Nuclear-Magnetic-Resonance Study of Self-Diffusion in a Bounded Medium

The investigation of the effects of diffusion in a magnetic field gradient on the spin-echo experiment in nuclear magnetic resonance (NMR) is extended to small samples in which the diffusion is bounded, or restricted. From the point of view of NMR, bounded diffusion means that the spin dephasing time $T_2\gg t_x$, the average time for a molecule to diffuse once across a sample width $a$. A more realistic criterion is that $a$ is small enough or that the diffusion coefficient $D$ is large enough that the quantity $\frac{\gamma G{a}^3}{D}$ is about equal to or less than 1, where $G$ is a linear magnetic field gradient and $\gamma$ is the nuclear gyromagnetic ration. An effective self-diffusion coefficient $D^{\prime }\mathrm{}\left(t\right)=-12\mathrm{} \frac{\ln \left[\frac{M(t,G)\mathrm{}}{M(t,0\mathrm{})\mathrm{}}\right]}{{\gamma }^2{G}^2{t}^3}$ is defined from the Hahn spin-echo experiment, where $t=2\mathrm{}\tau$ is the time of the echo, and $M(t,G)\mathrm{}$ is the echo amplitude. For infinite samples, $D^{\prime }=D$, the true self-diffusion coefficient. However, when $t_x\ll T_2$, then $\frac{D^{\prime }}{D}<1\mathrm{}$ and $D^{\prime }$ depends on $t$. The measurement of $D^{\prime }$ is made by holding the times of an echo, $t=2\mathrm{}\tau$, constant and varying $G$. Experimental data are presented on $D^{\prime }\mathrm{}\left(t\right)\mathrm{}$ for four values of $a$ and values of $\frac{\gamma G{a}^3}{D}$ which range from being much greater than unity to less than unity. Results of the Carr-Purcell experiment are also presented and briefly discussed. A comparison of data from the spin-echo experiment is made with a theoretical calculation of $D^{\prime }\mathrm{}\left(t\right)\mathrm{}$ which uses Torrey's modification of the Bloch equations and requires that boundary conditions be satisfied. Results are compared with the theory developed by Robertson. A universal curve for $\frac{D^{\prime }}{D}$ versus $\frac{t}{t_x}$ is plotted, illustrating that $D^{\prime }$ is independent of $G$. It is shown that the reduced rate of decay of the echo envelope in the case of bounded diffusion is, in effect, a motional-narrowing phenomenon.