Understanding Carbon Nanotubes From Basics to Applications
This volume presents the foundations of carbon nanotube science including the most recent developments and the prospects for technological applications. Each chapter begins with a tutorial introduction to the relevant interdisciplinary topics from physics
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Abstract. Much of our knowledge on the atomic structure of a material system comes from experiments based on the interaction of radiation with the atoms. The present chapter is devoted to elastic interactions of radiations in a broad sense – electromagnetic waves and particles (electrons or neutrons) – with matter. Each elementary interaction process is called a scattering. An elastic scattering process can only modify the direction of the wave vector of the radiation, not its energy. A well-known example is the Rayleigh scattering on an electromagnetic wave by a polarizable object, which must be small on the wavelength scale. Then, the incident wave excites an electric dipole in the object, which oscillates in time with the frequency ω of the incident electric field and radiates a wave in all the directions. This radiated wave is the scattered radiation. Diffraction takes place when a wave is coherently scattered by many centers. Maxima of interferences arising in certain directions between the many scattered waves are linked to the spatial distribution of the diffusion centers. In transmission electron microscopy (TEM), electrons are diffracted by the electrostatic potential of the atoms. The transmitted electrons are used to construct an image of the scattering potential. In scanning tunneling microscopy (STM), electrons are elastically scattered by the potential barrier between a sharp tip and the surface of the sample. The tunneling current going across the barrier gives an information on the surface electronic density of states. All these techniques are reviewed in Sect. 3.1, and are illustrated with examples taken from graphene-based materials (Sect. 3.2) and nanotubes (Sect. 3.3).
3.1 Basic Theories 3.1.1 Kinematic Theory of Diffraction The principle of diffraction is that waves scattered from a collection of scattering centers interfere constructively in some directions. If one wants to gain information on the microscopic structure of a piece of matter from diffraction, the scattering objects must be the molecules, the atoms, or the nuclei of the atoms from which the sample is made of. It is therefore important to Ph. Lambin et al.: Structural Analysis by Elastic Scattering Techniques, Lect. Notes Phys. 677, 131–198 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com
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work with radiations that have a wavelength of the order of or smaller than the interatomic distances. For electromagnetic waves, this means X-rays. For non-relativistic particles √ with energy E0 , the wavelength is given by the de Broglie relation, λ = h/ 2mE0 . Thermal neutrons (E0 = 25 meV) for instance have λ = 0.182 nm. Electrons of 50 eV energy have about the same wavelength, but they are strongly inelastically scattered when traveling into a solid. These low-energy electrons are suitable for backscattering experiments, such as LEED which is surface diffraction. In transmission experiments, one has to work with electrons in the 100 keV range. For these high-energy electrons, the relativistic expre
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