Second-order quadrupole interaction

A characteristic broadening of the central transition can be observed in the case of strong quadrupole interaction (second-order quadrupole broadening). The second moment, Mf of the central transition with respect to its center of gravity is given by [12] ... 

The line widths and shapes to be expected for cubic crystals containing point defects have been derived by Cohen and Reif for both first and second order quadrupole interaction 97). In particular, for point defect concentrations greater than about 0.1 (in terms of probability /, of a lattice site being occupied by a defect) distributed in a random fashion over various possible lattice sites, the second order interaction gives rise to a lopsided central component whose shape is given by (97)... 

Figure 2.8. Energy level diagram of a spin- /2 system showing the Zeeman interaction and the first- and second-order quadrupole perturbation of the energy levels.

MAS of second-order quadrupole effects. The same type of rotations using Wigner rotation matrices can be applied to the second-order effects as for the first-order interaction which leads to a second-order quadrupole energy of interaction such that... 

Figure 2.15. Effect of a distribution of interactions on the second-order quadrupole powder lineshape of the central transition in detail with the mean interaction Xq = 2 MHz and -ri = 0 at a Larmor frequency of 80 MHz showing A. no distribution, B. a Gaussian distribution of isotropic chemical shifts with FWHM = 0.17 A, C. a Gaussian distribution of the quadrupole interaction of 340A and D. both the chemical shift and quadrupole interactions distributed, (A = 2344Hz).

Simulation of the complete DOR spectrum (centreband plus the spinning sidebands) will yield the NMR interaction parameters (Sun et al. 1992, Cochon and Amoureux 1993, Amoureux and Cochon 1993). However, it is most usual to perform the experiment to give improved resolution and simply quote the measured peak position which appears at the sum of the isotropic chemical and second-order quadrupole shifts. DOR experiments at more than one applied magnetic field will allow these different contributions to be separated and hence provide an estimate of the quadrupole interaction via the combined quadrupole effect parameter Pq... 

Overtone spectroscopy developed for irradiates the sample at approximately twice the Larmor frequency (Tycko and Opella 1987). If the quadrupole interaction is sufficiently large that second-order quadrupole effects are significant, the (— 1 1) transition becomes weakly allowed. In powders the spectmm is still structured, allowing the interactions to be deduced, but is narrowed by a factor of 8vq/xq. 

Most separated local field experiments concentrate on the CSA. In speetra of halfinteger nuclei with / > 1/2, the CSA is often much less important than the second-order quadrupole interaction. Static or MAS spectra display a typical quadrupole powder pattern from which the quadrupole coupling constant and the asymmetry parameter can be determined. These quadrupole parameters can then be used to determine the principal components of the electric field gradient tensor, which are related to the local environment of the quadrupole nucleus. Knowledge of the local geometry is extremely valuable, and correlation of the quadrupole and dipole information of OH groups provides useful insights into structure. 

The second-order quadrupole interaction depends on the Wigner rotation matrices, which become time-dependent when the sample is rotated about an angle 6 with respect to B. The average second-order quadrupolar shift then depends on the Legendre polynomials... 

Table 6a. The contribution to the isotropic shift by the second-order quadrupole interaction of different spin quantum numbers...

To determine if the features of the spectrum of Figure la indicate the existence of two or more lines, as simulated, or arise from a single NMR line with a lineshape due to second order quadrupole interactions, sodium-23 MASNMR spectra of a Na-Y zeolite... 

Plotted on the ppm scale, spectral features arising from second order quadrupole interaction effects would increase by a factor of 1.9, for the 96 MHz spectrum and 3.4, for the 76 MHz spectrum, when compared with the 132 MHz spectrum. Inspection of the spectra in Figure lb indicates that this is not occurring, establishing that the spectral features are associated with separate sodium-23 MASNMR lines. This result can be more clearly observed for the partially exchanged, hydrated Y zeolites discussed below. 

Although MAS is very widely applied to non-integer spin quadrupolar nuclei to probe atomic-scale structure in solids, such as distinguishing AlO and AlOg environments [21]. simple MAS about a single axis cannot produce a completely averaged isotropic spectrum. As the second-order quadrupole interaction contains both... 

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