NMR survey of reflected Brownian motion

Restricted diffusion is a common feature of many physicochemical, biological, and industrial processes. Nuclear magnetic resonance techniques are often used to survey the atomic or molecular motion in confining media by applying inhomogeneous magnetic fields to encode the trajectories of spin-bearing particles. The diversity and complexity of diffusive NMR phenomena, observed in experiments, result from the specific properties of reflected Brownian motion. Here the focus is on the mathematical aspects of this stochastic process, their physical interpretations, and their practical applications. The main achievements in this field, from Hahn’s discovery of spin echoes to present-day research, are presented in a unified mathematical language. A long-standing problem of restricted diffusion under arbitrary magnetic field is reformulated in terms of multiple correlation functions of the reflected Brownian motion. Many classical results are retrieved, extended, and critically discussed.

Figure 1

Schematic illustration of echo formation. In a static field $B_0$ along the $z$ axis, the spins precess around this axis with the Larmor frequency ${\omega }_0=\gamma B_0$, and their magnetizations are directed along $z$. (a) At time $t=0$, one applies a periodic magnetic field $B_1$ (rf pulse) rotating in the transverse plane $xy$ with frequency ${\omega }_0$. (b) Still precessing, each magnetization is turning, linearly in time, toward the transverse plane. (c) When $t={\tau }_{\pi }∕2$ (here $\gamma B_1{\tau }_{\pi }=\pi$), the magnetizations lie in the transverse plane, and the periodic magnetic field ceases. (d) During the period $\Delta T=T∕2-{\tau }_{\pi }$, the magnetizations slowly relax to the longitudinal direction (axis $z$). At $t=T∕2-{\tau }_{\pi }∕2$, another rf pulse is applied. (e) Acting during the time ${\tau }_{\pi }$, it inverts the longitudinal direction of the magnetizations. (f) When this 180° rf pulse is turned off, the slow relaxation during the subsequent time period $\Delta T$ returns the magnetizations into the transverse plane (f). Moreover, these magnetizations turn out to be partially refocused so that they form a macroscopic signal (an echo) at $t=T-{\tau }_{\pi }∕2$. To get a simpler view of the action of the rf pulses, one could consider the coordinate frame rotating around the $z$ axis with the frequency ${\omega }_0$. Throughout this review, the rf pulses are assumed to be very short allowing their durations (${\tau }_{\pi }∕2$ and ${\tau }_{\pi }$) to be neglected.

Figure 2

Typical spin-echo sequences. (a) Formation of a spin echo at time $T$ by application of a 180° rf pulse in steady gradient profile (Hahn experiment). (b) Periodic repetition of 180° rf pulses creates a train of spin echoes (Carr-Purcell experiment). (c) Pulsed gradient spin-echo formation (Stejskal-Tanner experiment).

Figure 3

Random-walk simulations of reflected Brownian motion confined inside a circle of radius $L$, for three values of the ratio $p=DT∕L^2$: (a) 0.1, (b) 1, (c) 10. In all cases, diffusion is started from the center.

Figure 4

Several effective temporal profiles $f\left(t\right)$: (a) steady, (b) CPMG, (c) Stejskal-Tanner rectangular, (d) trapezoidal, (e) periodic, and (f) stimulated echo. Profiles (a)–(c) correspond to the gradient shapes shown in Fig. 2 [for convenience, the rectangular profile (c) was shifted to the left]. The case (e) is related to a frequency modulated magnetic field. The 180° rf pulse is already taken into account for (a)–(e) by changing the sign of the effective profile. The stimulated echo profile (f) represents a steady gradient with three 90° rf pulses applied at times 0, $\delta$, and $1-\delta$. In this case, the echo shown corresponds to the coherence pathway $\{1,0,-1\}$.

Figure 5

(Color online) The second moment $\mathbb{E}\{{\varphi }^2∕2\}$ (solid line) as a function of $p$ for the restricted diffusion between two parallel planes (in a slab) under a steady linear magnetic-field gradient. For small $p$, this moment is compared to its leading term in the slow diffusion regime with (circles) and without (dashed line) the correction term in Eq. (97). One clearly sees the importance of this correction.

Figure 6

(Color online) The second moment $\mathbb{E}\{{\varphi }^2∕2\}$ (solid line) as a function of $p$ for the restricted diffusion between two parallel planes (in a slab) under steady linear magnetic-field gradient. For large $p$, the second moment is compared to its leading term in the motional narrowing regime with (circles) and without (dashed line) correction term in Eq. (125).

Figure 7

(Color online) Apparent diffusion coefficient as a function of $p$ for a cylindrical cell. The slow-diffusion correction (161) and the interpolation formula (166) are compared to the spin-echo measurements in a borosilicate glass cell by 131. Experimental data shown have been provided by Dr. M. E. Hayden.

Figure 8

(Color online) Macroscopic signal as a function of $q\delta$ is found analytically within the narrow-pulse approximation (circles), and numerically by the MCF approach for $\delta =0.01$ (solid line) and $\delta =0.1$ (dashed line). The physical parameters ($D=2.3\times {10}^{-9}{\mathrm{m}}^2∕\mathrm{s}$, $L=1.6\times {10}^{-5}\mathrm{m}$, $T=0.22\mathrm{s}$) are from 56. One sees that the approximate relation (173) is not applicable for $\delta =0.1$ since the condition $\delta p⪡1$ fails (here $p\simeq 2$).

Figure 9

(Color online) Illustration of a drastic deviation from the Gaussian behavior at high gradient intensity. On a logarithmic scale, experimental data for free diffusion (when the gradient applied along unrestricted direction) fall onto the straight line $-q^{2}p∕12$ as expected. In contrast, experimental data for restricted diffusion follow the theoretical relation (180). A small deviation from this behavior is mainly caused by surface relaxation as confirmed by numerical analysis. Experimental data have been provided by Dr. M. D. Hürlimann.

Figure 10

(Color online) Transition between different regimes of restricted diffusion in a slab under a linear gradient (on the left) and parabolic (on the right) magnetic fields (see description in the text).

Figure 11

Example constructed by 108 of two nonisometric domains with the identical Laplace operator spectra.