Numerical Solution of the Equation Governing Nuclear Magnetic Spin-Lattice Relaxation in a Paramagnetic-Spin-Doped Insulator

Abstract

A numerical solution of the equation $\frac{\partial M^z}{\partial t}=D\left(\frac{1}{r}\frac{{\partial }^2}{\partial r^2}\left(r{M}^z\right)\right)-\frac{(M^z-M_0^z)\overline{C}}{r^6}$ governing nuclear relaxation in a paramagnetic-spin-doped insulator has been obtained. The results are expressed in terms of $\overline{m}\mathrm{}\left(t\right)=\frac{\int b^R[M_0^z-M^z\mathrm{}(t\left)\right]r^2\mathrm{dr}}{\int b^R{M}_0^z{r}^2\mathrm{dr}},$ where $M^z\mathrm{}\left(0\right)=0\mathrm{}$, $b$ is the so-called "diffusion barrier" and ${\left(\frac{4\pi R^3}{3}\right)}^{-1\mathrm{}}$ equals the paramagnetic-spin concentration. Simple analytic forms for the long-time exponential decay of $\overline{m}\mathrm{}\left(t\right)\mathrm{}$ are obtained for either $D$ or $\overline{C}$ dominating the relaxation process. Graphical solutions for the intermediate regions are also obtained. The short-time nonexponential solution of $\overline{m}\mathrm{}\left(t\right)\mathrm{}$ is discussed.