The spatially dependent Larmor frequency,

ω (

r ), at position

r under a spatially linear field gradient,

g , is expressed as follows:

$$\omega \left( {\bf{r}} \right){\text{ }} = {\text{ }}\gamma \left( {H_0 + {\bf{g}}\,{\text{ }}{\bf{r}}} \right){\text{ }} = {\text{ }}\omega _{0{\text{ }}} + {\text{ }}\gamma {\text{ }}{\bf{g}}\,{\text{ }}{\bf{r}},$$

\rm(A1

where

H _{0} is the externally applied magnetic field and

g r = 0 at position

r = 0. When the duration of the field gradient is δ, the difference between the phase angle at

r (

φ(r) ) and that at r = 0 is

$$\phi \left( {\text{r}} \right){\text{ }} = {\text{ }}\gamma \ {\text{ }}{\bf{g}}\,{\text{ }}{\bf{r}}{\text{ \ }}\delta.$$

(A2)

The distance in the

g direction where

φ(r) = 2π is

$$q^{ - {\text{1}}} {\text{ }} = {\text{ 2}}\pi {\text{ }}/{\text{ }}\gamma \ {\text{ }}g{\text{\, }}\delta $$

(A3)

q ^{–1} is the length scale with the field gradient. For example, q ^{–1} is 235 μm when g = 10 G/cm and δ = 1 ms. When the sample size or the size of the detection area is several times larger than q ^{–1} , the total signal intensity vanishes because of dephasing.

For measurements of

D , a second field gradient is applied to rephase the dephased magnetization. Figure

A1 shows a typical pulse sequence with two pulsed field gradients (PFGs) having a rectangular shape along the z axis and the dephasing and rephasing behaviors of the magnetization when the individual spins do not change their positions at interval Δ between the two PFGs [

9 ,

10 ]. In Fig.

A1 , vectors on the z axis represent the isochromats, which are the sums of spin moments under the same magnetic fields, corresponding to individual positions. In Fig.

A1(a) , the isochromats are aligned along the y axis by an r.f. π/2 pulse. Under the first PFG, the isochromats precess at an angular velocity of γ

gr , corresponding to individual positions in the rotating frame (Fig.

A1(b) ). At the end of the first PFG, the isochromats are spirally twisted by a pitch of

q ^{–1} (Fig.

A1(c) ). The application of an r.f. π pulse along the y axis rotates the individual isochromats along the y axis by 180°, creating a mirror-symmetric arrangement of the isochromats with respect to the y-z plane (Fig.

A1(d) ). Under the second PFG, the individual isochromats precess at the same angular velocity as that under the first (Fig.

A1(e) ). At the end of the second PFG, the isochromats are aligned along the y axis (Fig.

A1(f) ). When the nucleus has displacement Δz in the z direction during Δ, it has a phase angular shift of

Fig. A1 Pulse sequence of pulsed field gradient spin-echo (PFGSE) NMR, and dephasing and rephasing behavior of the magnetization

$$\phi (\Delta z) = 2\pi \frac{{\Delta z}} {{q^{ - 1} }} = \gamma g\delta \Delta z.$$

(A4)

The echo signal intensity,

I( 2

τ,gδ) , at 2

τ is proportional to the vector sum of the isochromats in the sample, and is therefore expressed as follows:

$$\begin{array}{ll}I(2\tau, g\delta) = I(2\tau, 0) \int\int \cos\ (\phi(\Delta Z))\rho({\rm \bf{r}}) p({\rm \bf{r}}, \Delta z) d{\rm {\bf r}}d\Delta z,\end{array}$$

(A5)

where

ρ (

r ) is the density of the nucleus (constant for a homogeneous sample),

p (

r ,Δz) is the probability of displacement during Δ for the nucleus at

r , and

I (2

τ ,0) is the total signal intensity without PFG, expressed as

$$I(2\tau, 0) = I(0, 0) \exp (-2 \tau /T_2),$$

(A6)

where

I (0,0) is the initial signal intensity immediately after the r.f. π/2 pulse. For free diffusion in an isotropic medium,

p (

r ,Δz) has a Gaussian distribution:

$$p(r, \Delta z) = (4\pi D \Delta)^{-1/2} \exp \left(-\frac{\Delta z^2}{4 D \Delta}\right),$$

(A7)

where

D is the diffusion coefficient. Taking the diffusion during

δ into account,

I (2

τ ,

gδ ) is rewritten as follows:

$$I(2\tau,g \delta) = I(0) \exp \left(-(rg \gamma g \delta)^2 D \left(\Delta - \frac{\delta}{3}\right)\right),$$

(A8)

In common measurements,

gδ is varied under constant Δ, where τ is also constant. In this case, Eq. (A8) can be rewritten as follows:

$$I(2\tau,g \delta) = I(0) \exp \left(-(rg \gamma g \delta)^2 D \left(\Delta - \frac{\delta}{3}\right)\right),$$

(A9)

where I (0) is I (2τ,gδ ) with constant τ and gδ = 0. It should be noted that I (0) decays on T _{2} relaxation.