21 July 1994

Dear Dr. Shapiro,

Our postdoc, Lee Z. Wang, has been doing some calculations on diffusion in the presence of barriers. In an attempt to understand his work, I have gained some understanding of this problem which is what I want to talk about. The hope is that some of this is new to some of the readership and will be useful.

As has been known for a long time, the problem is to figure out how the pulsed gradient spin echo (PGSE) NMR signal is affected by the diffusion of spins in the presence of barriers. This subject is even more important than in the past because of the emergence of imaging which allows the measurements of inhomogeneous diffusion coefficients with various anisotropies, leading to information on microscopic structures through which flow and diffusion take place. This all goes back to the original work of Stejskal, Cotts, and others which speaks well of those pioneers. A detailed up-to-date summary is in the excellent book: Paul Callaghan, Principles of Nuclear Magnetic Resonance Microscopv, Oxford, 1991.

In the usual PGSE experiment, magnetic field gradient pulses are applied for time d on either side of the 180-degree pulse. The conventional notation also calls the interval between the gradient pulses D (so that d £ D ), the diffusion coefficient D, and the dimension of the restricted space a. There are three regimes into which the problem can be (and has been) divided. The first is the so-called free diffusion limit, D D<< a², in which the spins hardly diffuse in the longest time of interest D and, thus, the diffusing spins do not, on average, encounter the barriers. The second is the so-called restricted diffusion case in which D D>>a² but d D<<a². This means that D D is now large enough for the spins, on average, to encounter the barriers many times during D but not during the gradient pulse d . The last condition is the so-called short pulse approximation in which d <<D . The final case is for d D>>a² (which implies D D>>a² because d £ D ), i.e., the spins diffuse so much that, on average, they encounter the barriers many times even during d . For this case, the signal attenuation caused by diffusion presumably has a major contribution during the gradient pulse. We call this case rapid diffusion.

There is a minor semantic difficulty here. Free diffusion is exactly the opposite of rapid diffusion, i.e., the former is the case where the diffusion is so small, the time D so short, or the barrier spacing so large, that the barriers are effectively nonexistent. Maybe limited diffusion is a better term. The term restricted diffusion is not exactly appropriate either because the barriers restrict the spins in both the restricted and rapid diffusion cases whereas the usual restricted diffusion does not include rapid diffusion.

Lee used a stochastic theory of random motion with a Gaussian phase distribution assumption to derive a general expression that is not limited to the so-called short pulse gradient approximation, d D<<a². The resulting expression goes to sensible limits. Specifically, the attenuation exponent a, defined by the expression for the signal amplitude S=S₀exp(-a), becomes, to within a multiplicative constant

                g²g²d²D(D-d/3), DD<<a², free diffusion,

                a²g²g²d², DD>>a² but dD<<a², restricted diffusion,

                and a⁴g²g²d/D, dD>>a², rapid diffusion.

The free and the restricted diffusion limits agree with the usual well-known expressions [due to Stejskal, et al., and discussed in various parts of Callaghan, for example] while the rapid diffusion expression goes, in the limit d Þ D , to Neuman's result [J. Chem. Phys. 60, 4508-4511 (1974)] for nonpulsed, i.e., cw, gradient experiments.

The physical pictures for the first two cases are well known. Without going into detail, a keeps increasing for increasing D and D for free diffusion because (DD )^0.5 is equal to the diffusion distance and the attenuation is proportional to the width of the phase distribution. The expression for restricted diffusion, for which D D>>a², is obtained by replacing DD , the square of the diffusion distance, by a², the square of the barrier distance, in the free diffusion expression.a is independent of D and D because the numerous encounters with the walls during D makes the average coordinates the same for all the spins.

For the rapid diffusion case in which substantial diffusion takes place during the gradient, the dephasing of the spins depends not on the initial and final phases of the spins, as in the first two cases, but on the time integrals of their positions (which are proportional to their frequencies) during d , as shown by Douglass and McCall [J. Phys. Chem. 62, 1102 (1958)] for the cw gradient experiment. Neuman provides a helpful insight for the d -dependence by considering spins jumping between the positions corresponding to the two barriers. He shows that the phase distribution becomes Gaussian after a large number of wall encounters. The spins at the extremes in the distribution are those that get "stuck" at the barriers and their phase differences increase linearly with time d .

Irv Lowe has pointed out to me that the rapid diffusion limit can be obtained by considering the dephasing of the spins in the presence of the field gradient as a T₂ process where signal should decay as exp(-d /T₂). Slichter (on p. 154 of the 1963 edition and pp.212-213 in the 1990 edition) shows that 1/T₂=g ²H^2t for spins that jump between two fields ± H and stays at each value for time t . In our case, H=ag and t =a²/D so the signal exponent becomes -d /T₂=-d g ²g²a⁴/D which is the desired result.

At first sight, I had a problem thinking about the 1/D dependence for rapid diffusion. However, it is clear that as D increases with fixed d , the phase distribution narrows because all "phase paths" become more similar as the number of wall encounters become large in the same amount of time. Furthermore, diffusion during a gradient pulse makes the spin phases more similar, immediately after the pulse. For infinitely large D, the phases will peak at the average value. Inbetween the gradients, the spins will behave as though they never saw any gradient pulses, so the mechanism that gave rise to attenuation in the free and restricted diffusion regimes goes away, too.

Callaghan points out that diffusion "imaging" has no resolution limit in the usual sense. This is true but apparently not in the fast diffusion regime because we need to overcome this "peaking" of the spin phases during the gradient pulses in order to measure the attenuation caused by diffusion and this will require stronger and stronger gradients! So, it seems that we cannot put one over on nature.

I am grateful to my colleagues in the lab who offered much (most?) of the insights here. They are Arvind Caprihan, Irv Lowe, Dean Kuethe, Allen Waggoner, and Lee Wang. I welcome further inputs. Lee's derivation, which started all this, will be submitted for publication soon.

Eiichi Fukushima