Theory of saturation and double resonance in ESR spectra. V. The average ENDOR and ELDOR lines

The general theory for the analysis of saturation and double resonance effects in the ESR spectra of dilute solutions of free radicals developed in the earlier papers in this series, has been solved approximately to obtain relatively simple formulas for the general case of free radicals with many nuclei. The solution is based on an expansion linear in ${\mathrm{b\; \equiv \; W}}_n{\mathrm{/W}}_e$ (where W_n and W_e are, respectively, the lattice‐induced pure nuclear‐spin and electron‐spin flip relaxation rates), but it is asymptotically valid for Heisenberg and (polarized) chemical exchange processes. Also, coherence effects are neglected. The resulting expressions for ENDOR take the simple form of a single average (saturated) Lorentzian for each distinct ENDOR line obtained from a particular group of equivalent nuclei; while for ELDOR one similarly obtains an average ELDOR line shape. Expressions are given for the average saturation parameters that are needed in terms of the various relaxation mechanisms. The range of validity of these solutions is analyzed and discussed. In the case of equivalent nuclei of spins of 1/2, an interesting result is that the relative ENDOR enhancement of the ESR signal is in lowest order, approximately directly proportional to the number of equivalent spins (I=1/2) under certain conditions such as when the NMR resonance is not significantly saturated. This result, when applicable, can prove quite useful for analytical purposes. It is, however, no longer valid for ENDOR signals from groups of equivalent nuclei for which $\text{I>1/2}$ , and it must be modified, even for I=1/2, if lattice‐induced cross transitions involving combined electronic and nuclear spin transitions are more important than pure W_n, and these considerations are discussed. Effects of having equivalent nuclei which are not completely equivalent are also discussed. Details regarding the structure and simplifying symmetry considerations of the general solutions are also given.