Calculating the diffraction of electrons or X-rays by carbon nanotubes

EUROPHYSICS LETTERS 10 August 1996 Europhys. Lett., 35 (5), pp. 355-360 (1996) Calculating the diffraction of electrons or X-rays by carbon nanotubes A. A. Lucas, V. Bruyninckx and Ph. Lambin Department of Physics, Facultés Universitaires Notre-Dame de la Paix 61 rue de Bruxelles, B-5000 Namur, Belgium (received 9 April 1996; accepted 20 June 1996) PACS. 61.14Dc Theories of diffraction and scattering. PACS. 61.16Bg Transmission and scanning electron microscopy. PACS. 61.46+w Solid clusters (including fullerenes) and nanoparticles. Abstract. The theory developed by Cochran, Crick and Vand for the diffraction of X-rays by helical biological molecules has been used to compute the diffraction patterns of monolayer or multilayer carbon nanotubes. By describing a single-wall chiral tubule as a set of regular carbon helices, its exact, kinematical diffraction amplitude is obtained in closed form. The theory is illustrated by computer simulations which faithfully reproduce the observed intensity patterns of transmission electron diffraction from multilayer nanotubes. Introduction. Carbon nanotubes have been synthesized recently by the carbon-arc method of fullerene production [1]-[3] or by hydrocarbon cracking on a metal catalyst [4]. The morphology of nanotubes from the two methods is similar and has been studied by Transmission Electron Microscopy (TEM) and Diffraction (TED) [2], [5], [6]. The interpretation of TED patterns has been based on geometrical considerations in the reciprocal space of a nanotube but a quantitative theory of diffraction intensities is still lacking. Here, we report on a theory of electron or X-ray diffraction by a nanotube in the first Born approximation. While generally valid for X-rays, this approximation turns out to be reliable also for TED on nanotubes, on account of the small scattering power of carbon atoms for fast electrons [7]. Let (L, M) be the pair of integers which completely specifies what we shall call a tubule, that is a monolayer nanotube obtained by rolling up a graphene sheet, as illustrated in fig. 1 [8]. We observe that such a tubule can be considered as being made of L equidistant zig-zag pairs of helices, as shown in fig. 1 b). The diffraction amplitude of the complete tubule is the sum of the successive helix-pair amplitudes. Since the latter differ only by constant phase factors, the summation amounts to an exact geometrical series. The amplitude of a single carbon helix itself can be written down from the theory established by Cochran, Crick and Vand (CCV) [9] for X-ray diffraction by helical biological molecules [9]-[11]. The diffraction formula for a complete tubule can then be exploited to obtain the diffraction of a nanotube containing any number of tubule layers of arbitrary chirality. Computer simulations based on the present theory are given in the third section. c Les Editions de Physique

356 EUROPHYSICS LETTERS Fig. 1. Fig. 2. Fig. 1. a) Definition, in a graphene plane, of a tubule (L, M) of circumference C and chiral angle α. The figure illustrates the (L = 10, M = 4) nanotube, which has the same helicity as the (25, 10) tubule considered in fig. 2. b) Rolled-up graphene layer obtained by superposing (L, M) to (0, 0) of fig. 1 a). A zig-zag pair of carbon helices has been outlined. Fig. 2. Simulated diffraction pattern of a (L = 25, M = 10) tubule at normal incidence. projection of the tubule axis is the vertical symmetry axis of the streak pattern. The Diffraction theory. The diffraction amplitude of a circular, monoatomic helix wound around the z-axis is given by the CCV formula [9] A(k) =f(k) j exp[ik R j ] n,m A n,m (k), (1) A(k) =f(k) [ ( n δ k z 2π P + m )] [ [ ( J n (k r)exp i n ψ k ϕ+ π ) ]] +k z z. (2) p 2 n,m R j are the atomic positions along the helix, f(k) is the carbon atomic scattering factor for electrons or for X-rays [12], n, m are two integers, k =(k x,k y,k z ) is the wave vector transfer, k = kx 2 + ky, 2 ψ k =tan 1 (k y /k x ), J n is the n-th order cylindrical Bessel function and (r, ϕ, z) are the cylindrical coordinates of the initial atom of the helix of radius r, pitch P and atomic repeat distance p in the z-direction. The δ-function in eq. (2) means that the diffraction pattern of the helix is organized in a set of discrete layer lines [9] at altitudes ( n ) k z =2π P +m 2πl p T, (3) where l labels the layer lines and T is the true helix period. In a carbon tubule, irrespective of its chirality, the ratio P/p is always rational [7], [8] so that the true helix (and tubule) period T is the smallest common multiple of P and p. We now consider an (L, M) tubule as made up of L helical zig-zag rows of atoms and add up their amplitudes (fig. 1 b)). The two helices of each zig-zag pair are shifted from each other by the screw operation (z 1,ϕ 1 ) while the L equidistant rows are shifted from one to the next by

A. A. LUCAS et al.: CALCULATING THE DIFFRACTION OF ELECTRONS ETC. 357 the screw operation (z 0,ϕ 0 ). These operations are uniquely determined by the tubule (L, M) integers via [7] z 0 = 3M 2R d, ϕ 0 = 2L + M (L M) R 2 π; z 1 = 2R d, ϕ 1 = L + M R 2 π, (4) where R = L 2 + LM + M 2 and d 1.4 Å is the C-C bond distance. Every operation (z i,ϕ i ) results in multiplying (2) by phase factors exp[i( nϕ i + k z z i )]. Hence, the total amplitude of a complete (L, M) tubule is given by A (L,M) (k) = n,m a n,m (k) [ 1 + exp[i( nϕ 1 + k z z 1 )] ] 1 exp[i( nϕ 0 + k z z 0 )L] 1 exp[i( nϕ 0 + k z z 0 )]. (5) The effect of the last factor in (5) is to restrict the layer lines of nonzero intensity to a discrete set which produces a hexagonal distribution of diffraction spots (see the simulations below). A nanotube made of N coaxial tubules (L i,m i ) separated by the 3.4 Å graphite spacing [2] will have the total diffraction amplitude A nanotube (k) = N A (Li,M i)(k). (6) The diffraction intensities are the square modulus of the amplitudes (5) or (6). i=1 Computer simulations. Figure 2 shows a simulated diffraction pattern, at normal incidence, of a (L = 25, M = 10) tubule of radius r =12Åand chiral angle α =16.1. The atomic scattering amplitude f(k) of carbon (for 300 kev electrons typically used in TED) was taken from Doyle and Turner [12]. There are hexagonal sets of spots on visible first- and second-order diffraction circles. On the first-order circle, the two hexagons are rotated by ±α with respect to the symmetrical position (i.e. with hexagon vertices at 12 and 6 o clock) which would belong to an achiral (L, 0) tubule. One hexagon arises from diffraction by the upstream, hemi-cylindrical part of the tubule, the other hexagon from the downstream part. The spots are streaks elongated perpendicular to the tubule axis and have modulated intensities fading away from the axis. The streaking is a chirping phenomenon in wave vector space which derives from the real-space chirping of the honeycomb lattice parameter, as seen by electron waves, around the tubule circumference. The intensity modulations of the streaks reflect the oscillations of the Bessel functions in (2) and arise from interferences between the waves scattered by the two tubule edges. An equivalent interpretation of the streak modulations was provided by Iijima and Ichihashi [2] for their observation of the TED pattern of single (16, 2) tubule. When several tubules are coaxially assembled in a nanotube, the gross geometrical features of the diffraction pattern of each tubule are approximately conserved: there are twice as many hexagonal sets of streaking spots as there are tubules of distinct nonzero chiral angles, while achiral tubules give rise to one single degenerate hexagon of streaking spots on each diffraction circle. Using eqs. (5) and (6) for the total amplitude, the simulated diffraction at normal incidence of a nanotube made of 3 achiral and 4 chiral tubules (of 12 chiral angle) has been calculated. The result is shown in fig. 3 and compared with the experimental pattern reported by Iijima [2]. Although the intensities are not additive (only the amplitudes are), the multiple hexagonal organization is indeed conserved. There are 3 hexagonal sets of streaking spots on each diffraction circle. The mm2 symmetric hexagon arises from the achiral, parallel tubules in the nanotube. The other pair of hexagons is due to diffraction by the remaining tubules having about the same chiral angles α 12. The streaking of each spot and its fading

358 EUROPHYSICS LETTERS Fig. 3. Fig. 4. Fig. 3. a) Bright-field TEM image of a 7-layer nanotube and b) its observed diffraction pattern (reproduced courtesy of Dr. S. Iijima [2]); c) is a simulated TED pattern for a model 7-layer nanotube made of 3 coaxial achiral tubules of coordinates (29, 0), (38, 0), (47, 0) and 4 coaxial chiral tubules of about 12 chiral angle and of coordinates (48, 13), (55, 16), (63, 17) and (70, 20) in the (L, M) notation of fig. 1. The seven successive layers have radii increasing by about 3.4 Å. The small vertical splitting of the chiral streaks is caused by the slightly different chiral angles of the 4 chiral tubules. Fig. 4. Simulated intensity distribution along the zeroth-layer line produced by the 7-layer nanotube of fig. 3. away from the nanotube axis are clearly observed in the experimental pattern of fig. 3 b). The streak modulation, however, is barely visible on some spots but has been clearly demonstrated in other nanotube TED patterns taken at higher resolution [5]-[7]. Most of the details of the observed pattern (fig. 3 b)) are reproduced by the simulation (fig. 3 c)). No effort has been made to optimize the sequence of chiral angles nor the relative registry of the successive tubules but we are confident that a search through the chirality and registry space of the nanotube will lead to a sequence which further improves the agreement between observed and calculated patterns [7]. In addition to the hexagonal spots, a set of regularly spaced spots, labelled (0002m), appears on the zeroth-layer line. These are generated by the new characteristic length c 0 = 3.4 Å separating the successive tubules [2]. They can be understood from our eq. (4) in which the Bessel functions in (2) can be replaced by their large-argument approximation, ( J n (k r j ) cos k r j n π 2 π )/ k r j π/2, k r i 1. (7) 4 This implies constructive interferences whenever k = m2π/c 0 since the tubule radii r i differ by multiples of c 0. The (0002m) spots result from electron diffraction by the double grating

A. A. LUCAS et al.: CALCULATING THE DIFFRACTION OF ELECTRONS ETC. 359 formed by the graphene layers parallel to the beam on each side of the nanotube (fig. 3 a)). The two nanotube walls in turn produce weak inteferences which should appear as a modulated diffuse intensity between the (0002m) spots, as in the classic Young two-slit experiment. These modulations are not resolved in fig. 3 b) but have been seen clearly in [13]. The calculated intensity distribution along the zeroth-layer line is shown in fig. 4 where the (0002m) peaks are clearly dominant over the intervening, diffuse intensities whose modulation is caused by the double-slit interferences just discussed. Discussion. This work has presented simulated diffraction patterns which show that the new theory is able to account for all the qualitative features of observed TED patterns as reported in the literature. A detailed comparison between observed and computed diffraction intensities will require experimental intensity measurements which are rarely performed in TED work. We wish to point out several opportunities offered by the exploitation of our theoretical approach. One is the simulation of diffraction at nonnormal incidences. This is useful for investigating the chirality of individual tubules in a nanotube [7], [14], [15]. Another opportunity is to compute the diffraction by nanotubes containing defects such as atomic vacancies or even voids. Minority defects are readily taken into account by subtracting the amplitudes of the missing atoms. The present theory is applicable to other helical structures of light elements. One is the carbon fibers having the structure of conical scrolls [16] which are also describable as a set of regular carbon helices. Other examples are the BN and BCN nanotubes recently discovered [15], [17]. Disulfide of heavy metals (MoS 2, WS 2,...) are known to form nanotubes [15] but for these compounds the present theory will be useful only for X-ray diffraction since the kinematical approximation breaks down for electron scattering by heavy atoms. Certain extensions of the CCV theory are required for applications to coiled carbon nanotubes [4] or to cylindrical scrolls imbedded in straight nanotubes [18]. These extensions will be considered in a detailed version of the present report [7]. Additional Remark. While this paper was being published, a paper by Qin L. C., J. Mater. Res., 9 (1994) 2450, came to our attention in which a theoretical treatment similar to ours for the electron diffraction by monolayer nanotubes was proposed. Although the helical scheme used by Qin to describe the nanotube is different from ours, the end result is equivalent. *** We thank Dr. S. Iijima for allowing us to reproduce the micrographs shown in fig. 3 a), b). We have greatly benefited from discussions with Profs. S. Amelinckx, J. Van Landuyt, G. Van Tendeloo, Mr. D. Bernaerts, Prof. J. B. Nagy and Dr. A. Fonseca. We are also grateful to M. Mathot for technical help. The authors are grateful to the following supporting agencies: the Walloon Government, the Belgian National Science Foundation, the Belgian Ministry of Sciences and the European Commission HCM program. REFERENCES [1] Krätschmer W., Lamb L. D., Fostiropoulos K. and Huffman D. R., Nature, 347 (1990) 354. [2] Iijima S., Nature, 354 (1991) 56; Iijima S. and Ichihashi T., Nature, 363 (1993) 603. [3] Ebbesen T. W. and AjayanP.M.,Nature, 358 (1992) 220.

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