Azimuthal mean orientation

The angle a describes the orientation of a vector orbital, but it is important to understand the meaning of this orientation. It is the azimuthal mean orientation of an ensemble of orbitals within the sampled volume. The underlying distribution of orientations within the ensemble usually cannot be determined. This consideration... 

The azimuthal mean gives rise to an ambiguous quantity — the magic angle of 54.7°. If the vector orbital is tilted at 54.7°, there will be no variation in / with 0. Conversely, if there is no variation in I with 0, Equation 4.2.1 will reveal that a = 54.7°. An infinite number of orientation distributions can result in an azimuthal mean orientation of 54.7°, but the most important of these is complete disorder. If there is no variation in I with 0, one may speculate but not prove that a sample is disordered. 

Data for one full year (1964) for Nashville, Tennessee, and Knoxville, Tennessee, 265 km (165 mi) apart, were compared to determine the extent to which the frequencies of various parameters were similar. Knoxville is located in an area with mountainous ridges oriented southwest-northeast Nashville is situated in a comparahvely flat area. The data available are the number of hours during which each of 36 wind directions (every 10° azimuth) occurred, the average wind speed for each direction, the number of hours of each Pasquill stability class for each direchon, and the mean annual wind speed. 

Each unit of structure in the oriented polymer will also be considered to possess transverse isotropy. Its orientation can therefore be defined by polar and azimuthal angles (0, tp), but the condition of transverse isotropy for the whole sample means that the observed second moment will depend only on functions of 0 (in fact, P200 and P400) the functions involving (p taking fixed average values. 

The expression for harmonic factors is complex and is defined azimuthally by means of a Lagrange function. Sample orientation in routine powder diffraction experiment is fixed and so is the corresponding harmonic factor k y), which simplifies Eq, 2.82 to ... 

In samples with cylindrical symmetry, is obtained from the XRD patterns from the azimuthal intensity distributions /(a) of suitably chosen reflections, where a is the angle between a crystal axis and the deformation direction (external reference). For example, a meridional reflection can be used to calculate the orientation of the chain axis with respect to the draw direction. A degree of orientation,/, of a crystal axis with respect to the fiber axis is calculated from the intensity distribution by first calculating the mean cos

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Fig. 1.8 Experiment geometry, Ewald sphere, and reciprocal space. The stimple is placed in the center of the sphere. Its structure is defined in the real-space coordinates, r = (ri, r2, r3>. The origin of reciprocal space, s = (s 1, 2,. 3). is attached to the surface of the Ewtdd sphere. The orientation of its axes is strictly fixed to the orientation of the sample space r. Each point on a plane detector (for clarity shifted to a position in front of the sphere) ctin be addressed by the two coordinates scattering angle, 20, and azimuthed angle, (p. The mapping of these two coordinates into reciprocal space is mediated by means of the surface of the Ewald sphere [60]...

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