The Bloch Equations for an AB System and the Design of Spin-State-Selective RF Pulses for Coupled Spin Systems

Optimized AM/FM pulses are described which excite an arbitrary coherence of a coupled spin system with optimal intensity. These pulses may become important for creating maximum signal intensity in MQC-filtered and polarization-transfer experiments in HRNMR and in vivo spectroscopy. For an AB system, for example, a pure DQC or ZQC state is obtained (100%) compared to maximal 50% with a conventional π/2-π/2 MQC pulse sequence. The design and numerical optimization of these pulses, with pulse times of about 1/2J (J is the coupling constant), are described. Examples of three of these spin-state-selective pulses (SSSP) for an AB system are given; simulations and experiments confirm their expected performance. To get some insight in the relations between the expectation values of the several coherences under the influence of Zeeman, J-coupling, and RF terms, Bloch-like equations for a proton AB system are derived, with neglection of relaxation. A system of 15 first-order coupled differential equations is found. A solution to these equations is given for the evolution of the zero-quantum coherence and longitudinal magnetization.