Solid State Nuclear Magnetic Resonance (NMR) is perhaps the only spectroscopic technique that allows experimentalists to manipulate the spin systems they are interested in. Of particular interest are nuclei with spins greater than ½, or quadrupolar nuclei, as they constitute over 70% of the magnetically active spins. Two of the important mathematical tools used in the theory of studying NMR are representation theory together with perturbation theory. We will use both these tools to describe the underlying mathematical theory for quadrupolar nuclei. The theory shows that for non-symmetric satellite transitions in half-integer quadrupolar nuclei, perturbation effects up to third-order feature in the NMR spectra. We will also use irreducible representations to analyze experiments conducted on various spin systems and discuss ways to design new ones. Another topic that will also be explored is the theory of rotary resonance in half-integer quadrupolar nuclei. The theory explains why techniques like FASTER (FAster Spinning gives Transfer Enhancement at Rotary resonance) improve the efficiency of symmetric multiple quantum experiments.
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