Theory of Nuclear Magnetic Resonance Detected by Nuclear Radiations

A comprehensive theoretical description is given for the effects of magnetic resonance on the angular distribution of radiation emitted from oriented nuclear states. The formulation is made in a general way. It may be applied to an ensemble of nuclei oriented by any method: For example, nuclear reactions, angular correlations, and low-temperature nuclear orientation may be treated. In fact the theory can also be applied to optical-double-resonance experiments. Statistical tensors are defined to describe nuclear crientation in the "resonant" state. Interactions of the oriented ensemble with extranuclear fields are then considered, and the effect of a radio-frequency (rf) field on the angular distribution of radiation is given. Two formulations are given for the "pure magnetic" case, for which numerical calculations were done. One employs an angular correlation formalism, following the evolution of the density matrix in the laboratory frame $S$, while the other is more closely related to conventional NMR. In the latter approach the transformation into the frame $S^{\prime \prime \prime }$, wherein the statistical tensors are time-invariant, is described in terms of a "generalized torque equation" governing the motion of a unit vector along the symmetry axis in $S^{\prime \prime \prime }$. Both formulations are exact. Time-dependent distribution functions are worked out in detail, with both fixed and random-phase angles between the rf field and the initial symmetry direction. Fast oscillations due to the constant magnetic field are modulated by slow oscillations due to the rf field. Time-integral curves were calculated. These show great sensitivity to the rank of the relevant statistical tensor, to geometry, and to the phase of the rf field. Multipole structure is predicted for certain geometries, with the resonance line showing a number of maxima equal to the rank of the statistical tensor. Under certain conditions two types of asymmetry are observable. A "transient" asymmetry appears for low-rf-field values: This asymmetry is sensitive to the sign of the nuclear moment, but it disappears in high-rf fields. Odd-rank statistical tensors can also give response functions with "persistent" asymmetry that remains at high-rf fields. This is a parity effect and is not sensitive to the sign of the nuclear moment. Effects of relaxation are also discussed briefly.