Analytic considerations in the theory of NMR microscopy

Nuclear magnetic resonance microscopy poses fundamental as well as practical challenges. The Torrey–Bloch equation obeyed by the magnetization density is basic to this field. We discuss exact solutions of the equation from a generalization that addresses the motion of the nuclear spins towards an attractive center with an arbitrary time dependence in the attractive (harmonic) potential. The solutions provide an analytic way to interpret observations made with gradient magnetic fields of arbitrary (finite) pulse duration and to study the simultaneous effect of spin diffusion and time-dependent magnetic fields.