Exact product operator evolution of weakly coupled spin-12 I(m)S(n) systems during arbitrary RF irradiation of the I...

In this article, we consider the evolution of weakly coupled I(m)S(n) systems of spin-12 nuclei during arbitrary RF irradiation of the I spins. Exact solutions are presented for the time dependence of the density operator in terms of its constituent product operator components for a complete set of initial states derived from polarization of either the I or the S spin. The solutions extend the range of applications that are accessible to the product operator formalism and its associated vector picture of nuclear spin evolution. This marriage of quantum mechanics and a literal vector description of spin dynamics during RF irradiation supports physical intuition and has led to simple pulses for selective coherence transfer, among other new applications. The evolution of initial states that are free of transverse S-spin components can be described by classical precession of the I-spin components about effective fields defined by the interaction between the coupling and RF fields. Although there is no analogue involving classical rotations for the evolution of initial states composed of S(x) or S(y), a vector description is still possible, and the solutions completely characterize the nature of J-coupling modulation during RF pulses. We emphasize the Cartesian product operator basis in the present treatment, but the solutions are readily obtained in any other basis that might prove suitable in analyzing an experiment. For a system of N coupled spins, standard exact methods involving diagonalization and multiplication of the 2(N) x 2(N) matrices that represent the system require on the order of (2(N))(3) operations to calculate the system response to a general RF waveform at each point in the time domain. By contrast, the efficiency of the present method scales linearly with the number of spins. Since the formalism presented also accommodates the absence of either RF irradiation or the coupling, the solutions provide an efficient means of general pulse sequence simulation, encompassing any combination of arbitrary RF waveforms, delays, and coherence gradients.