Azimuthal diffraction

Fig. 11. Simulated diffraction space of a chiral (40, 5) SWCNT. (a) Normal incidence diffraction pattern with 2mm symmetry (b),(c),(d) and (e) four sections of diffraction space at the levels indicated by arrows. Note the absence of azimuthal dependence of the intensity. The radii of the dark circles are given by the zeros of the sums of Bessel functions [17].

Several sections of the diffraction space of a chiral SWCNT (40, 5) are reproduced in Fig. 11. In Fig. 11(a) the normal incidence pattern is shown note the 2mm symmetry. The sections = constant exhibit bright circles having radii corresponding to the maxima of the Bessel functions in Eq.(7). The absence of azimuthal dependence of the intensity is consistent with the point group symmetry of diffraction space, which reflects the symmetry of direct space i.e. the infinite chiral tube as well as the corresponding diffraction space exhibit a rotation axis of infinite multiplicity parallel to the tube axis. 

The quantitative assessment of the degree of crystallite orientation by x-ray examination is not free of ambiguity. From a comparative analysis [23] in which results obtained from the consideration of (105) and from three different variations of equatorial reflection were compared, the conclusion was that the first procedure can lead to underrated results, i.e., to the underestimation of the orientation. However, it can be assumed that this does not result from an incorrect procedure, but from ignoring the fact that the adjacent (105) reflex can overlap. The absence of the plate effect of the orientation is characteristic of the orientation of crystallites in PET fibers. The evidence of this absence is the nearly identical azimuthal intensity distributions of the diffracted radiation in the reflexes originating from different families of lattice planes. The lack of the plate effect of orientation in the case of PET fiber stretching has to do with the rod mechanism of the crystallite orientation. 

For the simplest of mirrors, circular apertures, the effects of diffraction cause the diffraction-limited image to be an Airy pattern, and for large distances this pattern falls as 1/0 and will be azimuthally symmetric. 

Fig. 4.1 Reflection high-energy electron diffraction (RHEED) patterns corresponding to the best epitaxial conditions of CdSe deposited on a (111) InP face (thickness 85 nm). (a) Observation under the (112) azimuth (b) observation under the (110) azimuth. (With kind permission from Springer Science-l-Business Media [6])...

Fig. 2. Reflection high energy electron diffraction (RHEED) patterns taken from [110] azimuth, (a) Low-temperature grown GaAs at 250°C. (b) (Ga,Mn)As at 250°C. (c) I70°C, and (d) 320°C (Shen ei al. 1997a).

Several attempts to induce orientation by mechanical treatment have been reviewed 6). Trans-polyacetylene is not easily drawn but the m-rich material can be drawn to a draw ratio of above 3, with an increase in density to about 70% of the close-packed value. More recently Lugli et al. 377) reported a version of Shirakawa polyacetylene which can be drawn to a draw ratio of up to 8. The initial polymer is a m-rich material produced on a Ti-based catalyst of undisclosed composition and having an initial density of 0.9 g cm-3. On stretching, the density rises to 1.1 g cm-3 and optical and ir measurements show very high levels of dichroism. The (110) X-ray diffraction peak showed an azimuthal width of 11°. The unoriented material yields at 50 MPa while the oriented film breaks at a stress of 150 MPa. The oriented material, when iodine-doped, was 10 times as conductive (2000 S cm-1) as the unstretched film. By drawing polyacetylene as polymerized from solution in silicone oil, Basescu et al.15,16) were able to induce very high levels of orientation and a room temperature conductivity, after doping with iodine, of up to 1.5 x 10s S cm-1. 

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