General behavior of completely anisotropic ESR hyperfine centers under slow orientational diffusion motion

Based on the solution of an isotropic orientational diffusion equation, a restricted ensemble-averaging process has been formulated for completely anisotropic electron-spin-resonance (ESR) centers undergoing slow orientational diffusion motion in a restricted angular interval during their spin-spin relaxation time. By employing this averaging process, we have calculated the slow-tumbling motional linewidth and line shift of completely asymmetric ESR hyperfine centers ($g_x$≠$g_y$≠$g_z$, $A_x$≠$A_y$≠$A_z$) for arbitrary orientations of their symmetric axes with respect to the ESR external magnetic field B→. For B→ parallel to the ESR (x,y,z) principal axes, both the motional linewidth and line shift depend on the associated motional correlation time τ as ${\tau }^{\mathrm{-}1/2}$. When B→ is oriented in the vicinity of 45° with the principal z axis, the motional linewidth is proportional to ${\tau }^{\mathrm{-}1/3}$. There are ‘‘magic’’ orientations of B→ for which the motional line shift vanishes identically. It will be discussed that the analytical expressions obtained for the motional linewidth and line shift for B→ parallel to the principal axes enable one to determine reliable experimental motional correlation times from slow-tumbling ESR hyperfine spectra in polycrystalline-amorphous substances.