Polar alignment tensor

In the principal axis frame of the alignment tensor, A, the dipolar coupling between two nuclei, P and Q, as a function of the polar coordinates, 6 and tp, is given by... 

Figure 4.45 Illustration of the content of equ. (4.90) which describes the angular distribution of Auger electrons (eb) in coincidence with the preceding photoelectron (ea). The data refer to 2p3/2 ionization of magnesium by linearly polarized photons of 80 eV and subsequent L3-M1M1 Auger decay, with emission of both electrons in a plane perpendicular to the photon beam direction. The alignment tensor a,

Polar alignment resulting from this process establishes C ov symmetry in the medium. There are two unique directions in the point group, the polar direction X and the direction perpendicular and cylindrically symmetric with respect to the X direction. When the common assumption that frequencies and the Cartesian axes along which they are applied can be freely interchanged in lossless media is invoked, only two nonequivalent tensor components exist. 

Fig. 1 Illustrations of relationships between RDC intemuclei vector AB and an arbitrary molecular frame (a) and the alignment tensor frame (b). A protein molecule, carrying spin nuclei A and B, is represented using an ellipsoid. Bq is the external field. is the instantaneous angle between the intemuclei vector AB and Bq. Px,y,z specify the projection angles of Bg onto each axis of a molecular frame. Polar angle 9 and azimuth angle (j> are spherical coordinates of the vector AB in the alignment tensor frame...

A stress-induced alignment can also be detected in Raman experiments. The sensitivity of a vibration to the polarization of the incident and scattered light in a Raman experiment is determined by the polarizability tensor for the vibration. Even in the absence of polarization information, IR absorption or Raman measurements made in the presence of stress can be used to detect a preferential alignment of a defect by the effect the alignment has on the relative intensities of the stress-split-components of a vibrational band. 

It is interesting to note that formula (64) is applicable to two cylindrically symmetric particles, such as between two nanotubes [43, 61], if fhe applied field is the only source of radiation. If fhis is fhe case, only the principal axis (diagonal elements) of the polarizability tensors contribute, corresponding to the component aligned in the same direction as the laser polarization. 

Several factors contribute to the field-induced structural anisotropy that leads to optical anisotropy and hence to birefringence. All involve the particles polarization by the field and the partial alignment of their resultant dipole moments parallel to E. The resultant dipole moment / of a particle is the vector sum of its permanent and induced dipole moments. At the molecular level, electronic and atomic polarization occurs, the extent of which depends on the nature and symmetry of the molecule and on its polarizabilities (a and ax) along the parallel and perpendicular directions relative to the electric field or, for cylindrical symmetry, along the molecular axes a and b (a and a ). Naturally, the concept of the polarizability tensor is applicable to an assembly of molecules as a whole, e.g., a colloidal particle, as well. For such systems, and also for macromolecules and polyelectrolytes in an insulating medium, interfacial polarization may also have a major or even dominant contribution to the resultant dipole moment. 

Whenever a body is subject to an electric field and the body is non-spherical, there exists the possibility of creating a torque. See Figure 4A. Even for spherical bodies, if the polarization is a tensor not parallel to the field there exists the possibility for a torque to arise. For example, early experiments by Griffin and StowelF showed that Euglena (nonspherical) would align parallel to the applied field at low frequencies, or at very high... 

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