Random axial anisotropy

In these sperimagnetic structures, the subnetwork magnetisations are reduced. The influence of the exchange interactions and a random axial anisotropy on the average magnetic moment can be described using the Hamiltonian (Harris et al. 1973)... 

It should be noted that these types of spectra are expected only for quadrupolar nuclei of semiconductors in non-cubic axially-symmetric forms such as the WZ structure cubic forms such as ZB or rocksalt structures ideally lack any anisotropy, and the ST peaks overlap the CT peak. However, defects in such cubic structures can produce EFGs that have random orientations, and the resulting ST are spread out over a wide range. 

Fig. 10. Typical derivative curve for randomly oriented radicals showing the effect of r-anisotropy. a, Three principal g-values, b, Axially symmetric 0-tensor.

Notice that what counts is not necessarily the symmetry of the molecule but the symmetry of the site at which the nucleus is located. Thus, if we consider an octahedral metal hexafluoride MFg undergoing a random rotation, the metal will not experience any anisotropy relaxation effects but the fluorines will because they reside at sites having axial symmetry. 

Chemical-shift anisotropy is very sensitive to molecular structure and dynamics. Each nucleus can be pictured as being surrounded by an ellipsoidal chemical-shift field, A, arising from the influences of neighboring spins, as described by Eq. (4). If the molecules in the sample have no preferred orientational order, these tensors will be randomly distributed, and the line-shape is predictable. If the shielding is equivalent in all directions = (5yy = zi, A is spherical), a symmetric peak, like shown that in Fig. 29a, will be observed at qjso, which is defined in Eq. (5). Axial symmetry = Gyy A is, more or less, football-shaped) results in a powder pattern like that shown in Fig. 29b. In this case, the tensor elements may be labeled CTy ) and (g x and Gj ). If there is no symmetry in the chemical-shift field (gxx is a flattened football), then the... 

The conventional, and very convenient, index to describe the random motion associated with thermal processes is the correlation time, r. This index measures the time scale over which noticeable motion occurs. In the limit of fast motion, i.e., short correlation times, such as occur in normal motionally averaged liquids, the well known theory of Bloembergen, Purcell and Pound (BPP) allows calculation of the correlation time when a minimum is observed in a plot of relaxation time (inverse) temperature. However, the motions relevant to the region of a glass-to-rubber transition are definitely not of the fast or motionally averaged variety, so that BPP-type theories are not applicable. Recently, Lee and Tang developed an analytical theory for the slow orientational dynamic behavior of anisotropic ESR hyperfine and fine-structure centers. The theory holds for slow correlation times and is therefore applicable to the onset of polymer chain motions. Lee s theory was generalized to enable calculation of slow motion orientational correlation times from resolved NMR quadrupole spectra, as reported by Lee and Shet and it has now been expressed in terms of resolved NMR chemical shift anisotropy. It is this latter formulation of Lee s theory that shall be used to analyze our experimental results in what follows. The results of the theory are summarized below for the case of axially symmetric chemical shift anisotropy. 

These parameters can have values between 0 and 1 that correspond to random and axial spatial arrangement, respectively, in a matter similar to the nematic order parameter in liquid crystal polymers (both expressed in the literature as S) (Mitchell et al. 1987 Pople and Mitchell 1997 Lacey et al. 1998 Andersen and Mitchell 2013). Theoretical predictions and experimental work on nanorods in two and three dimensions have indicated that the electrical percolation takes its lowest value in the case of isotropic (S = 0) or slightly isotropic (S 0.1-0.2) system. As the level of anisotropy increases the percolation threshold takes ever higher values that are correlated with the level of alignment of the filler in the system (large values of S) (Mutiso and Winey 2012 White et al. 2009 Behnam et al. 2007). 

To determine which of the ENDOR lines are associated with methyl protons, we make use of the characteristic line shapes observed in frozen solutions. In frozen solution the anisotropic hf interaction of randomly oriented molecules leads to line broadening with a concomitant reduction in signal intensity. The situation for methyl protons (in the P-position) is more favorable, since they have a relatively small anisotropy. Furthermore, they have axially symmetric hf-tensors, which give rise to a characteristic line shape shown in Fig. 2 [23,24]. This characteristic shape... 

The theoretical model of the IR linear dichroism suggests a single-axis orientation of a molecular assembly with axial (cylindrical) synunetry. This means that the molecules are located with their longest axis in the direction, which induces the anisotropy, determined by the director n, and the side groups are randomly fixed or rotate along the length of this axis. Under these conditions, the dipole moments of transition caused by the normal vibrations form conical surfaces with director n (Figure 1.2) [2,3,6-8]. 

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