The creation, interconversion and observation of states of zero-, single- and double-quantum order in the N.M.R. spectroscopy of spin-1 nuclei using non-selective r.f. pulses

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Abstract
The response of spin-1 nuclei to sequences of high-power, non-selective r.f. pulses is calculated in detail using the Redfield density matrix formalism and Wigner rotation matrices. The spin system considered is subject to both a large, static Zeeman interaction and a small, first order, secular quadrupole coupling. It is shown that an in-phase pulse pair, 90° x -τ-90° x , applied such that the second pulse occurs at a zero-crossing of the on-resonance free-induction decay signal following the first pulse, creates a state of pure double quantum coherence and is thus equivalent to the soft double-quantum 90° pulse described by Pines and co-workers [6–10, 23]. In addition, in the presence of a resonance offset, Δ, it is shown that this state may be created either by an appropriate choice of Δ or by using the modified pulse sequence, 90° x -τ/2-180° x -τ/2-90° x . The response of the state of double-quantum coherence to resonance offsets and subsequent r.f. pulses is also investigated in some detail. In particular the effect of a general pulse, Θα, applied to the system after the double-quantum coherence has evolved for a period under the influence of a resonance offset, Δ, is calculated and the conditions required to effect total transfer of the order back into transverse magnetization (single-quantum coherence) are evaluated. Similarly, the conditions which effect a transfer of the double-quantum coherence into a mixed state of quadrupole (zero-quantum) and double-quantum order are discussed. The results clearly show the relationships between the phases of the r.f. pulses, the magnetization vectors, and the state of double-quantum coherence necessary to effect these transfers and thus lead naturally to a set of pictorial models for these processes. These involve consideration of both single-quantum and double-quantum frames of reference and agreement between the models and theory is quantitative, even to the derivation of operators corresponding to the cartesian components of the double-quantum coherence vector in the double-quantum frame. These operators are identical to those described by Vega and Pines [9] and Wokaun and Ernst [13]. Experimental verification of the main features of the theory is presented in the form of measurements on 2H nuclei in the magnetically orientable mesophase of the caesium perfluoro-octanoate/D2O system.