Theory of Linewidths in Electron Spin Resonance Spectra

A general theory of the linewidths in the electron spin resonance spectra of dilute solutions of free radicals has been developed in terms of the relaxation-matrix theory of Bloch, Redfield, and Ayant. In contrast to previous theories, it is shown that a composite line arising from a set of degenerate nuclear-spin states should, in general, consist of a sum of superimposed lines of Lorentzian shape with different widths rather than a single line with an over-all Lorentzian shape. A single Lorentzian line is still obtained, however, as a limiting case when the variation of the widths of the different components of a composite line is small compared to the average width. Although the non-Lorentzian shape of a composite line is often difficult to observe experimentally, a number of other observable properties are predicted by the present development that are outside the scope of the previous theories. For example, linewidth effects resulting from differences in the widths of the separate components of a composite line are predicted that explain the alternation in the linewidths from one hyperfine line to another recently observed in the ESR spectra in certain free radicals. The detailed form of the relaxation matrix is presented for intramolecular anisotropic and isotropic electron—nuclear dipolar interactions, quadrupole interactions, and g-tensor relaxations. Modulations of the spin density and hyperfine splittings are included, as are internal motions, and a number of cross terms between the different relaxation mechanisms arise. In general the relaxation matrix of a composite line contains significant off-diagonal elements, and the determination of the linewidths requires the evaluation of the eigenvalues of the matrix. Problems involving rapid chemical exchange, or modulation by jumps to a small number of sites, can be treated by the relaxation-matrix theory and, under special restrictions, by either the modified Bloch equations or the Anderson theory of motional narrowing. When applicable, these latter procedures can be used over the entire range of exchange rates, while the relaxation-matrix theory is limited to fast rates only.