Diffraction by DNA, carbon nanotubes and other helical nanostructures

Abstract

This review discusses the diffraction patterns of x-rays or electrons scattered by fibres of helical biological molecules and by carbon nanotubes (CNTs) from the unified point of view of the Fourier-Bessel transform of an atomic helix. This paper is intended for scientists who are not professional crystallographers. X-ray fibre diffraction patterns of Pauling's protein α-helix and of Crick and Pauling's protein coiled-coil are revisited. This is followed by a non-technical comparison between the historic x-ray diffraction patterns of the A and B conformations of DNA, which were crucial for the discovery of the double helix. The qualitative analysis of the diffraction images is supported by novel optical simulation experiments designed to pinpoint the gross structural informational content of the patterns. The spectacular helical structure of the tobacco mosaic virus determined by Rosalind Franklin and co-workers will then be described as an early example of the great power of x-ray crystallography in determining the structure of a large biomolecular edifice. After these mostly historical and didactic case studies, this paper will consider electron diffraction and transmission electron microscopy of CNTs of great current interest, focusing particularly on recent data obtained for single-wall, double-wall and scrolled nanotubes. Several points of convergence between the interpretations of the diffraction patterns of biological helices and CNTs will be emphasized.

Metal nanoparticles reveal the organization of single-walled carbon nanotubes in bundles

“…34 The intensity distribution I(q) in the diffraction pattern of a single SWCNT perpendicular to the ber axis depends on the tube diameter according to I(q) $ (J 0 (qr)) 2 , with J 0 being the zero-order Bessel function. 34,35 Fig. 3d shows the intensity distribution perpendicular to the bundle axis determined from the experimental data in comparison with calculations of the Bessel distributions for SWCNTs with 1.4 nm and 1.5 nm diameters.…”

Robust Bessel-function-based method for determination of the(n,m)indices of single-walled carbon nanotubes by electron diffraction

“…The result coincides with that determined from the same diffraction pattern but by an independent technique. 24 It is worth remarking that the same nanotube was previously determined to be a ͑21,9͒ metallic nanotube 22,25 by the preliminary method based on ͑D 0 , ␣͒ determination. 10 A direct comparison shows that the difference between the chiral angles of the ͑23,10͒ tube and the ͑21,9͒ tube is just 0.19°, but the diameters differ from each other by 10%.…”

“…Figure 1 shows a simulated EDP of a ͑12,7͒ chiral SWCNT in the normal-incidence condition. ͑If not specified, the electron beam is hereafter assumed to be in normal incidence.͒ According to the kinematical diffraction theory of carbon nanotubes, [18][19][20][21][22] an electron diffraction pattern of a SWCNT is the superposition of two sets of hexagonal patterns which are produced by the honeycomb lattices on the "top" and the "bottom" graphene layers of the nanotube, respectively. The two sets of patterns are mirrored about the tube axis.…”

De la simple hélice aux nanostructures tubulaires

“…Le formalisme ébauché dans l'encadré 2, développé ensuite pour des atomes régulièrement espacés sur une hélice puis pour une série d'hélices enroulées, permet de rendre parfaitement compte de cette image de diffraction [5]. Le lecteur intéressé peut se référer à l'excellent article d'Amand Lucas et Philippe Lambin [3]. Notons que l'on n'observe pas de manière évidente sur la fi gure 2a la croix de Saint-André sous-jacente à une hélice, décrite dans l'encadré 2, à cause des effets d'interférences entre les différentes hélices qui composent chaque nanotube.…”