Theoretical Description of Periodically Pulsed Nuclear Magnetic Double Resonance

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Abstract
Periodically pulsed nuclear magnetic double resonance is described by the Bloch equations for a single nucleus of spin 1 2 , and by the density matrix master equation for A m X n spin systems. In this experiment, the observing rf field H 1 and the strong rf field H 2 are periodically pulsed so that H 1 is on when H 2 is off, and off when H 2 is on. The H 1 and H 2 pulse intervals are each equal to τ seconds. The Bloch‐equation description illustrates the basic features of the experiment in terms of the macroscopic moment. This description provides a physical model for the production of double‐resonance signals in the limit of short τ . The density‐matrix analysis separates into two parts: the production of population changes by H 2; and the production of detected signals by H 1. The formalism is illustrated by the AX2 system, 1, 1, 2‐trichloroethane. Theoretical and experimental frequency‐sweep spectra are compared for values of τ ranging from 1.25 to 0.0315 sec. The spectra are dynamic in the sense that the detector output is periodic in time with a period of 2τ . The theoretical spectra are computed by assuming that the nuclear spin system is in a quasisteady state, and that the recorder pen swings between the maximum and minimum values of the detector output. The population changes caused by H 2 are complicated functions of τ . When τ = 1.25 sec , the spectra approximate periodically pulsed single resonance with the inclusion of the nuclear Overhauser effect. When τ = 0.0315 sec , the spectra approximate steady‐state double resonance corresponding to H 1 and H 2 having one‐half their amplitudes. The spectra for intermediate values of τ are complicated, and display a variety of unusual spectral features.

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