Spin diffusion in NMR in solids

The spin-flip diffusion equation for the local magnetization in solids is derived using Mori's equation of motion with the emphasis on the types of approximations involved. The spin-flip diffusion coefficient is expressed in terms of the local dipolar field. Comparison with other theories is also given. The effect of spin diffusion on nuclear magnetic resonance in disordered solids with a spatial distribution of resonance frequency $\ensuremath{\omega}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ and longitudinal spin-relaxation time ${T}_{1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ has been investigated in the limits of (a) rapid diffusion: $D{l}^{\ensuremath{-}2}\ensuremath{\gg}|\ensuremath{\Delta}A|$, and (b) slow diffusion: $D{l}^{\ensuremath{-}2}\ensuremath{\ll}|A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})|$ and $|\ensuremath{\Delta}A|$. $D$ is the spin diffusion coefficient, $A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{-}i\ensuremath{\omega}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ or ${T}_{1}{(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}^{\ensuremath{-}1}$, $|\ensuremath{\Delta}A|$ is the maximal variation in $A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, and $l$ is the average distance between extrema in $A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$. In the rapid diffusion limit, the case of $A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})={T}_{1}{(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}^{\ensuremath{-}1}$ and $A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{-}i\ensuremath{\omega}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ can be studied under the same treatment based on cumulant expansion method or second-order perturbation theory. Both yield the same results. The line shape is found to be a motionally narrowed Lorentzian whose width is determined by the spatial correlation of $\ensuremath{\omega}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$; in comparison with the result in the limit of $D\ensuremath{\rightarrow}\ensuremath{\infty}$, the longitudinal relaxation time of the sample as a whole is lengthened. The difference is determined by the spatial correlation of $\frac{1}{{T}_{1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}$. On the other hand in the slow diffusion limit, the case of $A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})={T}_{1}{(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}^{\ensuremath{-}1}$ and $A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{-}i\ensuremath{\omega}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ have to be studied separately. Hee selective summation of a perturbation expansion to an infinite order is involved. The procedure by which such expansion is summed depends on whether $A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is real [$A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})={T}_{1}{(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}^{\ensuremath{-}1}$] or imaginary [$A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{-}i\ensuremath{\omega}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$]. The individual absorption line is now broadened by spin diffisuion. In both limits, analytic expressions of the line broadening and the lengthening of the relaxation time are given. The differences between these results and those of $D=0 \mathrm{and} \ensuremath{\infty}$ yield information about the spatial distribution of $A(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$.