Theory of Saturation and Double-Resonance Effects in ESR Spectra

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Abstract
A theory of saturation in the electron spin resonance spectra of dilute solutions of free radicals has been developed in terms of the general Boltzmann equation for the density matrix given by Bloch and Redfield. It is shown, in contrast to the earlier theories, that, in general, a composite line arising from a set of degenerate nuclear spin states must be described by a generalized (matrix) saturated Lorentzian rather than a single line of over‐all saturated Lorentzian shape. The general properties of this solution are discussed in detail for the case when electron—nuclear dipolar interactions are the only important nuclear‐spin‐dependent relaxation mechanism. A particularly simple situation exists when similar nuclei are completely equivalent. Then each composite line consists of a superposition of saturated Lorentzians, and the linewidths and saturation parameters of each component depend only on the values of the total spin and of the z component of spin of the associated configuration of the completely equivalent nuclei. Thus it would be possible to saturate some components to a greater extent than others. These simple components may become coupled together in a complex fashion when the time‐dependent fluctuations of the dipolar interactions for equivalent nuclei are no longer identical, which is often the case for aromatic‐ring protons. Further coupling may be expected from the effects of nuclear‐quadrupole interactions and from intermolecular‐exchange phenomena and are only qualitatively discussed. In the absence of saturation the present theory reduces to the linewidth theory of Freed and Fraenkel. Detailed expressions are obtained for steady‐state electron—nuclear double‐resonance (ENDOR) effects on radicals containing completely equivalent sets of nuclei. The saturating fields can lead to coherence and induced‐transition effects. It is shown how the latter, under appropriate conditions, can result in enhancement of saturated ESR spectra. It appears necessary for enhancements that lattice‐induced nuclear spin transitions should be comparable in magnitude to the lattice‐induced electron spin transitions, or else cross transitions involving both nuclear and electron spins must be important in the relaxation process. Coherence effects involving both electron and nuclear spin transitions should be unimportant when the main contribution to the ESR linewidths results from secular processes and is nuclear spin independent. This is also the condition for NMR linewidths to be significantly narrower than ESR linewidths. Complications still arise from the coherence and linewidth coupling effects of the different nuclear transitions being excited. A brief discussion of enhancement by a transient ``heating'' effect is also given.