Diffusion Effects in the Inhomogeneously Broadened Case: High-Temperature Saturation of theF-Center Electron Spin Resonance

Abstract

An experimental and theoretical study is made of the effects of a random spectral diffusion process on the saturation behavior of a normally inhomogeneously broadened resonance. The random type of spectral diffusion process results, for example, if a spin diffuses on a set of sites having a distribution of local fields giving the inhomogeneous width but such that local fields at adjacent sites are uncorrelated. The calculation shows a transition in saturation properties to those characteristic of a homogeneously broadened resonance as the quantity $\ensuremath{\beta}=\frac{{\ensuremath{\omega}}_{D}({\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{D})}{{\ensuremath{\omega}}_{1}{{\ensuremath{\omega}}_{2}}^{*}}$ approaches unity, where ${\ensuremath{\omega}}_{1}$, ${\ensuremath{\omega}}_{D}$, and ${{\ensuremath{\omega}}_{2}}^{*}$ represent, respectively, the spin-lattice relaxation rate, the spectral diffusion rate, and the inhomogeneous width. A study of the transition behavior can yield values of ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{D}$ in the transition range. The analysis is shown to apply in the transition range 470 to 550\ifmmode^\circ\else\textdegree\fi{}C for the KCl $F$ center. Analysis of the transition saturation data yields a value of ${\ensuremath{\omega}}_{1}$ which agrees with the expression of Feldman, Warren, and Castle, which for absolute temperatures $T$ large compared with 210\ifmmode^\circ\else\textdegree\fi{}K becomes ${\ensuremath{\omega}}_{1}=3.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}1}{T}^{2}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. The spectral diffusion rate in zone-refined samples is given by ${\ensuremath{\omega}}_{D}=12{\ensuremath{\nu}}_{0} \mathrm{exp} (\ensuremath{-}\frac{{E}_{m}}{\mathrm{kT}})$, where ${\ensuremath{\nu}}_{0}=3.7\ifmmode\times\else\texttimes\fi{}{10}^{15}\ifmmode\times\else\texttimes\fi{}{10}^{\ifmmode\pm\else\textpm\fi{}1.2}$ ${\mathrm{sec}}^{\ensuremath{-}1}$, and ${E}_{m}=1.6\ifmmode\pm\else\textpm\fi{}0.2$ eV. The spectral diffusion is interpreted as resulting from diffusion of the $F$ center in [110] steps of length $\sqrt{2}a$, where $a$ is the interionic distance, with attempt frequency ${\ensuremath{\nu}}_{0}$ and motion energy ${E}_{m}$. This process does not account for the diffusion coefficient of the $F$ center, which results from diffusion of ionized electrons to anion vacancies, and which is limited in the case of dense coloration by charge compensating vacancy diffusion.