Quadrupolar interaction second-order

Zeeznan+quadrupolar interaction, first order and second order... 

However, in the presence of a strong quadrupolar interaction, the second-order cross-terms between the quadrupolar interaction and the CSA [27, 28] or the heteronuclear dipolar interaction [29, 30] also have to be taken into account. For instance, upon including the CSA in the expression of the time-dependent Hamiltonian in (9), the second-order term becomes... 

The first term in (24) was already calculated in (9). The second term can be neglected, as the CSA is usually too weak to require the second-order treatment. The last term is the second-order cross-term between the two interactions. A complete review of the second-order cross-terms and their effect on high-resolution solid-state NMR powder spectra of quadrupolar nuclei was recently published by Ashbrook et al. [31]. 

Let us calculate the frequencies of transitions between Zeeman eigenstates s) and r), assuming that the nuclei are only subjected to an isotropic chemical shift and the first- and second-order quadrupolar interaction. As seen in Sect. 2.1, the Hamiltonian that governs the spin system in the frame of the Zeeman interaction (the rotating frame) is... 

The second term in (29) is the anisotropic, orientation-dependent frequency shift due to the first-order quadrupolar interaction... 

The last two terms in (29) are the anisotropic frequency shifts due to the second-and fourth-rank parts of the second-order quadrupolar interaction ... 

In the following section, we explain the basic protocols used for removing the second-order quadrupolar broadening based on the refocusing of the second-order quadrupolar interaction. These protocols rely on mechanical reorientation of the rotor axis (DAS) or use a combination of sample spinning and rf manipulation of the spins evolution (MQMAS and STMAS). Experimental aspects of these methods, as well as methods for data processing and analysis, are described in Sects. 5.3 and 5.4. 

The simultaneous cancelation of both anisotropic parts of the second-order quadrupolar interaction occurs if at the end of the evolution period x the anisotropic component of the phase is null ... 

It has also been shown that the technique referred to as multiple-quantum variable angle spinning (MQVAS) can be used for the purpose of separating the CSA and quadrupolar lineshapes [155]. In this experiment, a judicious choice of the rotor orientation is fjRL = 70.12°, which leads to cancelation of the fourth-rank term of the second-order interaction (Fig. 2). Thus, the refocusing ratio k for the quadrupolar interaction depends on the ratio of the second-rank terms... 

The nucleus is quadrupolar (spin 7/2, natural abundance 99.76%), and thus, the spectra can be affected by both the first- and second-order quadru-pole interaction, though the second-order broadening is generally not the largest source of line broadening in these materials. In general, three major anisotropic interactions influence the line shapes seen in the NMR spectra of solid samples (i) the qua-... 

Hq is the quadmpolar interaction. It is only present for nuclei with I>Vz, which are around 70% of active NMR nuclei. This interaction is also orientahon-dependent however, only the first-order quadrupolar interaction can be averaged out by spinning the sample around the magic angle (MAS)-the second-order quadrupolar interaction cannot be leveled out by this technique, resulting in very broad line shapes or sometimes in invisible sites for SS NMR [109]. In the last... 

Corrected for second order quadrupolar interactions, relative to 1.0 M aqueous LiCl. 

---