Static and Dynamic Frequency Shifts in Electron Spin Resonance Spectra

Abstract

A detailed theoretical analysis has been made of the factors which affect the positions of the hyperfine lines in electron spin resonance spectra. The deviations from the usual high‐field‐approximation formulas are found to depend on the relaxation and line‐broadening interactions [${\mathcal{H}}_1\mathrm{(t)}$ ] as well as on the well‐known corrections to the high‐field approximation that result from the exact solution of the time‐independent static Hamiltonian [${\mathcal{H}}_o$ ]. The time‐dependent processes ${\mathcal{H}}_1\mathrm{(t)}$ make a direct dynamic contribution to the frequency shifts, and in addition they alter the frequency spectrum of the static Hamiltonian. This latter effect arises because the dynamics of the system are properly determined by the total Hamiltonian ${\mathcal{H}}_o+{\mathcal{H}}_1\mathrm{(t)}$ , not just by the time‐independent part ${\mathcal{H}}_o$ . The theory is developed in terms of the Redfield—Abragam relaxation‐matrix formulation, but simple models are also treated by a limited but more easily handled procedure based on the modified Bloch equations. The most interesting results predicted by the theory occur when there is a large alternating linewidth effect arising from modulations of the isotropic hyperfine splittings. Usually, the dynamic frequency shifts are less than the linewidths, but, when there is a large alternating linewidth effect, the dynamic shift is large compared to the width of the narrow lines in the spectrum even though it is still small compared to the width of the broad lines. The alternating linewidth phenomenon also changes the static frequency shift, e.g., in the case of two nuclei of spin I=1 it causes the mean relative static shift in the positions of the sharp lines to be a factor of 3 greater than in the absence of an alternating linewidth effect. The theory indicates that experimental studies of the frequency shifts and linewidths in appropriate systems should make it possible to determine both the magnitude of the mean‐square fluctuations in hyperfine splittings and the correlation time characteristic of the fluctuating motions. The experimental determination of these two quantities, which appear only as adjustable parameters in previous theories, should make it possible to obtain considerable information about the inter‐ and intramolecular factors which influence the instantaneous structure of free radicals. It has been rigorously shown that, when the width variations among a group of degenerate or nearly degenerate transitions are small, the net shift is determined by the average value of the shifts, both static and dynamic, of all the components involved in the set of degenerate transitions. Unresolved shifts from proton splittings (or from other nuclei with spin I=½) usually cause all the hyperfine lines to be shifted by the same amount, independent of whether or not the width variations are small. Normally, the shifts arising from the effects of tumbling motions on the intramolecular, anisotropic, electron—nuclear, magnetic‐dipole interactions are small, as are those arising from the cross term between the dipolar and g‐tensor interactions.