Second Moment of theCr3+EPR Line in Ruby Broadened by Strong Hyperfine Interactions

Abstract

An alternative approach to the Van Vleck formalism for calculating the second moment of an EPR line in the case of strong (super) hyperfine interactions is described. The calculation uses probability theory to arrive at the following expression for the second moment of the EPR line: $〈\ensuremath{\Delta}{E}^{2}〉={(2I+1)}^{\ensuremath{-}1}\ensuremath{\Sigma}\stackrel{N}{i=1}\ensuremath{\Sigma}\stackrel{2I+1}{j=1}\ensuremath{\Sigma}\stackrel{2I+1}{k=1}{[{\ensuremath{\epsilon}}_{a}({j}_{i})\ensuremath{-}{\ensuremath{\epsilon}}_{b}({k}_{i})]}^{2}\ifmmode\cdot\else\textperiodcentered\fi{}{P}_{i}(j, k),$ where ${\ensuremath{\epsilon}}_{a}({j}_{i})$ and ${\ensuremath{\epsilon}}_{b}({k}_{i})$ are the exact energy levels of the $i\mathrm{th}$ nucleus in the presence of the paramagnetic ion, when the ion is in its ${j}_{i}\mathrm{th}$ initial and ${k}_{i}\mathrm{th}$ final state, respectively, and $I$ is the spin of the nucleus. The ${P}_{i}(j, k)$ are the "transition" probabilities for the $i\mathrm{th}$ nucleus in the ${j}_{i}\mathrm{th}$ level of the initial spin states to be found in the ${k}_{i}\mathrm{th}$ level of the final nuclear-spin states after the ion transition. The paramagnetic ion strongly interacts with $N$ nuclear spins. The anisotropic EPR-line width variation of dilute Verneuil ruby has been analyzed, using this formulation, in terms of the (super) hyperfine interaction with the surrounding aluminum nuclei. The calculation shows that most of the linewidth (70 to 80%) is caused by the hyperfine interaction. The line shape is also calculated by a Monte Carlo technique and is found to be Gaussian.