Direct Observation of Nuclear Spin Diffusion in Real Space

Images directly visualizing the spatial spin-diffusion process are reported. The measurements were performed using a magnetic resonance force microscope. The field gradient associated with the force-detection experiment is large enough to affect the spin dynamics and a modified kinetics of the spin-diffusion process is observed. The effects of the gradient were compensated for by a pulse scheme and a pure Zeeman diffusion rate constant of $D=(6.2\pm 0.7)\times {10}^{-12}{\mathrm{cm}}^2/\mathrm{s}$ in ${\mathrm{CaF}}_2$ was observed.

Figure 1

Pulse sequence for spin-diffusion measurements. The first hyperbolic secant pulse inverts the magnetization over half of the field of view later acquired (step function). During the mixing time spin diffusion can proceed, optionally under the influence of $n$ equally spaced refocusing pulses. The prepulse reduces artifacts from direct rf excitation of the cantilever. The mechanical detection is preceded by a Hadamard encoding step to increase the detection sensitivity [21] and finally obtained under triangular frequency sweeps with a period matched to the cantilever eigenfrequency for resonant excitation.

Figure 2

(a) Spin diffusion in ${\mathrm{CaF}}_2$ and (b) Inversion-driven spin diffusion for evolution times of 0, 15.5, and 64.5 s with 0, 15, and 63 inversions [black circles, blue (dark gray) boxes, and red (light gray) triangles, respectively]. The solid curves are simulations. Note that the overall amplitude of the experimental data was normalized to match the simulations, compensating for drift (largest scaling factor 1.4). The effects of spin diffusion are much stronger in (b) than in (a). The dashed lines in part (a) are simulations of the experiment without spin-lattice relaxation for 15.5, 64.5, and 600 s mixing time, respectively. (c) Dependence of the effective spin-diffusion rate $D_{\mathrm{eff}}$ on the number of inversion pulses during a constant mixing time of 15.5 s. The black curve is a simulation fitted to the experimental data [red (gray) circles]. With more pulses the spins diffusion is inhibited less by the gradient.

Figure 3

Time evolution of the inverse dipolar spin temperature ${\beta }_d$ during the experiment described in Fig. 2. Part (a) shows the evolution without inversions. The white line emphasizes the spatial scan at 15.5 s for which the Zeeman polarization is shown in Fig. 2a. The step in the magnetization initially polarizes the dipolar reservoir. This blocks the diffusion of Zeeman polarization. Further time evolution is dictated by the diffusion of dipolar order. Part (b) shows the inverse dipolar spin temperature when the magnetization is inverted approximately every second (b). This inversion reverses the direction of the magnetization gradient and now the prior buildup of dipolar energy drives further Zeeman diffusion.