Tensor anisotropy

Figure 1 The principal sources of structural data are the NOEs, which give information on the spatial proximity d of protons coupling constants, which give information on dihedral angles < i and residual dipolar couplings, which give information on the relative orientation 0 of a bond vector with respect to the molecule (to the magnetic anisotropy tensor or an alignment tensor). Protons are shown as spheres. The dashed line indicates a coordinate system rigidly attached to the molecule.

The question whether or not stirred tank flow is locally isotropic, may be investigated with the help of a LES which resolves a great deal of the Reynolds stresses. To this end, the Reynolds stress data are best presented in terms of the so-called anisotropy tensor and its invariants Ah A2< and A3. 

The anisotropy tensor is related to the turbulent stresses, of course, and is defined as... 

It is worth mentioning that parameter p is insensitive, to first order approximation, to modulation of the residue-specific 15N chemical shift anisotropy tensor and/or dipolar interaction, as the (d2 + c2) term in the R) / R ratio is canceled out. The noncollinearity of the CSA and dipolar tensors will require corrections to Eqs. (10) and (12) for high degrees of rotational anisotropy (D /D > 1.5), as described in detail in Ref. [22]. 

The existence of molecules often creates permanent intramolecular optical anisotropy. The optical anisotropy of the liquid is then due to fluctuations in the orientations of the molecules or molecular subunits. If we assign a symmetric traceless anisotropy tensor a to each molecule or molecular subunit in the scattering volume, then the relaxation function for collective optical anisotropy fluctuations can be expressed as... 

From the lJ residual dipolar coupling the molecular magnetic anisotropy tensor is obtained, which differs from the metal contribution by an extent which depends on the magnetic anisotropy of the diamagnetic part. For example, in cytochrome b the diamagnetic, the paramagnetic and the total susceptibility anisotropy values are A Xax = —0.8, 2.8, 2.20 x 10 32 m3, respectively, and Axrh = 0.1, —1.1, — 1.34 x 10-32 m3, respectively [60]. The corresponding tensors sum up as expected. 

Ab Initio Calculations of 31P NMR Chemical Shielding Anisotropy Tensors in Phosphates The Effect of Geometry on Shielding... 

Influence of Geometry on Chemical Shielding Anisotropy Tensor... 

As seen from Fig. 15, in the case of unprotonated aromatic para carbons, a phenyl ring yr-flip does not change the orientation of the chemical shift anisotropy tensor components and, thus, this yr-flip does not affect the NMR response of such para carbons, whereas phenyl ring oscillations do. 

In contrast, for protonated ortho and meta aromatic carbons, both phenyl ring yr-flips and oscillations affect the orientation of the chemical shift anisotropy tensor components and, consequently, their NMR response. The change in the amplitude of the oscillations with temperature for protonated and unprotonated aromatic carbons is shown in Fig. 16. For the two types of carbons the amplitudes of the oscillations are quite similar. 

Fig. 15 Effect of a phenyl ring yr-flip on the orientation of the chemical shift anisotropy tensor components for unprotonated and protonated aromatic carbons...

Fig. 152 Geometry of the chemical shift anisotropy tensors of C = O groups in MGIM units...

The auxiliary quantities bik, ejfc and fik are introduced in Chapter 7 to take into account the effect of the induced anisotropy of medium on the dynamics of a single macromolecule in the system. In limits of applicability of the above equations (a2 C i /x, V7 2> 1), the quantities are defined in terms of the anisotropy tensors j3ji and eki as... 

In linear case, the dependence of the tensors (3ik and elk on the anisotropy tensor can be neglected, and all the above quantities become the unit matrixes. 

Thus, one can see that the single-mode approximation allows us to describe linear viscoelastic behaviour, while the characteristic quantities are the same quantities that were derived in Chapter 6. To consider non-linear effects, one must refer to equations (9.52) and (9.53) and retain the dependence of the relaxation equations on the anisotropy tensor. 

The quantity rj in set (9.58) represents the shear viscosity coefficient and depends on the invariant of the anisotropy tensor in the same way as the relaxation time i... 

Fig. 3. Example spectra from the one-dimensional dipolar-shift experiment taken from reference 7. (a) (Top) Experimental l3C chemical shift anisotropy powder pattern for Ru(C5H5)2 and (below) for comparison, the dipolar shift l3C spectrum for the same compound, (b) Calculated dipolar-shift lineshapes for different angles (indicated) between the lH-13C dipolar and chemical shift anisotropy tensor principal z-axes.

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