Chemical shielding anisotropy tensors

Champmartin D and Rubini P 1996 Determination of the 0-17 quadrupolar coupling constant and of the C-13 chemical shielding tensor anisotropy of the CO groups of pentane-2,4-dione and beta-diketonate complexes in solution. NMR relaxation study/norg. Chem. 35 179-83... 

Here, ak is the isotropic chemical shift referenced in ppm from the carrier frequency co0, SkSA is the anisotropy and tfk SA the asymmetry of the chemical-shielding tensor, here also expressed in ppm. Note that for heteronuclear cases different reference frequencies co0 are chosen for different nuclei (doubly rotating frame of reference). The two Euler angles ak and pk describe the orientation of the chemical-shielding tensor with respect to the laboratory-fixed frame of reference. The anisotropy dkSA defines the width and the asymmetry t]kSA the shape of the powder line shape (see Fig. 11.1a). 

It is a well-known fact that the magnitudes of the peptide backbone " N, and amide chemical shielding tensors and their orientations relative to the peptide planes are quite similar for all amino acids. Although the details of the tensor parameters may vary, this feature is fundamental for powerful new techniques exploiting residual anisotropies in liquid-state NMR of partially aligned samples, and for the general utility... 

One potential problem with chemical shift anisotropy lineshape analysis (or indeed analysis of lineshapes arising from any nuclear spin interaction) is that the analysis results in a description of the angular reorientation of the chemical-shielding tensor during the motion, not the molecule. To convert this information into details of how the molecule moves, we need to know how the chemical-shielding tensor (or other interaction tensor) is oriented in the molecular frame. A further possible complication with the analysis is that it may not be possible to achieve an experiment temperature at which the motion is completely quenched, and thus it may not be possible to directly measure the principal values of the interaction tensor, i.e. anisotropy, asymmetry and isotropic component. If the motion is complex, lack of certainty about the input tensor parameters leads to an ambiguous lineshape analysis, with several (or even many) possible fits to the experimental data. 

The determination of polymer structure at the atomic level is possible by analyzing orientation-dependent NMR interactions such as dipole-dipole, quadrupole and chemical shielding anisotropy as mentioned above. The outline of the atomic coordinate determination for oriented protein fibers used here is described more fully in Ref. [30]. The chemical shielding anisotropy (CSA) interaction for N nucleus in an amide (peptide) plane can be interpreted with the chemical shielding tensor transformation as shown in Fig. 8.3. [31, 32]. 

These slow motions which induce a change in the orientation of the chemical shielding tensor of a particular spin can also be studied by spin echo spectroscopy [47]. The experiment consists of producing and digitizing a whole spin-echo train while simultaneously decoupling the protons in a static sample. The Fourier transform of the train gives a spectrum in which the powder pattern of the chemical shift anisotropy is split into a number... 

The quantity a is known as the chemical shielding tensor associated with that particular nuclear site. The tensorial character of a implies that B/oc is in general in a direction different from that of Bo, which reflects the anisotropy of the molecular environment of the considered nucleus. As this is a purely magnetic interaction, analogous to the Zeeman one, the Hamiltonian Hcs is given by ... 

Up until this point, we have implicitly assumed time independence of the interactions and their corresponding Hamiltonian operators. This assumption is valid for a rigid stationary sample. However, when the sample is spun rapidly, each of the internal Hamiltonians becomes time-dependent. For example, Jfz,cs becomes time-dependent when there is chemical shielding anisotropy due to the fact that the orientations of the chemical shielding tensors relative to the applied magnetic field change as the sample... 

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