Orientational dependence, anisotropic interactions

Anisotropic interactions, orientational dependence, 26f Anisotropic rotation, polyformal spin relaxation, local motion, 70,78t Anisotropy, chemical shift (CSA), solid sample NMR, 22 Antiphase components, undesirable INEPT properties, 106 Areas, integrated, quantitative NMR studies, 1371... 

The effect of anisotropic interactions, orientation-dependent interactions in particular, which is responsible for the stability of the nematic phase to some degree, is prevalent in all mixtures. This question has been assigned an important place in the theory of low-molecular-weight liquid crystals of Melw and Saupe [49]. In the review of Flory s woric in [30], it was emphasized that although orientation-dependent interactions in polymers containing phenylene units, for example, can cause stabilization of the liquid-crystalline state, the asymmetry of the molecular shape is undoubtedly the dominant molecular characteristic responsible for the liquid-crystalline state in such systems. 

The most important interactions between the unpaired electron and the magnetic nuclei can be of two kinds dipole-dipole (anisotropic) interactions, that depend on the molecular orientation with respect to the external field, and Fermi contact or isotropic interactions (spherically symmetric). The latter are purely quantum mechanical, and arise when there is a non-zero probability of finding the electron at a particular magnetic nucleus. Hence, the interaction (coupling) will in principle be larger the more s-character there is in the singly occupied molecular orbital on the particular atom (JV). 

The second term in (29) is the anisotropic, orientation-dependent frequency shift due to the first-order quadrupolar interaction... 

The lines in an EPR spectrum can be split by interaction of the electron spin with the nuclear magnetic moment of atoms on which the unpaired electron is located (Parish, 1990). Only atoms with nuclear spin (I) nonzero exhibit this type of interaction, which can be of two types (1) contact interaction that is isotropic and results from the delocalization of the unpaired electron onto the nucleus and (2) dipolar interaction between electron spin and the nucleus. In the second case, the interaction is dependent on orientation and, therefore, anisotropic (Campbell and Dwek, 1984). 

The coefficients a and b (see Table 3.2) take into account the restrictions in spin dimensionality. For a = b= 1, the Heisenberg model with isotropic exchange interaction and isotropic susceptibility results. The combination of a = 1 and b = 0 yields the strictly anisotropic Ising model, in which the orientation of the spins is restricted to the z-axis. Consequently, the susceptibility is strongly orientation dependent and one needs to differentiate between x" in the direction of the z-axis ( easy axis ) and x perpendicular to z. The molar susceptibilities are then related as... 

In anisotropic materials, the electronic bonds may have different polarizabilities for different directions (you may think of different, orientation-dependent spring constants for the electronic harmonic oscillator). Remembering that only the E-vector of the light interacts with the electrons, we may use polarized light to test the polarizability of the material in different directions, lno is one of the most important electro-optic materials and we use it as an example. The common notations are shown in Figure 4.7. If the E-vector is in plane with the surface of the crystal, the wave is called a te wave. In this example, the te wave would experience the ordinary index na of LiNbOs (nG 2.20). If we rotate the polarization by 90°, the E-ve ctor will be vertical to the surface and the wave is called tm. In lno, it will experience the extraordinary index ne 2.29. Therefore these two differently polarized waves will propagate with different phase velocities v c/n. In the example of Figure 4.7, the te mode is faster than the tm mode. 

On the other hand, when rotational motion is slow, or when the symmetry of the complex is less than cubic, as in Mn2+ complexes with macromolecules, Au)Xr is often greater than one, the anisotropic interactions are incompletely averaged and EPR spectra similar to those for randomly oriented solid samples are observed. In these cases the spectra depend upon the angular relationships between the magnetic field vector and the crystal field axis of the ion. Moreover, when the symmetry of the manganese ion complexes deviate greatly from cubic, the EPR spectra depend upon the sharing of spectral intensity between the normal and forbidden (AMS = + 1, Amp = + 1) transitions. 

A common characteristic of the relevant spin interactions is that they are anisotropic and can be described by second-rank tensors. The resulting orientation-dependent NMR frequency is of the following form [1,9] ... 

Consider a number n of stiff polymer components (here stiff is used to mean semiflexible ) and define orientation-dependent ideal and interacting response (n x n) matrices X0(Q, u, u ) and X(Q, u, u ) respectively. In this case, orientational correlations have to be included in addition to the usual isotropic ones. Doi et al. [36-38] have developed the theory for solutions of stiff homopolymers. Their formalism is applied in Appendices C and D to multicomponent blend mixtures of stiff polymers without and with the incompressibility condition respectively. The interaction potentials comprise anisotropic (also called nematic) contributions as well as the usual isotropic ones ... 

Since the electron is not localized at one position in space. Equation 1.35 must be averaged over the electron probabihty distribution funcbon. H is averaged to zero when the electron cloud is spherical (as in s orbitals) and comes to a finite value for axially symmetric orbitals. The magnitude of the anisotropic hyperfine interaction then depends on the orientation of the paramagnehc system with respect to the external field. 

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