic response of the nanotube valence electrons. The angular distributions of channeled protons are generated by a computer simulation method using the numerical solution of the proton equations of motion in the transverse plane. Our analysis shows that the image force induces additional extrema in the proton deflection functions, giving rise to the rainbow maxima in the angular distributions that do not exist when the proton is subject to the atomic force only. These maxima could be used to probe the total interaction potential in proton channeling in nanotubes. Ó 2006 Elsevier B.V. All rights reserved. PACS: 61.85.+p; 41.75.Ht; 61.82.Rx Keywords: Nanotubes; Channeling; Dynamic polarization; Rainbows
1. Introduction It is well-known that meteorological rainbows appear as a consequence of photon scattering from water droplets [1,2]. Rainbows also occur in nucleus–nucleus collisions [3–5], atom or ion collisions with atoms or molecules [6], electron–molecule collisions [7], atom or electron scattering from crystal surfaces [8,9], and ion channeling in crystals [10,11]. We mention here recent studies of the rainbow effect in grazing atom scattering from metal surfaces under channeling conditions [12]. In those studies, precise measurements of the well-defined maxima in the angular distributions of scattered atoms, attributed to the rainbow effect, gave detailed information on the interaction potential of the *
Corresponding author. Tel./fax: +381 11 2455041. E-mail address: [email protected] (D. Borka).
0168-583X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.11.102
atoms with the metal surfaces. In addition, we mention here the theory of crystal rainbows [13], which was proven to be the proper theory of ion channeling in thin crystals. It has been applied successfully to ion channeling in short carbon nanotubes as well [14–17]. The reported simulations of ion channeling in carbon nanotubes [14–20] gave virtually no information on the effect of dynamic polarization of the nanotube valence electrons, which contribute to the ion energy loss and give rise to the image force acting on the ion [21]. This force attracts the ion to the nanotube walls. Unlike a crystal channel, the relatively large ‘‘empty’’ space inside a nanotube provides conditions similar to those in grazing atomsurface scattering, where the image force plays an important role [22]. This force was studied for cylindrical channels in solids [23], where it was shown to affect the angular distributions of highly charged ions transmitted through the channels [24]. The importance of the image force has been
132
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demonstrated recently in a simulation of ion channeling in a single-wall nanotube [25], where a significant redistribution of the ion flux after several oscillations of the ions between the nanotube walls has been predicted. In analogy to the grazing atom-surface scattering under channeling conditions [12], we expect that measurements of the positions of the rainbow maxima in the angular distributions of ions channeled in short carbon nanotubes can provide detailed information on the corresponding interaction potential. With that in mind, we have recently studied the rainbow effect with short (11, 9) single-wall carbon nanotubes and found that the image interaction gives rise to additional rainbow singularities in the angular distributions, which would be absent if ion channeling were governed only by the repulsive interaction with the nanotube wall [26]. The present work is an extension of our previous study of single-wall nanotubes [26] to the case of double-wall nanotubes. In particular, we are interested in the effects of the dynamic polarization interaction on ion trajectories in between the constituent nanotubes of a double-wall carbon nanotube. We investigate here the influence of the dynamic polarization effect on the angular distribution of protons channeled in (6, 4)@(11, 9) double-wall carbon nanotubes. The proton velocities are 3 and 5 a.u., corresponding to energies of 0.223 and 0.621 MeV, while the nanotube lengths are 0.1 and 0.2 lm. The image force acting on the ion has already been calculated for the doublewall [27] and multi-wall carbon nanotubes [28]. In those cases, the electrostatic coupling between the nanotube walls leads to a rich dependence of the image force on ion velocity [27,28]. 2. Theory The system we investigate is a proton moving along a (6, 4)@(11, 9) double-wall carbon nanotube, i.e. inside the (6, 4) constituent nanotube and in between the (6, 4) and (11, 9) constituent nanotubes. We assume that the axes of the constituent nanotubes coincide and are aligned along the z-axis of a Cartesian coordinate system with the origin placed in the entrance plane of the nanotube. The initial proton velocity vector, V~0 , is taken to be parallel to the z-axis. We also assume that the repulsive interaction between the proton and the nanotube atoms can be treated classically, and use the Doyle–Turner expression for the proton–nanotube atom interaction potential [29] averaged axially [30] and azimuthally [31]. The corresponding force repels the proton from the nanotube walls. The dynamic polarization interaction is treated by a two-dimensional (2D) 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 azimuthal averaging [27], consistent with the treatment of the repulsive interaction. We note that, while the axial averaging (the continuum approximation) is a standard procedure in ion channeling [30], the azimuthal averaging of the interaction is performed because the constituent
nanotubes are chiral. It has been shown that the 2D hydrodynamic model of the dynamic response of the electron gas provides a good approximation for both the ion energy loss due to the collective electron excitations, and the image force acting on the ion [21]. The angular distribution of transmitted protons is generated by the computer simulation method. The components of the proton impact parameter, x0 and y0, are chosen randomly within the uniform distribution in the region of the entrance plane defined by the outer constituent nanotube. The protons with impact parameters inside the rings defined by ½Rin a; Rin þ a and ½Rout a; Rout , where Rin and Rout are the radii of the inner and outer con1=3 stituent nanotubes, respectively, a ¼ ½9p2 =ð128Z 2 Þ a0 is the nanotube atom screening radius, and a0 is the Bohr radius, are treated as if they are backscattered, and are disregarded. The proton equations of motion in the transverse plane are solved numerically. Since the bond length of two nanotube atoms is 0.144 nm [32], we find Rin ¼ 0:346 nm and Rout ¼ 0:689 nm. The initial number of protons is 2 883 443. The components of the proton scattering angle, i.e. of the deflection function, Hx and Hy, are obtained via the expressions Hx ¼ V x =V 0 and Hy ¼ V y =V 0 , where Vx and Vy are the transverse components of the final proton velocity vector, V~. It has been demonstrated that proton channeling in nanotubes can be analyzed successfully via the corresponding mapping of the impact parameter plane, the x0y0 plane, to the scattering angle plane, the HxHy plane. However, in the case under consideration, the total interaction potential, which is the sum of the repulsive and attractive interaction potentials, is axially symmetric. This means that the analysis of the mapping is reduced to the analysis of 1=2 the scattering angle, H ¼ ðH2x þ H2y Þ , as a function of 1=2 the impact parameter, r0 ¼ ðx20 þ y 20 Þ . Therefore, we take that y ¼ 0, and analyze the deflection function Hx ðx0 Þ only. The extrema of this function are the rainbow extrema, and the corresponding singularities appearing in the angular distribution of channeled protons are the rainbow singularities [2]. 3. Results and discussion Fig. 1 gives the repulsive, attractive and total interaction potentials of a proton moving along the (6, 4)@(11, 9) double-wall carbon nanotube as a function of the distance of the proton from the nanotube axis, r, for the proton velocities V 0 ¼ 3 and 5 a.u. The vertical dotted lines separate the regions inside the inner constituent nanotube, r < Rin , in between the inner and outer constituent nanotubes, Rin < r < Rout , and outside the outer constituent nanotube, r > Rout . We show in Fig. 2(a) the deflection functions of protons channeled in the nanotube for the total interaction potential, i.e. with both the repulsive and attractive interaction potentials, and for the repulsive interaction potential only,
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Fig. 1. The interaction potentials of a proton channeled in a (6, 4)@(11, 9) double-wall carbon nanotube as a function of the distance of the proton from the nanotube axis for the proton velocities V 0 ¼ 3 and 5 a.u. The two vertical dotted lines represent the radii of the constituent nanotubes (Rin ¼ 6:54 a.u. and Rout ¼ 13:01 a.u.). The dotted curves give the repulsive interaction potential, the thick and thin dashed curves the attractive interaction potential for the proton velocities of 3 and 5 a.u., respectively, and the thick and thin solid curves the total interaction potential for the two proton velocities.
i.e. without the attractive interaction potential, for a proton velocity of 3 a.u. and nanotube length of 0.1 lm. There are 12 rainbow extrema. It is clear that the two extrema designated by 1 are due to the action of the attractive force, and that the other 10 extrema, i.e. the four extrema designated by 2, the two extrema designated by 3 and the four extrema designated by 4, are due to the joint action of the repulsive and attractive forces. Fig. 2(b) shows the corresponding angular distributions of channeled protons for the total interaction potential and the repulsive interaction potential only. There are four rainbow singularities. The singularity designated by 1 is attributed to the image force acting on the proton, and singularities 2, 3 and 4 to both the atomic and image forces. We give in Fig. 3(a) the deflection functions of protons channeled in the nanotube for the total interaction potential and the repulsive interaction potential only for a proton velocity of 3 a.u. and nanotube length of 0.2 lm. There are 24 rainbow extrema. It is clear that the two extrema designated by 1a, the two extrema designated by 1b and the two extrema designated by 2 are due to the action of the image force acting on the proton, and that the other 18 extrema, i.e. the two extrema designated by 3a, the four extrema designated by 3b, the four extrema designated by 4, the two extrema designated by 5, the four extrema designated by 6 and the two extrema designated by 7 are due to the joint action of the atomic and image forces. Fig. 3(b) gives the corresponding angular distributions of channeled protons for the total interaction potential and the repulsive interaction potential only. There are seven rainbow singularities. The singularities designated by 1
Fig. 2. (a) The deflection functions of protons channeled in a (6, 4)@(11, 9) double-wall carbon nanotube for the total interaction potential – solid curves, and without the attractive interaction potential – dashed curves. The vertical dotted lines represent the walls of the constituent nanotubes. (b) The corresponding angular distributions of channeled protons for the total interaction potential – solid curve, and without the attractive interaction potential – dashed curve. The proton velocity is V 0 ¼ 3 a.u. and the nanotube length is L ¼ 0:1 lm.
(1a and 1b) and 2 are attributed to the image force, and singularities 3 (3a and 3b), 4, 5, 6 and 7 to both the atomic and image forces. We calculated also the deflection functions of protons moving outside the (6, 4)@(11, 9) double-wall carbon nanotube for the total interaction potential and the repulsive interaction potential alone for a proton velocity of 3 a.u. and nanotube lengths of 0.1 and 0.2 lm. Only the extrema due to the image force acting on the proton were found. The results are very similar to the analogous results obtained earlier with the protons moving inside the (11, 9) single-wall carbon nanotubes [26]. This is explained by the facts that both the repulsive and attractive interaction potentials outside the (6, 4)@(11, 9) double-wall carbon nanotube are similar to those on either side of the (11, 9) single-wall carbon nanotube, and that for the shorter (11, 9) nanotubes only the extrema due to the image force appear.
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Fig. 3. (a) The deflection functions of protons channeled in a (6, 4)@(11, 9) double-wall carbon nanotube for the total interaction potential – solid curves, and without the attractive interaction potential – dashed curves. The vertical dotted lines represent the walls of the constituent nanotubes. (b) The corresponding angular distributions of channeled protons for the total interaction potential – solid curve, and without the attractive interaction potential – dashed curve. The proton velocity is V 0 ¼ 3 a.u. and the nanotube length is L ¼ 0:2 lm.
Fig. 4(a) and (b) show the proton trajectories corresponding to the extrema of the deflection functions for the total interaction potential shown in Figs. 2(a) and 3(a), respectively. We call them the rainbow proton trajectories [26]. Our analysis shows that these trajectories can be placed into five classes. The nth class, n ¼1–5, includes the rainbow proton trajectories with n proton deflections within the total interaction potential well. The corresponding rainbows are the rainbows of the nth order. One can see in Fig. 4(a) that the proton trajectories designated by 1 and 3, corresponding to rainbow extrema 1 and 3 in Fig. 2(a), belong to the first class, that trajectory 2, corresponding to extremum 2 in Fig. 2(a), belongs to the second class, and that trajectory 4, corresponding to extremum 4 in Fig. 2(a), belongs to the third class. Similarly, one can see in Fig. 4(b) that the rainbows corresponding to the proton trajectories designated by 2 and 3a are the primary rainbows, that the rainbows corresponding to trajectories
1a, 1b and 5 are the secondary rainbows, that the rainbows corresponding to trajectories 3b and 7 are the tertiary rainbows, that the rainbow corresponding to trajectory 4 is the rainbow of the fourth order, and that the rainbow corresponding to trajectory 6 is the rainbow of the fifth order. The analysis of the deflection functions and angular distributions of protons channeled in the nanotubes for the total interaction potential and the repulsive interaction potential only for a proton velocity of 5 a.u. and nanotube lengths of 0.1 and 0.2 lm showed that the influence of the dynamic polarization effect on the angular distributions was hardly visible. This is explained by the fact that the image force acting on the proton weakens considerably when one changes the proton velocity from 3 to 5 a.u. [26]. Also, the analysis of the deflection functions of protons channeled inside the (6, 4) constituent nanotube and in between the (6, 4) and (11, 9) constituent nanotubes showed that the influence of the dynamic polarization effect
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positions of the rainbow singularities in the angular distributions resulting from the repulsive forces alone. These effects are qualitatively similar to those found in our previous simulations of the rainbow effect in (11, 9) single-wall carbon nanotubes [26]. On the other hand, it is shown here that the influence of the dynamic polarization effect on the deflection functions of protons channeled between the constituent nanotubes of the (6, 4)@(11, 9) double-wall carbon nanotube is much weaker than its influence on the deflection functions of protons channeled inside the (6, 4) constituent nanotube. In particular, the inclusion of the dynamical polarization effect in our simulations produces no additional rainbow singularities in the angular distributions of protons channeled between the constituent nanotubes. This novel result can be rationalized by the fact that the space between the constituent nanotubes is too narrow to allow for a full development of the long-range effects in the image interaction. These observations may prove useful in designing future studies of the rainbow effect in multi-wall carbon nanotubes. Finally, we also show that the proton trajectories corresponding to the rainbow maxima in the angular distributions of channeled protons in question can be classified depending on the number of proton deflections within the total interaction potential well. The number of classes appearing for the proton velocity of 3 a.u. and the nanotube lengths of 0.1 and 0.2 lm was five, i.e. we found the primary, secondary and tertiary rainbows as well as rainbows of the fourth and fifth orders. References Fig. 4. (a) The proton trajectories corresponding to the extrema of the deflection function for the total interaction potential shown in Fig. 2(a). The proton velocity is V 0 ¼ 3 a.u. and the nanotube length is L ¼ 0:1 lm. (b) The proton trajectories corresponding to the extrema of the deflection function for the total interaction potential shown in Fig. 3(a). The proton velocity is V 0 ¼ 3 a.u. and the nanotube length is L ¼ 0:2 lm. The horizontal dotted lines represent the upper and lower walls of the inner constituent nanotube, and the upper wall of the outer constituent nanotube.
in between the constituent nanotubes is much weaker than inside the (6, 4) constituent nanotube. 4. Conclusions We report here on the influence of the dynamic polarization effect, or the image force, on the angular distributions of protons channeled in (6, 4)@(11, 9) double-wall carbon nanotubes. The proton velocities are 3 and 5 a.u. while the nanotube lengths are 0.1 and 0.2 lm. It is shown that the dynamic polarization effect introduces additional extrema in the deflection functions for protons channeled inside the (6, 4) constituent nanotube. As a result, the corresponding angular distributions of channeled protons contain additional rainbow singularities, which are not present when proton channeling is governed by the repulsive forces alone. Also, the image force acting on protons changes the
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