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Angular distributions of high energy protons channeled in long (10, 10) single-wall carbon nanotubes
Nuclear Instruments and Methods in Physics Research B 267 (2009) 2365–2368
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Angular distributions of high energy protons channeled in long (10, 10) single-wall carbon nanotubes S. Petrovic´ *, D. Borka, I. Telecˇki, N. Neškovic´ Laboratory of Physics (010), Vincˇa Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 3 March 2009 Received in revised form 27 April 2009 Available online 18 May 2009 PACS: 78.70.g 61.85.+p Keywords: Nanotubes Channeling
a b s t r a c t In this work we study the angular distributions of 1 GeV protons channeled in long (10, 10) single-wall carbon nanotubes. The nanotube length, L, is varied between 10 and 100 lm. The angular distributions of channeled protons are obtained using the numerical solution of the proton equations of motion in the transverse plane and the Monte Carlo method. The effects of proton energy loss and scattering angle dispersion caused by its collisions with the nanotube electrons are taken into account. Analysis shows that for L < 30 lm, the transverse structure of the nanotube could be deduced from the angular distribution. For L P 40 lm, the angular distribution contains the concentric circular ridges whose number increases and the average distance between them decreases when L increases. A possible application of the obtained results for characterization of carbon nanotubes is discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Axial crystal ion channeling is the ion motion along axial crystal channels. The channeling effect can happen if the angle between the initial ion velocity vector and the channel axis is smaller than the critical angle for channeling, which depends on the ion, its velocity and the crystal (channel). The motion of ion along the channel is explained by its repulsion from the atomic strings defining the channel, which are result of a correlated series of collisions of the ion with the atoms in the atomic strings. During the process the angle between the ion velocity vector and the channel axis remains small (smaller than the critical angle). An analogous explanation is valid for planar channeling, in which the ions move along the planar crystal channels. Axial and planar crystal channeling effects are described in detail in the review article of Gemmell [1]. Carbon nanotubes were discovered by Iijima in 1991 [2]. One can describe them as graphite crystallographic planes, with carbon atoms at the (two-dimensional) hexagonal lattice sites, rolled up into cylinders. The diameters of nanotubes are of the order of a nanometer and their lengths can be more than a hundred micrometers. They have remarkable geometrical and physical properties [3], and are considered as very promising new materials [4,5]. Soon after carbon nanotubes were discovered, Klimov and Letokhov [6] foresaw the effect of channeling of positively charged * Corresponding author. Tel.: +381 11 244 7700; fax: +381 11 244 7963. E-mail address: [email protected] (S. Petrovic´). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.05.002
particles in them, and discussed emission of hard X-rays by the channeled particles. After that, a number of theoretical groups have studied ion channeling in carbon nanotubes [7–12]. Greenenko and Shulga investigated in detail the motion of channeled protons in a bundle of (10, 10) single-wall carbon nanotubes (SWCNTs) [13]. Recently, the effect of ion channeling in nanotubes was reviewed by Miškovic´ [14]. All the above mentioned studies were theoretical. Although some experimental progress for observing the ion channeling effect in carbon nanotubes has been reported [15], to the authors’ best knowledge direct experimental evidence of the ion channeling effect in carbon nanotubes is still lacking. The main problem for achieving this goal is how to prepare a sample with the straight and uniform SWCNTs. The most promising method for such a preparation is the growth of SWCNTs inside the holes made in the dielectric media [15,16]. Recently, Chai et al. reported experimental observation of the channeling effect of 300 keV electrons through multi-wall carbon nanotubes (MWCNTs) [17]. They used a single MWCNT encapsulated in a carbon fiber coating, placed inside a transmission electron microscope. Petrovic´ et al. [18,19] and Neškovic´ et al. [20] have demonstrated that the rainbow effect enables the full explanation of the angular distributions of 1 GeV protons transmitted through the short straight and bent bundles of (10, 10) SWCNTs. Recently, Petrovic´ et al. [21] investigated channeling of 1 GeV protons in long (11, 9) SWCNTs. They showed that the angular distributions of channeled protons were characterized by a ring structure, with the number of rings increasing and the average distance between
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them decreasing in a regular and predictable way when the nanotube length increases. For large nanotube lengths, the angular distribution becomes equilibrated. Here we study the angular distributions of 1 GeV protons channeled in long (10, 10) SWCNTs. The nanotube length is chosen between 10 and 100 lm. This is the first detailed study of the angular distributions of channeled ions in long (10, 10) SWCNTs. The motivation for this work is to try to answer what are the similarities and differences between the angular distributions of 1 GeV protons channeled through long chiral (11, 9) [21] and achiral (10, 10) SWCNTs, and how these facts could be used for the characterization of the carbon nanotubes? The angular distribution of channeled protons is generated by the Monte Carlo method using the numerical solution of the proton equations of motion in the transverse plane. The proton energy loss and scattering angle dispersion caused by its collisions with the nanotube electrons are taken into account. We also include the divergence of the incident proton beam.
dE 4pZ 21 e4 2me c2 v 2 ne ln b2 ; ¼ 2 dz me v hxe
ð3Þ
where xe ¼ ð4pe2 ne =me Þ1=2 and ne ¼ DU th =4p; me is the electron mass, v the proton velocity, c2 ¼ 1=ð1 b2 Þ, b ¼ v =c and c the speed of light; xe is the angular frequency of the proton induced oscillations of the electron gas of the nanotube, and ne ¼ ne ðx; yÞ is the average (along the z axis) density of the electron gas [1,25]. The proton scattering angle dispersion caused by its collisions with the nanotube electrons is given by the expression:
dX2e me dE ; ¼ 2 2 dz dz m v
ð4Þ
where m is the relativistic proton mass [1,25]. The dispersions of the transverse components pffiffiffi of the proton scattering angle, Hx and Hy, are Xex ¼ Xey ¼ Xe = 2. Since the proton motion in the transverse plane is non-relativistic, the corresponding equations of motion are made relativistically correct by using the relativistic proton mass instead of its rest mass.
2. Theory The system we investigate is a relativistic high energy proton moving through a (10, 10) SWCNT. The z axis is taken to be parallel to the nanotube axis and the origin lies in its entrance plane, which is the impact parameter plane. From the channeling physics point of view, a (10, 10) SWCNT can be treated as one axial channel, which consists of 40 straight atomic strings parallel to its axis (which is also the channel axis). Accordingly, the ion channeling effect in carbon nanotubes has the same origin as in the case of axial crystal channels, i.e. it is a consequence of a correlated series of collisions of ion with the atoms in the atomic strings. We employ the continuum approximation [22], and use Molière’s expression for the proton–nanotube atom interaction potential [18–21],
VðrÞ ¼
Z 1 Z 2 e2 ½0:35 expðbrÞ þ 0:55 expð4brÞ þ 0:10 r expð20brÞ;
ð1Þ
where Z1 and Z2 are the atomic numbers of the proton and nanotube atom, respectively, e is the elementary charge, r is the distance between the proton and nanotube atom, b = 0.3/a, a ¼ ½9p2 =ð128Z 2 Þ1=3 a0 is the nanotube atom screening radius, and a0 is the Bohr radius. The effect of thermal vibrations of the nanotube atoms is included in the calculation. It is done by averaging the continuum interaction potential of the proton and ith atomic string, Ui, over the transversal displacements of the nanotube atoms assuming that they are small, independent and described by the Gaussian distribution function [23,24], 2 U th i ðx; yÞ ¼ U i ðx; yÞ þ ðrth =2ÞDU i ðx; yÞ;
ð2Þ
where U th i is the continuum interaction potentials of the proton and ith atomic string of the nanotube with the effect of thermal vibrations included, D @ xx þ @ yy , x and y are the transverse components of the proton position and rth is the standard deviation of the distribution, i.e. the one-dimensional thermal vibration amplitude of the nanotube atoms. It should be noted that the use of expression (2) is justified when the distance between the ion and the atomic string is large compared to the one-dimensional thermal vibration amplitude, which always holds for the case under the consideration. The proton–nanotube continuum interaction potential, U th , is the sum of the continuum interaction potentials of the proton and atomic strings. For the specific electronic proton energy loss we use the expression
3. Results and discussion As already stated, here we study the angular distributions of 1 GeV protons channeled in long (10, 10) SWCNTs. The nanotube length, L, is varied between 10 and 100 lm. Since the nanotube atom bond length is 0.14 nm, the nanotube radius is 0.67 nm and the thickness of one atomic layer, d, is 0.24 nm [3]. The one-dimensional thermal vibration amplitude of the nanotube atoms is estimated, using the Debye approximation, to be rth = 0.0053 nm [26], which is much less then the screening radius of the carbon atom, a = 0.026 nm. The proton impact parameters are chosen randomly within the (two-dimensional) uniform distribution in the circular region of the entrance plane defined by the nanotube radius. The proton whose impact parameter happens to be inside one of the circles around the atomic strings of the nanotube having the radius equal to a is treated as if it is backscattered, and is disregarded. Also, if the proton, during its motion through the nanotube, appears inside one of these circles or outside the circular region defined by the nanotube radius, or if its scattering angle is larger then the critical angle for channeling, wc ¼ ð2Z 1 Z 2 e2 =EdÞ1=2 , it is treated as if it is dechanneled, and is disregarded too. The number of channeled protons used in the calculation was 3 500 000. The divergence of the incident proton beam is set at Xb ¼ 0:1wc . Fig. 1(a)–(d) shows angular distributions of 1 GeV protons channeled in 10, 30, 50 and 100 lm long nanotubes, respectively. The sizes of a bin along the Hx and Hy axes are 6.4 lrad. For L = 10 lm, the angular distribution is characterized by 20 maxima lying on lines U ¼ tan1 ðHy =Hx Þ ¼ ð2n þ 3Þp=20, n = 0– 19, and 20 maxima lying on lines U ¼ ð2n þ 1Þp=20, n = 0–19, in the peripheral region of the scattering angle plane, and a maximum at the origin. The analysis shows that the observed 40 maxima carry the information about the transverse structure of the nanotube, which is composed of 20 pairs of straight atomic strings parallel to its axis [18,19]. In the case in which L = 30 lm, the angular distribution is characterized by 20 maxima lying on lines U ¼ ð2n þ 3Þp=20, n = 0–19, in the peripheral region of the scattering angle plane, a concentric circular ridge with 20 maxima lying on lines U ¼ ð2n þ 1Þp=20, n = 0–19, a concentric circular ridge with no azimuthal asymmetry in the central region of the scattering angle plane, and a maximum at the origin. As in the case of L = 10 lm, the transverse structure of the nanotube could be deduced from the angular distribution. The analysis shows that for L P 40 lm, the angular distributions of
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Fig. 1. Angular distributions of 1 GeV protons channeled in (10, 10) SWCNTs of length: (a) 10 lm, (b) 30 lm, (c) 50 lm and (d) 100 lm.
channeled protons carry no information about the transverse structure of the nanotube. They contain only the concentric circular ridges with no azimuthal asymmetry and the maxima at the origin. For L = 50 lm, the angular distribution contains five concentric circular ridges and a maximum at the origin. The distances between the ridges are close to each other, and the average distance between them is smaller than in the case of L = 30 lm. The analysis shows that the trend of increase of the number of ridges and decrease of the average distance between them when L increases also holds for L > 50 lm. For L P 80 lm, the ridges become less pronounced, due to the effect of proton scattering angle dispersion caused by its collisions with the nanotube electrons. Finally, for L = 100 lm, the angular distribution becomes close to a bellshaped one, with nine small ridges. Fig. 2(a) and (b) show the dependences of the number of the concentric circular ridges appearing in the angular distributions of channeled protons, N, and the average distance between them, d, on the variable L, for between 40 and 100 lm, respectively.
The analysis shows that the linear function f1 ¼ a1 L þ b1 is a good fit to the dependence of the number of ridges, N, on the variable L. The corresponding values for the fitting parameters are: a1 = 1.96 lm1 and b1 = 0.07. Also, for the dependence of the average distance between the ridges, d, on the variable L, the analysis shows that the function f2 ¼ a2 expðL=b2 Þ þ a3 expðL=b3 Þ can provide an excellent fit to this dependence. The corresponding values for the fitting parameters are: a2 = 0.1170 mrad, b2 = 24.40 lm, a3 = 0.02811 mrad and b3 = 184.9 lm. Those fitting curves are also shown in Fig. 2(a) and (b). It is interesting to compare the results presented here with the ones presented in our previous work on channeling of 1 GeV protons in long (11, 9) SWCNTs [21]. The concentric circular ridges in the angular distribution of channeled protons in the (10, 10) case correspond to the rings in the angular distributions in the (11, 9) case. In both cases the number of ridges and the number of rings increase linearly with the length of the nanotubes. Also, the average distance between the ridges, in the (10, 10) case, changes with L exponentially in the same way as the average dis-
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Fig. 2. Dependences of: (a) the number of the circular ridges and (b) the average distance between them for the angular distributions of 1 GeV protons channeled in (10, 10) SWCNTs on their length. The full lines represent the fitting curves.
tance between the rings, in the (11, 9) case. This opens a possibility of employing the angular distributions of protons channeled in long carbon nanotubes for their characterization. Namely, it seems that each type of nanotube for each length has a characteristic pattern of concentric circular ridges, what could be checked experimentally. However, an easier way to check this possibility experimentally would be to change the proton energy instead of L. This method for the characterization of carbon nanotubes would have been complementary to the method proposed in our previous work [18], that employs the angular distribution of channeled protons in the case of very short rather that long (10, 10) SWCNTs. 4. Conclusions The angular distributions of 1 GeV protons channeled in long (10, 10) SWCNTs is studied here. The obtained results show that for L < 30 lm, the transverse structure of the nanotube could be deduced from the angular distribution of channeled protons. For L P 40 lm, the angular distribution is characterized by the concentric circular ridges whose number increases linearly and the distance between them decreases exponentially when L increases. The comparison between the angular distributions of 1 GeV protons channeled in long (10, 10) and (11, 9) SWCNTs opens a possibility of employing the angular distributions of protons channeled in long carbon nanotubes for their characterization. References [1] D.S. Gemmell, Rev. Mod. Phys. 46 (1974) 129. [2] S. Iijima, Nature 354 (1991) 56.
[3] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 2001. [4] Z. Yao, H.W.Ch. Postma, L. Balents, C. Dekker, Nature 402 (1999) 273. [5] R.H. Baughman, A.A. Zakhidov, W.A. de Heer, Science 297 (2002) 787. [6] V.V. Klimov, V.S. Letokhov, Phys. Lett. A 222 (1996) 424. [7] L.A. Gevorgian, K.A. Ispirian, R.K. Ispirian, Nucl. Instr. and Meth. B 145 (1998) 155. [8] N.K. Zhevago, V.I. Glebov, Phys. Lett. A 250 (1998) 360; N.K. Zhevago, V.I. Glebov, Phys. Lett. A 310 (2003) 301. [9] V.M. Biryukov, S. Bellucci, Phys. Lett. B 542 (2002) 111; S. Bellucci, V.M. Biryukov, Yu.A. Chesnokov, V. Guidi, W. Scandale, Nucl. Instr. and Meth. B 203 (2003) 236. [10] A.V. Krasheninnikov, K. Nordlund, Phys. Rev. B 71 (2005) 245408. [11] L.-P. Zheng, Z.-Y. Zhu, Y. Li, D.-Z. Zhu, H.-H. Xia, J. Phys. Chem. C 112 (2008) 15204; L.-P. Zheng, Z.-Y. Zhu, Y. Li, D.-Z. Zhu, H.-H. Xia, Nucl. Instr. and Meth. B 266 (2008) 849. [12] C.S. Moura, L. Amaral, J. Phys. Chem. B 109 (2005) 13515; C.S. Moura, L. Amaral, Carbon 45 (2007) 1802. [13] A.A. Greenenko, N.F. Shulga, Nucl. Instr. and Meth. B 205 (2003) 767. [14] Z.L. Miškovic´, Rad. Eff. Def. Sol. 162 (2007) 185. [15] Z. Zhu, D. Zhu, R. Lu, Z. Xu, W. Zhang, H. Xia, Proceedings of the International Conference on Charged and Neutral Particles Channeling Phenomena, SPIE, Bellingham, Washington, 2005, p. 597413. [16] A.S. Berdinsky, P.S. Alegaonkar, H.C. Lee, J.S. Jung, J.H. Han, J.B. Yoo, D. Fink, L.T. Chadderton, Nano 2 (2007) 59. [17] G. Chai, H. Heinrich, L. Chowa, T. Schenkel, Appl. Phys. Lett. 91 (2007) 103101. [18] S. Petrovic´, D. Borka, N. Neškovic´, Eur. Phys. J. B 44 (2005) 41. [19] S. Petrovic´, D. Borka, N. Neškovic´, Nucl. Instr. and Meth. B 234 (2005) 78. [20] N. Neškovic´, S. Petrovic´, D. Borka, Nucl. Instr. and Meth. B 230 (2005) 106. [21] S. Petrovic´, I. Telecˇki, D. Borka, N. Neškovic´, Phys. Lett. A 372 (2008) 6003. [22] J. Lindhard, K. Dan. Vidensk. Selsk., Mat. Fys. Medd. 34 (14) (1965) 1. [23] N. Neškovic´, Phys. Rev. B 33 (1986) 6030. [24] B.R. Appleton, S. Erginsoy, W.M. Gibson, Phys. Rev. 161 (1967) 330. [25] S. Petrovic´, S. Korica, M. Kokkoris, N. Neškovic´, Nucl. Instr. and Meth. B 193 (2002) 152. [26] J. Hone, B. Batlogg, Z. Benes, A.T. Johnson, J.E. Fischer, Science 289 (2000) 1730.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Angular distributions of high energy protons channeled in long (10, 10) single-wall carbon nanotubes S. Petrovic´ *, D. Borka, I. Telecˇki, N. Neškovic´ Laboratory of Physics (010), Vincˇa Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 3 March 2009 Received in revised form 27 April 2009 Available online 18 May 2009 PACS: 78.70.g 61.85.+p Keywords: Nanotubes Channeling
a b s t r a c t In this work we study the angular distributions of 1 GeV protons channeled in long (10, 10) single-wall carbon nanotubes. The nanotube length, L, is varied between 10 and 100 lm. The angular distributions of channeled protons are obtained using the numerical solution of the proton equations of motion in the transverse plane and the Monte Carlo method. The effects of proton energy loss and scattering angle dispersion caused by its collisions with the nanotube electrons are taken into account. Analysis shows that for L < 30 lm, the transverse structure of the nanotube could be deduced from the angular distribution. For L P 40 lm, the angular distribution contains the concentric circular ridges whose number increases and the average distance between them decreases when L increases. A possible application of the obtained results for characterization of carbon nanotubes is discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Axial crystal ion channeling is the ion motion along axial crystal channels. The channeling effect can happen if the angle between the initial ion velocity vector and the channel axis is smaller than the critical angle for channeling, which depends on the ion, its velocity and the crystal (channel). The motion of ion along the channel is explained by its repulsion from the atomic strings defining the channel, which are result of a correlated series of collisions of the ion with the atoms in the atomic strings. During the process the angle between the ion velocity vector and the channel axis remains small (smaller than the critical angle). An analogous explanation is valid for planar channeling, in which the ions move along the planar crystal channels. Axial and planar crystal channeling effects are described in detail in the review article of Gemmell [1]. Carbon nanotubes were discovered by Iijima in 1991 [2]. One can describe them as graphite crystallographic planes, with carbon atoms at the (two-dimensional) hexagonal lattice sites, rolled up into cylinders. The diameters of nanotubes are of the order of a nanometer and their lengths can be more than a hundred micrometers. They have remarkable geometrical and physical properties [3], and are considered as very promising new materials [4,5]. Soon after carbon nanotubes were discovered, Klimov and Letokhov [6] foresaw the effect of channeling of positively charged * Corresponding author. Tel.: +381 11 244 7700; fax: +381 11 244 7963. E-mail address: [email protected] (S. Petrovic´). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.05.002
particles in them, and discussed emission of hard X-rays by the channeled particles. After that, a number of theoretical groups have studied ion channeling in carbon nanotubes [7–12]. Greenenko and Shulga investigated in detail the motion of channeled protons in a bundle of (10, 10) single-wall carbon nanotubes (SWCNTs) [13]. Recently, the effect of ion channeling in nanotubes was reviewed by Miškovic´ [14]. All the above mentioned studies were theoretical. Although some experimental progress for observing the ion channeling effect in carbon nanotubes has been reported [15], to the authors’ best knowledge direct experimental evidence of the ion channeling effect in carbon nanotubes is still lacking. The main problem for achieving this goal is how to prepare a sample with the straight and uniform SWCNTs. The most promising method for such a preparation is the growth of SWCNTs inside the holes made in the dielectric media [15,16]. Recently, Chai et al. reported experimental observation of the channeling effect of 300 keV electrons through multi-wall carbon nanotubes (MWCNTs) [17]. They used a single MWCNT encapsulated in a carbon fiber coating, placed inside a transmission electron microscope. Petrovic´ et al. [18,19] and Neškovic´ et al. [20] have demonstrated that the rainbow effect enables the full explanation of the angular distributions of 1 GeV protons transmitted through the short straight and bent bundles of (10, 10) SWCNTs. Recently, Petrovic´ et al. [21] investigated channeling of 1 GeV protons in long (11, 9) SWCNTs. They showed that the angular distributions of channeled protons were characterized by a ring structure, with the number of rings increasing and the average distance between
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them decreasing in a regular and predictable way when the nanotube length increases. For large nanotube lengths, the angular distribution becomes equilibrated. Here we study the angular distributions of 1 GeV protons channeled in long (10, 10) SWCNTs. The nanotube length is chosen between 10 and 100 lm. This is the first detailed study of the angular distributions of channeled ions in long (10, 10) SWCNTs. The motivation for this work is to try to answer what are the similarities and differences between the angular distributions of 1 GeV protons channeled through long chiral (11, 9) [21] and achiral (10, 10) SWCNTs, and how these facts could be used for the characterization of the carbon nanotubes? The angular distribution of channeled protons is generated by the Monte Carlo method using the numerical solution of the proton equations of motion in the transverse plane. The proton energy loss and scattering angle dispersion caused by its collisions with the nanotube electrons are taken into account. We also include the divergence of the incident proton beam.
dE 4pZ 21 e4 2me c2 v 2 ne ln b2 ; ¼ 2 dz me v hxe
ð3Þ
where xe ¼ ð4pe2 ne =me Þ1=2 and ne ¼ DU th =4p; me is the electron mass, v the proton velocity, c2 ¼ 1=ð1 b2 Þ, b ¼ v =c and c the speed of light; xe is the angular frequency of the proton induced oscillations of the electron gas of the nanotube, and ne ¼ ne ðx; yÞ is the average (along the z axis) density of the electron gas [1,25]. The proton scattering angle dispersion caused by its collisions with the nanotube electrons is given by the expression:
dX2e me dE ; ¼ 2 2 dz dz m v
ð4Þ
where m is the relativistic proton mass [1,25]. The dispersions of the transverse components pffiffiffi of the proton scattering angle, Hx and Hy, are Xex ¼ Xey ¼ Xe = 2. Since the proton motion in the transverse plane is non-relativistic, the corresponding equations of motion are made relativistically correct by using the relativistic proton mass instead of its rest mass.
2. Theory The system we investigate is a relativistic high energy proton moving through a (10, 10) SWCNT. The z axis is taken to be parallel to the nanotube axis and the origin lies in its entrance plane, which is the impact parameter plane. From the channeling physics point of view, a (10, 10) SWCNT can be treated as one axial channel, which consists of 40 straight atomic strings parallel to its axis (which is also the channel axis). Accordingly, the ion channeling effect in carbon nanotubes has the same origin as in the case of axial crystal channels, i.e. it is a consequence of a correlated series of collisions of ion with the atoms in the atomic strings. We employ the continuum approximation [22], and use Molière’s expression for the proton–nanotube atom interaction potential [18–21],
VðrÞ ¼
Z 1 Z 2 e2 ½0:35 expðbrÞ þ 0:55 expð4brÞ þ 0:10 r expð20brÞ;
ð1Þ
where Z1 and Z2 are the atomic numbers of the proton and nanotube atom, respectively, e is the elementary charge, r is the distance between the proton and nanotube atom, b = 0.3/a, a ¼ ½9p2 =ð128Z 2 Þ1=3 a0 is the nanotube atom screening radius, and a0 is the Bohr radius. The effect of thermal vibrations of the nanotube atoms is included in the calculation. It is done by averaging the continuum interaction potential of the proton and ith atomic string, Ui, over the transversal displacements of the nanotube atoms assuming that they are small, independent and described by the Gaussian distribution function [23,24], 2 U th i ðx; yÞ ¼ U i ðx; yÞ þ ðrth =2ÞDU i ðx; yÞ;
ð2Þ
where U th i is the continuum interaction potentials of the proton and ith atomic string of the nanotube with the effect of thermal vibrations included, D @ xx þ @ yy , x and y are the transverse components of the proton position and rth is the standard deviation of the distribution, i.e. the one-dimensional thermal vibration amplitude of the nanotube atoms. It should be noted that the use of expression (2) is justified when the distance between the ion and the atomic string is large compared to the one-dimensional thermal vibration amplitude, which always holds for the case under the consideration. The proton–nanotube continuum interaction potential, U th , is the sum of the continuum interaction potentials of the proton and atomic strings. For the specific electronic proton energy loss we use the expression
3. Results and discussion As already stated, here we study the angular distributions of 1 GeV protons channeled in long (10, 10) SWCNTs. The nanotube length, L, is varied between 10 and 100 lm. Since the nanotube atom bond length is 0.14 nm, the nanotube radius is 0.67 nm and the thickness of one atomic layer, d, is 0.24 nm [3]. The one-dimensional thermal vibration amplitude of the nanotube atoms is estimated, using the Debye approximation, to be rth = 0.0053 nm [26], which is much less then the screening radius of the carbon atom, a = 0.026 nm. The proton impact parameters are chosen randomly within the (two-dimensional) uniform distribution in the circular region of the entrance plane defined by the nanotube radius. The proton whose impact parameter happens to be inside one of the circles around the atomic strings of the nanotube having the radius equal to a is treated as if it is backscattered, and is disregarded. Also, if the proton, during its motion through the nanotube, appears inside one of these circles or outside the circular region defined by the nanotube radius, or if its scattering angle is larger then the critical angle for channeling, wc ¼ ð2Z 1 Z 2 e2 =EdÞ1=2 , it is treated as if it is dechanneled, and is disregarded too. The number of channeled protons used in the calculation was 3 500 000. The divergence of the incident proton beam is set at Xb ¼ 0:1wc . Fig. 1(a)–(d) shows angular distributions of 1 GeV protons channeled in 10, 30, 50 and 100 lm long nanotubes, respectively. The sizes of a bin along the Hx and Hy axes are 6.4 lrad. For L = 10 lm, the angular distribution is characterized by 20 maxima lying on lines U ¼ tan1 ðHy =Hx Þ ¼ ð2n þ 3Þp=20, n = 0– 19, and 20 maxima lying on lines U ¼ ð2n þ 1Þp=20, n = 0–19, in the peripheral region of the scattering angle plane, and a maximum at the origin. The analysis shows that the observed 40 maxima carry the information about the transverse structure of the nanotube, which is composed of 20 pairs of straight atomic strings parallel to its axis [18,19]. In the case in which L = 30 lm, the angular distribution is characterized by 20 maxima lying on lines U ¼ ð2n þ 3Þp=20, n = 0–19, in the peripheral region of the scattering angle plane, a concentric circular ridge with 20 maxima lying on lines U ¼ ð2n þ 1Þp=20, n = 0–19, a concentric circular ridge with no azimuthal asymmetry in the central region of the scattering angle plane, and a maximum at the origin. As in the case of L = 10 lm, the transverse structure of the nanotube could be deduced from the angular distribution. The analysis shows that for L P 40 lm, the angular distributions of
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Fig. 1. Angular distributions of 1 GeV protons channeled in (10, 10) SWCNTs of length: (a) 10 lm, (b) 30 lm, (c) 50 lm and (d) 100 lm.
channeled protons carry no information about the transverse structure of the nanotube. They contain only the concentric circular ridges with no azimuthal asymmetry and the maxima at the origin. For L = 50 lm, the angular distribution contains five concentric circular ridges and a maximum at the origin. The distances between the ridges are close to each other, and the average distance between them is smaller than in the case of L = 30 lm. The analysis shows that the trend of increase of the number of ridges and decrease of the average distance between them when L increases also holds for L > 50 lm. For L P 80 lm, the ridges become less pronounced, due to the effect of proton scattering angle dispersion caused by its collisions with the nanotube electrons. Finally, for L = 100 lm, the angular distribution becomes close to a bellshaped one, with nine small ridges. Fig. 2(a) and (b) show the dependences of the number of the concentric circular ridges appearing in the angular distributions of channeled protons, N, and the average distance between them, d, on the variable L, for between 40 and 100 lm, respectively.
The analysis shows that the linear function f1 ¼ a1 L þ b1 is a good fit to the dependence of the number of ridges, N, on the variable L. The corresponding values for the fitting parameters are: a1 = 1.96 lm1 and b1 = 0.07. Also, for the dependence of the average distance between the ridges, d, on the variable L, the analysis shows that the function f2 ¼ a2 expðL=b2 Þ þ a3 expðL=b3 Þ can provide an excellent fit to this dependence. The corresponding values for the fitting parameters are: a2 = 0.1170 mrad, b2 = 24.40 lm, a3 = 0.02811 mrad and b3 = 184.9 lm. Those fitting curves are also shown in Fig. 2(a) and (b). It is interesting to compare the results presented here with the ones presented in our previous work on channeling of 1 GeV protons in long (11, 9) SWCNTs [21]. The concentric circular ridges in the angular distribution of channeled protons in the (10, 10) case correspond to the rings in the angular distributions in the (11, 9) case. In both cases the number of ridges and the number of rings increase linearly with the length of the nanotubes. Also, the average distance between the ridges, in the (10, 10) case, changes with L exponentially in the same way as the average dis-
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Fig. 2. Dependences of: (a) the number of the circular ridges and (b) the average distance between them for the angular distributions of 1 GeV protons channeled in (10, 10) SWCNTs on their length. The full lines represent the fitting curves.
tance between the rings, in the (11, 9) case. This opens a possibility of employing the angular distributions of protons channeled in long carbon nanotubes for their characterization. Namely, it seems that each type of nanotube for each length has a characteristic pattern of concentric circular ridges, what could be checked experimentally. However, an easier way to check this possibility experimentally would be to change the proton energy instead of L. This method for the characterization of carbon nanotubes would have been complementary to the method proposed in our previous work [18], that employs the angular distribution of channeled protons in the case of very short rather that long (10, 10) SWCNTs. 4. Conclusions The angular distributions of 1 GeV protons channeled in long (10, 10) SWCNTs is studied here. The obtained results show that for L < 30 lm, the transverse structure of the nanotube could be deduced from the angular distribution of channeled protons. For L P 40 lm, the angular distribution is characterized by the concentric circular ridges whose number increases linearly and the distance between them decreases exponentially when L increases. The comparison between the angular distributions of 1 GeV protons channeled in long (10, 10) and (11, 9) SWCNTs opens a possibility of employing the angular distributions of protons channeled in long carbon nanotubes for their characterization. References [1] D.S. Gemmell, Rev. Mod. Phys. 46 (1974) 129. [2] S. Iijima, Nature 354 (1991) 56.
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