Spin Diffusion in Inhomogeneously Broadened Systems

The theory of cross-relaxation within inhomogeneous spin systems (spin diffusion) is discussed. The qualitative theory of Bloembergen, Shapiro, Pershan, and Artman is generalized. For the special case where the exchange- and concentration-independent contribution to the cross-relaxation linewidth are zero, the diffusion time is given by $\tau =T_{12}{〈{{\nu }_c}^2〉}^{-\frac{3}{2}}{\left({T_2}^{*}\right)}^{-3\mathrm{}}$. An exact theory of cross-relaxation in the limit $T_2\gg {T_2}^{*}$ is also developed and spin-diffusion is shown to be governed by an integro-differential equation. The conditions under which an ordinary diffusion equation describes spin diffusion is examined and the diffusion constant derived. The expression for $\tau$ given above is verified by this theory.