Spin-Lattice Relaxation in Periodically Perturbed Systems

Spin-lattice relaxation theory is examined, in the perturbation-theory regime, for nuclear spin systems which experience time-dependent forces not only from fluctuating spin-lattice interactions but also from externally produced periodic disturbances. These include strong steady rf fields (spin-locking and the Lee-Goldburg experiments), amplitude and phase-modulated rf fields (pulsed narrowing experiments), and modulation of the secular dipolar fields through rotation of the sample. It is shown for relatively slow molecular motions ($T_2>\tau \gg {{\omega }_0}^{-1\mathrm{}}$, where $T_2$ is the relaxation time, $\tau$ the correlation time, and ${\omega }_0$ the rf frequency) that uncoupled Bloch equations can be obtained for suitably defined components of the magnetization. The spin-lattice relaxation times, which limit the degree of narrowing that can be obtained, depend in every case on Fourier components of the thermal motion in the neighborhood of the frequency of the periodic disturbance, and are of the same order of magnitude for all the experiments. Various narrowing experiments on substances having molecular motion are discussed.