Stretching of carbon-carbon bonds in a 0.7 nm diameter carbon nanotube studied by electron diffraction

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1 PHYSICAL REVIEW B 70, (2004) Stretching of carbon-carbon bonds in a 0.7 nm diameter carbon nanotube studied by electron diffraction Kaori Hirahara* and Shunji Bandow Department of Materials Science and Engineering, 21st Century COE, Meijo University, Nagoya , Japan Hiromichi Kataura Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology, Tsukuba , Japan Mathieu Kociak Laboratoire de Physique des Solides, Bâtiment 510, Université Paris-Sud, CNRS UMR8502, F Orsay, France Sumio Iijima Department of Materials Science and Engineering, 21st Century COE, Meijo University, Nagoya , Japan, NEC Corporation, 34 Miyukigaoka, Tsukuba, Ibaraki , Japan, and Research Center for Advanced Carbon Materials, National Institute of Advanced Industrial Science and Technology, Tsukuba , Japan (Received 17 June 2004; published 19 November 2004) We studied a carbon-carbon bond-length of single-wall carbon nanotube with the diameter in the class of 0.7 nm by the electron diffraction. The nanotube, used in this study, was grown as an inner tube of a double wall carbon nanotube (DWNT) from peapod. Taking advantage of the use of DWNT, we can accurately determine the direction of incident electron beam, chiral indices, and both intensities and positions of diffraction spots. Due to careful analyses of these electron diffraction data, it was found that the carbon-carbon bond-length for 0.7 nm diameter class of an inner tube is stretched by 0.9±0.1% as compared with that for 1.4 nm outer tube. This is experimental evidence indicating a stretching of carbon-carbon bond in the small diameter tube. DOI: /PhysRevB PACS number(s): Lj, c, Tp, De I. INTRODUCTION A model of a rolled-up graphene sheet is normally used to explain the structure of a carbon nanotube. 1 For extremely thin single-wall carbon nanotubes (SWNTs), especially around nm in diameter, their electronic structure cannot be simply described by ideal sp 2 bonded character due to large curvature of the graphene sheet. 2,3 This fact suggests a possibility of a small change in carbon-carbon C-C bond length for a small diameter tube, and such a change in bond length will modify the electronic structure. Although the theoretically predicted C- C bond length for nanotubes is nm, 2,4 this value has not been examined experimentally. In the present study, we report C-C bond-length of SWNTs with diameters less than 1 nm by using electron diffraction (ED). ED is a powerful method for structural characterization applicable to individual carbon nanotubes 1,5 9 other than the x-ray diffraction and scanning tunneling microscopy, which have been used for the analyses of nanotube structures Peapods derived double wall carbon nanotubes (DWNTs) were used in the present experiments. Reasons of using DWNTs are indicated as follows: First, the inner tubes with 0.7 nm diameter are found to be more stable under electron beam irradiation than free-standing SWNTs. Secondly and most essentially, the relative value of the C-C bond length of the inner tube ( nm diameter) can be accurately determined by referring to that of the outer tube ( nm diameter). 19 Coexistence of two SWNTs in a DWNT enables us to calibrate accurately the diffraction data as proposed in this report. II. EXPERIMENT Images of high-resolution transmission electron microscope (HRTEM: JEOL JEM-2010F) and ED patterns were recorded from an isolated DWNT. For taking ED patterns, we converged the electron beam to a size of about 5 10 nm on the specimen with an acceleration voltage of 120 kv and reduced the dose of the electron beam as low as possible in order to minimize electron beam irradiation damage. Typical exposure time was less than 3 s, where the doses of electron beam on the specimen were e/nm pa/nm 2 and e/nm pa/nm 2, respectively, for recording high-resolution images and ED patterns. Electron beam irradiation damage of nanotubes was carefully avoided in recording HRTEM images, and we confirmed that no appreciable damages were recognized under such an exposure condition. III. EXPERIMENTAL RESULTS A. ED pattern of a DWNT We first determined the chiral indices of a DWNT by analyzing its ED patterns. The incident electron beam direction against the tube axis was estimated by accurate measurements on the positions and intensities of diffraction spots /2004/70(20)/205422(7)/$ The American Physical Society

2 HIRAHARA et al. PHYSICAL REVIEW B 70, (2004) FIG. 1. Electron diffraction from an isolated double-wall carbon nanotube (DWNT) (a) and schematic of hk diffractions (b). Mean diameter D and interlayer distance d can be measured on the equatorial line, and they are, respectively, 1.16±0.2 nm and 0.37±0.04 nm. One layer line between a pair of arrows is perpendicular to the tube axis. Schematic of hk diffraction rods corresponded to the first peaks of layer lines in (a) is in (b). Apparent chiral angles for inner inner and outer tubes outer can be measured, respectively, to be 30.0 ±1 and 18.1 ±0.2. Then, we obtained a relative value of the C-C bond length for a thin inner tube by referring to that for a outer tube. An ED pattern taken from an isolated DWNT is shown in Fig. 1(a) and its schematic in Fig. 1(b). A pair of chiral indices of this DWNT was determined with the aid of simulations, by following the procedure described in Refs The ED pattern of a DWNT is a superposition of those from the inner and outer tubes. Now, the diffracted amplitude from a single tube is a set of diffused diffraction spots perpendicular to the tube axis, which are called layer lines. The layer lines of a SWNT originate in the diffractions from both top and bottom parts of a rolled graphene sheet perpendicular to the incident electron beam. If the inner and outer tubes of a DWNT are not registered in their orientation, the layer lines of the two tubes do not overlap. Therefore, a single DWNT should give four sets (two pairs) of diffraction patterns. The analysis of layer lines can be carried out in the same way as proposed for SWNT, 6,7 except for the center layer, called equatorial line, in which the amplitudes of two tubes overlap. The amplitude along one layer line is described by the Bessel function, which represents a pseudo-oscillation with a pseudoperiod inversely proportional to the tube diameter. Series of the layer lines modified by the oscillation are clearly seen in the experimental ED pattern [Fig. 1(a)], where a layer line is pointed by a pair of arrows. Therefore, we can assign each layer line to the inner or outer tube by measuring the period of oscillation in the layer line. In Fig. 1(a), two types of oscillations with different periods can be seen as indicated, respectively, by open and solid arrow-heads. Lengths of the diffused rods (first peak maxima of layer lines) noted by open arrow heads are shorter than those noted by solid arrow heads. This difference directly indicates that the openarrowed layer lines are from the large outer tube and solidarrowed ones are from the small inner tube. The first peak maxima of a set of layer lines associate with hk diffraction spots, and have a sixfold symmetry. Here we employed two-dimensional Miller indices hk for hexagonal lattice to represent each layer line instead of threedimensional ones hki0, since a graphene sheet forming a SWNT has two-dimensional hexagonal lattice [Fig. 1(b)]. Twelve of the open diffused rods represent (10) type of diffraction spots from the outer tube, which makes a pair of hexagons as pointed by dotted lines. The solid rods are responsible for the diffraction from the inner tube and only two (10) type of spots are recognized at corners of the other hexagon indicated by the solid line. 23 Asymmetric intensity distributions, namely, 01 and 01 spots being strong, while 10 and 1 0 spots very weak (marked by in the figure), are caused by a tilt of the inner tube from the normal incidence of electron beam. Precise measurement of the tilt angle is needed to determine the C-C bond length as described later. B. Determination of chiral indices The rotation of each hexagon against the tube axis is related to the chiral angle, 6,7 which can be measured independently. In order to minimize the error, we actually measured 2 for the outer tube [see definition of 2 in Fig. 1(b)], where is equal to 30. A measured value of the chiral angle for the outer tube outer was estimated to be 18.1±0.2. For the inner tube, inner was measured directly as 30.0± 1.0. On the other hand, the mean diameter D, which is the average diameter of inner and outer tubes, and the interlayer distance d can be deduced from the periods of the oscillation of the equatorial line. The oscillation is describable by the product of two oscillations with periods of D and d, 24 so that these values in Fig. 1(a) became 1.16± 0.2 nm and 0.37± 0.04 nm, respectively. From these measured values of outer, inner, D, and d, we uniquely determined the chiral indices of inner and outer tubes of a DWNT, recorded in Fig. 1, being (5, 5) and (14, 6), respectively. This DWNT was designated as (5, 5)@(14, 6). C. Tilt and rotation of the DWNT against the electron beam To determine the C- C bond-length, we need to know more accurate positions and intensities of each diffraction

3 STRETCHING OF CARBON-CARBON BONDS IN A 0.7- PHYSICAL REVIEW B 70, (2004) FIG. 2. Schematic of a (5, 5)@(14, 6) DWNT (a), intensity profiles of the parts of equatorial lines (b) and tilt angle dependence of intensity ratio between 11 and 1 1 spots I 1 1 /I 11 of (5, 5) tube (c). We define the tilt angle and rotation angle as indicated in (a). In(b), the top is taken from the enclosed region on the experimental diffraction pattern inserted, and those in the bottom are simulated patterns obtained by rotating the DWNT around the tube axis in the range between Curve in (c) is calculated under the fixed value of at 4.5 and obtained by integrating the whole of first peaks of the layer lines. spot. Diffraction intensities from the (5, 5) inner tube is extremely sensitive to a small change in the incident beam direction with respect to the tube axis, since the atom density for the (5, 5) armchair tube is not uniform in the circumferential direction as illustrated in the right of Fig. 2(a). Although the effect of diffraction intensities to determine the chiral indices is small, it is essential to deduce rotation angle and tilt angle [see Fig. 2(a)] by the following way in order to minimize the experimental error. According to the diffraction theory, 21 the diffraction intensity of the equatorial line does not depend on but changes as a function of. We therefore simulated a series of the ED patterns for a (5, 5)@(14, 6) DWNT as a function of in the range between 0 and 18. Intensity profiles of some portions of equatorial lines in the experimental and simulated ED patterns are shown in Fig. 2(b). Comparing peak positions and intensities of the simulated patterns with those of the experimental one, we found that the experimental data fit well the simulated pattern when =4.5±1. Furthermore, we optimized by simulating a series of ED patterns as a function of for a fixed value of 4.5, and found that the diffraction intensity for (5, 5) inner tube changes critically on a minute change in on layer lines. For example, diffraction intensity of 11 spot indicates a much faster decrease than that of 1 1 with increasing, as shown in Fig. 2(c). Since these two spots are crystallographically equivalent, they should have the same intensity if the tube axis is exactly perpendicular to the electron beam =0. However, when a lattice plane is slightly tilted against the electron beam, the Ewald-sphere does not necessarily cross the same parts of equivalent diffraction disks in the reciprocal space, 25 so that the diffraction intensities differ. The experimental value of I 1 1 /I 11 was estimated to be 1.4 within an error range of +0.2 and Referring to the calculated intensity ratio [see Fig. 2(c)], we obtained a value =2.5±0.5 which gives the best match on the intensity ratio between experiment and calculation. In addition, asymmetry of diffraction intensities observed in (10) type spots for the inner tube can be explained by setting =4.5 and =2.5. On the experimental pattern shown in Fig. 1, intensities of 10 and 1 0 spots are very weak, while 01 and 01 spots are clearly visible. In the reciprocal space, a (10) layer line for (5, 5) tube corresponds to a part of ringlike intensity distribution normal to the tube axis, and this ring has tenfolded rotation symmetry in the diffraction intensity. At =4.5 and =2.5, Ewald-sphere cut the ring at node and loop parts, which leads to asymmetric intensities of (10) type reflections. Details for such an extinction role appearing in (5, 5) tube are described in the Appendix by using simulations of reciprocal lattice for (5, 5)@(14, 6). D. Measurement of C-C bond length of inner- and outer-tubes From (11) lattice spacings d 11 for (5, 5) inner tube and (14, 6) outer tubes, we examined relative C-C bond-length of inner tube a inner with respect to that of outer tube a outer. Prior to analyze the bond length, it should be noted that the diffraction spots were broaden in the radial direction because of the curved nature of a graphene-sheet forming nanotube and also a small number of carbon atoms contributing to the diffraction. The broadened nature of the diffraction spots that causes the positions of first peak maxima of layer lines for nanotube are not exactly the same as those for a planar graphene. Therefore, the distance from the center to a peak maximum of a hk diffraction spot of the tube does not directly correspond to the interplanar spacings d hk. 26 On the other hand, distances between layer lines can be measured accurately due to sufficient number of carbon atoms contributing to the diffraction other than those in the radial direction. By the above reasons, we measured distances between 11 and 1 1 layer lines from the line profiles of the ED pattern in the axial direction. Figure 3 shows such line profiles of experimental and simulated ED patterns. Line profiles along A-A and B-B [Fig. 3(a)], which are defined by integrating the intensities of diffracted rods along the direction perpendicular to the tube axis, are shown in Figs. 3(c)-A and -B, respectively. In profile A, two types of peaks corresponding to (10) and (11) reflections of (14, 6) outer-tube are existed as pointed in the figure. The other pair of peaks in profile B is a set of (11) reflections of (5, 5) inner-tube. A simulated ED pattern and a line profile from the enclosed region are shown in Figs. 3(b) and 3(d), respectively. After normalizing the calculated ED pattern by using peak positions of the (14, 6) tube, shown in profile A, we compared the (11) peak positions of (5, 5) tube in the experimental pattern with that in the simulated one. Top and bottom panels in Fig. 4(a) are the enlarged line profiles around 11 peaks, respectively, for outer and inner

4 HIRAHARA et al. PHYSICAL REVIEW B 70, (2004) FIG. 3. Line profiles of experimental and simulated electron diffraction patterns taken for DWNT. A part of the experimental pattern in Fig. 1(a) is in (a). Simulated pattern of (5, 5)@(14, 6) at =2.5 and =4.5 is in (b). Intensity profiles along A-A and B-B in (a) are shown in (c), and that from the enclosed region in (b) is in (d). These profiles are taken along the axial direction and obtained by integrating in the direction perpendicular to the tube axis. tubes. By assuming the same C-C bond length associated with inner and outer tubes a inner =a outer, experimental 11 peak position for (5, 5) inner tube is shifted substantially by 0.9± 0.1% [see Fig. 4(b)]. This shift is attributable to the fact that the spacing d 11 for (5, 5) inner tube is greater than that for (14, 6) outer tube. For the sake of clarity, we computed some ED patterns by changing the value of a inner for a fixed value of a outer [see Figs. 4(c) and 4(d)], and found that a calculated diffraction profile with a inner /a outer =1.009 matches well the experimental profile. This means that a inner is stretched by 0.9% with respect to a outer. IV. DISCUSSION Here we consider the reason of stretching in d 11 for inner tube. Two possibilities are discussed below on the basis of a hexagonal ring associated with carbon atoms. One is that each hexagon in the (5, 5) tube is distorted by modifying the bond angles without changing the bond length. The other is that the C-C bond itself is stretched. In the former case, if the projections of hexagonal rings are stretched by 0.9% along the axial direction with respect to direction, the tube should shrink by about 2% in the radial direction. In the experimental ED pattern, however, we could not find such shrinkage. This fact suggests that the larger d 11 of (5, 5) tube should be originated in the stretching of C-C bond length. FIG. 4. Expanded intensity profiles around 11 diffraction spots of (14, 6) outer and (5, 5) inner tubes. Experimental diffraction profiles are in (a). Simulated diffraction profiles by changing the ratio of C-C bond-lengths of inner tube a inner and outer tube a outer are in (b), (c), and (d) with ratios in the figure. We find that the profile (c) matches well the experimental results shown in (a). Next, we consider two experimental problems with the accuracy on the measurement on the C-C bond length. The first problem is that the true positions of the layer lines are changed critically on. These positions are modified by 1/cos in the axial direction, since the Ewald sphere obliquely crosses the hexagonal lattice in the reciprocal space. Generally, accurate determination of such a small is difficult, 7,21 but the difficulty was overcome as presented above. Furthermore, we calculated that the modification of the layer-line positions at =2.5 may cause an error proportional to 1/cos =0.001, which error is negligibly small as compared with the 11 peak shift measured experimentally. The other problem is the accuracy in calibration on the ED patterns. The accuracy for two different samples, recorded independently, is generally in the order of 1%. The calibration on ED patterns for isolated SWNTs needs another pattern as the external standard, which may cause small random errors induced by a slight change in the experimental conditions. Such experiments cannot be adequate enough for discussing a delicate change in the C-C bond length. A single DWNT, however, gives simultaneously two independent ED patterns recorded under exactly the same experimental conditions. Therefore, we can circumvent the experimental problems stated above. Of course, the value of a outer cannot be determined with high accuracy but provides a useful refer

5 STRETCHING OF CARBON-CARBON BONDS IN A 0.7- PHYSICAL REVIEW B 70, (2004) ence standard for calibration on d 11 of inner tube in DWNT. The present result demonstrates experimental evidence, indicating that the C-C bond is stretched in the small diameter tube. According to the Raman spectroscopy performed on peapod derived DWNTs by Kataura et al. 16 and Bandow et al., 27 the downshift on the C-C stretching mode vibration frequency for nanotubes with diameters less than 0.7 nm was observed. The softening of the vibration frequency for a 0.7 nm tube in the Raman spectrum probably associates with slightly longer C- C bond length. This indication matches well the present result found in the electron diffraction study. V. SUMMARY We succeeded to record the ED pattern from a 0.7 nm diameter carbon nanotube formed as an inner tube of a DWNT. From the analysis of the diffraction spots, a set of chiral indices of DWNT was determined as (5, 5)@(14, 6). In the determination of the chiral indices, we scanned and in order to simulate the experimental diffraction intensities and positions accurately, and found =4.5 and =2.5 to reproduce exactly the experimental diffraction spots. As a result, we found that the spacing d 11 for (5, 5) inner tube was larger by 0.9% than that for (14, 6) outer tube. This is experimental evidence indicating that the C-C bond-length in a 0.7 nm nanotube is relatively stretched as compared with that in the large-diameter nanotubes. ACKNOWLEDGMENTS K.H. benefited from fruitful conversations with T. Miyake and his colleagues, S. Okada, and T. Yumura. M.K. thanks L. Henrard and Ph. Lambin for providing the FORTRAN code of DIFFRACT. H.K. and S.B. thank, respectively, the Grant-in- Aid for Scientific Research (A) No and (C), No from the Ministry of Education, Culture, Sports, Science and Technology of Japan. This work is supported by the U.S. Office of Naval Research (ONR-N ) and Japan Science and Technology Corporation. APPENDIX In this work, we optimized tilt angle on the basis of asymmetric diffraction intensities of crystallographically equivalent diffraction spots as shown in Fig. 2. Here we demonstrate a simulation of reciprocal space for a (5, 5)@(14, 6) DWNT and explain the reason why such asymmetric intensities can be seen. FIG. 5. Simulations of the reciprocal space for (5, 5)@(14, 6). Simulated electron diffraction pattern for (5, 5)@(14, 6) at =2.5 with =4.5 is in (a). Disklike distributions of (10) and (11) reciprocal lattice for (5, 5) inner tube are shown, respectively, in (b) and (c), which are normal to the tube axis. At =4.5, a plane perpendicular to the vector k 0, which is the wave vector of incident electron beam, passes the disks along the dotted line as shown in (b) and (c). Enlarged patterns around the center regions of (b) and (c) are, respectively, in (d) and (e), indicating that the Ewald-sphere does not pass the center of disks due to the curvature of the Ewald sphere. In addition, when the tube is inclined against the electron beam, the cross section of the layer line and the Ewald sphere apart from the center with a function of. The distance between the Ewald sphere and the center of the hk disk causes asymmetric diffraction intensities for the crystallographically equivalent reflections. Diffraction intensity profiles of (10) and (11) disks along the intersection of the Ewald sphere at =4.5 with various values of are shown in (f). The profiles of (10) reflection at the values of =2.5 and 2.5 are, respectively, equal to those along A 1 -A 1 and A 2 -A 2 in (a). Furthermore, the profiles of (11) reflection at =2.5 and 2.5 are, respectively, equal to those along B 1 -B 1 and B 2 -B 2 in (a)

6 HIRAHARA et al. PHYSICAL REVIEW B 70, (2004) Figure 5(a) shows a simulated ED pattern of (5, 5)@(14, 6) at =2.5 with =4.5. The Bragg condition is systematically represented by the intersection of the reciprocal lattice cut by the Ewald sphere. Here we consider a reciprocal lattice of (5, 5) inner tube and examined where the Ewald sphere cuts it. Distributions of simulated diffraction intensity for (10) and (11) reflections in the direction perpendicular to the tube axis are shown, respectively, in Figs. 5(b) and 5(c). We can see that these diffraction disks have ten fold symmetry, which leads that the diffraction intensity on the ED pattern changes with respect to. At =4.5, a plane perpendicular to the vector k 0, which is the wave vector of incident electron beam, cut the disks along the dotted lines as shown in Figs. 5(b) and 5(c). Since the Ewald sphere has its own curvature, the locus of the sphere is slightly apart from such dotted lines. The radius of the Ewald sphere k 0 is about 288 nm 1 at an acceleration voltage of 120 kv. If the tube is accurately normal to k 0 =0, the Ewald sphere passes the positions separated by and nm 1 from the centers of (10) and (11) disks, respectively, as shown in Figs. 5(d) and 5(e) [these are enlarged patterns in Figs. 5(b) and 5(c) around the center parts]. If the tube inclines against the electron beam, the cross lines shift as a function of. For (10) reciprocal disk, the Ewald sphere passes the line apart by nm 1 from the center of the disk when =2.5, and by nm 1 from the center when = 2.5. Here the Bragg conditions at =2.5 and 2.5 correspond, respectively, to those for 10 (and 01) and 01 (and 1 0) reflections. Intensity profiles at the intersection between the Ewald sphere and the (10) diffraction disk at a fixed value of =4.5 with various values of are shown in Fig. 5(f). Here the profiles at =2.5 and 2.5 are, respectively, equal to those along A 1 A 1 and A 2 A 2 in Fig. 5(a). In Figs. 5(d) and 5(f), we can see that the Ewald sphere at =2.5 or 2.5 crosses the node and loop parts marked by and O in Fig. 5(d), which give asymmetric intensity profile in (10) type reflection pattern. On the other hand, for =0, symmetric intensity profile can be expected as indicated by the dotted line in Fig. 5(f). For (11) reflection as shown in Fig. 5(e), the locus of the Ewald sphere is apart by nm 1 from the center of the disk when =2.5, and by nm 1 from the center when = 2.5. The difference in cutting positions causes the minute difference of diffraction intensities between the 11 and 1 1 reflections as shown in Fig. 5(f). Therefore, we could optimize by examining the diffraction intensities of equivalent diffraction spots with aid of simulations of ED patterns as a function of. As already reported in Ref. 6, hk reflections in an ED pattern of single nanotube has been recognized to have 2mm symmetry with respect to the directions parallel and perpendicular to the tube axis, since the 2mm symmetry originates in the diffractions from top and bottom parts of a rolled graphene-sheet. A series of the present work for ED of nanotubes, however, indicates that ED patterns of several isolated single nanotubes have no longer 2mm symmetry in the strict sense of the word. If the tube has a certain chirality with relatively small translation vector, 28 and the tube axis is precisely normal to the electron beam, hk reflection set has exactly the 2mm symmetry. Therefore, it should be stressed here that the breakdown of 2mm symmetry is often found in the nanotubes having relatively high symmetry with their atomistic structure in the cross section normal to the axial direction like achiral tubes, and such a high-symmetric structure often leads the extinction role for diffraction spots as shown in Fig. 5. *Author to whom correspondence should be addressed; electronic address: kaori h@ccmfs.meijo-u.ac.jp 1 S. Iijima, Nature (London) 354, 57(1991). 2 K. Kanamitsu and S. Saito, J. Phys. Soc. Jpn. 71, 483 (2002). 3 X. Blasé, Lorin X. Benedict, Eric L. Shirley, and Steven G. Louie, Phys. Rev. Lett. 72, 1878 (1994). 4 R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998), p X. F. Zhang, X. B. Zhang, G. Van Tendeloo, S. Amelinckx, M. Op de Beeck, and J. Van Landuyt, J. Cryst. Growth 130, 368 (1993). 6 S. Iijima and T. Ichihashi, Nature (London) 363, 603 (1993). 7 L.-C. 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7 STRETCHING OF CARBON-CARBON BONDS IN A 0.7- PHYSICAL REVIEW B 70, (2004) B 32, 457 (2003). 22 M. Kociak, K. Suenaga, K. Hirahara, Y. Saito, T. Nakahira, and S. Iijima, Phys. Rev. Lett. 89, (2002). 23 Although a pair of hexagons should appear for each nanotube, we only observed one hexagon for the inner tube in ED pattern, which is typical for an achiral tube diffraction. 24 A part of DWNT, where the electron beam is parallel to the graphene sheet, is regarded as the convolution of two double slits with spacings of D and d, respectively. Equatorial line is represented by the Fourier transform of this region. 25 P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. J. Whelan, Electron Microscopy of Thin Crystals (Butterworths, London, 1965), p Here, exactly speaking, interplanar spacing means interlinear spacing, because of the two-dimensional graphene sheet. 27 S. Bandow, G. Chen, G. U. Sumanasekera, R. Gupta, M. Yudasaka, S. Iijima, and P. C. Eklund, Phys. Rev. B 66, (2002). 28 Length of a primitive translation vector T of a nanotube, which is normal to the chiral vector C h, is defined by T =a C-C 3 n 2 +nm+m 2 /d R, where d R is the greatest common divisor for the values of 2n+m and n+2m. If T is relatively small, the tube often has high symmetry normal to the tube axis which leads discrete rotation