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

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 $\beta =\frac{{\omega }_D({\omega }_1+{\omega }_D)}{{\omega }_1{{\omega }_2}^{*}}$ approaches unity, where ${\omega }_1$, ${\omega }_D$, and ${{\omega }_2}^{*}$ 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 ${\omega }_1$ and ${\omega }_D$ in the transition range. The analysis is shown to apply in the transition range 470 to 550°C for the KCl $F$ center. Analysis of the transition saturation data yields a value of ${\omega }_1$ which agrees with the expression of Feldman, Warren, and Castle, which for absolute temperatures $T$ large compared with 210°K becomes ${\omega }_1=3.5\mathrm{}\times {10}^{-1\mathrm{}}{T}^2$ ${\sec }^{-1\mathrm{}}$. The spectral diffusion rate in zone-refined samples is given by ${\omega }_D=12\mathrm{}{\nu }_0 \exp (-\frac{E_m}{\mathrm{kT}})$, where ${\nu }_0=3.7\mathrm{}\times {10}^{15}\times {10}^{\pm 1.2}$ ${\sec }^{-1\mathrm{}}$, and $E_m=1.6\mathrm{}\pm 0.2$ 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 ${\nu }_0$ 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.