Electron diffraction from cylindrical nanotubes
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Electron diffraction intensities from cylindrical objects can be conveniently analyzed using Bessel functions. Analytic formulas and geometry of the diffraction patterns from cylindrical carbon nanotubes are presented in general forms in terms of structural parameters, such as the pitch angle and the radius of a tubule. As an example the Fraunhofer diffraction pattern from a graphitic tubule of structure [18,2] has been simulated to illustrate the characteristics of such diffraction patterns. The validity of the projection approximation is also discussed.
I. INTRODUCTION The recent experimental advance in the making of carbon nanotubes1 in laboratories has been most interesting in materials science2 because of their specific physical properties, such as high-strength mechanical properties,3 sensitive electronic properties,4 etc. All these physical properties are very sensitive to the atomic structures. Though it has been known that the tubes are made of graphene sheets, they can still have various geometries in the form of spiral tubes. As a consequence, for instance, the atomic structure of the caps in a closed tubule is directly related to the spiral geometry.5 Due to the very small size of the nanotubes, an electron probe offers a unique tool in characterizing the atomic structures in diffraction experiments. It is so partly because fast electrons (10 keV to 1 MeV in energy) have very small wavelengths and partly because the interaction between the incident electrons with the scattering atoms is much stronger compared to other radiations, such as x-rays or neutrons. Though high-resolution electron microscopy can provide information on the size (radii of the tubes), it is difficult to obtain any structural information along the axial direction, which has been shown both by experimental micrographs1 and numerical simulations,6 such as the periodicity, pitch length, pitch angles, etc. On the other hand, electron diffraction patterns contain a great deal of structural information, which sometimes is more easily accessible than in the case of images which are more sensitive to the imaging parameters of the microscope. Most recently Iijima and Ichihashi7 have reported structural characterizations of single-shell carbon nanotubes of about 1 nm in diameter using high-resolution electron microscopy and electron diffraction. The geom2450 http://journals.cambridge.org
J. Mater. Res., Vol. 9, No. 9, Sep 1994 Downloaded: 11 Mar 2015
etry of the electron diffraction patterns from the carbon nanotubes was also discussed in Ref. 8 using constructed reciprocal spaces of the tubules. In the present paper electron diffraction patterns from such nanotubes are analyzed. Analytic expressions for the diffraction intensities are given. The geometry of diffraction patterns is also discussed in terms of structural parameters of the nanotubes. As an example, the characteristics of the diffraction pattern from a single layer graphitic tubule of diameter 1.33 nm are presented to illustrate the results. II. DIFFRACTION FROM CYLINDRICAL OB
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