Anisotropic Rotational Diffusion and Electron Spin Resonance Linewidths

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Abstract
The recent general theory of ESR linewidths for dilute solutions of free radicals developed by Freed and Fraenkel is extended to cover the possibility that the rotational motions of the free radicals are anisotropic. The analysis is greatly simplified by expressing the solution of the rotational‐diffusion equation as an expansion in Wigner rotation matrices. Only the spectral‐density functions in the linewidth expressions given by Freed and Fraenkel are changed by the inclusion of anisotropic rotational diffusion. These spectral‐density functions are found to depend upon five different relaxation times which are expressible in terms of the three principal values of the diffusion tensor ${ \germ{ R \germ} $}. It is shown how an analysis of the ESR linewidth effects, which vary from one hyperfine component to another, may be used in conjunction with calculations of the anisotropic electron—nuclear dipolar interactions to estimate the principal values of ${ \germ{ R \germ} $}. The experimental linewidth study on the para‐dinitrobenzene anion radical is reanalyzed, and it is shown that an anisotropic rotational‐diffusion mechanism cannot by itself explain the anomalous absence of any significant quadratic linewidth dependence on the total Z component of spin of the ring protons. While an order of magnitude estimate of ${ \germ{ R \germ} $} was made, further information permitting a quantitative understanding of the anomalous result (presumably due to fluctuations in isotropic hyperfine interactions) would be necessary to obtain a meaningful estimate of the principal values of ${ \germ{ R \germ} $} for this radical. The physical significance of ${ \germ{ R \germ} $} in the approximation of a Brownian particle as well as in terms of intermolecular potentials is discussed.