Comparison and use of vector and quantum representations of J-coupled spin evolution in an IS spin system during RF irradiation of one spin
A comprehensive survey is provided of the analytical expressions for the orthogonal product operator states arising from any initial state of an IS J-coupled spin system during arbitrary RF irradiation of one spin. These equations exactly characterize the effect of J coupling during the application of the RF field. The survey differentiates two kinds of spin rotation, classical and nonclassical, where the second kind comprises any interconversion that includes the transverse two-spin coherence states, 2S(y)I(x) or 2S(y)I(y), as initial, transient, or final states, and the first kind comprises all other rotations. Classical rotations are defined as linear rotations of the nuclear spin magnetization vectors around effective fields and there is an exact correspondence between the resulting vector model and the quantum mechanical (QM) equations at all RF field strengths. The effect of scalar coupling can be neglected for B(1) > 5J. Nonclassical rotations are nonlinear in time for a constant RF field. At high field (B(1) > 50J), the effect of J modulation is negligible, and the rotation of magnetizations is classical to a very good approximation. At intermediate strengths (5J < B(1) < 50J), a semi-classical vector model of I-spin irradiation is applicable in which the J-coupled precession of the S spins is determined from a reduced coupling constant, but the effect of the S spins on the I spins is ignored (this model has previously been used to determine the effect of coupling during adiabatic pulses). At lower powers, the exact QM-derived equations must be used for nonclassical rotations, but continuous pictorial descriptions of the rotation of magnetizations determined from the vector sum of the product operator states are helpful in designing novel NMR applications. At all powers it is proven that the instantaneous reduced coupling constant acting on the S spins is J cosφ, where φ is the polar angle of the I-spin magnetizations, thus establishing the central tenet of the semi-classical model applicable at moderate power. Several spinstate transformations that combine the effects of RF and scalar coupling to produce the overall rotation can be generated in 100% yield using low power irradiation. Analogous transformations are also available using classical rotations and, in combination with their nonclassical counterparts, form a general class of frequency-selective pulses that we call J pulses. Any combination of a 90 degrees pulse and a consecutive (2J)(-1) delay period can be replaced with a J pulse, and some initial approaches to designing shaped J pulses with improved offset profiles are explored. Copyright 2000 Academic Press.