Nuclear-Spin-Diffusion Relaxation to a Finite Density of Paramagnetic Impurity Ions

The effect of a finite concentration of paramagnetic impurity ions on nuclear-spin-diffusion relaxation is studied theoretically and experimentally. It is found that in the spin-diffusion limited case, which is the one of greatest interest here, the equation for the relaxation rate can be modified in such a way as to permit a moderately simple description of the relaxation for all normal ion densities. The modification consists in the replacement of certain exponents in the usual relaxation-rate expression for small ion densities by other exponents which are determined numerically when a single parameter has been specified. This parameter is $x=N^{\frac{1}{3}}{\left(\frac{C}{D}\right)}^{\frac{1}{4}}$, where $N$ is the concentration of the ions, $\frac{C}{r^6}$ is the transition probability of the nuclei due to the dipole-dipole interaction between an ion and a nucleus, and $D$ is the spin-diffusion coefficient. It is shown also that the amount of nonexponential behavior in the relaxation recovery depends on this same parameter, and that the effects of finite ion density on the way in which $T_1$ depends on ion concentration, frequency, and temperature can be studied at the same time as the nonexponential behavior. Applications are made both to results of pulse NMR experiments which are reported here for the first time and to the results of experiments that have been reported by other workers.