Quadrupolar axially symmetric

One notes that in general, if a non-dipolar medium contains molecular fields, its Kerr constant (200) is not zero. As an example, we take quadrupolar axially symmetric molecules, for which the molecular field is of the form ... 

In the case of quadrupolar, axially symmetric molecules, it reduces to ... 

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. 

The NMR experiments were performed using the quadrupolar echo pulse sequence 7i/2x—Ti—7i/2y—T2—acquisition with phase-cycling and quadrature detection. A Bruker MSL 400 spectrometer was used for the high pressure studies operating at a resonance frequency of 61.4 MHz. In the liquid-crystalline phase, perdeuterated lipids display NMR spectra, which are superpositions of axially symmetric quadrupolar powder patterns of all C-D bonds.From the sharp edges, the quadrupolar splittings... 

Fig. 4. Quadrupolar powder patterns (a) Spin NMR powder pattern showing that the central -)<- ) transition is broadened only by dipolar coupling, chemical shift anisotropy, and the second-order quadrupolar interactions, (b) Spin 1 NMR powder pattern for a nucleus in an axially symmetric electric field gradient (see text). The central doublet corresponds to 6 = 90° in Eq. (10). The other features of low intensity correspond to 6 = 0° and 6 = 180°. (c) Theoretical line shape of the ) - -) transition of a quadrupolar nuclear spin in a powder with fast magic-angle spinning for different values of the asymmetry parameter t (IS) ...

The powder pattern for an / = 1 quadrupolar interaction can be obtained by reflecting the axially symmetric chemical shielding powder pattern about its average value,53 as shown in Fig. 8. This follows from Eqs (42a) and (42b) having the same functional form as that of Eq. (24). From Eq. (42b) we obtain... 

The ZnC>4 site of ZnO is axially symmetric, which is manifest in the quadrupolar lineshape with 77 = 0. Commercial ZnS samples are often mixtures... 

Because dipolar interaction is related to the intemuclear vector, which is the direct description of the molecular framework, the relation between the orientation distribution function obtained from a dipolar-DECODER spectrum and the molecular frame distribution is simpler than that from CSA-DECODER and quadrupolar-DECODER spectrum. One example of the 2D dipolar-DECODER experimental spectrum is shown in Fig. 19. The change of the orientation distribution caused by the deformation can be measured by the difference of the spectra before and after deformation. Of course, since the dipolar interaction tensor is always axially symmetric, the information content of a dipolar-DECODER spectrum is decreased. 

In this equation e is the charge on the electron, / is the spin quantum number of the nucleus, and Av is the separation between adjacent lines of the 2/-component multiplet. This spectrum arises because the quadrupolar interaction pushes some energy levels up and some down the transitions between adjacent energy levels (recall that the only allowed transitions are between adjacent levels) are shifted by amounts proportional to mz. The center of gravity of the spectrum is not affected to first order. Equation (15.6) applies only to axially symmetric electric field gradient tensors, such as those experienced by 2H in a typical C-D bond. The... 

Fig. 2. Energy level diagrams for some quadrupolar nuclei placed in an axially symmetric electric field gradient...

Theoretical expressions for spin-lattice relaxation of 2H nuclei (determined by locally axially symmetric quadrupolar interactions modulated by molecular motions) can be derived for specific dynamic processes, allowing the correct dynamic model to be established by comparison of theoretical and experimental results [34,35]. In addition, T, anisotropy effects, which can be revealed using a modified inversion recovery experiment, can also be informative with regard to establishing the dynamic model [34,35]. 

If there is some molecular motion with characteristic times on the order of ICT sec, the NMR spectrum will no longer have the Pake doublet lineshape discussed earlier. For example, in gel-phase bilayers a perdeuterated lipid acyl chain will have a broad, relatively featureless spectmm, as shown in Fig. 3. These spectra do not lend themselves to easy analysis The molecular motion in the membrane is not rapid enough to be axially symmetric (see the description of the fluid bilayer below) on the NMR time scale but is fast enough to influence the average value of the quadrupolar interaction and thus the splittings of the individual labels. 

Solution of a dipolar liquid in a quadrupolar solvent. Let us assume the solvent (component 1) to be non-dipolar, and the solute (component 2) as consisting of dipolar molecules. Moreover, for the sake of simplicity, let the molecides of both components be axially symmetric. Equation (227) now yields, on ne ecting fluctuations and non-linear polarizability ... 

This important equation governs, like an equation of motion, the time development of the system under a Hamiltonian H. It yields eight coupled differential equations for the coefficients c,(r). The resulting solutions for various Hamiltonians resemble rotations in an eight-dimensional space spanned by the eight nontrivial basis operators. With the convenient basis set of p, [14, 16, 68], the time evolution under Zeeman interaction can be visualized as a precession in the pi-pi, Ps-Pe Pi P planes. The axially symmetric quadrupolar interaction, on the other hand, mixes between pi and pe and between p and ps. Therefore, evolution under quadrupolar interaction does not lead to precession of magnetization within the x-y plane, but rotates it out of this plane into a not directly accessible order and back again. 

To understand why so-called magic-angle spinning (MAS) is so successful as a means of line narrowing, it is first necessary to recognise that the CSA, dipolar, and first-order quadrupolar interaction all have basically the same orientational dependence for an axially symmetric tensor (this is always the case for the dipolar interaction, and corresponds to a CSA or first-order quadrupolar interaction with a zero asymmetry parameter), the orientationally dependent part of the frequency of a particular crystalHte can be expressed in the form... 

To first order, the CT for half-integer quadrupolar nuclei is not perturbed by the quadrupolar interaction (Equation (9)) however, second-order perturbation theory is typically required to properly describe the line shape and position of the CT. For the case where the EFG tensor is axially symmetric (i.e., ijq = 0.0) and in the absence of anisotropic magnetic shielding, the frequencies, vq(1I2, —112), for the CT of a stationary powder sample are given by " ... 

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