- Email: [email protected]
On the measurement of helicity of carbon nanotubes
ultramicroscopy Ultramicroscopy 67 (1997) 181 189
ELSEVIER
On the measurement of helicity of carbon nanotubes L.C. Qin*, T. Ichihashi, S. Iijima NEC Corporation, Fundamental Research Laboratories, 34 Miyukigaoka, Tsukuba, lbaraki 305, Japan Received 8 July 1996; accepted 16 October 1996
Abstract For a single-walled cylindrical carbon nanotube, its diameter and helicity determine the atomic structure of the nanotube except its handedness. An analysis has been given of the relationship between the true helicity of a single-walled cylindrical carbon nanotube and the apparent semi-splitting angle measured in an electron diffraction pattern from the tubule. Depending on the order of the Bessel function that dominates the diffraction intensity of the specific layer line on which the measurement was made, a difference between the two angles can be as large as 70%. This deviation is caused by the cylindrical curvature of the diffracting nanotube. The prefactors that relate these two quantities have also been deduced to account for the cylindrical effect. Using the proposed cylindrical correction factors, the indices of two carbon nanotubes were determined to be [12, l ] and [31, 13], respectively. The reciprocal space structure of cylindrical carbon nanotubes, for both single- and multi-walled, has been described as well based on the analytic expression for structure factors. PACS: 61.14.Dc; 61.48. + c Kewvords: Carbon nanotube; Electron diffraction; Helicity
1. Introduction The report of the discovery of c a r b o n nanotubes [1] in soot, p r o d u c e d in arc-discharge evaporators, has inspired m a n y new ideas and has since led to m a n y new research topics related to this new allotrope of carbon. In simple and idealistic terms, carbon nanotubes are hollow cylindrical tubules that are c o m p o s e d of concentric graphitic shells with
* Corresponding author. E-mail: [email protected].
diameters on the scale of nanometers. They have been found to exist in both multiwalled [1] and single-walled forms [2, 3]. In most cases they are also helical, as demonstrated by their electron diffraction geometry [1, 2]. Theoretical calculations of the electronic structure of single-walled carbon nanotubes [4, 5] indicate that such a n a n o t u b e can be either semiconducting or metallic, depending on the diameter and helicity of the tubule. It is therefore very i m p o r t a n t to establish a procedure to measure the diameter and helicity of a c a r b o n n a n o t u b e in order to determine its atomic structure and to characterize its electronic properties.
0304-3991/97/$17.00 ~ 1997 Elsevier Science B.V. All rights reserved Pll S0304-399 1(96)00095-2
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L.C. Qin et al. / UItramicroscopy 67 (1997) 18l 189
The atomic structure of a carbon nanotube can be completely determined by specifying three independent parameters, such as its diameter, its helicity, and its handedness. Among the three, diameter and helicity are most important because they can have a large number of possible values and it is these two parameters that determine many of the potentially useful physical properties of a carbon nanotube. The handedness of a tubule seems not to be crucial to many of the important properties of the tubule. In general, the handedness of a helical structure has two possibilities: left-handed or right-handed. Recently it has been suggested that dark-field electron microscopy [6] can be used to identify the handedness of helical tubules following the technique first proposed by Finch [7]. The determination of the atomic structure of a single-walled carbon nanotube therefore is reduced to the measurement of two parameters: diameter and helical angle. The diameter of a carbon nanotube can be measured directly by high-resolution electron microscopy in real space. Accurate measurement can be achieved when the magnification of electron microscope images is calibrated against a well established scale, such as the graphitic (0 0 2) lattice spacing, which is well known to be 0.34nm. However, the measurement of helical angles is not simple and straightforward. The popular method, which was first introduced by Iijima [1] and has since been widely employed in measuring the helical angles of carbon nanotubes [8, 9], is an approximate one. In this method, the curvature of carbon nanotubes is neglected, and the helical angle is approximated by the apparent semi-splitting angle that is measured in the corresponding experimental electron diffraction patterns. On the other hand, using the helical diffraction theory developed for discovering the double helix structure of DNA [-10], it has been shown that the electron diffraction intensity distribution from carbon nanotubes can be analytically expressed in terms of Bessel functions while the cylindricality of the tubules was taken into account [11]. In the electron diffraction patterns, where diffraction intensity maxima are located on well separated lines (layer lines), each layer line is practically determined by only a single Bessel function of certain order. Since the cylindrical curvature influences the diffraction
geometry, in particular when the diameter of the tubule is small, it is of great interest as well as practical importance to examine to what extent the approximation holds in order to improve the accuracy and to assess the reliability of measurements. In this paper, the relationship is established between the true helical angle ~ of a single-walled cylindrical carbon nanotube and the apparent semi-splitting angle 0 experimentally measured in the corresponding electron diffraction patterns using analytic formulas. It is shown that, depending on the order of the Bessel function which dominates the diffraction intensity of the specific layer line on which the splitting angle is actually measured, the difference between the two angles can be as large as 70%. A correction formula is deduced to account for the effect of cylindricality. As an example, the crystallographic indices of a 0.9 nm diameter single-walled carbon nanotube is determined using the new procedure. The reciprocal space structure of helical carbon nanotubes, for both single- and multi-walled, are also described using Bessel functions. The effect of tilt, the measurement of the helicities of multi-walled carbon nanotubes, and the effect of orientational degeneracy are discussed as well.
2. Structural description The structure of a single-walled cylindrical carbon nanotube can be completely described in radial projection using two indices that specify both the perimeter and the helicity of the tubule once the coordinate system is clearly defined. On a twodimensional network structure of graphene, the two basis vectors are chosen as shown in Fig. la, where the origin is chosen at an atomic site, and the basis vectors, al and a2, also terminate at atomic sites with an inter-angle of 120 °. For this orientation, the corresponding electron diffraction geometry is schematically given in Fig. lb, where the basis vectors a* and a~, are also given. A cylindrical nanotube can be formed by rolling the graphene sheet about an axis perpendicular to the perimeter. The perimeter can be defined by two integers, [-u, v], that specify a lattice vector. The helical angle ~ is defined as the angle between the
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183
Fig. 2. Indexing scheme of carbon nanotubes. Shown is [I 8, 2].
(a)
structures, and negative values are for left-handed structures. Assuming that the axial direction is [U, V], the indices can then be calculated to a constant factor for a given perimeter [u, v] U / V = (u - 2v)/(2u - v).
(b) Fig. 1. (a) Structure of graphene with basis vectors a~ and a2; [b) its corresponding diffraction pattern.
basis vector al (or its equivalent) and the perimeter vector I-u, v]. With a lattice such defined, the diameter of a tubule l-u, v] can be calculated by d = ao (u 2 + vz - u v ) X / z / n ,
(1)
where ao = 0.245 nm is the lattice constant of graphene; and its helical angle is = c o s - ' { ( 2 u - v)/[2(u 2 + u 2 -- UU)I/2]}.
(2)
It should be noted that the helical angle becomes degenerate when it exceeds the 60 ° range due to hexagonal symmetry of the graphene structure. When the angle is limited to be within the range of ( - 3 0 c~,30°], positive values are for right-handed
(3)
As an illustration, the radial projection of an [18, 2] tubule is schematically shown in Fig. 2. For this specific tubule, the perimeter is designated as a, and this tubule has a two-fold rotation axis. Its diameter is d = 1.33 nm, its helical angle ~ = 5.8', and its axial direction is [14, 34] with a periodicity of c = 7.25 nm along the tubule axis. For the reason of simplicity and clarity, only half of the periodicity c/2 is shown in the figure.
3. Structure factor of a single-walled carbon nanotube For a system composed of scattering atoms located at positions rj, the total scattering amplitude F(q) can be expressed as F(q) = ~ fj exp(2rt iq • r j ) , J
(4)
whereJi is the atomic scattering amplitude for atom j and q the scattering vector with magnitude given
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L.C Qin et al. / Ultramicroscopy 67 (1997) 181 189
by q = 2 [sin(O/2)]/2,
(5)
in which O and 2 are the total scattering angle and electron wavelength, respectively. For a simple case, where all atoms are positioned at the surface of a cylinder to form a single-walled carbon nanotube of radius ro, the total scattering amplitude normalized to the atomic scattering amplitude of carbon is [11]
(a)
(b)
F(R, cb, 1) = ~ exp[in(~b + rt/2)] J,,(2xroR) n
x ~ exp[i(n~bj + 2~Izj/c)],
(6)
J
where (R, 4~, l) are the polar coordinates in reciprocal space, (ro, qSj,zi) the polar coordinates of atomic positions in real space, and J,,(2rcroR) is the Bessel function of order n. The summation is done over all possible n values determined by the selection rule [11].
(c) Fig. 3. Relationship between ~, the true helical angle of a single helix, and 0~, the apparent semi-splitting angle measured in diffraction patterns from layer line l: (a) single helix with pitch length c; (b) corresponding diffraction geometry defining the apparent semi-splitting angle 0t measured on layer line l; (c) radial projection of the single helix where ro is the radius of the helix.
4. Cylindrical correction The correction due to the cylindricality of a nanotube is illustrated in Fig. 3. For simplicity but without loss of generality, only a single helix is considered here. For a single helix line of diameter ro and pitch length c, shown in Fig. 3a, the scattering amplitude equals a single Bessel function
F(R, ~, 1) = roJl(2~zroR)exp[i(q~ + ~/2)1],
(7)
and the reflection intensity is
I(R, cb, l) = IF(R, ~,
1)[2 =
[roJ~(2xroR)]2.
(8)
For a given apparent semi-splitting angle 0t measured in the diffraction pattern using layer line l, as schematically illustrated in Fig. 3b and c, we can obtain the following relationship: tan(03 = R~c/l = (uJl) tan(c0,
(9)
where Rt is the radial distance measured in the diffraction pattern, uz the value at which J~ (u) assumes its first maximum and c~ = t a n - l[c/(2rtro)],
(10)
is the true helical angle of the helix (Fig. 3c). Therefore we have deduced the prefactor, udl, that relates the apparent semi-splitting angle 0t measured in electron diffraction patterns and the true helical angle ~: l = 1: ul = 1.7,
ut/l = 1.7,
1=2:
uz=3.0,
ut/l=l.5,
1=4:
u4=5.3,
ul/l=l.3.
When the line index 1 increases, the value of ut approaches the value of 1 and the prefactor udl approaches unity. On the other hand, when the helical angle is held constant, the increase of diameter of a tubule will introduce higher order of rotational symmetry to the structure. In turn, higher orders of rotational symmetry will make the dominating Bessel function to be of higher orders. Therefore, when the diameter of a tubule increases, the difference between the apparent semi-splitting angle 0 and the true helical angle c~will get smaller. This is consistent with the experimental fact that as the diameter of a tubule increases, the cylindricality
L.C. Qin et al. / Ultramicroscopy 67 (1997) 181 189
185
of the tubule plays a lesser role in affecting the corresponding diffraction intensity distribution. Another situation is that when the helical angle is large, the reflection intensity will be dominated by Bessel functions of higher orders as the order of rotational symmetry increases, and this in turn makes the prefactors ut/I closer to unity.
5. Examples As an example to illustrate the relationship, Fraunhofer diffraction of the [18, 2] tubule shown in Fig. 2 was computed numerically. The diffraction pattern is given in Fig. 4. From the pattern, we can measure that the apparent semi-splitting angle 01 is about 9 °, which is about 50% larger than the true helical angle (7 = 5.85) of the structure. Noting that the measurement is obtained from the layer line dominated by the Bessel function of order n = 2, as governed by the selection rule for this tubule which has a two-fold rotational axis [11], this 50% difference was well expected. Fig. 5 shows the experimental electron diffraction pattern from a 0.9 nm diameter single-walled
Fig. 5. Experimental electron diffraction pattern of a 0.9 nm diameter single-walled carbon nanotube. 0t is the apparent semi-splitting angle which is measured to be about 7 from the pattern.
carbon nanotube [12]. The apparent semi-splitting angle measured from the experimental electron diffraction pattern is about 7'. Combined with the diameter, the two-integer indices were determined to be [12, 1], which is a semiconductor. For this tubule, its true helical angle is 4.3 °. This value is about 61% of the apparent semi-splitting angle measured in the experimental electron diffraction pattern.
6. Discussion 6.1. Reciprocal space structure of carbon nanotubes
Fig. 4. Simulated Fraunhofer diffraction pattern of an [18, 2] carbon nanotube. Here 20 gives the apparent total splitting angle. The arrowheads indicate the layers lines of given indices.
Using the analytic formulas describing the scattering amplitudes from nanotubes, it is quite straightforward now to construct the reciprocal space for helical tubules. This method gives more quantitative and rigorous expressions for carbon nanotubes than the geometric method [13], though the latter may appear to be more descriptive. In general terms, the reciprocal space of a single-walled carbon nanotube can be described as composed of a set of concentric rings equally-spaced with
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L.C Qin et al./ Ultramicroscopy 67 (1997) 181 189
a periodicity of c* = 1/c, where c is the periodicity of the tubule along its axial direction. The radii of the rings, however, are different, each of which determined by a specific Bessel function that predominantly determines the scattering intensity distribution for this layer line. The rings are of finite content in contrast with the 5-function like sharpness of Bragg peaks from crystals. Although each layer line is in general contributed by a large number of Bessel functions of different orders, only the one with the lowest order actually dominates, as the peak intensities of higher order Bessel functions decrease rapidly with increasing orders. An electron diffraction pattern obtained in experiment is a section of the reciprocal space. The sectioning plane passes through the origin of the reciprocal space. When the tubule is lying horizontally, viz., the tubule axis is perpendicular to the incident electron beam, the sectioning plane is parallel to the tubule axis. Under this condition, the separation of the Bessel peaks reaches its maximum.
tItI
Fig. 6. Change of diffraction geometry when a tubule is tilted by angle [3 with respect to the horizontal axis.
6.2. Measurement from tilted tubules When the tubule is tilted with respect to the incident electron beam, as schematically illustrated in Fig. 6, the reciprocal space of the tubule will be tilted accordingly, just like the situation where the reciprocal space structure is attached to the tubule rigidly. In this case, the separation of the two Bessel bands, or equivalently, the apparent semi-splitting angle measured in electron diffraction patterns, will differ from its maximum value. Assuming that the inclination angle is fl, the relationship between the maximum apparent semi-splitting angle 0 and the actual apparent semi-splitting angle 0' becomes tan(0'/2) = [-tan2(0/2) COS2 fl - - sin 2 fl] 1/2,
(1 l)
which is a nonlinear relationship [14]. The layer line spacing AZ* will also be enlarged in the corresponding electron diffraction patterns if the tubule is tilted. In this case, the measured layer line spacing would be AZ* = AZ*/cos fl,
(12)
where AZ* denotes the layer line spacing with zero tilt.
6.3. Diffraction from multi-walled tubules Apart from the simplistic single-walled carbon nanotubes, very often carbon nanotubes are found to be multi-walled. For a multi-walled carbon nanotube, there are loosely two constraints on the stacking of the graphene shells. On the one hand, like in graphite, an ordered stacking of the sequence ABAB... may be preferred in order to lower the total energy of the system. On the other hand, when cylindrical tubules were formed, such an ordered stacking will make it impossible for neighboring shells to form concentric tubules and spaced at the graphitic spacing, i.e., 0.34 nm. Therefore, deviations from the ideal arrangement are expected. They can be: (a) multiple helicities; (b) non-concentric shelling; and (c) the inter-shell spacing is different from that in graphite. All these circumstances have been experimentally observed and reported. When a multi-walled carbon nanotube has multiple helicities but is concentric, in principle the helicity of each shell can be determined from diffraction geometry as long as high-resolution
L.C. Qin et al. / Ultramicroscopy 67 (1997) 181-189
AZ f
187
,Z
~ I i
I
k ""--'(
f
~
X ~
X ~ Y
Y
(ai)
(bi)
.. •
[loo1 , ,/,[loo1
*[110 •
•
• O•~
t
110]~ • ° •
Oo
•
•
•
•
•
•
(aii)
•
•
00
O0
•
(bii)
Fig. 7. Two sets of helical geometry with different helicities but giving rise to the same apparent semi-splitting angle in experimental measurement. Noting that the diffraction patterns are oriented at different directions with respect to the tubule axis. (a) [1 0 0]* oriented, giving a true helical angle ~ (corresponding to the experimental electron diffraction pattern given in Fig. 5); and (b) [1 1 0]* oriented, corresponding to a true helical angle 30 ° - c~ (an experimental electron diffraction pattern is given in Fig. 8).
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L.C. Qin et al. / Ultramicroseopy 67 (1997) 181-189
Fig. 8. Experimental electron diffraction pattern showing that the graphitic [1 l 0]* reflections are nearly parallel to the tubule axis. This orientation indicates that the helical angle lies in the range 15 '~ ~< ~ ~< 30" as explained in Fig. 7.
experimental electron diffraction patterns are obtainable with good signal/noise ratios. The structure factor of multi-walled carbon nanotubes can be expressed as a coherent sum of the structure factors of all individual single-walled nanotubes, each of which is given by Eq. (6). For such a case, the electron diffraction intensity maxima from each individual shell will not overlap due to the different diameters of the constituent shells. However, given the small number of scattering atoms in each shell, practically it may be very difficult to recognize each set of peaks from the mixture. There are some alternative methods that have been suggested to measure the helicities using dark-field electron microscopy and high-resolution electron microscopy. But these methods give poor helicity values, though they may provide information about the handedness of a tubule.
6.4. Orientational degeneracy It should also be noted that, although the helical angle of a tubule falls within the range of 60 ° due to the hexagonal lattice symmetry of graphene, the apparent semi-angle obtained falls only within the 15 ° range in any experimental measurement. This is because the largest splitting angle between two
hexagonal spot patterns is 30 °. However, there are two distinguishably different situations, as explained by Fig. 7, where two sets of helical geometry are depicted. In Fig. 7a, the helical angle falls within the range 0 ~ < ~ < 1 5 °, while in Fig. 7b the helical angle ~ falls within the range of 15 ° ~< ~ ~< 30 °. Combining the two situations, the practical measurement of the helical angle will give rise to values within 0 ~< ~ ~< 30 °, while the other possible values ( - 3 0 ° < ~ < 0 °) give rise to the same diffraction intensity distribution and they correspond to the opposite handedness. But in both cases, the apparent semi-splitting angles measured in experimental electron diffraction patterns would appear to be the same. It may also be worthwhile to note that the two diffraction patterns are actually oriented at different directions with respect to the tubule axis. In terms of the indices of the reciprocal lattice of graphene, in Fig. 7a, the tubule axis is nearly parallel to [1 0 0]*, while in Fig. 7b it is nearly parallel to [1 1 0]*. The apparent semi-splitting angles were measured on these reflection spots for the two non-equivalent geometries, respectively. Both cases have been observed in experiments. For example, Fig. 5 shows a situation where the splitting angle is measured on [1 0 0]* reflections, which gives rise to a helical angle of 4.3 °. While in
L.C. Qin et al. / Ultramicroscopy 67 (1997) 181 189
Fig. 8, the splitting angle is m o r e conveniently measured using the [1 1 0]* reflections, which suggest that the diffracting tubule can be indexed as [31, 13] when the d i a m e t e r of the t u b u l e d = 2.1 n m [12] is also i n c o r p o r a t e d into the d e d u c t i o n of the tubule structure, F o r this tubule, it has a helical angle of 24.7', while the a p p a r e n t semi-splitting angle m e a sured in the diffraction p a t t e r n is a b o u t 6 ° .
189
Acknowledgements T h e a u t h o r s wish to t h a n k N E D O I n t e r n a t i o n a l Joint Research G r a n t for p a r t i a l financial s u p p o r t .
References Eli S. lijima, Nature 354 (1991) 56.
7. Conclusions Electron diffraction p a t t e r n s can be used to m e a s u r e the helical angles of c a r b o n n a n o t u b e s within the range of 0 ~< ~ ~< 15 ° w h e n the g r a p h i t i c [1 0 0]* reflections lie a l o n g the t u b u l e axis a n d within the range 15 ° ~< ~ ~< 30 ° when the g r a p h i t i c [1 1 0]* reflections lie a l o n g the t u b u l e axis. T h e true helicity of a t u b u l e can be d e t e r m i n e d m o r e a c c u r a t e l y when a cylindrical c o r r e c t i o n is a p p l i e d to c o r r e l a t e the e x p e r i m e n t a l l y m e a s u r e d a p p a r e n t semi-splitting angle a n d the true helical angle. T h e cylindrical c o r r e c t i o n s are necessary in the determ i n a t i o n of helicities of n a n o t u b e s with small helical angles, since the difference between the true helical angle a n d the e x p e r i m e n t a l l y m e a s u r e d a p p a r e n t semi-splitting angle can be as large as 70%.
[2] S. Iijima and T. Ichihashi, Nature 363 (1993) 603. [3] D.S. Bethune, C.H. Kiang, M.S. de Vries, G. Gorman, R. Savay, J. Vazquez and R. Beyers, Nature 363(19931 605. [4] N. Hamada, S. Sawada and A. Oshiyama, Phys. Rev. Lett. 68 (1992) 1579. E5] R. Saito, M. Fujita, G. Dresselhaus and M.S. Dresselhaus, Phys. Rev. B 46 (1992) 1804. [6] L. Margulis, P. Dluzewski, Y. Feldman and R. Tenne, J. Microsc. 181 (1996)68. [71 J.T. Finch, J. Mol. Biol. 66 (1972) 291. [8] X.F. Zhang, X.B. Zhang, G. Van Tendeloo, S. Amelinckx, M. Op de Beeck and J. Van Landuyt, J. Crystal Growth 130 (1993) 368. [9] M. Liu and J.M. Cowley, Carbon 32 (t994t 393. [10] L.C. Qin, J. Mater. Res. 9 (1994) 2450. [11] A. Klug, F.H.C. Crick and H.W. Wyckoft\ Acta Crystallogr. 11 (1958) 199. [12] S. Iijima, MRS Bull. 19 (11) (1994) 43. [13] X.B. Zhang, X.F. Zhang, S. Amelinckx, G. Van Tendeloo and J. Van Landuyt, Ultramicroscopy 54 t1994) 237. [14] X.B. Zhang and S. Amelinckx, Carbon 32 t1994) 1537.
ELSEVIER
On the measurement of helicity of carbon nanotubes L.C. Qin*, T. Ichihashi, S. Iijima NEC Corporation, Fundamental Research Laboratories, 34 Miyukigaoka, Tsukuba, lbaraki 305, Japan Received 8 July 1996; accepted 16 October 1996
Abstract For a single-walled cylindrical carbon nanotube, its diameter and helicity determine the atomic structure of the nanotube except its handedness. An analysis has been given of the relationship between the true helicity of a single-walled cylindrical carbon nanotube and the apparent semi-splitting angle measured in an electron diffraction pattern from the tubule. Depending on the order of the Bessel function that dominates the diffraction intensity of the specific layer line on which the measurement was made, a difference between the two angles can be as large as 70%. This deviation is caused by the cylindrical curvature of the diffracting nanotube. The prefactors that relate these two quantities have also been deduced to account for the cylindrical effect. Using the proposed cylindrical correction factors, the indices of two carbon nanotubes were determined to be [12, l ] and [31, 13], respectively. The reciprocal space structure of cylindrical carbon nanotubes, for both single- and multi-walled, has been described as well based on the analytic expression for structure factors. PACS: 61.14.Dc; 61.48. + c Kewvords: Carbon nanotube; Electron diffraction; Helicity
1. Introduction The report of the discovery of c a r b o n nanotubes [1] in soot, p r o d u c e d in arc-discharge evaporators, has inspired m a n y new ideas and has since led to m a n y new research topics related to this new allotrope of carbon. In simple and idealistic terms, carbon nanotubes are hollow cylindrical tubules that are c o m p o s e d of concentric graphitic shells with
* Corresponding author. E-mail: [email protected].
diameters on the scale of nanometers. They have been found to exist in both multiwalled [1] and single-walled forms [2, 3]. In most cases they are also helical, as demonstrated by their electron diffraction geometry [1, 2]. Theoretical calculations of the electronic structure of single-walled carbon nanotubes [4, 5] indicate that such a n a n o t u b e can be either semiconducting or metallic, depending on the diameter and helicity of the tubule. It is therefore very i m p o r t a n t to establish a procedure to measure the diameter and helicity of a c a r b o n n a n o t u b e in order to determine its atomic structure and to characterize its electronic properties.
0304-3991/97/$17.00 ~ 1997 Elsevier Science B.V. All rights reserved Pll S0304-399 1(96)00095-2
182
L.C. Qin et al. / UItramicroscopy 67 (1997) 18l 189
The atomic structure of a carbon nanotube can be completely determined by specifying three independent parameters, such as its diameter, its helicity, and its handedness. Among the three, diameter and helicity are most important because they can have a large number of possible values and it is these two parameters that determine many of the potentially useful physical properties of a carbon nanotube. The handedness of a tubule seems not to be crucial to many of the important properties of the tubule. In general, the handedness of a helical structure has two possibilities: left-handed or right-handed. Recently it has been suggested that dark-field electron microscopy [6] can be used to identify the handedness of helical tubules following the technique first proposed by Finch [7]. The determination of the atomic structure of a single-walled carbon nanotube therefore is reduced to the measurement of two parameters: diameter and helical angle. The diameter of a carbon nanotube can be measured directly by high-resolution electron microscopy in real space. Accurate measurement can be achieved when the magnification of electron microscope images is calibrated against a well established scale, such as the graphitic (0 0 2) lattice spacing, which is well known to be 0.34nm. However, the measurement of helical angles is not simple and straightforward. The popular method, which was first introduced by Iijima [1] and has since been widely employed in measuring the helical angles of carbon nanotubes [8, 9], is an approximate one. In this method, the curvature of carbon nanotubes is neglected, and the helical angle is approximated by the apparent semi-splitting angle that is measured in the corresponding experimental electron diffraction patterns. On the other hand, using the helical diffraction theory developed for discovering the double helix structure of DNA [-10], it has been shown that the electron diffraction intensity distribution from carbon nanotubes can be analytically expressed in terms of Bessel functions while the cylindricality of the tubules was taken into account [11]. In the electron diffraction patterns, where diffraction intensity maxima are located on well separated lines (layer lines), each layer line is practically determined by only a single Bessel function of certain order. Since the cylindrical curvature influences the diffraction
geometry, in particular when the diameter of the tubule is small, it is of great interest as well as practical importance to examine to what extent the approximation holds in order to improve the accuracy and to assess the reliability of measurements. In this paper, the relationship is established between the true helical angle ~ of a single-walled cylindrical carbon nanotube and the apparent semi-splitting angle 0 experimentally measured in the corresponding electron diffraction patterns using analytic formulas. It is shown that, depending on the order of the Bessel function which dominates the diffraction intensity of the specific layer line on which the splitting angle is actually measured, the difference between the two angles can be as large as 70%. A correction formula is deduced to account for the effect of cylindricality. As an example, the crystallographic indices of a 0.9 nm diameter single-walled carbon nanotube is determined using the new procedure. The reciprocal space structure of helical carbon nanotubes, for both single- and multi-walled, are also described using Bessel functions. The effect of tilt, the measurement of the helicities of multi-walled carbon nanotubes, and the effect of orientational degeneracy are discussed as well.
2. Structural description The structure of a single-walled cylindrical carbon nanotube can be completely described in radial projection using two indices that specify both the perimeter and the helicity of the tubule once the coordinate system is clearly defined. On a twodimensional network structure of graphene, the two basis vectors are chosen as shown in Fig. la, where the origin is chosen at an atomic site, and the basis vectors, al and a2, also terminate at atomic sites with an inter-angle of 120 °. For this orientation, the corresponding electron diffraction geometry is schematically given in Fig. lb, where the basis vectors a* and a~, are also given. A cylindrical nanotube can be formed by rolling the graphene sheet about an axis perpendicular to the perimeter. The perimeter can be defined by two integers, [-u, v], that specify a lattice vector. The helical angle ~ is defined as the angle between the
L.C. Qin et al. / Ultramicroscopy 67 (1997) 181-189
183
Fig. 2. Indexing scheme of carbon nanotubes. Shown is [I 8, 2].
(a)
structures, and negative values are for left-handed structures. Assuming that the axial direction is [U, V], the indices can then be calculated to a constant factor for a given perimeter [u, v] U / V = (u - 2v)/(2u - v).
(b) Fig. 1. (a) Structure of graphene with basis vectors a~ and a2; [b) its corresponding diffraction pattern.
basis vector al (or its equivalent) and the perimeter vector I-u, v]. With a lattice such defined, the diameter of a tubule l-u, v] can be calculated by d = ao (u 2 + vz - u v ) X / z / n ,
(1)
where ao = 0.245 nm is the lattice constant of graphene; and its helical angle is = c o s - ' { ( 2 u - v)/[2(u 2 + u 2 -- UU)I/2]}.
(2)
It should be noted that the helical angle becomes degenerate when it exceeds the 60 ° range due to hexagonal symmetry of the graphene structure. When the angle is limited to be within the range of ( - 3 0 c~,30°], positive values are for right-handed
(3)
As an illustration, the radial projection of an [18, 2] tubule is schematically shown in Fig. 2. For this specific tubule, the perimeter is designated as a, and this tubule has a two-fold rotation axis. Its diameter is d = 1.33 nm, its helical angle ~ = 5.8', and its axial direction is [14, 34] with a periodicity of c = 7.25 nm along the tubule axis. For the reason of simplicity and clarity, only half of the periodicity c/2 is shown in the figure.
3. Structure factor of a single-walled carbon nanotube For a system composed of scattering atoms located at positions rj, the total scattering amplitude F(q) can be expressed as F(q) = ~ fj exp(2rt iq • r j ) , J
(4)
whereJi is the atomic scattering amplitude for atom j and q the scattering vector with magnitude given
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by q = 2 [sin(O/2)]/2,
(5)
in which O and 2 are the total scattering angle and electron wavelength, respectively. For a simple case, where all atoms are positioned at the surface of a cylinder to form a single-walled carbon nanotube of radius ro, the total scattering amplitude normalized to the atomic scattering amplitude of carbon is [11]
(a)
(b)
F(R, cb, 1) = ~ exp[in(~b + rt/2)] J,,(2xroR) n
x ~ exp[i(n~bj + 2~Izj/c)],
(6)
J
where (R, 4~, l) are the polar coordinates in reciprocal space, (ro, qSj,zi) the polar coordinates of atomic positions in real space, and J,,(2rcroR) is the Bessel function of order n. The summation is done over all possible n values determined by the selection rule [11].
(c) Fig. 3. Relationship between ~, the true helical angle of a single helix, and 0~, the apparent semi-splitting angle measured in diffraction patterns from layer line l: (a) single helix with pitch length c; (b) corresponding diffraction geometry defining the apparent semi-splitting angle 0t measured on layer line l; (c) radial projection of the single helix where ro is the radius of the helix.
4. Cylindrical correction The correction due to the cylindricality of a nanotube is illustrated in Fig. 3. For simplicity but without loss of generality, only a single helix is considered here. For a single helix line of diameter ro and pitch length c, shown in Fig. 3a, the scattering amplitude equals a single Bessel function
F(R, ~, 1) = roJl(2~zroR)exp[i(q~ + ~/2)1],
(7)
and the reflection intensity is
I(R, cb, l) = IF(R, ~,
1)[2 =
[roJ~(2xroR)]2.
(8)
For a given apparent semi-splitting angle 0t measured in the diffraction pattern using layer line l, as schematically illustrated in Fig. 3b and c, we can obtain the following relationship: tan(03 = R~c/l = (uJl) tan(c0,
(9)
where Rt is the radial distance measured in the diffraction pattern, uz the value at which J~ (u) assumes its first maximum and c~ = t a n - l[c/(2rtro)],
(10)
is the true helical angle of the helix (Fig. 3c). Therefore we have deduced the prefactor, udl, that relates the apparent semi-splitting angle 0t measured in electron diffraction patterns and the true helical angle ~: l = 1: ul = 1.7,
ut/l = 1.7,
1=2:
uz=3.0,
ut/l=l.5,
1=4:
u4=5.3,
ul/l=l.3.
When the line index 1 increases, the value of ut approaches the value of 1 and the prefactor udl approaches unity. On the other hand, when the helical angle is held constant, the increase of diameter of a tubule will introduce higher order of rotational symmetry to the structure. In turn, higher orders of rotational symmetry will make the dominating Bessel function to be of higher orders. Therefore, when the diameter of a tubule increases, the difference between the apparent semi-splitting angle 0 and the true helical angle c~will get smaller. This is consistent with the experimental fact that as the diameter of a tubule increases, the cylindricality
L.C. Qin et al. / Ultramicroscopy 67 (1997) 181 189
185
of the tubule plays a lesser role in affecting the corresponding diffraction intensity distribution. Another situation is that when the helical angle is large, the reflection intensity will be dominated by Bessel functions of higher orders as the order of rotational symmetry increases, and this in turn makes the prefactors ut/I closer to unity.
5. Examples As an example to illustrate the relationship, Fraunhofer diffraction of the [18, 2] tubule shown in Fig. 2 was computed numerically. The diffraction pattern is given in Fig. 4. From the pattern, we can measure that the apparent semi-splitting angle 01 is about 9 °, which is about 50% larger than the true helical angle (7 = 5.85) of the structure. Noting that the measurement is obtained from the layer line dominated by the Bessel function of order n = 2, as governed by the selection rule for this tubule which has a two-fold rotational axis [11], this 50% difference was well expected. Fig. 5 shows the experimental electron diffraction pattern from a 0.9 nm diameter single-walled
Fig. 5. Experimental electron diffraction pattern of a 0.9 nm diameter single-walled carbon nanotube. 0t is the apparent semi-splitting angle which is measured to be about 7 from the pattern.
carbon nanotube [12]. The apparent semi-splitting angle measured from the experimental electron diffraction pattern is about 7'. Combined with the diameter, the two-integer indices were determined to be [12, 1], which is a semiconductor. For this tubule, its true helical angle is 4.3 °. This value is about 61% of the apparent semi-splitting angle measured in the experimental electron diffraction pattern.
6. Discussion 6.1. Reciprocal space structure of carbon nanotubes
Fig. 4. Simulated Fraunhofer diffraction pattern of an [18, 2] carbon nanotube. Here 20 gives the apparent total splitting angle. The arrowheads indicate the layers lines of given indices.
Using the analytic formulas describing the scattering amplitudes from nanotubes, it is quite straightforward now to construct the reciprocal space for helical tubules. This method gives more quantitative and rigorous expressions for carbon nanotubes than the geometric method [13], though the latter may appear to be more descriptive. In general terms, the reciprocal space of a single-walled carbon nanotube can be described as composed of a set of concentric rings equally-spaced with
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a periodicity of c* = 1/c, where c is the periodicity of the tubule along its axial direction. The radii of the rings, however, are different, each of which determined by a specific Bessel function that predominantly determines the scattering intensity distribution for this layer line. The rings are of finite content in contrast with the 5-function like sharpness of Bragg peaks from crystals. Although each layer line is in general contributed by a large number of Bessel functions of different orders, only the one with the lowest order actually dominates, as the peak intensities of higher order Bessel functions decrease rapidly with increasing orders. An electron diffraction pattern obtained in experiment is a section of the reciprocal space. The sectioning plane passes through the origin of the reciprocal space. When the tubule is lying horizontally, viz., the tubule axis is perpendicular to the incident electron beam, the sectioning plane is parallel to the tubule axis. Under this condition, the separation of the Bessel peaks reaches its maximum.
tItI
Fig. 6. Change of diffraction geometry when a tubule is tilted by angle [3 with respect to the horizontal axis.
6.2. Measurement from tilted tubules When the tubule is tilted with respect to the incident electron beam, as schematically illustrated in Fig. 6, the reciprocal space of the tubule will be tilted accordingly, just like the situation where the reciprocal space structure is attached to the tubule rigidly. In this case, the separation of the two Bessel bands, or equivalently, the apparent semi-splitting angle measured in electron diffraction patterns, will differ from its maximum value. Assuming that the inclination angle is fl, the relationship between the maximum apparent semi-splitting angle 0 and the actual apparent semi-splitting angle 0' becomes tan(0'/2) = [-tan2(0/2) COS2 fl - - sin 2 fl] 1/2,
(1 l)
which is a nonlinear relationship [14]. The layer line spacing AZ* will also be enlarged in the corresponding electron diffraction patterns if the tubule is tilted. In this case, the measured layer line spacing would be AZ* = AZ*/cos fl,
(12)
where AZ* denotes the layer line spacing with zero tilt.
6.3. Diffraction from multi-walled tubules Apart from the simplistic single-walled carbon nanotubes, very often carbon nanotubes are found to be multi-walled. For a multi-walled carbon nanotube, there are loosely two constraints on the stacking of the graphene shells. On the one hand, like in graphite, an ordered stacking of the sequence ABAB... may be preferred in order to lower the total energy of the system. On the other hand, when cylindrical tubules were formed, such an ordered stacking will make it impossible for neighboring shells to form concentric tubules and spaced at the graphitic spacing, i.e., 0.34 nm. Therefore, deviations from the ideal arrangement are expected. They can be: (a) multiple helicities; (b) non-concentric shelling; and (c) the inter-shell spacing is different from that in graphite. All these circumstances have been experimentally observed and reported. When a multi-walled carbon nanotube has multiple helicities but is concentric, in principle the helicity of each shell can be determined from diffraction geometry as long as high-resolution
L.C. Qin et al. / Ultramicroscopy 67 (1997) 181-189
AZ f
187
,Z
~ I i
I
k ""--'(
f
~
X ~
X ~ Y
Y
(ai)
(bi)
.. •
[loo1 , ,/,[loo1
*[110 •
•
• O•~
t
110]~ • ° •
Oo
•
•
•
•
•
•
(aii)
•
•
00
O0
•
(bii)
Fig. 7. Two sets of helical geometry with different helicities but giving rise to the same apparent semi-splitting angle in experimental measurement. Noting that the diffraction patterns are oriented at different directions with respect to the tubule axis. (a) [1 0 0]* oriented, giving a true helical angle ~ (corresponding to the experimental electron diffraction pattern given in Fig. 5); and (b) [1 1 0]* oriented, corresponding to a true helical angle 30 ° - c~ (an experimental electron diffraction pattern is given in Fig. 8).
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Fig. 8. Experimental electron diffraction pattern showing that the graphitic [1 l 0]* reflections are nearly parallel to the tubule axis. This orientation indicates that the helical angle lies in the range 15 '~ ~< ~ ~< 30" as explained in Fig. 7.
experimental electron diffraction patterns are obtainable with good signal/noise ratios. The structure factor of multi-walled carbon nanotubes can be expressed as a coherent sum of the structure factors of all individual single-walled nanotubes, each of which is given by Eq. (6). For such a case, the electron diffraction intensity maxima from each individual shell will not overlap due to the different diameters of the constituent shells. However, given the small number of scattering atoms in each shell, practically it may be very difficult to recognize each set of peaks from the mixture. There are some alternative methods that have been suggested to measure the helicities using dark-field electron microscopy and high-resolution electron microscopy. But these methods give poor helicity values, though they may provide information about the handedness of a tubule.
6.4. Orientational degeneracy It should also be noted that, although the helical angle of a tubule falls within the range of 60 ° due to the hexagonal lattice symmetry of graphene, the apparent semi-angle obtained falls only within the 15 ° range in any experimental measurement. This is because the largest splitting angle between two
hexagonal spot patterns is 30 °. However, there are two distinguishably different situations, as explained by Fig. 7, where two sets of helical geometry are depicted. In Fig. 7a, the helical angle falls within the range 0 ~ < ~ < 1 5 °, while in Fig. 7b the helical angle ~ falls within the range of 15 ° ~< ~ ~< 30 °. Combining the two situations, the practical measurement of the helical angle will give rise to values within 0 ~< ~ ~< 30 °, while the other possible values ( - 3 0 ° < ~ < 0 °) give rise to the same diffraction intensity distribution and they correspond to the opposite handedness. But in both cases, the apparent semi-splitting angles measured in experimental electron diffraction patterns would appear to be the same. It may also be worthwhile to note that the two diffraction patterns are actually oriented at different directions with respect to the tubule axis. In terms of the indices of the reciprocal lattice of graphene, in Fig. 7a, the tubule axis is nearly parallel to [1 0 0]*, while in Fig. 7b it is nearly parallel to [1 1 0]*. The apparent semi-splitting angles were measured on these reflection spots for the two non-equivalent geometries, respectively. Both cases have been observed in experiments. For example, Fig. 5 shows a situation where the splitting angle is measured on [1 0 0]* reflections, which gives rise to a helical angle of 4.3 °. While in
L.C. Qin et al. / Ultramicroscopy 67 (1997) 181 189
Fig. 8, the splitting angle is m o r e conveniently measured using the [1 1 0]* reflections, which suggest that the diffracting tubule can be indexed as [31, 13] when the d i a m e t e r of the t u b u l e d = 2.1 n m [12] is also i n c o r p o r a t e d into the d e d u c t i o n of the tubule structure, F o r this tubule, it has a helical angle of 24.7', while the a p p a r e n t semi-splitting angle m e a sured in the diffraction p a t t e r n is a b o u t 6 ° .
189
Acknowledgements T h e a u t h o r s wish to t h a n k N E D O I n t e r n a t i o n a l Joint Research G r a n t for p a r t i a l financial s u p p o r t .
References Eli S. lijima, Nature 354 (1991) 56.
7. Conclusions Electron diffraction p a t t e r n s can be used to m e a s u r e the helical angles of c a r b o n n a n o t u b e s within the range of 0 ~< ~ ~< 15 ° w h e n the g r a p h i t i c [1 0 0]* reflections lie a l o n g the t u b u l e axis a n d within the range 15 ° ~< ~ ~< 30 ° when the g r a p h i t i c [1 1 0]* reflections lie a l o n g the t u b u l e axis. T h e true helicity of a t u b u l e can be d e t e r m i n e d m o r e a c c u r a t e l y when a cylindrical c o r r e c t i o n is a p p l i e d to c o r r e l a t e the e x p e r i m e n t a l l y m e a s u r e d a p p a r e n t semi-splitting angle a n d the true helical angle. T h e cylindrical c o r r e c t i o n s are necessary in the determ i n a t i o n of helicities of n a n o t u b e s with small helical angles, since the difference between the true helical angle a n d the e x p e r i m e n t a l l y m e a s u r e d a p p a r e n t semi-splitting angle can be as large as 70%.
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