Multiple-Pulse Nuclear Magnetic Resonance Transients in Solids

Abstract

The response of a spin system is calculated when a pair of 90° rf pulses is applied to a set of static identical interacting nuclei, initially polarized in an external static magnetic field. For pulse spacings the order of the spin-spin relaxation time, a "solid echo" is predicted. This effect is strongly dependent on the relative phasing of the two pulses and is maximized for a 90° phase shift. Extending the work of Powles and Strange, it is shown that the second moment of the nuclear resonance absorption line can be obtained from the solid echo in a straightforward manner, and to a predictable accuracy. A general expression is derived for the principal error term arising in the estimation of the second moment by the solid-echo technique and is applicable to a system of static interacting nuclei of any spin $I$. Preliminary experimental data shows the presence of solid echoes in powdered aluminum ($I=\frac{5}{2}$). An experimental estimate of the second moment gives $\Delta M_2=9.5\mathrm{}\pm 0.2$ ${\mathrm{G}}^2$ at 297°K. The effect of two closely spaced rf 90° pulses has also been calculated for a system of static interacting spins composed of two magnetic species. The rf pulses are assumed to interact with one species only. Some new and interesting effects are predicted, especially in the case when the two pulses are coherent. Unlike a single-spin species where this pulse combination would give zero signal, the presence of the second magnetic ingredient gives rise to a signal the initial slope of which is proportional to the secondmoment contribution of the nonresonant spins. Direct measurement of this "cross second moment" should be very valuable, particularly when scalar interactions are present as well as the dipolar interaction. The automatic removal of the resonant spin contribution to the total second moment would tend to increase the accuracy of a scalar coupling constant determination, particularly if the resonant spin term were dominant. Preliminary experiments on a single crystal of NaF show general qualitative agreement with the predictions. Also calculated is the double-pulse response of a single magnetic species with half-integral spin which has both a dipolar and quadrupolar interaction. The system treated is one of well resolved quadrupole satellites. The rf is assumed to interact with the central transition only. Kambe and Ollom have calculated the second moment of the steady-state absorption line of the central transition due to dipolar broadening in the case of well-resolved quadrupole structure. In the present work, it is shown that, as might be expected, the second moment as derived from the free induction decay, when the central line only is pulsed, is in agreement with that of Kambe and Ollom. If a second pulse is applied to the system, in phase with the first, a nonzero signal is predicted, even though this is a single-spin species. It is shown that the growth of this signal is characterized by only part of the dipolar interaction, and a second moment which can be extracted is analogous to the "cross second moment" of a two-spin-species system. When a scalar interaction is present as well as the dipolar term, the nontrivial fact is shown that for two pulses the interaction measured is no longer a simple fraction of the steady-state second moment. The scalar coupling constants and the dipolar lattice sums are shown to be combined in a different way in each case, so that a double-pulse experiment will yield new information on the spin system. This should certainly help in estimating the scalar coupling constants further than just nearest neighbors.